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+    {
+        "type": "text",
+        "text": "A REAL- WORLD WEBAGENT WITH PLANNING, LONG CONTEXT UNDERSTANDING, AND PROGRAM SYNTHESIS",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Izzeddin Gur $^{1*}$  Hiroki Furuta $^{1,2*†}$  Austin Huang $^{1}$  Mustafa Safdari $^{1}$  Yutaka Matsuo $^{2}$  Douglas Eck $^{1}$  Aleksandra Faust $^{1}$",
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+    {
+        "type": "text",
+        "text": "<sup>1</sup>Google DeepMind, <sup>2</sup>The University of Tokyo  \nizzeddin@google.com, furuta@weblab.t.u-tokyo.ac.jp",
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+    {
+        "type": "text",
+        "text": "ABSTRACT",
+        "text_level": 1,
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+        "text": "Pre-trained large language models (LLMs) have recently achieved better generalization and sample efficiency in autonomous web automation. However, the performance on real-world websites has still suffered from (1) open domaininess, (2) limited context length, and (3) lack of inductive bias on HTML. We introduce WebAgent, an LLM-driven agent that learns from self-experience to complete tasks on real websites following natural language instructions. WebAgent plans ahead by decomposing instructions into sub-instructions, summarizes long HTML documents into task-relevant snippets, and acts on websites via Python programs generated from those. We design WebAgent with Flan-U-PaLM, for grounded code generation, and HTML-T5, a new pre-trained LLM for long HTML documents using local and global attention mechanisms and a mixture of long-span denoising objectives, for planning and summarization. We empirically demonstrate that our modular recipe improves the success on real websites by over  $50\\%$ , and that HTML-T5 is the best model to solve various HTML understanding tasks; achieving  $18.7\\%$  higher success rate than the prior method on MiniWoB web automation benchmark, and SoTA performance on Mind2Web, an offline task planning evaluation.",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Large language models (LLM) (Brown et al., 2020; Chowdhery et al., 2022; OpenAI, 2023) can solve a variety of natural language tasks, such as arithmetic, commonsense, logical reasoning, question answering, text generation (Brown et al., 2020; Kojima et al., 2022; Wei et al., 2022), and even interactive decision making tasks (Ahn et al., 2022; Yao et al., 2022b). Recently, LLMs have also demonstrated success in autonomous web navigation by controlling computers or browsers to follow natural language instructions through multi-step reasoning and decision making (Furuta et al., 2023; Gur et al., 2022; Kim et al., 2023).",
+        "bbox": [
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+        "text": "However, web automation on real-world websites has still suffered from (1) the lack of pre-defined action space, (2) much longer HTML documents than simulated observations, and (3) the absence of domain-specific knowledge for understanding HTML documents (Figure 1). Considering the open-endedness of real-world websites and the complexity of instructions, defining appropriate action spaces in advance is challenging. In addition, although several works have argued that recent LLMs with instruction-finetuning or reinforcement learning from human feedback improve HTML understanding and web automation accuracy (Furuta et al., 2023; Kim et al., 2023), their architectures are not always suitable to process real-world HTML documents; as presented in Figure 2, HTML tokens of real websites are much longer than those of simulators, and most LLMs have shorter context lengths than the average HTML tokens in real websites. It is prohibitively costly to treat such long documents as inputs directly, or adopt prior techniques such as text-XPath alignment (Li et al., 2021b) or text-HTML token separation (Wang et al., 2022a). To prioritize broader task generalization and model-size scaling, such domain knowledge for HTML documents is ignored in recent LLMs.",
+        "bbox": [
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "page_footnote",
+        "text": "\\*Equal Contribution.  $\\dagger$  Work done as Student Researcher at Google.",
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+        "page_idx": 0
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+    {
+        "type": "page_number",
+        "text": "1",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/39eb50e4f17095f2f001b2f9831de1fa590a3e008341c90c90d990424da2412b.jpg",
+        "image_caption": [
+            "Figure 1: Challenges in real-world web automation. Recent language model agents (Furuta et al., 2023; Gur et al., 2022; Kim et al., 2023; Yao et al., 2022b) can navigate simulated websites (Shi et al., 2017; Yao et al., 2022a), where the agents manipulate pre-defied actions and receive simplified HTML documents that are easy to parse. In contrast, language model agents continue to face challenges in navigating real-world websites, where they must interact with dynamic environments, handle open-ended actions (actions that cannot be pre-determined), and process lengthy HTML documents containing significant amounts of task-irrelevant information."
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "In this work, we introduce WebAgent, an LLM-driven autonomous agent that learns from self-experience to complete user instructions on real websites by combining canonical web actions in a program space (Figure 3). WebAgent (i) plans sub-instructions for each step by decomposing natural language instructions, (ii) summarizes long HTML documents into task-relevant snippets based on the plan, and (iii) acts via programming on real websites by grounding sub-instructions and HTML snippets into executable Python codes. We combine two LLMs to form WebAgent: newly introduced HTML-T5, a domain-expert pre-trained language model, for task planning and conditional HTML summarization and Flan-U-PaLM (Chowdhery et al., 2022; Chung et al., 2022) for grounded code generation. HTML-T5 has an encoder-decoder architecture and is specialized to capture the structure of long HTML documents better by adopting local and global attention mechanisms (Guo et al., 2022). It is",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "pre-trained using a mixture of long-span denoising objective (Tay et al., 2022) on a large-scale HTML corpus extracted from CommonCrawl. To ground language model agents into real websites, we introduce self-experience supervision, where the domain-expert language models are finetuned with data generated by scripted planning/summarization and self-generated programming.",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/292bfb3753779e5220a522cbe2a88efdf71e18b2c7d0fb464b76b3b5de2e6d05.jpg",
+        "image_caption": [
+            "Figure 2: Statistics of HTML tokens among real websites. Compared to simulator (about 0.5K tokens on average), HTML tokens of real websites are much longer (from 7K to 14K), which takes up the context length of large language models. As preprocessing, we remove the irrelevant tags (e.g. <script>, <meta>) and keep necessary attributes (e.g. id, type, value)."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Existing LLM-driven agents often solve decision making tasks with a single LLM conditioned on different prompts per role (Kim et al., 2023; Sun et al., 2023; Zheng et al., 2023), which is, however, not enough for real-world tasks whose complexity is higher than that of simulators. The empirical evaluations reveal that our method incorporating self-bootstrapped specialist language models improves HTML understanding and grounding, and achieves better generalization than single LLM agent. In real-world web automation, WebAgent significantly increases the success rate by  $50\\%$ , and error analysis emphasizes that coupling task planning with HTML summarization in specialized language models is essential for task success. Moreover, HTML-T5 not only works as a core module for WebAgent but also achieves strong results by itself on the web-based tasks. On MiniWoB++ (Liu et al., 2018; Shi et al., 2017), HTML-T5 achieves  $18.7\\%$  higher success than previous language model agent (Gur et al., 2022) while also outperforming competitive baselines, such as naive local-global attention models (Guo et al., 2022) and its instruction-finetuned ones (Chung et al., 2022). On the Mind2Web (Deng et al., 2023), an offline task planning dataset, HTML-T5 achieves SoTA performance among Synapse (Zheng et al., 2023) with GPT-3.5, and MindAct with FLan-T5-XL and GPT-4 (OpenAI, 2023). In summary, our key contributions are:",
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+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- We propose WebAgent, integration of two modular LLMs under self-supervision for real-world web automation. The domain-expert language model handles planning and HTML summarization, and a generalist language model generates executable Python programs.",
+            "- We newly introduce HTML-T5 - a language model with local-global attention mechanism that is pre-trained with a mixture of long-span denoising objective on a large-scale HTML corpus, curated from CommonCrawl, to capture the syntax and semantics of HTML better."
+        ],
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            173,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "page_number",
+        "text": "2",
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+    {
+        "type": "text",
+        "text": "- WebAgent notably improves the success rate by over  $50\\%$  in real websites. When fine-tuned on downstream demonstrations, HTML-T5 itself outperforms prior language model agent by  $18.7\\%$  in MiniWoB++, and achieves SoTA performance in Mind2Web, even surpassing GPT-4.",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "2 RELATED WORKS",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Web Automation Web automation is a sequential decision making task where agents manipulate browsers following given instructions (Shi et al., 2017), such as form filling (Diaz et al., 2013) or information retrieval (Adolphs et al., 2022) through the sequence of computer actions (Li et al., 2020; Mazumder & Riva, 2020; Shvo et al., 2021). Prior works have realized the web automation via reinforcement learning (Gur et al., 2019; Humphreys et al., 2022; Jia et al., 2019; Shaw et al., 2023), finetuned (Furuta et al., 2023; Gur et al., 2022) or prompted LLMs (Kim et al., 2023; Sun et al., 2023; Yao et al., 2022b; Zheng et al., 2023) on the simulated websites (Shi et al., 2017; Toyama et al., 2021; Yao et al., 2022a). However, there are still huge gaps",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "between simplified simulators and real web environments; for instance, the average tokens for HTML pages are about 15 times larger (Figure 2), and pre-defined action space for specific websites is a strong assumption that may harm the generalization to out-of-distribution web pages or instructions.",
+        "bbox": [
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+            439
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/008f120bf43ed7db0faf65fe70ffdc4cf8a5b7042f7dbc9fedd7124cb2b75175.jpg",
+        "image_caption": [
+            "Figure 3: WebAgent is a combination of LLMs: HTML-T5 for planning and summarization, and Flan-U-PaLM for grounded program synthesis. It is better suited for the real-world tasks; open domain action space, complex natural language instructions, and long HTML documents. See Appendix D for examples."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            171,
+            818,
+            290
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "MindAct (Deng et al., 2023) could be the most relevant work, where finetuned language model summarizes the raw HTML document into task-relevant snippets, and another model predicts the web actions in a multi-choice QA format. While MindAct also combines several language models, it has just adopted DeBERTa (He et al., 2021) and Flan-T5 (Chung et al., 2022) for summarization and actor modules, and evaluated it on the offline dataset. In contrast, we design HTML-T5, specialized for web-based tasks, to handle long HTML documents. WebAgent leverages HTML-T5 finetuned with self-experience for summarization and planning, and Flan-U-PaLM as a capable programmer, which enables it to generate open-ended web actions and to act on online real-world websites.",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Program Synthesis In addition to common LLMs (Brown et al., 2020; Chowdhery et al., 2022; Touvron et al., 2023), several works have proposed programming-focused language models (Chen et al., 2021a; Feng et al., 2020; Li et al., 2022; Wang et al., 2021) and their benchmarks (Austin et al., 2021; Hendrycks et al., 2021a; Lu et al., 2021). Another line of work has investigated the tool augmentation of LLMs (Parisi et al., 2022) by decoding API calls (Schick et al., 2023) or Python snippets to be parsed with the interpreter (Gao et al., 2023). Most works deal with the program synthesis on the static dataset, except for the attempts in robotics (Liang et al., 2023) and game (Trivedi et al., 2022; Wang et al., 2023a), where LLMs output Python or JavaScript snippets to command the agents. Similarly, we leverage the ability of code generation as an open-ended action space for web-based agents to manipulate the real website, and demonstrate LLMs can sequentially decode Python selenium codes considering the given sub-instructions and HTML in the prompts.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "See extended related works on document understanding and LLM for task planning in Appendix B.",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 WEBAGENT",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "WebAgent is a new architecture that combines two LLMs to achieve efficient real-world web automation. HTML-T5, a domain-expert LLM, is responsible for predicting the next sub-instruction (planning) and generating related HTML snippets (summarization). Flan-U-PaLM (540B) (Chowdhery et al., 2022; Chung et al., 2022), is prompted to generate executable Python programs based on the planning and summarization provided by HTML-T5 (Figure 3). This modular two-stage approach enables WebAgent to effectively navigate and process HTML documents.",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Workflow Users initiate natural language interactions with a clear intent, such as apartment searching. Upon receiving the initial request, HTML-T5 formulates a \"go to <URL>\" sub-instruction, triggering Flan-U-PaLM to generate a corresponding Python program that navigates to the specified website. The raw HTML content of the newly opened page is extracted and fed into HTML-T5 along with the",
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "page_number",
+        "text": "3",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/68d40e222ee1de14fe894538971ddde15bb4c63c76e00a9c98e17b3e1a31de01.jpg",
+        "image_caption": [
+            "Figure 4: HTML-T5 consists of (1) local and global attention mechanisms (Ainslie et al., 2020; Guo et al., 2022) and (2) a mixture of denoising objectives (Tay et al., 2022) with longer-span corruption on large-scale HTML corpus. The local and global attention mechanisms are suitable for the hierarchical tree structures of HTML documents. Because of the sparsity of content tokens in HTML, short mean span length (e.g.  $\\mu = 3$ ), often used in prior works (Raffel et al., 2020), only masks less meaningful chunks. Employing longer span length (e.g.  $\\mu = 8$ ) helps pre-trained language models to capture the syntax and semantics of HTML better."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "user's instruction and previous planning steps. This information is utilized to predict the next subinstruction and identify relevant reference IDs for extractive HTML summarization. Flan-U-PaLM, in turn, generates a Python program based on these sub-instructions and combined HTML snippets. This iterative process of planning, summarization, and program synthesis continues until a designated end-of-episode sub-instruction is predicted or the maximum number of iterations is reached.",
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+    {
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+        "text": "3.1 HTML-T5",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Prior research has shown that general-purpose LLMs, such as T5 (Raffel et al., 2020), Flan-T5 (Chung et al., 2022), and InstructGPT (Ouyang et al., 2022), can effectively navigate web environments (Furuta et al., 2023; Gur et al., 2022; Kim et al., 2023). However, unlike specialist transformer models (Li et al., 2021b; Wang et al., 2022a; Zhao et al., 2022), these general-purpose LLMs do not fully utilize the HTML-specific information that could otherwise lead to better understanding of HTML content. To address this limitation, we introduce HTML-T5, a pre-trained encoder-decoder language model specifically designed for HTML-based web automation tasks. HTML-T5 carefully merges the generalist and specialist characteristics of language models. It processes HTML in a text-to-text manner and employs local and global attention mechanisms (Ainslie et al., 2020) in the encoder to capture the hierarchical structure of long HTML inputs. HTML-T5 is pre-trained on a large-scale HTML corpus curated from CommonCrawl using a mixture of long-span denoising objectives (Tay et al., 2022), and then finetuned it for each downstream task. For WebAgent, we employ the self-experience supervision approach to align the model with real websites.",
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+        "text": "Model Architecture Unlike natural language, HTML documents possess an explicit hierarchical structure. This structure comprises elements such as <input>, <label>, and <button>, along with their associated attributes like class, label, and id. These elements are defined locally and combined hierarchically to create HTML documents. To model this inherent hierarchy, we replace the common dense attention (Vaswani et al., 2017) with local and global attention mechanisms (Ainslie et al., 2020). Local attention restricts each token to only attend to neighboring tokens within a window. Additionally, transient global attention allows each token to attend to tokens beyond its immediate window. This is achieved through the aggregation and normalization of token representations within each window, resulting in a global memory representation. Figure 4 describes the concepts of HTML-T5; leaf elements in HTML (green) could be processed by local attention, and internal elements (purple) could be compressed into transient global attention, which naturally fits the hierarchical structure of HTML. Following LongT5 (Guo et al., 2022), we use dense attention in the decoder.",
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+        "text": "Pre-Training with Mixture of Long-Span Denoising Our pre-training approach for HTML-T5 utilizes a span denoising objective. This involves randomly masking spans of tokens within an HTML document, with span lengths drawn from a Gaussian distribution with a mean of  $\\mu$ . The objective is then to predict the masked spans using the remaining tokens in the HTML document (Raffel",
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+        "text": "Published as a conference paper at ICLR 2024",
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+        "text": "4",
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+    {
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+        "img_path": "images/c0d379122078615db84d4188636f792c9b1241b57d9a2090c5ced0276d4f3e66.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 1: Success rate of real-world web automation on real estate, social media and map websites. The score stands for the percentage of covered attributes specified in given instructions. WebAgent, with language model modules for planning and summarization, achieves the best success (65%, 70%, 80%, respectively), surpassing other baselines, such as a single Flan-U-PaLM, that with a planning language model (Flan-U-PaLM+P), and that with a summarization language model (Flan-U-PaLM+S). Without language model modules, prompted Flan-U-PaLM plans in an open-loop manner (Plan:  $\\times$ ) and regular-expression-based retrieval summarizes HTML inputs ( $\\mathbf{Sum}$ :  $\\times$ ). The results imply that self-experience supervision notably improves the performance, and task planning should be learned by finetuning domain language models for closed-loop planning, rather than by prompting single LLM for open-loop planning. The error analysis describes the ratio across three types of errors in (real-estate)/(social-media)/(map) domains, which also points out that better adaptive planner to decompose the given instructions would contribute to further improvements of WebAgent."
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"2\">Modules</td><td colspan=\"2\">real-estate</td><td colspan=\"2\">social-media</td><td colspan=\"2\">map</td><td colspan=\"3\">Error Ratio (%)</td></tr><tr><td>Plan</td><td>Sum</td><td>Success</td><td>Score</td><td>Success</td><td>Score</td><td>Success</td><td>Score</td><td>Program</td><td>Plan</td><td>Sum</td></tr><tr><td>Flan-U-PaLM</td><td>✘</td><td>✘</td><td>10.0</td><td>55.3</td><td>20.0</td><td>25.0</td><td>10.0</td><td>51.3</td><td>36/88/11</td><td>38/0/78</td><td>26/12/11</td></tr><tr><td>Flan-U-PaLM+P</td><td>✓</td><td>✘</td><td>50.0</td><td>79.5</td><td>20.0</td><td>38.3</td><td>30.0</td><td>73.8</td><td>39/65/14</td><td>56/30/29</td><td>5/5/57</td></tr><tr><td>Flan-U-PaLM+S</td><td>✘</td><td>✓</td><td>0.0</td><td>45.7</td><td>25.0</td><td>62.1</td><td>15.0</td><td>46.3</td><td>30/67/0</td><td>40/13/100</td><td>30/20/0</td></tr><tr><td>WebAgent</td><td>✓</td><td>✓</td><td>65.0</td><td>87.6</td><td>70.0</td><td>85.8</td><td>80.0</td><td>93.8</td><td>20/33/25</td><td>70/50/50</td><td>10/17/25</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "et al., 2020; Tay et al., 2022; Ainslie et al., 2023). While a span length of  $\\mu = 3$  is commonly used, such short spans often mask less meaningful chunks in HTML documents, such as  $< / ,\\mathrm{id} =$  or  $n>$  (Figure 4), where the signal-to-noise ratio can be significantly lower than natural language text. In contrast, longer spans can contain more semantically meaningful chunks, such as <form class=\"or type=\"submit\">. Our empirical findings indicate that setting  $\\mu \\in \\{8,64\\}$  yields the optimal mixture for HTML documents (Section 4.2).",
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+        "type": "text",
+        "text": "We adopt 4096 input sequence length and 910 output sequence length during pre-training. In total,  $15\\%$  of input tokens are randomly masked in the denoising objective. For the pre-training dataset, we collect 100 WARC files (April 2019) from the CommonCrawl corpus and remove the non-Unicode or alphanumeric-only HTML documents. We then extract subtrees around  $\\langle \\text{label} \\rangle$  elements that have a special attribute called for that associates the corresponding label with a unique input element in the same HTML document. This pre-processing step improves the quality of the pre-training corpus by focusing only on HTML that is relevant for instruction following and grounding. Our final dataset has 3.41M examples. We pre-train HTML-T5 for 100K iterations following the practice in other T5 models (Chung et al., 2022; Lester et al., 2021). See Appendix C for further details.",
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+    {
+        "type": "text",
+        "text": "3.2 SELF-EXPERIENCE SUPERVISION FOR ALIGNMENT WITH REAL WEBSITES",
+        "text_level": 1,
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+        "text": "Gathering example demonstrations for LLMs to understand websites poses a significant obstacle in real-world web automation. While humans can effortlessly execute instruction following on actual websites, manually annotating every planning, summarization, and program synthesis step as detailed above is impractical. To address this issue, we propose self-experience supervision, a semi-supervised approach that necessitates minimal human involvement. In this method, manually curated scripts generate planning and summarization steps, while Flan-U-PaLM is tasked with generating Python programs. Our WebAgent aligns domain-specific language models, such as HTML-T5, with these self-gathered real-world experiences through fine-tuning (Wang et al., 2022b). This enables the generalization and alignment of agents to complex real-world tasks.",
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+        "text": "Instruction Templates We maintain a collection of instruction templates that incorporate placeholders such as \"Show me the way from <start> to <goal> by <n-th> <transportation> at map website\". We sample instructions by randomly assigning values to placeholders from pre-defined key-value pairs.",
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+        "text": "Scripted Planning and Prompted Programming We employ a rule-based parser to decompose instructions into sequences of sub-instructions; corresponding reference IDs are retrieved from HTML using regular expressions. At each step of the process, Flan-U-PaLM is provided with the sub-instruction and the associated HTML snippets to generate navigation programs that are executed through Selenium WebDriver. The success of recorded demonstrations varies, and automating success criteria for real-world tasks remains challenging. To refine the learning experience, we utilize environmental feedback to eliminate critical failures, such as program execution errors, retriever errors, and clearly erroneous URL prefixes (Ni et al., 2023).",
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+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+    {
+        "type": "page_number",
+        "text": "5",
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+    {
+        "type": "text",
+        "text": "map: Show me the way from San Jose to Mountain View by 2nd Cycling at map website?",
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/26812782f9335a66950e4217971de5ad7b58954c90a245c51d5a453bdb827222.jpg",
+        "image_caption": [
+            "Figure 5: Example episodes of real-world web automation in map domain. Considering the given instruction and HTML, WebAgent predicts the next sub-instruction and task-relevant snippet, and then synthesizes the Python script (gray), while treating the sub-instruction as a comment in the script. See Appendix H for extended figure."
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Finetuning for Planning and Summarization HTML-T5, a core component of WebAgent, is fine-tuned using self-experience demonstrations gathered through instruction sampling, scripted planning, and prompted program synthesis, as detailed earlier. It utilizes task instructions (e.g. please search 2 bedroom and  $2+$  bathroom houses in new york, ny with a max price of $7500 on real estate website), sub-instruction histories (e.g. go to real estate website, type in new york into search, click on search, click on price, click on max rent), and raw HTML as inputs. Subsequently, it generates the next sub-instruction (e.g. type in 7500 into max rent) and extracts the relevant data-ref attributes used for retrieving HTML snippets. Section 4.1 demonstrates the significance of integrating HTML summarization into sub-instruction prediction for enhancing real-world web automation performance.",
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+    {
+        "type": "text",
+        "text": "3.3 GROUNDED PROGRAM SYNTHESIS",
+        "text_level": 1,
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+    {
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+        "text": "Web automation on real-world websites faces challenges due to the open-ended action spaces, unlike simplified simulators (Shi et al., 2017; Yao et al., 2022a). In contrast to previous approaches (Gur et al., 2019; Humphreys et al., 2022; Jia et al., 2019; Liu et al., 2018), real-world web agents cannot pre-define a categorical action space to specify the interactive elements on the websites. To address this open-domain challenge, we introduce the act via programming paradigm in web automation by utilizing the conditional code generation capabilities of LLMs (Chen et al., 2021a; Liang et al., 2023). Provided with few-shot generic examples (such as selecting checkboxes, entering text into inputs, clicking on options, and scrolling etc.) for program generation, the next sub-instruction, and the extracted HTML snippet from HTML-T5, Flan-U-PaLM (Chowdhery et al., 2022; Chung et al., 2022) decodes an Python program (Figure 3) executable with Selenium WebDriver, a library for browser automation. This conditional program synthesis requires LLMs to not only generate code to follow natural language instructions but also understand the semantics and functionality of HTML elements. We provide several Python snippet examples generated by Flan-U-PaLM as follows (sub-instructions are treated as comments in the script):",
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+        "type": "code",
+        "sub_type": "code",
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+        "code_body": "1 # Type in walnut creek, ca into search  \n2 driver.find_element(By.css_selector, '[data-ref=\"175\"').clear()  \n3 driver.find_element(By.css_selector, '[data-ref=\"175\"').send_keys(\"walnut creek, ca\")  \n4  \n5 # Submit the search  \n6 driver.find_element(By.css_selector, '[data-ref=\"175\"').submit()  \n7  \n8 # Click on the apartments  \n9 driver.find_element(By.css_selector, '[data-ref=\"572\"').click()  \n10  \n11 # Scroll down housing type by 200px  \n12 driver.executeScript('getScrollParent.document.querySelector \"#type-of-housing\").scrollBy({top: 200})')",
+        "guess_lang": "txt",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTAL RESULTS",
+        "text_level": 1,
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+        "text": "To study how a modular combination of LLMs under self-supervision enables real-world web automation by overcoming open-endedness and long context documents, we execute instruction-following tasks on real websites (Section 4.1). In Appendix F, we also test WebAgent on WebSRC (Chen et al., 2021b), a static HTML comprehension benchmark, compared to prior transformer models specialized for structured documents (Li et al., 2021b; Zhao et al., 2022). In addition, we quantify the performance of HTML-T5 itself on simulated web benchmark, MiniWoB++, and offline task planning benchmark, Mind2Web (Section 4.2).",
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+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+        "text": "6",
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+    {
+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Architectures</td><td>Attention Type</td><td>L = 2048</td><td>L = 4096</td></tr><tr><td>Flan-T5-Base</td><td>Dense</td><td>34.0%</td><td>35.3%</td></tr><tr><td>Long-T5-Base</td><td>Local</td><td>43.4%</td><td>44.0%</td></tr><tr><td>Long-T5-Base</td><td>Local &amp; Global</td><td>53.1%</td><td>53.6%</td></tr></table>",
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+        "table_footnote": [
+            "Table 2: (Left) Architecture comparison on MiniWoB++ 12K dataset (Liu et al., 2018) with average success rate over 56 tasks. Local and global attention matches to the hierarchical tree structure of HTML, and then improves the success rate by over  $18\\%$ , compared to the instruction-finetuned dense attentions (Chung et al., 2022; Furuta et al., 2023). (Right) HTML-denoising comparison with different mixtures of span length (Raffel et al., 2020; Tay et al., 2022). We use LongT5-Base models for pre-training. HTML-denoising generally improves the performance on offline task planning on real estate website and MiniWoB benchmark. Especially, using longer span lengths  $(\\mu \\in \\{8,6\\})$  outperforms other choices, including the popular configuration in natural language domain  $(\\mu \\in \\{3,8,64\\} + \\text{Prefix LM objective})$ , which can reduce the less meaningful prediction from shorter spans (e.g.  $\\mu = 3$ ), and inject the structural bias of HTML better."
+        ],
+        "table_body": "<table><tr><td>Span Length μ</td><td>real-estate</td><td>MiniWoB++</td></tr><tr><td>(no HTML-denoising)</td><td>78.07</td><td>53.8%</td></tr><tr><td>3,8,64,Prefix</td><td>80.56</td><td>55.2%</td></tr><tr><td>3,8,64</td><td>80.56</td><td>55.4%</td></tr><tr><td>8,64</td><td>82.46</td><td>57.0%</td></tr><tr><td>8,32,64</td><td>82.16</td><td>55.6%</td></tr><tr><td>8,64,96</td><td>81.29</td><td>53.6%</td></tr><tr><td>16,64</td><td>79.97</td><td>55.2%</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.1 REAL-WORLD WEB AUTOMATION",
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+        "text": "Evaluation Methodology We first evaluate WebAgent with the real-world navigation performance under human supervision, at real estate website (a platform for housing), social media website (a network of communities), and map website. These three websites have different properties. real-estate requires long-horizon planning (about 20 steps per episode) for complex formfilling with a few page transitions (at least 2 pages), and social-media needs shorter plans (about 10 steps per episode) with many page transitions (at least 4 pages) by selecting appropriate hyperlinks on the page. map is the easiest domain with shorter plans and a few page transitions. WebAgent receives natural language instructions (e.g. Can you search for a studio bedroom,  $1+$  bathroom apartments in oroville, ca for corporate housing on real estate website?, or Could you present the most new thread of Python community filtered by Tutorial tag on social media website?), and acts via planning, summarizing by HTML-T5, and then programming by Flan-U-PaLM (Figure 5). Through the self-experience supervision process, we curate 260 episodes on real estate website, 230 episodes on social media website, and 410 episodes on map website to finetune HTML-T5.",
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+    {
+        "type": "text",
+        "text": "We prepare 20 different natural language instructions (see Appendix G for the full list), and measure the success rate and score for the evaluation. The score represents the percentage of required attributes covered during the episode (Yao et al., 2022a); for instance, (1) apartments for (2) corporate housing with (3) studio bedroom and (4)  $1+$  bathroom located in (5) oroville, ca, can be specified in the instruction. When the agents could search the housing satisfying (1), (2), (5) and not (3), (4), the score is  $60 (= 100 \\times 3/5)$ . If the agents achieve 100 score, that episode will mark as success.",
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+    {
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+        "text": "Results For comparison, we prepare three baselines, consisting of language model modules and a single LLM conditioned on different prompts per role, such as Flan-U-PaLM (Chung et al., 2022), that with a planning language model (Flan-U-PaLM+P), and that with a summarization language model (Flan-U-PaLM+S). If they do not use language model modules, prompted Flan-U-PaLM plans in an open-loop manner (Plan:  $\\times$ ), and regular-expression-based retrieval summarizes given raw HTML (Sum:  $\\times$ ). Table 1 shows that by leveraging planning and summarization language model modules, WebAgent achieves best  $65\\%$  success and  $87.6\\%$  score on real-estate,  $70\\%$  success and  $85.8\\%$  score on social-media, and  $80\\%$  success and  $93.8\\%$  score on map, significantly outperforming single Flan-U-PaLM, or with partial language model modules (most of those achieve about  $10 - 30\\%$  success). This result suggests that self-experience supervision notably improves the performance, and closed-loop planning grounded on HTML observations via finetuned domain language models is more suitable for open-ended web automation than open-loop planning with few-shot LLMs. This trend is remarkable in real-estate (even Flan-U-PaLM+P achieves  $50\\%$  success), where the longer planning horizon is needed to fulfill instructions. We also observe that coupling sub-instruction prediction with HTML summarization in language model modules plays a critical role in task success. The development of more capable planning modules to decompose the given instructions adaptively and accurately could help WebAgent improve the performance further.",
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+    {
+        "type": "text",
+        "text": "Error Analysis We also analyze the reason of failures by categorizing them into programming, planning, and summarization errors (Table 1). Programming error does not satisfy the given sub-instructions or HTML snippet. Planning error predicts sub-instructions conflicting with user instruc",
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+    },
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+        "text": "7",
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+    {
+        "type": "table",
+        "img_path": "images/462d8b5c4b12ad07e600b031d4b48db0944699b962d105877e3fc35a0e224b46.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td rowspan=\"2\">Train</td><td colspan=\"4\">Cross-Task</td><td colspan=\"4\">Cross-Website</td><td colspan=\"4\">Cross-Domain</td></tr><tr><td>Ele. Acc</td><td>Op. F1</td><td>Step SR</td><td>SR</td><td>Ele. Acc</td><td>Op. F1</td><td>Step SR</td><td>SR</td><td>Ele. Acc</td><td>Op. F1</td><td>Step SR</td><td>SR</td></tr><tr><td>Synapse (GPT-3.5)</td><td>ICL</td><td>34.4</td><td>-</td><td>30.6</td><td>2.0</td><td>28.8</td><td>-</td><td>23.4</td><td>1.1</td><td>29.4</td><td>-</td><td>25.9</td><td>1.6</td></tr><tr><td>MindAct (Flan-T5-XL)</td><td>SL</td><td>55.1</td><td>75.7</td><td>52.0</td><td>5.2</td><td>42.0</td><td>65.2</td><td>38.9</td><td>5.1</td><td>42.1</td><td>66.5</td><td>39.6</td><td>2.9</td></tr><tr><td>MindAct (GPT-4)</td><td>ICL</td><td>41.6</td><td>60.6</td><td>36.2</td><td>2.0</td><td>35.8</td><td>51.1</td><td>30.1</td><td>2.0</td><td>37.1</td><td>46.5</td><td>26.4</td><td>2.0</td></tr><tr><td>HTML-T5-XL (ours)</td><td>SL</td><td>60.6</td><td>81.7</td><td>57.8</td><td>10.3</td><td>47.6</td><td>71.9</td><td>42.9</td><td>5.6</td><td>50.2</td><td>74.9</td><td>48.3</td><td>5.1</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 4: Offline action prediction performance in Mind2Web dataset. We leverage the cached candidate generation results and direct QA formulation by following Deng et al. (2023). HTML-T5 significantly outperforms MindAct with Flan-T5 or GPT-4, and Synapse (Zheng et al., 2023) with GPT-3.5, across task/website/domain generalization in terms of all the metrics (element accuracy, operation F1, and success rates).",
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+        "text": "tions, and summarization error fails to extract the relevant HTML snippets for given sub-instructions. From the website perspective, the failures on real-estate concentrate in planning because of its long-horizon nature. map also fails in planning when confusing starting point and destination. In contrast, social-media tends to fail in programming due to the ambiguous sub-instructions or summarization including redundant hyperlinks, which results in transiting wrong pages or clicking unexecutable elements. From the method perspective, WebAgent often fails in planning by predicting incorrect sub-instructions (for instance, in real-estate, WebAgent generates incorrect plans in  $70\\%$  of failure episodes), while other baselines more fail in programming or summarization steps. This observation indicates that, through the self-experience supervision, the ratio of programming and summarization errors has decreased while the fundamental difficulty of planning, which requires consistent and accurate prediction over long horizon without error accumulation, still remains.",
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+        "type": "text",
+        "text": "4.2 ABLATION OF HTML-T5",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In addition to the evaluation as WebAgent system, we extensively examine HTML-T5 about (i) the generalization to other websites with Mind2Web (Deng et al., 2023), (ii) the performance on MiniWoB++, a standard web automation benchmark (Liu et al., 2018; Shi et al., 2017), and (iii) its architecture and pre-training objective. We adopt 16K tokens for the context window unless otherwise mentioned. We present results on offline task planning, and description generation (Gur et al., 2022) to test HTML understanding on static dataset in Appendix I.",
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+        "text": "Mind2Web Mind2Web (Deng et al., 2023) is an action-annotated real-world dataset with over 2K instructions collected from 137 websites. It provides action prediction tasks that measure the generalization of LLMs across the tasks, websites, and their domains (e.g. travel, shopping). Similar to real-world evaluation, the input is a set of HTML snippets, a task instruction, and an action history. The output comprises a target element to interact with, along with the operation, such as click, type, or select an option. We finetune HTML-T5-XL with the training dataset. The performance is evaluated with element accuracy,",
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+        "type": "text",
+        "text": "operation F1, and step success rate that cares for both element and operation correctness. Table 4 reveals that HTML-T5 significantly outperforms baselines with Flan-T5-XL or GPT-4 (OpenAI, 2023) across task/website/domain generalization, which increases element accuracy by  $5 - 8\\%$  , operation F1 by  $6 - 8\\%$  , and step success rate by  $4 - 8\\%$  . This highlights that HTML-T5 can handle real-world web automation tasks better and shows generalization beyond our real-world evaluation with 3 websites.",
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+    {
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+        "text": "MiniWoB++ We here evaluate HTML-T5 on 56 simulated tasks in MiniWoB++ using 100 evaluation episodes per task. Inputs are analogous to real-world evaluation, utilizing HTML documents, while outputs are adhering to a pre-defined format by the simulator such as click(ref = X). We finetune HTML-T5 with 12K human demonstrations (Liu et al., 2018), and compare the average success rate to prior supervised-learned agents (Gur et al., 2022; Humphreys et al., 2022), LongT5, and its instruction-finetuned variants (Chung et al., 2022)  $^{1}$ . Table 3 shows that HTML-T5-XL significantly",
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+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [
+            "Table 3: Average success rate of MiniWoB++ with 56 tasks. We use 12K demonstrations and compare HTML-T5 among supervised-finetuned methods. HTML-T5-XL outperforms WebN-T5-XL (Gurel al., 2022), the prior best method, by  $18.7\\%$ . HTML-denoising also yields better the success rate than instruction tuned ones. Finetuned HTML-T5 with 347K episodes (Furuta et al., 2023) outperforms Flan-T5-XXL (11B parameters) even with 3B parameters, which gets closer to SoTA with GPT-3.5. See Appendix K for the detailed results.",
+            "element and operation correctness. Table 4 rees with Flan-T5-XL or GPT-4 (OpenAI, 2023) eases element accuracy by  $5 - 8\\%$  , operation F1 nts that HTML-T5 can handle real-world web and our real-world evaluation with 3 websites."
+        ],
+        "table_body": "<table><tr><td>Models</td><td>Data</td><td>Success</td><td>Diff.</td></tr><tr><td>SoTA (Zheng et al., 2023)</td><td>-</td><td>99.2%</td><td>-</td></tr><tr><td>CC-Net</td><td>2.4M</td><td>32.0%</td><td>-</td></tr><tr><td>WebN-T5-XL</td><td>12K</td><td>48.4%</td><td>-</td></tr><tr><td>LongT5-Base</td><td></td><td>53.8%</td><td>0.0</td></tr><tr><td>LongT5-Large</td><td>12K</td><td>56.3%</td><td>0.0</td></tr><tr><td>LongT5-XL</td><td></td><td>60.4%</td><td>0.0</td></tr><tr><td>Flan-LongT5-Base</td><td></td><td>54.1%</td><td>+0.3</td></tr><tr><td>Flan-LongT5-Large</td><td>12K</td><td>56.1%</td><td>-0.2</td></tr><tr><td>Flan-LongT5-XL</td><td></td><td>61.1%</td><td>+0.7</td></tr><tr><td>HTML-T5-Base (ours)</td><td></td><td>57.0%</td><td>+3.2</td></tr><tr><td>HTML-T5-Large (ours)</td><td>12K</td><td>60.8%</td><td>+4.5</td></tr><tr><td>HTML-T5-XL (ours)</td><td></td><td>67.1%</td><td>+6.7</td></tr><tr><td>Flan-T5-XL</td><td>347K</td><td>75.5%</td><td>-</td></tr><tr><td>Flan-T5-XXL</td><td></td><td>79.0%</td><td>-</td></tr><tr><td>HTML-T5-XL (ours)</td><td>347K</td><td>85.6%</td><td>-</td></tr></table>",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+    {
+        "type": "page_footnote",
+        "text": "<sup>1</sup>We finetune LongT5 models with Flan dataset released by Chung et al. (2022). See Appendix J.",
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+    {
+        "type": "page_number",
+        "text": "8",
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+    {
+        "type": "text",
+        "text": "outperforms WebN-T5, the prior best model, by  $18.7\\%$ . Notably, we demonstrate HTML-denoising consistently improves the performance on top of LongT5 in all the model sizes, better than instruction-finetuning introduced in prior work (Furuta et al., 2023). Furthermore, we finetune HTML-T5-XL with 347K demonstrations from Furuta et al. (2023), which performs better than 11B-parameter Flant5-XXL even with 3B parameters, achieving  $85.6\\%$  success. These prove we successfully incorporate domain knowledge on HTML comprehension for web automation into pre-trained language models.",
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+    {
+        "type": "text",
+        "text": "Architecture and Objective We hypothesize that local and global attention mechanisms can capture the hierarchical structures of HTML documents better than dense attention. We compare the web automation performance among 56 MiniWoB++ tasks (Gur et al., 2022), by finetuning HTML-T5 with public 12K-episode dataset (Liu et al., 2018). We adopt 2048 and 4096 tokens as input length and prepare Base-size architectures. Table 2 (left) reveals that the combination of local and global attentions achieves the superior success rate by over  $18\\%$  compared to the instruction-finetuned dense attentions (Chung et al., 2022; Raffel et al., 2020) and local attention only. Surprisingly, local attention only still surpasses the dense attention by about  $9\\%$ , which suggests local relation between elements and attributes in HTML are essential for web tasks.",
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+    {
+        "type": "text",
+        "text": "As for pre-training objective in Table 2 (right), HTML-denoising generally improves the performance on offline task planning on real estate website and MiniWoB. Especially, using only longer span lengths  $(\\mu \\in \\{8,64\\})$  outperforms other choices, including the popular configuration in natural language domain  $(\\mu \\in \\{3,8,64\\} +\\text{Prefix LM objective})$ , which can reduce the less meaningful prediction from shorter spans (e.g.  $\\mu = 3$ ), and inject the structural bias of HTML into language models better. See Appendix I.2 for further results with model scaling.",
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+    {
+        "type": "text",
+        "text": "5 DISCUSSION AND LIMITATION",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Modular Approach with Specialist Language Models We demonstrate it is beneficial to divide web automation into planning, HTML summarization, and code generation, and to combine domain-expert language models aligned with self-experience data. Such modular approaches have also been adopted to support the inference of LLMs (Xu et al., 2023), multimodal tasks (Zeng et al., 2022), and robotics (Ahn et al., 2022), which, however, might cause additional computational costs and latency.",
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+    {
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+        "text": "Broad Generalization across the Internet Because open-loop planning with prompted Flan-U-PaLM achieves at most  $10 - 30\\%$  success, we have demonstrated that self-experience supervision on real websites is essential for planning modules. As we demonstrated in Mind2Web, our method could generalize across the internet if we have enough data. It would be expected to collect demonstrations at scale and align larger domain-expert models with them in future works.",
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+    {
+        "type": "text",
+        "text": "Feedback for Program Synthesis We leverage Flan-U-PaLM with 540B parameters, as a capable program synthesis module via few-shot prompting. Such a large model, however, makes it challenging to reflect the feedback about the errors in generated code, compared to smaller models. We leave it as future direction to incorporate the feedback for program synthesis into larger language models.",
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+    {
+        "type": "text",
+        "text": "Evaluation for Real-world Web Automation Beyond the simulated web environments (Shi et al., 2017; Yao et al., 2022a), we have exhibited WebAgent can follow given complex and sometimes ambiguous instructions on real estate, social media and map websites. On the other hand, it is costly to evaluate the performance of autonomous agents in the real world. Automated evaluation with minimal human intervention would be helpful for the scalable development of real-world web agents.",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We build a system for real-world web automation, combining HTML-T5 for planning and HTML summarization and Flan-U-PaLM for grounded program synthesis. Our proposed WebAgent achieves around  $70 - 80\\%$  success on real websites via self-experience supervision, outperforming single LLM approach by over  $50\\%$ , which suggests dividing the sequence of sub-problems with multiple language models can increase the entire task success. We also propose a scalable recipe for HTML-specialized language models where we train local and global attention mechanisms with a mixture of long-span denoising objectives to capture the hierarchical structures of HTML documents. HTML-T5 not only plays an essential role in WebAgent but also can achieve the best results on a variety of HTML-based benchmarks such as Mind2Web and MiniWoB++. We hope our work contributes to getting us one-step closer to the practical deployment of autonomous web agent systems.",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+    },
+    {
+        "type": "page_number",
+        "text": "9",
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+    {
+        "type": "text",
+        "text": "ETHICS STATEMENT",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This paper presents encouraging evidence of autonomous agents' potential for deployment on real websites, extending beyond simulated environments. In the foreseeable future, this technology could lead to the development of sophisticated AI assistant tools for computers and smartphones, enhancing productivity and accessibility for society.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "While we recognize the promising aspects of autonomous agents, we must also consider the potential for misuse and unintended consequences in their development. As our proposed system is based on LLMs, there is a risk of prompt injection. The improper use of web automation could pose cybersecurity threats and expose users to scams. To mitigate these risks, it is crucial for researchers, policymakers, and industry stakeholders to collaborate on establishing guidelines and regulations for the development of autonomous agents. Additionally, security research focused on LLM agents will become an essential domain for society.",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS",
+        "text_level": 1,
+        "bbox": [
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+            311,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "We thank Heiga Zen, Yingjie Miao, Yusuke Iwasawa, Joshua Ainslie, Santiago Ontonon, Quoc V. Le, Zoubin Ghahramani, Jeff Dean, Tris Warkentin for the supports and advises on this work. Hiroki Furuta was supported by JSPS KAKENHI Grant Number JP22J21582.",
+        "bbox": [
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+    {
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+        "text": "Published as a conference paper at ICLR 2024",
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+    },
+    {
+        "type": "page_number",
+        "text": "16",
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+    {
+        "type": "text",
+        "text": "APPENDIX",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "A NOTE FOR REAL-WORLD EVALUATION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            135,
+            532,
+            151
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "The development of autonomous agents should consider the security and safety aspects. In the real website evaluation, we have carefully conducted the experiments under human supervision in case undesired behaviors happen. We use Selenium WebDriver<sup>2</sup>, a popular library for browser automation, and limit the access per second not to stress the server. We have anonymized the real websites we tested on for safety and privacy concerns.",
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+    {
+        "type": "text",
+        "text": "B EXTENDED RELATED WORKS",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Document Understanding Understanding structural documents has been a practical challenge for transformer-based language models. Prior works employ layout-informed tokens (Xu et al., 2019) or even multimodal tokens from visual inputs (Appalaraju et al., 2021; Li et al., 2021a;c). Especially, for the documents written in markup languages, text-XPath alignment (Li et al., 2021b), token separation between text and HTML (Wang et al., 2022a), or extra topological information of HTML (Zhao et al., 2022) are proposed to leverage their syntax better. On the other hand, such a domain knowledge conflicts with recent generalist and scaling trends around LLMs (Anil et al., 2023; OpenAI, 2023). Because web agents require the instruction-conditioned HTML understanding, it also would be desirable to reconcile specialist aspects for HTML documents with generalist capabilities for natural language tasks. In this work, we design HTML-T5 to incorporate the structural bias of HTML by combining local-global attention for the encoder and a mixture of long-span denoising, while it can solve instruction-following better in downstream web-based tasks.",
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+    {
+        "type": "text",
+        "text": "LLM for Task Planning The prior knowledge of commonsense in LLMs has allowed us to leverage them for a variety of task planning. For instance, Huang et al. (2022) propose LLM agent that generates natural language plans in an open-loop manner. Nottingham et al. (2023) and Wang et al. (2023b) perform sequential closed-loop planning on MineCraft. Singh et al. (2022) decode robotic plans with pythonic text, and several works incorporate planning definition and domain language into the outputs (Liu et al., 2023; Silver et al., 2023; Valmeekam et al., 2023). On the other hand, our WebAgent leverages finetuned specialist language models and performs closed-loop planning coupled with HTML summarization by decomposing given instructions. We empirically prove that our system is superior to open-loop planning with a single generalist LLM with prompting.",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+    },
+    {
+        "type": "footer",
+        "text": "2https://www.selenium.dev/",
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+        "text": "17",
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+    {
+        "type": "text",
+        "text": "C IMPLEMENTATION DETAILS OF HTML-T5",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We use the implementation of local and global attentions released by Guo et al. (2022)  $^{3}$ . Following Guo et al. (2022), we set the local radius to  $r = 127$ , and block size for transient global attention to  $k = 16$ . For the pre-training objective, similar to Tay et al. (2022), we construct the mixtures and then use long mean span lengths:  $\\mu \\in \\{8,64\\}$ , and all the denoising ratio (percentage of masked tokens in the input sequence) is set to 0.15. We adopt 4096 input sequence length and 910 output sequence length during the pre-training. The batch size for training is set to 128. We train the models with 100K iterations following other pre-training strategies for T5 families (Chung et al., 2022; Lester et al., 2021). We leverage SeqIO (Roberts et al., 2022) and T5X (Roberts et al., 2022) library to manage the training pipeline. We also use SentencePiece (Kudo & Richardson, 2018) with 32K tokens from C4 dataset (Raffel et al., 2020) as a tokenizer. During the downstream finetuning, we adopt 16K tokens for the context window unless otherwise mentioned. We have used cloud TPU-v3, which has a 32 GiB HBM memory space, with 128 cores for the experiments.",
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+    },
+    {
+        "type": "text",
+        "text": "For the dataset, we prepare 100 WARC files (April 2019) from CommonCrawl<sup>4</sup>, and pre-process the raw HTML by removing non-Unicode and alphanumeric documents and extracting subtrees around <label> elements that have for attribute, to reduce the noise in training corpus, which results in about 3.41M examples (Table 5).",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/7fd6cf739a084a3e5b5cc4f233dfdb36c0a7270db79023960e95f4261455c534.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"># of examples</td><td colspan=\"3\"># of tokens</td></tr><tr><td>Average</td><td>90th</td><td>Max</td></tr><tr><td>3.41M</td><td>1020</td><td>4566</td><td>7627</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Table 5: Statistics of CommonCrawl HTML corpus for self-supervised denoising pre-training of HTML-T5. Input lengths are measured in tokens by Kudo & Richardson (2018).",
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+    },
+    {
+        "type": "text",
+        "text": "D WEBAGENT EXAMPLE FLOW IN REAL-ESTATE WEBSITE",
+        "text_level": 1,
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+        ],
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+    },
+    {
+        "type": "image",
+        "img_path": "images/88fc5fcee87490b47b5253d1f3eb1a8bb982422cc47b34b196f0900168fada52.jpg",
+        "image_caption": [
+            "Figure 6: An example flow with planning, summarization, and grounded program synthesis in the real estate website. HTML-T5 iteratively predicts a decomposed sub-instruction and task-relevant snippet (orange) in a closed-loop manner, conditioning on the HTML documents, instruction (yellow), and history of past predictions (green). Flan-U-PaLM is prompted with sub-instruction and snippet (orange) to decode python programs (blue)."
+        ],
+        "image_footnote": [],
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+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+    },
+    {
+        "type": "page_footnote",
+        "text": "<sup>3</sup>https://github.com/google-research/longt5",
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+    },
+    {
+        "type": "page_footnote",
+        "text": "4https://commoncrawl.org/",
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+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "page_number",
+        "text": "18",
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+    },
+    {
+        "type": "text",
+        "text": "E REAL-WORLD WEB AUTOMATION WITH DIFFERENT GENERALIST LLMS",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We compare different generalist LLMs as a module of WebAgent among model-size variants (Flan-PaLM-8B, Flan-PaLM-62B, Flan-U-PaLM-540B), and publicly accessible LLM (gpt-3.5-turbo). We test those models on map website following the same 20 instructions in Appendix G. The results in Figure 7 imply that the performance of Flan-U-PaLM-540B and gpt-3.5-turbo are the same (80% success, 93.8% score), and Flan-PaLM-62B (60% success, 86.3% score) is lower than Flan-U-PaLM-540B, which is caused by the inaccurate program synthesis. In addition, Flan-PaLM-8B could not generate proper programs at all. We believe that any LLM with sufficient program synthesis capabilities could be integrated into WebAgent, including Flan-U-PaLM-540B.",
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+    },
+    {
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+        "img_path": "images/916e50d74c020f9cb0d236afab32dc1ed65a7d2c884e2096b5a230912f277f9f.jpg",
+        "image_caption": [
+            "Figure 7: Performance (left) and error analysis (right) of real-world web automation with different generalist LLMs. We compare model-size variants (Flan-PaLM-8B, Flan-PaLM-62B, Flan-U-PaLM-540B) and public LLM (gpt-3.5-turbo) on map website."
+        ],
+        "image_footnote": [
+            "Flan-PaLM-8B Flan-PaLM-62B Flan-U-PaLM-540B gpt-3.5-turbo"
+        ],
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+        ],
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+    },
+    {
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+        "img_path": "images/46597798c14ef302facae1af40771a7f89def6ac32be30e6a2cb521024bf3eac.jpg",
+        "image_caption": [],
+        "image_footnote": [],
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+    },
+    {
+        "type": "text",
+        "text": "F WEBSRC:STATIC HTML COMPREHENSION",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "To emphasize the advantage of our modular approach, we test WebAgent on a static website comprehension benchmark, WebSRC (Chen et al., 2021b), which is a contextual QA dataset with HTML documents. The questions require an understanding of the spatial and logical structure of websites, and the answers are either text span on HTML or yes/no. For the comprehensive evaluation, WebSRC has three different types of websites, KV, Comparison, and Table. KV task is a value extraction from the attribute key. Comparison task has several entities with the same attributes. Table task requires a structural understanding with header columns and values in the row. We finetune HTML-T5 for snippet extraction to predict data-ref corresponding to the answer and use dev set for the evaluation.",
+        "bbox": [
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+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "As did in real-world web automation, HTML-T5 first predicts data-ref attribute of task-relevant snippet from the input HTML document. To make sure there is enough context, we extract the snippet from the predicted element to the two-level-up via XPath. If it exceeds the context length of Flan-U-PaLM, we limit it into parent elements. If it still does not work, we truncate the end of extracted snippet to fit within the token budget. Because snippet extraction in table structure often loses the context to solve question-answering, we just truncate HTML documents for Table tasks. Flan-U-PaLM predicts the answers seeing 5-shot examples.",
+        "bbox": [
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+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "As shown in Table 6, single LLM, such as Flan-U-PaLM or HTML-T5, has struggled to the limited context length or model capacity. In contrast, WebAgent, our LLM-collaborative approach, enhances the performance from both single generalist and specialist LLMs, and shows competitive results with strong baselines. This demonstrates that modular LLMs work complementarily to each other. Figure 8 presents the performance comparison on different types of websites (KV, Comparison, Table) among MarkupLM (Li et al., 2021b), TIE (Zhao et al., 2022), and WebAgent. WebAgent is better at Comparison tasks, but inferior to structural understanding for KV and Table tasks, compared to other baselines, which suggest that generalist LLMs are still not suitable for recognizing structural data such as table.",
+        "bbox": [
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+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "page_number",
+        "text": "19",
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+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/ab044febc4916d8d39054cd7a0f69aa0a1a4734f3c6d41477f739b7295982854.jpg",
+        "image_caption": [
+            "Figure 8: The performance comparison on different types of websites in WebSRC dev set."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "image",
+        "img_path": "images/ebfb855adef0edfd3f212c3985046599db77f08845be4188ad76bdbb01c67deb.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "table",
+        "img_path": "images/e62923af5df1b07880da83058794ac564ed9b2e6242aa8b7b3012f5ef4667a81.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Models</td><td>EM</td><td>F1</td></tr><tr><td>T-PLM (Chen et al., 2021b)</td><td>61.67</td><td>69.85</td></tr><tr><td>H-PLM (Chen et al., 2021b)</td><td>70.12</td><td>74.14</td></tr><tr><td>V-PLM (Chen et al., 2021b)</td><td>73.22</td><td>76.16</td></tr><tr><td>MarkupLM-Large (Li et al., 2021b)</td><td>74.43</td><td>80.54</td></tr><tr><td>TIE-Large (Zhao et al., 2022)</td><td>81.66</td><td>86.24</td></tr><tr><td>Flan-U-PaLM</td><td>40.01</td><td>47.56</td></tr><tr><td>HTML-T5-Large</td><td>73.09</td><td>76.66</td></tr><tr><td>HTML-T5-XL</td><td>74.73</td><td>78.73</td></tr><tr><td>WebAgent</td><td>75.50</td><td>85.75</td></tr><tr><td>WebAgent (oracle)</td><td>76.91</td><td>86.64</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Table 6: Evaluation on WebSRC (Chen et al., 2021b) with dev set. WebAgent, our collaborative LLMs, enhances the performance from both single generalist (Flan-U-PaLM) or specialist LLMs (HTML-T5). WebAgent (oracle) uses oracle snippets that are guaranteed to include the answers, instead of those predicted by finetuned HTML-T5.",
+        "bbox": [
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+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "page_number",
+        "text": "20",
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+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "G LIST OF LANGUAGE INSTRUCTIONS FOR REAL-WORLD WEB AUTOMATION",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "real-estate",
+        "text_level": 1,
+        "bbox": [
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+            132,
+            285,
+            143
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1. can you search for a studio bedroom,  $1+$  bathroom houses in escondido, ca for corporate housing and price less than 12100 on real estate website.",
+            "2. can you find me a studio bedroom,  $1+$  bathroom townhomes in hollywood, ca and price less than 14600 on real estate website.",
+            "3. can you search for a studio bedroom, 1+ bathroom condos in inglewood, ca for senior housing and price less than 8700 on real estate website.",
+            "4. I would like to search for a studio bedroom,  $1+$  bathroom houses in compton, ca and price more than 1200 for corporate housing on real estate website.",
+            "5. can you search for a studio bedroom,  $1+$  bathroom apartments in oroville, ca for corporate housing on real estate website.",
+            "6. find me a studio bedroom,  $1+$  bathroom houses in modesto, ca on real estate website.",
+            "7. can you search for a studio bedroom,  $1+$  bathroom condos in redwood city, ca for student and price more than 1900 on real estate website.",
+            "8. find me a 1 bedroom condos in Santa clara, ca and price between 1600 and 7400 on real estate website.",
+            "9. find me a 1 bedroom,  $3+$  bathroom apartments in martinez, ca with min price 1800 on real estate website.",
+            "10. can you find me a 2 bedroom,  $2+$  bathroom townhomes in concord, ca and price more than 600 on real estate website.",
+            "11. can you find me a studio bedroom,  $2+$  bathroom apartments in san diego, ca and price less than 9300 on real estate website.",
+            "12. find me a studio bedroom houses in novato, ca and price between 1500 and 6700 on real estate website.",
+            "13. can you find me a studio bedroom, any bathroom townhomes in petaluma, ca and price more than 1000 on real estate website.",
+            "14. search for a 1 bedroom apartments in modesto, ca and price more than 1000 on real estate website.",
+            "15. find me a 1 bedroom,  $2+$  bathroom apartments in watts, ca for senior housing less than 6300 on real estate website.",
+            "16. can you find me a 1 bedroom houses in victorville, ca that have dog friendly, furnished and price more than 700 on real estate website.",
+            "17. I need a 2 bedroom, any bathroom condos in inglewood, ca and price more than 1000 on real estate website.",
+            "18. find me a 2 bedroom,  $2+$  bathroom apartments in livermore, ca on real estate website.",
+            "19. can you find me a 2 bedroom apartments in Santa clara, ca that has parking and price less than 10300 on real estate website.",
+            "20. can you search for a 2 bedroom condos in oakland, ca on real estate website."
+        ],
+        "bbox": [
+            169,
+            152,
+            825,
+            595
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "social-media",
+        "text_level": 1,
+        "bbox": [
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+            609,
+            294,
+            621
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1. Show me the most hot thread in r/google at social media website.",
+            "2. Can you point out the most hot thread in r/learnpython at social media website.",
+            "3. Could you check the 1st hot thread in r/artificial at social media website.",
+            "4. Can I check the most hot thread in Taiwan on social media website.",
+            "5. Show me the first new thread in r/facebook at social media website.",
+            "6. Present the most new thread of r/Python filtered by Tutorial flair on social media website.",
+            "7. Could you check the 1st new thread in r/facebook at social media website.",
+            "8. I want to read the 1st hot thread from r/Python tagged as Daily Thread at social media website.",
+            "9. Present the most hot thread of r/google filtered by Info | Mod Post flair on social media website.",
+            "10. Show me the most new thread in r/learnmachinelearning filtered by Help flair at social media website.",
+            "11. Can you point out the first hot thread in r/deeplearning at social media website.",
+            "12. Could you check the 1st hot thread in r/machinelearningnews at social media website.",
+            "13. Present the most hot thread of r/artificial filtered by News flair on social media website.",
+            "14. Please find me the first hot thread in r/facebook at social media website.",
+            "15. Present the most new thread of r/machinelearningnews filtered by Startup News flair on social media website.",
+            "16. Show me the most hot thread in r/artificial filtered by AI Art flair at social media website.",
+            "17. Could you check the first new thread in r/facebook at social media website.",
+            "18. I want to read the most top thread from r/google tagged as Info | Mod Post at social media website.",
+            "19. Show me the most new thread in r/startups filtered by Share Your Startup flair at social media website.",
+            "20. Could you check the 2nd new thread in r/facebook at social media website."
+        ],
+        "bbox": [
+            169,
+            628,
+            825,
+            921
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "page_number",
+        "text": "21",
+        "bbox": [
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+            946,
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+            959
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "map",
+        "text_level": 1,
+        "bbox": [
+            171,
+            104,
+            205,
+            118
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1. Show me the way from San Jose to Mountain View by 2nd Cycling at map website.",
+            "2. Please show me the way from The Painted Ladies to San Francisco Zoo with 3rd Best option at map website.",
+            "3. Could you tell me the path from California Academy of Sciences to de Young Museum by 1st Transit at map website.",
+            "4. Could you tell me the way from Union Square to The Painted Ladies with 2nd Cycling option at map website.",
+            "5. Please present the way from Chappell Hayes Observation Tower to San Jose with 2nd Walking option at map website.",
+            "6. Please present the path from Jack London Square to Emeryville by 2nd Cycling at map website.",
+            "7. I'd like to move The Midway from Children's Fairyland by 1st Cycling at map website.",
+            "8. I'd like to move Chase Center from San Francisco - Oakland Bay Bridge with 2nd Transit option at map website.",
+            "9. I want to move Pier 39 from Berkeley by 3rd Cycling at map website.",
+            "10. I want to go to Emeryville from Mountain View with 2nd Cycling option at map website.",
+            "11. Can you point out the way from San Mateo to Stanford University by 2nd Cycling at map website.",
+            "12. Could you point out the way from Palace of Fine Arts to UC Berkeley by 1st Cycling at map website.",
+            "13. Point out the way from The Painted Ladies to San Francisco Museum of Modern Art by 2nd Driving at map website.",
+            "14. Could you find the path from Union Square to Palo Alto by 1st Cycling at map website.",
+            "15. Please check the way from San Jose to San Jose Mineta International Airport with 1st Walking at map website.",
+            "16. Check the path from San Francisco Zoo to Berkeley with 1st Cycling at map website.",
+            "17. I'd like to check Parking Lots along the way from Stanford University to The Painted Ladies with Best option at map website.",
+            "18. Check Gas stations along the way from de Young Museum to Oakland with Driving option at map website.",
+            "19. Please show me Hotels along the way from Palace of Fine Arts to Berkeley by Transit at map website.",
+            "20. Check Gas stations along the way from Bay Area Discovery Museum to Santa Cruz with Best option at map website."
+        ],
+        "bbox": [
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+            525
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "H EXAMPLE EPISODE IN REAL-WORLD WEB AUTOMATION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            547,
+            687,
+            563
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
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+            47
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "page_number",
+        "text": "22",
+        "bbox": [
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+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "image",
+        "img_path": "images/81e6d9229d40e0f549a16d94f91441de1e3686b68edeef85c4e8ff8bdec041bf.jpg",
+        "image_caption": [
+            "map: Show me the way from San Jose to Mountain View by 2nd Cycling at map website?",
+            "Figure 9: Example episodes of real-world web automation in map domain."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            383,
+            63,
+            607,
+            917
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "page_number",
+        "text": "23",
+        "bbox": [
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+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "I EXTENSIVE ABLATION OF HTML-T5",
+        "text_level": 1,
+        "bbox": [
+            171,
+            102,
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+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "I.1 DATASET AND INITIALIZATION",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "To test our recipe described in Section 2.1, we compare the different dataset and model initialization for pre-training on downstream task performances; offline task planning on real-estate and average success rate on MiniWoB with 12K dataset. We use Base-size models for the experiments. For HTML-denoising, we prepare the corpus from CommonCrawl with (Extracted) or without (Raw) subtree extraction around label elements on the documents. We also compare the initialization of base architectures before HTML-denoising; from scratch or with pre-trained models on PEGASUS objective (Zhang et al., 2020) that is a masked important sentence prediction from long-context paragraph. Table 7 reveals that snippet extraction on HTML corpus improves downstream performances since such a pre-processing can reduce the noise in raw HTML. Moreover, initialization with PEGASUS pre-trained weights is essential for HTML-T5, because of the long-context and instruction-following nature of HTML-based tasks.",
+        "bbox": [
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+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "table",
+        "img_path": "images/e72f0a66b5e4c844fcfdcb078b029ff38d0aff81f8a62c9d62bfd223455d1f22.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>CC-HTML</td><td>PEGASUS</td><td>real-estate</td><td>MiniWoB++</td></tr><tr><td>Raw</td><td>✓</td><td>80.56</td><td>56.7%</td></tr><tr><td>Extracted</td><td>×</td><td>67.11</td><td>49.1%</td></tr><tr><td>Extracted</td><td>✓</td><td>82.46</td><td>57.0%</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "I.2 OFFLINE EVALUATION ON TASK PLANNING WITH MODEL SCALING",
+        "text_level": 1,
+        "bbox": [
+            171,
+            537,
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+            551
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "We compare the offline task planning performance between HTML-T5 and LongT5 (without HTML-denosing) with different model sizes; with Base (220M parameters), Large (770M parameters), and XL (3B parameters). As described in Section 3.1, the models predict the next sub-instructions in a closed-loop manner considering the current HTML observations, user instructions, and previous sub-instruction histories as inputs. For offline task planning evaluation, we use the demonstrations on real-estate website; preparing 130 demonstrations and splitting them into train  $(90\\%)$  and test splits  $(10\\%)$ . We report the best per-step exact match accuracy in test set.",
+        "bbox": [
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+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "Table 8 shows that HTML-T5 outperforms LongT5 on the accuracy of sub-instruction prediction, which demonstrates that HTML-denoising pre-training captures the structural bias of HTML better without sacrificing the ability to understand natural language instructions. This also implies that our proposed HTML-denoising can scale to larger-size models consistently.",
+        "bbox": [
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+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "table",
+        "img_path": "images/b6a2098fa72611736d26b491a3accdd118be85e7ba4ac9da15586a02cc8b6473.jpg",
+        "table_caption": [
+            "Table 7: Ablations of HTML-T5-Base on dataset quality and initialization. We evaluate offline task planning on real-estate and average success rate on MiniWoB with 12K dataset. For HTML-denoising, we prepare HTML corpus from CommonCrawl with (Extracted) or without (Raw) subtree extraction around label elements. We also compare the pre-training of base architectures with PEGASUS objective (Zhang et al., 2020) before HTML-denoising. The results imply that PEGASUS pre-training is critical for the architectures and preprocessing with subtree extraction improves the downstream performance on HTML-based tasks."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Models</td><td>real-estate</td><td>Diff.</td></tr><tr><td>LongT5-Base</td><td>78.07</td><td>0.0</td></tr><tr><td>LongT5-Large</td><td>82.89</td><td>0.0</td></tr><tr><td>LongT5-XL</td><td>81.29</td><td>0.0</td></tr><tr><td>HTML-T5-Base</td><td>82.46</td><td>+4.39</td></tr><tr><td>HTML-T5-Large</td><td>83.63</td><td>+0.74</td></tr><tr><td>HTML-T5-XL</td><td>83.92</td><td>+2.63</td></tr></table>",
+        "bbox": [
+            351,
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+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "Table 8: Accuracy of offline evaluation on task planning. We leverage the demonstrations in real-estate websites. Compared to original LongT5, and as we scale model size, HTML-T5 improves the accuracy of sub-instruction prediction.",
+        "bbox": [
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+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "page_number",
+        "text": "24",
+        "bbox": [
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+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "I.3 DESCRIPTION GENERATION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            103,
+            406,
+            118
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "We also investigate the capability of HTML-T5 on static HTML comprehension tasks, as well as interactive decision making tasks. We use Description Generation benchmark (Gur et al., 2022), where the models generate the textual description of elements, typically used for accessibility purposes and annotated with a special attribute in the HTML schema known as for. We evaluate the understanding the structure of HTML as it would appear to a user, despite not having access to the rendered website directly.",
+        "bbox": [
+            169,
+            128,
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+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "We compare LaMDA (Thoppilan et al., 2022), T5, LongT5, and HTML-T5 with respect to accuracy, BLEU (Papineni et al., 2002), and ROUGE-1 (Lin, 2004) score. As shown in Table 9, local and global attention mechanisms, underlying between LongT5 and HTML-T5, could almost solve the benchmark by improving the previous best performance by over  $10\\%$ , with still improved performance as model size increases. Compared to the effect of local-global attention, HTML-T5 marginally improves against LongT5, which emphasizes that local and global attentions are critical to capture the hierarchical structure of HTML documents.",
+        "bbox": [
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+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "table",
+        "img_path": "images/35326e793ea4643db518c7985bda6eade5dc3fd51e65af27e37ff968e6aecd83.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Models</td><td colspan=\"3\">Dev</td><td colspan=\"3\">Test</td></tr><tr><td>Accuracy</td><td>BLEU</td><td>ROUGE-1</td><td>Accuracy</td><td>BLEU</td><td>ROUGE-1</td></tr><tr><td>LaMDA-1B (Gur et al., 2022)</td><td>83.3</td><td>87.5</td><td>90.2</td><td>84.3</td><td>88.6</td><td>91.2</td></tr><tr><td>T5-Large (Gur et al., 2022)</td><td>83.2</td><td>90.2</td><td>90.5</td><td>84.3</td><td>91.7</td><td>91.5</td></tr><tr><td>T5-XL (Gur et al., 2022)</td><td>84.0</td><td>90.8</td><td>90.9</td><td>85.2</td><td>92.1</td><td>91.9</td></tr><tr><td>LongT5-Base</td><td>96.4</td><td>98.0</td><td>98.5</td><td>95.6</td><td>97.4</td><td>98.2</td></tr><tr><td>LongT5-Large</td><td>98.1</td><td>98.9</td><td>99.2</td><td>97.7</td><td>98.5</td><td>99.0</td></tr><tr><td>LongT5-XL</td><td>98.4</td><td>99.1</td><td>99.3</td><td>98.5</td><td>99.2</td><td>99.3</td></tr><tr><td>HTML-T5-Base</td><td>96.5</td><td>98.1</td><td>98.6</td><td>95.9</td><td>97.5</td><td>98.3</td></tr><tr><td>HTML-T5-Large</td><td>98.1</td><td>98.9</td><td>99.2</td><td>97.7</td><td>98.3</td><td>99.1</td></tr><tr><td>HTML-T5-XL</td><td>98.4</td><td>99.0</td><td>99.3</td><td>98.9</td><td>99.4</td><td>99.5</td></tr></table>",
+        "bbox": [
+            207,
+            327,
+            792,
+            484
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "Table 9: Results of Description Generation benchmark (Gur et al., 2022). We compare LaMDA (Thoppilan et al., 2022), T5, LongT5, and HTML-T5 with respect to accuracy, BLEU, and ROUGE-1 scores. The results demonstrate that local and global attention mechanisms, shared modules between LongT5 and HTML-T5, could almost completely solve the benchmark by improving the previous best performance by over  $10\\%$ . HTML-T5 slightly outperforms LongT5.",
+        "bbox": [
+            169,
+            491,
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+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "page_number",
+        "text": "25",
+        "bbox": [
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+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "J FLAN-LONGT5",
+        "text_level": 1,
+        "bbox": [
+            171,
+            102,
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+            118
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "In the web automation literature (Furuta et al., 2023; Kim et al., 2023), instruction-finetuned LLMs have great success in HTML comprehension and improve the task success. For the comparison to HTML-denosing, we prepare the instruction-finetuned LongT5 (i.e. Flan-LongT5) by leveraging Flan dataset released by Chung et al. (2022). We finetuned the pre-trained LongT5 with 100K iterations and picked up the best checkpoints.",
+        "bbox": [
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+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "As a sanity check of instruction-tuning, we evaluate Flan-LongT5 with few-shot/zero-shot settings on CoT benchmark (GSM8K (Cobbe et al., 2021), StrategyQA (Geva et al., 2021), SVAMP (Patel et al., 2021), Asdiv (Miao et al., 2021), CommonsenseQA (Talmor et al., 2019)), BigBench-Hard (BBH) (Suzgun et al., 2022), and MMLU (Hendrycks et al., 2021b) as tested in Longpre et al. (2023). We reevaluate the performance of Flan-T5, using official checkpoints  $^{5}$ . We also check the performance of Flan-LongT5 on downstream summarization tasks, originally evaluated on LongT5 (Guo et al., 2022). We use arXiv (Cohan et al., 2018), PubMed (Cohan et al., 2018), BigPatent (Sharma et al., 2019), Multi-News (Fabbri et al., 2019), MediaSum (Zhu et al., 2021), CNN / Daily Mail (Nallapati et al., 2016) dataset for the evaluation, measuring the performance with ROUGE-1/2/L metrics.",
+        "bbox": [
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+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "Table 10 shows that we have successfully replicated the LongT5 version of instruction-finetuned language models. Flan-LongT5 achieves competitive results to original Flan-T5; for instance, Flan-LongT5-Large (36.64) outperforms Flan-T5-Large (35.25), but Flan-LongT5-XL (39.05) is still behind Flan-T5-XL (43.03) on average. This might be caused by the training instability of XL-size models (Guo et al., 2022). Because, unlike HTML-T5 on HTML-based tasks, reasoning tasks do not have long-context or hierarchical syntax, it is not surprising for Flan-LongT5 not to outperform Flan-T5. Table 11 also demonstrates that we have successfully conducted instruction-tuning without losing the capability of long text summarization.",
+        "bbox": [
+            169,
+            356,
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+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
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+            47
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "footer",
+        "text": "$^{5}$ https://github.com/google-research/t5x/blob/main/docs/models.md#flan-t5-checkpoints",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "page_number",
+        "text": "26",
+        "bbox": [
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+            960
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "table",
+        "img_path": "images/fe6ead2c0abfc36a13c12d8d3ec35576cf8b18243e0631d2d6096adae397649c.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 10: Performance of Flan-LongT5 on reasoning tasks. We reevaluate the performance of Flan-T5 (Chung et al., 2022), using official checkpoints. Flan-LongT5 achieves competitive results to original Flan-T5."
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\">Models</td><td colspan=\"2\">CoT</td><td colspan=\"2\">MMLU</td><td colspan=\"2\">BBH</td><td colspan=\"2\">BBH-CoT</td><td colspan=\"2\">Avg.</td></tr><tr><td>Zero</td><td>Few</td><td>Zero</td><td>Few</td><td>Zero</td><td>Few</td><td>Zero</td><td>Few</td><td>CoT</td><td>Direct</td></tr><tr><td>Flan-T5-Large</td><td>35.14</td><td>40.03</td><td>40.68</td><td>45.12</td><td>25.90</td><td>37.48</td><td>26.17</td><td>31.45</td><td>33.20</td><td>37.29</td></tr><tr><td>Flan-T5-XL</td><td>51.74</td><td>52.64</td><td>50.76</td><td>52.40</td><td>26.09</td><td>40.96</td><td>34.12</td><td>35.62</td><td>43.53</td><td>42.55</td></tr><tr><td>Flan-LongT5-Large</td><td>44.78</td><td>45.34</td><td>38.44</td><td>40.03</td><td>28.67</td><td>34.67</td><td>29.38</td><td>31.85</td><td>37.84</td><td>35.45</td></tr><tr><td>Flan-LongT5-XL</td><td>48.78</td><td>50.02</td><td>43.44</td><td>44.74</td><td>26.53</td><td>37.77</td><td>29.09</td><td>32.01</td><td>39.97</td><td>38.12</td></tr></table>",
+        "bbox": [
+            174,
+            232,
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+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "table",
+        "img_path": "images/b7c47b9b48bcc626a490d20b3db147f718e7b53f02e6ae021d9656c5b5363c71.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 11: Performance of Flan-LongT5 on downstream summarization tasks, compared to LongT5 (Guo et al., 2022). We measure the performance with ROUGE-1/2/L metrics."
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\">Models</td><td colspan=\"3\">arXiv</td><td colspan=\"3\">PubMed</td><td colspan=\"3\">BigPatent</td><td colspan=\"3\">MultiNews</td><td colspan=\"3\">MediaSum</td><td colspan=\"3\">CNN / Daily Mail</td></tr><tr><td>R-1</td><td>R-2</td><td>R-L</td><td>R-1</td><td>R-2</td><td>R-L</td><td>R-1</td><td>R-2</td><td>R-L</td><td>R-1</td><td>R-2</td><td>R-L</td><td>R-1</td><td>R-2</td><td>R-L</td><td>R-1</td><td>R-2</td><td>R-L</td></tr><tr><td>LongT5-Large</td><td>48.28</td><td>21.63</td><td>44.11</td><td>49.98</td><td>24.69</td><td>46.46</td><td>70.38</td><td>56.81</td><td>62.73</td><td>47.18</td><td>18.44</td><td>24.18</td><td>35.54</td><td>19.04</td><td>32.20</td><td>42.49</td><td>20.51</td><td>40.18</td></tr><tr><td>LongT5-XL</td><td>48.35</td><td>21.92</td><td>44.27</td><td>50.23</td><td>24.76</td><td>46.67</td><td>76.87</td><td>66.06</td><td>70.76</td><td>48.17</td><td>19.43</td><td>24.94</td><td>36.15</td><td>19.66</td><td>32.80</td><td>43.94</td><td>21.40</td><td>41.28</td></tr><tr><td>Flan-LongT5-Large</td><td>48.52</td><td>22.00</td><td>44.46</td><td>50.46</td><td>25.08</td><td>46.96</td><td>70.53</td><td>57.13</td><td>63.02</td><td>47.76</td><td>18.99</td><td>24.52</td><td>35.71</td><td>19.18</td><td>32.33</td><td>43.13</td><td>20.89</td><td>37.28</td></tr><tr><td>Flan-LongT5-XL</td><td>48.37</td><td>21.75</td><td>44.22</td><td>50.23</td><td>24.75</td><td>46.73</td><td>76.31</td><td>65.17</td><td>70.01</td><td>48.19</td><td>19.47</td><td>24.80</td><td>36.16</td><td>19.75</td><td>32.81</td><td>43.46</td><td>21.00</td><td>37.34</td></tr></table>",
+        "bbox": [
+            382,
+            79,
+            514,
+            916
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            174,
+            32,
+            477,
+            47
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "page_number",
+        "text": "27",
+        "bbox": [
+            490,
+            948,
+            506,
+            959
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "table",
+        "img_path": "images/77e5ff9425ef9f21dae079cbc384a317ac669836ac0e2aee759e0ac319fb5f5e.jpg",
+        "table_caption": [
+            "K PER-TASK PERFORMANCE ON MINIWOB++"
+        ],
+        "table_footnote": [
+            "Table 12: Per-task average success rate on 56 tasks from MiniWoB++. We refer to Furuta et al. (2023) and Gur et al. (2022) for the baseline performances."
+        ],
+        "table_body": "<table><tr><td>Task</td><td>HTML-T5-XL (347K)</td><td>HTML-T5-XL (12K)</td><td>Plan-T5-XL (347K)</td><td>WebN-T5-XL (12K)</td></tr><tr><td>book-flight</td><td>0.99</td><td>0.00</td><td>0.48</td><td>0.00</td></tr><tr><td>choose-date</td><td>0.16</td><td>0.03</td><td>0.08</td><td>0.00</td></tr><tr><td>choose-date-easy</td><td>1.00</td><td>0.28</td><td>1.00</td><td>0.03</td></tr><tr><td>choose-date-medium</td><td>0.56</td><td>0.14</td><td>0.57</td><td>0.00</td></tr><tr><td>choose-list</td><td>0.22</td><td>0.19</td><td>0.16</td><td>0.26</td></tr><tr><td>click-button</td><td>1.00</td><td>0.92</td><td>0.98</td><td>1.00</td></tr><tr><td>click-button-sequence</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>click-checkboxes</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.96</td></tr><tr><td>click-checkboxes-large</td><td>0.90</td><td>0.94</td><td>0.98</td><td>0.22</td></tr><tr><td>click-checkboxes-soft</td><td>0.99</td><td>0.64</td><td>1.00</td><td>0.54</td></tr><tr><td>click-checkboxes-transfer</td><td>1.00</td><td>1.00</td><td>0.99</td><td>0.63</td></tr><tr><td>click-collapsible</td><td>1.00</td><td>0.41</td><td>1.00</td><td>0.00</td></tr><tr><td>click-collapsible-2</td><td>0.93</td><td>0.26</td><td>0.94</td><td>0.00</td></tr><tr><td>click-color</td><td>1.00</td><td>1.00</td><td>0.27</td><td>0.27</td></tr><tr><td>click-dialog</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>click-dialog-2</td><td>0.74</td><td>0.31</td><td>0.34</td><td>0.24</td></tr><tr><td>click-link</td><td>0.99</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>click-menu</td><td>0.37</td><td>0.26</td><td>0.41</td><td>0.37</td></tr><tr><td>click-option</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.87</td></tr><tr><td>click-pie</td><td>0.96</td><td>0.89</td><td>0.99</td><td>0.51</td></tr><tr><td>click-scroll-list</td><td>0.99</td><td>0.91</td><td>0.00</td><td>0.00</td></tr><tr><td>click-shades</td><td>0.00</td><td>0.05</td><td>0.00</td><td>0.00</td></tr><tr><td>click-shape</td><td>0.79</td><td>0.57</td><td>0.58</td><td>0.53</td></tr><tr><td>click-tab</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.74</td></tr><tr><td>click-tab-2</td><td>0.94</td><td>0.40</td><td>0.94</td><td>0.18</td></tr><tr><td>click-tab-2-hard</td><td>0.88</td><td>0.30</td><td>0.57</td><td>0.12</td></tr><tr><td>click-test</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>click-test-2</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>click-widget</td><td>1.00</td><td>0.94</td><td>1.00</td><td>1.00</td></tr><tr><td>count-shape</td><td>0.67</td><td>0.55</td><td>0.64</td><td>0.41</td></tr><tr><td>email-inbox</td><td>1.00</td><td>0.99</td><td>0.99</td><td>0.38</td></tr><tr><td>email-inbox-forward-nl</td><td>1.00</td><td>0.92</td><td>1.00</td><td>0.60</td></tr><tr><td>email-inbox-forward-nl-turk</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.33</td></tr><tr><td>email-inbox-nl-turk</td><td>0.99</td><td>0.76</td><td>0.92</td><td>0.23</td></tr><tr><td>enter-date</td><td>1.00</td><td>0.00</td><td>1.00</td><td>0.00</td></tr><tr><td>enter-password</td><td>1.00</td><td>0.99</td><td>1.00</td><td>0.97</td></tr><tr><td>enter-text</td><td>1.00</td><td>0.96</td><td>1.00</td><td>0.89</td></tr><tr><td>enter-text-dynamic</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.98</td></tr><tr><td>enter-time</td><td>1.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr><tr><td>focus-text</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>focus-text-2</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>grid-coordinate</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.49</td></tr><tr><td>guess-number</td><td>0.13</td><td>0.00</td><td>0.10</td><td>0.00</td></tr><tr><td>identify-shape</td><td>1.00</td><td>0.89</td><td>0.90</td><td>0.88</td></tr><tr><td>login-user</td><td>1.00</td><td>0.80</td><td>1.00</td><td>0.82</td></tr><tr><td>login-user-popup</td><td>1.00</td><td>0.63</td><td>0.97</td><td>0.72</td></tr><tr><td>multi-layouts</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.83</td></tr><tr><td>multi-orderings</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.88</td></tr><tr><td>navigate-tree</td><td>0.99</td><td>0.99</td><td>1.00</td><td>0.91</td></tr><tr><td>search-engine</td><td>0.93</td><td>0.55</td><td>0.59</td><td>0.34</td></tr><tr><td>social-media</td><td>0.99</td><td>0.93</td><td>0.99</td><td>0.21</td></tr><tr><td>social-media-all</td><td>0.31</td><td>0.84</td><td>0.09</td><td>0.00</td></tr><tr><td>social-media-some</td><td>0.89</td><td>0.60</td><td>0.39</td><td>0.02</td></tr><tr><td>tic-tac-toe</td><td>0.57</td><td>0.46</td><td>0.42</td><td>0.48</td></tr><tr><td>use-autocomplete</td><td>0.97</td><td>0.23</td><td>0.98</td><td>0.22</td></tr><tr><td>use-spinner</td><td>0.07</td><td>0.07</td><td>0.03</td><td>0.07</td></tr><tr><td>Average</td><td>0.856</td><td>0.655</td><td>0.755</td><td>0.484</td></tr></table>",
+        "bbox": [
+            184,
+            136,
+            810,
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+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            173,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "page_number",
+        "text": "28",
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+        ],
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+    }
+]
\ No newline at end of file
diff --git a/2024/A Real-World WebAgent with Planning, Long Context Understanding, and Program Synthesis/7b0b68ac-f5c6-4674-acc8-4425712f7a19_model.json b/2024/A Real-World WebAgent with Planning, Long Context Understanding, and Program Synthesis/7b0b68ac-f5c6-4674-acc8-4425712f7a19_model.json
new file mode 100644
index 0000000000000000000000000000000000000000..92abd15d8ea96fc94560068cd238ade51b42f223
--- /dev/null
+++ b/2024/A Real-World WebAgent with Planning, Long Context Understanding, and Program Synthesis/7b0b68ac-f5c6-4674-acc8-4425712f7a19_model.json	
@@ -0,0 +1,4194 @@
+[
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.1,
+                0.747,
+                0.172
+            ],
+            "angle": 0,
+            "content": "A REAL- WORLD WEBAGENT WITH PLANNING, LONG CONTEXT UNDERSTANDING, AND PROGRAM SYNTHESIS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.183,
+                0.194,
+                0.798,
+                0.224
+            ],
+            "angle": 0,
+            "content": "Izzeddin Gur\\(^{1*}\\) Hiroki Furuta\\(^{1,2*†}\\) Austin Huang\\(^{1}\\) Mustafa Safdari\\(^{1}\\) Yutaka Matsuo\\(^{2}\\) Douglas Eck\\(^{1}\\) Aleksandra Faust\\(^{1}\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.183,
+                0.224,
+                0.676,
+                0.254
+            ],
+            "angle": 0,
+            "content": "<sup>1</sup>Google DeepMind, <sup>2</sup>The University of Tokyo  \nizzeddin@google.com, furuta@weblab.t.u-tokyo.ac.jp"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.451,
+                0.29,
+                0.547,
+                0.305
+            ],
+            "angle": 0,
+            "content": "ABSTRACT"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.23,
+                0.321,
+                0.77,
+                0.545
+            ],
+            "angle": 0,
+            "content": "Pre-trained large language models (LLMs) have recently achieved better generalization and sample efficiency in autonomous web automation. However, the performance on real-world websites has still suffered from (1) open domaininess, (2) limited context length, and (3) lack of inductive bias on HTML. We introduce WebAgent, an LLM-driven agent that learns from self-experience to complete tasks on real websites following natural language instructions. WebAgent plans ahead by decomposing instructions into sub-instructions, summarizes long HTML documents into task-relevant snippets, and acts on websites via Python programs generated from those. We design WebAgent with Flan-U-PaLM, for grounded code generation, and HTML-T5, a new pre-trained LLM for long HTML documents using local and global attention mechanisms and a mixture of long-span denoising objectives, for planning and summarization. We empirically demonstrate that our modular recipe improves the success on real websites by over \\(50\\%\\), and that HTML-T5 is the best model to solve various HTML understanding tasks; achieving \\(18.7\\%\\) higher success rate than the prior method on MiniWoB web automation benchmark, and SoTA performance on Mind2Web, an offline task planning evaluation."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.174,
+                0.57,
+                0.338,
+                0.585
+            ],
+            "angle": 0,
+            "content": "1 INTRODUCTION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.601,
+                0.828,
+                0.7
+            ],
+            "angle": 0,
+            "content": "Large language models (LLM) (Brown et al., 2020; Chowdhery et al., 2022; OpenAI, 2023) can solve a variety of natural language tasks, such as arithmetic, commonsense, logical reasoning, question answering, text generation (Brown et al., 2020; Kojima et al., 2022; Wei et al., 2022), and even interactive decision making tasks (Ahn et al., 2022; Yao et al., 2022b). Recently, LLMs have also demonstrated success in autonomous web navigation by controlling computers or browsers to follow natural language instructions through multi-step reasoning and decision making (Furuta et al., 2023; Gur et al., 2022; Kim et al., 2023)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.706,
+                0.827,
+                0.888
+            ],
+            "angle": 0,
+            "content": "However, web automation on real-world websites has still suffered from (1) the lack of pre-defined action space, (2) much longer HTML documents than simulated observations, and (3) the absence of domain-specific knowledge for understanding HTML documents (Figure 1). Considering the open-endedness of real-world websites and the complexity of instructions, defining appropriate action spaces in advance is challenging. In addition, although several works have argued that recent LLMs with instruction-finetuning or reinforcement learning from human feedback improve HTML understanding and web automation accuracy (Furuta et al., 2023; Kim et al., 2023), their architectures are not always suitable to process real-world HTML documents; as presented in Figure 2, HTML tokens of real websites are much longer than those of simulators, and most LLMs have shorter context lengths than the average HTML tokens in real websites. It is prohibitively costly to treat such long documents as inputs directly, or adopt prior techniques such as text-XPath alignment (Li et al., 2021b) or text-HTML token separation (Wang et al., 2022a). To prioritize broader task generalization and model-size scaling, such domain knowledge for HTML documents is ignored in recent LLMs."
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.193,
+                0.897,
+                0.468,
+                0.925
+            ],
+            "angle": 0,
+            "content": "\\*Equal Contribution. \\(\\dagger\\) Work done as Student Researcher at Google."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.504,
+                0.96
+            ],
+            "angle": 0,
+            "content": "1"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.24,
+                0.102,
+                0.761,
+                0.22
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.171,
+                0.222,
+                0.829,
+                0.299
+            ],
+            "angle": 0,
+            "content": "Figure 1: Challenges in real-world web automation. Recent language model agents (Furuta et al., 2023; Gur et al., 2022; Kim et al., 2023; Yao et al., 2022b) can navigate simulated websites (Shi et al., 2017; Yao et al., 2022a), where the agents manipulate pre-defied actions and receive simplified HTML documents that are easy to parse. In contrast, language model agents continue to face challenges in navigating real-world websites, where they must interact with dynamic environments, handle open-ended actions (actions that cannot be pre-determined), and process lengthy HTML documents containing significant amounts of task-irrelevant information."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.308,
+                0.553,
+                0.558
+            ],
+            "angle": 0,
+            "content": "In this work, we introduce WebAgent, an LLM-driven autonomous agent that learns from self-experience to complete user instructions on real websites by combining canonical web actions in a program space (Figure 3). WebAgent (i) plans sub-instructions for each step by decomposing natural language instructions, (ii) summarizes long HTML documents into task-relevant snippets based on the plan, and (iii) acts via programming on real websites by grounding sub-instructions and HTML snippets into executable Python codes. We combine two LLMs to form WebAgent: newly introduced HTML-T5, a domain-expert pre-trained language model, for task planning and conditional HTML summarization and Flan-U-PaLM (Chowdhery et al., 2022; Chung et al., 2022) for grounded code generation. HTML-T5 has an encoder-decoder architecture and is specialized to capture the structure of long HTML documents better by adopting local and global attention mechanisms (Guo et al., 2022). It is"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.557,
+                0.825,
+                0.614
+            ],
+            "angle": 0,
+            "content": "pre-trained using a mixture of long-span denoising objective (Tay et al., 2022) on a large-scale HTML corpus extracted from CommonCrawl. To ground language model agents into real websites, we introduce self-experience supervision, where the domain-expert language models are finetuned with data generated by scripted planning/summarization and self-generated programming."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.574,
+                0.31,
+                0.811,
+                0.438
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.559,
+                0.441,
+                0.828,
+                0.556
+            ],
+            "angle": 0,
+            "content": "Figure 2: Statistics of HTML tokens among real websites. Compared to simulator (about 0.5K tokens on average), HTML tokens of real websites are much longer (from 7K to 14K), which takes up the context length of large language models. As preprocessing, we remove the irrelevant tags (e.g. <script>, <meta>) and keep necessary attributes (e.g. id, type, value)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.62,
+                0.827,
+                0.829
+            ],
+            "angle": 0,
+            "content": "Existing LLM-driven agents often solve decision making tasks with a single LLM conditioned on different prompts per role (Kim et al., 2023; Sun et al., 2023; Zheng et al., 2023), which is, however, not enough for real-world tasks whose complexity is higher than that of simulators. The empirical evaluations reveal that our method incorporating self-bootstrapped specialist language models improves HTML understanding and grounding, and achieves better generalization than single LLM agent. In real-world web automation, WebAgent significantly increases the success rate by \\(50\\%\\), and error analysis emphasizes that coupling task planning with HTML summarization in specialized language models is essential for task success. Moreover, HTML-T5 not only works as a core module for WebAgent but also achieves strong results by itself on the web-based tasks. On MiniWoB++ (Liu et al., 2018; Shi et al., 2017), HTML-T5 achieves \\(18.7\\%\\) higher success than previous language model agent (Gur et al., 2022) while also outperforming competitive baselines, such as naive local-global attention models (Guo et al., 2022) and its instruction-finetuned ones (Chung et al., 2022). On the Mind2Web (Deng et al., 2023), an offline task planning dataset, HTML-T5 achieves SoTA performance among Synapse (Zheng et al., 2023) with GPT-3.5, and MindAct with FLan-T5-XL and GPT-4 (OpenAI, 2023). In summary, our key contributions are:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.18,
+                0.835,
+                0.826,
+                0.878
+            ],
+            "angle": 0,
+            "content": "- We propose WebAgent, integration of two modular LLMs under self-supervision for real-world web automation. The domain-expert language model handles planning and HTML summarization, and a generalist language model generates executable Python programs."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.18,
+                0.882,
+                0.826,
+                0.925
+            ],
+            "angle": 0,
+            "content": "- We newly introduce HTML-T5 - a language model with local-global attention mechanism that is pre-trained with a mixture of long-span denoising objective on a large-scale HTML corpus, curated from CommonCrawl, to capture the syntax and semantics of HTML better."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.18,
+                0.835,
+                0.826,
+                0.925
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.504,
+                0.96
+            ],
+            "angle": 0,
+            "content": "2"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.179,
+                0.104,
+                0.827,
+                0.149
+            ],
+            "angle": 0,
+            "content": "- WebAgent notably improves the success rate by over \\(50\\%\\) in real websites. When fine-tuned on downstream demonstrations, HTML-T5 itself outperforms prior language model agent by \\(18.7\\%\\) in MiniWoB++, and achieves SoTA performance in Mind2Web, even surpassing GPT-4."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.169,
+                0.356,
+                0.185
+            ],
+            "angle": 0,
+            "content": "2 RELATED WORKS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.201,
+                0.536,
+                0.395
+            ],
+            "angle": 0,
+            "content": "Web Automation Web automation is a sequential decision making task where agents manipulate browsers following given instructions (Shi et al., 2017), such as form filling (Diaz et al., 2013) or information retrieval (Adolphs et al., 2022) through the sequence of computer actions (Li et al., 2020; Mazumder & Riva, 2020; Shvo et al., 2021). Prior works have realized the web automation via reinforcement learning (Gur et al., 2019; Humphreys et al., 2022; Jia et al., 2019; Shaw et al., 2023), finetuned (Furuta et al., 2023; Gur et al., 2022) or prompted LLMs (Kim et al., 2023; Sun et al., 2023; Yao et al., 2022b; Zheng et al., 2023) on the simulated websites (Shi et al., 2017; Toyama et al., 2021; Yao et al., 2022a). However, there are still huge gaps"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.396,
+                0.827,
+                0.44
+            ],
+            "angle": 0,
+            "content": "between simplified simulators and real web environments; for instance, the average tokens for HTML pages are about 15 times larger (Figure 2), and pre-defined action space for specific websites is a strong assumption that may harm the generalization to out-of-distribution web pages or instructions."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.552,
+                0.172,
+                0.819,
+                0.291
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.543,
+                0.293,
+                0.828,
+                0.382
+            ],
+            "angle": 0,
+            "content": "Figure 3: WebAgent is a combination of LLMs: HTML-T5 for planning and summarization, and Flan-U-PaLM for grounded program synthesis. It is better suited for the real-world tasks; open domain action space, complex natural language instructions, and long HTML documents. See Appendix D for examples."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.444,
+                0.828,
+                0.556
+            ],
+            "angle": 0,
+            "content": "MindAct (Deng et al., 2023) could be the most relevant work, where finetuned language model summarizes the raw HTML document into task-relevant snippets, and another model predicts the web actions in a multi-choice QA format. While MindAct also combines several language models, it has just adopted DeBERTa (He et al., 2021) and Flan-T5 (Chung et al., 2022) for summarization and actor modules, and evaluated it on the offline dataset. In contrast, we design HTML-T5, specialized for web-based tasks, to handle long HTML documents. WebAgent leverages HTML-T5 finetuned with self-experience for summarization and planning, and Flan-U-PaLM as a capable programmer, which enables it to generate open-ended web actions and to act on online real-world websites."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.563,
+                0.827,
+                0.716
+            ],
+            "angle": 0,
+            "content": "Program Synthesis In addition to common LLMs (Brown et al., 2020; Chowdhery et al., 2022; Touvron et al., 2023), several works have proposed programming-focused language models (Chen et al., 2021a; Feng et al., 2020; Li et al., 2022; Wang et al., 2021) and their benchmarks (Austin et al., 2021; Hendrycks et al., 2021a; Lu et al., 2021). Another line of work has investigated the tool augmentation of LLMs (Parisi et al., 2022) by decoding API calls (Schick et al., 2023) or Python snippets to be parsed with the interpreter (Gao et al., 2023). Most works deal with the program synthesis on the static dataset, except for the attempts in robotics (Liang et al., 2023) and game (Trivedi et al., 2022; Wang et al., 2023a), where LLMs output Python or JavaScript snippets to command the agents. Similarly, we leverage the ability of code generation as an open-ended action space for web-based agents to manipulate the real website, and demonstrate LLMs can sequentially decode Python selenium codes considering the given sub-instructions and HTML in the prompts."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.722,
+                0.825,
+                0.737
+            ],
+            "angle": 0,
+            "content": "See extended related works on document understanding and LLM for task planning in Appendix B."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.752,
+                0.308,
+                0.767
+            ],
+            "angle": 0,
+            "content": "3 WEBAGENT"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.778,
+                0.827,
+                0.862
+            ],
+            "angle": 0,
+            "content": "WebAgent is a new architecture that combines two LLMs to achieve efficient real-world web automation. HTML-T5, a domain-expert LLM, is responsible for predicting the next sub-instruction (planning) and generating related HTML snippets (summarization). Flan-U-PaLM (540B) (Chowdhery et al., 2022; Chung et al., 2022), is prompted to generate executable Python programs based on the planning and summarization provided by HTML-T5 (Figure 3). This modular two-stage approach enables WebAgent to effectively navigate and process HTML documents."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.868,
+                0.828,
+                0.926
+            ],
+            "angle": 0,
+            "content": "Workflow Users initiate natural language interactions with a clear intent, such as apartment searching. Upon receiving the initial request, HTML-T5 formulates a \"go to <URL>\" sub-instruction, triggering Flan-U-PaLM to generate a corresponding Python program that navigates to the specified website. The raw HTML content of the newly opened page is extracted and fed into HTML-T5 along with the"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
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+                0.96
+            ],
+            "angle": 0,
+            "content": "3"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.272,
+                0.102,
+                0.725,
+                0.294
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.171,
+                0.297,
+                0.828,
+                0.377
+            ],
+            "angle": 0,
+            "content": "Figure 4: HTML-T5 consists of (1) local and global attention mechanisms (Ainslie et al., 2020; Guo et al., 2022) and (2) a mixture of denoising objectives (Tay et al., 2022) with longer-span corruption on large-scale HTML corpus. The local and global attention mechanisms are suitable for the hierarchical tree structures of HTML documents. Because of the sparsity of content tokens in HTML, short mean span length (e.g. \\(\\mu = 3\\)), often used in prior works (Raffel et al., 2020), only masks less meaningful chunks. Employing longer span length (e.g. \\(\\mu = 8\\)) helps pre-trained language models to capture the syntax and semantics of HTML better."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.393,
+                0.828,
+                0.465
+            ],
+            "angle": 0,
+            "content": "user's instruction and previous planning steps. This information is utilized to predict the next subinstruction and identify relevant reference IDs for extractive HTML summarization. Flan-U-PaLM, in turn, generates a Python program based on these sub-instructions and combined HTML snippets. This iterative process of planning, summarization, and program synthesis continues until a designated end-of-episode sub-instruction is predicted or the maximum number of iterations is reached."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.481,
+                0.293,
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+            ],
+            "angle": 0,
+            "content": "3.1 HTML-T5"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.508,
+                0.825,
+                0.689
+            ],
+            "angle": 0,
+            "content": "Prior research has shown that general-purpose LLMs, such as T5 (Raffel et al., 2020), Flan-T5 (Chung et al., 2022), and InstructGPT (Ouyang et al., 2022), can effectively navigate web environments (Furuta et al., 2023; Gur et al., 2022; Kim et al., 2023). However, unlike specialist transformer models (Li et al., 2021b; Wang et al., 2022a; Zhao et al., 2022), these general-purpose LLMs do not fully utilize the HTML-specific information that could otherwise lead to better understanding of HTML content. To address this limitation, we introduce HTML-T5, a pre-trained encoder-decoder language model specifically designed for HTML-based web automation tasks. HTML-T5 carefully merges the generalist and specialist characteristics of language models. It processes HTML in a text-to-text manner and employs local and global attention mechanisms (Ainslie et al., 2020) in the encoder to capture the hierarchical structure of long HTML inputs. HTML-T5 is pre-trained on a large-scale HTML corpus curated from CommonCrawl using a mixture of long-span denoising objectives (Tay et al., 2022), and then finetuned it for each downstream task. For WebAgent, we employ the self-experience supervision approach to align the model with real websites."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
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+            ],
+            "angle": 0,
+            "content": "Model Architecture Unlike natural language, HTML documents possess an explicit hierarchical structure. This structure comprises elements such as <input>, <label>, and <button>, along with their associated attributes like class, label, and id. These elements are defined locally and combined hierarchically to create HTML documents. To model this inherent hierarchy, we replace the common dense attention (Vaswani et al., 2017) with local and global attention mechanisms (Ainslie et al., 2020). Local attention restricts each token to only attend to neighboring tokens within a window. Additionally, transient global attention allows each token to attend to tokens beyond its immediate window. This is achieved through the aggregation and normalization of token representations within each window, resulting in a global memory representation. Figure 4 describes the concepts of HTML-T5; leaf elements in HTML (green) could be processed by local attention, and internal elements (purple) could be compressed into transient global attention, which naturally fits the hierarchical structure of HTML. Following LongT5 (Guo et al., 2022), we use dense attention in the decoder."
+        },
+        {
+            "type": "text",
+            "bbox": [
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+            ],
+            "angle": 0,
+            "content": "Pre-Training with Mixture of Long-Span Denoising Our pre-training approach for HTML-T5 utilizes a span denoising objective. This involves randomly masking spans of tokens within an HTML document, with span lengths drawn from a Gaussian distribution with a mean of \\(\\mu\\). The objective is then to predict the masked spans using the remaining tokens in the HTML document (Raffel"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
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+            "angle": 0,
+            "content": "4"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.175,
+                0.103,
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+            ],
+            "angle": 0,
+            "content": "<table><tr><td rowspan=\"2\"></td><td colspan=\"2\">Modules</td><td colspan=\"2\">real-estate</td><td colspan=\"2\">social-media</td><td colspan=\"2\">map</td><td colspan=\"3\">Error Ratio (%)</td></tr><tr><td>Plan</td><td>Sum</td><td>Success</td><td>Score</td><td>Success</td><td>Score</td><td>Success</td><td>Score</td><td>Program</td><td>Plan</td><td>Sum</td></tr><tr><td>Flan-U-PaLM</td><td>✘</td><td>✘</td><td>10.0</td><td>55.3</td><td>20.0</td><td>25.0</td><td>10.0</td><td>51.3</td><td>36/88/11</td><td>38/0/78</td><td>26/12/11</td></tr><tr><td>Flan-U-PaLM+P</td><td>✓</td><td>✘</td><td>50.0</td><td>79.5</td><td>20.0</td><td>38.3</td><td>30.0</td><td>73.8</td><td>39/65/14</td><td>56/30/29</td><td>5/5/57</td></tr><tr><td>Flan-U-PaLM+S</td><td>✘</td><td>✓</td><td>0.0</td><td>45.7</td><td>25.0</td><td>62.1</td><td>15.0</td><td>46.3</td><td>30/67/0</td><td>40/13/100</td><td>30/20/0</td></tr><tr><td>WebAgent</td><td>✓</td><td>✓</td><td>65.0</td><td>87.6</td><td>70.0</td><td>85.8</td><td>80.0</td><td>93.8</td><td>20/33/25</td><td>70/50/50</td><td>10/17/25</td></tr></table>"
+        },
+        {
+            "type": "table_footnote",
+            "bbox": [
+                0.17,
+                0.188,
+                0.828,
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+            ],
+            "angle": 0,
+            "content": "Table 1: Success rate of real-world web automation on real estate, social media and map websites. The score stands for the percentage of covered attributes specified in given instructions. WebAgent, with language model modules for planning and summarization, achieves the best success (65%, 70%, 80%, respectively), surpassing other baselines, such as a single Flan-U-PaLM, that with a planning language model (Flan-U-PaLM+P), and that with a summarization language model (Flan-U-PaLM+S). Without language model modules, prompted Flan-U-PaLM plans in an open-loop manner (Plan: \\(\\times\\)) and regular-expression-based retrieval summarizes HTML inputs (\\(\\mathbf{Sum}\\): \\(\\times\\)). The results imply that self-experience supervision notably improves the performance, and task planning should be learned by finetuning domain language models for closed-loop planning, rather than by prompting single LLM for open-loop planning. The error analysis describes the ratio across three types of errors in (real-estate)/(social-media)/(map) domains, which also points out that better adaptive planner to decompose the given instructions would contribute to further improvements of WebAgent."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.351,
+                0.827,
+                0.435
+            ],
+            "angle": 0,
+            "content": "et al., 2020; Tay et al., 2022; Ainslie et al., 2023). While a span length of \\(\\mu = 3\\) is commonly used, such short spans often mask less meaningful chunks in HTML documents, such as \\(< / ,\\mathrm{id} =\\) or \\(n>\\) (Figure 4), where the signal-to-noise ratio can be significantly lower than natural language text. In contrast, longer spans can contain more semantically meaningful chunks, such as <form class=\"or type=\"submit\">. Our empirical findings indicate that setting \\(\\mu \\in \\{8,64\\}\\) yields the optimal mixture for HTML documents (Section 4.2)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.441,
+                0.828,
+                0.568
+            ],
+            "angle": 0,
+            "content": "We adopt 4096 input sequence length and 910 output sequence length during pre-training. In total, \\(15\\%\\) of input tokens are randomly masked in the denoising objective. For the pre-training dataset, we collect 100 WARC files (April 2019) from the CommonCrawl corpus and remove the non-Unicode or alphanumeric-only HTML documents. We then extract subtrees around \\(\\langle \\text{label} \\rangle\\) elements that have a special attribute called for that associates the corresponding label with a unique input element in the same HTML document. This pre-processing step improves the quality of the pre-training corpus by focusing only on HTML that is relevant for instruction following and grounding. Our final dataset has 3.41M examples. We pre-train HTML-T5 for 100K iterations following the practice in other T5 models (Chung et al., 2022; Lester et al., 2021). See Appendix C for further details."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.59,
+                0.731,
+                0.604
+            ],
+            "angle": 0,
+            "content": "3.2 SELF-EXPERIENCE SUPERVISION FOR ALIGNMENT WITH REAL WEBSITES"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.618,
+                0.826,
+                0.744
+            ],
+            "angle": 0,
+            "content": "Gathering example demonstrations for LLMs to understand websites poses a significant obstacle in real-world web automation. While humans can effortlessly execute instruction following on actual websites, manually annotating every planning, summarization, and program synthesis step as detailed above is impractical. To address this issue, we propose self-experience supervision, a semi-supervised approach that necessitates minimal human involvement. In this method, manually curated scripts generate planning and summarization steps, while Flan-U-PaLM is tasked with generating Python programs. Our WebAgent aligns domain-specific language models, such as HTML-T5, with these self-gathered real-world experiences through fine-tuning (Wang et al., 2022b). This enables the generalization and alignment of agents to complex real-world tasks."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.75,
+                0.827,
+                0.807
+            ],
+            "angle": 0,
+            "content": "Instruction Templates We maintain a collection of instruction templates that incorporate placeholders such as \"Show me the way from <start> to <goal> by <n-th> <transportation> at map website\". We sample instructions by randomly assigning values to placeholders from pre-defined key-value pairs."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.813,
+                0.827,
+                0.926
+            ],
+            "angle": 0,
+            "content": "Scripted Planning and Prompted Programming We employ a rule-based parser to decompose instructions into sequences of sub-instructions; corresponding reference IDs are retrieved from HTML using regular expressions. At each step of the process, Flan-U-PaLM is provided with the sub-instruction and the associated HTML snippets to generate navigation programs that are executed through Selenium WebDriver. The success of recorded demonstrations varies, and automating success criteria for real-world tasks remains challenging. To refine the learning experience, we utilize environmental feedback to eliminate critical failures, such as program execution errors, retriever errors, and clearly erroneous URL prefixes (Ni et al., 2023)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.505,
+                0.96
+            ],
+            "angle": 0,
+            "content": "5"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.195,
+                0.102,
+                0.669,
+                0.113
+            ],
+            "angle": 0,
+            "content": "map: Show me the way from San Jose to Mountain View by 2nd Cycling at map website?"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.196,
+                0.113,
+                0.805,
+                0.242
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.171,
+                0.243,
+                0.828,
+                0.283
+            ],
+            "angle": 0,
+            "content": "Figure 5: Example episodes of real-world web automation in map domain. Considering the given instruction and HTML, WebAgent predicts the next sub-instruction and task-relevant snippet, and then synthesizes the Python script (gray), while treating the sub-instruction as a comment in the script. See Appendix H for extended figure."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.292,
+                0.828,
+                0.419
+            ],
+            "angle": 0,
+            "content": "Finetuning for Planning and Summarization HTML-T5, a core component of WebAgent, is fine-tuned using self-experience demonstrations gathered through instruction sampling, scripted planning, and prompted program synthesis, as detailed earlier. It utilizes task instructions (e.g. please search 2 bedroom and \\(2+\\) bathroom houses in new york, ny with a max price of $7500 on real estate website), sub-instruction histories (e.g. go to real estate website, type in new york into search, click on search, click on price, click on max rent), and raw HTML as inputs. Subsequently, it generates the next sub-instruction (e.g. type in 7500 into max rent) and extracts the relevant data-ref attributes used for retrieving HTML snippets. Section 4.1 demonstrates the significance of integrating HTML summarization into sub-instruction prediction for enhancing real-world web automation performance."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.436,
+                0.456,
+                0.451
+            ],
+            "angle": 0,
+            "content": "3.3 GROUNDED PROGRAM SYNTHESIS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.463,
+                0.828,
+                0.658
+            ],
+            "angle": 0,
+            "content": "Web automation on real-world websites faces challenges due to the open-ended action spaces, unlike simplified simulators (Shi et al., 2017; Yao et al., 2022a). In contrast to previous approaches (Gur et al., 2019; Humphreys et al., 2022; Jia et al., 2019; Liu et al., 2018), real-world web agents cannot pre-define a categorical action space to specify the interactive elements on the websites. To address this open-domain challenge, we introduce the act via programming paradigm in web automation by utilizing the conditional code generation capabilities of LLMs (Chen et al., 2021a; Liang et al., 2023). Provided with few-shot generic examples (such as selecting checkboxes, entering text into inputs, clicking on options, and scrolling etc.) for program generation, the next sub-instruction, and the extracted HTML snippet from HTML-T5, Flan-U-PaLM (Chowdhery et al., 2022; Chung et al., 2022) decodes an Python program (Figure 3) executable with Selenium WebDriver, a library for browser automation. This conditional program synthesis requires LLMs to not only generate code to follow natural language instructions but also understand the semantics and functionality of HTML elements. We provide several Python snippet examples generated by Flan-U-PaLM as follows (sub-instructions are treated as comments in the script):"
+        },
+        {
+            "type": "code",
+            "bbox": [
+                0.157,
+                0.666,
+                0.793,
+                0.774
+            ],
+            "angle": 0,
+            "content": "1 # Type in walnut creek, ca into search  \n2 driver.find_element(By.css_selector, '[data-ref=\"175\"').clear()  \n3 driver.find_element(By.css_selector, '[data-ref=\"175\"').send_keys(\"walnut creek, ca\")  \n4  \n5 # Submit the search  \n6 driver.find_element(By.css_selector, '[data-ref=\"175\"').submit()  \n7  \n8 # Click on the apartments  \n9 driver.find_element(By.css_selector, '[data-ref=\"572\"').click()  \n10  \n11 # Scroll down housing type by 200px  \n12 driver.executeScript('getScrollParent.document.querySelector \"#type-of-housing\").scrollBy({top: 200})')"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.795,
+                0.421,
+                0.81
+            ],
+            "angle": 0,
+            "content": "4 EXPERIMENTAL RESULTS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.827,
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+            "angle": 0,
+            "content": "To study how a modular combination of LLMs under self-supervision enables real-world web automation by overcoming open-endedness and long context documents, we execute instruction-following tasks on real websites (Section 4.1). In Appendix F, we also test WebAgent on WebSRC (Chen et al., 2021b), a static HTML comprehension benchmark, compared to prior transformer models specialized for structured documents (Li et al., 2021b; Zhao et al., 2022). In addition, we quantify the performance of HTML-T5 itself on simulated web benchmark, MiniWoB++, and offline task planning benchmark, Mind2Web (Section 4.2)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
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+                0.506,
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+            ],
+            "angle": 0,
+            "content": "6"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.187,
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+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Architectures</td><td>Attention Type</td><td>L = 2048</td><td>L = 4096</td></tr><tr><td>Flan-T5-Base</td><td>Dense</td><td>34.0%</td><td>35.3%</td></tr><tr><td>Long-T5-Base</td><td>Local</td><td>43.4%</td><td>44.0%</td></tr><tr><td>Long-T5-Base</td><td>Local &amp; Global</td><td>53.1%</td><td>53.6%</td></tr></table>"
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.536,
+                0.103,
+                0.825,
+                0.199
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Span Length μ</td><td>real-estate</td><td>MiniWoB++</td></tr><tr><td>(no HTML-denoising)</td><td>78.07</td><td>53.8%</td></tr><tr><td>3,8,64,Prefix</td><td>80.56</td><td>55.2%</td></tr><tr><td>3,8,64</td><td>80.56</td><td>55.4%</td></tr><tr><td>8,64</td><td>82.46</td><td>57.0%</td></tr><tr><td>8,32,64</td><td>82.16</td><td>55.6%</td></tr><tr><td>8,64,96</td><td>81.29</td><td>53.6%</td></tr><tr><td>16,64</td><td>79.97</td><td>55.2%</td></tr></table>"
+        },
+        {
+            "type": "table_footnote",
+            "bbox": [
+                0.17,
+                0.202,
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+            ],
+            "angle": 0,
+            "content": "Table 2: (Left) Architecture comparison on MiniWoB++ 12K dataset (Liu et al., 2018) with average success rate over 56 tasks. Local and global attention matches to the hierarchical tree structure of HTML, and then improves the success rate by over \\(18\\%\\), compared to the instruction-finetuned dense attentions (Chung et al., 2022; Furuta et al., 2023). (Right) HTML-denoising comparison with different mixtures of span length (Raffel et al., 2020; Tay et al., 2022). We use LongT5-Base models for pre-training. HTML-denoising generally improves the performance on offline task planning on real estate website and MiniWoB benchmark. Especially, using longer span lengths \\((\\mu \\in \\{8,6\\})\\) outperforms other choices, including the popular configuration in natural language domain \\((\\mu \\in \\{3,8,64\\} + \\text{Prefix LM objective})\\), which can reduce the less meaningful prediction from shorter spans (e.g. \\(\\mu = 3\\)), and inject the structural bias of HTML better."
+        },
+        {
+            "type": "title",
+            "bbox": [
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+            "angle": 0,
+            "content": "4.1 REAL-WORLD WEB AUTOMATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
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+            ],
+            "angle": 0,
+            "content": "Evaluation Methodology We first evaluate WebAgent with the real-world navigation performance under human supervision, at real estate website (a platform for housing), social media website (a network of communities), and map website. These three websites have different properties. real-estate requires long-horizon planning (about 20 steps per episode) for complex formfilling with a few page transitions (at least 2 pages), and social-media needs shorter plans (about 10 steps per episode) with many page transitions (at least 4 pages) by selecting appropriate hyperlinks on the page. map is the easiest domain with shorter plans and a few page transitions. WebAgent receives natural language instructions (e.g. Can you search for a studio bedroom, \\(1+\\) bathroom apartments in oroville, ca for corporate housing on real estate website?, or Could you present the most new thread of Python community filtered by Tutorial tag on social media website?), and acts via planning, summarizing by HTML-T5, and then programming by Flan-U-PaLM (Figure 5). Through the self-experience supervision process, we curate 260 episodes on real estate website, 230 episodes on social media website, and 410 episodes on map website to finetune HTML-T5."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.549,
+                0.825,
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+            ],
+            "angle": 0,
+            "content": "We prepare 20 different natural language instructions (see Appendix G for the full list), and measure the success rate and score for the evaluation. The score represents the percentage of required attributes covered during the episode (Yao et al., 2022a); for instance, (1) apartments for (2) corporate housing with (3) studio bedroom and (4) \\(1+\\) bathroom located in (5) oroville, ca, can be specified in the instruction. When the agents could search the housing satisfying (1), (2), (5) and not (3), (4), the score is \\(60 (= 100 \\times 3/5)\\). If the agents achieve 100 score, that episode will mark as success."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
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+            ],
+            "angle": 0,
+            "content": "Results For comparison, we prepare three baselines, consisting of language model modules and a single LLM conditioned on different prompts per role, such as Flan-U-PaLM (Chung et al., 2022), that with a planning language model (Flan-U-PaLM+P), and that with a summarization language model (Flan-U-PaLM+S). If they do not use language model modules, prompted Flan-U-PaLM plans in an open-loop manner (Plan: \\(\\times\\)), and regular-expression-based retrieval summarizes given raw HTML (Sum: \\(\\times\\)). Table 1 shows that by leveraging planning and summarization language model modules, WebAgent achieves best \\(65\\%\\) success and \\(87.6\\%\\) score on real-estate, \\(70\\%\\) success and \\(85.8\\%\\) score on social-media, and \\(80\\%\\) success and \\(93.8\\%\\) score on map, significantly outperforming single Flan-U-PaLM, or with partial language model modules (most of those achieve about \\(10 - 30\\%\\) success). This result suggests that self-experience supervision notably improves the performance, and closed-loop planning grounded on HTML observations via finetuned domain language models is more suitable for open-ended web automation than open-loop planning with few-shot LLMs. This trend is remarkable in real-estate (even Flan-U-PaLM+P achieves \\(50\\%\\) success), where the longer planning horizon is needed to fulfill instructions. We also observe that coupling sub-instruction prediction with HTML summarization in language model modules plays a critical role in task success. The development of more capable planning modules to decompose the given instructions adaptively and accurately could help WebAgent improve the performance further."
+        },
+        {
+            "type": "text",
+            "bbox": [
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+            "angle": 0,
+            "content": "Error Analysis We also analyze the reason of failures by categorizing them into programming, planning, and summarization errors (Table 1). Programming error does not satisfy the given sub-instructions or HTML snippet. Planning error predicts sub-instructions conflicting with user instruc"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
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+            ],
+            "angle": 0,
+            "content": "7"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.188,
+                0.102,
+                0.812,
+                0.166
+            ],
+            "angle": 0,
+            "content": "<table><tr><td rowspan=\"2\"></td><td rowspan=\"2\">Train</td><td colspan=\"4\">Cross-Task</td><td colspan=\"4\">Cross-Website</td><td colspan=\"4\">Cross-Domain</td></tr><tr><td>Ele. Acc</td><td>Op. F1</td><td>Step SR</td><td>SR</td><td>Ele. Acc</td><td>Op. F1</td><td>Step SR</td><td>SR</td><td>Ele. Acc</td><td>Op. F1</td><td>Step SR</td><td>SR</td></tr><tr><td>Synapse (GPT-3.5)</td><td>ICL</td><td>34.4</td><td>-</td><td>30.6</td><td>2.0</td><td>28.8</td><td>-</td><td>23.4</td><td>1.1</td><td>29.4</td><td>-</td><td>25.9</td><td>1.6</td></tr><tr><td>MindAct (Flan-T5-XL)</td><td>SL</td><td>55.1</td><td>75.7</td><td>52.0</td><td>5.2</td><td>42.0</td><td>65.2</td><td>38.9</td><td>5.1</td><td>42.1</td><td>66.5</td><td>39.6</td><td>2.9</td></tr><tr><td>MindAct (GPT-4)</td><td>ICL</td><td>41.6</td><td>60.6</td><td>36.2</td><td>2.0</td><td>35.8</td><td>51.1</td><td>30.1</td><td>2.0</td><td>37.1</td><td>46.5</td><td>26.4</td><td>2.0</td></tr><tr><td>HTML-T5-XL (ours)</td><td>SL</td><td>60.6</td><td>81.7</td><td>57.8</td><td>10.3</td><td>47.6</td><td>71.9</td><td>42.9</td><td>5.6</td><td>50.2</td><td>74.9</td><td>48.3</td><td>5.1</td></tr></table>"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.169,
+                0.827,
+                0.221
+            ],
+            "angle": 0,
+            "content": "Table 4: Offline action prediction performance in Mind2Web dataset. We leverage the cached candidate generation results and direct QA formulation by following Deng et al. (2023). HTML-T5 significantly outperforms MindAct with Flan-T5 or GPT-4, and Synapse (Zheng et al., 2023) with GPT-3.5, across task/website/domain generalization in terms of all the metrics (element accuracy, operation F1, and success rates)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.229,
+                0.828,
+                0.382
+            ],
+            "angle": 0,
+            "content": "tions, and summarization error fails to extract the relevant HTML snippets for given sub-instructions. From the website perspective, the failures on real-estate concentrate in planning because of its long-horizon nature. map also fails in planning when confusing starting point and destination. In contrast, social-media tends to fail in programming due to the ambiguous sub-instructions or summarization including redundant hyperlinks, which results in transiting wrong pages or clicking unexecutable elements. From the method perspective, WebAgent often fails in planning by predicting incorrect sub-instructions (for instance, in real-estate, WebAgent generates incorrect plans in \\(70\\%\\) of failure episodes), while other baselines more fail in programming or summarization steps. This observation indicates that, through the self-experience supervision, the ratio of programming and summarization errors has decreased while the fundamental difficulty of planning, which requires consistent and accurate prediction over long horizon without error accumulation, still remains."
+        },
+        {
+            "type": "title",
+            "bbox": [
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+                0.388,
+                0.393,
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+            ],
+            "angle": 0,
+            "content": "4.2 ABLATION OF HTML-T5"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.414,
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+            ],
+            "angle": 0,
+            "content": "In addition to the evaluation as WebAgent system, we extensively examine HTML-T5 about (i) the generalization to other websites with Mind2Web (Deng et al., 2023), (ii) the performance on MiniWoB++, a standard web automation benchmark (Liu et al., 2018; Shi et al., 2017), and (iii) its architecture and pre-training objective. We adopt 16K tokens for the context window unless otherwise mentioned. We present results on offline task planning, and description generation (Gur et al., 2022) to test HTML understanding on static dataset in Appendix I."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.574,
+                0.52,
+                0.741
+            ],
+            "angle": 0,
+            "content": "Mind2Web Mind2Web (Deng et al., 2023) is an action-annotated real-world dataset with over 2K instructions collected from 137 websites. It provides action prediction tasks that measure the generalization of LLMs across the tasks, websites, and their domains (e.g. travel, shopping). Similar to real-world evaluation, the input is a set of HTML snippets, a task instruction, and an action history. The output comprises a target element to interact with, along with the operation, such as click, type, or select an option. We finetune HTML-T5-XL with the training dataset. The performance is evaluated with element accuracy,"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.741,
+                0.827,
+                0.81
+            ],
+            "angle": 0,
+            "content": "operation F1, and step success rate that cares for both element and operation correctness. Table 4 reveals that HTML-T5 significantly outperforms baselines with Flan-T5-XL or GPT-4 (OpenAI, 2023) across task/website/domain generalization, which increases element accuracy by \\(5 - 8\\%\\) , operation F1 by \\(6 - 8\\%\\) , and step success rate by \\(4 - 8\\%\\) . This highlights that HTML-T5 can handle real-world web automation tasks better and shows generalization beyond our real-world evaluation with 3 websites."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.816,
+                0.826,
+                0.902
+            ],
+            "angle": 0,
+            "content": "MiniWoB++ We here evaluate HTML-T5 on 56 simulated tasks in MiniWoB++ using 100 evaluation episodes per task. Inputs are analogous to real-world evaluation, utilizing HTML documents, while outputs are adhering to a pre-defined format by the simulator such as click(ref = X). We finetune HTML-T5 with 12K human demonstrations (Liu et al., 2018), and compare the average success rate to prior supervised-learned agents (Gur et al., 2022; Humphreys et al., 2022), LongT5, and its instruction-finetuned variants (Chung et al., 2022) \\(^{1}\\). Table 3 shows that HTML-T5-XL significantly"
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.547,
+                0.401,
+                0.807,
+                0.594
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Models</td><td>Data</td><td>Success</td><td>Diff.</td></tr><tr><td>SoTA (Zheng et al., 2023)</td><td>-</td><td>99.2%</td><td>-</td></tr><tr><td>CC-Net</td><td>2.4M</td><td>32.0%</td><td>-</td></tr><tr><td>WebN-T5-XL</td><td>12K</td><td>48.4%</td><td>-</td></tr><tr><td>LongT5-Base</td><td></td><td>53.8%</td><td>0.0</td></tr><tr><td>LongT5-Large</td><td>12K</td><td>56.3%</td><td>0.0</td></tr><tr><td>LongT5-XL</td><td></td><td>60.4%</td><td>0.0</td></tr><tr><td>Flan-LongT5-Base</td><td></td><td>54.1%</td><td>+0.3</td></tr><tr><td>Flan-LongT5-Large</td><td>12K</td><td>56.1%</td><td>-0.2</td></tr><tr><td>Flan-LongT5-XL</td><td></td><td>61.1%</td><td>+0.7</td></tr><tr><td>HTML-T5-Base (ours)</td><td></td><td>57.0%</td><td>+3.2</td></tr><tr><td>HTML-T5-Large (ours)</td><td>12K</td><td>60.8%</td><td>+4.5</td></tr><tr><td>HTML-T5-XL (ours)</td><td></td><td>67.1%</td><td>+6.7</td></tr><tr><td>Flan-T5-XL</td><td>347K</td><td>75.5%</td><td>-</td></tr><tr><td>Flan-T5-XXL</td><td></td><td>79.0%</td><td>-</td></tr><tr><td>HTML-T5-XL (ours)</td><td>347K</td><td>85.6%</td><td>-</td></tr></table>"
+        },
+        {
+            "type": "table_footnote",
+            "bbox": [
+                0.527,
+                0.597,
+                0.827,
+                0.736
+            ],
+            "angle": 0,
+            "content": "Table 3: Average success rate of MiniWoB++ with 56 tasks. We use 12K demonstrations and compare HTML-T5 among supervised-finetuned methods. HTML-T5-XL outperforms WebN-T5-XL (Gurel al., 2022), the prior best method, by \\(18.7\\%\\). HTML-denoising also yields better the success rate than instruction tuned ones. Finetuned HTML-T5 with 347K episodes (Furuta et al., 2023) outperforms Flan-T5-XXL (11B parameters) even with 3B parameters, which gets closer to SoTA with GPT-3.5. See Appendix K for the detailed results."
+        },
+        {
+            "type": "table_footnote",
+            "bbox": [
+                0.527,
+                0.739,
+                0.827,
+                0.811
+            ],
+            "angle": 0,
+            "content": "element and operation correctness. Table 4 rees with Flan-T5-XL or GPT-4 (OpenAI, 2023) eases element accuracy by \\(5 - 8\\%\\) , operation F1 nts that HTML-T5 can handle real-world web and our real-world evaluation with 3 websites."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.527,
+                0.597,
+                0.827,
+                0.811
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.193,
+                0.91,
+                0.766,
+                0.925
+            ],
+            "angle": 0,
+            "content": "<sup>1</sup>We finetune LongT5 models with Flan dataset released by Chung et al. (2022). See Appendix J."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.504,
+                0.96
+            ],
+            "angle": 0,
+            "content": "8"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.104,
+                0.827,
+                0.189
+            ],
+            "angle": 0,
+            "content": "outperforms WebN-T5, the prior best model, by \\(18.7\\%\\). Notably, we demonstrate HTML-denoising consistently improves the performance on top of LongT5 in all the model sizes, better than instruction-finetuning introduced in prior work (Furuta et al., 2023). Furthermore, we finetune HTML-T5-XL with 347K demonstrations from Furuta et al. (2023), which performs better than 11B-parameter Flant5-XXL even with 3B parameters, achieving \\(85.6\\%\\) success. These prove we successfully incorporate domain knowledge on HTML comprehension for web automation into pre-trained language models."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.195,
+                0.828,
+                0.321
+            ],
+            "angle": 0,
+            "content": "Architecture and Objective We hypothesize that local and global attention mechanisms can capture the hierarchical structures of HTML documents better than dense attention. We compare the web automation performance among 56 MiniWoB++ tasks (Gur et al., 2022), by finetuning HTML-T5 with public 12K-episode dataset (Liu et al., 2018). We adopt 2048 and 4096 tokens as input length and prepare Base-size architectures. Table 2 (left) reveals that the combination of local and global attentions achieves the superior success rate by over \\(18\\%\\) compared to the instruction-finetuned dense attentions (Chung et al., 2022; Raffel et al., 2020) and local attention only. Surprisingly, local attention only still surpasses the dense attention by about \\(9\\%\\), which suggests local relation between elements and attributes in HTML are essential for web tasks."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.327,
+                0.828,
+                0.412
+            ],
+            "angle": 0,
+            "content": "As for pre-training objective in Table 2 (right), HTML-denoising generally improves the performance on offline task planning on real estate website and MiniWoB. Especially, using only longer span lengths \\((\\mu \\in \\{8,64\\})\\) outperforms other choices, including the popular configuration in natural language domain \\((\\mu \\in \\{3,8,64\\} +\\text{Prefix LM objective})\\), which can reduce the less meaningful prediction from shorter spans (e.g. \\(\\mu = 3\\)), and inject the structural bias of HTML into language models better. See Appendix I.2 for further results with model scaling."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.422,
+                0.46,
+                0.438
+            ],
+            "angle": 0,
+            "content": "5 DISCUSSION AND LIMITATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.455,
+                0.828,
+                0.527
+            ],
+            "angle": 0,
+            "content": "Modular Approach with Specialist Language Models We demonstrate it is beneficial to divide web automation into planning, HTML summarization, and code generation, and to combine domain-expert language models aligned with self-experience data. Such modular approaches have also been adopted to support the inference of LLMs (Xu et al., 2023), multimodal tasks (Zeng et al., 2022), and robotics (Ahn et al., 2022), which, however, might cause additional computational costs and latency."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.532,
+                0.828,
+                0.603
+            ],
+            "angle": 0,
+            "content": "Broad Generalization across the Internet Because open-loop planning with prompted Flan-U-PaLM achieves at most \\(10 - 30\\%\\) success, we have demonstrated that self-experience supervision on real websites is essential for planning modules. As we demonstrated in Mind2Web, our method could generalize across the internet if we have enough data. It would be expected to collect demonstrations at scale and align larger domain-expert models with them in future works."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.608,
+                0.827,
+                0.667
+            ],
+            "angle": 0,
+            "content": "Feedback for Program Synthesis We leverage Flan-U-PaLM with 540B parameters, as a capable program synthesis module via few-shot prompting. Such a large model, however, makes it challenging to reflect the feedback about the errors in generated code, compared to smaller models. We leave it as future direction to incorporate the feedback for program synthesis into larger language models."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.671,
+                0.829,
+                0.743
+            ],
+            "angle": 0,
+            "content": "Evaluation for Real-world Web Automation Beyond the simulated web environments (Shi et al., 2017; Yao et al., 2022a), we have exhibited WebAgent can follow given complex and sometimes ambiguous instructions on real estate, social media and map websites. On the other hand, it is costly to evaluate the performance of autonomous agents in the real world. Automated evaluation with minimal human intervention would be helpful for the scalable development of real-world web agents."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.753,
+                0.321,
+                0.768
+            ],
+            "angle": 0,
+            "content": "6 CONCLUSION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.785,
+                0.828,
+                0.926
+            ],
+            "angle": 0,
+            "content": "We build a system for real-world web automation, combining HTML-T5 for planning and HTML summarization and Flan-U-PaLM for grounded program synthesis. Our proposed WebAgent achieves around \\(70 - 80\\%\\) success on real websites via self-experience supervision, outperforming single LLM approach by over \\(50\\%\\), which suggests dividing the sequence of sub-problems with multiple language models can increase the entire task success. We also propose a scalable recipe for HTML-specialized language models where we train local and global attention mechanisms with a mixture of long-span denoising objectives to capture the hierarchical structures of HTML documents. HTML-T5 not only plays an essential role in WebAgent but also can achieve the best results on a variety of HTML-based benchmarks such as Mind2Web and MiniWoB++. We hope our work contributes to getting us one-step closer to the practical deployment of autonomous web agent systems."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.506,
+                0.96
+            ],
+            "angle": 0,
+            "content": "9"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.103,
+                0.347,
+                0.119
+            ],
+            "angle": 0,
+            "content": "ETHICS STATEMENT"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.135,
+                0.825,
+                0.193
+            ],
+            "angle": 0,
+            "content": "This paper presents encouraging evidence of autonomous agents' potential for deployment on real websites, extending beyond simulated environments. In the foreseeable future, this technology could lead to the development of sophisticated AI assistant tools for computers and smartphones, enhancing productivity and accessibility for society."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.198,
+                0.827,
+                0.298
+            ],
+            "angle": 0,
+            "content": "While we recognize the promising aspects of autonomous agents, we must also consider the potential for misuse and unintended consequences in their development. As our proposed system is based on LLMs, there is a risk of prompt injection. The improper use of web automation could pose cybersecurity threats and expose users to scams. To mitigate these risks, it is crucial for researchers, policymakers, and industry stakeholders to collaborate on establishing guidelines and regulations for the development of autonomous agents. Additionally, security research focused on LLM agents will become an essential domain for society."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.312,
+                0.329,
+                0.325
+            ],
+            "angle": 0,
+            "content": "ACKNOWLEDGMENTS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.335,
+                0.828,
+                0.38
+            ],
+            "angle": 0,
+            "content": "We thank Heiga Zen, Yingjie Miao, Yusuke Iwasawa, Joshua Ainslie, Santiago Ontonon, Quoc V. Le, Zoubin Ghahramani, Jeff Dean, Tris Warkentin for the supports and advises on this work. Hiroki Furuta was supported by JSPS KAKENHI Grant Number JP22J21582."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.399,
+                0.289,
+                0.414
+            ],
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+            "content": "Shunyu Yao, Jeffrey Zhao, Dian Yu, Nan Du, Izhak Shafran, Karthik Narasimhan, and Yuan Cao. React: Synergizing reasoning and acting in language models. arXiv preprint arXiv:2210.03629, 2022b."
+        },
+        {
+            "type": "ref_text",
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+            ],
+            "angle": 0,
+            "content": "Andy Zeng, Maria Attarian, Brian Ichter, Krzysztof Choromanski, Adrian Wong, Stefan Welker, Federico Tombari, Aveek Purohit, Michael Ryoo, Vikas Sindhwani, Johnny Lee, Vincent Vanhoucke, and Pete Florence. Socratic models: Composing zero-shot multimodal reasoning with language. arXiv preprint arXiv:2204.00598, 2022."
+        },
+        {
+            "type": "ref_text",
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+            "content": "Jingqing Zhang, Yao Zhao, Mohammad Saleh, and Peter J. Liu. Pegasus: Pre-training with extracted gap-sentences for abstractive summarization. In International Conference on Machine Learning, 2020."
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
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+            ],
+            "angle": 0,
+            "content": "Zihan Zhao, Lu Chen, Ruisheng Cao, Hongshen Xu, Xingyu Chen, and Kai Yu. TIE: Topological information enhanced structural reading comprehension on web pages. In Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pp. 1808-1821, 2022."
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
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+            ],
+            "angle": 0,
+            "content": "Longtao Zheng, Rundong Wang, and Bo An. Synapse: Leveraging few-shot exemplars for human-level computer control. arXiv preprint arXiv:2306.07863, 2023."
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
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+            ],
+            "angle": 0,
+            "content": "Chenguang Zhu, Yang Liu, Jie Mei, and Michael Zeng. Mediasum: A large-scale media interview dataset for dialogue summarization. arXiv preprint arXiv:2103.06410, 2021."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.173,
+                0.103,
+                0.827,
+                0.856
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.491,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "16"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.103,
+                0.265,
+                0.119
+            ],
+            "angle": 0,
+            "content": "APPENDIX"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.136,
+                0.533,
+                0.152
+            ],
+            "angle": 0,
+            "content": "A NOTE FOR REAL-WORLD EVALUATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.168,
+                0.828,
+                0.239
+            ],
+            "angle": 0,
+            "content": "The development of autonomous agents should consider the security and safety aspects. In the real website evaluation, we have carefully conducted the experiments under human supervision in case undesired behaviors happen. We use Selenium WebDriver<sup>2</sup>, a popular library for browser automation, and limit the access per second not to stress the server. We have anonymized the real websites we tested on for safety and privacy concerns."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.258,
+                0.458,
+                0.274
+            ],
+            "angle": 0,
+            "content": "B EXTENDED RELATED WORKS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.29,
+                0.828,
+                0.457
+            ],
+            "angle": 0,
+            "content": "Document Understanding Understanding structural documents has been a practical challenge for transformer-based language models. Prior works employ layout-informed tokens (Xu et al., 2019) or even multimodal tokens from visual inputs (Appalaraju et al., 2021; Li et al., 2021a;c). Especially, for the documents written in markup languages, text-XPath alignment (Li et al., 2021b), token separation between text and HTML (Wang et al., 2022a), or extra topological information of HTML (Zhao et al., 2022) are proposed to leverage their syntax better. On the other hand, such a domain knowledge conflicts with recent generalist and scaling trends around LLMs (Anil et al., 2023; OpenAI, 2023). Because web agents require the instruction-conditioned HTML understanding, it also would be desirable to reconcile specialist aspects for HTML documents with generalist capabilities for natural language tasks. In this work, we design HTML-T5 to incorporate the structural bias of HTML by combining local-global attention for the encoder and a mixture of long-span denoising, while it can solve instruction-following better in downstream web-based tasks."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.463,
+                0.829,
+                0.591
+            ],
+            "angle": 0,
+            "content": "LLM for Task Planning The prior knowledge of commonsense in LLMs has allowed us to leverage them for a variety of task planning. For instance, Huang et al. (2022) propose LLM agent that generates natural language plans in an open-loop manner. Nottingham et al. (2023) and Wang et al. (2023b) perform sequential closed-loop planning on MineCraft. Singh et al. (2022) decode robotic plans with pythonic text, and several works incorporate planning definition and domain language into the outputs (Liu et al., 2023; Silver et al., 2023; Valmeekam et al., 2023). On the other hand, our WebAgent leverages finetuned specialist language models and performs closed-loop planning coupled with HTML summarization by decomposing given instructions. We empirically prove that our system is superior to open-loop planning with a single generalist LLM with prompting."
+        },
+        {
+            "type": "footer",
+            "bbox": [
+                0.193,
+                0.91,
+                0.424,
+                0.924
+            ],
+            "angle": 0,
+            "content": "2https://www.selenium.dev/"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.491,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "17"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.103,
+                0.563,
+                0.119
+            ],
+            "angle": 0,
+            "content": "C IMPLEMENTATION DETAILS OF HTML-T5"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.134,
+                0.828,
+                0.302
+            ],
+            "angle": 0,
+            "content": "We use the implementation of local and global attentions released by Guo et al. (2022) \\(^{3}\\). Following Guo et al. (2022), we set the local radius to \\(r = 127\\), and block size for transient global attention to \\(k = 16\\). For the pre-training objective, similar to Tay et al. (2022), we construct the mixtures and then use long mean span lengths: \\(\\mu \\in \\{8,64\\}\\), and all the denoising ratio (percentage of masked tokens in the input sequence) is set to 0.15. We adopt 4096 input sequence length and 910 output sequence length during the pre-training. The batch size for training is set to 128. We train the models with 100K iterations following other pre-training strategies for T5 families (Chung et al., 2022; Lester et al., 2021). We leverage SeqIO (Roberts et al., 2022) and T5X (Roberts et al., 2022) library to manage the training pipeline. We also use SentencePiece (Kudo & Richardson, 2018) with 32K tokens from C4 dataset (Raffel et al., 2020) as a tokenizer. During the downstream finetuning, we adopt 16K tokens for the context window unless otherwise mentioned. We have used cloud TPU-v3, which has a 32 GiB HBM memory space, with 128 cores for the experiments."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.308,
+                0.825,
+                0.365
+            ],
+            "angle": 0,
+            "content": "For the dataset, we prepare 100 WARC files (April 2019) from CommonCrawl<sup>4</sup>, and pre-process the raw HTML by removing non-Unicode and alphanumeric documents and extracting subtrees around <label> elements that have for attribute, to reduce the noise in training corpus, which results in about 3.41M examples (Table 5)."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.361,
+                0.377,
+                0.638,
+                0.437
+            ],
+            "angle": 0,
+            "content": "<table><tr><td rowspan=\"2\"># of examples</td><td colspan=\"3\"># of tokens</td></tr><tr><td>Average</td><td>90th</td><td>Max</td></tr><tr><td>3.41M</td><td>1020</td><td>4566</td><td>7627</td></tr></table>"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.171,
+                0.444,
+                0.826,
+                0.471
+            ],
+            "angle": 0,
+            "content": "Table 5: Statistics of CommonCrawl HTML corpus for self-supervised denoising pre-training of HTML-T5. Input lengths are measured in tokens by Kudo & Richardson (2018)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.5,
+                0.691,
+                0.515
+            ],
+            "angle": 0,
+            "content": "D WEBAGENT EXAMPLE FLOW IN REAL-ESTATE WEBSITE"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.208,
+                0.532,
+                0.797,
+                0.822
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.171,
+                0.827,
+                0.825,
+                0.88
+            ],
+            "angle": 0,
+            "content": "Figure 6: An example flow with planning, summarization, and grounded program synthesis in the real estate website. HTML-T5 iteratively predicts a decomposed sub-instruction and task-relevant snippet (orange) in a closed-loop manner, conditioning on the HTML documents, instruction (yellow), and history of past predictions (green). Flan-U-PaLM is prompted with sub-instruction and snippet (orange) to decode python programs (blue)."
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.191,
+                0.896,
+                0.563,
+                0.91
+            ],
+            "angle": 0,
+            "content": "<sup>3</sup>https://github.com/google-research/longt5"
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.193,
+                0.911,
+                0.413,
+                0.924
+            ],
+            "angle": 0,
+            "content": "4https://commoncrawl.org/"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.191,
+                0.896,
+                0.563,
+                0.924
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "page_number",
+            "bbox": [
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+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "18"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.171,
+                0.103,
+                0.816,
+                0.119
+            ],
+            "angle": 0,
+            "content": "E REAL-WORLD WEB AUTOMATION WITH DIFFERENT GENERALIST LLMS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.134,
+                0.828,
+                0.259
+            ],
+            "angle": 0,
+            "content": "We compare different generalist LLMs as a module of WebAgent among model-size variants (Flan-PaLM-8B, Flan-PaLM-62B, Flan-U-PaLM-540B), and publicly accessible LLM (gpt-3.5-turbo). We test those models on map website following the same 20 instructions in Appendix G. The results in Figure 7 imply that the performance of Flan-U-PaLM-540B and gpt-3.5-turbo are the same (80% success, 93.8% score), and Flan-PaLM-62B (60% success, 86.3% score) is lower than Flan-U-PaLM-540B, which is caused by the inaccurate program synthesis. In addition, Flan-PaLM-8B could not generate proper programs at all. We believe that any LLM with sufficient program synthesis capabilities could be integrated into WebAgent, including Flan-U-PaLM-540B."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.177,
+                0.271,
+                0.499,
+                0.431
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.5,
+                0.27,
+                0.828,
+                0.431
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_footnote",
+            "bbox": [
+                0.2,
+                0.435,
+                0.805,
+                0.449
+            ],
+            "angle": 0,
+            "content": "Flan-PaLM-8B Flan-PaLM-62B Flan-U-PaLM-540B gpt-3.5-turbo"
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.171,
+                0.454,
+                0.825,
+                0.495
+            ],
+            "angle": 0,
+            "content": "Figure 7: Performance (left) and error analysis (right) of real-world web automation with different generalist LLMs. We compare model-size variants (Flan-PaLM-8B, Flan-PaLM-62B, Flan-U-PaLM-540B) and public LLM (gpt-3.5-turbo) on map website."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.516,
+                0.578,
+                0.531
+            ],
+            "angle": 0,
+            "content": "F WEBSRC:STATIC HTML COMPREHENSION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.547,
+                0.828,
+                0.673
+            ],
+            "angle": 0,
+            "content": "To emphasize the advantage of our modular approach, we test WebAgent on a static website comprehension benchmark, WebSRC (Chen et al., 2021b), which is a contextual QA dataset with HTML documents. The questions require an understanding of the spatial and logical structure of websites, and the answers are either text span on HTML or yes/no. For the comprehensive evaluation, WebSRC has three different types of websites, KV, Comparison, and Table. KV task is a value extraction from the attribute key. Comparison task has several entities with the same attributes. Table task requires a structural understanding with header columns and values in the row. We finetune HTML-T5 for snippet extraction to predict data-ref corresponding to the answer and use dev set for the evaluation."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.679,
+                0.828,
+                0.779
+            ],
+            "angle": 0,
+            "content": "As did in real-world web automation, HTML-T5 first predicts data-ref attribute of task-relevant snippet from the input HTML document. To make sure there is enough context, we extract the snippet from the predicted element to the two-level-up via XPath. If it exceeds the context length of Flan-U-PaLM, we limit it into parent elements. If it still does not work, we truncate the end of extracted snippet to fit within the token budget. Because snippet extraction in table structure often loses the context to solve question-answering, we just truncate HTML documents for Table tasks. Flan-U-PaLM predicts the answers seeing 5-shot examples."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.784,
+                0.828,
+                0.909
+            ],
+            "angle": 0,
+            "content": "As shown in Table 6, single LLM, such as Flan-U-PaLM or HTML-T5, has struggled to the limited context length or model capacity. In contrast, WebAgent, our LLM-collaborative approach, enhances the performance from both single generalist and specialist LLMs, and shows competitive results with strong baselines. This demonstrates that modular LLMs work complementarily to each other. Figure 8 presents the performance comparison on different types of websites (KV, Comparison, Table) among MarkupLM (Li et al., 2021b), TIE (Zhao et al., 2022), and WebAgent. WebAgent is better at Comparison tasks, but inferior to structural understanding for KV and Table tasks, compared to other baselines, which suggest that generalist LLMs are still not suitable for recognizing structural data such as table."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "19"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.172,
+                0.199,
+                0.497,
+                0.381
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.498,
+                0.199,
+                0.824,
+                0.381
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.231,
+                0.391,
+                0.766,
+                0.406
+            ],
+            "angle": 0,
+            "content": "Figure 8: The performance comparison on different types of websites in WebSRC dev set."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.33,
+                0.609,
+                0.67,
+                0.778
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Models</td><td>EM</td><td>F1</td></tr><tr><td>T-PLM (Chen et al., 2021b)</td><td>61.67</td><td>69.85</td></tr><tr><td>H-PLM (Chen et al., 2021b)</td><td>70.12</td><td>74.14</td></tr><tr><td>V-PLM (Chen et al., 2021b)</td><td>73.22</td><td>76.16</td></tr><tr><td>MarkupLM-Large (Li et al., 2021b)</td><td>74.43</td><td>80.54</td></tr><tr><td>TIE-Large (Zhao et al., 2022)</td><td>81.66</td><td>86.24</td></tr><tr><td>Flan-U-PaLM</td><td>40.01</td><td>47.56</td></tr><tr><td>HTML-T5-Large</td><td>73.09</td><td>76.66</td></tr><tr><td>HTML-T5-XL</td><td>74.73</td><td>78.73</td></tr><tr><td>WebAgent</td><td>75.50</td><td>85.75</td></tr><tr><td>WebAgent (oracle)</td><td>76.91</td><td>86.64</td></tr></table>"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.171,
+                0.784,
+                0.828,
+                0.825
+            ],
+            "angle": 0,
+            "content": "Table 6: Evaluation on WebSRC (Chen et al., 2021b) with dev set. WebAgent, our collaborative LLMs, enhances the performance from both single generalist (Flan-U-PaLM) or specialist LLMs (HTML-T5). WebAgent (oracle) uses oracle snippets that are guaranteed to include the answers, instead of those predicted by finetuned HTML-T5."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.51,
+                0.96
+            ],
+            "angle": 0,
+            "content": "20"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.103,
+                0.825,
+                0.119
+            ],
+            "angle": 0,
+            "content": "G LIST OF LANGUAGE INSTRUCTIONS FOR REAL-WORLD WEB AUTOMATION"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.174,
+                0.133,
+                0.286,
+                0.145
+            ],
+            "angle": 0,
+            "content": "real-estate"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.177,
+                0.153,
+                0.826,
+                0.178
+            ],
+            "angle": 0,
+            "content": "1. can you search for a studio bedroom, \\(1+\\) bathroom houses in escondido, ca for corporate housing and price less than 12100 on real estate website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.176,
+                0.18,
+                0.825,
+                0.206
+            ],
+            "angle": 0,
+            "content": "2. can you find me a studio bedroom, \\(1+\\) bathroom townhomes in hollywood, ca and price less than 14600 on real estate website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.176,
+                0.208,
+                0.825,
+                0.233
+            ],
+            "angle": 0,
+            "content": "3. can you search for a studio bedroom, 1+ bathroom condos in inglewood, ca for senior housing and price less than 8700 on real estate website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.176,
+                0.235,
+                0.825,
+                0.261
+            ],
+            "angle": 0,
+            "content": "4. I would like to search for a studio bedroom, \\(1+\\) bathroom houses in compton, ca and price more than 1200 for corporate housing on real estate website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.176,
+                0.262,
+                0.825,
+                0.287
+            ],
+            "angle": 0,
+            "content": "5. can you search for a studio bedroom, \\(1+\\) bathroom apartments in oroville, ca for corporate housing on real estate website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.176,
+                0.289,
+                0.698,
+                0.302
+            ],
+            "angle": 0,
+            "content": "6. find me a studio bedroom, \\(1+\\) bathroom houses in modesto, ca on real estate website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.176,
+                0.304,
+                0.825,
+                0.329
+            ],
+            "angle": 0,
+            "content": "7. can you search for a studio bedroom, \\(1+\\) bathroom condos in redwood city, ca for student and price more than 1900 on real estate website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.176,
+                0.331,
+                0.802,
+                0.345
+            ],
+            "angle": 0,
+            "content": "8. find me a 1 bedroom condos in Santa clara, ca and price between 1600 and 7400 on real estate website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.176,
+                0.346,
+                0.818,
+                0.36
+            ],
+            "angle": 0,
+            "content": "9. find me a 1 bedroom, \\(3+\\) bathroom apartments in martinez, ca with min price 1800 on real estate website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.361,
+                0.825,
+                0.386
+            ],
+            "angle": 0,
+            "content": "10. can you find me a 2 bedroom, \\(2+\\) bathroom townhomes in concord, ca and price more than 600 on real estate website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.388,
+                0.825,
+                0.413
+            ],
+            "angle": 0,
+            "content": "11. can you find me a studio bedroom, \\(2+\\) bathroom apartments in san diego, ca and price less than 9300 on real estate website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.415,
+                0.806,
+                0.429
+            ],
+            "angle": 0,
+            "content": "12. find me a studio bedroom houses in novato, ca and price between 1500 and 6700 on real estate website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.43,
+                0.825,
+                0.455
+            ],
+            "angle": 0,
+            "content": "13. can you find me a studio bedroom, any bathroom townhomes in petaluma, ca and price more than 1000 on real estate website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.457,
+                0.778,
+                0.47
+            ],
+            "angle": 0,
+            "content": "14. search for a 1 bedroom apartments in modesto, ca and price more than 1000 on real estate website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.472,
+                0.825,
+                0.496
+            ],
+            "angle": 0,
+            "content": "15. find me a 1 bedroom, \\(2+\\) bathroom apartments in watts, ca for senior housing less than 6300 on real estate website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.499,
+                0.825,
+                0.524
+            ],
+            "angle": 0,
+            "content": "16. can you find me a 1 bedroom houses in victorville, ca that have dog friendly, furnished and price more than 700 on real estate website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.526,
+                0.826,
+                0.54
+            ],
+            "angle": 0,
+            "content": "17. I need a 2 bedroom, any bathroom condos in inglewood, ca and price more than 1000 on real estate website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.541,
+                0.701,
+                0.554
+            ],
+            "angle": 0,
+            "content": "18. find me a 2 bedroom, \\(2+\\) bathroom apartments in livermore, ca on real estate website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.556,
+                0.825,
+                0.581
+            ],
+            "angle": 0,
+            "content": "19. can you find me a 2 bedroom apartments in Santa clara, ca that has parking and price less than 10300 on real estate website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.583,
+                0.649,
+                0.596
+            ],
+            "angle": 0,
+            "content": "20. can you search for a 2 bedroom condos in oakland, ca on real estate website."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.171,
+                0.153,
+                0.826,
+                0.596
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.61,
+                0.295,
+                0.622
+            ],
+            "angle": 0,
+            "content": "social-media"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.177,
+                0.63,
+                0.583,
+                0.643
+            ],
+            "angle": 0,
+            "content": "1. Show me the most hot thread in r/google at social media website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.176,
+                0.645,
+                0.663,
+                0.658
+            ],
+            "angle": 0,
+            "content": "2. Can you point out the most hot thread in r/learnpython at social media website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.176,
+                0.66,
+                0.624,
+                0.673
+            ],
+            "angle": 0,
+            "content": "3. Could you check the 1st hot thread in r/artificial at social media website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.176,
+                0.674,
+                0.595,
+                0.687
+            ],
+            "angle": 0,
+            "content": "4. Can I check the most hot thread in Taiwan on social media website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.176,
+                0.689,
+                0.595,
+                0.701
+            ],
+            "angle": 0,
+            "content": "5. Show me the first new thread in r/facebook at social media website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.176,
+                0.703,
+                0.723,
+                0.716
+            ],
+            "angle": 0,
+            "content": "6. Present the most new thread of r/Python filtered by Tutorial flair on social media website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.176,
+                0.718,
+                0.634,
+                0.731
+            ],
+            "angle": 0,
+            "content": "7. Could you check the 1st new thread in r/facebook at social media website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.176,
+                0.733,
+                0.756,
+                0.746
+            ],
+            "angle": 0,
+            "content": "8. I want to read the 1st hot thread from r/Python tagged as Daily Thread at social media website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.176,
+                0.747,
+                0.761,
+                0.76
+            ],
+            "angle": 0,
+            "content": "9. Present the most hot thread of r/google filtered by Info | Mod Post flair on social media website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.762,
+                0.797,
+                0.775
+            ],
+            "angle": 0,
+            "content": "10. Show me the most new thread in r/learnmachinelearning filtered by Help flair at social media website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.777,
+                0.663,
+                0.79
+            ],
+            "angle": 0,
+            "content": "11. Can you point out the first hot thread in r/deeplearning at social media website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.792,
+                0.703,
+                0.805
+            ],
+            "angle": 0,
+            "content": "12. Could you check the 1st hot thread in r/machinelearningnews at social media website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.807,
+                0.711,
+                0.819
+            ],
+            "angle": 0,
+            "content": "13. Present the most hot thread of r/artificial filtered by News flair on social media website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.821,
+                0.621,
+                0.833
+            ],
+            "angle": 0,
+            "content": "14. Please find me the first hot thread in r/facebook at social media website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.835,
+                0.826,
+                0.848
+            ],
+            "angle": 0,
+            "content": "15. Present the most new thread of r/machinelearningnews filtered by Startup News flair on social media website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.85,
+                0.724,
+                0.863
+            ],
+            "angle": 0,
+            "content": "16. Show me the most hot thread in r/artificial filtered by AI Art flair at social media website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.865,
+                0.641,
+                0.877
+            ],
+            "angle": 0,
+            "content": "17. Could you check the first new thread in r/facebook at social media website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.879,
+                0.778,
+                0.892
+            ],
+            "angle": 0,
+            "content": "18. I want to read the most top thread from r/google tagged as Info | Mod Post at social media website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.894,
+                0.798,
+                0.907
+            ],
+            "angle": 0,
+            "content": "19. Show me the most new thread in r/startups filtered by Share Your Startup flair at social media website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.909,
+                0.64,
+                0.922
+            ],
+            "angle": 0,
+            "content": "20. Could you check the 2nd new thread in r/facebook at social media website."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.171,
+                0.63,
+                0.826,
+                0.922
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "21"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.106,
+                0.207,
+                0.119
+            ],
+            "angle": 0,
+            "content": "map"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.176,
+                0.125,
+                0.69,
+                0.138
+            ],
+            "angle": 0,
+            "content": "1. Show me the way from San Jose to Mountain View by 2nd Cycling at map website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.175,
+                0.141,
+                0.826,
+                0.154
+            ],
+            "angle": 0,
+            "content": "2. Please show me the way from The Painted Ladies to San Francisco Zoo with 3rd Best option at map website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.175,
+                0.155,
+                0.825,
+                0.18
+            ],
+            "angle": 0,
+            "content": "3. Could you tell me the path from California Academy of Sciences to de Young Museum by 1st Transit at map website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.176,
+                0.183,
+                0.825,
+                0.208
+            ],
+            "angle": 0,
+            "content": "4. Could you tell me the way from Union Square to The Painted Ladies with 2nd Cycling option at map website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.176,
+                0.211,
+                0.825,
+                0.237
+            ],
+            "angle": 0,
+            "content": "5. Please present the way from Chappell Hayes Observation Tower to San Jose with 2nd Walking option at map website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.176,
+                0.239,
+                0.762,
+                0.252
+            ],
+            "angle": 0,
+            "content": "6. Please present the path from Jack London Square to Emeryville by 2nd Cycling at map website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.176,
+                0.254,
+                0.712,
+                0.267
+            ],
+            "angle": 0,
+            "content": "7. I'd like to move The Midway from Children's Fairyland by 1st Cycling at map website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.176,
+                0.269,
+                0.825,
+                0.293
+            ],
+            "angle": 0,
+            "content": "8. I'd like to move Chase Center from San Francisco - Oakland Bay Bridge with 2nd Transit option at map website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.176,
+                0.297,
+                0.609,
+                0.31
+            ],
+            "angle": 0,
+            "content": "9. I want to move Pier 39 from Berkeley by 3rd Cycling at map website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.312,
+                0.723,
+                0.325
+            ],
+            "angle": 0,
+            "content": "10. I want to go to Emeryville from Mountain View with 2nd Cycling option at map website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.328,
+                0.777,
+                0.341
+            ],
+            "angle": 0,
+            "content": "11. Can you point out the way from San Mateo to Stanford University by 2nd Cycling at map website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.343,
+                0.795,
+                0.355
+            ],
+            "angle": 0,
+            "content": "12. Could you point out the way from Palace of Fine Arts to UC Berkeley by 1st Cycling at map website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.358,
+                0.825,
+                0.382
+            ],
+            "angle": 0,
+            "content": "13. Point out the way from The Painted Ladies to San Francisco Museum of Modern Art by 2nd Driving at map website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.385,
+                0.713,
+                0.398
+            ],
+            "angle": 0,
+            "content": "14. Could you find the path from Union Square to Palo Alto by 1st Cycling at map website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.401,
+                0.825,
+                0.425
+            ],
+            "angle": 0,
+            "content": "15. Please check the way from San Jose to San Jose Mineta International Airport with 1st Walking at map website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.428,
+                0.7,
+                0.442
+            ],
+            "angle": 0,
+            "content": "16. Check the path from San Francisco Zoo to Berkeley with 1st Cycling at map website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.444,
+                0.825,
+                0.469
+            ],
+            "angle": 0,
+            "content": "17. I'd like to check Parking Lots along the way from Stanford University to The Painted Ladies with Best option at map website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.471,
+                0.826,
+                0.485
+            ],
+            "angle": 0,
+            "content": "18. Check Gas stations along the way from de Young Museum to Oakland with Driving option at map website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.487,
+                0.798,
+                0.5
+            ],
+            "angle": 0,
+            "content": "19. Please show me Hotels along the way from Palace of Fine Arts to Berkeley by Transit at map website."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.502,
+                0.825,
+                0.526
+            ],
+            "angle": 0,
+            "content": "20. Check Gas stations along the way from Bay Area Discovery Museum to Santa Cruz with Best option at map website."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.171,
+                0.125,
+                0.826,
+                0.526
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.548,
+                0.688,
+                0.564
+            ],
+            "angle": 0,
+            "content": "H EXAMPLE EPISODE IN REAL-WORLD WEB AUTOMATION"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "22"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.034,
+                0.479,
+                0.048
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.365,
+                0.422,
+                0.382,
+                0.913
+            ],
+            "angle": 270,
+            "content": "map: Show me the way from San Jose to Mountain View by 2nd Cycling at map website?"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.384,
+                0.064,
+                0.608,
+                0.919
+            ],
+            "angle": 270,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.618,
+                0.342,
+                0.635,
+                0.682
+            ],
+            "angle": 270,
+            "content": "Figure 9: Example episodes of real-world web automation in map domain."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.491,
+                0.949,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "23"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.103,
+                0.517,
+                0.119
+            ],
+            "angle": 0,
+            "content": "I EXTENSIVE ABLATION OF HTML-T5"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.14,
+                0.429,
+                0.154
+            ],
+            "angle": 0,
+            "content": "I.1 DATASET AND INITIALIZATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.168,
+                0.828,
+                0.321
+            ],
+            "angle": 0,
+            "content": "To test our recipe described in Section 2.1, we compare the different dataset and model initialization for pre-training on downstream task performances; offline task planning on real-estate and average success rate on MiniWoB with 12K dataset. We use Base-size models for the experiments. For HTML-denoising, we prepare the corpus from CommonCrawl with (Extracted) or without (Raw) subtree extraction around label elements on the documents. We also compare the initialization of base architectures before HTML-denoising; from scratch or with pre-trained models on PEGASUS objective (Zhang et al., 2020) that is a masked important sentence prediction from long-context paragraph. Table 7 reveals that snippet extraction on HTML corpus improves downstream performances since such a pre-processing can reduce the noise in raw HTML. Moreover, initialization with PEGASUS pre-trained weights is essential for HTML-T5, because of the long-context and instruction-following nature of HTML-based tasks."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.312,
+                0.339,
+                0.688,
+                0.41
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>CC-HTML</td><td>PEGASUS</td><td>real-estate</td><td>MiniWoB++</td></tr><tr><td>Raw</td><td>✓</td><td>80.56</td><td>56.7%</td></tr><tr><td>Extracted</td><td>×</td><td>67.11</td><td>49.1%</td></tr><tr><td>Extracted</td><td>✓</td><td>82.46</td><td>57.0%</td></tr></table>"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.171,
+                0.417,
+                0.828,
+                0.495
+            ],
+            "angle": 0,
+            "content": "Table 7: Ablations of HTML-T5-Base on dataset quality and initialization. We evaluate offline task planning on real-estate and average success rate on MiniWoB with 12K dataset. For HTML-denoising, we prepare HTML corpus from CommonCrawl with (Extracted) or without (Raw) subtree extraction around label elements. We also compare the pre-training of base architectures with PEGASUS objective (Zhang et al., 2020) before HTML-denoising. The results imply that PEGASUS pre-training is critical for the architectures and preprocessing with subtree extraction improves the downstream performance on HTML-based tasks."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.538,
+                0.681,
+                0.552
+            ],
+            "angle": 0,
+            "content": "I.2 OFFLINE EVALUATION ON TASK PLANNING WITH MODEL SCALING"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.567,
+                0.827,
+                0.667
+            ],
+            "angle": 0,
+            "content": "We compare the offline task planning performance between HTML-T5 and LongT5 (without HTML-denosing) with different model sizes; with Base (220M parameters), Large (770M parameters), and XL (3B parameters). As described in Section 3.1, the models predict the next sub-instructions in a closed-loop manner considering the current HTML observations, user instructions, and previous sub-instruction histories as inputs. For offline task planning evaluation, we use the demonstrations on real-estate website; preparing 130 demonstrations and splitting them into train \\((90\\%)\\) and test splits \\((10\\%)\\). We report the best per-step exact match accuracy in test set."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.672,
+                0.827,
+                0.729
+            ],
+            "angle": 0,
+            "content": "Table 8 shows that HTML-T5 outperforms LongT5 on the accuracy of sub-instruction prediction, which demonstrates that HTML-denoising pre-training captures the structural bias of HTML better without sacrificing the ability to understand natural language instructions. This also implies that our proposed HTML-denoising can scale to larger-size models consistently."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.352,
+                0.748,
+                0.647,
+                0.86
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Models</td><td>real-estate</td><td>Diff.</td></tr><tr><td>LongT5-Base</td><td>78.07</td><td>0.0</td></tr><tr><td>LongT5-Large</td><td>82.89</td><td>0.0</td></tr><tr><td>LongT5-XL</td><td>81.29</td><td>0.0</td></tr><tr><td>HTML-T5-Base</td><td>82.46</td><td>+4.39</td></tr><tr><td>HTML-T5-Large</td><td>83.63</td><td>+0.74</td></tr><tr><td>HTML-T5-XL</td><td>83.92</td><td>+2.63</td></tr></table>"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.171,
+                0.867,
+                0.827,
+                0.906
+            ],
+            "angle": 0,
+            "content": "Table 8: Accuracy of offline evaluation on task planning. We leverage the demonstrations in real-estate websites. Compared to original LongT5, and as we scale model size, HTML-T5 improves the accuracy of sub-instruction prediction."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "24"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.104,
+                0.407,
+                0.119
+            ],
+            "angle": 0,
+            "content": "I.3 DESCRIPTION GENERATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.13,
+                0.825,
+                0.214
+            ],
+            "angle": 0,
+            "content": "We also investigate the capability of HTML-T5 on static HTML comprehension tasks, as well as interactive decision making tasks. We use Description Generation benchmark (Gur et al., 2022), where the models generate the textual description of elements, typically used for accessibility purposes and annotated with a special attribute in the HTML schema known as for. We evaluate the understanding the structure of HTML as it would appear to a user, despite not having access to the rendered website directly."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.22,
+                0.827,
+                0.318
+            ],
+            "angle": 0,
+            "content": "We compare LaMDA (Thoppilan et al., 2022), T5, LongT5, and HTML-T5 with respect to accuracy, BLEU (Papineni et al., 2002), and ROUGE-1 (Lin, 2004) score. As shown in Table 9, local and global attention mechanisms, underlying between LongT5 and HTML-T5, could almost solve the benchmark by improving the previous best performance by over \\(10\\%\\), with still improved performance as model size increases. Compared to the effect of local-global attention, HTML-T5 marginally improves against LongT5, which emphasizes that local and global attentions are critical to capture the hierarchical structure of HTML documents."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.208,
+                0.328,
+                0.794,
+                0.486
+            ],
+            "angle": 0,
+            "content": "<table><tr><td rowspan=\"2\">Models</td><td colspan=\"3\">Dev</td><td colspan=\"3\">Test</td></tr><tr><td>Accuracy</td><td>BLEU</td><td>ROUGE-1</td><td>Accuracy</td><td>BLEU</td><td>ROUGE-1</td></tr><tr><td>LaMDA-1B (Gur et al., 2022)</td><td>83.3</td><td>87.5</td><td>90.2</td><td>84.3</td><td>88.6</td><td>91.2</td></tr><tr><td>T5-Large (Gur et al., 2022)</td><td>83.2</td><td>90.2</td><td>90.5</td><td>84.3</td><td>91.7</td><td>91.5</td></tr><tr><td>T5-XL (Gur et al., 2022)</td><td>84.0</td><td>90.8</td><td>90.9</td><td>85.2</td><td>92.1</td><td>91.9</td></tr><tr><td>LongT5-Base</td><td>96.4</td><td>98.0</td><td>98.5</td><td>95.6</td><td>97.4</td><td>98.2</td></tr><tr><td>LongT5-Large</td><td>98.1</td><td>98.9</td><td>99.2</td><td>97.7</td><td>98.5</td><td>99.0</td></tr><tr><td>LongT5-XL</td><td>98.4</td><td>99.1</td><td>99.3</td><td>98.5</td><td>99.2</td><td>99.3</td></tr><tr><td>HTML-T5-Base</td><td>96.5</td><td>98.1</td><td>98.6</td><td>95.9</td><td>97.5</td><td>98.3</td></tr><tr><td>HTML-T5-Large</td><td>98.1</td><td>98.9</td><td>99.2</td><td>97.7</td><td>98.3</td><td>99.1</td></tr><tr><td>HTML-T5-XL</td><td>98.4</td><td>99.0</td><td>99.3</td><td>98.9</td><td>99.4</td><td>99.5</td></tr></table>"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.171,
+                0.492,
+                0.825,
+                0.557
+            ],
+            "angle": 0,
+            "content": "Table 9: Results of Description Generation benchmark (Gur et al., 2022). We compare LaMDA (Thoppilan et al., 2022), T5, LongT5, and HTML-T5 with respect to accuracy, BLEU, and ROUGE-1 scores. The results demonstrate that local and global attention mechanisms, shared modules between LongT5 and HTML-T5, could almost completely solve the benchmark by improving the previous best performance by over \\(10\\%\\). HTML-T5 slightly outperforms LongT5."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "25"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.103,
+                0.334,
+                0.119
+            ],
+            "angle": 0,
+            "content": "J FLAN-LONGT5"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.134,
+                0.827,
+                0.207
+            ],
+            "angle": 0,
+            "content": "In the web automation literature (Furuta et al., 2023; Kim et al., 2023), instruction-finetuned LLMs have great success in HTML comprehension and improve the task success. For the comparison to HTML-denosing, we prepare the instruction-finetuned LongT5 (i.e. Flan-LongT5) by leveraging Flan dataset released by Chung et al. (2022). We finetuned the pre-trained LongT5 with 100K iterations and picked up the best checkpoints."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.211,
+                0.828,
+                0.351
+            ],
+            "angle": 0,
+            "content": "As a sanity check of instruction-tuning, we evaluate Flan-LongT5 with few-shot/zero-shot settings on CoT benchmark (GSM8K (Cobbe et al., 2021), StrategyQA (Geva et al., 2021), SVAMP (Patel et al., 2021), Asdiv (Miao et al., 2021), CommonsenseQA (Talmor et al., 2019)), BigBench-Hard (BBH) (Suzgun et al., 2022), and MMLU (Hendrycks et al., 2021b) as tested in Longpre et al. (2023). We reevaluate the performance of Flan-T5, using official checkpoints \\(^{5}\\). We also check the performance of Flan-LongT5 on downstream summarization tasks, originally evaluated on LongT5 (Guo et al., 2022). We use arXiv (Cohan et al., 2018), PubMed (Cohan et al., 2018), BigPatent (Sharma et al., 2019), Multi-News (Fabbri et al., 2019), MediaSum (Zhu et al., 2021), CNN / Daily Mail (Nallapati et al., 2016) dataset for the evaluation, measuring the performance with ROUGE-1/2/L metrics."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.357,
+                0.828,
+                0.471
+            ],
+            "angle": 0,
+            "content": "Table 10 shows that we have successfully replicated the LongT5 version of instruction-finetuned language models. Flan-LongT5 achieves competitive results to original Flan-T5; for instance, Flan-LongT5-Large (36.64) outperforms Flan-T5-Large (35.25), but Flan-LongT5-XL (39.05) is still behind Flan-T5-XL (43.03) on average. This might be caused by the training instability of XL-size models (Guo et al., 2022). Because, unlike HTML-T5 on HTML-based tasks, reasoning tasks do not have long-context or hierarchical syntax, it is not surprising for Flan-LongT5 not to outperform Flan-T5. Table 11 also demonstrates that we have successfully conducted instruction-tuning without losing the capability of long text summarization."
+        },
+        {
+            "type": "footer",
+            "bbox": [
+                0.172,
+                0.897,
+                0.767,
+                0.925
+            ],
+            "angle": 0,
+            "content": "\\(^{5}\\)https://github.com/google-research/t5x/blob/main/docs/models.md#flan-t5-checkpoints"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.51,
+                0.961
+            ],
+            "angle": 0,
+            "content": "26"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.175,
+                0.034,
+                0.478,
+                0.048
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.175,
+                0.233,
+                0.307,
+                0.79
+            ],
+            "angle": 270,
+            "content": "<table><tr><td rowspan=\"2\">Models</td><td colspan=\"2\">CoT</td><td colspan=\"2\">MMLU</td><td colspan=\"2\">BBH</td><td colspan=\"2\">BBH-CoT</td><td colspan=\"2\">Avg.</td></tr><tr><td>Zero</td><td>Few</td><td>Zero</td><td>Few</td><td>Zero</td><td>Few</td><td>Zero</td><td>Few</td><td>CoT</td><td>Direct</td></tr><tr><td>Flan-T5-Large</td><td>35.14</td><td>40.03</td><td>40.68</td><td>45.12</td><td>25.90</td><td>37.48</td><td>26.17</td><td>31.45</td><td>33.20</td><td>37.29</td></tr><tr><td>Flan-T5-XL</td><td>51.74</td><td>52.64</td><td>50.76</td><td>52.40</td><td>26.09</td><td>40.96</td><td>34.12</td><td>35.62</td><td>43.53</td><td>42.55</td></tr><tr><td>Flan-LongT5-Large</td><td>44.78</td><td>45.34</td><td>38.44</td><td>40.03</td><td>28.67</td><td>34.67</td><td>29.38</td><td>31.85</td><td>37.84</td><td>35.45</td></tr><tr><td>Flan-LongT5-XL</td><td>48.78</td><td>50.02</td><td>43.44</td><td>44.74</td><td>26.53</td><td>37.77</td><td>29.09</td><td>32.01</td><td>39.97</td><td>38.12</td></tr></table>"
+        },
+        {
+            "type": "table_footnote",
+            "bbox": [
+                0.32,
+                0.101,
+                0.354,
+                0.922
+            ],
+            "angle": 270,
+            "content": "Table 10: Performance of Flan-LongT5 on reasoning tasks. We reevaluate the performance of Flan-T5 (Chung et al., 2022), using official checkpoints. Flan-LongT5 achieves competitive results to original Flan-T5."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.383,
+                0.08,
+                0.515,
+                0.917
+            ],
+            "angle": 270,
+            "content": "<table><tr><td rowspan=\"2\">Models</td><td colspan=\"3\">arXiv</td><td colspan=\"3\">PubMed</td><td colspan=\"3\">BigPatent</td><td colspan=\"3\">MultiNews</td><td colspan=\"3\">MediaSum</td><td colspan=\"3\">CNN / Daily Mail</td></tr><tr><td>R-1</td><td>R-2</td><td>R-L</td><td>R-1</td><td>R-2</td><td>R-L</td><td>R-1</td><td>R-2</td><td>R-L</td><td>R-1</td><td>R-2</td><td>R-L</td><td>R-1</td><td>R-2</td><td>R-L</td><td>R-1</td><td>R-2</td><td>R-L</td></tr><tr><td>LongT5-Large</td><td>48.28</td><td>21.63</td><td>44.11</td><td>49.98</td><td>24.69</td><td>46.46</td><td>70.38</td><td>56.81</td><td>62.73</td><td>47.18</td><td>18.44</td><td>24.18</td><td>35.54</td><td>19.04</td><td>32.20</td><td>42.49</td><td>20.51</td><td>40.18</td></tr><tr><td>LongT5-XL</td><td>48.35</td><td>21.92</td><td>44.27</td><td>50.23</td><td>24.76</td><td>46.67</td><td>76.87</td><td>66.06</td><td>70.76</td><td>48.17</td><td>19.43</td><td>24.94</td><td>36.15</td><td>19.66</td><td>32.80</td><td>43.94</td><td>21.40</td><td>41.28</td></tr><tr><td>Flan-LongT5-Large</td><td>48.52</td><td>22.00</td><td>44.46</td><td>50.46</td><td>25.08</td><td>46.96</td><td>70.53</td><td>57.13</td><td>63.02</td><td>47.76</td><td>18.99</td><td>24.52</td><td>35.71</td><td>19.18</td><td>32.33</td><td>43.13</td><td>20.89</td><td>37.28</td></tr><tr><td>Flan-LongT5-XL</td><td>48.37</td><td>21.75</td><td>44.22</td><td>50.23</td><td>24.75</td><td>46.73</td><td>76.31</td><td>65.17</td><td>70.01</td><td>48.19</td><td>19.47</td><td>24.80</td><td>36.16</td><td>19.75</td><td>32.81</td><td>43.46</td><td>21.00</td><td>37.34</td></tr></table>"
+        },
+        {
+            "type": "table_footnote",
+            "bbox": [
+                0.527,
+                0.111,
+                0.544,
+                0.914
+            ],
+            "angle": 270,
+            "content": "Table 11: Performance of Flan-LongT5 on downstream summarization tasks, compared to LongT5 (Guo et al., 2022). We measure the performance with ROUGE-1/2/L metrics."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.491,
+                0.949,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "27"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.173,
+                0.103,
+                0.58,
+                0.119
+            ],
+            "angle": 0,
+            "content": "K PER-TASK PERFORMANCE ON MINIWOB++"
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.186,
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+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Task</td><td>HTML-T5-XL (347K)</td><td>HTML-T5-XL (12K)</td><td>Plan-T5-XL (347K)</td><td>WebN-T5-XL (12K)</td></tr><tr><td>book-flight</td><td>0.99</td><td>0.00</td><td>0.48</td><td>0.00</td></tr><tr><td>choose-date</td><td>0.16</td><td>0.03</td><td>0.08</td><td>0.00</td></tr><tr><td>choose-date-easy</td><td>1.00</td><td>0.28</td><td>1.00</td><td>0.03</td></tr><tr><td>choose-date-medium</td><td>0.56</td><td>0.14</td><td>0.57</td><td>0.00</td></tr><tr><td>choose-list</td><td>0.22</td><td>0.19</td><td>0.16</td><td>0.26</td></tr><tr><td>click-button</td><td>1.00</td><td>0.92</td><td>0.98</td><td>1.00</td></tr><tr><td>click-button-sequence</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>click-checkboxes</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.96</td></tr><tr><td>click-checkboxes-large</td><td>0.90</td><td>0.94</td><td>0.98</td><td>0.22</td></tr><tr><td>click-checkboxes-soft</td><td>0.99</td><td>0.64</td><td>1.00</td><td>0.54</td></tr><tr><td>click-checkboxes-transfer</td><td>1.00</td><td>1.00</td><td>0.99</td><td>0.63</td></tr><tr><td>click-collapsible</td><td>1.00</td><td>0.41</td><td>1.00</td><td>0.00</td></tr><tr><td>click-collapsible-2</td><td>0.93</td><td>0.26</td><td>0.94</td><td>0.00</td></tr><tr><td>click-color</td><td>1.00</td><td>1.00</td><td>0.27</td><td>0.27</td></tr><tr><td>click-dialog</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>click-dialog-2</td><td>0.74</td><td>0.31</td><td>0.34</td><td>0.24</td></tr><tr><td>click-link</td><td>0.99</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>click-menu</td><td>0.37</td><td>0.26</td><td>0.41</td><td>0.37</td></tr><tr><td>click-option</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.87</td></tr><tr><td>click-pie</td><td>0.96</td><td>0.89</td><td>0.99</td><td>0.51</td></tr><tr><td>click-scroll-list</td><td>0.99</td><td>0.91</td><td>0.00</td><td>0.00</td></tr><tr><td>click-shades</td><td>0.00</td><td>0.05</td><td>0.00</td><td>0.00</td></tr><tr><td>click-shape</td><td>0.79</td><td>0.57</td><td>0.58</td><td>0.53</td></tr><tr><td>click-tab</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.74</td></tr><tr><td>click-tab-2</td><td>0.94</td><td>0.40</td><td>0.94</td><td>0.18</td></tr><tr><td>click-tab-2-hard</td><td>0.88</td><td>0.30</td><td>0.57</td><td>0.12</td></tr><tr><td>click-test</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>click-test-2</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>click-widget</td><td>1.00</td><td>0.94</td><td>1.00</td><td>1.00</td></tr><tr><td>count-shape</td><td>0.67</td><td>0.55</td><td>0.64</td><td>0.41</td></tr><tr><td>email-inbox</td><td>1.00</td><td>0.99</td><td>0.99</td><td>0.38</td></tr><tr><td>email-inbox-forward-nl</td><td>1.00</td><td>0.92</td><td>1.00</td><td>0.60</td></tr><tr><td>email-inbox-forward-nl-turk</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.33</td></tr><tr><td>email-inbox-nl-turk</td><td>0.99</td><td>0.76</td><td>0.92</td><td>0.23</td></tr><tr><td>enter-date</td><td>1.00</td><td>0.00</td><td>1.00</td><td>0.00</td></tr><tr><td>enter-password</td><td>1.00</td><td>0.99</td><td>1.00</td><td>0.97</td></tr><tr><td>enter-text</td><td>1.00</td><td>0.96</td><td>1.00</td><td>0.89</td></tr><tr><td>enter-text-dynamic</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.98</td></tr><tr><td>enter-time</td><td>1.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr><tr><td>focus-text</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>focus-text-2</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>grid-coordinate</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.49</td></tr><tr><td>guess-number</td><td>0.13</td><td>0.00</td><td>0.10</td><td>0.00</td></tr><tr><td>identify-shape</td><td>1.00</td><td>0.89</td><td>0.90</td><td>0.88</td></tr><tr><td>login-user</td><td>1.00</td><td>0.80</td><td>1.00</td><td>0.82</td></tr><tr><td>login-user-popup</td><td>1.00</td><td>0.63</td><td>0.97</td><td>0.72</td></tr><tr><td>multi-layouts</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.83</td></tr><tr><td>multi-orderings</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.88</td></tr><tr><td>navigate-tree</td><td>0.99</td><td>0.99</td><td>1.00</td><td>0.91</td></tr><tr><td>search-engine</td><td>0.93</td><td>0.55</td><td>0.59</td><td>0.34</td></tr><tr><td>social-media</td><td>0.99</td><td>0.93</td><td>0.99</td><td>0.21</td></tr><tr><td>social-media-all</td><td>0.31</td><td>0.84</td><td>0.09</td><td>0.00</td></tr><tr><td>social-media-some</td><td>0.89</td><td>0.60</td><td>0.39</td><td>0.02</td></tr><tr><td>tic-tac-toe</td><td>0.57</td><td>0.46</td><td>0.42</td><td>0.48</td></tr><tr><td>use-autocomplete</td><td>0.97</td><td>0.23</td><td>0.98</td><td>0.22</td></tr><tr><td>use-spinner</td><td>0.07</td><td>0.07</td><td>0.03</td><td>0.07</td></tr><tr><td>Average</td><td>0.856</td><td>0.655</td><td>0.755</td><td>0.484</td></tr></table>"
+        },
+        {
+            "type": "table_footnote",
+            "bbox": [
+                0.172,
+                0.747,
+                0.825,
+                0.775
+            ],
+            "angle": 0,
+            "content": "Table 12: Per-task average success rate on 56 tasks from MiniWoB++. We refer to Furuta et al. (2023) and Gur et al. (2022) for the baseline performances."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "28"
+        }
+    ]
+]
\ No newline at end of file
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+# A REAL- WORLD WEBAGENT WITH PLANNING, LONG CONTEXT UNDERSTANDING, AND PROGRAM SYNTHESIS
+
+Izzeddin Gur $^{1*}$  Hiroki Furuta $^{1,2*†}$  Austin Huang $^{1}$  Mustafa Safdari $^{1}$  Yutaka Matsuo $^{2}$  Douglas Eck $^{1}$  Aleksandra Faust $^{1}$
+
+<sup>1</sup>Google DeepMind, <sup>2</sup>The University of Tokyo  
+izzeddin@google.com, furuta@weblab.t.u-tokyo.ac.jp
+
+# ABSTRACT
+
+Pre-trained large language models (LLMs) have recently achieved better generalization and sample efficiency in autonomous web automation. However, the performance on real-world websites has still suffered from (1) open domaininess, (2) limited context length, and (3) lack of inductive bias on HTML. We introduce WebAgent, an LLM-driven agent that learns from self-experience to complete tasks on real websites following natural language instructions. WebAgent plans ahead by decomposing instructions into sub-instructions, summarizes long HTML documents into task-relevant snippets, and acts on websites via Python programs generated from those. We design WebAgent with Flan-U-PaLM, for grounded code generation, and HTML-T5, a new pre-trained LLM for long HTML documents using local and global attention mechanisms and a mixture of long-span denoising objectives, for planning and summarization. We empirically demonstrate that our modular recipe improves the success on real websites by over  $50\%$ , and that HTML-T5 is the best model to solve various HTML understanding tasks; achieving  $18.7\%$  higher success rate than the prior method on MiniWoB web automation benchmark, and SoTA performance on Mind2Web, an offline task planning evaluation.
+
+# 1 INTRODUCTION
+
+Large language models (LLM) (Brown et al., 2020; Chowdhery et al., 2022; OpenAI, 2023) can solve a variety of natural language tasks, such as arithmetic, commonsense, logical reasoning, question answering, text generation (Brown et al., 2020; Kojima et al., 2022; Wei et al., 2022), and even interactive decision making tasks (Ahn et al., 2022; Yao et al., 2022b). Recently, LLMs have also demonstrated success in autonomous web navigation by controlling computers or browsers to follow natural language instructions through multi-step reasoning and decision making (Furuta et al., 2023; Gur et al., 2022; Kim et al., 2023).
+
+However, web automation on real-world websites has still suffered from (1) the lack of pre-defined action space, (2) much longer HTML documents than simulated observations, and (3) the absence of domain-specific knowledge for understanding HTML documents (Figure 1). Considering the open-endedness of real-world websites and the complexity of instructions, defining appropriate action spaces in advance is challenging. In addition, although several works have argued that recent LLMs with instruction-finetuning or reinforcement learning from human feedback improve HTML understanding and web automation accuracy (Furuta et al., 2023; Kim et al., 2023), their architectures are not always suitable to process real-world HTML documents; as presented in Figure 2, HTML tokens of real websites are much longer than those of simulators, and most LLMs have shorter context lengths than the average HTML tokens in real websites. It is prohibitively costly to treat such long documents as inputs directly, or adopt prior techniques such as text-XPath alignment (Li et al., 2021b) or text-HTML token separation (Wang et al., 2022a). To prioritize broader task generalization and model-size scaling, such domain knowledge for HTML documents is ignored in recent LLMs.
+
+![](images/39eb50e4f17095f2f001b2f9831de1fa590a3e008341c90c90d990424da2412b.jpg)  
+Figure 1: Challenges in real-world web automation. Recent language model agents (Furuta et al., 2023; Gur et al., 2022; Kim et al., 2023; Yao et al., 2022b) can navigate simulated websites (Shi et al., 2017; Yao et al., 2022a), where the agents manipulate pre-defied actions and receive simplified HTML documents that are easy to parse. In contrast, language model agents continue to face challenges in navigating real-world websites, where they must interact with dynamic environments, handle open-ended actions (actions that cannot be pre-determined), and process lengthy HTML documents containing significant amounts of task-irrelevant information.
+
+In this work, we introduce WebAgent, an LLM-driven autonomous agent that learns from self-experience to complete user instructions on real websites by combining canonical web actions in a program space (Figure 3). WebAgent (i) plans sub-instructions for each step by decomposing natural language instructions, (ii) summarizes long HTML documents into task-relevant snippets based on the plan, and (iii) acts via programming on real websites by grounding sub-instructions and HTML snippets into executable Python codes. We combine two LLMs to form WebAgent: newly introduced HTML-T5, a domain-expert pre-trained language model, for task planning and conditional HTML summarization and Flan-U-PaLM (Chowdhery et al., 2022; Chung et al., 2022) for grounded code generation. HTML-T5 has an encoder-decoder architecture and is specialized to capture the structure of long HTML documents better by adopting local and global attention mechanisms (Guo et al., 2022). It is
+
+pre-trained using a mixture of long-span denoising objective (Tay et al., 2022) on a large-scale HTML corpus extracted from CommonCrawl. To ground language model agents into real websites, we introduce self-experience supervision, where the domain-expert language models are finetuned with data generated by scripted planning/summarization and self-generated programming.
+
+![](images/292bfb3753779e5220a522cbe2a88efdf71e18b2c7d0fb464b76b3b5de2e6d05.jpg)  
+Figure 2: Statistics of HTML tokens among real websites. Compared to simulator (about 0.5K tokens on average), HTML tokens of real websites are much longer (from 7K to 14K), which takes up the context length of large language models. As preprocessing, we remove the irrelevant tags (e.g. <script>, <meta>) and keep necessary attributes (e.g. id, type, value).
+
+Existing LLM-driven agents often solve decision making tasks with a single LLM conditioned on different prompts per role (Kim et al., 2023; Sun et al., 2023; Zheng et al., 2023), which is, however, not enough for real-world tasks whose complexity is higher than that of simulators. The empirical evaluations reveal that our method incorporating self-bootstrapped specialist language models improves HTML understanding and grounding, and achieves better generalization than single LLM agent. In real-world web automation, WebAgent significantly increases the success rate by  $50\%$ , and error analysis emphasizes that coupling task planning with HTML summarization in specialized language models is essential for task success. Moreover, HTML-T5 not only works as a core module for WebAgent but also achieves strong results by itself on the web-based tasks. On MiniWoB++ (Liu et al., 2018; Shi et al., 2017), HTML-T5 achieves  $18.7\%$  higher success than previous language model agent (Gur et al., 2022) while also outperforming competitive baselines, such as naive local-global attention models (Guo et al., 2022) and its instruction-finetuned ones (Chung et al., 2022). On the Mind2Web (Deng et al., 2023), an offline task planning dataset, HTML-T5 achieves SoTA performance among Synapse (Zheng et al., 2023) with GPT-3.5, and MindAct with FLan-T5-XL and GPT-4 (OpenAI, 2023). In summary, our key contributions are:
+
+- We propose WebAgent, integration of two modular LLMs under self-supervision for real-world web automation. The domain-expert language model handles planning and HTML summarization, and a generalist language model generates executable Python programs.  
+- We newly introduce HTML-T5 - a language model with local-global attention mechanism that is pre-trained with a mixture of long-span denoising objective on a large-scale HTML corpus, curated from CommonCrawl, to capture the syntax and semantics of HTML better.
+
+- WebAgent notably improves the success rate by over  $50\%$  in real websites. When fine-tuned on downstream demonstrations, HTML-T5 itself outperforms prior language model agent by  $18.7\%$  in MiniWoB++, and achieves SoTA performance in Mind2Web, even surpassing GPT-4.
+
+# 2 RELATED WORKS
+
+Web Automation Web automation is a sequential decision making task where agents manipulate browsers following given instructions (Shi et al., 2017), such as form filling (Diaz et al., 2013) or information retrieval (Adolphs et al., 2022) through the sequence of computer actions (Li et al., 2020; Mazumder & Riva, 2020; Shvo et al., 2021). Prior works have realized the web automation via reinforcement learning (Gur et al., 2019; Humphreys et al., 2022; Jia et al., 2019; Shaw et al., 2023), finetuned (Furuta et al., 2023; Gur et al., 2022) or prompted LLMs (Kim et al., 2023; Sun et al., 2023; Yao et al., 2022b; Zheng et al., 2023) on the simulated websites (Shi et al., 2017; Toyama et al., 2021; Yao et al., 2022a). However, there are still huge gaps
+
+between simplified simulators and real web environments; for instance, the average tokens for HTML pages are about 15 times larger (Figure 2), and pre-defined action space for specific websites is a strong assumption that may harm the generalization to out-of-distribution web pages or instructions.
+
+![](images/008f120bf43ed7db0faf65fe70ffdc4cf8a5b7042f7dbc9fedd7124cb2b75175.jpg)  
+Figure 3: WebAgent is a combination of LLMs: HTML-T5 for planning and summarization, and Flan-U-PaLM for grounded program synthesis. It is better suited for the real-world tasks; open domain action space, complex natural language instructions, and long HTML documents. See Appendix D for examples.
+
+MindAct (Deng et al., 2023) could be the most relevant work, where finetuned language model summarizes the raw HTML document into task-relevant snippets, and another model predicts the web actions in a multi-choice QA format. While MindAct also combines several language models, it has just adopted DeBERTa (He et al., 2021) and Flan-T5 (Chung et al., 2022) for summarization and actor modules, and evaluated it on the offline dataset. In contrast, we design HTML-T5, specialized for web-based tasks, to handle long HTML documents. WebAgent leverages HTML-T5 finetuned with self-experience for summarization and planning, and Flan-U-PaLM as a capable programmer, which enables it to generate open-ended web actions and to act on online real-world websites.
+
+Program Synthesis In addition to common LLMs (Brown et al., 2020; Chowdhery et al., 2022; Touvron et al., 2023), several works have proposed programming-focused language models (Chen et al., 2021a; Feng et al., 2020; Li et al., 2022; Wang et al., 2021) and their benchmarks (Austin et al., 2021; Hendrycks et al., 2021a; Lu et al., 2021). Another line of work has investigated the tool augmentation of LLMs (Parisi et al., 2022) by decoding API calls (Schick et al., 2023) or Python snippets to be parsed with the interpreter (Gao et al., 2023). Most works deal with the program synthesis on the static dataset, except for the attempts in robotics (Liang et al., 2023) and game (Trivedi et al., 2022; Wang et al., 2023a), where LLMs output Python or JavaScript snippets to command the agents. Similarly, we leverage the ability of code generation as an open-ended action space for web-based agents to manipulate the real website, and demonstrate LLMs can sequentially decode Python selenium codes considering the given sub-instructions and HTML in the prompts.
+
+See extended related works on document understanding and LLM for task planning in Appendix B.
+
+# 3 WEBAGENT
+
+WebAgent is a new architecture that combines two LLMs to achieve efficient real-world web automation. HTML-T5, a domain-expert LLM, is responsible for predicting the next sub-instruction (planning) and generating related HTML snippets (summarization). Flan-U-PaLM (540B) (Chowdhery et al., 2022; Chung et al., 2022), is prompted to generate executable Python programs based on the planning and summarization provided by HTML-T5 (Figure 3). This modular two-stage approach enables WebAgent to effectively navigate and process HTML documents.
+
+Workflow Users initiate natural language interactions with a clear intent, such as apartment searching. Upon receiving the initial request, HTML-T5 formulates a "go to <URL>" sub-instruction, triggering Flan-U-PaLM to generate a corresponding Python program that navigates to the specified website. The raw HTML content of the newly opened page is extracted and fed into HTML-T5 along with the
+
+![](images/68d40e222ee1de14fe894538971ddde15bb4c63c76e00a9c98e17b3e1a31de01.jpg)  
+Figure 4: HTML-T5 consists of (1) local and global attention mechanisms (Ainslie et al., 2020; Guo et al., 2022) and (2) a mixture of denoising objectives (Tay et al., 2022) with longer-span corruption on large-scale HTML corpus. The local and global attention mechanisms are suitable for the hierarchical tree structures of HTML documents. Because of the sparsity of content tokens in HTML, short mean span length (e.g.  $\mu = 3$ ), often used in prior works (Raffel et al., 2020), only masks less meaningful chunks. Employing longer span length (e.g.  $\mu = 8$ ) helps pre-trained language models to capture the syntax and semantics of HTML better.
+
+user's instruction and previous planning steps. This information is utilized to predict the next subinstruction and identify relevant reference IDs for extractive HTML summarization. Flan-U-PaLM, in turn, generates a Python program based on these sub-instructions and combined HTML snippets. This iterative process of planning, summarization, and program synthesis continues until a designated end-of-episode sub-instruction is predicted or the maximum number of iterations is reached.
+
+# 3.1 HTML-T5
+
+Prior research has shown that general-purpose LLMs, such as T5 (Raffel et al., 2020), Flan-T5 (Chung et al., 2022), and InstructGPT (Ouyang et al., 2022), can effectively navigate web environments (Furuta et al., 2023; Gur et al., 2022; Kim et al., 2023). However, unlike specialist transformer models (Li et al., 2021b; Wang et al., 2022a; Zhao et al., 2022), these general-purpose LLMs do not fully utilize the HTML-specific information that could otherwise lead to better understanding of HTML content. To address this limitation, we introduce HTML-T5, a pre-trained encoder-decoder language model specifically designed for HTML-based web automation tasks. HTML-T5 carefully merges the generalist and specialist characteristics of language models. It processes HTML in a text-to-text manner and employs local and global attention mechanisms (Ainslie et al., 2020) in the encoder to capture the hierarchical structure of long HTML inputs. HTML-T5 is pre-trained on a large-scale HTML corpus curated from CommonCrawl using a mixture of long-span denoising objectives (Tay et al., 2022), and then finetuned it for each downstream task. For WebAgent, we employ the self-experience supervision approach to align the model with real websites.
+
+Model Architecture Unlike natural language, HTML documents possess an explicit hierarchical structure. This structure comprises elements such as <input>, <label>, and <button>, along with their associated attributes like class, label, and id. These elements are defined locally and combined hierarchically to create HTML documents. To model this inherent hierarchy, we replace the common dense attention (Vaswani et al., 2017) with local and global attention mechanisms (Ainslie et al., 2020). Local attention restricts each token to only attend to neighboring tokens within a window. Additionally, transient global attention allows each token to attend to tokens beyond its immediate window. This is achieved through the aggregation and normalization of token representations within each window, resulting in a global memory representation. Figure 4 describes the concepts of HTML-T5; leaf elements in HTML (green) could be processed by local attention, and internal elements (purple) could be compressed into transient global attention, which naturally fits the hierarchical structure of HTML. Following LongT5 (Guo et al., 2022), we use dense attention in the decoder.
+
+Pre-Training with Mixture of Long-Span Denoising Our pre-training approach for HTML-T5 utilizes a span denoising objective. This involves randomly masking spans of tokens within an HTML document, with span lengths drawn from a Gaussian distribution with a mean of  $\mu$ . The objective is then to predict the masked spans using the remaining tokens in the HTML document (Raffel
+
+<table><tr><td rowspan="2"></td><td colspan="2">Modules</td><td colspan="2">real-estate</td><td colspan="2">social-media</td><td colspan="2">map</td><td colspan="3">Error Ratio (%)</td></tr><tr><td>Plan</td><td>Sum</td><td>Success</td><td>Score</td><td>Success</td><td>Score</td><td>Success</td><td>Score</td><td>Program</td><td>Plan</td><td>Sum</td></tr><tr><td>Flan-U-PaLM</td><td>✘</td><td>✘</td><td>10.0</td><td>55.3</td><td>20.0</td><td>25.0</td><td>10.0</td><td>51.3</td><td>36/88/11</td><td>38/0/78</td><td>26/12/11</td></tr><tr><td>Flan-U-PaLM+P</td><td>✓</td><td>✘</td><td>50.0</td><td>79.5</td><td>20.0</td><td>38.3</td><td>30.0</td><td>73.8</td><td>39/65/14</td><td>56/30/29</td><td>5/5/57</td></tr><tr><td>Flan-U-PaLM+S</td><td>✘</td><td>✓</td><td>0.0</td><td>45.7</td><td>25.0</td><td>62.1</td><td>15.0</td><td>46.3</td><td>30/67/0</td><td>40/13/100</td><td>30/20/0</td></tr><tr><td>WebAgent</td><td>✓</td><td>✓</td><td>65.0</td><td>87.6</td><td>70.0</td><td>85.8</td><td>80.0</td><td>93.8</td><td>20/33/25</td><td>70/50/50</td><td>10/17/25</td></tr></table>
+
+Table 1: Success rate of real-world web automation on real estate, social media and map websites. The score stands for the percentage of covered attributes specified in given instructions. WebAgent, with language model modules for planning and summarization, achieves the best success (65%, 70%, 80%, respectively), surpassing other baselines, such as a single Flan-U-PaLM, that with a planning language model (Flan-U-PaLM+P), and that with a summarization language model (Flan-U-PaLM+S). Without language model modules, prompted Flan-U-PaLM plans in an open-loop manner (Plan:  $\times$ ) and regular-expression-based retrieval summarizes HTML inputs ( $\mathbf{Sum}$ :  $\times$ ). The results imply that self-experience supervision notably improves the performance, and task planning should be learned by finetuning domain language models for closed-loop planning, rather than by prompting single LLM for open-loop planning. The error analysis describes the ratio across three types of errors in (real-estate)/(social-media)/(map) domains, which also points out that better adaptive planner to decompose the given instructions would contribute to further improvements of WebAgent.
+
+et al., 2020; Tay et al., 2022; Ainslie et al., 2023). While a span length of  $\mu = 3$  is commonly used, such short spans often mask less meaningful chunks in HTML documents, such as  $< / ,\mathrm{id} =$  or  $n>$  (Figure 4), where the signal-to-noise ratio can be significantly lower than natural language text. In contrast, longer spans can contain more semantically meaningful chunks, such as <form class="or type="submit">. Our empirical findings indicate that setting  $\mu \in \{8,64\}$  yields the optimal mixture for HTML documents (Section 4.2).
+
+We adopt 4096 input sequence length and 910 output sequence length during pre-training. In total,  $15\%$  of input tokens are randomly masked in the denoising objective. For the pre-training dataset, we collect 100 WARC files (April 2019) from the CommonCrawl corpus and remove the non-Unicode or alphanumeric-only HTML documents. We then extract subtrees around  $\langle \text{label} \rangle$  elements that have a special attribute called for that associates the corresponding label with a unique input element in the same HTML document. This pre-processing step improves the quality of the pre-training corpus by focusing only on HTML that is relevant for instruction following and grounding. Our final dataset has 3.41M examples. We pre-train HTML-T5 for 100K iterations following the practice in other T5 models (Chung et al., 2022; Lester et al., 2021). See Appendix C for further details.
+
+# 3.2 SELF-EXPERIENCE SUPERVISION FOR ALIGNMENT WITH REAL WEBSITES
+
+Gathering example demonstrations for LLMs to understand websites poses a significant obstacle in real-world web automation. While humans can effortlessly execute instruction following on actual websites, manually annotating every planning, summarization, and program synthesis step as detailed above is impractical. To address this issue, we propose self-experience supervision, a semi-supervised approach that necessitates minimal human involvement. In this method, manually curated scripts generate planning and summarization steps, while Flan-U-PaLM is tasked with generating Python programs. Our WebAgent aligns domain-specific language models, such as HTML-T5, with these self-gathered real-world experiences through fine-tuning (Wang et al., 2022b). This enables the generalization and alignment of agents to complex real-world tasks.
+
+Instruction Templates We maintain a collection of instruction templates that incorporate placeholders such as "Show me the way from <start> to <goal> by <n-th> <transportation> at map website". We sample instructions by randomly assigning values to placeholders from pre-defined key-value pairs.
+
+Scripted Planning and Prompted Programming We employ a rule-based parser to decompose instructions into sequences of sub-instructions; corresponding reference IDs are retrieved from HTML using regular expressions. At each step of the process, Flan-U-PaLM is provided with the sub-instruction and the associated HTML snippets to generate navigation programs that are executed through Selenium WebDriver. The success of recorded demonstrations varies, and automating success criteria for real-world tasks remains challenging. To refine the learning experience, we utilize environmental feedback to eliminate critical failures, such as program execution errors, retriever errors, and clearly erroneous URL prefixes (Ni et al., 2023).
+
+map: Show me the way from San Jose to Mountain View by 2nd Cycling at map website?
+
+![](images/26812782f9335a66950e4217971de5ad7b58954c90a245c51d5a453bdb827222.jpg)  
+Figure 5: Example episodes of real-world web automation in map domain. Considering the given instruction and HTML, WebAgent predicts the next sub-instruction and task-relevant snippet, and then synthesizes the Python script (gray), while treating the sub-instruction as a comment in the script. See Appendix H for extended figure.
+
+Finetuning for Planning and Summarization HTML-T5, a core component of WebAgent, is fine-tuned using self-experience demonstrations gathered through instruction sampling, scripted planning, and prompted program synthesis, as detailed earlier. It utilizes task instructions (e.g. please search 2 bedroom and  $2+$  bathroom houses in new york, ny with a max price of $7500 on real estate website), sub-instruction histories (e.g. go to real estate website, type in new york into search, click on search, click on price, click on max rent), and raw HTML as inputs. Subsequently, it generates the next sub-instruction (e.g. type in 7500 into max rent) and extracts the relevant data-ref attributes used for retrieving HTML snippets. Section 4.1 demonstrates the significance of integrating HTML summarization into sub-instruction prediction for enhancing real-world web automation performance.
+
+# 3.3 GROUNDED PROGRAM SYNTHESIS
+
+Web automation on real-world websites faces challenges due to the open-ended action spaces, unlike simplified simulators (Shi et al., 2017; Yao et al., 2022a). In contrast to previous approaches (Gur et al., 2019; Humphreys et al., 2022; Jia et al., 2019; Liu et al., 2018), real-world web agents cannot pre-define a categorical action space to specify the interactive elements on the websites. To address this open-domain challenge, we introduce the act via programming paradigm in web automation by utilizing the conditional code generation capabilities of LLMs (Chen et al., 2021a; Liang et al., 2023). Provided with few-shot generic examples (such as selecting checkboxes, entering text into inputs, clicking on options, and scrolling etc.) for program generation, the next sub-instruction, and the extracted HTML snippet from HTML-T5, Flan-U-PaLM (Chowdhery et al., 2022; Chung et al., 2022) decodes an Python program (Figure 3) executable with Selenium WebDriver, a library for browser automation. This conditional program synthesis requires LLMs to not only generate code to follow natural language instructions but also understand the semantics and functionality of HTML elements. We provide several Python snippet examples generated by Flan-U-PaLM as follows (sub-instructions are treated as comments in the script):
+
+```txt
+1 # Type in walnut creek, ca into search  
+2 driver.find_element(By.css_selector, '[data-ref="175"').clear()  
+3 driver.find_element(By.css_selector, '[data-ref="175"').send_keys("walnut creek, ca")  
+4  
+5 # Submit the search  
+6 driver.find_element(By.css_selector, '[data-ref="175"').submit()  
+7  
+8 # Click on the apartments  
+9 driver.find_element(By.css_selector, '[data-ref="572"').click()  
+10  
+11 # Scroll down housing type by 200px  
+12 driver.executeScript('getScrollParent.document.querySelector "#type-of-housing").scrollBy({top: 200})')
+```
+
+# 4 EXPERIMENTAL RESULTS
+
+To study how a modular combination of LLMs under self-supervision enables real-world web automation by overcoming open-endedness and long context documents, we execute instruction-following tasks on real websites (Section 4.1). In Appendix F, we also test WebAgent on WebSRC (Chen et al., 2021b), a static HTML comprehension benchmark, compared to prior transformer models specialized for structured documents (Li et al., 2021b; Zhao et al., 2022). In addition, we quantify the performance of HTML-T5 itself on simulated web benchmark, MiniWoB++, and offline task planning benchmark, Mind2Web (Section 4.2).
+
+<table><tr><td>Architectures</td><td>Attention Type</td><td>L = 2048</td><td>L = 4096</td></tr><tr><td>Flan-T5-Base</td><td>Dense</td><td>34.0%</td><td>35.3%</td></tr><tr><td>Long-T5-Base</td><td>Local</td><td>43.4%</td><td>44.0%</td></tr><tr><td>Long-T5-Base</td><td>Local &amp; Global</td><td>53.1%</td><td>53.6%</td></tr></table>
+
+<table><tr><td>Span Length μ</td><td>real-estate</td><td>MiniWoB++</td></tr><tr><td>(no HTML-denoising)</td><td>78.07</td><td>53.8%</td></tr><tr><td>3,8,64,Prefix</td><td>80.56</td><td>55.2%</td></tr><tr><td>3,8,64</td><td>80.56</td><td>55.4%</td></tr><tr><td>8,64</td><td>82.46</td><td>57.0%</td></tr><tr><td>8,32,64</td><td>82.16</td><td>55.6%</td></tr><tr><td>8,64,96</td><td>81.29</td><td>53.6%</td></tr><tr><td>16,64</td><td>79.97</td><td>55.2%</td></tr></table>
+
+Table 2: (Left) Architecture comparison on MiniWoB++ 12K dataset (Liu et al., 2018) with average success rate over 56 tasks. Local and global attention matches to the hierarchical tree structure of HTML, and then improves the success rate by over  $18\%$ , compared to the instruction-finetuned dense attentions (Chung et al., 2022; Furuta et al., 2023). (Right) HTML-denoising comparison with different mixtures of span length (Raffel et al., 2020; Tay et al., 2022). We use LongT5-Base models for pre-training. HTML-denoising generally improves the performance on offline task planning on real estate website and MiniWoB benchmark. Especially, using longer span lengths  $(\mu \in \{8,6\})$  outperforms other choices, including the popular configuration in natural language domain  $(\mu \in \{3,8,64\} + \text{Prefix LM objective})$ , which can reduce the less meaningful prediction from shorter spans (e.g.  $\mu = 3$ ), and inject the structural bias of HTML better.
+
+# 4.1 REAL-WORLD WEB AUTOMATION
+
+Evaluation Methodology We first evaluate WebAgent with the real-world navigation performance under human supervision, at real estate website (a platform for housing), social media website (a network of communities), and map website. These three websites have different properties. real-estate requires long-horizon planning (about 20 steps per episode) for complex formfilling with a few page transitions (at least 2 pages), and social-media needs shorter plans (about 10 steps per episode) with many page transitions (at least 4 pages) by selecting appropriate hyperlinks on the page. map is the easiest domain with shorter plans and a few page transitions. WebAgent receives natural language instructions (e.g. Can you search for a studio bedroom,  $1+$  bathroom apartments in oroville, ca for corporate housing on real estate website?, or Could you present the most new thread of Python community filtered by Tutorial tag on social media website?), and acts via planning, summarizing by HTML-T5, and then programming by Flan-U-PaLM (Figure 5). Through the self-experience supervision process, we curate 260 episodes on real estate website, 230 episodes on social media website, and 410 episodes on map website to finetune HTML-T5.
+
+We prepare 20 different natural language instructions (see Appendix G for the full list), and measure the success rate and score for the evaluation. The score represents the percentage of required attributes covered during the episode (Yao et al., 2022a); for instance, (1) apartments for (2) corporate housing with (3) studio bedroom and (4)  $1+$  bathroom located in (5) oroville, ca, can be specified in the instruction. When the agents could search the housing satisfying (1), (2), (5) and not (3), (4), the score is  $60 (= 100 \times 3/5)$ . If the agents achieve 100 score, that episode will mark as success.
+
+Results For comparison, we prepare three baselines, consisting of language model modules and a single LLM conditioned on different prompts per role, such as Flan-U-PaLM (Chung et al., 2022), that with a planning language model (Flan-U-PaLM+P), and that with a summarization language model (Flan-U-PaLM+S). If they do not use language model modules, prompted Flan-U-PaLM plans in an open-loop manner (Plan:  $\times$ ), and regular-expression-based retrieval summarizes given raw HTML (Sum:  $\times$ ). Table 1 shows that by leveraging planning and summarization language model modules, WebAgent achieves best  $65\%$  success and  $87.6\%$  score on real-estate,  $70\%$  success and  $85.8\%$  score on social-media, and  $80\%$  success and  $93.8\%$  score on map, significantly outperforming single Flan-U-PaLM, or with partial language model modules (most of those achieve about  $10 - 30\%$  success). This result suggests that self-experience supervision notably improves the performance, and closed-loop planning grounded on HTML observations via finetuned domain language models is more suitable for open-ended web automation than open-loop planning with few-shot LLMs. This trend is remarkable in real-estate (even Flan-U-PaLM+P achieves  $50\%$  success), where the longer planning horizon is needed to fulfill instructions. We also observe that coupling sub-instruction prediction with HTML summarization in language model modules plays a critical role in task success. The development of more capable planning modules to decompose the given instructions adaptively and accurately could help WebAgent improve the performance further.
+
+Error Analysis We also analyze the reason of failures by categorizing them into programming, planning, and summarization errors (Table 1). Programming error does not satisfy the given sub-instructions or HTML snippet. Planning error predicts sub-instructions conflicting with user instruc
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Train</td><td colspan="4">Cross-Task</td><td colspan="4">Cross-Website</td><td colspan="4">Cross-Domain</td></tr><tr><td>Ele. Acc</td><td>Op. F1</td><td>Step SR</td><td>SR</td><td>Ele. Acc</td><td>Op. F1</td><td>Step SR</td><td>SR</td><td>Ele. Acc</td><td>Op. F1</td><td>Step SR</td><td>SR</td></tr><tr><td>Synapse (GPT-3.5)</td><td>ICL</td><td>34.4</td><td>-</td><td>30.6</td><td>2.0</td><td>28.8</td><td>-</td><td>23.4</td><td>1.1</td><td>29.4</td><td>-</td><td>25.9</td><td>1.6</td></tr><tr><td>MindAct (Flan-T5-XL)</td><td>SL</td><td>55.1</td><td>75.7</td><td>52.0</td><td>5.2</td><td>42.0</td><td>65.2</td><td>38.9</td><td>5.1</td><td>42.1</td><td>66.5</td><td>39.6</td><td>2.9</td></tr><tr><td>MindAct (GPT-4)</td><td>ICL</td><td>41.6</td><td>60.6</td><td>36.2</td><td>2.0</td><td>35.8</td><td>51.1</td><td>30.1</td><td>2.0</td><td>37.1</td><td>46.5</td><td>26.4</td><td>2.0</td></tr><tr><td>HTML-T5-XL (ours)</td><td>SL</td><td>60.6</td><td>81.7</td><td>57.8</td><td>10.3</td><td>47.6</td><td>71.9</td><td>42.9</td><td>5.6</td><td>50.2</td><td>74.9</td><td>48.3</td><td>5.1</td></tr></table>
+
+Table 4: Offline action prediction performance in Mind2Web dataset. We leverage the cached candidate generation results and direct QA formulation by following Deng et al. (2023). HTML-T5 significantly outperforms MindAct with Flan-T5 or GPT-4, and Synapse (Zheng et al., 2023) with GPT-3.5, across task/website/domain generalization in terms of all the metrics (element accuracy, operation F1, and success rates).
+
+tions, and summarization error fails to extract the relevant HTML snippets for given sub-instructions. From the website perspective, the failures on real-estate concentrate in planning because of its long-horizon nature. map also fails in planning when confusing starting point and destination. In contrast, social-media tends to fail in programming due to the ambiguous sub-instructions or summarization including redundant hyperlinks, which results in transiting wrong pages or clicking unexecutable elements. From the method perspective, WebAgent often fails in planning by predicting incorrect sub-instructions (for instance, in real-estate, WebAgent generates incorrect plans in  $70\%$  of failure episodes), while other baselines more fail in programming or summarization steps. This observation indicates that, through the self-experience supervision, the ratio of programming and summarization errors has decreased while the fundamental difficulty of planning, which requires consistent and accurate prediction over long horizon without error accumulation, still remains.
+
+# 4.2 ABLATION OF HTML-T5
+
+In addition to the evaluation as WebAgent system, we extensively examine HTML-T5 about (i) the generalization to other websites with Mind2Web (Deng et al., 2023), (ii) the performance on MiniWoB++, a standard web automation benchmark (Liu et al., 2018; Shi et al., 2017), and (iii) its architecture and pre-training objective. We adopt 16K tokens for the context window unless otherwise mentioned. We present results on offline task planning, and description generation (Gur et al., 2022) to test HTML understanding on static dataset in Appendix I.
+
+Mind2Web Mind2Web (Deng et al., 2023) is an action-annotated real-world dataset with over 2K instructions collected from 137 websites. It provides action prediction tasks that measure the generalization of LLMs across the tasks, websites, and their domains (e.g. travel, shopping). Similar to real-world evaluation, the input is a set of HTML snippets, a task instruction, and an action history. The output comprises a target element to interact with, along with the operation, such as click, type, or select an option. We finetune HTML-T5-XL with the training dataset. The performance is evaluated with element accuracy,
+
+operation F1, and step success rate that cares for both element and operation correctness. Table 4 reveals that HTML-T5 significantly outperforms baselines with Flan-T5-XL or GPT-4 (OpenAI, 2023) across task/website/domain generalization, which increases element accuracy by  $5 - 8\%$  , operation F1 by  $6 - 8\%$  , and step success rate by  $4 - 8\%$  . This highlights that HTML-T5 can handle real-world web automation tasks better and shows generalization beyond our real-world evaluation with 3 websites.
+
+MiniWoB++ We here evaluate HTML-T5 on 56 simulated tasks in MiniWoB++ using 100 evaluation episodes per task. Inputs are analogous to real-world evaluation, utilizing HTML documents, while outputs are adhering to a pre-defined format by the simulator such as click(ref = X). We finetune HTML-T5 with 12K human demonstrations (Liu et al., 2018), and compare the average success rate to prior supervised-learned agents (Gur et al., 2022; Humphreys et al., 2022), LongT5, and its instruction-finetuned variants (Chung et al., 2022)  $^{1}$ . Table 3 shows that HTML-T5-XL significantly
+
+<table><tr><td>Models</td><td>Data</td><td>Success</td><td>Diff.</td></tr><tr><td>SoTA (Zheng et al., 2023)</td><td>-</td><td>99.2%</td><td>-</td></tr><tr><td>CC-Net</td><td>2.4M</td><td>32.0%</td><td>-</td></tr><tr><td>WebN-T5-XL</td><td>12K</td><td>48.4%</td><td>-</td></tr><tr><td>LongT5-Base</td><td></td><td>53.8%</td><td>0.0</td></tr><tr><td>LongT5-Large</td><td>12K</td><td>56.3%</td><td>0.0</td></tr><tr><td>LongT5-XL</td><td></td><td>60.4%</td><td>0.0</td></tr><tr><td>Flan-LongT5-Base</td><td></td><td>54.1%</td><td>+0.3</td></tr><tr><td>Flan-LongT5-Large</td><td>12K</td><td>56.1%</td><td>-0.2</td></tr><tr><td>Flan-LongT5-XL</td><td></td><td>61.1%</td><td>+0.7</td></tr><tr><td>HTML-T5-Base (ours)</td><td></td><td>57.0%</td><td>+3.2</td></tr><tr><td>HTML-T5-Large (ours)</td><td>12K</td><td>60.8%</td><td>+4.5</td></tr><tr><td>HTML-T5-XL (ours)</td><td></td><td>67.1%</td><td>+6.7</td></tr><tr><td>Flan-T5-XL</td><td>347K</td><td>75.5%</td><td>-</td></tr><tr><td>Flan-T5-XXL</td><td></td><td>79.0%</td><td>-</td></tr><tr><td>HTML-T5-XL (ours)</td><td>347K</td><td>85.6%</td><td>-</td></tr></table>
+
+Table 3: Average success rate of MiniWoB++ with 56 tasks. We use 12K demonstrations and compare HTML-T5 among supervised-finetuned methods. HTML-T5-XL outperforms WebN-T5-XL (Gurel al., 2022), the prior best method, by  $18.7\%$ . HTML-denoising also yields better the success rate than instruction tuned ones. Finetuned HTML-T5 with 347K episodes (Furuta et al., 2023) outperforms Flan-T5-XXL (11B parameters) even with 3B parameters, which gets closer to SoTA with GPT-3.5. See Appendix K for the detailed results.  
+element and operation correctness. Table 4 rees with Flan-T5-XL or GPT-4 (OpenAI, 2023) eases element accuracy by  $5 - 8\%$  , operation F1 nts that HTML-T5 can handle real-world web and our real-world evaluation with 3 websites.
+
+outperforms WebN-T5, the prior best model, by  $18.7\%$ . Notably, we demonstrate HTML-denoising consistently improves the performance on top of LongT5 in all the model sizes, better than instruction-finetuning introduced in prior work (Furuta et al., 2023). Furthermore, we finetune HTML-T5-XL with 347K demonstrations from Furuta et al. (2023), which performs better than 11B-parameter Flant5-XXL even with 3B parameters, achieving  $85.6\%$  success. These prove we successfully incorporate domain knowledge on HTML comprehension for web automation into pre-trained language models.
+
+Architecture and Objective We hypothesize that local and global attention mechanisms can capture the hierarchical structures of HTML documents better than dense attention. We compare the web automation performance among 56 MiniWoB++ tasks (Gur et al., 2022), by finetuning HTML-T5 with public 12K-episode dataset (Liu et al., 2018). We adopt 2048 and 4096 tokens as input length and prepare Base-size architectures. Table 2 (left) reveals that the combination of local and global attentions achieves the superior success rate by over  $18\%$  compared to the instruction-finetuned dense attentions (Chung et al., 2022; Raffel et al., 2020) and local attention only. Surprisingly, local attention only still surpasses the dense attention by about  $9\%$ , which suggests local relation between elements and attributes in HTML are essential for web tasks.
+
+As for pre-training objective in Table 2 (right), HTML-denoising generally improves the performance on offline task planning on real estate website and MiniWoB. Especially, using only longer span lengths  $(\mu \in \{8,64\})$  outperforms other choices, including the popular configuration in natural language domain  $(\mu \in \{3,8,64\} +\text{Prefix LM objective})$ , which can reduce the less meaningful prediction from shorter spans (e.g.  $\mu = 3$ ), and inject the structural bias of HTML into language models better. See Appendix I.2 for further results with model scaling.
+
+# 5 DISCUSSION AND LIMITATION
+
+Modular Approach with Specialist Language Models We demonstrate it is beneficial to divide web automation into planning, HTML summarization, and code generation, and to combine domain-expert language models aligned with self-experience data. Such modular approaches have also been adopted to support the inference of LLMs (Xu et al., 2023), multimodal tasks (Zeng et al., 2022), and robotics (Ahn et al., 2022), which, however, might cause additional computational costs and latency.
+
+Broad Generalization across the Internet Because open-loop planning with prompted Flan-U-PaLM achieves at most  $10 - 30\%$  success, we have demonstrated that self-experience supervision on real websites is essential for planning modules. As we demonstrated in Mind2Web, our method could generalize across the internet if we have enough data. It would be expected to collect demonstrations at scale and align larger domain-expert models with them in future works.
+
+Feedback for Program Synthesis We leverage Flan-U-PaLM with 540B parameters, as a capable program synthesis module via few-shot prompting. Such a large model, however, makes it challenging to reflect the feedback about the errors in generated code, compared to smaller models. We leave it as future direction to incorporate the feedback for program synthesis into larger language models.
+
+Evaluation for Real-world Web Automation Beyond the simulated web environments (Shi et al., 2017; Yao et al., 2022a), we have exhibited WebAgent can follow given complex and sometimes ambiguous instructions on real estate, social media and map websites. On the other hand, it is costly to evaluate the performance of autonomous agents in the real world. Automated evaluation with minimal human intervention would be helpful for the scalable development of real-world web agents.
+
+# 6 CONCLUSION
+
+We build a system for real-world web automation, combining HTML-T5 for planning and HTML summarization and Flan-U-PaLM for grounded program synthesis. Our proposed WebAgent achieves around  $70 - 80\%$  success on real websites via self-experience supervision, outperforming single LLM approach by over  $50\%$ , which suggests dividing the sequence of sub-problems with multiple language models can increase the entire task success. We also propose a scalable recipe for HTML-specialized language models where we train local and global attention mechanisms with a mixture of long-span denoising objectives to capture the hierarchical structures of HTML documents. HTML-T5 not only plays an essential role in WebAgent but also can achieve the best results on a variety of HTML-based benchmarks such as Mind2Web and MiniWoB++. We hope our work contributes to getting us one-step closer to the practical deployment of autonomous web agent systems.
+
+# ETHICS STATEMENT
+
+This paper presents encouraging evidence of autonomous agents' potential for deployment on real websites, extending beyond simulated environments. In the foreseeable future, this technology could lead to the development of sophisticated AI assistant tools for computers and smartphones, enhancing productivity and accessibility for society.
+
+While we recognize the promising aspects of autonomous agents, we must also consider the potential for misuse and unintended consequences in their development. As our proposed system is based on LLMs, there is a risk of prompt injection. The improper use of web automation could pose cybersecurity threats and expose users to scams. To mitigate these risks, it is crucial for researchers, policymakers, and industry stakeholders to collaborate on establishing guidelines and regulations for the development of autonomous agents. Additionally, security research focused on LLM agents will become an essential domain for society.
+
+# ACKNOWLEDGMENTS
+
+We thank Heiga Zen, Yingjie Miao, Yusuke Iwasawa, Joshua Ainslie, Santiago Ontonon, Quoc V. Le, Zoubin Ghahramani, Jeff Dean, Tris Warkentin for the supports and advises on this work. Hiroki Furuta was supported by JSPS KAKENHI Grant Number JP22J21582.
+
+# REFERENCES
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+
+# APPENDIX
+
+# A NOTE FOR REAL-WORLD EVALUATION
+
+The development of autonomous agents should consider the security and safety aspects. In the real website evaluation, we have carefully conducted the experiments under human supervision in case undesired behaviors happen. We use Selenium WebDriver<sup>2</sup>, a popular library for browser automation, and limit the access per second not to stress the server. We have anonymized the real websites we tested on for safety and privacy concerns.
+
+# B EXTENDED RELATED WORKS
+
+Document Understanding Understanding structural documents has been a practical challenge for transformer-based language models. Prior works employ layout-informed tokens (Xu et al., 2019) or even multimodal tokens from visual inputs (Appalaraju et al., 2021; Li et al., 2021a;c). Especially, for the documents written in markup languages, text-XPath alignment (Li et al., 2021b), token separation between text and HTML (Wang et al., 2022a), or extra topological information of HTML (Zhao et al., 2022) are proposed to leverage their syntax better. On the other hand, such a domain knowledge conflicts with recent generalist and scaling trends around LLMs (Anil et al., 2023; OpenAI, 2023). Because web agents require the instruction-conditioned HTML understanding, it also would be desirable to reconcile specialist aspects for HTML documents with generalist capabilities for natural language tasks. In this work, we design HTML-T5 to incorporate the structural bias of HTML by combining local-global attention for the encoder and a mixture of long-span denoising, while it can solve instruction-following better in downstream web-based tasks.
+
+LLM for Task Planning The prior knowledge of commonsense in LLMs has allowed us to leverage them for a variety of task planning. For instance, Huang et al. (2022) propose LLM agent that generates natural language plans in an open-loop manner. Nottingham et al. (2023) and Wang et al. (2023b) perform sequential closed-loop planning on MineCraft. Singh et al. (2022) decode robotic plans with pythonic text, and several works incorporate planning definition and domain language into the outputs (Liu et al., 2023; Silver et al., 2023; Valmeekam et al., 2023). On the other hand, our WebAgent leverages finetuned specialist language models and performs closed-loop planning coupled with HTML summarization by decomposing given instructions. We empirically prove that our system is superior to open-loop planning with a single generalist LLM with prompting.
+
+# C IMPLEMENTATION DETAILS OF HTML-T5
+
+We use the implementation of local and global attentions released by Guo et al. (2022)  $^{3}$ . Following Guo et al. (2022), we set the local radius to  $r = 127$ , and block size for transient global attention to  $k = 16$ . For the pre-training objective, similar to Tay et al. (2022), we construct the mixtures and then use long mean span lengths:  $\mu \in \{8,64\}$ , and all the denoising ratio (percentage of masked tokens in the input sequence) is set to 0.15. We adopt 4096 input sequence length and 910 output sequence length during the pre-training. The batch size for training is set to 128. We train the models with 100K iterations following other pre-training strategies for T5 families (Chung et al., 2022; Lester et al., 2021). We leverage SeqIO (Roberts et al., 2022) and T5X (Roberts et al., 2022) library to manage the training pipeline. We also use SentencePiece (Kudo & Richardson, 2018) with 32K tokens from C4 dataset (Raffel et al., 2020) as a tokenizer. During the downstream finetuning, we adopt 16K tokens for the context window unless otherwise mentioned. We have used cloud TPU-v3, which has a 32 GiB HBM memory space, with 128 cores for the experiments.
+
+For the dataset, we prepare 100 WARC files (April 2019) from CommonCrawl<sup>4</sup>, and pre-process the raw HTML by removing non-Unicode and alphanumeric documents and extracting subtrees around <label> elements that have for attribute, to reduce the noise in training corpus, which results in about 3.41M examples (Table 5).
+
+<table><tr><td rowspan="2"># of examples</td><td colspan="3"># of tokens</td></tr><tr><td>Average</td><td>90th</td><td>Max</td></tr><tr><td>3.41M</td><td>1020</td><td>4566</td><td>7627</td></tr></table>
+
+Table 5: Statistics of CommonCrawl HTML corpus for self-supervised denoising pre-training of HTML-T5. Input lengths are measured in tokens by Kudo & Richardson (2018).
+
+# D WEBAGENT EXAMPLE FLOW IN REAL-ESTATE WEBSITE
+
+![](images/88fc5fcee87490b47b5253d1f3eb1a8bb982422cc47b34b196f0900168fada52.jpg)  
+Figure 6: An example flow with planning, summarization, and grounded program synthesis in the real estate website. HTML-T5 iteratively predicts a decomposed sub-instruction and task-relevant snippet (orange) in a closed-loop manner, conditioning on the HTML documents, instruction (yellow), and history of past predictions (green). Flan-U-PaLM is prompted with sub-instruction and snippet (orange) to decode python programs (blue).
+
+# E REAL-WORLD WEB AUTOMATION WITH DIFFERENT GENERALIST LLMS
+
+We compare different generalist LLMs as a module of WebAgent among model-size variants (Flan-PaLM-8B, Flan-PaLM-62B, Flan-U-PaLM-540B), and publicly accessible LLM (gpt-3.5-turbo). We test those models on map website following the same 20 instructions in Appendix G. The results in Figure 7 imply that the performance of Flan-U-PaLM-540B and gpt-3.5-turbo are the same (80% success, 93.8% score), and Flan-PaLM-62B (60% success, 86.3% score) is lower than Flan-U-PaLM-540B, which is caused by the inaccurate program synthesis. In addition, Flan-PaLM-8B could not generate proper programs at all. We believe that any LLM with sufficient program synthesis capabilities could be integrated into WebAgent, including Flan-U-PaLM-540B.
+
+Figure 7: Performance (left) and error analysis (right) of real-world web automation with different generalist LLMs. We compare model-size variants (Flan-PaLM-8B, Flan-PaLM-62B, Flan-U-PaLM-540B) and public LLM (gpt-3.5-turbo) on map website.  
+![](images/916e50d74c020f9cb0d236afab32dc1ed65a7d2c884e2096b5a230912f277f9f.jpg)  
+Flan-PaLM-8B Flan-PaLM-62B Flan-U-PaLM-540B gpt-3.5-turbo
+
+![](images/46597798c14ef302facae1af40771a7f89def6ac32be30e6a2cb521024bf3eac.jpg)
+
+# F WEBSRC:STATIC HTML COMPREHENSION
+
+To emphasize the advantage of our modular approach, we test WebAgent on a static website comprehension benchmark, WebSRC (Chen et al., 2021b), which is a contextual QA dataset with HTML documents. The questions require an understanding of the spatial and logical structure of websites, and the answers are either text span on HTML or yes/no. For the comprehensive evaluation, WebSRC has three different types of websites, KV, Comparison, and Table. KV task is a value extraction from the attribute key. Comparison task has several entities with the same attributes. Table task requires a structural understanding with header columns and values in the row. We finetune HTML-T5 for snippet extraction to predict data-ref corresponding to the answer and use dev set for the evaluation.
+
+As did in real-world web automation, HTML-T5 first predicts data-ref attribute of task-relevant snippet from the input HTML document. To make sure there is enough context, we extract the snippet from the predicted element to the two-level-up via XPath. If it exceeds the context length of Flan-U-PaLM, we limit it into parent elements. If it still does not work, we truncate the end of extracted snippet to fit within the token budget. Because snippet extraction in table structure often loses the context to solve question-answering, we just truncate HTML documents for Table tasks. Flan-U-PaLM predicts the answers seeing 5-shot examples.
+
+As shown in Table 6, single LLM, such as Flan-U-PaLM or HTML-T5, has struggled to the limited context length or model capacity. In contrast, WebAgent, our LLM-collaborative approach, enhances the performance from both single generalist and specialist LLMs, and shows competitive results with strong baselines. This demonstrates that modular LLMs work complementarily to each other. Figure 8 presents the performance comparison on different types of websites (KV, Comparison, Table) among MarkupLM (Li et al., 2021b), TIE (Zhao et al., 2022), and WebAgent. WebAgent is better at Comparison tasks, but inferior to structural understanding for KV and Table tasks, compared to other baselines, which suggest that generalist LLMs are still not suitable for recognizing structural data such as table.
+
+![](images/ab044febc4916d8d39054cd7a0f69aa0a1a4734f3c6d41477f739b7295982854.jpg)  
+Figure 8: The performance comparison on different types of websites in WebSRC dev set.
+
+![](images/ebfb855adef0edfd3f212c3985046599db77f08845be4188ad76bdbb01c67deb.jpg)
+
+<table><tr><td>Models</td><td>EM</td><td>F1</td></tr><tr><td>T-PLM (Chen et al., 2021b)</td><td>61.67</td><td>69.85</td></tr><tr><td>H-PLM (Chen et al., 2021b)</td><td>70.12</td><td>74.14</td></tr><tr><td>V-PLM (Chen et al., 2021b)</td><td>73.22</td><td>76.16</td></tr><tr><td>MarkupLM-Large (Li et al., 2021b)</td><td>74.43</td><td>80.54</td></tr><tr><td>TIE-Large (Zhao et al., 2022)</td><td>81.66</td><td>86.24</td></tr><tr><td>Flan-U-PaLM</td><td>40.01</td><td>47.56</td></tr><tr><td>HTML-T5-Large</td><td>73.09</td><td>76.66</td></tr><tr><td>HTML-T5-XL</td><td>74.73</td><td>78.73</td></tr><tr><td>WebAgent</td><td>75.50</td><td>85.75</td></tr><tr><td>WebAgent (oracle)</td><td>76.91</td><td>86.64</td></tr></table>
+
+Table 6: Evaluation on WebSRC (Chen et al., 2021b) with dev set. WebAgent, our collaborative LLMs, enhances the performance from both single generalist (Flan-U-PaLM) or specialist LLMs (HTML-T5). WebAgent (oracle) uses oracle snippets that are guaranteed to include the answers, instead of those predicted by finetuned HTML-T5.
+
+# G LIST OF LANGUAGE INSTRUCTIONS FOR REAL-WORLD WEB AUTOMATION
+
+# real-estate
+
+1. can you search for a studio bedroom,  $1+$  bathroom houses in escondido, ca for corporate housing and price less than 12100 on real estate website.  
+2. can you find me a studio bedroom,  $1+$  bathroom townhomes in hollywood, ca and price less than 14600 on real estate website.  
+3. can you search for a studio bedroom, 1+ bathroom condos in inglewood, ca for senior housing and price less than 8700 on real estate website.  
+4. I would like to search for a studio bedroom,  $1+$  bathroom houses in compton, ca and price more than 1200 for corporate housing on real estate website.  
+5. can you search for a studio bedroom,  $1+$  bathroom apartments in oroville, ca for corporate housing on real estate website.  
+6. find me a studio bedroom,  $1+$  bathroom houses in modesto, ca on real estate website.  
+7. can you search for a studio bedroom,  $1+$  bathroom condos in redwood city, ca for student and price more than 1900 on real estate website.  
+8. find me a 1 bedroom condos in Santa clara, ca and price between 1600 and 7400 on real estate website.  
+9. find me a 1 bedroom,  $3+$  bathroom apartments in martinez, ca with min price 1800 on real estate website.  
+10. can you find me a 2 bedroom,  $2+$  bathroom townhomes in concord, ca and price more than 600 on real estate website.  
+11. can you find me a studio bedroom,  $2+$  bathroom apartments in san diego, ca and price less than 9300 on real estate website.  
+12. find me a studio bedroom houses in novato, ca and price between 1500 and 6700 on real estate website.  
+13. can you find me a studio bedroom, any bathroom townhomes in petaluma, ca and price more than 1000 on real estate website.  
+14. search for a 1 bedroom apartments in modesto, ca and price more than 1000 on real estate website.  
+15. find me a 1 bedroom,  $2+$  bathroom apartments in watts, ca for senior housing less than 6300 on real estate website.  
+16. can you find me a 1 bedroom houses in victorville, ca that have dog friendly, furnished and price more than 700 on real estate website.  
+17. I need a 2 bedroom, any bathroom condos in inglewood, ca and price more than 1000 on real estate website.  
+18. find me a 2 bedroom,  $2+$  bathroom apartments in livermore, ca on real estate website.  
+19. can you find me a 2 bedroom apartments in Santa clara, ca that has parking and price less than 10300 on real estate website.  
+20. can you search for a 2 bedroom condos in oakland, ca on real estate website.
+
+# social-media
+
+1. Show me the most hot thread in r/google at social media website.  
+2. Can you point out the most hot thread in r/learnpython at social media website.  
+3. Could you check the 1st hot thread in r/artificial at social media website.  
+4. Can I check the most hot thread in Taiwan on social media website.  
+5. Show me the first new thread in r/facebook at social media website.  
+6. Present the most new thread of r/Python filtered by Tutorial flair on social media website.  
+7. Could you check the 1st new thread in r/facebook at social media website.  
+8. I want to read the 1st hot thread from r/Python tagged as Daily Thread at social media website.  
+9. Present the most hot thread of r/google filtered by Info | Mod Post flair on social media website.  
+10. Show me the most new thread in r/learnmachinelearning filtered by Help flair at social media website.  
+11. Can you point out the first hot thread in r/deeplearning at social media website.  
+12. Could you check the 1st hot thread in r/machinelearningnews at social media website.  
+13. Present the most hot thread of r/artificial filtered by News flair on social media website.  
+14. Please find me the first hot thread in r/facebook at social media website.  
+15. Present the most new thread of r/machinelearningnews filtered by Startup News flair on social media website.  
+16. Show me the most hot thread in r/artificial filtered by AI Art flair at social media website.  
+17. Could you check the first new thread in r/facebook at social media website.  
+18. I want to read the most top thread from r/google tagged as Info | Mod Post at social media website.  
+19. Show me the most new thread in r/startups filtered by Share Your Startup flair at social media website.  
+20. Could you check the 2nd new thread in r/facebook at social media website.
+
+# map
+
+1. Show me the way from San Jose to Mountain View by 2nd Cycling at map website.  
+2. Please show me the way from The Painted Ladies to San Francisco Zoo with 3rd Best option at map website.  
+3. Could you tell me the path from California Academy of Sciences to de Young Museum by 1st Transit at map website.  
+4. Could you tell me the way from Union Square to The Painted Ladies with 2nd Cycling option at map website.  
+5. Please present the way from Chappell Hayes Observation Tower to San Jose with 2nd Walking option at map website.  
+6. Please present the path from Jack London Square to Emeryville by 2nd Cycling at map website.  
+7. I'd like to move The Midway from Children's Fairyland by 1st Cycling at map website.  
+8. I'd like to move Chase Center from San Francisco - Oakland Bay Bridge with 2nd Transit option at map website.  
+9. I want to move Pier 39 from Berkeley by 3rd Cycling at map website.  
+10. I want to go to Emeryville from Mountain View with 2nd Cycling option at map website.  
+11. Can you point out the way from San Mateo to Stanford University by 2nd Cycling at map website.  
+12. Could you point out the way from Palace of Fine Arts to UC Berkeley by 1st Cycling at map website.  
+13. Point out the way from The Painted Ladies to San Francisco Museum of Modern Art by 2nd Driving at map website.  
+14. Could you find the path from Union Square to Palo Alto by 1st Cycling at map website.  
+15. Please check the way from San Jose to San Jose Mineta International Airport with 1st Walking at map website.  
+16. Check the path from San Francisco Zoo to Berkeley with 1st Cycling at map website.  
+17. I'd like to check Parking Lots along the way from Stanford University to The Painted Ladies with Best option at map website.  
+18. Check Gas stations along the way from de Young Museum to Oakland with Driving option at map website.  
+19. Please show me Hotels along the way from Palace of Fine Arts to Berkeley by Transit at map website.  
+20. Check Gas stations along the way from Bay Area Discovery Museum to Santa Cruz with Best option at map website.
+
+# H EXAMPLE EPISODE IN REAL-WORLD WEB AUTOMATION
+
+![](images/81e6d9229d40e0f549a16d94f91441de1e3686b68edeef85c4e8ff8bdec041bf.jpg)  
+map: Show me the way from San Jose to Mountain View by 2nd Cycling at map website?  
+Figure 9: Example episodes of real-world web automation in map domain.
+
+# I EXTENSIVE ABLATION OF HTML-T5
+
+# I.1 DATASET AND INITIALIZATION
+
+To test our recipe described in Section 2.1, we compare the different dataset and model initialization for pre-training on downstream task performances; offline task planning on real-estate and average success rate on MiniWoB with 12K dataset. We use Base-size models for the experiments. For HTML-denoising, we prepare the corpus from CommonCrawl with (Extracted) or without (Raw) subtree extraction around label elements on the documents. We also compare the initialization of base architectures before HTML-denoising; from scratch or with pre-trained models on PEGASUS objective (Zhang et al., 2020) that is a masked important sentence prediction from long-context paragraph. Table 7 reveals that snippet extraction on HTML corpus improves downstream performances since such a pre-processing can reduce the noise in raw HTML. Moreover, initialization with PEGASUS pre-trained weights is essential for HTML-T5, because of the long-context and instruction-following nature of HTML-based tasks.
+
+<table><tr><td>CC-HTML</td><td>PEGASUS</td><td>real-estate</td><td>MiniWoB++</td></tr><tr><td>Raw</td><td>✓</td><td>80.56</td><td>56.7%</td></tr><tr><td>Extracted</td><td>×</td><td>67.11</td><td>49.1%</td></tr><tr><td>Extracted</td><td>✓</td><td>82.46</td><td>57.0%</td></tr></table>
+
+# I.2 OFFLINE EVALUATION ON TASK PLANNING WITH MODEL SCALING
+
+We compare the offline task planning performance between HTML-T5 and LongT5 (without HTML-denosing) with different model sizes; with Base (220M parameters), Large (770M parameters), and XL (3B parameters). As described in Section 3.1, the models predict the next sub-instructions in a closed-loop manner considering the current HTML observations, user instructions, and previous sub-instruction histories as inputs. For offline task planning evaluation, we use the demonstrations on real-estate website; preparing 130 demonstrations and splitting them into train  $(90\%)$  and test splits  $(10\%)$ . We report the best per-step exact match accuracy in test set.
+
+Table 8 shows that HTML-T5 outperforms LongT5 on the accuracy of sub-instruction prediction, which demonstrates that HTML-denoising pre-training captures the structural bias of HTML better without sacrificing the ability to understand natural language instructions. This also implies that our proposed HTML-denoising can scale to larger-size models consistently.
+
+Table 7: Ablations of HTML-T5-Base on dataset quality and initialization. We evaluate offline task planning on real-estate and average success rate on MiniWoB with 12K dataset. For HTML-denoising, we prepare HTML corpus from CommonCrawl with (Extracted) or without (Raw) subtree extraction around label elements. We also compare the pre-training of base architectures with PEGASUS objective (Zhang et al., 2020) before HTML-denoising. The results imply that PEGASUS pre-training is critical for the architectures and preprocessing with subtree extraction improves the downstream performance on HTML-based tasks.  
+
+<table><tr><td>Models</td><td>real-estate</td><td>Diff.</td></tr><tr><td>LongT5-Base</td><td>78.07</td><td>0.0</td></tr><tr><td>LongT5-Large</td><td>82.89</td><td>0.0</td></tr><tr><td>LongT5-XL</td><td>81.29</td><td>0.0</td></tr><tr><td>HTML-T5-Base</td><td>82.46</td><td>+4.39</td></tr><tr><td>HTML-T5-Large</td><td>83.63</td><td>+0.74</td></tr><tr><td>HTML-T5-XL</td><td>83.92</td><td>+2.63</td></tr></table>
+
+Table 8: Accuracy of offline evaluation on task planning. We leverage the demonstrations in real-estate websites. Compared to original LongT5, and as we scale model size, HTML-T5 improves the accuracy of sub-instruction prediction.
+
+# I.3 DESCRIPTION GENERATION
+
+We also investigate the capability of HTML-T5 on static HTML comprehension tasks, as well as interactive decision making tasks. We use Description Generation benchmark (Gur et al., 2022), where the models generate the textual description of elements, typically used for accessibility purposes and annotated with a special attribute in the HTML schema known as for. We evaluate the understanding the structure of HTML as it would appear to a user, despite not having access to the rendered website directly.
+
+We compare LaMDA (Thoppilan et al., 2022), T5, LongT5, and HTML-T5 with respect to accuracy, BLEU (Papineni et al., 2002), and ROUGE-1 (Lin, 2004) score. As shown in Table 9, local and global attention mechanisms, underlying between LongT5 and HTML-T5, could almost solve the benchmark by improving the previous best performance by over  $10\%$ , with still improved performance as model size increases. Compared to the effect of local-global attention, HTML-T5 marginally improves against LongT5, which emphasizes that local and global attentions are critical to capture the hierarchical structure of HTML documents.
+
+<table><tr><td rowspan="2">Models</td><td colspan="3">Dev</td><td colspan="3">Test</td></tr><tr><td>Accuracy</td><td>BLEU</td><td>ROUGE-1</td><td>Accuracy</td><td>BLEU</td><td>ROUGE-1</td></tr><tr><td>LaMDA-1B (Gur et al., 2022)</td><td>83.3</td><td>87.5</td><td>90.2</td><td>84.3</td><td>88.6</td><td>91.2</td></tr><tr><td>T5-Large (Gur et al., 2022)</td><td>83.2</td><td>90.2</td><td>90.5</td><td>84.3</td><td>91.7</td><td>91.5</td></tr><tr><td>T5-XL (Gur et al., 2022)</td><td>84.0</td><td>90.8</td><td>90.9</td><td>85.2</td><td>92.1</td><td>91.9</td></tr><tr><td>LongT5-Base</td><td>96.4</td><td>98.0</td><td>98.5</td><td>95.6</td><td>97.4</td><td>98.2</td></tr><tr><td>LongT5-Large</td><td>98.1</td><td>98.9</td><td>99.2</td><td>97.7</td><td>98.5</td><td>99.0</td></tr><tr><td>LongT5-XL</td><td>98.4</td><td>99.1</td><td>99.3</td><td>98.5</td><td>99.2</td><td>99.3</td></tr><tr><td>HTML-T5-Base</td><td>96.5</td><td>98.1</td><td>98.6</td><td>95.9</td><td>97.5</td><td>98.3</td></tr><tr><td>HTML-T5-Large</td><td>98.1</td><td>98.9</td><td>99.2</td><td>97.7</td><td>98.3</td><td>99.1</td></tr><tr><td>HTML-T5-XL</td><td>98.4</td><td>99.0</td><td>99.3</td><td>98.9</td><td>99.4</td><td>99.5</td></tr></table>
+
+Table 9: Results of Description Generation benchmark (Gur et al., 2022). We compare LaMDA (Thoppilan et al., 2022), T5, LongT5, and HTML-T5 with respect to accuracy, BLEU, and ROUGE-1 scores. The results demonstrate that local and global attention mechanisms, shared modules between LongT5 and HTML-T5, could almost completely solve the benchmark by improving the previous best performance by over  $10\%$ . HTML-T5 slightly outperforms LongT5.
+
+# J FLAN-LONGT5
+
+In the web automation literature (Furuta et al., 2023; Kim et al., 2023), instruction-finetuned LLMs have great success in HTML comprehension and improve the task success. For the comparison to HTML-denosing, we prepare the instruction-finetuned LongT5 (i.e. Flan-LongT5) by leveraging Flan dataset released by Chung et al. (2022). We finetuned the pre-trained LongT5 with 100K iterations and picked up the best checkpoints.
+
+As a sanity check of instruction-tuning, we evaluate Flan-LongT5 with few-shot/zero-shot settings on CoT benchmark (GSM8K (Cobbe et al., 2021), StrategyQA (Geva et al., 2021), SVAMP (Patel et al., 2021), Asdiv (Miao et al., 2021), CommonsenseQA (Talmor et al., 2019)), BigBench-Hard (BBH) (Suzgun et al., 2022), and MMLU (Hendrycks et al., 2021b) as tested in Longpre et al. (2023). We reevaluate the performance of Flan-T5, using official checkpoints  $^{5}$ . We also check the performance of Flan-LongT5 on downstream summarization tasks, originally evaluated on LongT5 (Guo et al., 2022). We use arXiv (Cohan et al., 2018), PubMed (Cohan et al., 2018), BigPatent (Sharma et al., 2019), Multi-News (Fabbri et al., 2019), MediaSum (Zhu et al., 2021), CNN / Daily Mail (Nallapati et al., 2016) dataset for the evaluation, measuring the performance with ROUGE-1/2/L metrics.
+
+Table 10 shows that we have successfully replicated the LongT5 version of instruction-finetuned language models. Flan-LongT5 achieves competitive results to original Flan-T5; for instance, Flan-LongT5-Large (36.64) outperforms Flan-T5-Large (35.25), but Flan-LongT5-XL (39.05) is still behind Flan-T5-XL (43.03) on average. This might be caused by the training instability of XL-size models (Guo et al., 2022). Because, unlike HTML-T5 on HTML-based tasks, reasoning tasks do not have long-context or hierarchical syntax, it is not surprising for Flan-LongT5 not to outperform Flan-T5. Table 11 also demonstrates that we have successfully conducted instruction-tuning without losing the capability of long text summarization.
+
+<table><tr><td rowspan="2">Models</td><td colspan="2">CoT</td><td colspan="2">MMLU</td><td colspan="2">BBH</td><td colspan="2">BBH-CoT</td><td colspan="2">Avg.</td></tr><tr><td>Zero</td><td>Few</td><td>Zero</td><td>Few</td><td>Zero</td><td>Few</td><td>Zero</td><td>Few</td><td>CoT</td><td>Direct</td></tr><tr><td>Flan-T5-Large</td><td>35.14</td><td>40.03</td><td>40.68</td><td>45.12</td><td>25.90</td><td>37.48</td><td>26.17</td><td>31.45</td><td>33.20</td><td>37.29</td></tr><tr><td>Flan-T5-XL</td><td>51.74</td><td>52.64</td><td>50.76</td><td>52.40</td><td>26.09</td><td>40.96</td><td>34.12</td><td>35.62</td><td>43.53</td><td>42.55</td></tr><tr><td>Flan-LongT5-Large</td><td>44.78</td><td>45.34</td><td>38.44</td><td>40.03</td><td>28.67</td><td>34.67</td><td>29.38</td><td>31.85</td><td>37.84</td><td>35.45</td></tr><tr><td>Flan-LongT5-XL</td><td>48.78</td><td>50.02</td><td>43.44</td><td>44.74</td><td>26.53</td><td>37.77</td><td>29.09</td><td>32.01</td><td>39.97</td><td>38.12</td></tr></table>
+
+Table 10: Performance of Flan-LongT5 on reasoning tasks. We reevaluate the performance of Flan-T5 (Chung et al., 2022), using official checkpoints. Flan-LongT5 achieves competitive results to original Flan-T5.
+
+<table><tr><td rowspan="2">Models</td><td colspan="3">arXiv</td><td colspan="3">PubMed</td><td colspan="3">BigPatent</td><td colspan="3">MultiNews</td><td colspan="3">MediaSum</td><td colspan="3">CNN / Daily Mail</td></tr><tr><td>R-1</td><td>R-2</td><td>R-L</td><td>R-1</td><td>R-2</td><td>R-L</td><td>R-1</td><td>R-2</td><td>R-L</td><td>R-1</td><td>R-2</td><td>R-L</td><td>R-1</td><td>R-2</td><td>R-L</td><td>R-1</td><td>R-2</td><td>R-L</td></tr><tr><td>LongT5-Large</td><td>48.28</td><td>21.63</td><td>44.11</td><td>49.98</td><td>24.69</td><td>46.46</td><td>70.38</td><td>56.81</td><td>62.73</td><td>47.18</td><td>18.44</td><td>24.18</td><td>35.54</td><td>19.04</td><td>32.20</td><td>42.49</td><td>20.51</td><td>40.18</td></tr><tr><td>LongT5-XL</td><td>48.35</td><td>21.92</td><td>44.27</td><td>50.23</td><td>24.76</td><td>46.67</td><td>76.87</td><td>66.06</td><td>70.76</td><td>48.17</td><td>19.43</td><td>24.94</td><td>36.15</td><td>19.66</td><td>32.80</td><td>43.94</td><td>21.40</td><td>41.28</td></tr><tr><td>Flan-LongT5-Large</td><td>48.52</td><td>22.00</td><td>44.46</td><td>50.46</td><td>25.08</td><td>46.96</td><td>70.53</td><td>57.13</td><td>63.02</td><td>47.76</td><td>18.99</td><td>24.52</td><td>35.71</td><td>19.18</td><td>32.33</td><td>43.13</td><td>20.89</td><td>37.28</td></tr><tr><td>Flan-LongT5-XL</td><td>48.37</td><td>21.75</td><td>44.22</td><td>50.23</td><td>24.75</td><td>46.73</td><td>76.31</td><td>65.17</td><td>70.01</td><td>48.19</td><td>19.47</td><td>24.80</td><td>36.16</td><td>19.75</td><td>32.81</td><td>43.46</td><td>21.00</td><td>37.34</td></tr></table>
+
+Table 11: Performance of Flan-LongT5 on downstream summarization tasks, compared to LongT5 (Guo et al., 2022). We measure the performance with ROUGE-1/2/L metrics.
+
+K PER-TASK PERFORMANCE ON MINIWOB++  
+
+<table><tr><td>Task</td><td>HTML-T5-XL (347K)</td><td>HTML-T5-XL (12K)</td><td>Plan-T5-XL (347K)</td><td>WebN-T5-XL (12K)</td></tr><tr><td>book-flight</td><td>0.99</td><td>0.00</td><td>0.48</td><td>0.00</td></tr><tr><td>choose-date</td><td>0.16</td><td>0.03</td><td>0.08</td><td>0.00</td></tr><tr><td>choose-date-easy</td><td>1.00</td><td>0.28</td><td>1.00</td><td>0.03</td></tr><tr><td>choose-date-medium</td><td>0.56</td><td>0.14</td><td>0.57</td><td>0.00</td></tr><tr><td>choose-list</td><td>0.22</td><td>0.19</td><td>0.16</td><td>0.26</td></tr><tr><td>click-button</td><td>1.00</td><td>0.92</td><td>0.98</td><td>1.00</td></tr><tr><td>click-button-sequence</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>click-checkboxes</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.96</td></tr><tr><td>click-checkboxes-large</td><td>0.90</td><td>0.94</td><td>0.98</td><td>0.22</td></tr><tr><td>click-checkboxes-soft</td><td>0.99</td><td>0.64</td><td>1.00</td><td>0.54</td></tr><tr><td>click-checkboxes-transfer</td><td>1.00</td><td>1.00</td><td>0.99</td><td>0.63</td></tr><tr><td>click-collapsible</td><td>1.00</td><td>0.41</td><td>1.00</td><td>0.00</td></tr><tr><td>click-collapsible-2</td><td>0.93</td><td>0.26</td><td>0.94</td><td>0.00</td></tr><tr><td>click-color</td><td>1.00</td><td>1.00</td><td>0.27</td><td>0.27</td></tr><tr><td>click-dialog</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>click-dialog-2</td><td>0.74</td><td>0.31</td><td>0.34</td><td>0.24</td></tr><tr><td>click-link</td><td>0.99</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>click-menu</td><td>0.37</td><td>0.26</td><td>0.41</td><td>0.37</td></tr><tr><td>click-option</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.87</td></tr><tr><td>click-pie</td><td>0.96</td><td>0.89</td><td>0.99</td><td>0.51</td></tr><tr><td>click-scroll-list</td><td>0.99</td><td>0.91</td><td>0.00</td><td>0.00</td></tr><tr><td>click-shades</td><td>0.00</td><td>0.05</td><td>0.00</td><td>0.00</td></tr><tr><td>click-shape</td><td>0.79</td><td>0.57</td><td>0.58</td><td>0.53</td></tr><tr><td>click-tab</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.74</td></tr><tr><td>click-tab-2</td><td>0.94</td><td>0.40</td><td>0.94</td><td>0.18</td></tr><tr><td>click-tab-2-hard</td><td>0.88</td><td>0.30</td><td>0.57</td><td>0.12</td></tr><tr><td>click-test</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>click-test-2</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>click-widget</td><td>1.00</td><td>0.94</td><td>1.00</td><td>1.00</td></tr><tr><td>count-shape</td><td>0.67</td><td>0.55</td><td>0.64</td><td>0.41</td></tr><tr><td>email-inbox</td><td>1.00</td><td>0.99</td><td>0.99</td><td>0.38</td></tr><tr><td>email-inbox-forward-nl</td><td>1.00</td><td>0.92</td><td>1.00</td><td>0.60</td></tr><tr><td>email-inbox-forward-nl-turk</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.33</td></tr><tr><td>email-inbox-nl-turk</td><td>0.99</td><td>0.76</td><td>0.92</td><td>0.23</td></tr><tr><td>enter-date</td><td>1.00</td><td>0.00</td><td>1.00</td><td>0.00</td></tr><tr><td>enter-password</td><td>1.00</td><td>0.99</td><td>1.00</td><td>0.97</td></tr><tr><td>enter-text</td><td>1.00</td><td>0.96</td><td>1.00</td><td>0.89</td></tr><tr><td>enter-text-dynamic</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.98</td></tr><tr><td>enter-time</td><td>1.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr><tr><td>focus-text</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>focus-text-2</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>grid-coordinate</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.49</td></tr><tr><td>guess-number</td><td>0.13</td><td>0.00</td><td>0.10</td><td>0.00</td></tr><tr><td>identify-shape</td><td>1.00</td><td>0.89</td><td>0.90</td><td>0.88</td></tr><tr><td>login-user</td><td>1.00</td><td>0.80</td><td>1.00</td><td>0.82</td></tr><tr><td>login-user-popup</td><td>1.00</td><td>0.63</td><td>0.97</td><td>0.72</td></tr><tr><td>multi-layouts</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.83</td></tr><tr><td>multi-orderings</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.88</td></tr><tr><td>navigate-tree</td><td>0.99</td><td>0.99</td><td>1.00</td><td>0.91</td></tr><tr><td>search-engine</td><td>0.93</td><td>0.55</td><td>0.59</td><td>0.34</td></tr><tr><td>social-media</td><td>0.99</td><td>0.93</td><td>0.99</td><td>0.21</td></tr><tr><td>social-media-all</td><td>0.31</td><td>0.84</td><td>0.09</td><td>0.00</td></tr><tr><td>social-media-some</td><td>0.89</td><td>0.60</td><td>0.39</td><td>0.02</td></tr><tr><td>tic-tac-toe</td><td>0.57</td><td>0.46</td><td>0.42</td><td>0.48</td></tr><tr><td>use-autocomplete</td><td>0.97</td><td>0.23</td><td>0.98</td><td>0.22</td></tr><tr><td>use-spinner</td><td>0.07</td><td>0.07</td><td>0.03</td><td>0.07</td></tr><tr><td>Average</td><td>0.856</td><td>0.655</td><td>0.755</td><td>0.484</td></tr></table>
+
+Table 12: Per-task average success rate on 56 tasks from MiniWoB++. We refer to Furuta et al. (2023) and Gur et al. (2022) for the baseline performances.
\ No newline at end of file
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+                                    "content": " Austin Huang"
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+                            ]
+                        }
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+                        }
+                    ],
+                    "index": 4
+                },
+                {
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+                                    "content": "Pre-trained large language models (LLMs) have recently achieved better generalization and sample efficiency in autonomous web automation. However, the performance on real-world websites has still suffered from (1) open domaininess, (2) limited context length, and (3) lack of inductive bias on HTML. We introduce WebAgent, an LLM-driven agent that learns from self-experience to complete tasks on real websites following natural language instructions. WebAgent plans ahead by decomposing instructions into sub-instructions, summarizes long HTML documents into task-relevant snippets, and acts on websites via Python programs generated from those. We design WebAgent with Flan-U-PaLM, for grounded code generation, and HTML-T5, a new pre-trained LLM for long HTML documents using local and global attention mechanisms and a mixture of long-span denoising objectives, for planning and summarization. We empirically demonstrate that our modular recipe improves the success on real websites by over "
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+                                    "content": ", and that HTML-T5 is the best model to solve various HTML understanding tasks; achieving "
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+                                    "type": "text",
+                                    "content": " higher success rate than the prior method on MiniWoB web automation benchmark, and SoTA performance on Mind2Web, an offline task planning evaluation."
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+                                    "content": "1 INTRODUCTION"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Large language models (LLM) (Brown et al., 2020; Chowdhery et al., 2022; OpenAI, 2023) can solve a variety of natural language tasks, such as arithmetic, commonsense, logical reasoning, question answering, text generation (Brown et al., 2020; Kojima et al., 2022; Wei et al., 2022), and even interactive decision making tasks (Ahn et al., 2022; Yao et al., 2022b). Recently, LLMs have also demonstrated success in autonomous web navigation by controlling computers or browsers to follow natural language instructions through multi-step reasoning and decision making (Furuta et al., 2023; Gur et al., 2022; Kim et al., 2023)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
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+                                    "type": "text",
+                                    "content": "However, web automation on real-world websites has still suffered from (1) the lack of pre-defined action space, (2) much longer HTML documents than simulated observations, and (3) the absence of domain-specific knowledge for understanding HTML documents (Figure 1). Considering the open-endedness of real-world websites and the complexity of instructions, defining appropriate action spaces in advance is challenging. In addition, although several works have argued that recent LLMs with instruction-finetuning or reinforcement learning from human feedback improve HTML understanding and web automation accuracy (Furuta et al., 2023; Kim et al., 2023), their architectures are not always suitable to process real-world HTML documents; as presented in Figure 2, HTML tokens of real websites are much longer than those of simulators, and most LLMs have shorter context lengths than the average HTML tokens in real websites. It is prohibitively costly to treat such long documents as inputs directly, or adopt prior techniques such as text-XPath alignment (Li et al., 2021b) or text-HTML token separation (Wang et al., 2022a). To prioritize broader task generalization and model-size scaling, such domain knowledge for HTML documents is ignored in recent LLMs."
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+                            ]
+                        }
+                    ],
+                    "index": 8
+                }
+            ],
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+                                    "content": "\\*Equal Contribution. "
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+                                    "content": " Work done as Student Researcher at Google."
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+                                            "content": "Figure 1: Challenges in real-world web automation. Recent language model agents (Furuta et al., 2023; Gur et al., 2022; Kim et al., 2023; Yao et al., 2022b) can navigate simulated websites (Shi et al., 2017; Yao et al., 2022a), where the agents manipulate pre-defied actions and receive simplified HTML documents that are easy to parse. In contrast, language model agents continue to face challenges in navigating real-world websites, where they must interact with dynamic environments, handle open-ended actions (actions that cannot be pre-determined), and process lengthy HTML documents containing significant amounts of task-irrelevant information."
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+                                    "content": "In this work, we introduce WebAgent, an LLM-driven autonomous agent that learns from self-experience to complete user instructions on real websites by combining canonical web actions in a program space (Figure 3). WebAgent (i) plans sub-instructions for each step by decomposing natural language instructions, (ii) summarizes long HTML documents into task-relevant snippets based on the plan, and (iii) acts via programming on real websites by grounding sub-instructions and HTML snippets into executable Python codes. We combine two LLMs to form WebAgent: newly introduced HTML-T5, a domain-expert pre-trained language model, for task planning and conditional HTML summarization and Flan-U-PaLM (Chowdhery et al., 2022; Chung et al., 2022) for grounded code generation. HTML-T5 has an encoder-decoder architecture and is specialized to capture the structure of long HTML documents better by adopting local and global attention mechanisms (Guo et al., 2022). It is"
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+                                            "content": "Figure 2: Statistics of HTML tokens among real websites. Compared to simulator (about 0.5K tokens on average), HTML tokens of real websites are much longer (from 7K to 14K), which takes up the context length of large language models. As preprocessing, we remove the irrelevant tags (e.g. <script>, <meta>) and keep necessary attributes (e.g. id, type, value)."
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+                                    "content": "Existing LLM-driven agents often solve decision making tasks with a single LLM conditioned on different prompts per role (Kim et al., 2023; Sun et al., 2023; Zheng et al., 2023), which is, however, not enough for real-world tasks whose complexity is higher than that of simulators. The empirical evaluations reveal that our method incorporating self-bootstrapped specialist language models improves HTML understanding and grounding, and achieves better generalization than single LLM agent. In real-world web automation, WebAgent significantly increases the success rate by "
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+                                    "content": ", and error analysis emphasizes that coupling task planning with HTML summarization in specialized language models is essential for task success. Moreover, HTML-T5 not only works as a core module for WebAgent but also achieves strong results by itself on the web-based tasks. On MiniWoB++ (Liu et al., 2018; Shi et al., 2017), HTML-T5 achieves "
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+                                    "content": " higher success than previous language model agent (Gur et al., 2022) while also outperforming competitive baselines, such as naive local-global attention models (Guo et al., 2022) and its instruction-finetuned ones (Chung et al., 2022). On the Mind2Web (Deng et al., 2023), an offline task planning dataset, HTML-T5 achieves SoTA performance among Synapse (Zheng et al., 2023) with GPT-3.5, and MindAct with FLan-T5-XL and GPT-4 (OpenAI, 2023). In summary, our key contributions are:"
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+                                    "content": "2 RELATED WORKS"
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+                                    "type": "text",
+                                    "content": "Web Automation Web automation is a sequential decision making task where agents manipulate browsers following given instructions (Shi et al., 2017), such as form filling (Diaz et al., 2013) or information retrieval (Adolphs et al., 2022) through the sequence of computer actions (Li et al., 2020; Mazumder & Riva, 2020; Shvo et al., 2021). Prior works have realized the web automation via reinforcement learning (Gur et al., 2019; Humphreys et al., 2022; Jia et al., 2019; Shaw et al., 2023), finetuned (Furuta et al., 2023; Gur et al., 2022) or prompted LLMs (Kim et al., 2023; Sun et al., 2023; Yao et al., 2022b; Zheng et al., 2023) on the simulated websites (Shi et al., 2017; Toyama et al., 2021; Yao et al., 2022a). However, there are still huge gaps"
+                                }
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+                {
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+                                    "type": "text",
+                                    "content": "between simplified simulators and real web environments; for instance, the average tokens for HTML pages are about 15 times larger (Figure 2), and pre-defined action space for specific websites is a strong assumption that may harm the generalization to out-of-distribution web pages or instructions."
+                                }
+                            ]
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+                                            "type": "text",
+                                            "content": "Figure 3: WebAgent is a combination of LLMs: HTML-T5 for planning and summarization, and Flan-U-PaLM for grounded program synthesis. It is better suited for the real-world tasks; open domain action space, complex natural language instructions, and long HTML documents. See Appendix D for examples."
+                                        }
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+                            ],
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+                                    "content": "MindAct (Deng et al., 2023) could be the most relevant work, where finetuned language model summarizes the raw HTML document into task-relevant snippets, and another model predicts the web actions in a multi-choice QA format. While MindAct also combines several language models, it has just adopted DeBERTa (He et al., 2021) and Flan-T5 (Chung et al., 2022) for summarization and actor modules, and evaluated it on the offline dataset. In contrast, we design HTML-T5, specialized for web-based tasks, to handle long HTML documents. WebAgent leverages HTML-T5 finetuned with self-experience for summarization and planning, and Flan-U-PaLM as a capable programmer, which enables it to generate open-ended web actions and to act on online real-world websites."
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+                                    "content": "Program Synthesis In addition to common LLMs (Brown et al., 2020; Chowdhery et al., 2022; Touvron et al., 2023), several works have proposed programming-focused language models (Chen et al., 2021a; Feng et al., 2020; Li et al., 2022; Wang et al., 2021) and their benchmarks (Austin et al., 2021; Hendrycks et al., 2021a; Lu et al., 2021). Another line of work has investigated the tool augmentation of LLMs (Parisi et al., 2022) by decoding API calls (Schick et al., 2023) or Python snippets to be parsed with the interpreter (Gao et al., 2023). Most works deal with the program synthesis on the static dataset, except for the attempts in robotics (Liang et al., 2023) and game (Trivedi et al., 2022; Wang et al., 2023a), where LLMs output Python or JavaScript snippets to command the agents. Similarly, we leverage the ability of code generation as an open-ended action space for web-based agents to manipulate the real website, and demonstrate LLMs can sequentially decode Python selenium codes considering the given sub-instructions and HTML in the prompts."
+                                }
+                            ]
+                        }
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+                                    "type": "text",
+                                    "content": "See extended related works on document understanding and LLM for task planning in Appendix B."
+                                }
+                            ]
+                        }
+                    ],
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+                                    "type": "text",
+                                    "content": "3 WEBAGENT"
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+                                    "content": "WebAgent is a new architecture that combines two LLMs to achieve efficient real-world web automation. HTML-T5, a domain-expert LLM, is responsible for predicting the next sub-instruction (planning) and generating related HTML snippets (summarization). Flan-U-PaLM (540B) (Chowdhery et al., 2022; Chung et al., 2022), is prompted to generate executable Python programs based on the planning and summarization provided by HTML-T5 (Figure 3). This modular two-stage approach enables WebAgent to effectively navigate and process HTML documents."
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+                                    "content": "Workflow Users initiate natural language interactions with a clear intent, such as apartment searching. Upon receiving the initial request, HTML-T5 formulates a \"go to <URL>\" sub-instruction, triggering Flan-U-PaLM to generate a corresponding Python program that navigates to the specified website. The raw HTML content of the newly opened page is extracted and fed into HTML-T5 along with the"
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+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                            "content": "Figure 4: HTML-T5 consists of (1) local and global attention mechanisms (Ainslie et al., 2020; Guo et al., 2022) and (2) a mixture of denoising objectives (Tay et al., 2022) with longer-span corruption on large-scale HTML corpus. The local and global attention mechanisms are suitable for the hierarchical tree structures of HTML documents. Because of the sparsity of content tokens in HTML, short mean span length (e.g. "
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+                                            "type": "text",
+                                            "content": "), often used in prior works (Raffel et al., 2020), only masks less meaningful chunks. Employing longer span length (e.g. "
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+                                            "type": "text",
+                                            "content": ") helps pre-trained language models to capture the syntax and semantics of HTML better."
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+                                    "type": "text",
+                                    "content": "user's instruction and previous planning steps. This information is utilized to predict the next subinstruction and identify relevant reference IDs for extractive HTML summarization. Flan-U-PaLM, in turn, generates a Python program based on these sub-instructions and combined HTML snippets. This iterative process of planning, summarization, and program synthesis continues until a designated end-of-episode sub-instruction is predicted or the maximum number of iterations is reached."
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+                                    "content": "3.1 HTML-T5"
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+                                    "content": "Prior research has shown that general-purpose LLMs, such as T5 (Raffel et al., 2020), Flan-T5 (Chung et al., 2022), and InstructGPT (Ouyang et al., 2022), can effectively navigate web environments (Furuta et al., 2023; Gur et al., 2022; Kim et al., 2023). However, unlike specialist transformer models (Li et al., 2021b; Wang et al., 2022a; Zhao et al., 2022), these general-purpose LLMs do not fully utilize the HTML-specific information that could otherwise lead to better understanding of HTML content. To address this limitation, we introduce HTML-T5, a pre-trained encoder-decoder language model specifically designed for HTML-based web automation tasks. HTML-T5 carefully merges the generalist and specialist characteristics of language models. It processes HTML in a text-to-text manner and employs local and global attention mechanisms (Ainslie et al., 2020) in the encoder to capture the hierarchical structure of long HTML inputs. HTML-T5 is pre-trained on a large-scale HTML corpus curated from CommonCrawl using a mixture of long-span denoising objectives (Tay et al., 2022), and then finetuned it for each downstream task. For WebAgent, we employ the self-experience supervision approach to align the model with real websites."
+                                }
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+                                    "type": "text",
+                                    "content": "Model Architecture Unlike natural language, HTML documents possess an explicit hierarchical structure. This structure comprises elements such as <input>, <label>, and <button>, along with their associated attributes like class, label, and id. These elements are defined locally and combined hierarchically to create HTML documents. To model this inherent hierarchy, we replace the common dense attention (Vaswani et al., 2017) with local and global attention mechanisms (Ainslie et al., 2020). Local attention restricts each token to only attend to neighboring tokens within a window. Additionally, transient global attention allows each token to attend to tokens beyond its immediate window. This is achieved through the aggregation and normalization of token representations within each window, resulting in a global memory representation. Figure 4 describes the concepts of HTML-T5; leaf elements in HTML (green) could be processed by local attention, and internal elements (purple) could be compressed into transient global attention, which naturally fits the hierarchical structure of HTML. Following LongT5 (Guo et al., 2022), we use dense attention in the decoder."
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+                                    "content": "Pre-Training with Mixture of Long-Span Denoising Our pre-training approach for HTML-T5 utilizes a span denoising objective. This involves randomly masking spans of tokens within an HTML document, with span lengths drawn from a Gaussian distribution with a mean of "
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+                                    "content": ". The objective is then to predict the masked spans using the remaining tokens in the HTML document (Raffel"
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+                                            "html": "<table><tr><td rowspan=\"2\"></td><td colspan=\"2\">Modules</td><td colspan=\"2\">real-estate</td><td colspan=\"2\">social-media</td><td colspan=\"2\">map</td><td colspan=\"3\">Error Ratio (%)</td></tr><tr><td>Plan</td><td>Sum</td><td>Success</td><td>Score</td><td>Success</td><td>Score</td><td>Success</td><td>Score</td><td>Program</td><td>Plan</td><td>Sum</td></tr><tr><td>Flan-U-PaLM</td><td>✘</td><td>✘</td><td>10.0</td><td>55.3</td><td>20.0</td><td>25.0</td><td>10.0</td><td>51.3</td><td>36/88/11</td><td>38/0/78</td><td>26/12/11</td></tr><tr><td>Flan-U-PaLM+P</td><td>✓</td><td>✘</td><td>50.0</td><td>79.5</td><td>20.0</td><td>38.3</td><td>30.0</td><td>73.8</td><td>39/65/14</td><td>56/30/29</td><td>5/5/57</td></tr><tr><td>Flan-U-PaLM+S</td><td>✘</td><td>✓</td><td>0.0</td><td>45.7</td><td>25.0</td><td>62.1</td><td>15.0</td><td>46.3</td><td>30/67/0</td><td>40/13/100</td><td>30/20/0</td></tr><tr><td>WebAgent</td><td>✓</td><td>✓</td><td>65.0</td><td>87.6</td><td>70.0</td><td>85.8</td><td>80.0</td><td>93.8</td><td>20/33/25</td><td>70/50/50</td><td>10/17/25</td></tr></table>",
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+                                            "type": "text",
+                                            "content": "Table 1: Success rate of real-world web automation on real estate, social media and map websites. The score stands for the percentage of covered attributes specified in given instructions. WebAgent, with language model modules for planning and summarization, achieves the best success (65%, 70%, 80%, respectively), surpassing other baselines, such as a single Flan-U-PaLM, that with a planning language model (Flan-U-PaLM+P), and that with a summarization language model (Flan-U-PaLM+S). Without language model modules, prompted Flan-U-PaLM plans in an open-loop manner (Plan: "
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+                                            "type": "text",
+                                            "content": ") and regular-expression-based retrieval summarizes HTML inputs ("
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+                                            "content": "). The results imply that self-experience supervision notably improves the performance, and task planning should be learned by finetuning domain language models for closed-loop planning, rather than by prompting single LLM for open-loop planning. The error analysis describes the ratio across three types of errors in (real-estate)/(social-media)/(map) domains, which also points out that better adaptive planner to decompose the given instructions would contribute to further improvements of WebAgent."
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+                                    "type": "text",
+                                    "content": "et al., 2020; Tay et al., 2022; Ainslie et al., 2023). While a span length of "
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " (Figure 4), where the signal-to-noise ratio can be significantly lower than natural language text. In contrast, longer spans can contain more semantically meaningful chunks, such as <form class=\"or type=\"submit\">. Our empirical findings indicate that setting "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\mu \\in \\{8,64\\}"
+                                },
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " yields the optimal mixture for HTML documents (Section 4.2)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
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+                                    "type": "text",
+                                    "content": "We adopt 4096 input sequence length and 910 output sequence length during pre-training. In total, "
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+                                    "type": "inline_equation",
+                                    "content": "15\\%"
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+                                    "type": "text",
+                                    "content": " of input tokens are randomly masked in the denoising objective. For the pre-training dataset, we collect 100 WARC files (April 2019) from the CommonCrawl corpus and remove the non-Unicode or alphanumeric-only HTML documents. We then extract subtrees around "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "\\langle \\text{label} \\rangle"
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+                                    "type": "text",
+                                    "content": " elements that have a special attribute called for that associates the corresponding label with a unique input element in the same HTML document. This pre-processing step improves the quality of the pre-training corpus by focusing only on HTML that is relevant for instruction following and grounding. Our final dataset has 3.41M examples. We pre-train HTML-T5 for 100K iterations following the practice in other T5 models (Chung et al., 2022; Lester et al., 2021). See Appendix C for further details."
+                                }
+                            ]
+                        }
+                    ],
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+                                    "type": "text",
+                                    "content": "3.2 SELF-EXPERIENCE SUPERVISION FOR ALIGNMENT WITH REAL WEBSITES"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
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+                                    "type": "text",
+                                    "content": "Gathering example demonstrations for LLMs to understand websites poses a significant obstacle in real-world web automation. While humans can effortlessly execute instruction following on actual websites, manually annotating every planning, summarization, and program synthesis step as detailed above is impractical. To address this issue, we propose self-experience supervision, a semi-supervised approach that necessitates minimal human involvement. In this method, manually curated scripts generate planning and summarization steps, while Flan-U-PaLM is tasked with generating Python programs. Our WebAgent aligns domain-specific language models, such as HTML-T5, with these self-gathered real-world experiences through fine-tuning (Wang et al., 2022b). This enables the generalization and alignment of agents to complex real-world tasks."
+                                }
+                            ]
+                        }
+                    ],
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+                                    "type": "text",
+                                    "content": "Instruction Templates We maintain a collection of instruction templates that incorporate placeholders such as \"Show me the way from <start> to <goal> by <n-th> <transportation> at map website\". We sample instructions by randomly assigning values to placeholders from pre-defined key-value pairs."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Scripted Planning and Prompted Programming We employ a rule-based parser to decompose instructions into sequences of sub-instructions; corresponding reference IDs are retrieved from HTML using regular expressions. At each step of the process, Flan-U-PaLM is provided with the sub-instruction and the associated HTML snippets to generate navigation programs that are executed through Selenium WebDriver. The success of recorded demonstrations varies, and automating success criteria for real-world tasks remains challenging. To refine the learning experience, we utilize environmental feedback to eliminate critical failures, such as program execution errors, retriever errors, and clearly erroneous URL prefixes (Ni et al., 2023)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                }
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+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
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+                                    "type": "text",
+                                    "content": "map: Show me the way from San Jose to Mountain View by 2nd Cycling at map website?"
+                                }
+                            ]
+                        }
+                    ],
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+                                            "type": "text",
+                                            "content": "Figure 5: Example episodes of real-world web automation in map domain. Considering the given instruction and HTML, WebAgent predicts the next sub-instruction and task-relevant snippet, and then synthesizes the Python script (gray), while treating the sub-instruction as a comment in the script. See Appendix H for extended figure."
+                                        }
+                                    ]
+                                }
+                            ],
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+                                    "type": "text",
+                                    "content": "Finetuning for Planning and Summarization HTML-T5, a core component of WebAgent, is fine-tuned using self-experience demonstrations gathered through instruction sampling, scripted planning, and prompted program synthesis, as detailed earlier. It utilizes task instructions (e.g. please search 2 bedroom and "
+                                },
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+                                    "type": "text",
+                                    "content": " bathroom houses in new york, ny with a max price of $7500 on real estate website), sub-instruction histories (e.g. go to real estate website, type in new york into search, click on search, click on price, click on max rent), and raw HTML as inputs. Subsequently, it generates the next sub-instruction (e.g. type in 7500 into max rent) and extracts the relevant data-ref attributes used for retrieving HTML snippets. Section 4.1 demonstrates the significance of integrating HTML summarization into sub-instruction prediction for enhancing real-world web automation performance."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
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+                                    "type": "text",
+                                    "content": "3.3 GROUNDED PROGRAM SYNTHESIS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
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+                                    "type": "text",
+                                    "content": "Web automation on real-world websites faces challenges due to the open-ended action spaces, unlike simplified simulators (Shi et al., 2017; Yao et al., 2022a). In contrast to previous approaches (Gur et al., 2019; Humphreys et al., 2022; Jia et al., 2019; Liu et al., 2018), real-world web agents cannot pre-define a categorical action space to specify the interactive elements on the websites. To address this open-domain challenge, we introduce the act via programming paradigm in web automation by utilizing the conditional code generation capabilities of LLMs (Chen et al., 2021a; Liang et al., 2023). Provided with few-shot generic examples (such as selecting checkboxes, entering text into inputs, clicking on options, and scrolling etc.) for program generation, the next sub-instruction, and the extracted HTML snippet from HTML-T5, Flan-U-PaLM (Chowdhery et al., 2022; Chung et al., 2022) decodes an Python program (Figure 3) executable with Selenium WebDriver, a library for browser automation. This conditional program synthesis requires LLMs to not only generate code to follow natural language instructions but also understand the semantics and functionality of HTML elements. We provide several Python snippet examples generated by Flan-U-PaLM as follows (sub-instructions are treated as comments in the script):"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
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+                                            "type": "text",
+                                            "content": "1 # Type in walnut creek, ca into search  \n2 driver.find_element(By.css_selector, '[data-ref=\"175\"').clear()  \n3 driver.find_element(By.css_selector, '[data-ref=\"175\"').send_keys(\"walnut creek, ca\")  \n4  \n5 # Submit the search  \n6 driver.find_element(By.css_selector, '[data-ref=\"175\"').submit()  \n7  \n8 # Click on the apartments  \n9 driver.find_element(By.css_selector, '[data-ref=\"572\"').click()  \n10  \n11 # Scroll down housing type by 200px  \n12 driver.executeScript('getScrollParent.document.querySelector \"#type-of-housing\").scrollBy({top: 200})')"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 7,
+                            "angle": 0,
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+                                    "content": "4 EXPERIMENTAL RESULTS"
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+                                    "content": "To study how a modular combination of LLMs under self-supervision enables real-world web automation by overcoming open-endedness and long context documents, we execute instruction-following tasks on real websites (Section 4.1). In Appendix F, we also test WebAgent on WebSRC (Chen et al., 2021b), a static HTML comprehension benchmark, compared to prior transformer models specialized for structured documents (Li et al., 2021b; Zhao et al., 2022). In addition, we quantify the performance of HTML-T5 itself on simulated web benchmark, MiniWoB++, and offline task planning benchmark, Mind2Web (Section 4.2)."
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+                                            "html": "<table><tr><td>Architectures</td><td>Attention Type</td><td>L = 2048</td><td>L = 4096</td></tr><tr><td>Flan-T5-Base</td><td>Dense</td><td>34.0%</td><td>35.3%</td></tr><tr><td>Long-T5-Base</td><td>Local</td><td>43.4%</td><td>44.0%</td></tr><tr><td>Long-T5-Base</td><td>Local &amp; Global</td><td>53.1%</td><td>53.6%</td></tr></table>",
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+                                            "type": "text",
+                                            "content": "Table 2: (Left) Architecture comparison on MiniWoB++ 12K dataset (Liu et al., 2018) with average success rate over 56 tasks. Local and global attention matches to the hierarchical tree structure of HTML, and then improves the success rate by over "
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+                                            "content": ", compared to the instruction-finetuned dense attentions (Chung et al., 2022; Furuta et al., 2023). (Right) HTML-denoising comparison with different mixtures of span length (Raffel et al., 2020; Tay et al., 2022). We use LongT5-Base models for pre-training. HTML-denoising generally improves the performance on offline task planning on real estate website and MiniWoB benchmark. Especially, using longer span lengths "
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+                                            "type": "inline_equation",
+                                            "content": "(\\mu \\in \\{8,6\\})"
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+                                            "type": "text",
+                                            "content": " outperforms other choices, including the popular configuration in natural language domain "
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+                                            "type": "inline_equation",
+                                            "content": "(\\mu \\in \\{3,8,64\\} + \\text{Prefix LM objective})"
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+                                            "type": "text",
+                                            "content": ", which can reduce the less meaningful prediction from shorter spans (e.g. "
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+                                            "content": "\\mu = 3"
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+                                            "type": "text",
+                                            "content": "), and inject the structural bias of HTML better."
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+                            "index": 3,
+                            "angle": 0,
+                            "type": "table_footnote"
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+                                    "type": "text",
+                                    "content": "4.1 REAL-WORLD WEB AUTOMATION"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
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+                                    "type": "text",
+                                    "content": "Evaluation Methodology We first evaluate WebAgent with the real-world navigation performance under human supervision, at real estate website (a platform for housing), social media website (a network of communities), and map website. These three websites have different properties. real-estate requires long-horizon planning (about 20 steps per episode) for complex formfilling with a few page transitions (at least 2 pages), and social-media needs shorter plans (about 10 steps per episode) with many page transitions (at least 4 pages) by selecting appropriate hyperlinks on the page. map is the easiest domain with shorter plans and a few page transitions. WebAgent receives natural language instructions (e.g. Can you search for a studio bedroom, "
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": " bathroom apartments in oroville, ca for corporate housing on real estate website?, or Could you present the most new thread of Python community filtered by Tutorial tag on social media website?), and acts via planning, summarizing by HTML-T5, and then programming by Flan-U-PaLM (Figure 5). Through the self-experience supervision process, we curate 260 episodes on real estate website, 230 episodes on social media website, and 410 episodes on map website to finetune HTML-T5."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
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+                {
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+                                    "type": "text",
+                                    "content": "We prepare 20 different natural language instructions (see Appendix G for the full list), and measure the success rate and score for the evaluation. The score represents the percentage of required attributes covered during the episode (Yao et al., 2022a); for instance, (1) apartments for (2) corporate housing with (3) studio bedroom and (4) "
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+                                    "type": "text",
+                                    "content": " bathroom located in (5) oroville, ca, can be specified in the instruction. When the agents could search the housing satisfying (1), (2), (5) and not (3), (4), the score is "
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+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "60 (= 100 \\times 3/5)"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ". If the agents achieve 100 score, that episode will mark as success."
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+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
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+                                    "type": "text",
+                                    "content": "Results For comparison, we prepare three baselines, consisting of language model modules and a single LLM conditioned on different prompts per role, such as Flan-U-PaLM (Chung et al., 2022), that with a planning language model (Flan-U-PaLM+P), and that with a summarization language model (Flan-U-PaLM+S). If they do not use language model modules, prompted Flan-U-PaLM plans in an open-loop manner (Plan: "
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+                                    "type": "text",
+                                    "content": "), and regular-expression-based retrieval summarizes given raw HTML (Sum: "
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+                                    "type": "text",
+                                    "content": "). Table 1 shows that by leveraging planning and summarization language model modules, WebAgent achieves best "
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+                                    "type": "text",
+                                    "content": " success and "
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+                                    "type": "text",
+                                    "content": " score on real-estate, "
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+                                    "content": " success and "
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": " score on social-media, and "
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+                                    "content": " success and "
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+                                    "type": "text",
+                                    "content": " score on map, significantly outperforming single Flan-U-PaLM, or with partial language model modules (most of those achieve about "
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+                                    "type": "inline_equation",
+                                    "content": "10 - 30\\%"
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+                                    "type": "text",
+                                    "content": " success). This result suggests that self-experience supervision notably improves the performance, and closed-loop planning grounded on HTML observations via finetuned domain language models is more suitable for open-ended web automation than open-loop planning with few-shot LLMs. This trend is remarkable in real-estate (even Flan-U-PaLM+P achieves "
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+                                    "type": "text",
+                                    "content": " success), where the longer planning horizon is needed to fulfill instructions. We also observe that coupling sub-instruction prediction with HTML summarization in language model modules plays a critical role in task success. The development of more capable planning modules to decompose the given instructions adaptively and accurately could help WebAgent improve the performance further."
+                                }
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+                                    "type": "text",
+                                    "content": "Error Analysis We also analyze the reason of failures by categorizing them into programming, planning, and summarization errors (Table 1). Programming error does not satisfy the given sub-instructions or HTML snippet. Planning error predicts sub-instructions conflicting with user instruc"
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+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                            "html": "<table><tr><td rowspan=\"2\"></td><td rowspan=\"2\">Train</td><td colspan=\"4\">Cross-Task</td><td colspan=\"4\">Cross-Website</td><td colspan=\"4\">Cross-Domain</td></tr><tr><td>Ele. Acc</td><td>Op. F1</td><td>Step SR</td><td>SR</td><td>Ele. Acc</td><td>Op. F1</td><td>Step SR</td><td>SR</td><td>Ele. Acc</td><td>Op. F1</td><td>Step SR</td><td>SR</td></tr><tr><td>Synapse (GPT-3.5)</td><td>ICL</td><td>34.4</td><td>-</td><td>30.6</td><td>2.0</td><td>28.8</td><td>-</td><td>23.4</td><td>1.1</td><td>29.4</td><td>-</td><td>25.9</td><td>1.6</td></tr><tr><td>MindAct (Flan-T5-XL)</td><td>SL</td><td>55.1</td><td>75.7</td><td>52.0</td><td>5.2</td><td>42.0</td><td>65.2</td><td>38.9</td><td>5.1</td><td>42.1</td><td>66.5</td><td>39.6</td><td>2.9</td></tr><tr><td>MindAct (GPT-4)</td><td>ICL</td><td>41.6</td><td>60.6</td><td>36.2</td><td>2.0</td><td>35.8</td><td>51.1</td><td>30.1</td><td>2.0</td><td>37.1</td><td>46.5</td><td>26.4</td><td>2.0</td></tr><tr><td>HTML-T5-XL (ours)</td><td>SL</td><td>60.6</td><td>81.7</td><td>57.8</td><td>10.3</td><td>47.6</td><td>71.9</td><td>42.9</td><td>5.6</td><td>50.2</td><td>74.9</td><td>48.3</td><td>5.1</td></tr></table>",
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+                                    "type": "text",
+                                    "content": "Table 4: Offline action prediction performance in Mind2Web dataset. We leverage the cached candidate generation results and direct QA formulation by following Deng et al. (2023). HTML-T5 significantly outperforms MindAct with Flan-T5 or GPT-4, and Synapse (Zheng et al., 2023) with GPT-3.5, across task/website/domain generalization in terms of all the metrics (element accuracy, operation F1, and success rates)."
+                                }
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+                    ],
+                    "index": 2
+                },
+                {
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+                                    "type": "text",
+                                    "content": "tions, and summarization error fails to extract the relevant HTML snippets for given sub-instructions. From the website perspective, the failures on real-estate concentrate in planning because of its long-horizon nature. map also fails in planning when confusing starting point and destination. In contrast, social-media tends to fail in programming due to the ambiguous sub-instructions or summarization including redundant hyperlinks, which results in transiting wrong pages or clicking unexecutable elements. From the method perspective, WebAgent often fails in planning by predicting incorrect sub-instructions (for instance, in real-estate, WebAgent generates incorrect plans in "
+                                },
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+                                    "type": "text",
+                                    "content": " of failure episodes), while other baselines more fail in programming or summarization steps. This observation indicates that, through the self-experience supervision, the ratio of programming and summarization errors has decreased while the fundamental difficulty of planning, which requires consistent and accurate prediction over long horizon without error accumulation, still remains."
+                                }
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+                                    "content": "4.2 ABLATION OF HTML-T5"
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+                                    "type": "text",
+                                    "content": "In addition to the evaluation as WebAgent system, we extensively examine HTML-T5 about (i) the generalization to other websites with Mind2Web (Deng et al., 2023), (ii) the performance on MiniWoB++, a standard web automation benchmark (Liu et al., 2018; Shi et al., 2017), and (iii) its architecture and pre-training objective. We adopt 16K tokens for the context window unless otherwise mentioned. We present results on offline task planning, and description generation (Gur et al., 2022) to test HTML understanding on static dataset in Appendix I."
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+                                    "content": "Mind2Web Mind2Web (Deng et al., 2023) is an action-annotated real-world dataset with over 2K instructions collected from 137 websites. It provides action prediction tasks that measure the generalization of LLMs across the tasks, websites, and their domains (e.g. travel, shopping). Similar to real-world evaluation, the input is a set of HTML snippets, a task instruction, and an action history. The output comprises a target element to interact with, along with the operation, such as click, type, or select an option. We finetune HTML-T5-XL with the training dataset. The performance is evaluated with element accuracy,"
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+                                    "content": "operation F1, and step success rate that cares for both element and operation correctness. Table 4 reveals that HTML-T5 significantly outperforms baselines with Flan-T5-XL or GPT-4 (OpenAI, 2023) across task/website/domain generalization, which increases element accuracy by "
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+                                    "type": "text",
+                                    "content": " . This highlights that HTML-T5 can handle real-world web automation tasks better and shows generalization beyond our real-world evaluation with 3 websites."
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+                                    "content": "MiniWoB++ We here evaluate HTML-T5 on 56 simulated tasks in MiniWoB++ using 100 evaluation episodes per task. Inputs are analogous to real-world evaluation, utilizing HTML documents, while outputs are adhering to a pre-defined format by the simulator such as click(ref = X). We finetune HTML-T5 with 12K human demonstrations (Liu et al., 2018), and compare the average success rate to prior supervised-learned agents (Gur et al., 2022; Humphreys et al., 2022), LongT5, and its instruction-finetuned variants (Chung et al., 2022) "
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+                                    "type": "text",
+                                    "content": ". Table 3 shows that HTML-T5-XL significantly"
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+                                            "html": "<table><tr><td>Models</td><td>Data</td><td>Success</td><td>Diff.</td></tr><tr><td>SoTA (Zheng et al., 2023)</td><td>-</td><td>99.2%</td><td>-</td></tr><tr><td>CC-Net</td><td>2.4M</td><td>32.0%</td><td>-</td></tr><tr><td>WebN-T5-XL</td><td>12K</td><td>48.4%</td><td>-</td></tr><tr><td>LongT5-Base</td><td></td><td>53.8%</td><td>0.0</td></tr><tr><td>LongT5-Large</td><td>12K</td><td>56.3%</td><td>0.0</td></tr><tr><td>LongT5-XL</td><td></td><td>60.4%</td><td>0.0</td></tr><tr><td>Flan-LongT5-Base</td><td></td><td>54.1%</td><td>+0.3</td></tr><tr><td>Flan-LongT5-Large</td><td>12K</td><td>56.1%</td><td>-0.2</td></tr><tr><td>Flan-LongT5-XL</td><td></td><td>61.1%</td><td>+0.7</td></tr><tr><td>HTML-T5-Base (ours)</td><td></td><td>57.0%</td><td>+3.2</td></tr><tr><td>HTML-T5-Large (ours)</td><td>12K</td><td>60.8%</td><td>+4.5</td></tr><tr><td>HTML-T5-XL (ours)</td><td></td><td>67.1%</td><td>+6.7</td></tr><tr><td>Flan-T5-XL</td><td>347K</td><td>75.5%</td><td>-</td></tr><tr><td>Flan-T5-XXL</td><td></td><td>79.0%</td><td>-</td></tr><tr><td>HTML-T5-XL (ours)</td><td>347K</td><td>85.6%</td><td>-</td></tr></table>",
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+                                            "content": "Table 3: Average success rate of MiniWoB++ with 56 tasks. We use 12K demonstrations and compare HTML-T5 among supervised-finetuned methods. HTML-T5-XL outperforms WebN-T5-XL (Gurel al., 2022), the prior best method, by "
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+                                            "content": ". HTML-denoising also yields better the success rate than instruction tuned ones. Finetuned HTML-T5 with 347K episodes (Furuta et al., 2023) outperforms Flan-T5-XXL (11B parameters) even with 3B parameters, which gets closer to SoTA with GPT-3.5. See Appendix K for the detailed results."
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+                                            "content": "element and operation correctness. Table 4 rees with Flan-T5-XL or GPT-4 (OpenAI, 2023) eases element accuracy by "
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+                                            "content": " , operation F1 nts that HTML-T5 can handle real-world web and our real-world evaluation with 3 websites."
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+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "type": "text",
+                                    "content": "<sup>1</sup>We finetune LongT5 models with Flan dataset released by Chung et al. (2022). See Appendix J."
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+                                    "content": "outperforms WebN-T5, the prior best model, by "
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+                                    "content": ". Notably, we demonstrate HTML-denoising consistently improves the performance on top of LongT5 in all the model sizes, better than instruction-finetuning introduced in prior work (Furuta et al., 2023). Furthermore, we finetune HTML-T5-XL with 347K demonstrations from Furuta et al. (2023), which performs better than 11B-parameter Flant5-XXL even with 3B parameters, achieving "
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+                                    "type": "text",
+                                    "content": " success. These prove we successfully incorporate domain knowledge on HTML comprehension for web automation into pre-trained language models."
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+                                    "content": "Architecture and Objective We hypothesize that local and global attention mechanisms can capture the hierarchical structures of HTML documents better than dense attention. We compare the web automation performance among 56 MiniWoB++ tasks (Gur et al., 2022), by finetuning HTML-T5 with public 12K-episode dataset (Liu et al., 2018). We adopt 2048 and 4096 tokens as input length and prepare Base-size architectures. Table 2 (left) reveals that the combination of local and global attentions achieves the superior success rate by over "
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+                                    "type": "text",
+                                    "content": " compared to the instruction-finetuned dense attentions (Chung et al., 2022; Raffel et al., 2020) and local attention only. Surprisingly, local attention only still surpasses the dense attention by about "
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+                                    "type": "text",
+                                    "content": ", which suggests local relation between elements and attributes in HTML are essential for web tasks."
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+                                    "type": "text",
+                                    "content": "As for pre-training objective in Table 2 (right), HTML-denoising generally improves the performance on offline task planning on real estate website and MiniWoB. Especially, using only longer span lengths "
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+                                    "type": "inline_equation",
+                                    "content": "(\\mu \\in \\{8,64\\})"
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+                                    "content": " outperforms other choices, including the popular configuration in natural language domain "
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+                                    "content": "(\\mu \\in \\{3,8,64\\} +\\text{Prefix LM objective})"
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+                                    "type": "text",
+                                    "content": ", which can reduce the less meaningful prediction from shorter spans (e.g. "
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+                                    "type": "text",
+                                    "content": "), and inject the structural bias of HTML into language models better. See Appendix I.2 for further results with model scaling."
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+                                    "type": "text",
+                                    "content": "5 DISCUSSION AND LIMITATION"
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+                                    "type": "text",
+                                    "content": "Modular Approach with Specialist Language Models We demonstrate it is beneficial to divide web automation into planning, HTML summarization, and code generation, and to combine domain-expert language models aligned with self-experience data. Such modular approaches have also been adopted to support the inference of LLMs (Xu et al., 2023), multimodal tasks (Zeng et al., 2022), and robotics (Ahn et al., 2022), which, however, might cause additional computational costs and latency."
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+                                    "type": "text",
+                                    "content": "Broad Generalization across the Internet Because open-loop planning with prompted Flan-U-PaLM achieves at most "
+                                },
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+                                    "type": "text",
+                                    "content": " success, we have demonstrated that self-experience supervision on real websites is essential for planning modules. As we demonstrated in Mind2Web, our method could generalize across the internet if we have enough data. It would be expected to collect demonstrations at scale and align larger domain-expert models with them in future works."
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+                                    "content": "Feedback for Program Synthesis We leverage Flan-U-PaLM with 540B parameters, as a capable program synthesis module via few-shot prompting. Such a large model, however, makes it challenging to reflect the feedback about the errors in generated code, compared to smaller models. We leave it as future direction to incorporate the feedback for program synthesis into larger language models."
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+                                    "content": "Evaluation for Real-world Web Automation Beyond the simulated web environments (Shi et al., 2017; Yao et al., 2022a), we have exhibited WebAgent can follow given complex and sometimes ambiguous instructions on real estate, social media and map websites. On the other hand, it is costly to evaluate the performance of autonomous agents in the real world. Automated evaluation with minimal human intervention would be helpful for the scalable development of real-world web agents."
+                                }
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+                                    "content": "6 CONCLUSION"
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+                    ],
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+                },
+                {
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+                                    "type": "text",
+                                    "content": "We build a system for real-world web automation, combining HTML-T5 for planning and HTML summarization and Flan-U-PaLM for grounded program synthesis. Our proposed WebAgent achieves around "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "70 - 80\\%"
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+                                    "type": "text",
+                                    "content": " success on real websites via self-experience supervision, outperforming single LLM approach by over "
+                                },
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+                                    "type": "text",
+                                    "content": ", which suggests dividing the sequence of sub-problems with multiple language models can increase the entire task success. We also propose a scalable recipe for HTML-specialized language models where we train local and global attention mechanisms with a mixture of long-span denoising objectives to capture the hierarchical structures of HTML documents. HTML-T5 not only plays an essential role in WebAgent but also can achieve the best results on a variety of HTML-based benchmarks such as Mind2Web and MiniWoB++. We hope our work contributes to getting us one-step closer to the practical deployment of autonomous web agent systems."
+                                }
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+                                    "content": "ETHICS STATEMENT"
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+                                    "type": "text",
+                                    "content": "This paper presents encouraging evidence of autonomous agents' potential for deployment on real websites, extending beyond simulated environments. In the foreseeable future, this technology could lead to the development of sophisticated AI assistant tools for computers and smartphones, enhancing productivity and accessibility for society."
+                                }
+                            ]
+                        }
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+                                    "content": "While we recognize the promising aspects of autonomous agents, we must also consider the potential for misuse and unintended consequences in their development. As our proposed system is based on LLMs, there is a risk of prompt injection. The improper use of web automation could pose cybersecurity threats and expose users to scams. To mitigate these risks, it is crucial for researchers, policymakers, and industry stakeholders to collaborate on establishing guidelines and regulations for the development of autonomous agents. Additionally, security research focused on LLM agents will become an essential domain for society."
+                                }
+                            ]
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+                                    "type": "text",
+                                    "content": "ACKNOWLEDGMENTS"
+                                }
+                            ]
+                        }
+                    ],
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+                },
+                {
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+                                    "type": "text",
+                                    "content": "We thank Heiga Zen, Yingjie Miao, Yusuke Iwasawa, Joshua Ainslie, Santiago Ontonon, Quoc V. Le, Zoubin Ghahramani, Jeff Dean, Tris Warkentin for the supports and advises on this work. Hiroki Furuta was supported by JSPS KAKENHI Grant Number JP22J21582."
+                                }
+                            ]
+                        }
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+                                    "content": "REFERENCES"
+                                }
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+                                        }
+                                    ]
+                                }
+                            ],
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+                        {
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+                                        }
+                                    ]
+                                }
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+                                        }
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+                                        }
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+                                            "content": "Longtao Zheng, Rundong Wang, and Bo An. Synapse: Leveraging few-shot exemplars for human-level computer control. arXiv preprint arXiv:2306.07863, 2023."
+                                        }
+                                    ]
+                                }
+                            ],
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+                                            "content": "Chenguang Zhu, Yang Liu, Jie Mei, and Michael Zeng. Mediasum: A large-scale media interview dataset for dialogue summarization. arXiv preprint arXiv:2103.06410, 2021."
+                                        }
+                                    ]
+                                }
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+                                    "type": "text",
+                                    "content": "A NOTE FOR REAL-WORLD EVALUATION"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
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+                                    "type": "text",
+                                    "content": "The development of autonomous agents should consider the security and safety aspects. In the real website evaluation, we have carefully conducted the experiments under human supervision in case undesired behaviors happen. We use Selenium WebDriver<sup>2</sup>, a popular library for browser automation, and limit the access per second not to stress the server. We have anonymized the real websites we tested on for safety and privacy concerns."
+                                }
+                            ]
+                        }
+                    ],
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+                },
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+                                    "type": "text",
+                                    "content": "B EXTENDED RELATED WORKS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Document Understanding Understanding structural documents has been a practical challenge for transformer-based language models. Prior works employ layout-informed tokens (Xu et al., 2019) or even multimodal tokens from visual inputs (Appalaraju et al., 2021; Li et al., 2021a;c). Especially, for the documents written in markup languages, text-XPath alignment (Li et al., 2021b), token separation between text and HTML (Wang et al., 2022a), or extra topological information of HTML (Zhao et al., 2022) are proposed to leverage their syntax better. On the other hand, such a domain knowledge conflicts with recent generalist and scaling trends around LLMs (Anil et al., 2023; OpenAI, 2023). Because web agents require the instruction-conditioned HTML understanding, it also would be desirable to reconcile specialist aspects for HTML documents with generalist capabilities for natural language tasks. In this work, we design HTML-T5 to incorporate the structural bias of HTML by combining local-global attention for the encoder and a mixture of long-span denoising, while it can solve instruction-following better in downstream web-based tasks."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
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+                        {
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "LLM for Task Planning The prior knowledge of commonsense in LLMs has allowed us to leverage them for a variety of task planning. For instance, Huang et al. (2022) propose LLM agent that generates natural language plans in an open-loop manner. Nottingham et al. (2023) and Wang et al. (2023b) perform sequential closed-loop planning on MineCraft. Singh et al. (2022) decode robotic plans with pythonic text, and several works incorporate planning definition and domain language into the outputs (Liu et al., 2023; Silver et al., 2023; Valmeekam et al., 2023). On the other hand, our WebAgent leverages finetuned specialist language models and performs closed-loop planning coupled with HTML summarization by decomposing given instructions. We empirically prove that our system is superior to open-loop planning with a single generalist LLM with prompting."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                }
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+                                    "type": "text",
+                                    "content": "C IMPLEMENTATION DETAILS OF HTML-T5"
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+                                    "content": "We use the implementation of local and global attentions released by Guo et al. (2022) "
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+                                    "type": "inline_equation",
+                                    "content": "^{3}"
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+                                    "type": "text",
+                                    "content": ". Following Guo et al. (2022), we set the local radius to "
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+                                    "type": "inline_equation",
+                                    "content": "r = 127"
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+                                    "type": "text",
+                                    "content": ", and block size for transient global attention to "
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+                                    "type": "inline_equation",
+                                    "content": "k = 16"
+                                },
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+                                    ],
+                                    "type": "text",
+                                    "content": ". For the pre-training objective, similar to Tay et al. (2022), we construct the mixtures and then use long mean span lengths: "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "\\mu \\in \\{8,64\\}"
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+                                    "type": "text",
+                                    "content": ", and all the denoising ratio (percentage of masked tokens in the input sequence) is set to 0.15. We adopt 4096 input sequence length and 910 output sequence length during the pre-training. The batch size for training is set to 128. We train the models with 100K iterations following other pre-training strategies for T5 families (Chung et al., 2022; Lester et al., 2021). We leverage SeqIO (Roberts et al., 2022) and T5X (Roberts et al., 2022) library to manage the training pipeline. We also use SentencePiece (Kudo & Richardson, 2018) with 32K tokens from C4 dataset (Raffel et al., 2020) as a tokenizer. During the downstream finetuning, we adopt 16K tokens for the context window unless otherwise mentioned. We have used cloud TPU-v3, which has a 32 GiB HBM memory space, with 128 cores for the experiments."
+                                }
+                            ]
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+                                    "type": "text",
+                                    "content": "For the dataset, we prepare 100 WARC files (April 2019) from CommonCrawl<sup>4</sup>, and pre-process the raw HTML by removing non-Unicode and alphanumeric documents and extracting subtrees around <label> elements that have for attribute, to reduce the noise in training corpus, which results in about 3.41M examples (Table 5)."
+                                }
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+                                            "html": "<table><tr><td rowspan=\"2\"># of examples</td><td colspan=\"3\"># of tokens</td></tr><tr><td>Average</td><td>90th</td><td>Max</td></tr><tr><td>3.41M</td><td>1020</td><td>4566</td><td>7627</td></tr></table>",
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+                                    "type": "text",
+                                    "content": "Table 5: Statistics of CommonCrawl HTML corpus for self-supervised denoising pre-training of HTML-T5. Input lengths are measured in tokens by Kudo & Richardson (2018)."
+                                }
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+                                    "type": "text",
+                                    "content": "D WEBAGENT EXAMPLE FLOW IN REAL-ESTATE WEBSITE"
+                                }
+                            ]
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+                                            "content": "Figure 6: An example flow with planning, summarization, and grounded program synthesis in the real estate website. HTML-T5 iteratively predicts a decomposed sub-instruction and task-relevant snippet (orange) in a closed-loop manner, conditioning on the HTML documents, instruction (yellow), and history of past predictions (green). Flan-U-PaLM is prompted with sub-instruction and snippet (orange) to decode python programs (blue)."
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+                                    "content": "<sup>3</sup>https://github.com/google-research/longt5"
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+                                    "type": "text",
+                                    "content": "E REAL-WORLD WEB AUTOMATION WITH DIFFERENT GENERALIST LLMS"
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+                                    "content": "We compare different generalist LLMs as a module of WebAgent among model-size variants (Flan-PaLM-8B, Flan-PaLM-62B, Flan-U-PaLM-540B), and publicly accessible LLM (gpt-3.5-turbo). We test those models on map website following the same 20 instructions in Appendix G. The results in Figure 7 imply that the performance of Flan-U-PaLM-540B and gpt-3.5-turbo are the same (80% success, 93.8% score), and Flan-PaLM-62B (60% success, 86.3% score) is lower than Flan-U-PaLM-540B, which is caused by the inaccurate program synthesis. In addition, Flan-PaLM-8B could not generate proper programs at all. We believe that any LLM with sufficient program synthesis capabilities could be integrated into WebAgent, including Flan-U-PaLM-540B."
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+                                            "content": "Flan-PaLM-8B Flan-PaLM-62B Flan-U-PaLM-540B gpt-3.5-turbo"
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+                                            "content": "Figure 7: Performance (left) and error analysis (right) of real-world web automation with different generalist LLMs. We compare model-size variants (Flan-PaLM-8B, Flan-PaLM-62B, Flan-U-PaLM-540B) and public LLM (gpt-3.5-turbo) on map website."
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+                                    "content": "F WEBSRC:STATIC HTML COMPREHENSION"
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+                                    "type": "text",
+                                    "content": "To emphasize the advantage of our modular approach, we test WebAgent on a static website comprehension benchmark, WebSRC (Chen et al., 2021b), which is a contextual QA dataset with HTML documents. The questions require an understanding of the spatial and logical structure of websites, and the answers are either text span on HTML or yes/no. For the comprehensive evaluation, WebSRC has three different types of websites, KV, Comparison, and Table. KV task is a value extraction from the attribute key. Comparison task has several entities with the same attributes. Table task requires a structural understanding with header columns and values in the row. We finetune HTML-T5 for snippet extraction to predict data-ref corresponding to the answer and use dev set for the evaluation."
+                                }
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+                                    "type": "text",
+                                    "content": "As did in real-world web automation, HTML-T5 first predicts data-ref attribute of task-relevant snippet from the input HTML document. To make sure there is enough context, we extract the snippet from the predicted element to the two-level-up via XPath. If it exceeds the context length of Flan-U-PaLM, we limit it into parent elements. If it still does not work, we truncate the end of extracted snippet to fit within the token budget. Because snippet extraction in table structure often loses the context to solve question-answering, we just truncate HTML documents for Table tasks. Flan-U-PaLM predicts the answers seeing 5-shot examples."
+                                }
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+                                    "type": "text",
+                                    "content": "As shown in Table 6, single LLM, such as Flan-U-PaLM or HTML-T5, has struggled to the limited context length or model capacity. In contrast, WebAgent, our LLM-collaborative approach, enhances the performance from both single generalist and specialist LLMs, and shows competitive results with strong baselines. This demonstrates that modular LLMs work complementarily to each other. Figure 8 presents the performance comparison on different types of websites (KV, Comparison, Table) among MarkupLM (Li et al., 2021b), TIE (Zhao et al., 2022), and WebAgent. WebAgent is better at Comparison tasks, but inferior to structural understanding for KV and Table tasks, compared to other baselines, which suggest that generalist LLMs are still not suitable for recognizing structural data such as table."
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+                                            "html": "<table><tr><td>Models</td><td>EM</td><td>F1</td></tr><tr><td>T-PLM (Chen et al., 2021b)</td><td>61.67</td><td>69.85</td></tr><tr><td>H-PLM (Chen et al., 2021b)</td><td>70.12</td><td>74.14</td></tr><tr><td>V-PLM (Chen et al., 2021b)</td><td>73.22</td><td>76.16</td></tr><tr><td>MarkupLM-Large (Li et al., 2021b)</td><td>74.43</td><td>80.54</td></tr><tr><td>TIE-Large (Zhao et al., 2022)</td><td>81.66</td><td>86.24</td></tr><tr><td>Flan-U-PaLM</td><td>40.01</td><td>47.56</td></tr><tr><td>HTML-T5-Large</td><td>73.09</td><td>76.66</td></tr><tr><td>HTML-T5-XL</td><td>74.73</td><td>78.73</td></tr><tr><td>WebAgent</td><td>75.50</td><td>85.75</td></tr><tr><td>WebAgent (oracle)</td><td>76.91</td><td>86.64</td></tr></table>",
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+                                    "content": "Table 6: Evaluation on WebSRC (Chen et al., 2021b) with dev set. WebAgent, our collaborative LLMs, enhances the performance from both single generalist (Flan-U-PaLM) or specialist LLMs (HTML-T5). WebAgent (oracle) uses oracle snippets that are guaranteed to include the answers, instead of those predicted by finetuned HTML-T5."
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+                                            "content": " bathroom houses in escondido, ca for corporate housing and price less than 12100 on real estate website."
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+                                            "content": " bathroom townhomes in hollywood, ca and price less than 14600 on real estate website."
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+                                            "content": " bathroom houses in compton, ca and price more than 1200 for corporate housing on real estate website."
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+                                            "content": " bathroom apartments in oroville, ca for corporate housing on real estate website."
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+                                            "content": " bathroom houses in modesto, ca on real estate website."
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+                                            "content": " bathroom condos in redwood city, ca for student and price more than 1900 on real estate website."
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+                                            "content": " bathroom townhomes in concord, ca and price more than 600 on real estate website."
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+                                            "content": " bathroom apartments in san diego, ca and price less than 9300 on real estate website."
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+                                            "content": "1. Show me the way from San Jose to Mountain View by 2nd Cycling at map website."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 2
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+                                            "type": "text",
+                                            "content": "2. Please show me the way from The Painted Ladies to San Francisco Zoo with 3rd Best option at map website."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 3
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+                                            "type": "text",
+                                            "content": "3. Could you tell me the path from California Academy of Sciences to de Young Museum by 1st Transit at map website."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 4
+                        },
+                        {
+                            "bbox": [
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+                                            "content": "4. Could you tell me the way from Union Square to The Painted Ladies with 2nd Cycling option at map website."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 5
+                        },
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+                                            "type": "text",
+                                            "content": "5. Please present the way from Chappell Hayes Observation Tower to San Jose with 2nd Walking option at map website."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 6
+                        },
+                        {
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+                                            "type": "text",
+                                            "content": "6. Please present the path from Jack London Square to Emeryville by 2nd Cycling at map website."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 7
+                        },
+                        {
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+                                201,
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+                                211
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+                                            "type": "text",
+                                            "content": "7. I'd like to move The Midway from Children's Fairyland by 1st Cycling at map website."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 8
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+                                            "type": "text",
+                                            "content": "8. I'd like to move Chase Center from San Francisco - Oakland Bay Bridge with 2nd Transit option at map website."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 9
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+                                            "type": "text",
+                                            "content": "9. I want to move Pier 39 from Berkeley by 3rd Cycling at map website."
+                                        }
+                                    ]
+                                }
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+                                            "type": "text",
+                                            "content": "10. I want to go to Emeryville from Mountain View with 2nd Cycling option at map website."
+                                        }
+                                    ]
+                                }
+                            ],
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+                                            "type": "text",
+                                            "content": "11. Can you point out the way from San Mateo to Stanford University by 2nd Cycling at map website."
+                                        }
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+                                            "type": "text",
+                                            "content": "12. Could you point out the way from Palace of Fine Arts to UC Berkeley by 1st Cycling at map website."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 13
+                        },
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+                                            "type": "text",
+                                            "content": "13. Point out the way from The Painted Ladies to San Francisco Museum of Modern Art by 2nd Driving at map website."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 14
+                        },
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+                                            "type": "text",
+                                            "content": "14. Could you find the path from Union Square to Palo Alto by 1st Cycling at map website."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 15
+                        },
+                        {
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+                                            "type": "text",
+                                            "content": "15. Please check the way from San Jose to San Jose Mineta International Airport with 1st Walking at map website."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 16
+                        },
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+                            "type": "text",
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+                                            "type": "text",
+                                            "content": "16. Check the path from San Francisco Zoo to Berkeley with 1st Cycling at map website."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 17
+                        },
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+                                            "type": "text",
+                                            "content": "17. I'd like to check Parking Lots along the way from Stanford University to The Painted Ladies with Best option at map website."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 18
+                        },
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+                                            "type": "text",
+                                            "content": "18. Check Gas stations along the way from de Young Museum to Oakland with Driving option at map website."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 19
+                        },
+                        {
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+                                            "type": "text",
+                                            "content": "19. Please show me Hotels along the way from Palace of Fine Arts to Berkeley by Transit at map website."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 20
+                        },
+                        {
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+                                            "type": "text",
+                                            "content": "20. Check Gas stations along the way from Bay Area Discovery Museum to Santa Cruz with Best option at map website."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 21
+                        }
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+                                    "type": "text",
+                                    "content": "H EXAMPLE EPISODE IN REAL-WORLD WEB AUTOMATION"
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+                                    "content": "I EXTENSIVE ABLATION OF HTML-T5"
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+                                    "content": "I.1 DATASET AND INITIALIZATION"
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+                                    "type": "text",
+                                    "content": "To test our recipe described in Section 2.1, we compare the different dataset and model initialization for pre-training on downstream task performances; offline task planning on real-estate and average success rate on MiniWoB with 12K dataset. We use Base-size models for the experiments. For HTML-denoising, we prepare the corpus from CommonCrawl with (Extracted) or without (Raw) subtree extraction around label elements on the documents. We also compare the initialization of base architectures before HTML-denoising; from scratch or with pre-trained models on PEGASUS objective (Zhang et al., 2020) that is a masked important sentence prediction from long-context paragraph. Table 7 reveals that snippet extraction on HTML corpus improves downstream performances since such a pre-processing can reduce the noise in raw HTML. Moreover, initialization with PEGASUS pre-trained weights is essential for HTML-T5, because of the long-context and instruction-following nature of HTML-based tasks."
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+                                            "html": "<table><tr><td>CC-HTML</td><td>PEGASUS</td><td>real-estate</td><td>MiniWoB++</td></tr><tr><td>Raw</td><td>✓</td><td>80.56</td><td>56.7%</td></tr><tr><td>Extracted</td><td>×</td><td>67.11</td><td>49.1%</td></tr><tr><td>Extracted</td><td>✓</td><td>82.46</td><td>57.0%</td></tr></table>",
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+                                    "content": "I.2 OFFLINE EVALUATION ON TASK PLANNING WITH MODEL SCALING"
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+                        }
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+                                    "content": "We compare the offline task planning performance between HTML-T5 and LongT5 (without HTML-denosing) with different model sizes; with Base (220M parameters), Large (770M parameters), and XL (3B parameters). As described in Section 3.1, the models predict the next sub-instructions in a closed-loop manner considering the current HTML observations, user instructions, and previous sub-instruction histories as inputs. For offline task planning evaluation, we use the demonstrations on real-estate website; preparing 130 demonstrations and splitting them into train "
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+                                    "content": "(90\\%)"
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+                                    "type": "text",
+                                    "content": " and test splits "
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+                                    "type": "inline_equation",
+                                    "content": "(10\\%)"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ". We report the best per-step exact match accuracy in test set."
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+                                    "type": "text",
+                                    "content": "Table 8 shows that HTML-T5 outperforms LongT5 on the accuracy of sub-instruction prediction, which demonstrates that HTML-denoising pre-training captures the structural bias of HTML better without sacrificing the ability to understand natural language instructions. This also implies that our proposed HTML-denoising can scale to larger-size models consistently."
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+                                            "content": "Table 7: Ablations of HTML-T5-Base on dataset quality and initialization. We evaluate offline task planning on real-estate and average success rate on MiniWoB with 12K dataset. For HTML-denoising, we prepare HTML corpus from CommonCrawl with (Extracted) or without (Raw) subtree extraction around label elements. We also compare the pre-training of base architectures with PEGASUS objective (Zhang et al., 2020) before HTML-denoising. The results imply that PEGASUS pre-training is critical for the architectures and preprocessing with subtree extraction improves the downstream performance on HTML-based tasks."
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+                                            "html": "<table><tr><td>Models</td><td>real-estate</td><td>Diff.</td></tr><tr><td>LongT5-Base</td><td>78.07</td><td>0.0</td></tr><tr><td>LongT5-Large</td><td>82.89</td><td>0.0</td></tr><tr><td>LongT5-XL</td><td>81.29</td><td>0.0</td></tr><tr><td>HTML-T5-Base</td><td>82.46</td><td>+4.39</td></tr><tr><td>HTML-T5-Large</td><td>83.63</td><td>+0.74</td></tr><tr><td>HTML-T5-XL</td><td>83.92</td><td>+2.63</td></tr></table>",
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+                                    "content": "Table 8: Accuracy of offline evaluation on task planning. We leverage the demonstrations in real-estate websites. Compared to original LongT5, and as we scale model size, HTML-T5 improves the accuracy of sub-instruction prediction."
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+                                    "content": "I.3 DESCRIPTION GENERATION"
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+                                    "type": "text",
+                                    "content": "We also investigate the capability of HTML-T5 on static HTML comprehension tasks, as well as interactive decision making tasks. We use Description Generation benchmark (Gur et al., 2022), where the models generate the textual description of elements, typically used for accessibility purposes and annotated with a special attribute in the HTML schema known as for. We evaluate the understanding the structure of HTML as it would appear to a user, despite not having access to the rendered website directly."
+                                }
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+                                    "content": "We compare LaMDA (Thoppilan et al., 2022), T5, LongT5, and HTML-T5 with respect to accuracy, BLEU (Papineni et al., 2002), and ROUGE-1 (Lin, 2004) score. As shown in Table 9, local and global attention mechanisms, underlying between LongT5 and HTML-T5, could almost solve the benchmark by improving the previous best performance by over "
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+                                    "type": "text",
+                                    "content": ", with still improved performance as model size increases. Compared to the effect of local-global attention, HTML-T5 marginally improves against LongT5, which emphasizes that local and global attentions are critical to capture the hierarchical structure of HTML documents."
+                                }
+                            ]
+                        }
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+                                            "html": "<table><tr><td rowspan=\"2\">Models</td><td colspan=\"3\">Dev</td><td colspan=\"3\">Test</td></tr><tr><td>Accuracy</td><td>BLEU</td><td>ROUGE-1</td><td>Accuracy</td><td>BLEU</td><td>ROUGE-1</td></tr><tr><td>LaMDA-1B (Gur et al., 2022)</td><td>83.3</td><td>87.5</td><td>90.2</td><td>84.3</td><td>88.6</td><td>91.2</td></tr><tr><td>T5-Large (Gur et al., 2022)</td><td>83.2</td><td>90.2</td><td>90.5</td><td>84.3</td><td>91.7</td><td>91.5</td></tr><tr><td>T5-XL (Gur et al., 2022)</td><td>84.0</td><td>90.8</td><td>90.9</td><td>85.2</td><td>92.1</td><td>91.9</td></tr><tr><td>LongT5-Base</td><td>96.4</td><td>98.0</td><td>98.5</td><td>95.6</td><td>97.4</td><td>98.2</td></tr><tr><td>LongT5-Large</td><td>98.1</td><td>98.9</td><td>99.2</td><td>97.7</td><td>98.5</td><td>99.0</td></tr><tr><td>LongT5-XL</td><td>98.4</td><td>99.1</td><td>99.3</td><td>98.5</td><td>99.2</td><td>99.3</td></tr><tr><td>HTML-T5-Base</td><td>96.5</td><td>98.1</td><td>98.6</td><td>95.9</td><td>97.5</td><td>98.3</td></tr><tr><td>HTML-T5-Large</td><td>98.1</td><td>98.9</td><td>99.2</td><td>97.7</td><td>98.3</td><td>99.1</td></tr><tr><td>HTML-T5-XL</td><td>98.4</td><td>99.0</td><td>99.3</td><td>98.9</td><td>99.4</td><td>99.5</td></tr></table>",
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+                                    "content": "Table 9: Results of Description Generation benchmark (Gur et al., 2022). We compare LaMDA (Thoppilan et al., 2022), T5, LongT5, and HTML-T5 with respect to accuracy, BLEU, and ROUGE-1 scores. The results demonstrate that local and global attention mechanisms, shared modules between LongT5 and HTML-T5, could almost completely solve the benchmark by improving the previous best performance by over "
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+                                    "content": ". HTML-T5 slightly outperforms LongT5."
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+                                    "content": "In the web automation literature (Furuta et al., 2023; Kim et al., 2023), instruction-finetuned LLMs have great success in HTML comprehension and improve the task success. For the comparison to HTML-denosing, we prepare the instruction-finetuned LongT5 (i.e. Flan-LongT5) by leveraging Flan dataset released by Chung et al. (2022). We finetuned the pre-trained LongT5 with 100K iterations and picked up the best checkpoints."
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+                                    "content": "As a sanity check of instruction-tuning, we evaluate Flan-LongT5 with few-shot/zero-shot settings on CoT benchmark (GSM8K (Cobbe et al., 2021), StrategyQA (Geva et al., 2021), SVAMP (Patel et al., 2021), Asdiv (Miao et al., 2021), CommonsenseQA (Talmor et al., 2019)), BigBench-Hard (BBH) (Suzgun et al., 2022), and MMLU (Hendrycks et al., 2021b) as tested in Longpre et al. (2023). We reevaluate the performance of Flan-T5, using official checkpoints "
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+                                    "content": ". We also check the performance of Flan-LongT5 on downstream summarization tasks, originally evaluated on LongT5 (Guo et al., 2022). We use arXiv (Cohan et al., 2018), PubMed (Cohan et al., 2018), BigPatent (Sharma et al., 2019), Multi-News (Fabbri et al., 2019), MediaSum (Zhu et al., 2021), CNN / Daily Mail (Nallapati et al., 2016) dataset for the evaluation, measuring the performance with ROUGE-1/2/L metrics."
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+                                    "content": "Table 10 shows that we have successfully replicated the LongT5 version of instruction-finetuned language models. Flan-LongT5 achieves competitive results to original Flan-T5; for instance, Flan-LongT5-Large (36.64) outperforms Flan-T5-Large (35.25), but Flan-LongT5-XL (39.05) is still behind Flan-T5-XL (43.03) on average. This might be caused by the training instability of XL-size models (Guo et al., 2022). Because, unlike HTML-T5 on HTML-based tasks, reasoning tasks do not have long-context or hierarchical syntax, it is not surprising for Flan-LongT5 not to outperform Flan-T5. Table 11 also demonstrates that we have successfully conducted instruction-tuning without losing the capability of long text summarization."
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+                                            "html": "<table><tr><td rowspan=\"2\">Models</td><td colspan=\"2\">CoT</td><td colspan=\"2\">MMLU</td><td colspan=\"2\">BBH</td><td colspan=\"2\">BBH-CoT</td><td colspan=\"2\">Avg.</td></tr><tr><td>Zero</td><td>Few</td><td>Zero</td><td>Few</td><td>Zero</td><td>Few</td><td>Zero</td><td>Few</td><td>CoT</td><td>Direct</td></tr><tr><td>Flan-T5-Large</td><td>35.14</td><td>40.03</td><td>40.68</td><td>45.12</td><td>25.90</td><td>37.48</td><td>26.17</td><td>31.45</td><td>33.20</td><td>37.29</td></tr><tr><td>Flan-T5-XL</td><td>51.74</td><td>52.64</td><td>50.76</td><td>52.40</td><td>26.09</td><td>40.96</td><td>34.12</td><td>35.62</td><td>43.53</td><td>42.55</td></tr><tr><td>Flan-LongT5-Large</td><td>44.78</td><td>45.34</td><td>38.44</td><td>40.03</td><td>28.67</td><td>34.67</td><td>29.38</td><td>31.85</td><td>37.84</td><td>35.45</td></tr><tr><td>Flan-LongT5-XL</td><td>48.78</td><td>50.02</td><td>43.44</td><td>44.74</td><td>26.53</td><td>37.77</td><td>29.09</td><td>32.01</td><td>39.97</td><td>38.12</td></tr></table>",
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+                                            "content": "Table 10: Performance of Flan-LongT5 on reasoning tasks. We reevaluate the performance of Flan-T5 (Chung et al., 2022), using official checkpoints. Flan-LongT5 achieves competitive results to original Flan-T5."
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+                                            "html": "<table><tr><td rowspan=\"2\">Models</td><td colspan=\"3\">arXiv</td><td colspan=\"3\">PubMed</td><td colspan=\"3\">BigPatent</td><td colspan=\"3\">MultiNews</td><td colspan=\"3\">MediaSum</td><td colspan=\"3\">CNN / Daily Mail</td></tr><tr><td>R-1</td><td>R-2</td><td>R-L</td><td>R-1</td><td>R-2</td><td>R-L</td><td>R-1</td><td>R-2</td><td>R-L</td><td>R-1</td><td>R-2</td><td>R-L</td><td>R-1</td><td>R-2</td><td>R-L</td><td>R-1</td><td>R-2</td><td>R-L</td></tr><tr><td>LongT5-Large</td><td>48.28</td><td>21.63</td><td>44.11</td><td>49.98</td><td>24.69</td><td>46.46</td><td>70.38</td><td>56.81</td><td>62.73</td><td>47.18</td><td>18.44</td><td>24.18</td><td>35.54</td><td>19.04</td><td>32.20</td><td>42.49</td><td>20.51</td><td>40.18</td></tr><tr><td>LongT5-XL</td><td>48.35</td><td>21.92</td><td>44.27</td><td>50.23</td><td>24.76</td><td>46.67</td><td>76.87</td><td>66.06</td><td>70.76</td><td>48.17</td><td>19.43</td><td>24.94</td><td>36.15</td><td>19.66</td><td>32.80</td><td>43.94</td><td>21.40</td><td>41.28</td></tr><tr><td>Flan-LongT5-Large</td><td>48.52</td><td>22.00</td><td>44.46</td><td>50.46</td><td>25.08</td><td>46.96</td><td>70.53</td><td>57.13</td><td>63.02</td><td>47.76</td><td>18.99</td><td>24.52</td><td>35.71</td><td>19.18</td><td>32.33</td><td>43.13</td><td>20.89</td><td>37.28</td></tr><tr><td>Flan-LongT5-XL</td><td>48.37</td><td>21.75</td><td>44.22</td><td>50.23</td><td>24.75</td><td>46.73</td><td>76.31</td><td>65.17</td><td>70.01</td><td>48.19</td><td>19.47</td><td>24.80</td><td>36.16</td><td>19.75</td><td>32.81</td><td>43.46</td><td>21.00</td><td>37.34</td></tr></table>",
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+                                            "html": "<table><tr><td>Task</td><td>HTML-T5-XL (347K)</td><td>HTML-T5-XL (12K)</td><td>Plan-T5-XL (347K)</td><td>WebN-T5-XL (12K)</td></tr><tr><td>book-flight</td><td>0.99</td><td>0.00</td><td>0.48</td><td>0.00</td></tr><tr><td>choose-date</td><td>0.16</td><td>0.03</td><td>0.08</td><td>0.00</td></tr><tr><td>choose-date-easy</td><td>1.00</td><td>0.28</td><td>1.00</td><td>0.03</td></tr><tr><td>choose-date-medium</td><td>0.56</td><td>0.14</td><td>0.57</td><td>0.00</td></tr><tr><td>choose-list</td><td>0.22</td><td>0.19</td><td>0.16</td><td>0.26</td></tr><tr><td>click-button</td><td>1.00</td><td>0.92</td><td>0.98</td><td>1.00</td></tr><tr><td>click-button-sequence</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>click-checkboxes</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.96</td></tr><tr><td>click-checkboxes-large</td><td>0.90</td><td>0.94</td><td>0.98</td><td>0.22</td></tr><tr><td>click-checkboxes-soft</td><td>0.99</td><td>0.64</td><td>1.00</td><td>0.54</td></tr><tr><td>click-checkboxes-transfer</td><td>1.00</td><td>1.00</td><td>0.99</td><td>0.63</td></tr><tr><td>click-collapsible</td><td>1.00</td><td>0.41</td><td>1.00</td><td>0.00</td></tr><tr><td>click-collapsible-2</td><td>0.93</td><td>0.26</td><td>0.94</td><td>0.00</td></tr><tr><td>click-color</td><td>1.00</td><td>1.00</td><td>0.27</td><td>0.27</td></tr><tr><td>click-dialog</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>click-dialog-2</td><td>0.74</td><td>0.31</td><td>0.34</td><td>0.24</td></tr><tr><td>click-link</td><td>0.99</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>click-menu</td><td>0.37</td><td>0.26</td><td>0.41</td><td>0.37</td></tr><tr><td>click-option</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.87</td></tr><tr><td>click-pie</td><td>0.96</td><td>0.89</td><td>0.99</td><td>0.51</td></tr><tr><td>click-scroll-list</td><td>0.99</td><td>0.91</td><td>0.00</td><td>0.00</td></tr><tr><td>click-shades</td><td>0.00</td><td>0.05</td><td>0.00</td><td>0.00</td></tr><tr><td>click-shape</td><td>0.79</td><td>0.57</td><td>0.58</td><td>0.53</td></tr><tr><td>click-tab</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.74</td></tr><tr><td>click-tab-2</td><td>0.94</td><td>0.40</td><td>0.94</td><td>0.18</td></tr><tr><td>click-tab-2-hard</td><td>0.88</td><td>0.30</td><td>0.57</td><td>0.12</td></tr><tr><td>click-test</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>click-test-2</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>click-widget</td><td>1.00</td><td>0.94</td><td>1.00</td><td>1.00</td></tr><tr><td>count-shape</td><td>0.67</td><td>0.55</td><td>0.64</td><td>0.41</td></tr><tr><td>email-inbox</td><td>1.00</td><td>0.99</td><td>0.99</td><td>0.38</td></tr><tr><td>email-inbox-forward-nl</td><td>1.00</td><td>0.92</td><td>1.00</td><td>0.60</td></tr><tr><td>email-inbox-forward-nl-turk</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.33</td></tr><tr><td>email-inbox-nl-turk</td><td>0.99</td><td>0.76</td><td>0.92</td><td>0.23</td></tr><tr><td>enter-date</td><td>1.00</td><td>0.00</td><td>1.00</td><td>0.00</td></tr><tr><td>enter-password</td><td>1.00</td><td>0.99</td><td>1.00</td><td>0.97</td></tr><tr><td>enter-text</td><td>1.00</td><td>0.96</td><td>1.00</td><td>0.89</td></tr><tr><td>enter-text-dynamic</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.98</td></tr><tr><td>enter-time</td><td>1.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr><tr><td>focus-text</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>focus-text-2</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td>grid-coordinate</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.49</td></tr><tr><td>guess-number</td><td>0.13</td><td>0.00</td><td>0.10</td><td>0.00</td></tr><tr><td>identify-shape</td><td>1.00</td><td>0.89</td><td>0.90</td><td>0.88</td></tr><tr><td>login-user</td><td>1.00</td><td>0.80</td><td>1.00</td><td>0.82</td></tr><tr><td>login-user-popup</td><td>1.00</td><td>0.63</td><td>0.97</td><td>0.72</td></tr><tr><td>multi-layouts</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.83</td></tr><tr><td>multi-orderings</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.88</td></tr><tr><td>navigate-tree</td><td>0.99</td><td>0.99</td><td>1.00</td><td>0.91</td></tr><tr><td>search-engine</td><td>0.93</td><td>0.55</td><td>0.59</td><td>0.34</td></tr><tr><td>social-media</td><td>0.99</td><td>0.93</td><td>0.99</td><td>0.21</td></tr><tr><td>social-media-all</td><td>0.31</td><td>0.84</td><td>0.09</td><td>0.00</td></tr><tr><td>social-media-some</td><td>0.89</td><td>0.60</td><td>0.39</td><td>0.02</td></tr><tr><td>tic-tac-toe</td><td>0.57</td><td>0.46</td><td>0.42</td><td>0.48</td></tr><tr><td>use-autocomplete</td><td>0.97</td><td>0.23</td><td>0.98</td><td>0.22</td></tr><tr><td>use-spinner</td><td>0.07</td><td>0.07</td><td>0.03</td><td>0.07</td></tr><tr><td>Average</td><td>0.856</td><td>0.655</td><td>0.755</td><td>0.484</td></tr></table>",
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+                                            "content": "Table 12: Per-task average success rate on 56 tasks from MiniWoB++. We refer to Furuta et al. (2023) and Gur et al. (2022) for the baseline performances."
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diff --git a/2024/ASID_ Active Exploration for System Identification in Robotic Manipulation/22c89e0b-1caf-4875-86e0-f3b9397a36cc_content_list.json b/2024/ASID_ Active Exploration for System Identification in Robotic Manipulation/22c89e0b-1caf-4875-86e0-f3b9397a36cc_content_list.json
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+++ b/2024/ASID_ Active Exploration for System Identification in Robotic Manipulation/22c89e0b-1caf-4875-86e0-f3b9397a36cc_content_list.json	
@@ -0,0 +1,2380 @@
+[
+    {
+        "type": "text",
+        "text": "ASID: ACTIVE EXPLORATION FOR SYSTEM IDENTIFICATION IN ROBOTIC MANIPULATION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            99,
+            823,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Marius Memmel, Andrew Wagenvaker, Chuning Zhu, Dieter Fox, Abhishek Gupta",
+        "bbox": [
+            179,
+            169,
+            769,
+            184
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Paul G. Allen School of Computer Science & Engineering",
+        "bbox": [
+            183,
+            185,
+            566,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "University of Washington",
+        "bbox": [
+            183,
+            199,
+            352,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Seattle, WA 98195, USA",
+        "bbox": [
+            183,
+            212,
+            349,
+            224
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "{memmelma,ajwagen,zchuning,fox,abhgupta}@cs.washington.edu",
+        "bbox": [
+            183,
+            226,
+            750,
+            239
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT",
+        "text_level": 1,
+        "bbox": [
+            450,
+            277,
+            545,
+            291
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Model-free control strategies such as reinforcement learning have shown the ability to learn control strategies without requiring an accurate model or simulator of the world. While this is appealing due to the lack of modeling requirements, such methods can be sample inefficient, making them impractical in many real-world domains. On the other hand, model-based control techniques leveraging accurate simulators can circumvent these challenges and use a large amount of cheap simulation data to learn controllers that can effectively transfer to the real world. The challenge with such model-based techniques is the requirement for an extremely accurate simulation, requiring both the specification of appropriate simulation assets and physical parameters. This requires considerable human effort to design for every environment being considered. In this work, we propose a learning system that can leverage a small amount of real-world data to autonomously refine a simulation model and then plan an accurate control strategy that can be deployed in the real world. Our approach critically relies on utilizing an initial (possibly inaccurate) simulator to design effective exploration policies that, when deployed in the real world, collect high-quality data. We demonstrate the efficacy of this paradigm in identifying articulation, mass, and other physical parameters in several challenging robotic manipulation tasks, and illustrate that only a small amount of real-world data can allow for effective sim-to-real transfer. Project website at https://weirdlabuw.github.io/asid",
+        "bbox": [
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+            306,
+            769,
+            583
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION",
+        "text_level": 1,
+        "bbox": [
+            173,
+            608,
+            338,
+            623
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Controlling robots to perform dynamic, goal-directed behavior in the real world is challenging. Reinforcement Learning (RL) has emerged as a promising technique to learn such behaviors without requiring known models of the environment, instead relying on data sampled directly from the environment (Schulman et al., 2017a; Haarnoja et al., 2018). In principle, these techniques can be deployed in new environments with a minimal amount of human effort, and allow for continual improvement of behavior. Such techniques have been shown to successfully learn complex behaviors in a variety of scenarios, ranging from table-top manipulation (Yu et al., 2020) to locomotion (Hwangbo et al., 2019) and even dexterous manipulation (Zhu et al., 2019).",
+        "bbox": [
+            169,
+            638,
+            485,
+            875
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/51fb0e85ce1afcff040acbcb66b612d5ddd50ffe92b5f417ff20aac8a157502f.jpg",
+        "image_caption": [
+            "Figure 1: ASD: A depiction of our proposed process of active exploration for system identification, from learning exploration policies to real-world deployment."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            496,
+            607,
+            823,
+            821
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "However, these capabilities come at a cost, requiring full access to the environment in order to design reset mechanisms and reward functions. Such requirements necessitate training these methods in carefully controlled and often curated environments, limiting their applicability. Moreover, de",
+        "bbox": [
+            169,
+            881,
+            823,
+            925
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "page_number",
+        "text": "1",
+        "bbox": [
+            493,
+            948,
+            503,
+            959
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ploying tabula-asa reinforcement learning methods in the real world often requires a prohibitively large number of samples, which may be impractical to collect.",
+        "bbox": [
+            174,
+            103,
+            821,
+            131
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "One approach to circumvent these challenges is to rely on simulators to cheaply generate large amounts of data and use RL to train a policy. However, directly deploying policies trained in simulation in the real world is often ineffective due to the discrepancy between the simulation and the real world, the so-called sim2real gap. For example, the physical properties and parameters of the simulation might diverge from the real world, rendering a simulation-trained policy useless in reality.",
+        "bbox": [
+            174,
+            138,
+            821,
+            208
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Taking inspiration from system identification, we argue that the key to effective sim2real transfer is an initial round of exploration in the real world to learn an effective simulator. We propose a generic pipeline for sim2real transfer, Active Exploration for System IDentification (ASID), which decouples exploration and exploitation: (1) exploration in real to collect informative data of unknown parameters, (2) refinement of the simulation parameters using data collected in real, (3) policy training on the updated simulator to accomplish the goal tasks. Our exploration procedure is motivated by work in theoretical statistics and seeks to induce trajectories corresponding to large Fisher information, thereby providing maximally informative observations. By using our initial round of exploration to obtain accurate estimates of the parameters in the real world, we show that in many cases, the policies trained in step (3) successfully transfer to real in a zero-shot fashion, even in settings where training a policy in sim without additional knowledge of real would fail.",
+        "bbox": [
+            174,
+            215,
+            821,
+            368
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A key insight in our approach is that, while a policy trained in sim to accomplish the goal task may not effectively transfer, strategies that explore effectively in sim often also explore effectively in real. As an example, say our goal task is to hit a ball to a particular location with a robotic arm and assume the mass of the ball is unknown. If we train a policy in sim to hit the ball without knowledge of the mass, when deployed in real it will almost certainly fail, as the force at which it should strike the ball depends critically on the (unknown) mass. To learn the mass, however, essentially any contact between the ball and the arm suffices. Achieving some contact between the ball and the arm requires a significantly less precise motion, and indeed, does not require any prior knowledge of the mass. We can therefore train a policy in sim that learns to effectively explore—hit the ball in any direction—and deploy this in real to collect information on the true parameters, ultimately allowing us to obtain a higher-fidelity simulator that does allow sim2real transfer on the goal task.",
+        "bbox": [
+            174,
+            375,
+            821,
+            527
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We are particularly interested in the application of our pipeline to modern robotic settings and evaluate ASD on four tasks: sphere manipulation, laptop articulation, rod balancing, and shuffleboard. We show that in all settings, our approach is able to effectively identify unknown parameters of the real environment (e.g. geometry, articulation, center of mass, and physical parameters like mass, friction, or stiffness), and using this knowledge, learn a policy in sim for the goal task that successfully transfers to real. In all cases, by deploying effective exploration policies trained in simulation, we require only a very small amount of data from real—typically a single episode of data suffices.",
+        "bbox": [
+            174,
+            534,
+            821,
+            631
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK",
+        "text_level": 1,
+        "bbox": [
+            174,
+            652,
+            343,
+            667
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "System Identification: Our work is closely related to the field of system identification (Åström & Eykhoff, 1971; Söderström & Stoica, 1989; Ljung, 1998; Schon et al., 2011; Menda et al., 2020), which studies how to learn a model of the system dynamics efficiently. A large body of work, stretching back decades, has explored how inputs to a system should be chosen to most effectively learn the system's parameters (Mehra, 1974; 1976; Goodwin & Payne, 1977; Hjalmarsson et al., 1996; Lindqvist & Hjalmarsson, 2001; Gerencsér & Hjalmarsson, 2005; Rojas et al., 2007; Gevers et al., 2009; Gerencsér et al., 2009; Manchester, 2010; Rojas et al., 2011; Bombois et al., 2011; Hägg et al., 2013; Wagenmaker & Jamieson, 2020; Wagenmaker et al., 2021; Mania et al., 2022; Wagenmaker et al., 2023) or how to deal with partial observability (Schön et al., 2011; Menda et al., 2020). Similar to our exploration strategy, many of these works choose their inputs to maximize some function of the Fisher information matrix. A primary novelty of our approach is to use a simulator to learn effective exploration policies, and to apply our method to modern, real-world robotics tasks—indeed, our work can be seen as bridging the gap between classical work on system identification and modern sim2real techniques. While the aforementioned works are primarily theoretical, recent work has studied the application of such methods to a variety of real-world settings like active identification of physics parameters (Xu et al., 2019; Kumar et al., 2019; Mavrakis et al., 2020; Gao et al., 2020; 2022) or kinematic structure (Mo et al., 2021; Wang et al., 2022; Nie et al., 2022; Hsu",
+        "bbox": [
+            174,
+            686,
+            821,
+            922
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            173,
+            32,
+            478,
+            46
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "page_number",
+        "text": "2",
+        "bbox": [
+            493,
+            949,
+            503,
+            959
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "et al., 2023) through object-centric primitives. Another line of recent work aims to learn the parameters of the simulator to ultimately train a downstream policy on the learned parameters, and therefore apply task-specific policies for data collection (Zhu et al., 2018; Chebotar et al., 2019; Huang et al., 2023; Ren et al., 2023) or exploration policies that minimize its regret (Liang et al., 2020). The majority of these works, however, do not consider running an exploration policy that targets learning the unknown parameters, do not address solving downstream tasks, or rely on techniques that do not scale effectively to more complex tasks.",
+        "bbox": [
+            169,
+            103,
+            826,
+            202
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Simulation-to-Reality Transfer: Transferring learned policies from sim2real has shown to be successful in challenging tasks like dexterous manipulation (OpenAI et al., 2018; Handa et al., 2022; Chen et al., 2022), locomotion (Rudin et al., 2022), agile drone flight (Sadeghi & Levine, 2016) or contact rich assembly tasks Tang et al. (2023), yet challenges remain due to the sim2real gap. To deal with the gap, Domain Randomization (DR) (Tobin et al., 2017) trains policies over a distribution of environments in simulation, hoping for the real world to be represented among them. Subsequent works adaptively change the environment distribution (Muratore et al., 2019; Mehta et al., 2020) and incorporate real data (Chebotar et al., 2019; Ramos et al., 2019; Duan et al., 2023; Chen et al., 2023; Ma et al., 2023; Torne et al., 2024). While similar to our approach, these methods do not perform targeted exploration in real to update the simulator parameters. Other approaches seek to infer and adapt to simulation parameters during deployment (Kumar et al., 2021; Qi et al., 2023; Margolis et al., 2023), leverage offline data (Richards et al., 2021; Bose et al., 2024), or adapt online (Sinha et al., 2022); in contrast, we do not learn such an online adaptation strategy, but rather a better simulator. Finally, a commonly applied strategy is to train a policy in sim and then fine-tune in the real environment (Julian et al., 2021; Smith et al., 2022; Nakamoto et al., 2023); in contrast, we are interested in the (more challenging) regime where a direct transfer is not likely to give any learning signal to fine-tune from.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Model-Based RL: Model-based RL (MBRL) aims to solve the RL problem by learning a model of the dynamics, and using this model to either plan or solve a policy optimization problem (Deisenroth & Rasmussen, 2011; Williams et al., 2017; Nagabandi et al., 2018; Chua et al., 2018; Janner et al., 2019; Hafner et al., 2019; 2020; Janner et al., 2022; Zhu et al., 2023). While our approach is model-based in some sense, the majority of work in MBRL focuses on fully learned dynamic models; in contrast, our \"model\" is our simulator, and we aim to learn only a very small number of parameters, which can be much more sample-efficient. Furthermore, MBRL methods typically do not perform explicit exploration, while a key piece of our approach is a targeted exploration procedure. The MBRL works we are aware of which do rely on targeted exploration (Shyam et al., 2019; Pathak et al., 2019) typically rely on fully learned dynamic models and apply somewhat different exploration strategies, which we show in Appendix A.3.2 can perform significantly worse.",
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+    {
+        "type": "text",
+        "text": "3 PRELIMINARIES",
+        "text_level": 1,
+        "bbox": [
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+        "text": "We formulate our decision-making setting as Markov Decision Processes (MDPs). An MDP is defined as a tuple  $M^{\\star} = (\\mathcal{S},\\mathcal{A},\\{P_h^{\\star}\\}_{h = 1}^H,P_0,\\{r_h\\}_{h = 1}^H)$ , where  $\\mathcal{S}$  is the set of states,  $\\mathcal{A}$  the set of actions,  $P_{h}:S\\times \\mathcal{A}\\to \\triangle_{\\mathcal{S}}$  the transition kernel,  $P_0\\in \\triangle_S$  the initial state distribution, and  $r_h:\\mathcal{S}\\times \\mathcal{A}\\rightarrow \\mathbb{R}$  the reward function. We consider the episodic setting. At the beginning of an episode, the environment samples a state  $s_1\\sim P_0$ . The agent observes this state, plays some action  $a_1\\in \\mathcal{A}$ , and transitions to state  $s_2\\sim P_1(\\cdot |s_1,a_1)$ , receiving reward  $r_1(s_1,a_1)$ . After  $H$  steps, the environment resets and the process repeats. Our primary goal is to learn a policy  $\\pi$ -a mapping from states to actions—that maximizes reward in the true environment. We denote the value of a policy by  $V_0^\\pi \\coloneqq \\mathbb{E}_{M^\\star ,\\pi}[\\sum_{h = 1}^H r_h(s_h,a_h)]$ , where the expectation is over trajectories induced playing policy  $\\pi$  on MDP  $M^{\\star}$ . We think of the reward  $r$  as encoding our downstream task, and our end goal is to find a policy that solves our task, maximizing  $V_0^\\pi$ . We denote such policies as  $\\pi_{\\mathrm{task}}$ .",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the sim2real setting considered in this work, we assume that the reward is known, but that the dynamics of the real environment,  $P^{\\star} = \\{P_h^{\\star}\\}_{h=1}^H$ , are initially unknown. However, we assume that they belong to some known parametric family  $\\mathcal{P} := \\{P_{\\theta} : \\theta \\in \\Theta\\}$ , so that there exists some  $\\theta^{\\star} \\in \\Theta$  such that  $P^{\\star} = P_{\\theta^{\\star}}$ . Here we take  $\\theta$  to be some unknown parameter (for example, mass, friction, etc.), and  $P_{\\theta}$  the dynamics under parameter  $\\theta$  (which we might know from physics, first principles, etc.). For any  $\\theta$  and policy  $\\pi$ , the dynamics  $P_{\\theta}$  induce a distribution over state-action trajectories,  $\\tau = (s_1, a_1, s_2, \\dots, s_H, a_H)$ , which we denote by  $p_{\\theta}(\\cdot \\mid \\pi)$ . We can think of",
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+        "page_idx": 2
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "page_number",
+        "text": "3",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/c7767ffb66fa23029347339202edfbbb03daee2f781f15b3ae65ee20aef2538a.jpg",
+        "image_caption": [
+            "Figure 2: Overview of ASD: (1) Train an exploration policy  $\\pi_{\\mathrm{exp}}$  that maximizes the Fisher information, leveraging the vast amount of cheap simulation data. (2) Roll out  $\\pi_{\\mathrm{exp}}$  in real to collect informative data that can be used to (3) run system identification to identify physics parameters and reconstruct, e.g., geometric, collision, and kinematic properties. (4) Train a task-specific policy  $\\pi_{\\mathrm{task}}$  in the updated simulator and (5) zero-shot transfer  $\\pi_{\\mathrm{task}}$  to the real world."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "our simulator as instantiating  $p_{\\theta}(\\cdot \\mid \\pi)$ —we assume our simulator is able to accurately mimic the dynamics of an MDP with parameter  $\\pmb{\\theta}$  under policy  $\\pi$ , generating samples  $\\pmb{\\tau} \\sim p_{\\theta}(\\cdot \\mid \\pi)$ .",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In addition, we also assume that samples from our simulator are effectively \"free\"—for any  $\\theta$  and policy  $\\pi$ , we can generate as many trajectories  $\\tau \\sim p_{\\theta}(\\cdot \\mid \\pi)$  as we wish. Given this, it is possible to find the optimal policy under  $\\theta$  by simply running any standard RL algorithm in simulation. With knowledge of the true parameters  $\\theta^{\\star}$ , we can then easily find the optimal policy in real by sampling trajectories from the simulated environment with parameter  $\\theta^{\\star}$ . It follows that, if we can identify the true parameter  $\\theta^{\\star}$  in real, we can solve the goal task.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We consider the following learning protocol:",
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+    {
+        "type": "list",
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+        "list_items": [
+            "1. Learner chooses exploration policy  $\\pi_{\\mathrm{exp}}$  and plays it in real for a single episode, generating trajectory  $\\pmb{\\tau}_{\\mathrm{real}} \\sim p_{\\pmb{\\theta}^*}(\\cdot \\mid \\pi_{\\mathrm{exp}})$ .",
+            "2. Using  $\\pmb{\\tau}_{\\mathrm{real}}$  and the simulator in any way they wish, the learner obtains some policy  $\\pi_{\\mathrm{task}}$ .",
+            "3. Learner deploys  $\\pi_{\\mathrm{task}}$  in real and suffers loss  $\\max_{\\pi} V_0^{\\pi} - V_0^{\\pi_{\\mathrm{task}}}$ ."
+        ],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The goal of the learner is then to learn as much useful information as possible about the real environment from a single episode of interaction and use this information to obtain a policy that can solve the task in real as effectively as possible.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Parameter Estimation and Fisher Information: The Fisher information matrix plays a key role in the choice of our exploration policy,  $\\pi_{\\mathrm{exp}}$ . Recall that, for a distribution  $p_{\\theta}$ , satisfying certain regularity conditions, the Fisher information matrix is defined as:",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {I} (\\pmb {\\theta}) := \\mathbb {E} _ {\\pmb {\\tau} \\sim p _ {\\pmb {\\theta}}} \\left[ \\nabla_ {\\pmb {\\theta}} \\log p _ {\\pmb {\\theta}} (\\pmb {\\tau}) \\cdot \\nabla_ {\\pmb {\\theta}} \\log p _ {\\pmb {\\theta}} (\\pmb {\\tau}) ^ {\\top} \\right].\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Assume that we have access to data  $\\mathfrak{D} = (\\pmb{\\tau}_t)_{t=1}^T$ , where  $\\pmb{\\tau}_t \\sim p_{\\pmb{\\theta}^\\star}$  for  $t = 1, \\dots, T$ , and let  $\\widehat{\\pmb{\\theta}}(\\mathfrak{D})$  denote some unbiased estimator of  $\\pmb{\\theta}^\\star$ . Then the Cramer-Rao lower bound (see e.g. Pronzato & Pázman (2013)) states that, under certain regularity conditions, the covariance of  $\\widehat{\\pmb{\\theta}}(\\mathfrak{D})$  satisfies:",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbb {E} _ {\\mathfrak {D} \\sim p _ {\\theta^ {\\star}}} [ (\\widehat {\\boldsymbol {\\theta}} (\\mathfrak {D}) - \\boldsymbol {\\theta} ^ {\\star}) (\\widehat {\\boldsymbol {\\theta}} (\\mathfrak {D}) - \\boldsymbol {\\theta} ^ {\\star}) ^ {\\top} ] \\succeq T ^ {- 1} \\cdot \\mathcal {I} (\\boldsymbol {\\theta} ^ {\\star}) ^ {- 1}.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "From this it follows that the Fisher information serves as a lower bound on the mean-squared error:",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbb {E} _ {\\mathfrak {D} \\sim p _ {\\boldsymbol {\\theta} ^ {\\star}}} [ \\| \\widehat {\\boldsymbol {\\theta}} (\\mathfrak {D}) - \\boldsymbol {\\theta} ^ {\\star} \\| _ {2} ^ {2} ] = \\operatorname {t r} \\left(\\mathbb {E} _ {\\mathfrak {D} \\sim p _ {\\boldsymbol {\\theta} ^ {\\star}}} \\left[ \\left(\\widehat {\\boldsymbol {\\theta}} (\\mathfrak {D}) - \\boldsymbol {\\theta} ^ {\\star}\\right) \\left(\\widehat {\\boldsymbol {\\theta}} (\\mathfrak {D}) - \\boldsymbol {\\theta} ^ {\\star}\\right) ^ {\\top} \\right]\\right) \\geq T ^ {- 1} \\cdot \\operatorname {t r} \\left(\\mathcal {I} \\left(\\boldsymbol {\\theta} ^ {\\star}\\right) ^ {- 1}\\right). \\tag {1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This is in general tight—for example, the maximum likelihood estimator satisfies (1) with equality as  $T \\to \\infty$  (Van der Vaart, 2000). The Fisher information thus serves as a fundamental lower bound on parameter estimation error, a key motivation for our exploration procedure.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 ASID: TARGETED EXPLORATION FOR TEST-TIME SIMULATION CONSTRUCTION, IDENTIFICATION, AND POLICY OPTIMIZATION",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we present our proposed approach, ASD, a three-stage pipeline illustrated in Figure 2. We describe each component of ASD in the following.",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+    {
+        "type": "page_number",
+        "text": "4",
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+    {
+        "type": "text",
+        "text": "4.1 EXPLORATION VIA FISHER INFORMATION MAXIMIZATION",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "As motivated in Section 3, to learn a policy effectively accomplishing our task, it suffices to accurately identify  $\\theta^{\\star}$ . In the exploration phase, step 1 in our learning protocol, our goal is to then play an exploration policy  $\\pi_{\\mathrm{exp}}$  which generates a trajectory on the real environment that provides as much information on  $\\theta^{\\star}$  as possible. Following Section 3, the Fisher information gives a quantification of the usefulness of the data collected, which motivates our approach.",
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+    },
+    {
+        "type": "text",
+        "text": "In our setting, the distribution over trajectories generated during exploration in real,  $\\pmb{\\tau}_{\\mathrm{real}} \\sim p_{\\pmb{\\theta}^*}(\\cdot \\mid \\pi_{\\mathrm{exp}})$ , depends on the exploration policy,  $\\pi_{\\mathrm{exp}}$ , being played. As the Fisher information depends on the data distribution, it too scales with the choice of exploration policy:",
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {I} \\left(\\boldsymbol {\\theta} ^ {\\star}, \\pi_ {\\exp}\\right) := \\mathbb {E} _ {\\boldsymbol {\\tau} \\sim p _ {\\boldsymbol {\\theta} ^ {\\star}} (\\cdot | \\pi_ {\\exp})} \\left[ \\nabla_ {\\boldsymbol {\\theta}} \\log p _ {\\boldsymbol {\\theta} ^ {\\star}} (\\boldsymbol {\\tau} \\mid \\pi_ {\\exp}) \\cdot \\nabla_ {\\boldsymbol {\\theta}} \\log p _ {\\boldsymbol {\\theta} ^ {\\star}} (\\boldsymbol {\\tau} \\mid \\pi_ {\\exp}) ^ {\\top} \\right].\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            233,
+            253,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Following (1), if we collect trajectories by playing  $\\pi_{\\mathrm{exp}}$  and set  $\\widehat{\\pmb{\\theta}}$  to any unbiased estimator of  $\\pmb{\\theta}^{\\star}$  on these trajectories, the mean-squared error of  $\\widehat{\\pmb{\\theta}}$  will be lower bounded by  $\\operatorname{tr}(\\mathcal{I}(\\pmb{\\theta}^{\\star}, \\pi_{\\mathrm{exp}})^{-1})$ . The optimal exploration policy—the exploration policy which allows for the smallest estimation error—is, therefore, the policy which solves<sup>1</sup>",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\arg \\min  _ {\\pi} \\operatorname {t r} \\left(\\mathcal {I} \\left(\\boldsymbol {\\theta} ^ {\\star}, \\pi\\right) ^ {- 1}\\right). \\tag {2}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As an intuitive justification for this choice of exploration policy, note that the Fisher information is defined in terms of the gradient of the log-likelihood with respect to the unknown parameter. Thus, if playing some  $\\pi_{\\mathrm{exp}}$  makes the Fisher information \"large\", making  $\\operatorname{tr}(\\mathcal{I}(\\pmb{\\theta}^{\\star}, \\pi_{\\mathrm{exp}})^{-1})$  small, this suggests  $\\pi_{\\mathrm{exp}}$  induces trajectories that are very sensitive to the unknown parameters, i.e., trajectory that are significantly more likely under one set of parameters than another. By exploring to maximize the Fisher information, we, therefore, will collect trajectories that are maximally informative about the unknown parameters, since we will observe trajectories much more likely under one set of parameters than another. Motivated by this, we therefore seek to play a policy during exploration that solves (2).",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Implementing Fisher Information Maximization: In practice, several issues arise in solving (2), which we address here. First, the form of  $\\mathcal{I}(\\theta, \\pi)$  can be quite complicated, depending on the structure of  $p_{\\theta}(\\cdot \\mid \\pi)$ , and it may not be possible to efficiently obtain a solution to (2). To address this, we make a simplifying assumption on the dynamics, that our next state,  $s_{h+1}$ , evolves as:",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\ns _ {h + 1} = f _ {\\boldsymbol {\\theta}} \\left(s _ {h}, a _ {h}\\right) + w _ {h}, \\tag {3}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where  $s_h$  and  $a_h$  are the current state and action,  $w_h \\sim \\mathcal{N}(0, \\sigma_w^2 \\cdot I)$  is Gaussian process noise, and  $f_\\theta$  are the nominal dynamics. Under these dynamics, the Fisher information matrix reduces to",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {I} (\\pmb {\\theta}, \\pi) = \\sigma_ {w} ^ {- 2} \\cdot \\mathbb {E} _ {p _ {\\pmb {\\theta}} (\\cdot | \\pi)} \\left[ \\sum_ {h = 1} ^ {H} \\nabla_ {\\pmb {\\theta}} f _ {\\pmb {\\theta}} (s _ {h}, a _ {h}) \\cdot \\nabla_ {\\pmb {\\theta}} f _ {\\pmb {\\theta}} (s _ {h}, a _ {h}) ^ {\\top} \\right].\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We argue that solving (2) with this form of  $\\mathcal{I}(\\theta, \\pi)$  is a very intuitive objective, even in cases when the dynamics may not follow (3) exactly. Indeed, this suggests that during exploration, we should aim to reach states for which the dynamics  $f_{\\theta}$  have a large gradient with respect to  $\\theta$ —states for which the next state predicted by the dynamics is very sensitive to  $\\theta$ . In such states, observing the next state gives us a significant amount of information on  $\\theta$ , allowing us to accurately identify  $\\theta^{\\star}$ .",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "A second challenge in solving (2) is that we do not know the true parameter  $\\theta^{\\star}$ , which the optimization (2) depends on. To circumvent this, we rely on domain randomization in choosing our exploration policy, solving instead:",
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\pi_ {\\exp} = \\arg \\min  _ {\\pi} \\mathbb {E} _ {\\boldsymbol {\\theta} \\sim q _ {0}} [ \\operatorname {t r} (\\mathcal {I} (\\boldsymbol {\\theta}, \\pi) ^ {- 1}) ] \\tag {4}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "for some distribution over parameters  $q_0$ . While this is only an approximation of (2), in practice we find that this approximation yields effective exploration policies since, as described in Section 1, in many cases exploration policies require only a coarse model of the dynamics, and can therefore often be learned without precise knowledge of the unknown parameters.",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "A final challenge is that, in general, we may not have access to a differentiable simulator, and our dynamics themselves may not be differentiable. In such cases,  $\\nabla_{\\theta}f_{\\theta}(s_h,a_h)$  is unknown or",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "page_footnote",
+        "text": "<sup>1</sup>In the experiment design literature, this is known as an A-optimal experiment design (Pukelsheim, 2006).",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "page_number",
+        "text": "5",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "undefined, and the above approach cannot be applied. As a simple solution to this, we rely on a finite-differences approximation to the gradient, which still provides an effective measure of how sensitive the next state is to the unknown parameter. In practice, to solve (4) and obtain an exploration policy, we rely on standard policy optimization algorithms, such as PPO (Schulman et al., 2017b).",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 SYSTEM IDENTIFICATION",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "ASID runs the exploration policy  $\\pi_{\\mathrm{exp}}$  (Section 4.1) in the real environment to generate a single trajectory  $\\pmb{\\tau}_{\\mathrm{real}} \\sim p_{\\pmb{\\theta}^{\\star}}(\\cdot \\mid \\pi_{\\mathrm{exp}})$ . In the system identification phase, ASID then updates the simulator parameters using the collected trajectory. The goal is to find a distribution over simulator parameters that yield trajectories that match  $\\pmb{\\tau}_{\\mathrm{real}}$  as closely as possible. In particular, we wish to find some distribution over simulation parameters,  $q_{\\phi}$ , which minimizes:",
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbb {E} _ {\\boldsymbol {\\theta} \\sim q _ {\\phi}} \\big [ \\mathbb {E} _ {\\boldsymbol {\\tau} _ {\\mathrm {s i m}} \\sim p _ {\\boldsymbol {\\theta}} (\\cdot | \\mathcal {A} (\\boldsymbol {\\tau} _ {\\mathrm {r e a l}}))} \\big [ \\| \\boldsymbol {\\tau} _ {\\mathrm {r e a l}} - \\boldsymbol {\\tau} _ {\\mathrm {s i m}} \\| _ {2} ^ {2} \\big ] \\big ]\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+            284,
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+            303
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where  $p_{\\theta}(\\cdot \\mid \\mathcal{A}(\\tau_{\\mathrm{real}}))$  denotes the distribution over trajectories generated by the simulator with parameter  $\\theta$ , and playing the same sequence of actions as were played in  $\\tau_{\\mathrm{real}}$ . In practice, we apply REPS (Peters et al., 2010) for the simulation and Cross Entropy Method (CEM) for the real-world experiments. We stress that the ASD framework is generic, and other black-box optimization algorithms could be used instead.",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.3 SOLVING THE DOWNSTREAM TASK",
+        "text_level": 1,
+        "bbox": [
+            171,
+            400,
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+            414
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "After exploration and identification, the simulator can include information about the kinematic tree, position, orientation, or size of the object of interest, and the physical parameters of the real environment. With such a high-fidelity simulator, we aim to solve the downstream tasks entirely in simulation and transfer the learned policies  $\\pi_{\\mathrm{task}}$  to the real world in a zero-shot fashion. As with the system identification stage, ASD does not assume a particular method for solving the downstream task and any policy optimization algorithm can be used.",
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+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENTAL EVALUATION",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In our experimental evaluation, we aim to answer the following questions:",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1. Does ASD's exploration strategy yield sufficient information to identify unknown parameters?",
+            "2. Are downstream task policies learned via ASD successful when using the identified parameters?",
+            "3. Does the paradigm suggested by ASD transfer to performing tasks in the real world?"
+        ],
+        "bbox": [
+            171,
+            593,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We conduct our experimental evaluation in two scenarios. First, we conduct empirical experiments entirely in simulation (Todorov et al., 2012) to validate the behavior of the exploration, system identification, and downstream task modules of ASD. This involves treating a true simulation environment as the real-world environment and then a reconstruction of this simulation environment as the approximate constructed simulation. Policies learned in this approximate simulation can then be evaluated back in the original simulation environment to judge efficacy. Second, we apply this to two real-world manipulation tasks that are inherently dependent on accurate parameter identification, showing the ability of ASD to function in the real world, using real-world data.",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.1 ABLATIONS AND BASELINE COMPARISONS",
+        "text_level": 1,
+        "bbox": [
+            171,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "We compare ASD with several baselines and ablations for various portions of the pipeline.",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Exploration: To understand the importance of targeted exploration via Fisher information maximization we compare with two baselines. First, we compare with the naive approach of using data from a random policy for system identification that is not performing any targeted exploration. Secondly, we compare with Kumar et al. (2019), which aims to generate exploration trajectories that maximize mutual information with the parameters of interest. This method essentially learns a parameter inference network and then rewards exploration policies for minimizing its error.",
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
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+            47
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "page_number",
+        "text": "6",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/e92362ec669732740f6b8e7db813ec48adedc8d75b7c3f174cf56f5f2a34fe8b.jpg",
+        "image_caption": [
+            "(a) Rod balancing"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/697374f7e8c5fa0ed31e21a4d9f836dc63132e30c3abc6213cfba7507e1b4b0e.jpg",
+        "image_caption": [
+            "(b) Sphere manipulation"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            336,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/41f14a30bb866aa5e82f7a4e9ceb5dd22fb75be525a1022560ec047bc20c27b2.jpg",
+        "image_caption": [
+            "(c) Multi-friction",
+            "Figure 3: Depiction of environments in simulation"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            500,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/e311c93d59e4a2efe58624d13a6fd03b5bc085ac31599cf8f90d3276d909b669.jpg",
+        "image_caption": [
+            "(d) Articulation"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "System Identification: To evaluate the impact of system identification methods in this pipeline, we compare the effectiveness of using optimization-based system identification, e.g., (Peters et al., 2010; Memmel et al., 2022), with an end-to-end learned module (Kumar et al., 2019). This comparison shows the stability and extrapolation benefits of ASD over completely data-driven modeling techniques. In particular, we evaluate ASD with the optimization-based system identification replaced by the learned estimation module from Kumar et al. (2019) (ASID + estimator).",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Downstream Policy Learning: Since the eventual goal is to solve the downstream task, we finally compare how effective it is to learn downstream policies in simulation as opposed to an uninformed Domain Randomization (DR) over the parameters, in contrast to the targeted and identified parameters that stem from the ASD pipeline.",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.2 SIMULATED TASK DESCRIPTIONS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            419,
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+            431
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Sphere Manipulation: We consider two sphere manipulation tasks where physical parameters like rolling friction, object stiffness, and tangential surface friction are unknown and make a significant impact on policy performance: 1) striking a golf ball to reach a goal with unknown system friction (Figure 3b), 2) striking a golf ball in an environment with multiple patches that experience different surface friction (Figure 3c). In each scenario, the position of objects and the physical parameters are varied across evaluations. In this setting, we train downstream tasks with PPO.",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Rod Balancing: Balancing, or dynamic stacking of objects critically depends on an accurate estimate of the inertia parameters. We construct a rod balancing task where the agent can interact with a rod object (Figure 3a) to identify its physical parameters, in this case varying the distribution of mass along the rod. Once the parameters are identified, the rod must be balanced by placing it on a ledge (Figure 5 (left)). The error is measured by the tilting angle of the rod after placement, a task that requires accurate estimation of system parameters. In this case, we use the CEM to optimize downstream policies.",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Articulation: To stress generality, we consider environments that don't simply identify physical parameters like mass or friction but also the structure of the kinematic tree such as articulation and joint positioning. We consider an environment involving an articulated laptop model with a binary environment parameter representing whether articulation is present or not (Figure 3d).",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.3 DOES ASID LEARN EFFECTIVE EXPLORATION BEHAVIOR?",
+        "text_level": 1,
+        "bbox": [
+            171,
+            724,
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+            739
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To evaluate the exploration behavior quantitatively, we consider the case of multiple unknown parameters, where learning each parameter requires exploration in a different region. In particular, we compare the exploration policy learned by our method with the exploration method of Kumar et al. (2019), on the multi-friction sphere manipulation environment illustrated in Figure 3c where the surface exhibits three different friction parameters (indicated by the grass, sand, and gravel textures). We initialize the golf ball in the leftmost region (grass)—to explore, the arm must strike the ball to other regions to identify their friction parameters. We plot a heat map of the regions visited by the ball during exploration in Figure 4. As can be seen, our approach achieves roughly uniform coverage over the entire space, learning to strike the ball into each region, and illustrating that our method is able to effectively handle complex exploration tasks that involve exciting multiple parameters. In contrast, the approach of Kumar et al. (2019) does not effectively move the sphere to regions different from the starting region.",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            173,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "page_number",
+        "text": "7",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/48a4730105a54b9c086cc4b824dd62ee7e0e40670971203a69ff22e45cb1e532.jpg",
+        "table_caption": [
+            "Table 1: Downstream task results in simulation: Random exploration fails in tasks where directed exploration is required, e.g., striking a sphere or finding the inertia of a rod. When placing the rod with a single action, domain randomization (DR) cannot solve the task without knowing the physical parameters. Learned system identification (Kumar et al. (2019) and  $\\mathrm{ASID} +$  estimator) doesn't generalize to unseen trajectories and becomes far less effective than optimization-based system identification (cf.  $\\mathrm{ASID} + \\mathrm{SysID}$ )."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Task</td><td colspan=\"3\">Rod Balancing</td><td>Sphere Striking</td></tr><tr><td>Metric</td><td colspan=\"3\">Tilt angle in degree°</td><td>Success Rate in % ↑</td></tr><tr><td>Parameter</td><td>Inertia (left)</td><td>Inertia (middle)</td><td>Inertia (right)</td><td>Friction ~ [1.1, 1.5]</td></tr><tr><td>Random exploration</td><td>12.44 ± 19.6</td><td>4.20 ± 6.5</td><td>15.34 ± 15.9</td><td>10.62 ± 4.3</td></tr><tr><td>Kumar et al. (2019)</td><td>13.70 ± 9.3</td><td>2.82 ± 2.7</td><td>15.26 ± 9.8</td><td>9.50 ± 2.4</td></tr><tr><td>DR</td><td>26.69 ± 7.0</td><td>13.05 ± 7.3</td><td>1.13 ± 1.3</td><td>8.75 ± 1.5</td></tr><tr><td>ASID + estimator</td><td>17.73 ± 13.1</td><td>4.65 ± 5.1</td><td>9.99 ± 6.8</td><td>11.00 ± 5.2</td></tr><tr><td>ASID + SysID (ours)</td><td>0.00 ± 0.0</td><td>0.72 ± 1.0</td><td>0.00 ± 0.0</td><td>28.00 ± 9.7</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We further analyze the exploration behavior in the articulation environment (Figure 3d). Here our exploration policy interacts with the laptop  $80\\%$  of the time as opposed to  $20\\%$  for naïve baselines. Appendix A.2.2 shows that models trained to predict articulation from interaction data, e.g., Ditto (Jiang et al., 2022), can infer joint and part-level geometry from the collected data.",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.4 HOW DOES ASID PERFORM",
+        "text_level": 1,
+        "bbox": [
+            171,
+            397,
+            403,
+            411
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "QUANTITATIVELY IN SIMULATION ON DOWNSSTREAM TASKS?",
+        "text_level": 1,
+        "bbox": [
+            171,
+            411,
+            594,
+            425
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To quantitatively assess the benefits of ASD over baselines and ablations, we evaluate the performance of ASD with respect to the baselines in simulation on downstream evaluation for the rod balancing (Figure 3a) and sphere manipulation (Figure 3c) task. Results are given in Table 1. Observing the performance on the rod environment, it becomes clear that previous methods struggle with one-shot tasks. When the inertia is not properly identified, CEM predicts the wrong action with high certainty, causing it to pick and place the rod at the wrong location. Domain randomization (DR) on the other hand tries to learn a policy over all possible mass distributions which in this case leads to a policy that always grasps the object at the center of the reference frame. The success here depends on \"getting lucky\" and sampling the correct action for the",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/0c40718841ba689244bec07cb2924d2a4c39d4648a06a4c5472109abf931b90c.jpg",
+        "image_caption": [
+            "(a) ASID"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            629,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/8b6010dee3c3fba7b8e3ef7f5d25d11e90b565e5ce7dd628abb5d5afae3f9a63.jpg",
+        "image_caption": [
+            "(b) Kumar et al. (2019)",
+            "Figure 4: Visitation frequency of the sphere when explored by different exploration policies on multi-friction (Figure 3c). ASD activates the sphere over a much larger area, thereby identifying parameters more accurately"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            629,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In the sphere environment, the distance rolled and bounce of the sphere are critically dependent on parameters such as friction and stiffness, and misidentifying these parameters can lead to significantly different trajectories and unsuccessful downstream behavior. This is reflected in the significantly higher success rate of ASD as compared to baselines that are uninformed of parameters and simply optimize for robust policies (DR), those that try to use learned estimators for exploration (Kumar et al., 2019) or using random exploration.",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.5 DOES ASID ALLOW FOR REAL-WORLD CONTROLLER SYNTHESIS USING MINIMAL REAL-WORLD DATA?",
+        "text_level": 1,
+        "bbox": [
+            171,
+            728,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We further evaluate ASD on real-world tasks, replicating the rod balancing task from simulation (Figure 5) and introducing a novel shuffleboard task (Figure 6). As in simulation, we use a Franka Emika Panda robot for exploration and task performance in the real world. We compute the object's position and orientation by color-threshold the pointclouds from two calibrated Intel RealSense D455 cameras—this approach could easily be replaced by a more sophisticated pose estimation system if desired.",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Rod Balancing: The goal of this task is to properly balance a rod with an unknown mass distribution by informatively exploring the environment in the real world and using the resulting data to identify the appropriate physics parameters in simulation. The optimized controller in simulation is then deployed to perform the task in the real world. In this case, the downstream task is picking the rod at a certain point along its axis and balancing it on a perch (Figure 1, Figure 5).",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "page_number",
+        "text": "8",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The policy executes the downstream task by a pick and place primitive parameterized by the exact pick point. We deploy ASD fully autonomously, executing exploration, system identification, downstream task training, and execution in an end-to-end fashion.",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The mass distribution in the rod is varied and both the inertia and the environment friction must be inferred. While ASD correctly identifies the inertia of the rod most of the time, we find that a center of mass close to the middle of the rod causes ambiguities that hurt our system identification process causing the simulation to not be accurate enough for zero-shot transfer. Overall, ASD solves the rod-balancing task 6/9 times across varying mass distribution and pose while a domain-randomization policy trained without any environment interaction fails to solve it at all (Table 2).",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Shuffleboard: This task is a scaled-down version of the popular bar game tabletop shuffle-",
+        "bbox": [
+            169,
+            381,
+            483,
+            409
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/97ead37ff4566571d10a86f2f5d049351f0c7ce96358a8e6e13d9e8dd95e06bd.jpg",
+        "image_caption": [
+            "Figure 5: Real-world Rod Balancing: Simulation setup for training exploration and downstream task policies (left). Successful execution of autonomous real-world rod balancing with skewed mass (right)."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            498,
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+            205
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/54acf3f7512ce9d910ff1d6f7d25bdab059e9cc0eba7c237c035d191e8756879.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/aed5e0a4572fc2e2088b158f2898545b1fec35df1786ff5ea5554f9f3363dc9b.jpg",
+        "table_caption": [
+            "Table 2: Downstream task results in real: ASD successfully balances the rod while domain randomization (DR) fails catastrophically."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Task</td><td colspan=\"3\">Rod Balancing</td></tr><tr><td>Inertia</td><td>left</td><td>middle</td><td>right</td></tr><tr><td>DR</td><td>0/3</td><td>0/3</td><td>0/3</td></tr><tr><td>ASID (ours)</td><td>2/3</td><td>1/3</td><td>3/3</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "board where a puck must be shot to a target area on a slippery board. We closely follow the original game and pour wax (sand) on the board to decrease the surface friction. This modification makes the task especially difficult as the surface friction on the board changes slightly after each shot since the puck displaces the wax. The goal is to strike the puck to one of the target regions—one closer (yellow) and one further (blue) away from the robot (Figure 2, Figure 6). After exploring the scene, ASD estimates the sliding and torsional friction of the puck to account for the changing surface friction of the board. For executing the downstream task, a primitive positions the endeffector at a fixed distance to the puck and a policy predicts a force value that parameterizes a shot attempt.",
+        "bbox": [
+            169,
+            409,
+            823,
+            521
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Due to the changing surface friction, the domain randomization baseline struggles to shoot the puck to the desired zone. While it succeeds 3/10 times—probably because the surface friction was part of its training distribution—the other attempts fall short or overshoot the target. With its dedicated exploration phase, ASD can accurately adapt the simulation to the current conditions and lands the puck in the desired zone 7/10 times (Table 3).",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 DISCUSSION",
+        "text_level": 1,
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this work, we introduced a novel pipeline for performing real-world control by autonomously exploring the environment, using the collected data for system identification, and the resulting identified system for downstream policy optimization. In essence, this sim-to-real-to-sim-to-real pipeline allows for targeted test-time construction of simulation environments in a way that enables the performance of downstream tasks. We demonstrate how this",
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+    },
+    {
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+        "img_path": "images/f9b8c663ae723771e27fc999493f3fe12a12c45a0f4dbe4e7fd23cb988daca86.jpg",
+        "image_caption": [
+            "Figure 6: Real-world Shuffleboard: Simulation setup for training exploration and downstream task policies (left). Successful strike to the yellow zone (right)."
+        ],
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+        "table_caption": [
+            "Table 3: Downstream task results in real: ASID outperforms domain randomization (DR) on shooting the puck to the desired zones."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Task</td><td colspan=\"2\">Shuffleboard</td></tr><tr><td>Target zone</td><td>yellow (close)</td><td>blue (far)</td></tr><tr><td>DR</td><td>2/5</td><td>1/5</td></tr><tr><td>ASID (ours)</td><td>4/5</td><td>3/5</td></tr></table>",
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+        "type": "text",
+        "text": "type of identification requires autonomous and carefully directed exploration, and introduce a novel algorithm based on Fisher information maximization that is able to accomplish this directed exploration. The autonomously collected trajectories can then be paired with downstream optimization-based system identification and reconstruction algorithms for accurate simulation construction and downstream policy learning. We show the efficacy of this paradigm on multiple environments in simulation, as well as on rod balancing and shuffleboard, two challenging real-world tasks.",
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+        "text": "Published as a conference paper at ICLR 2024",
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+    {
+        "type": "text",
+        "text": "REFERENCES",
+        "text_level": 1,
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+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+        ],
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "page_number",
+        "text": "14",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A APPENDIX",
+        "text_level": 1,
+        "bbox": [
+            171,
+            102,
+            299,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.1 TASKDETAILS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            133,
+            323,
+            148
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.1.1 SPHERE MANIPULATION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            159,
+            403,
+            174
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In the sphere manipulation tasks, the observation space consists of endeffector position, sphere position, and robot joint angles. During the training of the exploration policy, we randomize the location of a sphere with  $r = 0.03$  to be between  $x \\in [0.430, 0.65]$ ,  $y \\in [-0.2, 0.23]$  for training and  $x \\in [0.55, 0.65]$ ,  $y \\in [-0.2, 0.23]$  for evaluation. Parameter ranges for training are 1) friction  $\\theta \\in [1e - 3, 5e - 3]$  and 2) patch friction  $\\theta_0, \\theta_1, \\theta_2 \\in [1e - 5, 1e - 3]$  and set fixed sphere and goal locations as well as parameters for the downstream task. For 1) we attach a paddle and limit the endeffector position such that the robot has to strike the sphere to the goal.",
+        "bbox": [
+            169,
+            183,
+            826,
+            282
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.1.2 ARTICULATION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            296,
+            339,
+            311
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In the case of the articulation environment, the parameter is binary and indicates whether articulation is present or not. During evaluation, articulation is always turned on and success is indicated by  $\\delta \\beta > 1e - 2$ . The initial laptop state is randomized over position  $x \\in [0.45, 0.65]$ ,  $y \\in [-0.1, 0.1]$ ,  $yaw \\in [0.00, 3.14]$  and lid angle  $\\beta \\in [1.2, 2.5]$ . The observation space contains the endeffector position, joint angles, position, orientation, and joint angle of the laptop.",
+        "bbox": [
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+            391
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.1.3 ROD BALANCING",
+        "text_level": 1,
+        "bbox": [
+            171,
+            406,
+            354,
+            420
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Similar to the sphere manipulation, we restrict the exploration policy to endeffector control with  $\\delta x, \\delta y$  and attach a peg to the Franka. The rod has dimensions  $0.04 \\times 0.3 \\times 0.04$  and initializes as  $x \\in [0.5, 0.6], y \\in [-0.2, 0.2], yaw \\in [0.00, 3.14]$",
+        "bbox": [
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+            430,
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+            474
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+            47
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "page_number",
+        "text": "15",
+        "bbox": [
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+            946,
+            508,
+            960
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.2 QUALITATIVE RESULTS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            104,
+            379,
+            118
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "A.2.1 EXPLORATION STRATEGIES LEARNED BY THE AGENT",
+        "text_level": 1,
+        "bbox": [
+            171,
+            131,
+            602,
+            145
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "We evaluate the exploration behavior qualitatively across multiple environments to understand whether the exploration behavior is meaningful. In the sphere environment, we observe the agent hitting the sphere multiple times when it bounces off the walls or stays within reach. The other environments also experience emergent behavior, as the agent executes a horizontal motion towards the top of the lid if the laptop is mostly open while an almost closed laptop causes it to push from top to bottom instead. Finally, when determining the rod's inertia, the policy pushes both sides to gain maximum information about the center of mass through the rotation motions in both directions. See Figure 7 for a visualization of the exploration for the sphere environment.",
+        "bbox": [
+            169,
+            157,
+            826,
+            270
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/88e67f76b3dcac6ed7fe3a4e058f7ddf1cb022d93485399766847d94e4a9b17c.jpg",
+        "image_caption": [
+            "(a)"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            301,
+            380,
+            393
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/b2c8249d46910b95a810fa027d282907f3608060d880abc644df4e85f57cd571.jpg",
+        "image_caption": [
+            "(b)",
+            "Figure 7: Absolute sphere displacement for different sphere starting locations. Zero means the sphere didn't get hit, higher numbers denote larger displacements. Initial endeffector position marked in red a) Random Coverage, b) PPO Coverage, c) Fisher Coverage, d) Legend."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            383,
+            301,
+            578,
+            393
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/a7a7ff719430050f1d379aa117982acee7b77226230dc727ec36f90e676311ed.jpg",
+        "image_caption": [
+            "(c)",
+            "(d)"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            581,
+            301,
+            810,
+            393
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "A.2.2 RECONSTRUCTING ARTICULATION FROM INTERACTION DATA",
+        "text_level": 1,
+        "bbox": [
+            171,
+            508,
+            656,
+            522
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/d86c3bd2c58a00e28a097c6008a5bfa708613a3ce450ddb9d779b82c8ca39d2e.jpg",
+        "image_caption": [
+            "Figure 8: Articulated object: before (left) and after (middle) exploration with ASD and part-level reconstruction with Ditto (Jiang et al., 2022): articulated  $\\blacksquare$  and static part  $\\blacksquare$  (right)."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            542,
+            339,
+            627
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/cc025e6a51187e73099ace305b85fbec9d46d5e729ea1d9f081f1e292970d4cf.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/b6ddffa2c57a5a4ae9f1223409d84cefbe36df9300c3304b72f3d0d3d5aa3c05.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            707,
+            546,
+            823,
+            625
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "A.3 COMPARISON TO MODEL-BASED APPROACHES",
+        "text_level": 1,
+        "bbox": [
+            171,
+            691,
+            539,
+            705
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "A.3.1 ENVIRONMENT SETUP",
+        "text_level": 1,
+        "bbox": [
+            171,
+            720,
+            390,
+            733
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/c7328bb6cec3777cc27662812a4da8d072f749241e6a6bd00ee4d48a9f238ff0.jpg",
+        "image_caption": [
+            "Figure 9: Environment with multiple spheres. The single red sphere is subject to changing friction values while the three blue spheres act as distractors."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            398,
+            753,
+            596,
+            869
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "page_number",
+        "text": "16",
+        "bbox": [
+            490,
+            948,
+            508,
+            959
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "A.3.2 MODEL-BASED EXPLORATION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            104,
+            442,
+            118
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "In this section, Figure 10, we compare the performance of our algorithm to that of the MAX algorithm (Shyam et al., 2019).",
+        "bbox": [
+            169,
+            127,
+            823,
+            157
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "MAX aims to cover the state space through multiple policies throughout training. Since it uses the disagreement between fully learned dynamics models, it gets distracted by novel states induced by the movement of all spheres (red and blue). In contrast, our method based on the Fisher information yields a single policy that seeks out the sphere affected by the changing physics parameters (red) and ignores the irrelevant spheres (blue) even if they lead to novel states.",
+        "bbox": [
+            169,
+            162,
+            826,
+            233
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/42cc6b51bf4a68edaf9d18fcb55084cd20c81045542b5b0c2f6c2edd500418d3.jpg",
+        "image_caption": [
+            "(a) MAX (checkpoints throughout training)"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            200,
+            263,
+            477,
+            422
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/28e5f461d8253e375093814e1f8991e08e868d30966e6170ba0cb4c6957de18a.jpg",
+        "image_caption": [
+            "(b) MAX (last checkpoints)"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            519,
+            263,
+            795,
+            421
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/d15278b0ba3718195425ac294d65e03120c64ef996de0ca1902a4da193d430b2.jpg",
+        "image_caption": [
+            "(c) Fisher information (checkpoints throughout training)"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            199,
+            469,
+            475,
+            627
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/124b7c407801a03b46742668c6e148ce184a9ac8dadf85813748a1b9cd5cc427.jpg",
+        "image_caption": [
+            "(d) Fisher information (last checkpoint)",
+            "Figure 10: Exploration MAX vs. Fisher information: Trajectories collected (30) from policy checkpoints throughout training or from the last policy checkpoint. While Model-based Active Exploration (MAX) (Shyam et al., 2019) explores the state space over the course of training, getting distracted by the novel states induced by the blue spheres  $\\blacksquare$  (c.f. fig. 10a), Fisher information-based exploration (ours) shows directed exploration (c.f. fig. 10d), moving towards the sphere with changing parameters (red  $\\blacksquare$ ) and yielding a single exploration policy."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            519,
+            479,
+            795,
+            638
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "page_number",
+        "text": "17",
+        "bbox": [
+            490,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "A.3.3 LEARNED STATE-TRANSITION MODELS VS. SIMULATORS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            104,
+            624,
+            118
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "We next illustrate the improvement using a simulator vs a fully learned dynamics model can give. We train a forward dynamics model (three layer MLP) on data generated both from MAX and our exploration procedure. When evaluated on out-of-distribution trajectories, i.e., trajectories not included in the training data, we find the model to be extremely inaccurate (see Figure 11). While the simulator extrapolates to unseen states and correctly predicts the movement of the sphere, the model hallucinates movement even when the end-effector does not interact with it at all! These findings make ASID preferable to a purely model-based approach.",
+        "bbox": [
+            169,
+            128,
+            826,
+            227
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/bfd00204d235cf2a3c9177a6a5c7a944551fcf20cb315ae56cf35d87e36cc208.jpg",
+        "image_caption": [
+            "(a) MAX (pushing actions)"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            200,
+            257,
+            472,
+            415
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/dfcecb9588b5fee519beeeaba891b25e722176fe38ca51c1775889130449ef55.jpg",
+        "image_caption": [
+            "(b) MAX (zero actions)"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            519,
+            257,
+            790,
+            415
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/64d2a64cfab890a9bb5440d13cf2b56bce855742c7e4e10b5e4b4363f7a13fce.jpg",
+        "image_caption": [
+            "(c) Fisher information (pushing actions)"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            199,
+            462,
+            470,
+            619
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/c90f5215d9859b8479126a23a040eb3c982b0408e8f6bba1c854e99bb458604e.jpg",
+        "image_caption": [
+            "(d) Fisher information (zero actions)",
+            "Figure 11: Learned models vs. simulator: Evaluation of a forward model trained on trajectories (30) from MAX (Shyam et al., 2019) (checkpoints throughout training) and Fisher information (last checkpoint). In contrast to our simulator, the learned model fails to predict the red sphere's trajectory accurately even under no contact scenarios (c.f. fig. 11b, fig. 11d)."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            519,
+            462,
+            790,
+            619
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "page_number",
+        "text": "18",
+        "bbox": [
+            490,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "A.4 POLICY GENERALIZATION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            104,
+            403,
+            118
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "We evaluate the generalization capabilities of our exploration policy for different sphere locations and parameter combinations seen and unseen during training (Figure 13). Since we train the policies using an arena setup, we remove the arena during this evaluation to be able to query sphere locations outside of it (Figure 12).",
+        "bbox": [
+            169,
+            128,
+            826,
+            186
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/0a66e7e74d783e56c6e303d23d942a3a67b0149d34f3648708a61fbd50b31157.jpg",
+        "image_caption": [
+            "Figure 12: Environment with free space. The red sphere is subject to changing parameter values and position is randomized."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            398,
+            198,
+            598,
+            314
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/0be412378c5a9e3a00a2a7339350ce0bc846481b2cf693a9e11d163976eddfe3.jpg",
+        "image_caption": [
+            "(a) Changing friction parameters (in distribution)"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            400,
+            472,
+            558
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/582062e3989358cdc7aa689d662b78257d4bac60b320c6b11641ce8c93a6c47e.jpg",
+        "image_caption": [
+            "(b) Changing mass parameter (out of distribution)",
+            "Figure 13: Generalization capabilities of an exploration policy trained on sphere friction for different sphere starting locations. The plot shows the success rate of pushing the sphere spawned at the corresponding x-y-location and evaluated over 5 different seeds, i.e., different physics parameter values. Initial endeffector position marked in red. Boxes denote sphere locations seen during training (purple box) and reachable area of the endeffector (orange box). Our policies experience surprising generalization capabilities to unseen sphere locations as long as they can be reached by the robot (cf.fig. 13a). Furthermore, the actions extrapolate to unseen physics parameters as long as they can be uncovered with similar interactions, e.g., pushing (cf.fig. 13b). This is the case because the policy does not have any information about the underlying physics parameters until it hits the sphere."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            500,
+            401,
+            792,
+            558
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "page_number",
+        "text": "19",
+        "bbox": [
+            490,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 18
+    }
+]
\ No newline at end of file
diff --git a/2024/ASID_ Active Exploration for System Identification in Robotic Manipulation/22c89e0b-1caf-4875-86e0-f3b9397a36cc_model.json b/2024/ASID_ Active Exploration for System Identification in Robotic Manipulation/22c89e0b-1caf-4875-86e0-f3b9397a36cc_model.json
new file mode 100644
index 0000000000000000000000000000000000000000..bbbae42bd51f4b05e2ec39838384b4e4136c3779
--- /dev/null
+++ b/2024/ASID_ Active Exploration for System Identification in Robotic Manipulation/22c89e0b-1caf-4875-86e0-f3b9397a36cc_model.json	
@@ -0,0 +1,3516 @@
+[
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.1,
+                0.825,
+                0.147
+            ],
+            "angle": 0,
+            "content": "ASID: ACTIVE EXPLORATION FOR SYSTEM IDENTIFICATION IN ROBOTIC MANIPULATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.181,
+                0.17,
+                0.77,
+                0.185
+            ],
+            "angle": 0,
+            "content": "Marius Memmel, Andrew Wagenvaker, Chuning Zhu, Dieter Fox, Abhishek Gupta"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.184,
+                0.186,
+                0.567,
+                0.199
+            ],
+            "angle": 0,
+            "content": "Paul G. Allen School of Computer Science & Engineering"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.184,
+                0.2,
+                0.354,
+                0.213
+            ],
+            "angle": 0,
+            "content": "University of Washington"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.184,
+                0.213,
+                0.35,
+                0.226
+            ],
+            "angle": 0,
+            "content": "Seattle, WA 98195, USA"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.184,
+                0.227,
+                0.75,
+                0.241
+            ],
+            "angle": 0,
+            "content": "{memmelma,ajwagen,zchuning,fox,abhgupta}@cs.washington.edu"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.451,
+                0.278,
+                0.547,
+                0.292
+            ],
+            "angle": 0,
+            "content": "ABSTRACT"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.23,
+                0.307,
+                0.77,
+                0.584
+            ],
+            "angle": 0,
+            "content": "Model-free control strategies such as reinforcement learning have shown the ability to learn control strategies without requiring an accurate model or simulator of the world. While this is appealing due to the lack of modeling requirements, such methods can be sample inefficient, making them impractical in many real-world domains. On the other hand, model-based control techniques leveraging accurate simulators can circumvent these challenges and use a large amount of cheap simulation data to learn controllers that can effectively transfer to the real world. The challenge with such model-based techniques is the requirement for an extremely accurate simulation, requiring both the specification of appropriate simulation assets and physical parameters. This requires considerable human effort to design for every environment being considered. In this work, we propose a learning system that can leverage a small amount of real-world data to autonomously refine a simulation model and then plan an accurate control strategy that can be deployed in the real world. Our approach critically relies on utilizing an initial (possibly inaccurate) simulator to design effective exploration policies that, when deployed in the real world, collect high-quality data. We demonstrate the efficacy of this paradigm in identifying articulation, mass, and other physical parameters in several challenging robotic manipulation tasks, and illustrate that only a small amount of real-world data can allow for effective sim-to-real transfer. Project website at https://weirdlabuw.github.io/asid"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.174,
+                0.609,
+                0.339,
+                0.624
+            ],
+            "angle": 0,
+            "content": "1 INTRODUCTION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.639,
+                0.486,
+                0.875
+            ],
+            "angle": 0,
+            "content": "Controlling robots to perform dynamic, goal-directed behavior in the real world is challenging. Reinforcement Learning (RL) has emerged as a promising technique to learn such behaviors without requiring known models of the environment, instead relying on data sampled directly from the environment (Schulman et al., 2017a; Haarnoja et al., 2018). In principle, these techniques can be deployed in new environments with a minimal amount of human effort, and allow for continual improvement of behavior. Such techniques have been shown to successfully learn complex behaviors in a variety of scenarios, ranging from table-top manipulation (Yu et al., 2020) to locomotion (Hwangbo et al., 2019) and even dexterous manipulation (Zhu et al., 2019)."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.498,
+                0.608,
+                0.825,
+                0.822
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.495,
+                0.833,
+                0.825,
+                0.873
+            ],
+            "angle": 0,
+            "content": "Figure 1: ASD: A depiction of our proposed process of active exploration for system identification, from learning exploration policies to real-world deployment."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.882,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "However, these capabilities come at a cost, requiring full access to the environment in order to design reset mechanisms and reward functions. Such requirements necessitate training these methods in carefully controlled and often curated environments, limiting their applicability. Moreover, de"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.495,
+                0.949,
+                0.504,
+                0.96
+            ],
+            "angle": 0,
+            "content": "1"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.034,
+                0.479,
+                0.047
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.175,
+                0.104,
+                0.823,
+                0.132
+            ],
+            "angle": 0,
+            "content": "ploying tabula-asa reinforcement learning methods in the real world often requires a prohibitively large number of samples, which may be impractical to collect."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.175,
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+                0.823,
+                0.209
+            ],
+            "angle": 0,
+            "content": "One approach to circumvent these challenges is to rely on simulators to cheaply generate large amounts of data and use RL to train a policy. However, directly deploying policies trained in simulation in the real world is often ineffective due to the discrepancy between the simulation and the real world, the so-called sim2real gap. For example, the physical properties and parameters of the simulation might diverge from the real world, rendering a simulation-trained policy useless in reality."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.175,
+                0.216,
+                0.823,
+                0.369
+            ],
+            "angle": 0,
+            "content": "Taking inspiration from system identification, we argue that the key to effective sim2real transfer is an initial round of exploration in the real world to learn an effective simulator. We propose a generic pipeline for sim2real transfer, Active Exploration for System IDentification (ASID), which decouples exploration and exploitation: (1) exploration in real to collect informative data of unknown parameters, (2) refinement of the simulation parameters using data collected in real, (3) policy training on the updated simulator to accomplish the goal tasks. Our exploration procedure is motivated by work in theoretical statistics and seeks to induce trajectories corresponding to large Fisher information, thereby providing maximally informative observations. By using our initial round of exploration to obtain accurate estimates of the parameters in the real world, we show that in many cases, the policies trained in step (3) successfully transfer to real in a zero-shot fashion, even in settings where training a policy in sim without additional knowledge of real would fail."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.175,
+                0.376,
+                0.823,
+                0.528
+            ],
+            "angle": 0,
+            "content": "A key insight in our approach is that, while a policy trained in sim to accomplish the goal task may not effectively transfer, strategies that explore effectively in sim often also explore effectively in real. As an example, say our goal task is to hit a ball to a particular location with a robotic arm and assume the mass of the ball is unknown. If we train a policy in sim to hit the ball without knowledge of the mass, when deployed in real it will almost certainly fail, as the force at which it should strike the ball depends critically on the (unknown) mass. To learn the mass, however, essentially any contact between the ball and the arm suffices. Achieving some contact between the ball and the arm requires a significantly less precise motion, and indeed, does not require any prior knowledge of the mass. We can therefore train a policy in sim that learns to effectively explore—hit the ball in any direction—and deploy this in real to collect information on the true parameters, ultimately allowing us to obtain a higher-fidelity simulator that does allow sim2real transfer on the goal task."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.175,
+                0.535,
+                0.823,
+                0.632
+            ],
+            "angle": 0,
+            "content": "We are particularly interested in the application of our pipeline to modern robotic settings and evaluate ASD on four tasks: sphere manipulation, laptop articulation, rod balancing, and shuffleboard. We show that in all settings, our approach is able to effectively identify unknown parameters of the real environment (e.g. geometry, articulation, center of mass, and physical parameters like mass, friction, or stiffness), and using this knowledge, learn a policy in sim for the goal task that successfully transfers to real. In all cases, by deploying effective exploration policies trained in simulation, we require only a very small amount of data from real—typically a single episode of data suffices."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.175,
+                0.653,
+                0.344,
+                0.669
+            ],
+            "angle": 0,
+            "content": "2 RELATED WORK"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.175,
+                0.687,
+                0.823,
+                0.923
+            ],
+            "angle": 0,
+            "content": "System Identification: Our work is closely related to the field of system identification (Åström & Eykhoff, 1971; Söderström & Stoica, 1989; Ljung, 1998; Schon et al., 2011; Menda et al., 2020), which studies how to learn a model of the system dynamics efficiently. A large body of work, stretching back decades, has explored how inputs to a system should be chosen to most effectively learn the system's parameters (Mehra, 1974; 1976; Goodwin & Payne, 1977; Hjalmarsson et al., 1996; Lindqvist & Hjalmarsson, 2001; Gerencsér & Hjalmarsson, 2005; Rojas et al., 2007; Gevers et al., 2009; Gerencsér et al., 2009; Manchester, 2010; Rojas et al., 2011; Bombois et al., 2011; Hägg et al., 2013; Wagenmaker & Jamieson, 2020; Wagenmaker et al., 2021; Mania et al., 2022; Wagenmaker et al., 2023) or how to deal with partial observability (Schön et al., 2011; Menda et al., 2020). Similar to our exploration strategy, many of these works choose their inputs to maximize some function of the Fisher information matrix. A primary novelty of our approach is to use a simulator to learn effective exploration policies, and to apply our method to modern, real-world robotics tasks—indeed, our work can be seen as bridging the gap between classical work on system identification and modern sim2real techniques. While the aforementioned works are primarily theoretical, recent work has studied the application of such methods to a variety of real-world settings like active identification of physics parameters (Xu et al., 2019; Kumar et al., 2019; Mavrakis et al., 2020; Gao et al., 2020; 2022) or kinematic structure (Mo et al., 2021; Wang et al., 2022; Nie et al., 2022; Hsu"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.495,
+                0.95,
+                0.504,
+                0.96
+            ],
+            "angle": 0,
+            "content": "2"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.104,
+                0.827,
+                0.203
+            ],
+            "angle": 0,
+            "content": "et al., 2023) through object-centric primitives. Another line of recent work aims to learn the parameters of the simulator to ultimately train a downstream policy on the learned parameters, and therefore apply task-specific policies for data collection (Zhu et al., 2018; Chebotar et al., 2019; Huang et al., 2023; Ren et al., 2023) or exploration policies that minimize its regret (Liang et al., 2020). The majority of these works, however, do not consider running an exploration policy that targets learning the unknown parameters, do not address solving downstream tasks, or rely on techniques that do not scale effectively to more complex tasks."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.174,
+                0.209,
+                0.827,
+                0.446
+            ],
+            "angle": 0,
+            "content": "Simulation-to-Reality Transfer: Transferring learned policies from sim2real has shown to be successful in challenging tasks like dexterous manipulation (OpenAI et al., 2018; Handa et al., 2022; Chen et al., 2022), locomotion (Rudin et al., 2022), agile drone flight (Sadeghi & Levine, 2016) or contact rich assembly tasks Tang et al. (2023), yet challenges remain due to the sim2real gap. To deal with the gap, Domain Randomization (DR) (Tobin et al., 2017) trains policies over a distribution of environments in simulation, hoping for the real world to be represented among them. Subsequent works adaptively change the environment distribution (Muratore et al., 2019; Mehta et al., 2020) and incorporate real data (Chebotar et al., 2019; Ramos et al., 2019; Duan et al., 2023; Chen et al., 2023; Ma et al., 2023; Torne et al., 2024). While similar to our approach, these methods do not perform targeted exploration in real to update the simulator parameters. Other approaches seek to infer and adapt to simulation parameters during deployment (Kumar et al., 2021; Qi et al., 2023; Margolis et al., 2023), leverage offline data (Richards et al., 2021; Bose et al., 2024), or adapt online (Sinha et al., 2022); in contrast, we do not learn such an online adaptation strategy, but rather a better simulator. Finally, a commonly applied strategy is to train a policy in sim and then fine-tune in the real environment (Julian et al., 2021; Smith et al., 2022; Nakamoto et al., 2023); in contrast, we are interested in the (more challenging) regime where a direct transfer is not likely to give any learning signal to fine-tune from."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.451,
+                0.828,
+                0.606
+            ],
+            "angle": 0,
+            "content": "Model-Based RL: Model-based RL (MBRL) aims to solve the RL problem by learning a model of the dynamics, and using this model to either plan or solve a policy optimization problem (Deisenroth & Rasmussen, 2011; Williams et al., 2017; Nagabandi et al., 2018; Chua et al., 2018; Janner et al., 2019; Hafner et al., 2019; 2020; Janner et al., 2022; Zhu et al., 2023). While our approach is model-based in some sense, the majority of work in MBRL focuses on fully learned dynamic models; in contrast, our \"model\" is our simulator, and we aim to learn only a very small number of parameters, which can be much more sample-efficient. Furthermore, MBRL methods typically do not perform explicit exploration, while a key piece of our approach is a targeted exploration procedure. The MBRL works we are aware of which do rely on targeted exploration (Shyam et al., 2019; Pathak et al., 2019) typically rely on fully learned dynamic models and apply somewhat different exploration strategies, which we show in Appendix A.3.2 can perform significantly worse."
+        },
+        {
+            "type": "title",
+            "bbox": [
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+            "angle": 0,
+            "content": "3 PRELIMINARIES"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
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+            ],
+            "angle": 0,
+            "content": "We formulate our decision-making setting as Markov Decision Processes (MDPs). An MDP is defined as a tuple \\(M^{\\star} = (\\mathcal{S},\\mathcal{A},\\{P_h^{\\star}\\}_{h = 1}^H,P_0,\\{r_h\\}_{h = 1}^H)\\), where \\(\\mathcal{S}\\) is the set of states, \\(\\mathcal{A}\\) the set of actions, \\(P_{h}:S\\times \\mathcal{A}\\to \\triangle_{\\mathcal{S}}\\) the transition kernel, \\(P_0\\in \\triangle_S\\) the initial state distribution, and \\(r_h:\\mathcal{S}\\times \\mathcal{A}\\rightarrow \\mathbb{R}\\) the reward function. We consider the episodic setting. At the beginning of an episode, the environment samples a state \\(s_1\\sim P_0\\). The agent observes this state, plays some action \\(a_1\\in \\mathcal{A}\\), and transitions to state \\(s_2\\sim P_1(\\cdot |s_1,a_1)\\), receiving reward \\(r_1(s_1,a_1)\\). After \\(H\\) steps, the environment resets and the process repeats. Our primary goal is to learn a policy \\(\\pi\\)-a mapping from states to actions—that maximizes reward in the true environment. We denote the value of a policy by \\(V_0^\\pi \\coloneqq \\mathbb{E}_{M^\\star ,\\pi}[\\sum_{h = 1}^H r_h(s_h,a_h)]\\), where the expectation is over trajectories induced playing policy \\(\\pi\\) on MDP \\(M^{\\star}\\). We think of the reward \\(r\\) as encoding our downstream task, and our end goal is to find a policy that solves our task, maximizing \\(V_0^\\pi\\). We denote such policies as \\(\\pi_{\\mathrm{task}}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.827,
+                0.828,
+                0.928
+            ],
+            "angle": 0,
+            "content": "In the sim2real setting considered in this work, we assume that the reward is known, but that the dynamics of the real environment, \\( P^{\\star} = \\{P_h^{\\star}\\}_{h=1}^H \\), are initially unknown. However, we assume that they belong to some known parametric family \\( \\mathcal{P} := \\{P_{\\theta} : \\theta \\in \\Theta\\} \\), so that there exists some \\( \\theta^{\\star} \\in \\Theta \\) such that \\( P^{\\star} = P_{\\theta^{\\star}} \\). Here we take \\( \\theta \\) to be some unknown parameter (for example, mass, friction, etc.), and \\( P_{\\theta} \\) the dynamics under parameter \\( \\theta \\) (which we might know from physics, first principles, etc.). For any \\( \\theta \\) and policy \\( \\pi \\), the dynamics \\( P_{\\theta} \\) induce a distribution over state-action trajectories, \\( \\tau = (s_1, a_1, s_2, \\dots, s_H, a_H) \\), which we denote by \\( p_{\\theta}(\\cdot \\mid \\pi) \\). We can think of"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.506,
+                0.96
+            ],
+            "angle": 0,
+            "content": "3"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.174,
+                0.101,
+                0.824,
+                0.219
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.171,
+                0.228,
+                0.825,
+                0.292
+            ],
+            "angle": 0,
+            "content": "Figure 2: Overview of ASD: (1) Train an exploration policy \\(\\pi_{\\mathrm{exp}}\\) that maximizes the Fisher information, leveraging the vast amount of cheap simulation data. (2) Roll out \\(\\pi_{\\mathrm{exp}}\\) in real to collect informative data that can be used to (3) run system identification to identify physics parameters and reconstruct, e.g., geometric, collision, and kinematic properties. (4) Train a task-specific policy \\(\\pi_{\\mathrm{task}}\\) in the updated simulator and (5) zero-shot transfer \\(\\pi_{\\mathrm{task}}\\) to the real world."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.306,
+                0.825,
+                0.336
+            ],
+            "angle": 0,
+            "content": "our simulator as instantiating \\( p_{\\theta}(\\cdot \\mid \\pi) \\)—we assume our simulator is able to accurately mimic the dynamics of an MDP with parameter \\( \\pmb{\\theta} \\) under policy \\( \\pi \\), generating samples \\( \\pmb{\\tau} \\sim p_{\\theta}(\\cdot \\mid \\pi) \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.341,
+                0.825,
+                0.424
+            ],
+            "angle": 0,
+            "content": "In addition, we also assume that samples from our simulator are effectively \"free\"—for any \\(\\theta\\) and policy \\(\\pi\\), we can generate as many trajectories \\(\\tau \\sim p_{\\theta}(\\cdot \\mid \\pi)\\) as we wish. Given this, it is possible to find the optimal policy under \\(\\theta\\) by simply running any standard RL algorithm in simulation. With knowledge of the true parameters \\(\\theta^{\\star}\\), we can then easily find the optimal policy in real by sampling trajectories from the simulated environment with parameter \\(\\theta^{\\star}\\). It follows that, if we can identify the true parameter \\(\\theta^{\\star}\\) in real, we can solve the goal task."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.432,
+                0.47,
+                0.447
+            ],
+            "angle": 0,
+            "content": "We consider the following learning protocol:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.457,
+                0.825,
+                0.488
+            ],
+            "angle": 0,
+            "content": "1. Learner chooses exploration policy \\(\\pi_{\\mathrm{exp}}\\) and plays it in real for a single episode, generating trajectory \\(\\pmb{\\tau}_{\\mathrm{real}} \\sim p_{\\pmb{\\theta}^*}(\\cdot \\mid \\pi_{\\mathrm{exp}})\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.489,
+                0.783,
+                0.505
+            ],
+            "angle": 0,
+            "content": "2. Using \\(\\pmb{\\tau}_{\\mathrm{real}}\\) and the simulator in any way they wish, the learner obtains some policy \\(\\pi_{\\mathrm{task}}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.507,
+                0.625,
+                0.525
+            ],
+            "angle": 0,
+            "content": "3. Learner deploys \\(\\pi_{\\mathrm{task}}\\) in real and suffers loss \\(\\max_{\\pi} V_0^{\\pi} - V_0^{\\pi_{\\mathrm{task}}}\\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.171,
+                0.457,
+                0.825,
+                0.525
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.534,
+                0.825,
+                0.577
+            ],
+            "angle": 0,
+            "content": "The goal of the learner is then to learn as much useful information as possible about the real environment from a single episode of interaction and use this information to obtain a policy that can solve the task in real as effectively as possible."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.591,
+                0.825,
+                0.634
+            ],
+            "angle": 0,
+            "content": "Parameter Estimation and Fisher Information: The Fisher information matrix plays a key role in the choice of our exploration policy, \\(\\pi_{\\mathrm{exp}}\\). Recall that, for a distribution \\(p_{\\theta}\\), satisfying certain regularity conditions, the Fisher information matrix is defined as:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.332,
+                0.639,
+                0.662,
+                0.659
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathcal {I} (\\pmb {\\theta}) := \\mathbb {E} _ {\\pmb {\\tau} \\sim p _ {\\pmb {\\theta}}} \\left[ \\nabla_ {\\pmb {\\theta}} \\log p _ {\\pmb {\\theta}} (\\pmb {\\tau}) \\cdot \\nabla_ {\\pmb {\\theta}} \\log p _ {\\pmb {\\theta}} (\\pmb {\\tau}) ^ {\\top} \\right].\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.664,
+                0.825,
+                0.711
+            ],
+            "angle": 0,
+            "content": "Assume that we have access to data \\(\\mathfrak{D} = (\\pmb{\\tau}_t)_{t=1}^T\\), where \\(\\pmb{\\tau}_t \\sim p_{\\pmb{\\theta}^\\star}\\) for \\(t = 1, \\dots, T\\), and let \\(\\widehat{\\pmb{\\theta}}(\\mathfrak{D})\\) denote some unbiased estimator of \\(\\pmb{\\theta}^\\star\\). Then the Cramer-Rao lower bound (see e.g. Pronzato & Pázman (2013)) states that, under certain regularity conditions, the covariance of \\(\\widehat{\\pmb{\\theta}}(\\mathfrak{D})\\) satisfies:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.31,
+                0.717,
+                0.685,
+                0.736
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbb {E} _ {\\mathfrak {D} \\sim p _ {\\theta^ {\\star}}} [ (\\widehat {\\boldsymbol {\\theta}} (\\mathfrak {D}) - \\boldsymbol {\\theta} ^ {\\star}) (\\widehat {\\boldsymbol {\\theta}} (\\mathfrak {D}) - \\boldsymbol {\\theta} ^ {\\star}) ^ {\\top} ] \\succeq T ^ {- 1} \\cdot \\mathcal {I} (\\boldsymbol {\\theta} ^ {\\star}) ^ {- 1}.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.741,
+                0.822,
+                0.756
+            ],
+            "angle": 0,
+            "content": "From this it follows that the Fisher information serves as a lower bound on the mean-squared error:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.185,
+                0.762,
+                0.825,
+                0.781
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbb {E} _ {\\mathfrak {D} \\sim p _ {\\boldsymbol {\\theta} ^ {\\star}}} [ \\| \\widehat {\\boldsymbol {\\theta}} (\\mathfrak {D}) - \\boldsymbol {\\theta} ^ {\\star} \\| _ {2} ^ {2} ] = \\operatorname {t r} \\left(\\mathbb {E} _ {\\mathfrak {D} \\sim p _ {\\boldsymbol {\\theta} ^ {\\star}}} \\left[ \\left(\\widehat {\\boldsymbol {\\theta}} (\\mathfrak {D}) - \\boldsymbol {\\theta} ^ {\\star}\\right) \\left(\\widehat {\\boldsymbol {\\theta}} (\\mathfrak {D}) - \\boldsymbol {\\theta} ^ {\\star}\\right) ^ {\\top} \\right]\\right) \\geq T ^ {- 1} \\cdot \\operatorname {t r} \\left(\\mathcal {I} \\left(\\boldsymbol {\\theta} ^ {\\star}\\right) ^ {- 1}\\right). \\tag {1}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.786,
+                0.825,
+                0.828
+            ],
+            "angle": 0,
+            "content": "This is in general tight—for example, the maximum likelihood estimator satisfies (1) with equality as \\( T \\to \\infty \\) (Van der Vaart, 2000). The Fisher information thus serves as a fundamental lower bound on parameter estimation error, a key motivation for our exploration procedure."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.171,
+                0.847,
+                0.736,
+                0.881
+            ],
+            "angle": 0,
+            "content": "4 ASID: TARGETED EXPLORATION FOR TEST-TIME SIMULATION CONSTRUCTION, IDENTIFICATION, AND POLICY OPTIMIZATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.896,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "In this section, we present our proposed approach, ASD, a three-stage pipeline illustrated in Figure 2. We describe each component of ASD in the following."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.505,
+                0.96
+            ],
+            "angle": 0,
+            "content": "4"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.104,
+                0.622,
+                0.119
+            ],
+            "angle": 0,
+            "content": "4.1 EXPLORATION VIA FISHER INFORMATION MAXIMIZATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.13,
+                0.825,
+                0.201
+            ],
+            "angle": 0,
+            "content": "As motivated in Section 3, to learn a policy effectively accomplishing our task, it suffices to accurately identify \\(\\theta^{\\star}\\). In the exploration phase, step 1 in our learning protocol, our goal is to then play an exploration policy \\(\\pi_{\\mathrm{exp}}\\) which generates a trajectory on the real environment that provides as much information on \\(\\theta^{\\star}\\) as possible. Following Section 3, the Fisher information gives a quantification of the usefulness of the data collected, which motivates our approach."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.207,
+                0.825,
+                0.25
+            ],
+            "angle": 0,
+            "content": "In our setting, the distribution over trajectories generated during exploration in real, \\(\\pmb{\\tau}_{\\mathrm{real}} \\sim p_{\\pmb{\\theta}^*}(\\cdot \\mid \\pi_{\\mathrm{exp}})\\), depends on the exploration policy, \\(\\pi_{\\mathrm{exp}}\\), being played. As the Fisher information depends on the data distribution, it too scales with the choice of exploration policy:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.235,
+                0.255,
+                0.76,
+                0.275
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathcal {I} \\left(\\boldsymbol {\\theta} ^ {\\star}, \\pi_ {\\exp}\\right) := \\mathbb {E} _ {\\boldsymbol {\\tau} \\sim p _ {\\boldsymbol {\\theta} ^ {\\star}} (\\cdot | \\pi_ {\\exp})} \\left[ \\nabla_ {\\boldsymbol {\\theta}} \\log p _ {\\boldsymbol {\\theta} ^ {\\star}} (\\boldsymbol {\\tau} \\mid \\pi_ {\\exp}) \\cdot \\nabla_ {\\boldsymbol {\\theta}} \\log p _ {\\boldsymbol {\\theta} ^ {\\star}} (\\boldsymbol {\\tau} \\mid \\pi_ {\\exp}) ^ {\\top} \\right].\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.282,
+                0.825,
+                0.343
+            ],
+            "angle": 0,
+            "content": "Following (1), if we collect trajectories by playing \\(\\pi_{\\mathrm{exp}}\\) and set \\(\\widehat{\\pmb{\\theta}}\\) to any unbiased estimator of \\(\\pmb{\\theta}^{\\star}\\) on these trajectories, the mean-squared error of \\(\\widehat{\\pmb{\\theta}}\\) will be lower bounded by \\(\\operatorname{tr}(\\mathcal{I}(\\pmb{\\theta}^{\\star}, \\pi_{\\mathrm{exp}})^{-1})\\). The optimal exploration policy—the exploration policy which allows for the smallest estimation error—is, therefore, the policy which solves<sup>1</sup>"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.41,
+                0.348,
+                0.825,
+                0.365
+            ],
+            "angle": 0,
+            "content": "\\[\n\\arg \\min  _ {\\pi} \\operatorname {t r} \\left(\\mathcal {I} \\left(\\boldsymbol {\\theta} ^ {\\star}, \\pi\\right) ^ {- 1}\\right). \\tag {2}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.371,
+                0.826,
+                0.496
+            ],
+            "angle": 0,
+            "content": "As an intuitive justification for this choice of exploration policy, note that the Fisher information is defined in terms of the gradient of the log-likelihood with respect to the unknown parameter. Thus, if playing some \\(\\pi_{\\mathrm{exp}}\\) makes the Fisher information \"large\", making \\(\\operatorname{tr}(\\mathcal{I}(\\pmb{\\theta}^{\\star}, \\pi_{\\mathrm{exp}})^{-1})\\) small, this suggests \\(\\pi_{\\mathrm{exp}}\\) induces trajectories that are very sensitive to the unknown parameters, i.e., trajectory that are significantly more likely under one set of parameters than another. By exploring to maximize the Fisher information, we, therefore, will collect trajectories that are maximally informative about the unknown parameters, since we will observe trajectories much more likely under one set of parameters than another. Motivated by this, we therefore seek to play a policy during exploration that solves (2)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.511,
+                0.825,
+                0.568
+            ],
+            "angle": 0,
+            "content": "Implementing Fisher Information Maximization: In practice, several issues arise in solving (2), which we address here. First, the form of \\(\\mathcal{I}(\\theta, \\pi)\\) can be quite complicated, depending on the structure of \\(p_{\\theta}(\\cdot \\mid \\pi)\\), and it may not be possible to efficiently obtain a solution to (2). To address this, we make a simplifying assumption on the dynamics, that our next state, \\(s_{h+1}\\), evolves as:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.411,
+                0.573,
+                0.825,
+                0.589
+            ],
+            "angle": 0,
+            "content": "\\[\ns _ {h + 1} = f _ {\\boldsymbol {\\theta}} \\left(s _ {h}, a _ {h}\\right) + w _ {h}, \\tag {3}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.596,
+                0.825,
+                0.626
+            ],
+            "angle": 0,
+            "content": "where \\( s_h \\) and \\( a_h \\) are the current state and action, \\( w_h \\sim \\mathcal{N}(0, \\sigma_w^2 \\cdot I) \\) is Gaussian process noise, and \\( f_\\theta \\) are the nominal dynamics. Under these dynamics, the Fisher information matrix reduces to"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.275,
+                0.631,
+                0.72,
+                0.657
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathcal {I} (\\pmb {\\theta}, \\pi) = \\sigma_ {w} ^ {- 2} \\cdot \\mathbb {E} _ {p _ {\\pmb {\\theta}} (\\cdot | \\pi)} \\left[ \\sum_ {h = 1} ^ {H} \\nabla_ {\\pmb {\\theta}} f _ {\\pmb {\\theta}} (s _ {h}, a _ {h}) \\cdot \\nabla_ {\\pmb {\\theta}} f _ {\\pmb {\\theta}} (s _ {h}, a _ {h}) ^ {\\top} \\right].\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.662,
+                0.825,
+                0.735
+            ],
+            "angle": 0,
+            "content": "We argue that solving (2) with this form of \\(\\mathcal{I}(\\theta, \\pi)\\) is a very intuitive objective, even in cases when the dynamics may not follow (3) exactly. Indeed, this suggests that during exploration, we should aim to reach states for which the dynamics \\(f_{\\theta}\\) have a large gradient with respect to \\(\\theta\\)—states for which the next state predicted by the dynamics is very sensitive to \\(\\theta\\). In such states, observing the next state gives us a significant amount of information on \\(\\theta\\), allowing us to accurately identify \\(\\theta^{\\star}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.739,
+                0.825,
+                0.782
+            ],
+            "angle": 0,
+            "content": "A second challenge in solving (2) is that we do not know the true parameter \\(\\theta^{\\star}\\), which the optimization (2) depends on. To circumvent this, we rely on domain randomization in choosing our exploration policy, solving instead:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.365,
+                0.787,
+                0.825,
+                0.805
+            ],
+            "angle": 0,
+            "content": "\\[\n\\pi_ {\\exp} = \\arg \\min  _ {\\pi} \\mathbb {E} _ {\\boldsymbol {\\theta} \\sim q _ {0}} [ \\operatorname {t r} (\\mathcal {I} (\\boldsymbol {\\theta}, \\pi) ^ {- 1}) ] \\tag {4}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.809,
+                0.826,
+                0.867
+            ],
+            "angle": 0,
+            "content": "for some distribution over parameters \\( q_0 \\). While this is only an approximation of (2), in practice we find that this approximation yields effective exploration policies since, as described in Section 1, in many cases exploration policies require only a coarse model of the dynamics, and can therefore often be learned without precise knowledge of the unknown parameters."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.872,
+                0.826,
+                0.902
+            ],
+            "angle": 0,
+            "content": "A final challenge is that, in general, we may not have access to a differentiable simulator, and our dynamics themselves may not be differentiable. In such cases, \\(\\nabla_{\\theta}f_{\\theta}(s_h,a_h)\\) is unknown or"
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.191,
+                0.91,
+                0.819,
+                0.925
+            ],
+            "angle": 0,
+            "content": "<sup>1</sup>In the experiment design literature, this is known as an A-optimal experiment design (Pukelsheim, 2006)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.504,
+                0.96
+            ],
+            "angle": 0,
+            "content": "5"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.825,
+                0.162
+            ],
+            "angle": 0,
+            "content": "undefined, and the above approach cannot be applied. As a simple solution to this, we rely on a finite-differences approximation to the gradient, which still provides an effective measure of how sensitive the next state is to the unknown parameter. In practice, to solve (4) and obtain an exploration policy, we rely on standard policy optimization algorithms, such as PPO (Schulman et al., 2017b)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.179,
+                0.394,
+                0.193
+            ],
+            "angle": 0,
+            "content": "4.2 SYSTEM IDENTIFICATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.206,
+                0.825,
+                0.277
+            ],
+            "angle": 0,
+            "content": "ASID runs the exploration policy \\(\\pi_{\\mathrm{exp}}\\) (Section 4.1) in the real environment to generate a single trajectory \\(\\pmb{\\tau}_{\\mathrm{real}} \\sim p_{\\pmb{\\theta}^{\\star}}(\\cdot \\mid \\pi_{\\mathrm{exp}})\\). In the system identification phase, ASID then updates the simulator parameters using the collected trajectory. The goal is to find a distribution over simulator parameters that yield trajectories that match \\(\\pmb{\\tau}_{\\mathrm{real}}\\) as closely as possible. In particular, we wish to find some distribution over simulation parameters, \\(q_{\\phi}\\), which minimizes:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.348,
+                0.285,
+                0.651,
+                0.304
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbb {E} _ {\\boldsymbol {\\theta} \\sim q _ {\\phi}} \\big [ \\mathbb {E} _ {\\boldsymbol {\\tau} _ {\\mathrm {s i m}} \\sim p _ {\\boldsymbol {\\theta}} (\\cdot | \\mathcal {A} (\\boldsymbol {\\tau} _ {\\mathrm {r e a l}}))} \\big [ \\| \\boldsymbol {\\tau} _ {\\mathrm {r e a l}} - \\boldsymbol {\\tau} _ {\\mathrm {s i m}} \\| _ {2} ^ {2} \\big ] \\big ]\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.312,
+                0.825,
+                0.382
+            ],
+            "angle": 0,
+            "content": "where \\(p_{\\theta}(\\cdot \\mid \\mathcal{A}(\\tau_{\\mathrm{real}}))\\) denotes the distribution over trajectories generated by the simulator with parameter \\(\\theta\\), and playing the same sequence of actions as were played in \\(\\tau_{\\mathrm{real}}\\). In practice, we apply REPS (Peters et al., 2010) for the simulation and Cross Entropy Method (CEM) for the real-world experiments. We stress that the ASD framework is generic, and other black-box optimization algorithms could be used instead."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.401,
+                0.462,
+                0.415
+            ],
+            "angle": 0,
+            "content": "4.3 SOLVING THE DOWNSTREAM TASK"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.427,
+                0.825,
+                0.511
+            ],
+            "angle": 0,
+            "content": "After exploration and identification, the simulator can include information about the kinematic tree, position, orientation, or size of the object of interest, and the physical parameters of the real environment. With such a high-fidelity simulator, we aim to solve the downstream tasks entirely in simulation and transfer the learned policies \\(\\pi_{\\mathrm{task}}\\) to the real world in a zero-shot fashion. As with the system identification stage, ASD does not assume a particular method for solving the downstream task and any policy optimization algorithm can be used."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.533,
+                0.455,
+                0.548
+            ],
+            "angle": 0,
+            "content": "5 EXPERIMENTAL EVALUATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.565,
+                0.661,
+                0.58
+            ],
+            "angle": 0,
+            "content": "In our experimental evaluation, we aim to answer the following questions:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.173,
+                0.594,
+                0.816,
+                0.608
+            ],
+            "angle": 0,
+            "content": "1. Does ASD's exploration strategy yield sufficient information to identify unknown parameters?"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.614,
+                0.825,
+                0.629
+            ],
+            "angle": 0,
+            "content": "2. Are downstream task policies learned via ASD successful when using the identified parameters?"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.634,
+                0.753,
+                0.649
+            ],
+            "angle": 0,
+            "content": "3. Does the paradigm suggested by ASD transfer to performing tasks in the real world?"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.172,
+                0.594,
+                0.825,
+                0.649
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.663,
+                0.826,
+                0.775
+            ],
+            "angle": 0,
+            "content": "We conduct our experimental evaluation in two scenarios. First, we conduct empirical experiments entirely in simulation (Todorov et al., 2012) to validate the behavior of the exploration, system identification, and downstream task modules of ASD. This involves treating a true simulation environment as the real-world environment and then a reconstruction of this simulation environment as the approximate constructed simulation. Policies learned in this approximate simulation can then be evaluated back in the original simulation environment to judge efficacy. Second, we apply this to two real-world manipulation tasks that are inherently dependent on accurate parameter identification, showing the ability of ASD to function in the real world, using real-world data."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.793,
+                0.515,
+                0.807
+            ],
+            "angle": 0,
+            "content": "5.1 ABLATIONS AND BASELINE COMPARISONS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.819,
+                0.77,
+                0.835
+            ],
+            "angle": 0,
+            "content": "We compare ASD with several baselines and ablations for various portions of the pipeline."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.84,
+                0.826,
+                0.926
+            ],
+            "angle": 0,
+            "content": "Exploration: To understand the importance of targeted exploration via Fisher information maximization we compare with two baselines. First, we compare with the naive approach of using data from a random policy for system identification that is not performing any targeted exploration. Secondly, we compare with Kumar et al. (2019), which aims to generate exploration trajectories that maximize mutual information with the parameters of interest. This method essentially learns a parameter inference network and then rewards exploration policies for minimizing its error."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.505,
+                0.96
+            ],
+            "angle": 0,
+            "content": "6"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.175,
+                0.102,
+                0.334,
+                0.186
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.202,
+                0.191,
+                0.31,
+                0.204
+            ],
+            "angle": 0,
+            "content": "(a) Rod balancing"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.338,
+                0.102,
+                0.498,
+                0.186
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.346,
+                0.191,
+                0.491,
+                0.204
+            ],
+            "angle": 0,
+            "content": "(b) Sphere manipulation"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.501,
+                0.102,
+                0.661,
+                0.186
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.529,
+                0.191,
+                0.632,
+                0.204
+            ],
+            "angle": 0,
+            "content": "(c) Multi-friction"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.664,
+                0.102,
+                0.825,
+                0.186
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.696,
+                0.191,
+                0.791,
+                0.204
+            ],
+            "angle": 0,
+            "content": "(d) Articulation"
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.339,
+                0.211,
+                0.658,
+                0.226
+            ],
+            "angle": 0,
+            "content": "Figure 3: Depiction of environments in simulation"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.248,
+                0.825,
+                0.332
+            ],
+            "angle": 0,
+            "content": "System Identification: To evaluate the impact of system identification methods in this pipeline, we compare the effectiveness of using optimization-based system identification, e.g., (Peters et al., 2010; Memmel et al., 2022), with an end-to-end learned module (Kumar et al., 2019). This comparison shows the stability and extrapolation benefits of ASD over completely data-driven modeling techniques. In particular, we evaluate ASD with the optimization-based system identification replaced by the learned estimation module from Kumar et al. (2019) (ASID + estimator)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.338,
+                0.825,
+                0.395
+            ],
+            "angle": 0,
+            "content": "Downstream Policy Learning: Since the eventual goal is to solve the downstream task, we finally compare how effective it is to learn downstream policies in simulation as opposed to an uninformed Domain Randomization (DR) over the parameters, in contrast to the targeted and identified parameters that stem from the ASD pipeline."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.42,
+                0.449,
+                0.433
+            ],
+            "angle": 0,
+            "content": "5.2 SIMULATED TASK DESCRIPTIONS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.449,
+                0.825,
+                0.533
+            ],
+            "angle": 0,
+            "content": "Sphere Manipulation: We consider two sphere manipulation tasks where physical parameters like rolling friction, object stiffness, and tangential surface friction are unknown and make a significant impact on policy performance: 1) striking a golf ball to reach a goal with unknown system friction (Figure 3b), 2) striking a golf ball in an environment with multiple patches that experience different surface friction (Figure 3c). In each scenario, the position of objects and the physical parameters are varied across evaluations. In this setting, we train downstream tasks with PPO."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.54,
+                0.825,
+                0.638
+            ],
+            "angle": 0,
+            "content": "Rod Balancing: Balancing, or dynamic stacking of objects critically depends on an accurate estimate of the inertia parameters. We construct a rod balancing task where the agent can interact with a rod object (Figure 3a) to identify its physical parameters, in this case varying the distribution of mass along the rod. Once the parameters are identified, the rod must be balanced by placing it on a ledge (Figure 5 (left)). The error is measured by the tilting angle of the rod after placement, a task that requires accurate estimation of system parameters. In this case, we use the CEM to optimize downstream policies."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.644,
+                0.825,
+                0.702
+            ],
+            "angle": 0,
+            "content": "Articulation: To stress generality, we consider environments that don't simply identify physical parameters like mass or friction but also the structure of the kinematic tree such as articulation and joint positioning. We consider an environment involving an articulated laptop model with a binary environment parameter representing whether articulation is present or not (Figure 3d)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.726,
+                0.613,
+                0.74
+            ],
+            "angle": 0,
+            "content": "5.3 DOES ASID LEARN EFFECTIVE EXPLORATION BEHAVIOR?"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.755,
+                0.827,
+                0.922
+            ],
+            "angle": 0,
+            "content": "To evaluate the exploration behavior quantitatively, we consider the case of multiple unknown parameters, where learning each parameter requires exploration in a different region. In particular, we compare the exploration policy learned by our method with the exploration method of Kumar et al. (2019), on the multi-friction sphere manipulation environment illustrated in Figure 3c where the surface exhibits three different friction parameters (indicated by the grass, sand, and gravel textures). We initialize the golf ball in the leftmost region (grass)—to explore, the arm must strike the ball to other regions to identify their friction parameters. We plot a heat map of the regions visited by the ball during exploration in Figure 4. As can be seen, our approach achieves roughly uniform coverage over the entire space, learning to strike the ball into each region, and illustrating that our method is able to effectively handle complex exploration tasks that involve exciting multiple parameters. In contrast, the approach of Kumar et al. (2019) does not effectively move the sphere to regions different from the starting region."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.504,
+                0.96
+            ],
+            "angle": 0,
+            "content": "7"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.171,
+                0.077,
+                0.825,
+                0.143
+            ],
+            "angle": 0,
+            "content": "Table 1: Downstream task results in simulation: Random exploration fails in tasks where directed exploration is required, e.g., striking a sphere or finding the inertia of a rod. When placing the rod with a single action, domain randomization (DR) cannot solve the task without knowing the physical parameters. Learned system identification (Kumar et al. (2019) and \\(\\mathrm{ASID} +\\) estimator) doesn't generalize to unseen trajectories and becomes far less effective than optimization-based system identification (cf. \\(\\mathrm{ASID} + \\mathrm{SysID}\\))."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.174,
+                0.153,
+                0.825,
+                0.281
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Task</td><td colspan=\"3\">Rod Balancing</td><td>Sphere Striking</td></tr><tr><td>Metric</td><td colspan=\"3\">Tilt angle in degree°</td><td>Success Rate in % ↑</td></tr><tr><td>Parameter</td><td>Inertia (left)</td><td>Inertia (middle)</td><td>Inertia (right)</td><td>Friction ~ [1.1, 1.5]</td></tr><tr><td>Random exploration</td><td>12.44 ± 19.6</td><td>4.20 ± 6.5</td><td>15.34 ± 15.9</td><td>10.62 ± 4.3</td></tr><tr><td>Kumar et al. (2019)</td><td>13.70 ± 9.3</td><td>2.82 ± 2.7</td><td>15.26 ± 9.8</td><td>9.50 ± 2.4</td></tr><tr><td>DR</td><td>26.69 ± 7.0</td><td>13.05 ± 7.3</td><td>1.13 ± 1.3</td><td>8.75 ± 1.5</td></tr><tr><td>ASID + estimator</td><td>17.73 ± 13.1</td><td>4.65 ± 5.1</td><td>9.99 ± 6.8</td><td>11.00 ± 5.2</td></tr><tr><td>ASID + SysID (ours)</td><td>0.00 ± 0.0</td><td>0.72 ± 1.0</td><td>0.00 ± 0.0</td><td>28.00 ± 9.7</td></tr></table>"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.296,
+                0.613,
+                0.38
+            ],
+            "angle": 0,
+            "content": "We further analyze the exploration behavior in the articulation environment (Figure 3d). Here our exploration policy interacts with the laptop \\(80\\%\\) of the time as opposed to \\(20\\%\\) for naïve baselines. Appendix A.2.2 shows that models trained to predict articulation from interaction data, e.g., Ditto (Jiang et al., 2022), can infer joint and part-level geometry from the collected data."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.398,
+                0.404,
+                0.412
+            ],
+            "angle": 0,
+            "content": "5.4 HOW DOES ASID PERFORM"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.412,
+                0.596,
+                0.426
+            ],
+            "angle": 0,
+            "content": "QUANTITATIVELY IN SIMULATION ON DOWNSSTREAM TASKS?"
+        },
+        {
+            "type": "text",
+            "bbox": [
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+                0.439,
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+            "content": "To quantitatively assess the benefits of ASD over baselines and ablations, we evaluate the performance of ASD with respect to the baselines in simulation on downstream evaluation for the rod balancing (Figure 3a) and sphere manipulation (Figure 3c) task. Results are given in Table 1. Observing the performance on the rod environment, it becomes clear that previous methods struggle with one-shot tasks. When the inertia is not properly identified, CEM predicts the wrong action with high certainty, causing it to pick and place the rod at the wrong location. Domain randomization (DR) on the other hand tries to learn a policy over all possible mass distributions which in this case leads to a policy that always grasps the object at the center of the reference frame. The success here depends on \"getting lucky\" and sampling the correct action for the"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.63,
+                0.293,
+                0.824,
+                0.375
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
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+            ],
+            "angle": 0,
+            "content": "(a) ASID"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.63,
+                0.398,
+                0.798,
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+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.656,
+                0.488,
+                0.796,
+                0.501
+            ],
+            "angle": 0,
+            "content": "(b) Kumar et al. (2019)"
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.624,
+                0.513,
+                0.825,
+                0.603
+            ],
+            "angle": 0,
+            "content": "Figure 4: Visitation frequency of the sphere when explored by different exploration policies on multi-friction (Figure 3c). ASD activates the sphere over a much larger area, thereby identifying parameters more accurately"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.626,
+                0.825,
+                0.71
+            ],
+            "angle": 0,
+            "content": "In the sphere environment, the distance rolled and bounce of the sphere are critically dependent on parameters such as friction and stiffness, and misidentifying these parameters can lead to significantly different trajectories and unsuccessful downstream behavior. This is reflected in the significantly higher success rate of ASD as compared to baselines that are uninformed of parameters and simply optimize for robust policies (DR), those that try to use learned estimators for exploration (Kumar et al., 2019) or using random exploration."
+        },
+        {
+            "type": "title",
+            "bbox": [
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+                0.729,
+                0.776,
+                0.756
+            ],
+            "angle": 0,
+            "content": "5.5 DOES ASID ALLOW FOR REAL-WORLD CONTROLLER SYNTHESIS USING MINIMAL REAL-WORLD DATA?"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.769,
+                0.825,
+                0.853
+            ],
+            "angle": 0,
+            "content": "We further evaluate ASD on real-world tasks, replicating the rod balancing task from simulation (Figure 5) and introducing a novel shuffleboard task (Figure 6). As in simulation, we use a Franka Emika Panda robot for exploration and task performance in the real world. We compute the object's position and orientation by color-threshold the pointclouds from two calibrated Intel RealSense D455 cameras—this approach could easily be replaced by a more sophisticated pose estimation system if desired."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.859,
+                0.825,
+                0.931
+            ],
+            "angle": 0,
+            "content": "Rod Balancing: The goal of this task is to properly balance a rod with an unknown mass distribution by informatively exploring the environment in the real world and using the resulting data to identify the appropriate physics parameters in simulation. The optimized controller in simulation is then deployed to perform the task in the real world. In this case, the downstream task is picking the rod at a certain point along its axis and balancing it on a perch (Figure 1, Figure 5)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
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+                0.949,
+                0.504,
+                0.96
+            ],
+            "angle": 0,
+            "content": "8"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.486,
+                0.189
+            ],
+            "angle": 0,
+            "content": "The policy executes the downstream task by a pick and place primitive parameterized by the exact pick point. We deploy ASD fully autonomously, executing exploration, system identification, downstream task training, and execution in an end-to-end fashion."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.195,
+                0.487,
+                0.376
+            ],
+            "angle": 0,
+            "content": "The mass distribution in the rod is varied and both the inertia and the environment friction must be inferred. While ASD correctly identifies the inertia of the rod most of the time, we find that a center of mass close to the middle of the rod causes ambiguities that hurt our system identification process causing the simulation to not be accurate enough for zero-shot transfer. Overall, ASD solves the rod-balancing task 6/9 times across varying mass distribution and pose while a domain-randomization policy trained without any environment interaction fails to solve it at all (Table 2)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.382,
+                0.485,
+                0.41
+            ],
+            "angle": 0,
+            "content": "Shuffleboard: This task is a scaled-down version of the popular bar game tabletop shuffle-"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.499,
+                0.102,
+                0.66,
+                0.207
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.662,
+                0.101,
+                0.824,
+                0.208
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.495,
+                0.218,
+                0.825,
+                0.271
+            ],
+            "angle": 0,
+            "content": "Figure 5: Real-world Rod Balancing: Simulation setup for training exploration and downstream task policies (left). Successful execution of autonomous real-world rod balancing with skewed mass (right)."
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.495,
+                0.281,
+                0.825,
+                0.32
+            ],
+            "angle": 0,
+            "content": "Table 2: Downstream task results in real: ASD successfully balances the rod while domain randomization (DR) fails catastrophically."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.528,
+                0.321,
+                0.791,
+                0.393
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Task</td><td colspan=\"3\">Rod Balancing</td></tr><tr><td>Inertia</td><td>left</td><td>middle</td><td>right</td></tr><tr><td>DR</td><td>0/3</td><td>0/3</td><td>0/3</td></tr><tr><td>ASID (ours)</td><td>2/3</td><td>1/3</td><td>3/3</td></tr></table>"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.41,
+                0.825,
+                0.522
+            ],
+            "angle": 0,
+            "content": "board where a puck must be shot to a target area on a slippery board. We closely follow the original game and pour wax (sand) on the board to decrease the surface friction. This modification makes the task especially difficult as the surface friction on the board changes slightly after each shot since the puck displaces the wax. The goal is to strike the puck to one of the target regions—one closer (yellow) and one further (blue) away from the robot (Figure 2, Figure 6). After exploring the scene, ASD estimates the sliding and torsional friction of the puck to account for the changing surface friction of the board. For executing the downstream task, a primitive positions the endeffector at a fixed distance to the puck and a policy predicts a force value that parameterizes a shot attempt."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.529,
+                0.486,
+                0.668
+            ],
+            "angle": 0,
+            "content": "Due to the changing surface friction, the domain randomization baseline struggles to shoot the puck to the desired zone. While it succeeds 3/10 times—probably because the surface friction was part of its training distribution—the other attempts fall short or overshoot the target. With its dedicated exploration phase, ASD can accurately adapt the simulation to the current conditions and lands the puck in the desired zone 7/10 times (Table 3)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
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+            ],
+            "angle": 0,
+            "content": "6 DISCUSSION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.694,
+                0.486,
+                0.832
+            ],
+            "angle": 0,
+            "content": "In this work, we introduced a novel pipeline for performing real-world control by autonomously exploring the environment, using the collected data for system identification, and the resulting identified system for downstream policy optimization. In essence, this sim-to-real-to-sim-to-real pipeline allows for targeted test-time construction of simulation environments in a way that enables the performance of downstream tasks. We demonstrate how this"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.499,
+                0.532,
+                0.66,
+                0.639
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.662,
+                0.533,
+                0.822,
+                0.64
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.495,
+                0.65,
+                0.825,
+                0.691
+            ],
+            "angle": 0,
+            "content": "Figure 6: Real-world Shuffleboard: Simulation setup for training exploration and downstream task policies (left). Successful strike to the yellow zone (right)."
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.495,
+                0.701,
+                0.825,
+                0.74
+            ],
+            "angle": 0,
+            "content": "Table 3: Downstream task results in real: ASID outperforms domain randomization (DR) on shooting the puck to the desired zones."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.51,
+                0.741,
+                0.808,
+                0.813
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Task</td><td colspan=\"2\">Shuffleboard</td></tr><tr><td>Target zone</td><td>yellow (close)</td><td>blue (far)</td></tr><tr><td>DR</td><td>2/5</td><td>1/5</td></tr><tr><td>ASID (ours)</td><td>4/5</td><td>3/5</td></tr></table>"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.833,
+                0.825,
+                0.917
+            ],
+            "angle": 0,
+            "content": "type of identification requires autonomous and carefully directed exploration, and introduce a novel algorithm based on Fisher information maximization that is able to accomplish this directed exploration. The autonomously collected trajectories can then be paired with downstream optimization-based system identification and reconstruction algorithms for accurate simulation construction and downstream policy learning. We show the efficacy of this paradigm on multiple environments in simulation, as well as on rod balancing and shuffleboard, two challenging real-world tasks."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
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+                0.504,
+                0.96
+            ],
+            "angle": 0,
+            "content": "9"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.175,
+                0.103,
+                0.289,
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+            ],
+            "angle": 0,
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+        {
+            "type": "ref_text",
+            "bbox": [
+                0.173,
+                0.324,
+                0.825,
+                0.368
+            ],
+            "angle": 0,
+            "content": "Henry Zhu, Abhishek Gupta, Aravind Rajeswaran, Sergey Levine, and Vikash Kumar. Dexterous manipulation with deep reinforcement learning: Efficient, general, and low-cost. In 2019 International Conference on Robotics and Automation (ICRA), pp. 3651-3657. IEEE, 2019."
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
+                0.173,
+                0.375,
+                0.825,
+                0.405
+            ],
+            "angle": 0,
+            "content": "Shaojun Zhu, Andrew Kimmel, Kostas Bekris, and Abdeslam Boullarias. Fast model identification via physics engines for data-efficient policy search. In *IJCAI*, 2018."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.173,
+                0.103,
+                0.826,
+                0.405
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.491,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "14"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.103,
+                0.3,
+                0.119
+            ],
+            "angle": 0,
+            "content": "A APPENDIX"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.134,
+                0.324,
+                0.149
+            ],
+            "angle": 0,
+            "content": "A.1 TASKDETAILS"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.16,
+                0.405,
+                0.175
+            ],
+            "angle": 0,
+            "content": "A.1.1 SPHERE MANIPULATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.184,
+                0.827,
+                0.283
+            ],
+            "angle": 0,
+            "content": "In the sphere manipulation tasks, the observation space consists of endeffector position, sphere position, and robot joint angles. During the training of the exploration policy, we randomize the location of a sphere with \\( r = 0.03 \\) to be between \\( x \\in [0.430, 0.65] \\), \\( y \\in [-0.2, 0.23] \\) for training and \\( x \\in [0.55, 0.65] \\), \\( y \\in [-0.2, 0.23] \\) for evaluation. Parameter ranges for training are 1) friction \\( \\theta \\in [1e - 3, 5e - 3] \\) and 2) patch friction \\( \\theta_0, \\theta_1, \\theta_2 \\in [1e - 5, 1e - 3] \\) and set fixed sphere and goal locations as well as parameters for the downstream task. For 1) we attach a paddle and limit the endeffector position such that the robot has to strike the sphere to the goal."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.297,
+                0.341,
+                0.312
+            ],
+            "angle": 0,
+            "content": "A.1.2 ARTICULATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.322,
+                0.825,
+                0.392
+            ],
+            "angle": 0,
+            "content": "In the case of the articulation environment, the parameter is binary and indicates whether articulation is present or not. During evaluation, articulation is always turned on and success is indicated by \\(\\delta \\beta > 1e - 2\\). The initial laptop state is randomized over position \\(x \\in [0.45, 0.65]\\), \\(y \\in [-0.1, 0.1]\\), \\(yaw \\in [0.00, 3.14]\\) and lid angle \\(\\beta \\in [1.2, 2.5]\\). The observation space contains the endeffector position, joint angles, position, orientation, and joint angle of the laptop."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.407,
+                0.356,
+                0.421
+            ],
+            "angle": 0,
+            "content": "A.1.3 ROD BALANCING"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.431,
+                0.825,
+                0.475
+            ],
+            "angle": 0,
+            "content": "Similar to the sphere manipulation, we restrict the exploration policy to endeffector control with \\(\\delta x, \\delta y\\) and attach a peg to the Franka. The rod has dimensions \\(0.04 \\times 0.3 \\times 0.04\\) and initializes as \\(x \\in [0.5, 0.6], y \\in [-0.2, 0.2], yaw \\in [0.00, 3.14]\\)"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.491,
+                0.948,
+                0.509,
+                0.961
+            ],
+            "angle": 0,
+            "content": "15"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.105,
+                0.38,
+                0.119
+            ],
+            "angle": 0,
+            "content": "A.2 QUALITATIVE RESULTS"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.132,
+                0.603,
+                0.146
+            ],
+            "angle": 0,
+            "content": "A.2.1 EXPLORATION STRATEGIES LEARNED BY THE AGENT"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.158,
+                0.827,
+                0.271
+            ],
+            "angle": 0,
+            "content": "We evaluate the exploration behavior qualitatively across multiple environments to understand whether the exploration behavior is meaningful. In the sphere environment, we observe the agent hitting the sphere multiple times when it bounces off the walls or stays within reach. The other environments also experience emergent behavior, as the agent executes a horizontal motion towards the top of the lid if the laptop is mostly open while an almost closed laptop causes it to push from top to bottom instead. Finally, when determining the rod's inertia, the policy pushes both sides to gain maximum information about the center of mass through the rotation motions in both directions. See Figure 7 for a visualization of the exploration for the sphere environment."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.185,
+                0.302,
+                0.382,
+                0.395
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.277,
+                0.408,
+                0.297,
+                0.421
+            ],
+            "angle": 0,
+            "content": "(a)"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.384,
+                0.302,
+                0.58,
+                0.395
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.474,
+                0.408,
+                0.495,
+                0.421
+            ],
+            "angle": 0,
+            "content": "(b)"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.582,
+                0.302,
+                0.811,
+                0.395
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.673,
+                0.408,
+                0.692,
+                0.421
+            ],
+            "angle": 0,
+            "content": "(c)"
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.787,
+                0.408,
+                0.807,
+                0.421
+            ],
+            "angle": 0,
+            "content": "(d)"
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.171,
+                0.433,
+                0.825,
+                0.476
+            ],
+            "angle": 0,
+            "content": "Figure 7: Absolute sphere displacement for different sphere starting locations. Zero means the sphere didn't get hit, higher numbers denote larger displacements. Initial endeffector position marked in red a) Random Coverage, b) PPO Coverage, c) Fisher Coverage, d) Legend."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.509,
+                0.658,
+                0.523
+            ],
+            "angle": 0,
+            "content": "A.2.2 RECONSTRUCTING ARTICULATION FROM INTERACTION DATA"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.174,
+                0.544,
+                0.341,
+                0.628
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.439,
+                0.545,
+                0.606,
+                0.628
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.708,
+                0.547,
+                0.824,
+                0.625
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.171,
+                0.638,
+                0.825,
+                0.665
+            ],
+            "angle": 0,
+            "content": "Figure 8: Articulated object: before (left) and after (middle) exploration with ASD and part-level reconstruction with Ditto (Jiang et al., 2022): articulated \\(\\blacksquare\\) and static part \\(\\blacksquare\\) (right)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.693,
+                0.54,
+                0.707
+            ],
+            "angle": 0,
+            "content": "A.3 COMPARISON TO MODEL-BASED APPROACHES"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.721,
+                0.391,
+                0.734
+            ],
+            "angle": 0,
+            "content": "A.3.1 ENVIRONMENT SETUP"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.4,
+                0.755,
+                0.598,
+                0.87
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.171,
+                0.88,
+                0.825,
+                0.909
+            ],
+            "angle": 0,
+            "content": "Figure 9: Environment with multiple spheres. The single red sphere is subject to changing friction values while the three blue spheres act as distractors."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.491,
+                0.949,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "16"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.105,
+                0.443,
+                0.119
+            ],
+            "angle": 0,
+            "content": "A.3.2 MODEL-BASED EXPLORATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.128,
+                0.825,
+                0.158
+            ],
+            "angle": 0,
+            "content": "In this section, Figure 10, we compare the performance of our algorithm to that of the MAX algorithm (Shyam et al., 2019)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.164,
+                0.827,
+                0.234
+            ],
+            "angle": 0,
+            "content": "MAX aims to cover the state space through multiple policies throughout training. Since it uses the disagreement between fully learned dynamics models, it gets distracted by novel states induced by the movement of all spheres (red and blue). In contrast, our method based on the Fisher information yields a single policy that seeks out the sphere affected by the changing physics parameters (red) and ignores the irrelevant spheres (blue) even if they lead to novel states."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.201,
+                0.264,
+                0.478,
+                0.424
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.209,
+                0.438,
+                0.468,
+                0.453
+            ],
+            "angle": 0,
+            "content": "(a) MAX (checkpoints throughout training)"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.52,
+                0.264,
+                0.797,
+                0.422
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.577,
+                0.438,
+                0.743,
+                0.453
+            ],
+            "angle": 0,
+            "content": "(b) MAX (last checkpoints)"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.2,
+                0.47,
+                0.476,
+                0.628
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.177,
+                0.643,
+                0.498,
+                0.67
+            ],
+            "angle": 0,
+            "content": "(c) Fisher information (checkpoints throughout training)"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.521,
+                0.481,
+                0.797,
+                0.64
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.542,
+                0.656,
+                0.778,
+                0.671
+            ],
+            "angle": 0,
+            "content": "(d) Fisher information (last checkpoint)"
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.171,
+                0.68,
+                0.825,
+                0.766
+            ],
+            "angle": 0,
+            "content": "Figure 10: Exploration MAX vs. Fisher information: Trajectories collected (30) from policy checkpoints throughout training or from the last policy checkpoint. While Model-based Active Exploration (MAX) (Shyam et al., 2019) explores the state space over the course of training, getting distracted by the novel states induced by the blue spheres \\(\\blacksquare\\) (c.f. fig. 10a), Fisher information-based exploration (ours) shows directed exploration (c.f. fig. 10d), moving towards the sphere with changing parameters (red \\(\\blacksquare\\)) and yielding a single exploration policy."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.491,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "17"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.105,
+                0.625,
+                0.119
+            ],
+            "angle": 0,
+            "content": "A.3.3 LEARNED STATE-TRANSITION MODELS VS. SIMULATORS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.129,
+                0.827,
+                0.228
+            ],
+            "angle": 0,
+            "content": "We next illustrate the improvement using a simulator vs a fully learned dynamics model can give. We train a forward dynamics model (three layer MLP) on data generated both from MAX and our exploration procedure. When evaluated on out-of-distribution trajectories, i.e., trajectories not included in the training data, we find the model to be extremely inaccurate (see Figure 11). While the simulator extrapolates to unseen states and correctly predicts the movement of the sphere, the model hallucinates movement even when the end-effector does not interact with it at all! These findings make ASID preferable to a purely model-based approach."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.201,
+                0.258,
+                0.473,
+                0.416
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.258,
+                0.43,
+                0.421,
+                0.445
+            ],
+            "angle": 0,
+            "content": "(a) MAX (pushing actions)"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.52,
+                0.258,
+                0.791,
+                0.416
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.589,
+                0.43,
+                0.731,
+                0.444
+            ],
+            "angle": 0,
+            "content": "(b) MAX (zero actions)"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.2,
+                0.463,
+                0.471,
+                0.621
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.22,
+                0.636,
+                0.457,
+                0.65
+            ],
+            "angle": 0,
+            "content": "(c) Fisher information (pushing actions)"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.521,
+                0.463,
+                0.791,
+                0.621
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.551,
+                0.636,
+                0.768,
+                0.649
+            ],
+            "angle": 0,
+            "content": "(d) Fisher information (zero actions)"
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.171,
+                0.661,
+                0.825,
+                0.718
+            ],
+            "angle": 0,
+            "content": "Figure 11: Learned models vs. simulator: Evaluation of a forward model trained on trajectories (30) from MAX (Shyam et al., 2019) (checkpoints throughout training) and Fisher information (last checkpoint). In contrast to our simulator, the learned model fails to predict the red sphere's trajectory accurately even under no contact scenarios (c.f. fig. 11b, fig. 11d)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.491,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "18"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.105,
+                0.405,
+                0.119
+            ],
+            "angle": 0,
+            "content": "A.4 POLICY GENERALIZATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.13,
+                0.827,
+                0.187
+            ],
+            "angle": 0,
+            "content": "We evaluate the generalization capabilities of our exploration policy for different sphere locations and parameter combinations seen and unseen during training (Figure 13). Since we train the policies using an arena setup, we remove the arena during this evaluation to be able to query sphere locations outside of it (Figure 12)."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.4,
+                0.199,
+                0.599,
+                0.315
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.171,
+                0.324,
+                0.825,
+                0.355
+            ],
+            "angle": 0,
+            "content": "Figure 12: Environment with free space. The red sphere is subject to changing parameter values and position is randomized."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.179,
+                0.401,
+                0.473,
+                0.559
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.192,
+                0.566,
+                0.484,
+                0.581
+            ],
+            "angle": 0,
+            "content": "(a) Changing friction parameters (in distribution)"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.501,
+                0.402,
+                0.793,
+                0.559
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.512,
+                0.566,
+                0.807,
+                0.581
+            ],
+            "angle": 0,
+            "content": "(b) Changing mass parameter (out of distribution)"
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.17,
+                0.591,
+                0.828,
+                0.732
+            ],
+            "angle": 0,
+            "content": "Figure 13: Generalization capabilities of an exploration policy trained on sphere friction for different sphere starting locations. The plot shows the success rate of pushing the sphere spawned at the corresponding x-y-location and evaluated over 5 different seeds, i.e., different physics parameter values. Initial endeffector position marked in red. Boxes denote sphere locations seen during training (purple box) and reachable area of the endeffector (orange box). Our policies experience surprising generalization capabilities to unseen sphere locations as long as they can be reached by the robot (cf.fig. 13a). Furthermore, the actions extrapolate to unseen physics parameters as long as they can be uncovered with similar interactions, e.g., pushing (cf.fig. 13b). This is the case because the policy does not have any information about the underlying physics parameters until it hits the sphere."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.491,
+                0.948,
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+# ASID: ACTIVE EXPLORATION FOR SYSTEM IDENTIFICATION IN ROBOTIC MANIPULATION
+
+Marius Memmel, Andrew Wagenvaker, Chuning Zhu, Dieter Fox, Abhishek Gupta
+
+Paul G. Allen School of Computer Science & Engineering
+
+University of Washington
+
+Seattle, WA 98195, USA
+
+{memmelma,ajwagen,zchuning,fox,abhgupta}@cs.washington.edu
+
+# ABSTRACT
+
+Model-free control strategies such as reinforcement learning have shown the ability to learn control strategies without requiring an accurate model or simulator of the world. While this is appealing due to the lack of modeling requirements, such methods can be sample inefficient, making them impractical in many real-world domains. On the other hand, model-based control techniques leveraging accurate simulators can circumvent these challenges and use a large amount of cheap simulation data to learn controllers that can effectively transfer to the real world. The challenge with such model-based techniques is the requirement for an extremely accurate simulation, requiring both the specification of appropriate simulation assets and physical parameters. This requires considerable human effort to design for every environment being considered. In this work, we propose a learning system that can leverage a small amount of real-world data to autonomously refine a simulation model and then plan an accurate control strategy that can be deployed in the real world. Our approach critically relies on utilizing an initial (possibly inaccurate) simulator to design effective exploration policies that, when deployed in the real world, collect high-quality data. We demonstrate the efficacy of this paradigm in identifying articulation, mass, and other physical parameters in several challenging robotic manipulation tasks, and illustrate that only a small amount of real-world data can allow for effective sim-to-real transfer. Project website at https://weirdlabuw.github.io/asid
+
+# 1 INTRODUCTION
+
+Controlling robots to perform dynamic, goal-directed behavior in the real world is challenging. Reinforcement Learning (RL) has emerged as a promising technique to learn such behaviors without requiring known models of the environment, instead relying on data sampled directly from the environment (Schulman et al., 2017a; Haarnoja et al., 2018). In principle, these techniques can be deployed in new environments with a minimal amount of human effort, and allow for continual improvement of behavior. Such techniques have been shown to successfully learn complex behaviors in a variety of scenarios, ranging from table-top manipulation (Yu et al., 2020) to locomotion (Hwangbo et al., 2019) and even dexterous manipulation (Zhu et al., 2019).
+
+![](images/51fb0e85ce1afcff040acbcb66b612d5ddd50ffe92b5f417ff20aac8a157502f.jpg)  
+Figure 1: ASD: A depiction of our proposed process of active exploration for system identification, from learning exploration policies to real-world deployment.
+
+However, these capabilities come at a cost, requiring full access to the environment in order to design reset mechanisms and reward functions. Such requirements necessitate training these methods in carefully controlled and often curated environments, limiting their applicability. Moreover, de
+
+ploying tabula-asa reinforcement learning methods in the real world often requires a prohibitively large number of samples, which may be impractical to collect.
+
+One approach to circumvent these challenges is to rely on simulators to cheaply generate large amounts of data and use RL to train a policy. However, directly deploying policies trained in simulation in the real world is often ineffective due to the discrepancy between the simulation and the real world, the so-called sim2real gap. For example, the physical properties and parameters of the simulation might diverge from the real world, rendering a simulation-trained policy useless in reality.
+
+Taking inspiration from system identification, we argue that the key to effective sim2real transfer is an initial round of exploration in the real world to learn an effective simulator. We propose a generic pipeline for sim2real transfer, Active Exploration for System IDentification (ASID), which decouples exploration and exploitation: (1) exploration in real to collect informative data of unknown parameters, (2) refinement of the simulation parameters using data collected in real, (3) policy training on the updated simulator to accomplish the goal tasks. Our exploration procedure is motivated by work in theoretical statistics and seeks to induce trajectories corresponding to large Fisher information, thereby providing maximally informative observations. By using our initial round of exploration to obtain accurate estimates of the parameters in the real world, we show that in many cases, the policies trained in step (3) successfully transfer to real in a zero-shot fashion, even in settings where training a policy in sim without additional knowledge of real would fail.
+
+A key insight in our approach is that, while a policy trained in sim to accomplish the goal task may not effectively transfer, strategies that explore effectively in sim often also explore effectively in real. As an example, say our goal task is to hit a ball to a particular location with a robotic arm and assume the mass of the ball is unknown. If we train a policy in sim to hit the ball without knowledge of the mass, when deployed in real it will almost certainly fail, as the force at which it should strike the ball depends critically on the (unknown) mass. To learn the mass, however, essentially any contact between the ball and the arm suffices. Achieving some contact between the ball and the arm requires a significantly less precise motion, and indeed, does not require any prior knowledge of the mass. We can therefore train a policy in sim that learns to effectively explore—hit the ball in any direction—and deploy this in real to collect information on the true parameters, ultimately allowing us to obtain a higher-fidelity simulator that does allow sim2real transfer on the goal task.
+
+We are particularly interested in the application of our pipeline to modern robotic settings and evaluate ASD on four tasks: sphere manipulation, laptop articulation, rod balancing, and shuffleboard. We show that in all settings, our approach is able to effectively identify unknown parameters of the real environment (e.g. geometry, articulation, center of mass, and physical parameters like mass, friction, or stiffness), and using this knowledge, learn a policy in sim for the goal task that successfully transfers to real. In all cases, by deploying effective exploration policies trained in simulation, we require only a very small amount of data from real—typically a single episode of data suffices.
+
+# 2 RELATED WORK
+
+System Identification: Our work is closely related to the field of system identification (Åström & Eykhoff, 1971; Söderström & Stoica, 1989; Ljung, 1998; Schon et al., 2011; Menda et al., 2020), which studies how to learn a model of the system dynamics efficiently. A large body of work, stretching back decades, has explored how inputs to a system should be chosen to most effectively learn the system's parameters (Mehra, 1974; 1976; Goodwin & Payne, 1977; Hjalmarsson et al., 1996; Lindqvist & Hjalmarsson, 2001; Gerencsér & Hjalmarsson, 2005; Rojas et al., 2007; Gevers et al., 2009; Gerencsér et al., 2009; Manchester, 2010; Rojas et al., 2011; Bombois et al., 2011; Hägg et al., 2013; Wagenmaker & Jamieson, 2020; Wagenmaker et al., 2021; Mania et al., 2022; Wagenmaker et al., 2023) or how to deal with partial observability (Schön et al., 2011; Menda et al., 2020). Similar to our exploration strategy, many of these works choose their inputs to maximize some function of the Fisher information matrix. A primary novelty of our approach is to use a simulator to learn effective exploration policies, and to apply our method to modern, real-world robotics tasks—indeed, our work can be seen as bridging the gap between classical work on system identification and modern sim2real techniques. While the aforementioned works are primarily theoretical, recent work has studied the application of such methods to a variety of real-world settings like active identification of physics parameters (Xu et al., 2019; Kumar et al., 2019; Mavrakis et al., 2020; Gao et al., 2020; 2022) or kinematic structure (Mo et al., 2021; Wang et al., 2022; Nie et al., 2022; Hsu
+
+et al., 2023) through object-centric primitives. Another line of recent work aims to learn the parameters of the simulator to ultimately train a downstream policy on the learned parameters, and therefore apply task-specific policies for data collection (Zhu et al., 2018; Chebotar et al., 2019; Huang et al., 2023; Ren et al., 2023) or exploration policies that minimize its regret (Liang et al., 2020). The majority of these works, however, do not consider running an exploration policy that targets learning the unknown parameters, do not address solving downstream tasks, or rely on techniques that do not scale effectively to more complex tasks.
+
+Simulation-to-Reality Transfer: Transferring learned policies from sim2real has shown to be successful in challenging tasks like dexterous manipulation (OpenAI et al., 2018; Handa et al., 2022; Chen et al., 2022), locomotion (Rudin et al., 2022), agile drone flight (Sadeghi & Levine, 2016) or contact rich assembly tasks Tang et al. (2023), yet challenges remain due to the sim2real gap. To deal with the gap, Domain Randomization (DR) (Tobin et al., 2017) trains policies over a distribution of environments in simulation, hoping for the real world to be represented among them. Subsequent works adaptively change the environment distribution (Muratore et al., 2019; Mehta et al., 2020) and incorporate real data (Chebotar et al., 2019; Ramos et al., 2019; Duan et al., 2023; Chen et al., 2023; Ma et al., 2023; Torne et al., 2024). While similar to our approach, these methods do not perform targeted exploration in real to update the simulator parameters. Other approaches seek to infer and adapt to simulation parameters during deployment (Kumar et al., 2021; Qi et al., 2023; Margolis et al., 2023), leverage offline data (Richards et al., 2021; Bose et al., 2024), or adapt online (Sinha et al., 2022); in contrast, we do not learn such an online adaptation strategy, but rather a better simulator. Finally, a commonly applied strategy is to train a policy in sim and then fine-tune in the real environment (Julian et al., 2021; Smith et al., 2022; Nakamoto et al., 2023); in contrast, we are interested in the (more challenging) regime where a direct transfer is not likely to give any learning signal to fine-tune from.
+
+Model-Based RL: Model-based RL (MBRL) aims to solve the RL problem by learning a model of the dynamics, and using this model to either plan or solve a policy optimization problem (Deisenroth & Rasmussen, 2011; Williams et al., 2017; Nagabandi et al., 2018; Chua et al., 2018; Janner et al., 2019; Hafner et al., 2019; 2020; Janner et al., 2022; Zhu et al., 2023). While our approach is model-based in some sense, the majority of work in MBRL focuses on fully learned dynamic models; in contrast, our "model" is our simulator, and we aim to learn only a very small number of parameters, which can be much more sample-efficient. Furthermore, MBRL methods typically do not perform explicit exploration, while a key piece of our approach is a targeted exploration procedure. The MBRL works we are aware of which do rely on targeted exploration (Shyam et al., 2019; Pathak et al., 2019) typically rely on fully learned dynamic models and apply somewhat different exploration strategies, which we show in Appendix A.3.2 can perform significantly worse.
+
+# 3 PRELIMINARIES
+
+We formulate our decision-making setting as Markov Decision Processes (MDPs). An MDP is defined as a tuple  $M^{\star} = (\mathcal{S},\mathcal{A},\{P_h^{\star}\}_{h = 1}^H,P_0,\{r_h\}_{h = 1}^H)$ , where  $\mathcal{S}$  is the set of states,  $\mathcal{A}$  the set of actions,  $P_{h}:S\times \mathcal{A}\to \triangle_{\mathcal{S}}$  the transition kernel,  $P_0\in \triangle_S$  the initial state distribution, and  $r_h:\mathcal{S}\times \mathcal{A}\rightarrow \mathbb{R}$  the reward function. We consider the episodic setting. At the beginning of an episode, the environment samples a state  $s_1\sim P_0$ . The agent observes this state, plays some action  $a_1\in \mathcal{A}$ , and transitions to state  $s_2\sim P_1(\cdot |s_1,a_1)$ , receiving reward  $r_1(s_1,a_1)$ . After  $H$  steps, the environment resets and the process repeats. Our primary goal is to learn a policy  $\pi$ -a mapping from states to actions—that maximizes reward in the true environment. We denote the value of a policy by  $V_0^\pi \coloneqq \mathbb{E}_{M^\star ,\pi}[\sum_{h = 1}^H r_h(s_h,a_h)]$ , where the expectation is over trajectories induced playing policy  $\pi$  on MDP  $M^{\star}$ . We think of the reward  $r$  as encoding our downstream task, and our end goal is to find a policy that solves our task, maximizing  $V_0^\pi$ . We denote such policies as  $\pi_{\mathrm{task}}$ .
+
+In the sim2real setting considered in this work, we assume that the reward is known, but that the dynamics of the real environment,  $P^{\star} = \{P_h^{\star}\}_{h=1}^H$ , are initially unknown. However, we assume that they belong to some known parametric family  $\mathcal{P} := \{P_{\theta} : \theta \in \Theta\}$ , so that there exists some  $\theta^{\star} \in \Theta$  such that  $P^{\star} = P_{\theta^{\star}}$ . Here we take  $\theta$  to be some unknown parameter (for example, mass, friction, etc.), and  $P_{\theta}$  the dynamics under parameter  $\theta$  (which we might know from physics, first principles, etc.). For any  $\theta$  and policy  $\pi$ , the dynamics  $P_{\theta}$  induce a distribution over state-action trajectories,  $\tau = (s_1, a_1, s_2, \dots, s_H, a_H)$ , which we denote by  $p_{\theta}(\cdot \mid \pi)$ . We can think of
+
+![](images/c7767ffb66fa23029347339202edfbbb03daee2f781f15b3ae65ee20aef2538a.jpg)  
+Figure 2: Overview of ASD: (1) Train an exploration policy  $\pi_{\mathrm{exp}}$  that maximizes the Fisher information, leveraging the vast amount of cheap simulation data. (2) Roll out  $\pi_{\mathrm{exp}}$  in real to collect informative data that can be used to (3) run system identification to identify physics parameters and reconstruct, e.g., geometric, collision, and kinematic properties. (4) Train a task-specific policy  $\pi_{\mathrm{task}}$  in the updated simulator and (5) zero-shot transfer  $\pi_{\mathrm{task}}$  to the real world.
+
+our simulator as instantiating  $p_{\theta}(\cdot \mid \pi)$ —we assume our simulator is able to accurately mimic the dynamics of an MDP with parameter  $\pmb{\theta}$  under policy  $\pi$ , generating samples  $\pmb{\tau} \sim p_{\theta}(\cdot \mid \pi)$ .
+
+In addition, we also assume that samples from our simulator are effectively "free"—for any  $\theta$  and policy  $\pi$ , we can generate as many trajectories  $\tau \sim p_{\theta}(\cdot \mid \pi)$  as we wish. Given this, it is possible to find the optimal policy under  $\theta$  by simply running any standard RL algorithm in simulation. With knowledge of the true parameters  $\theta^{\star}$ , we can then easily find the optimal policy in real by sampling trajectories from the simulated environment with parameter  $\theta^{\star}$ . It follows that, if we can identify the true parameter  $\theta^{\star}$  in real, we can solve the goal task.
+
+We consider the following learning protocol:
+
+1. Learner chooses exploration policy  $\pi_{\mathrm{exp}}$  and plays it in real for a single episode, generating trajectory  $\pmb{\tau}_{\mathrm{real}} \sim p_{\pmb{\theta}^*}(\cdot \mid \pi_{\mathrm{exp}})$ .  
+2. Using  $\pmb{\tau}_{\mathrm{real}}$  and the simulator in any way they wish, the learner obtains some policy  $\pi_{\mathrm{task}}$ .  
+3. Learner deploys  $\pi_{\mathrm{task}}$  in real and suffers loss  $\max_{\pi} V_0^{\pi} - V_0^{\pi_{\mathrm{task}}}$ .
+
+The goal of the learner is then to learn as much useful information as possible about the real environment from a single episode of interaction and use this information to obtain a policy that can solve the task in real as effectively as possible.
+
+Parameter Estimation and Fisher Information: The Fisher information matrix plays a key role in the choice of our exploration policy,  $\pi_{\mathrm{exp}}$ . Recall that, for a distribution  $p_{\theta}$ , satisfying certain regularity conditions, the Fisher information matrix is defined as:
+
+$$
+\mathcal {I} (\pmb {\theta}) := \mathbb {E} _ {\pmb {\tau} \sim p _ {\pmb {\theta}}} \left[ \nabla_ {\pmb {\theta}} \log p _ {\pmb {\theta}} (\pmb {\tau}) \cdot \nabla_ {\pmb {\theta}} \log p _ {\pmb {\theta}} (\pmb {\tau}) ^ {\top} \right].
+$$
+
+Assume that we have access to data  $\mathfrak{D} = (\pmb{\tau}_t)_{t=1}^T$ , where  $\pmb{\tau}_t \sim p_{\pmb{\theta}^\star}$  for  $t = 1, \dots, T$ , and let  $\widehat{\pmb{\theta}}(\mathfrak{D})$  denote some unbiased estimator of  $\pmb{\theta}^\star$ . Then the Cramer-Rao lower bound (see e.g. Pronzato & Pázman (2013)) states that, under certain regularity conditions, the covariance of  $\widehat{\pmb{\theta}}(\mathfrak{D})$  satisfies:
+
+$$
+\mathbb {E} _ {\mathfrak {D} \sim p _ {\theta^ {\star}}} [ (\widehat {\boldsymbol {\theta}} (\mathfrak {D}) - \boldsymbol {\theta} ^ {\star}) (\widehat {\boldsymbol {\theta}} (\mathfrak {D}) - \boldsymbol {\theta} ^ {\star}) ^ {\top} ] \succeq T ^ {- 1} \cdot \mathcal {I} (\boldsymbol {\theta} ^ {\star}) ^ {- 1}.
+$$
+
+From this it follows that the Fisher information serves as a lower bound on the mean-squared error:
+
+$$
+\mathbb {E} _ {\mathfrak {D} \sim p _ {\boldsymbol {\theta} ^ {\star}}} [ \| \widehat {\boldsymbol {\theta}} (\mathfrak {D}) - \boldsymbol {\theta} ^ {\star} \| _ {2} ^ {2} ] = \operatorname {t r} \left(\mathbb {E} _ {\mathfrak {D} \sim p _ {\boldsymbol {\theta} ^ {\star}}} \left[ \left(\widehat {\boldsymbol {\theta}} (\mathfrak {D}) - \boldsymbol {\theta} ^ {\star}\right) \left(\widehat {\boldsymbol {\theta}} (\mathfrak {D}) - \boldsymbol {\theta} ^ {\star}\right) ^ {\top} \right]\right) \geq T ^ {- 1} \cdot \operatorname {t r} \left(\mathcal {I} \left(\boldsymbol {\theta} ^ {\star}\right) ^ {- 1}\right). \tag {1}
+$$
+
+This is in general tight—for example, the maximum likelihood estimator satisfies (1) with equality as  $T \to \infty$  (Van der Vaart, 2000). The Fisher information thus serves as a fundamental lower bound on parameter estimation error, a key motivation for our exploration procedure.
+
+# 4 ASID: TARGETED EXPLORATION FOR TEST-TIME SIMULATION CONSTRUCTION, IDENTIFICATION, AND POLICY OPTIMIZATION
+
+In this section, we present our proposed approach, ASD, a three-stage pipeline illustrated in Figure 2. We describe each component of ASD in the following.
+
+# 4.1 EXPLORATION VIA FISHER INFORMATION MAXIMIZATION
+
+As motivated in Section 3, to learn a policy effectively accomplishing our task, it suffices to accurately identify  $\theta^{\star}$ . In the exploration phase, step 1 in our learning protocol, our goal is to then play an exploration policy  $\pi_{\mathrm{exp}}$  which generates a trajectory on the real environment that provides as much information on  $\theta^{\star}$  as possible. Following Section 3, the Fisher information gives a quantification of the usefulness of the data collected, which motivates our approach.
+
+In our setting, the distribution over trajectories generated during exploration in real,  $\pmb{\tau}_{\mathrm{real}} \sim p_{\pmb{\theta}^*}(\cdot \mid \pi_{\mathrm{exp}})$ , depends on the exploration policy,  $\pi_{\mathrm{exp}}$ , being played. As the Fisher information depends on the data distribution, it too scales with the choice of exploration policy:
+
+$$
+\mathcal {I} \left(\boldsymbol {\theta} ^ {\star}, \pi_ {\exp}\right) := \mathbb {E} _ {\boldsymbol {\tau} \sim p _ {\boldsymbol {\theta} ^ {\star}} (\cdot | \pi_ {\exp})} \left[ \nabla_ {\boldsymbol {\theta}} \log p _ {\boldsymbol {\theta} ^ {\star}} (\boldsymbol {\tau} \mid \pi_ {\exp}) \cdot \nabla_ {\boldsymbol {\theta}} \log p _ {\boldsymbol {\theta} ^ {\star}} (\boldsymbol {\tau} \mid \pi_ {\exp}) ^ {\top} \right].
+$$
+
+Following (1), if we collect trajectories by playing  $\pi_{\mathrm{exp}}$  and set  $\widehat{\pmb{\theta}}$  to any unbiased estimator of  $\pmb{\theta}^{\star}$  on these trajectories, the mean-squared error of  $\widehat{\pmb{\theta}}$  will be lower bounded by  $\operatorname{tr}(\mathcal{I}(\pmb{\theta}^{\star}, \pi_{\mathrm{exp}})^{-1})$ . The optimal exploration policy—the exploration policy which allows for the smallest estimation error—is, therefore, the policy which solves<sup>1</sup>
+
+$$
+\arg \min  _ {\pi} \operatorname {t r} \left(\mathcal {I} \left(\boldsymbol {\theta} ^ {\star}, \pi\right) ^ {- 1}\right). \tag {2}
+$$
+
+As an intuitive justification for this choice of exploration policy, note that the Fisher information is defined in terms of the gradient of the log-likelihood with respect to the unknown parameter. Thus, if playing some  $\pi_{\mathrm{exp}}$  makes the Fisher information "large", making  $\operatorname{tr}(\mathcal{I}(\pmb{\theta}^{\star}, \pi_{\mathrm{exp}})^{-1})$  small, this suggests  $\pi_{\mathrm{exp}}$  induces trajectories that are very sensitive to the unknown parameters, i.e., trajectory that are significantly more likely under one set of parameters than another. By exploring to maximize the Fisher information, we, therefore, will collect trajectories that are maximally informative about the unknown parameters, since we will observe trajectories much more likely under one set of parameters than another. Motivated by this, we therefore seek to play a policy during exploration that solves (2).
+
+Implementing Fisher Information Maximization: In practice, several issues arise in solving (2), which we address here. First, the form of  $\mathcal{I}(\theta, \pi)$  can be quite complicated, depending on the structure of  $p_{\theta}(\cdot \mid \pi)$ , and it may not be possible to efficiently obtain a solution to (2). To address this, we make a simplifying assumption on the dynamics, that our next state,  $s_{h+1}$ , evolves as:
+
+$$
+s _ {h + 1} = f _ {\boldsymbol {\theta}} \left(s _ {h}, a _ {h}\right) + w _ {h}, \tag {3}
+$$
+
+where  $s_h$  and  $a_h$  are the current state and action,  $w_h \sim \mathcal{N}(0, \sigma_w^2 \cdot I)$  is Gaussian process noise, and  $f_\theta$  are the nominal dynamics. Under these dynamics, the Fisher information matrix reduces to
+
+$$
+\mathcal {I} (\pmb {\theta}, \pi) = \sigma_ {w} ^ {- 2} \cdot \mathbb {E} _ {p _ {\pmb {\theta}} (\cdot | \pi)} \left[ \sum_ {h = 1} ^ {H} \nabla_ {\pmb {\theta}} f _ {\pmb {\theta}} (s _ {h}, a _ {h}) \cdot \nabla_ {\pmb {\theta}} f _ {\pmb {\theta}} (s _ {h}, a _ {h}) ^ {\top} \right].
+$$
+
+We argue that solving (2) with this form of  $\mathcal{I}(\theta, \pi)$  is a very intuitive objective, even in cases when the dynamics may not follow (3) exactly. Indeed, this suggests that during exploration, we should aim to reach states for which the dynamics  $f_{\theta}$  have a large gradient with respect to  $\theta$ —states for which the next state predicted by the dynamics is very sensitive to  $\theta$ . In such states, observing the next state gives us a significant amount of information on  $\theta$ , allowing us to accurately identify  $\theta^{\star}$ .
+
+A second challenge in solving (2) is that we do not know the true parameter  $\theta^{\star}$ , which the optimization (2) depends on. To circumvent this, we rely on domain randomization in choosing our exploration policy, solving instead:
+
+$$
+\pi_ {\exp} = \arg \min  _ {\pi} \mathbb {E} _ {\boldsymbol {\theta} \sim q _ {0}} [ \operatorname {t r} (\mathcal {I} (\boldsymbol {\theta}, \pi) ^ {- 1}) ] \tag {4}
+$$
+
+for some distribution over parameters  $q_0$ . While this is only an approximation of (2), in practice we find that this approximation yields effective exploration policies since, as described in Section 1, in many cases exploration policies require only a coarse model of the dynamics, and can therefore often be learned without precise knowledge of the unknown parameters.
+
+A final challenge is that, in general, we may not have access to a differentiable simulator, and our dynamics themselves may not be differentiable. In such cases,  $\nabla_{\theta}f_{\theta}(s_h,a_h)$  is unknown or
+
+undefined, and the above approach cannot be applied. As a simple solution to this, we rely on a finite-differences approximation to the gradient, which still provides an effective measure of how sensitive the next state is to the unknown parameter. In practice, to solve (4) and obtain an exploration policy, we rely on standard policy optimization algorithms, such as PPO (Schulman et al., 2017b).
+
+# 4.2 SYSTEM IDENTIFICATION
+
+ASID runs the exploration policy  $\pi_{\mathrm{exp}}$  (Section 4.1) in the real environment to generate a single trajectory  $\pmb{\tau}_{\mathrm{real}} \sim p_{\pmb{\theta}^{\star}}(\cdot \mid \pi_{\mathrm{exp}})$ . In the system identification phase, ASID then updates the simulator parameters using the collected trajectory. The goal is to find a distribution over simulator parameters that yield trajectories that match  $\pmb{\tau}_{\mathrm{real}}$  as closely as possible. In particular, we wish to find some distribution over simulation parameters,  $q_{\phi}$ , which minimizes:
+
+$$
+\mathbb {E} _ {\boldsymbol {\theta} \sim q _ {\phi}} \big [ \mathbb {E} _ {\boldsymbol {\tau} _ {\mathrm {s i m}} \sim p _ {\boldsymbol {\theta}} (\cdot | \mathcal {A} (\boldsymbol {\tau} _ {\mathrm {r e a l}}))} \big [ \| \boldsymbol {\tau} _ {\mathrm {r e a l}} - \boldsymbol {\tau} _ {\mathrm {s i m}} \| _ {2} ^ {2} \big ] \big ]
+$$
+
+where  $p_{\theta}(\cdot \mid \mathcal{A}(\tau_{\mathrm{real}}))$  denotes the distribution over trajectories generated by the simulator with parameter  $\theta$ , and playing the same sequence of actions as were played in  $\tau_{\mathrm{real}}$ . In practice, we apply REPS (Peters et al., 2010) for the simulation and Cross Entropy Method (CEM) for the real-world experiments. We stress that the ASD framework is generic, and other black-box optimization algorithms could be used instead.
+
+# 4.3 SOLVING THE DOWNSTREAM TASK
+
+After exploration and identification, the simulator can include information about the kinematic tree, position, orientation, or size of the object of interest, and the physical parameters of the real environment. With such a high-fidelity simulator, we aim to solve the downstream tasks entirely in simulation and transfer the learned policies  $\pi_{\mathrm{task}}$  to the real world in a zero-shot fashion. As with the system identification stage, ASD does not assume a particular method for solving the downstream task and any policy optimization algorithm can be used.
+
+# 5 EXPERIMENTAL EVALUATION
+
+In our experimental evaluation, we aim to answer the following questions:
+
+1. Does ASD's exploration strategy yield sufficient information to identify unknown parameters?  
+2. Are downstream task policies learned via ASD successful when using the identified parameters?  
+3. Does the paradigm suggested by ASD transfer to performing tasks in the real world?
+
+We conduct our experimental evaluation in two scenarios. First, we conduct empirical experiments entirely in simulation (Todorov et al., 2012) to validate the behavior of the exploration, system identification, and downstream task modules of ASD. This involves treating a true simulation environment as the real-world environment and then a reconstruction of this simulation environment as the approximate constructed simulation. Policies learned in this approximate simulation can then be evaluated back in the original simulation environment to judge efficacy. Second, we apply this to two real-world manipulation tasks that are inherently dependent on accurate parameter identification, showing the ability of ASD to function in the real world, using real-world data.
+
+# 5.1 ABLATIONS AND BASELINE COMPARISONS
+
+We compare ASD with several baselines and ablations for various portions of the pipeline.
+
+Exploration: To understand the importance of targeted exploration via Fisher information maximization we compare with two baselines. First, we compare with the naive approach of using data from a random policy for system identification that is not performing any targeted exploration. Secondly, we compare with Kumar et al. (2019), which aims to generate exploration trajectories that maximize mutual information with the parameters of interest. This method essentially learns a parameter inference network and then rewards exploration policies for minimizing its error.
+
+![](images/e92362ec669732740f6b8e7db813ec48adedc8d75b7c3f174cf56f5f2a34fe8b.jpg)  
+(a) Rod balancing
+
+![](images/697374f7e8c5fa0ed31e21a4d9f836dc63132e30c3abc6213cfba7507e1b4b0e.jpg)  
+(b) Sphere manipulation
+
+![](images/41f14a30bb866aa5e82f7a4e9ceb5dd22fb75be525a1022560ec047bc20c27b2.jpg)  
+(c) Multi-friction  
+Figure 3: Depiction of environments in simulation
+
+![](images/e311c93d59e4a2efe58624d13a6fd03b5bc085ac31599cf8f90d3276d909b669.jpg)  
+(d) Articulation
+
+System Identification: To evaluate the impact of system identification methods in this pipeline, we compare the effectiveness of using optimization-based system identification, e.g., (Peters et al., 2010; Memmel et al., 2022), with an end-to-end learned module (Kumar et al., 2019). This comparison shows the stability and extrapolation benefits of ASD over completely data-driven modeling techniques. In particular, we evaluate ASD with the optimization-based system identification replaced by the learned estimation module from Kumar et al. (2019) (ASID + estimator).
+
+Downstream Policy Learning: Since the eventual goal is to solve the downstream task, we finally compare how effective it is to learn downstream policies in simulation as opposed to an uninformed Domain Randomization (DR) over the parameters, in contrast to the targeted and identified parameters that stem from the ASD pipeline.
+
+# 5.2 SIMULATED TASK DESCRIPTIONS
+
+Sphere Manipulation: We consider two sphere manipulation tasks where physical parameters like rolling friction, object stiffness, and tangential surface friction are unknown and make a significant impact on policy performance: 1) striking a golf ball to reach a goal with unknown system friction (Figure 3b), 2) striking a golf ball in an environment with multiple patches that experience different surface friction (Figure 3c). In each scenario, the position of objects and the physical parameters are varied across evaluations. In this setting, we train downstream tasks with PPO.
+
+Rod Balancing: Balancing, or dynamic stacking of objects critically depends on an accurate estimate of the inertia parameters. We construct a rod balancing task where the agent can interact with a rod object (Figure 3a) to identify its physical parameters, in this case varying the distribution of mass along the rod. Once the parameters are identified, the rod must be balanced by placing it on a ledge (Figure 5 (left)). The error is measured by the tilting angle of the rod after placement, a task that requires accurate estimation of system parameters. In this case, we use the CEM to optimize downstream policies.
+
+Articulation: To stress generality, we consider environments that don't simply identify physical parameters like mass or friction but also the structure of the kinematic tree such as articulation and joint positioning. We consider an environment involving an articulated laptop model with a binary environment parameter representing whether articulation is present or not (Figure 3d).
+
+# 5.3 DOES ASID LEARN EFFECTIVE EXPLORATION BEHAVIOR?
+
+To evaluate the exploration behavior quantitatively, we consider the case of multiple unknown parameters, where learning each parameter requires exploration in a different region. In particular, we compare the exploration policy learned by our method with the exploration method of Kumar et al. (2019), on the multi-friction sphere manipulation environment illustrated in Figure 3c where the surface exhibits three different friction parameters (indicated by the grass, sand, and gravel textures). We initialize the golf ball in the leftmost region (grass)—to explore, the arm must strike the ball to other regions to identify their friction parameters. We plot a heat map of the regions visited by the ball during exploration in Figure 4. As can be seen, our approach achieves roughly uniform coverage over the entire space, learning to strike the ball into each region, and illustrating that our method is able to effectively handle complex exploration tasks that involve exciting multiple parameters. In contrast, the approach of Kumar et al. (2019) does not effectively move the sphere to regions different from the starting region.
+
+Table 1: Downstream task results in simulation: Random exploration fails in tasks where directed exploration is required, e.g., striking a sphere or finding the inertia of a rod. When placing the rod with a single action, domain randomization (DR) cannot solve the task without knowing the physical parameters. Learned system identification (Kumar et al. (2019) and  $\mathrm{ASID} +$  estimator) doesn't generalize to unseen trajectories and becomes far less effective than optimization-based system identification (cf.  $\mathrm{ASID} + \mathrm{SysID}$ ).  
+
+<table><tr><td>Task</td><td colspan="3">Rod Balancing</td><td>Sphere Striking</td></tr><tr><td>Metric</td><td colspan="3">Tilt angle in degree°</td><td>Success Rate in % ↑</td></tr><tr><td>Parameter</td><td>Inertia (left)</td><td>Inertia (middle)</td><td>Inertia (right)</td><td>Friction ~ [1.1, 1.5]</td></tr><tr><td>Random exploration</td><td>12.44 ± 19.6</td><td>4.20 ± 6.5</td><td>15.34 ± 15.9</td><td>10.62 ± 4.3</td></tr><tr><td>Kumar et al. (2019)</td><td>13.70 ± 9.3</td><td>2.82 ± 2.7</td><td>15.26 ± 9.8</td><td>9.50 ± 2.4</td></tr><tr><td>DR</td><td>26.69 ± 7.0</td><td>13.05 ± 7.3</td><td>1.13 ± 1.3</td><td>8.75 ± 1.5</td></tr><tr><td>ASID + estimator</td><td>17.73 ± 13.1</td><td>4.65 ± 5.1</td><td>9.99 ± 6.8</td><td>11.00 ± 5.2</td></tr><tr><td>ASID + SysID (ours)</td><td>0.00 ± 0.0</td><td>0.72 ± 1.0</td><td>0.00 ± 0.0</td><td>28.00 ± 9.7</td></tr></table>
+
+We further analyze the exploration behavior in the articulation environment (Figure 3d). Here our exploration policy interacts with the laptop  $80\%$  of the time as opposed to  $20\%$  for naïve baselines. Appendix A.2.2 shows that models trained to predict articulation from interaction data, e.g., Ditto (Jiang et al., 2022), can infer joint and part-level geometry from the collected data.
+
+# 5.4 HOW DOES ASID PERFORM
+
+# QUANTITATIVELY IN SIMULATION ON DOWNSSTREAM TASKS?
+
+To quantitatively assess the benefits of ASD over baselines and ablations, we evaluate the performance of ASD with respect to the baselines in simulation on downstream evaluation for the rod balancing (Figure 3a) and sphere manipulation (Figure 3c) task. Results are given in Table 1. Observing the performance on the rod environment, it becomes clear that previous methods struggle with one-shot tasks. When the inertia is not properly identified, CEM predicts the wrong action with high certainty, causing it to pick and place the rod at the wrong location. Domain randomization (DR) on the other hand tries to learn a policy over all possible mass distributions which in this case leads to a policy that always grasps the object at the center of the reference frame. The success here depends on "getting lucky" and sampling the correct action for the
+
+![](images/0c40718841ba689244bec07cb2924d2a4c39d4648a06a4c5472109abf931b90c.jpg)  
+(a) ASID
+
+![](images/8b6010dee3c3fba7b8e3ef7f5d25d11e90b565e5ce7dd628abb5d5afae3f9a63.jpg)  
+(b) Kumar et al. (2019)  
+Figure 4: Visitation frequency of the sphere when explored by different exploration policies on multi-friction (Figure 3c). ASD activates the sphere over a much larger area, thereby identifying parameters more accurately
+
+In the sphere environment, the distance rolled and bounce of the sphere are critically dependent on parameters such as friction and stiffness, and misidentifying these parameters can lead to significantly different trajectories and unsuccessful downstream behavior. This is reflected in the significantly higher success rate of ASD as compared to baselines that are uninformed of parameters and simply optimize for robust policies (DR), those that try to use learned estimators for exploration (Kumar et al., 2019) or using random exploration.
+
+# 5.5 DOES ASID ALLOW FOR REAL-WORLD CONTROLLER SYNTHESIS USING MINIMAL REAL-WORLD DATA?
+
+We further evaluate ASD on real-world tasks, replicating the rod balancing task from simulation (Figure 5) and introducing a novel shuffleboard task (Figure 6). As in simulation, we use a Franka Emika Panda robot for exploration and task performance in the real world. We compute the object's position and orientation by color-threshold the pointclouds from two calibrated Intel RealSense D455 cameras—this approach could easily be replaced by a more sophisticated pose estimation system if desired.
+
+Rod Balancing: The goal of this task is to properly balance a rod with an unknown mass distribution by informatively exploring the environment in the real world and using the resulting data to identify the appropriate physics parameters in simulation. The optimized controller in simulation is then deployed to perform the task in the real world. In this case, the downstream task is picking the rod at a certain point along its axis and balancing it on a perch (Figure 1, Figure 5).
+
+The policy executes the downstream task by a pick and place primitive parameterized by the exact pick point. We deploy ASD fully autonomously, executing exploration, system identification, downstream task training, and execution in an end-to-end fashion.
+
+The mass distribution in the rod is varied and both the inertia and the environment friction must be inferred. While ASD correctly identifies the inertia of the rod most of the time, we find that a center of mass close to the middle of the rod causes ambiguities that hurt our system identification process causing the simulation to not be accurate enough for zero-shot transfer. Overall, ASD solves the rod-balancing task 6/9 times across varying mass distribution and pose while a domain-randomization policy trained without any environment interaction fails to solve it at all (Table 2).
+
+Shuffleboard: This task is a scaled-down version of the popular bar game tabletop shuffle-
+
+![](images/97ead37ff4566571d10a86f2f5d049351f0c7ce96358a8e6e13d9e8dd95e06bd.jpg)  
+Figure 5: Real-world Rod Balancing: Simulation setup for training exploration and downstream task policies (left). Successful execution of autonomous real-world rod balancing with skewed mass (right).
+
+![](images/54acf3f7512ce9d910ff1d6f7d25bdab059e9cc0eba7c237c035d191e8756879.jpg)
+
+Table 2: Downstream task results in real: ASD successfully balances the rod while domain randomization (DR) fails catastrophically.  
+
+<table><tr><td>Task</td><td colspan="3">Rod Balancing</td></tr><tr><td>Inertia</td><td>left</td><td>middle</td><td>right</td></tr><tr><td>DR</td><td>0/3</td><td>0/3</td><td>0/3</td></tr><tr><td>ASID (ours)</td><td>2/3</td><td>1/3</td><td>3/3</td></tr></table>
+
+board where a puck must be shot to a target area on a slippery board. We closely follow the original game and pour wax (sand) on the board to decrease the surface friction. This modification makes the task especially difficult as the surface friction on the board changes slightly after each shot since the puck displaces the wax. The goal is to strike the puck to one of the target regions—one closer (yellow) and one further (blue) away from the robot (Figure 2, Figure 6). After exploring the scene, ASD estimates the sliding and torsional friction of the puck to account for the changing surface friction of the board. For executing the downstream task, a primitive positions the endeffector at a fixed distance to the puck and a policy predicts a force value that parameterizes a shot attempt.
+
+Due to the changing surface friction, the domain randomization baseline struggles to shoot the puck to the desired zone. While it succeeds 3/10 times—probably because the surface friction was part of its training distribution—the other attempts fall short or overshoot the target. With its dedicated exploration phase, ASD can accurately adapt the simulation to the current conditions and lands the puck in the desired zone 7/10 times (Table 3).
+
+# 6 DISCUSSION
+
+In this work, we introduced a novel pipeline for performing real-world control by autonomously exploring the environment, using the collected data for system identification, and the resulting identified system for downstream policy optimization. In essence, this sim-to-real-to-sim-to-real pipeline allows for targeted test-time construction of simulation environments in a way that enables the performance of downstream tasks. We demonstrate how this
+
+![](images/f9b8c663ae723771e27fc999493f3fe12a12c45a0f4dbe4e7fd23cb988daca86.jpg)  
+Figure 6: Real-world Shuffleboard: Simulation setup for training exploration and downstream task policies (left). Successful strike to the yellow zone (right).
+
+![](images/0e07bb54aaf6c5bdc8929af19ee9c0a283b511b7ec3610226617bc2a0d40ef50.jpg)
+
+Table 3: Downstream task results in real: ASID outperforms domain randomization (DR) on shooting the puck to the desired zones.  
+
+<table><tr><td>Task</td><td colspan="2">Shuffleboard</td></tr><tr><td>Target zone</td><td>yellow (close)</td><td>blue (far)</td></tr><tr><td>DR</td><td>2/5</td><td>1/5</td></tr><tr><td>ASID (ours)</td><td>4/5</td><td>3/5</td></tr></table>
+
+type of identification requires autonomous and carefully directed exploration, and introduce a novel algorithm based on Fisher information maximization that is able to accomplish this directed exploration. The autonomously collected trajectories can then be paired with downstream optimization-based system identification and reconstruction algorithms for accurate simulation construction and downstream policy learning. We show the efficacy of this paradigm on multiple environments in simulation, as well as on rod balancing and shuffleboard, two challenging real-world tasks.
+
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+
+# A APPENDIX
+
+# A.1 TASKDETAILS
+
+# A.1.1 SPHERE MANIPULATION
+
+In the sphere manipulation tasks, the observation space consists of endeffector position, sphere position, and robot joint angles. During the training of the exploration policy, we randomize the location of a sphere with  $r = 0.03$  to be between  $x \in [0.430, 0.65]$ ,  $y \in [-0.2, 0.23]$  for training and  $x \in [0.55, 0.65]$ ,  $y \in [-0.2, 0.23]$  for evaluation. Parameter ranges for training are 1) friction  $\theta \in [1e - 3, 5e - 3]$  and 2) patch friction  $\theta_0, \theta_1, \theta_2 \in [1e - 5, 1e - 3]$  and set fixed sphere and goal locations as well as parameters for the downstream task. For 1) we attach a paddle and limit the endeffector position such that the robot has to strike the sphere to the goal.
+
+# A.1.2 ARTICULATION
+
+In the case of the articulation environment, the parameter is binary and indicates whether articulation is present or not. During evaluation, articulation is always turned on and success is indicated by  $\delta \beta > 1e - 2$ . The initial laptop state is randomized over position  $x \in [0.45, 0.65]$ ,  $y \in [-0.1, 0.1]$ ,  $yaw \in [0.00, 3.14]$  and lid angle  $\beta \in [1.2, 2.5]$ . The observation space contains the endeffector position, joint angles, position, orientation, and joint angle of the laptop.
+
+# A.1.3 ROD BALANCING
+
+Similar to the sphere manipulation, we restrict the exploration policy to endeffector control with  $\delta x, \delta y$  and attach a peg to the Franka. The rod has dimensions  $0.04 \times 0.3 \times 0.04$  and initializes as  $x \in [0.5, 0.6], y \in [-0.2, 0.2], yaw \in [0.00, 3.14]$
+
+# A.2 QUALITATIVE RESULTS
+
+# A.2.1 EXPLORATION STRATEGIES LEARNED BY THE AGENT
+
+We evaluate the exploration behavior qualitatively across multiple environments to understand whether the exploration behavior is meaningful. In the sphere environment, we observe the agent hitting the sphere multiple times when it bounces off the walls or stays within reach. The other environments also experience emergent behavior, as the agent executes a horizontal motion towards the top of the lid if the laptop is mostly open while an almost closed laptop causes it to push from top to bottom instead. Finally, when determining the rod's inertia, the policy pushes both sides to gain maximum information about the center of mass through the rotation motions in both directions. See Figure 7 for a visualization of the exploration for the sphere environment.
+
+![](images/88e67f76b3dcac6ed7fe3a4e058f7ddf1cb022d93485399766847d94e4a9b17c.jpg)  
+(a)
+
+![](images/b2c8249d46910b95a810fa027d282907f3608060d880abc644df4e85f57cd571.jpg)  
+(b)  
+Figure 7: Absolute sphere displacement for different sphere starting locations. Zero means the sphere didn't get hit, higher numbers denote larger displacements. Initial endeffector position marked in red a) Random Coverage, b) PPO Coverage, c) Fisher Coverage, d) Legend.
+
+![](images/a7a7ff719430050f1d379aa117982acee7b77226230dc727ec36f90e676311ed.jpg)  
+(c)  
+(d)
+
+# A.2.2 RECONSTRUCTING ARTICULATION FROM INTERACTION DATA
+
+![](images/d86c3bd2c58a00e28a097c6008a5bfa708613a3ce450ddb9d779b82c8ca39d2e.jpg)  
+Figure 8: Articulated object: before (left) and after (middle) exploration with ASD and part-level reconstruction with Ditto (Jiang et al., 2022): articulated  $\blacksquare$  and static part  $\blacksquare$  (right).
+
+![](images/cc025e6a51187e73099ace305b85fbec9d46d5e729ea1d9f081f1e292970d4cf.jpg)
+
+![](images/b6ddffa2c57a5a4ae9f1223409d84cefbe36df9300c3304b72f3d0d3d5aa3c05.jpg)
+
+# A.3 COMPARISON TO MODEL-BASED APPROACHES
+
+# A.3.1 ENVIRONMENT SETUP
+
+![](images/c7328bb6cec3777cc27662812a4da8d072f749241e6a6bd00ee4d48a9f238ff0.jpg)  
+Figure 9: Environment with multiple spheres. The single red sphere is subject to changing friction values while the three blue spheres act as distractors.
+
+# A.3.2 MODEL-BASED EXPLORATION
+
+In this section, Figure 10, we compare the performance of our algorithm to that of the MAX algorithm (Shyam et al., 2019).
+
+MAX aims to cover the state space through multiple policies throughout training. Since it uses the disagreement between fully learned dynamics models, it gets distracted by novel states induced by the movement of all spheres (red and blue). In contrast, our method based on the Fisher information yields a single policy that seeks out the sphere affected by the changing physics parameters (red) and ignores the irrelevant spheres (blue) even if they lead to novel states.
+
+![](images/42cc6b51bf4a68edaf9d18fcb55084cd20c81045542b5b0c2f6c2edd500418d3.jpg)  
+(a) MAX (checkpoints throughout training)
+
+![](images/28e5f461d8253e375093814e1f8991e08e868d30966e6170ba0cb4c6957de18a.jpg)  
+(b) MAX (last checkpoints)
+
+![](images/d15278b0ba3718195425ac294d65e03120c64ef996de0ca1902a4da193d430b2.jpg)  
+(c) Fisher information (checkpoints throughout training)
+
+![](images/124b7c407801a03b46742668c6e148ce184a9ac8dadf85813748a1b9cd5cc427.jpg)  
+(d) Fisher information (last checkpoint)  
+Figure 10: Exploration MAX vs. Fisher information: Trajectories collected (30) from policy checkpoints throughout training or from the last policy checkpoint. While Model-based Active Exploration (MAX) (Shyam et al., 2019) explores the state space over the course of training, getting distracted by the novel states induced by the blue spheres  $\blacksquare$  (c.f. fig. 10a), Fisher information-based exploration (ours) shows directed exploration (c.f. fig. 10d), moving towards the sphere with changing parameters (red  $\blacksquare$ ) and yielding a single exploration policy.
+
+# A.3.3 LEARNED STATE-TRANSITION MODELS VS. SIMULATORS
+
+We next illustrate the improvement using a simulator vs a fully learned dynamics model can give. We train a forward dynamics model (three layer MLP) on data generated both from MAX and our exploration procedure. When evaluated on out-of-distribution trajectories, i.e., trajectories not included in the training data, we find the model to be extremely inaccurate (see Figure 11). While the simulator extrapolates to unseen states and correctly predicts the movement of the sphere, the model hallucinates movement even when the end-effector does not interact with it at all! These findings make ASID preferable to a purely model-based approach.
+
+![](images/bfd00204d235cf2a3c9177a6a5c7a944551fcf20cb315ae56cf35d87e36cc208.jpg)  
+(a) MAX (pushing actions)
+
+![](images/dfcecb9588b5fee519beeeaba891b25e722176fe38ca51c1775889130449ef55.jpg)  
+(b) MAX (zero actions)
+
+![](images/64d2a64cfab890a9bb5440d13cf2b56bce855742c7e4e10b5e4b4363f7a13fce.jpg)  
+(c) Fisher information (pushing actions)
+
+![](images/c90f5215d9859b8479126a23a040eb3c982b0408e8f6bba1c854e99bb458604e.jpg)  
+(d) Fisher information (zero actions)  
+Figure 11: Learned models vs. simulator: Evaluation of a forward model trained on trajectories (30) from MAX (Shyam et al., 2019) (checkpoints throughout training) and Fisher information (last checkpoint). In contrast to our simulator, the learned model fails to predict the red sphere's trajectory accurately even under no contact scenarios (c.f. fig. 11b, fig. 11d).
+
+# A.4 POLICY GENERALIZATION
+
+We evaluate the generalization capabilities of our exploration policy for different sphere locations and parameter combinations seen and unseen during training (Figure 13). Since we train the policies using an arena setup, we remove the arena during this evaluation to be able to query sphere locations outside of it (Figure 12).
+
+![](images/0a66e7e74d783e56c6e303d23d942a3a67b0149d34f3648708a61fbd50b31157.jpg)  
+Figure 12: Environment with free space. The red sphere is subject to changing parameter values and position is randomized.
+
+![](images/0be412378c5a9e3a00a2a7339350ce0bc846481b2cf693a9e11d163976eddfe3.jpg)  
+(a) Changing friction parameters (in distribution)
+
+![](images/582062e3989358cdc7aa689d662b78257d4bac60b320c6b11641ce8c93a6c47e.jpg)  
+(b) Changing mass parameter (out of distribution)  
+Figure 13: Generalization capabilities of an exploration policy trained on sphere friction for different sphere starting locations. The plot shows the success rate of pushing the sphere spawned at the corresponding x-y-location and evaluated over 5 different seeds, i.e., different physics parameter values. Initial endeffector position marked in red. Boxes denote sphere locations seen during training (purple box) and reachable area of the endeffector (orange box). Our policies experience surprising generalization capabilities to unseen sphere locations as long as they can be reached by the robot (cf.fig. 13a). Furthermore, the actions extrapolate to unseen physics parameters as long as they can be uncovered with similar interactions, e.g., pushing (cf.fig. 13b). This is the case because the policy does not have any information about the underlying physics parameters until it hits the sphere.
\ No newline at end of file
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+                                    "content": "Model-free control strategies such as reinforcement learning have shown the ability to learn control strategies without requiring an accurate model or simulator of the world. While this is appealing due to the lack of modeling requirements, such methods can be sample inefficient, making them impractical in many real-world domains. On the other hand, model-based control techniques leveraging accurate simulators can circumvent these challenges and use a large amount of cheap simulation data to learn controllers that can effectively transfer to the real world. The challenge with such model-based techniques is the requirement for an extremely accurate simulation, requiring both the specification of appropriate simulation assets and physical parameters. This requires considerable human effort to design for every environment being considered. In this work, we propose a learning system that can leverage a small amount of real-world data to autonomously refine a simulation model and then plan an accurate control strategy that can be deployed in the real world. Our approach critically relies on utilizing an initial (possibly inaccurate) simulator to design effective exploration policies that, when deployed in the real world, collect high-quality data. We demonstrate the efficacy of this paradigm in identifying articulation, mass, and other physical parameters in several challenging robotic manipulation tasks, and illustrate that only a small amount of real-world data can allow for effective sim-to-real transfer. Project website at https://weirdlabuw.github.io/asid"
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+                    "lines": [
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Taking inspiration from system identification, we argue that the key to effective sim2real transfer is an initial round of exploration in the real world to learn an effective simulator. We propose a generic pipeline for sim2real transfer, Active Exploration for System IDentification (ASID), which decouples exploration and exploitation: (1) exploration in real to collect informative data of unknown parameters, (2) refinement of the simulation parameters using data collected in real, (3) policy training on the updated simulator to accomplish the goal tasks. Our exploration procedure is motivated by work in theoretical statistics and seeks to induce trajectories corresponding to large Fisher information, thereby providing maximally informative observations. By using our initial round of exploration to obtain accurate estimates of the parameters in the real world, we show that in many cases, the policies trained in step (3) successfully transfer to real in a zero-shot fashion, even in settings where training a policy in sim without additional knowledge of real would fail."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
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+                                    "type": "text",
+                                    "content": "A key insight in our approach is that, while a policy trained in sim to accomplish the goal task may not effectively transfer, strategies that explore effectively in sim often also explore effectively in real. As an example, say our goal task is to hit a ball to a particular location with a robotic arm and assume the mass of the ball is unknown. If we train a policy in sim to hit the ball without knowledge of the mass, when deployed in real it will almost certainly fail, as the force at which it should strike the ball depends critically on the (unknown) mass. To learn the mass, however, essentially any contact between the ball and the arm suffices. Achieving some contact between the ball and the arm requires a significantly less precise motion, and indeed, does not require any prior knowledge of the mass. We can therefore train a policy in sim that learns to effectively explore—hit the ball in any direction—and deploy this in real to collect information on the true parameters, ultimately allowing us to obtain a higher-fidelity simulator that does allow sim2real transfer on the goal task."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
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+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "We are particularly interested in the application of our pipeline to modern robotic settings and evaluate ASD on four tasks: sphere manipulation, laptop articulation, rod balancing, and shuffleboard. We show that in all settings, our approach is able to effectively identify unknown parameters of the real environment (e.g. geometry, articulation, center of mass, and physical parameters like mass, friction, or stiffness), and using this knowledge, learn a policy in sim for the goal task that successfully transfers to real. In all cases, by deploying effective exploration policies trained in simulation, we require only a very small amount of data from real—typically a single episode of data suffices."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
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+                                    "type": "text",
+                                    "content": "2 RELATED WORK"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
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+                                    "type": "text",
+                                    "content": "System Identification: Our work is closely related to the field of system identification (Åström & Eykhoff, 1971; Söderström & Stoica, 1989; Ljung, 1998; Schon et al., 2011; Menda et al., 2020), which studies how to learn a model of the system dynamics efficiently. A large body of work, stretching back decades, has explored how inputs to a system should be chosen to most effectively learn the system's parameters (Mehra, 1974; 1976; Goodwin & Payne, 1977; Hjalmarsson et al., 1996; Lindqvist & Hjalmarsson, 2001; Gerencsér & Hjalmarsson, 2005; Rojas et al., 2007; Gevers et al., 2009; Gerencsér et al., 2009; Manchester, 2010; Rojas et al., 2011; Bombois et al., 2011; Hägg et al., 2013; Wagenmaker & Jamieson, 2020; Wagenmaker et al., 2021; Mania et al., 2022; Wagenmaker et al., 2023) or how to deal with partial observability (Schön et al., 2011; Menda et al., 2020). Similar to our exploration strategy, many of these works choose their inputs to maximize some function of the Fisher information matrix. A primary novelty of our approach is to use a simulator to learn effective exploration policies, and to apply our method to modern, real-world robotics tasks—indeed, our work can be seen as bridging the gap between classical work on system identification and modern sim2real techniques. While the aforementioned works are primarily theoretical, recent work has studied the application of such methods to a variety of real-world settings like active identification of physics parameters (Xu et al., 2019; Kumar et al., 2019; Mavrakis et al., 2020; Gao et al., 2020; 2022) or kinematic structure (Mo et al., 2021; Wang et al., 2022; Nie et al., 2022; Hsu"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                }
+            ],
+            "discarded_blocks": [
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
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+                                    "type": "text",
+                                    "content": "2"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                }
+            ],
+            "page_size": [
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+                792
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+        {
+            "para_blocks": [
+                {
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+                    ],
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+                                    "type": "text",
+                                    "content": "et al., 2023) through object-centric primitives. Another line of recent work aims to learn the parameters of the simulator to ultimately train a downstream policy on the learned parameters, and therefore apply task-specific policies for data collection (Zhu et al., 2018; Chebotar et al., 2019; Huang et al., 2023; Ren et al., 2023) or exploration policies that minimize its regret (Liang et al., 2020). The majority of these works, however, do not consider running an exploration policy that targets learning the unknown parameters, do not address solving downstream tasks, or rely on techniques that do not scale effectively to more complex tasks."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "Simulation-to-Reality Transfer: Transferring learned policies from sim2real has shown to be successful in challenging tasks like dexterous manipulation (OpenAI et al., 2018; Handa et al., 2022; Chen et al., 2022), locomotion (Rudin et al., 2022), agile drone flight (Sadeghi & Levine, 2016) or contact rich assembly tasks Tang et al. (2023), yet challenges remain due to the sim2real gap. To deal with the gap, Domain Randomization (DR) (Tobin et al., 2017) trains policies over a distribution of environments in simulation, hoping for the real world to be represented among them. Subsequent works adaptively change the environment distribution (Muratore et al., 2019; Mehta et al., 2020) and incorporate real data (Chebotar et al., 2019; Ramos et al., 2019; Duan et al., 2023; Chen et al., 2023; Ma et al., 2023; Torne et al., 2024). While similar to our approach, these methods do not perform targeted exploration in real to update the simulator parameters. Other approaches seek to infer and adapt to simulation parameters during deployment (Kumar et al., 2021; Qi et al., 2023; Margolis et al., 2023), leverage offline data (Richards et al., 2021; Bose et al., 2024), or adapt online (Sinha et al., 2022); in contrast, we do not learn such an online adaptation strategy, but rather a better simulator. Finally, a commonly applied strategy is to train a policy in sim and then fine-tune in the real environment (Julian et al., 2021; Smith et al., 2022; Nakamoto et al., 2023); in contrast, we are interested in the (more challenging) regime where a direct transfer is not likely to give any learning signal to fine-tune from."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Model-Based RL: Model-based RL (MBRL) aims to solve the RL problem by learning a model of the dynamics, and using this model to either plan or solve a policy optimization problem (Deisenroth & Rasmussen, 2011; Williams et al., 2017; Nagabandi et al., 2018; Chua et al., 2018; Janner et al., 2019; Hafner et al., 2019; 2020; Janner et al., 2022; Zhu et al., 2023). While our approach is model-based in some sense, the majority of work in MBRL focuses on fully learned dynamic models; in contrast, our \"model\" is our simulator, and we aim to learn only a very small number of parameters, which can be much more sample-efficient. Furthermore, MBRL methods typically do not perform explicit exploration, while a key piece of our approach is a targeted exploration procedure. The MBRL works we are aware of which do rely on targeted exploration (Shyam et al., 2019; Pathak et al., 2019) typically rely on fully learned dynamic models and apply somewhat different exploration strategies, which we show in Appendix A.3.2 can perform significantly worse."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
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+                                    "type": "text",
+                                    "content": "3 PRELIMINARIES"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
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+                    "angle": 0,
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "We formulate our decision-making setting as Markov Decision Processes (MDPs). An MDP is defined as a tuple "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M^{\\star} = (\\mathcal{S},\\mathcal{A},\\{P_h^{\\star}\\}_{h = 1}^H,P_0,\\{r_h\\}_{h = 1}^H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": ", where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
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+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{S}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": " is the set of states, "
+                                },
+                                {
+                                    "bbox": [
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+                                        525,
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+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{A}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": " the set of actions, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{h}:S\\times \\mathcal{A}\\to \\triangle_{\\mathcal{S}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": " the transition kernel, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_0\\in \\triangle_S"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": " the initial state distribution, and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r_h:\\mathcal{S}\\times \\mathcal{A}\\rightarrow \\mathbb{R}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": " the reward function. We consider the episodic setting. At the beginning of an episode, the environment samples a state "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s_1\\sim P_0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": ". The agent observes this state, plays some action "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "a_1\\in \\mathcal{A}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
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+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": ", and transitions to state "
+                                },
+                                {
+                                    "bbox": [
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+                                        506,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s_2\\sim P_1(\\cdot |s_1,a_1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": ", receiving reward "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r_1(s_1,a_1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": ". After "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": " steps, the environment resets and the process repeats. Our primary goal is to learn a policy "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": "-a mapping from states to actions—that maximizes reward in the true environment. We denote the value of a policy by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "V_0^\\pi \\coloneqq \\mathbb{E}_{M^\\star ,\\pi}[\\sum_{h = 1}^H r_h(s_h,a_h)]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": ", where the expectation is over trajectories induced playing policy "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": " on MDP "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M^{\\star}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": ". We think of the reward "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": " as encoding our downstream task, and our end goal is to find a policy that solves our task, maximizing "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "V_0^\\pi"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": ". We denote such policies as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi_{\\mathrm{task}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        506,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        654,
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "In the sim2real setting considered in this work, we assume that the reward is known, but that the dynamics of the real environment, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        654,
+                                        506,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P^{\\star} = \\{P_h^{\\star}\\}_{h=1}^H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        654,
+                                        506,
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+                                    ],
+                                    "type": "text",
+                                    "content": ", are initially unknown. However, we assume that they belong to some known parametric family "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        654,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{P} := \\{P_{\\theta} : \\theta \\in \\Theta\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        654,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": ", so that there exists some "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        654,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta^{\\star} \\in \\Theta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        654,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        654,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P^{\\star} = P_{\\theta^{\\star}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        654,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": ". Here we take "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        654,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        654,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": " to be some unknown parameter (for example, mass, friction, etc.), and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        654,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{\\theta}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        654,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": " the dynamics under parameter "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        654,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        654,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": " (which we might know from physics, first principles, etc.). For any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        654,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        654,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": " and policy "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        654,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        654,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": ", the dynamics "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        654,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{\\theta}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        654,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": " induce a distribution over state-action trajectories, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        654,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau = (s_1, a_1, s_2, \\dots, s_H, a_H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        654,
+                                        506,
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+                                    ],
+                                    "type": "text",
+                                    "content": ", which we denote by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        654,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p_{\\theta}(\\cdot \\mid \\pi)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        654,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": ". We can think of"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
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+                    "type": "header",
+                    "angle": 0,
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
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+                                    "type": "text",
+                                    "content": "3"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                }
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+                            "index": 1,
+                            "angle": 0,
+                            "type": "image_body"
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+                                            "type": "text",
+                                            "content": "Figure 2: Overview of ASD: (1) Train an exploration policy "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                180,
+                                                504,
+                                                231
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pi_{\\mathrm{exp}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                180,
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+                                                231
+                                            ],
+                                            "type": "text",
+                                            "content": " that maximizes the Fisher information, leveraging the vast amount of cheap simulation data. (2) Roll out "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                180,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pi_{\\mathrm{exp}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                180,
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+                                                231
+                                            ],
+                                            "type": "text",
+                                            "content": " in real to collect informative data that can be used to (3) run system identification to identify physics parameters and reconstruct, e.g., geometric, collision, and kinematic properties. (4) Train a task-specific policy "
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pi_{\\mathrm{task}}"
+                                        },
+                                        {
+                                            "bbox": [
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+                                                180,
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+                                                231
+                                            ],
+                                            "type": "text",
+                                            "content": " in the updated simulator and (5) zero-shot transfer "
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pi_{\\mathrm{task}}"
+                                        },
+                                        {
+                                            "bbox": [
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+                                                180,
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+                                                231
+                                            ],
+                                            "type": "text",
+                                            "content": " to the real world."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 2,
+                            "angle": 0,
+                            "type": "image_caption"
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
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+                        266
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "our simulator as instantiating "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
+                                        504,
+                                        266
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p_{\\theta}(\\cdot \\mid \\pi)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
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+                                        266
+                                    ],
+                                    "type": "text",
+                                    "content": "—we assume our simulator is able to accurately mimic the dynamics of an MDP with parameter "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
+                                        504,
+                                        266
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\theta}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
+                                        504,
+                                        266
+                                    ],
+                                    "type": "text",
+                                    "content": " under policy "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
+                                        504,
+                                        266
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
+                                        504,
+                                        266
+                                    ],
+                                    "type": "text",
+                                    "content": ", generating samples "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
+                                        504,
+                                        266
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\tau} \\sim p_{\\theta}(\\cdot \\mid \\pi)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
+                                        504,
+                                        266
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        270,
+                        504,
+                        335
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                270,
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+                                335
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
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+                                        335
+                                    ],
+                                    "type": "text",
+                                    "content": "In addition, we also assume that samples from our simulator are effectively \"free\"—for any "
+                                },
+                                {
+                                    "bbox": [
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+                                        270,
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+                                        335
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
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+                                        335
+                                    ],
+                                    "type": "text",
+                                    "content": " and policy "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
+                                        504,
+                                        335
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
+                                        504,
+                                        335
+                                    ],
+                                    "type": "text",
+                                    "content": ", we can generate as many trajectories "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
+                                        504,
+                                        335
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau \\sim p_{\\theta}(\\cdot \\mid \\pi)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
+                                        504,
+                                        335
+                                    ],
+                                    "type": "text",
+                                    "content": " as we wish. Given this, it is possible to find the optimal policy under "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
+                                        504,
+                                        335
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
+                                        504,
+                                        335
+                                    ],
+                                    "type": "text",
+                                    "content": " by simply running any standard RL algorithm in simulation. With knowledge of the true parameters "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
+                                        504,
+                                        335
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta^{\\star}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
+                                        504,
+                                        335
+                                    ],
+                                    "type": "text",
+                                    "content": ", we can then easily find the optimal policy in real by sampling trajectories from the simulated environment with parameter "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
+                                        504,
+                                        335
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta^{\\star}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
+                                        504,
+                                        335
+                                    ],
+                                    "type": "text",
+                                    "content": ". It follows that, if we can identify the true parameter "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
+                                        504,
+                                        335
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta^{\\star}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
+                                        504,
+                                        335
+                                    ],
+                                    "type": "text",
+                                    "content": " in real, we can solve the goal task."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        105,
+                        342,
+                        287,
+                        354
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "We consider the following learning protocol:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 9,
+                    "blocks": [
+                        {
+                            "bbox": [
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+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": "1. Learner chooses exploration policy "
+                                        },
+                                        {
+                                            "bbox": [
+                                                105,
+                                                361,
+                                                504,
+                                                386
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pi_{\\mathrm{exp}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                105,
+                                                361,
+                                                504,
+                                                386
+                                            ],
+                                            "type": "text",
+                                            "content": " and plays it in real for a single episode, generating trajectory "
+                                        },
+                                        {
+                                            "bbox": [
+                                                105,
+                                                361,
+                                                504,
+                                                386
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pmb{\\tau}_{\\mathrm{real}} \\sim p_{\\pmb{\\theta}^*}(\\cdot \\mid \\pi_{\\mathrm{exp}})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                105,
+                                                361,
+                                                504,
+                                                386
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 6
+                        },
+                        {
+                            "bbox": [
+                                104,
+                                387,
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+                                399
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "spans": [
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+                                            "bbox": [
+                                                104,
+                                                387,
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+                                                399
+                                            ],
+                                            "type": "text",
+                                            "content": "2. Using "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                387,
+                                                479,
+                                                399
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pmb{\\tau}_{\\mathrm{real}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                387,
+                                                479,
+                                                399
+                                            ],
+                                            "type": "text",
+                                            "content": " and the simulator in any way they wish, the learner obtains some policy "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                387,
+                                                479,
+                                                399
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pi_{\\mathrm{task}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                387,
+                                                479,
+                                                399
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 7
+                        },
+                        {
+                            "bbox": [
+                                105,
+                                401,
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+                                415
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                    ],
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+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": "3. Learner deploys "
+                                        },
+                                        {
+                                            "bbox": [
+                                                105,
+                                                401,
+                                                382,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pi_{\\mathrm{task}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                105,
+                                                401,
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+                                            ],
+                                            "type": "text",
+                                            "content": " in real and suffers loss "
+                                        },
+                                        {
+                                            "bbox": [
+                                                105,
+                                                401,
+                                                382,
+                                                415
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\max_{\\pi} V_0^{\\pi} - V_0^{\\pi_{\\mathrm{task}}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                105,
+                                                401,
+                                                382,
+                                                415
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 8
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
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+                        456
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "The goal of the learner is then to learn as much useful information as possible about the real environment from a single episode of interaction and use this information to obtain a policy that can solve the task in real as effectively as possible."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
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+                    ],
+                    "type": "text",
+                    "angle": 0,
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Parameter Estimation and Fisher Information: The Fisher information matrix plays a key role in the choice of our exploration policy, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        468,
+                                        504,
+                                        502
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+                                    "type": "inline_equation",
+                                    "content": "\\pi_{\\mathrm{exp}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        468,
+                                        504,
+                                        502
+                                    ],
+                                    "type": "text",
+                                    "content": ". Recall that, for a distribution "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        468,
+                                        504,
+                                        502
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p_{\\theta}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        468,
+                                        504,
+                                        502
+                                    ],
+                                    "type": "text",
+                                    "content": ", satisfying certain regularity conditions, the Fisher information matrix is defined as:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
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+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
+                                        203,
+                                        506,
+                                        405,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathcal {I} (\\pmb {\\theta}) := \\mathbb {E} _ {\\pmb {\\tau} \\sim p _ {\\pmb {\\theta}}} \\left[ \\nabla_ {\\pmb {\\theta}} \\log p _ {\\pmb {\\theta}} (\\pmb {\\tau}) \\cdot \\nabla_ {\\pmb {\\theta}} \\log p _ {\\pmb {\\theta}} (\\pmb {\\tau}) ^ {\\top} \\right].",
+                                    "image_path": "70f4b44385ee3c5736da64cd11ac7176783aeef9b904ed9ecda1edb76c1bce13.jpg"
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+                    "angle": 0,
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+                            "bbox": [
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+                                563
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "text",
+                                    "content": "Assume that we have access to data "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathfrak{D} = (\\pmb{\\tau}_t)_{t=1}^T"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "text",
+                                    "content": ", where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\tau}_t \\sim p_{\\pmb{\\theta}^\\star}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "text",
+                                    "content": " for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t = 1, \\dots, T"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "text",
+                                    "content": ", and let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\widehat{\\pmb{\\theta}}(\\mathfrak{D})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "text",
+                                    "content": " denote some unbiased estimator of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\theta}^\\star"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then the Cramer-Rao lower bound (see e.g. Pronzato & Pázman (2013)) states that, under certain regularity conditions, the covariance of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\widehat{\\pmb{\\theta}}(\\mathfrak{D})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfies:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
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+                        567,
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+                        582
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+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\mathbb {E} _ {\\mathfrak {D} \\sim p _ {\\theta^ {\\star}}} [ (\\widehat {\\boldsymbol {\\theta}} (\\mathfrak {D}) - \\boldsymbol {\\theta} ^ {\\star}) (\\widehat {\\boldsymbol {\\theta}} (\\mathfrak {D}) - \\boldsymbol {\\theta} ^ {\\star}) ^ {\\top} ] \\succeq T ^ {- 1} \\cdot \\mathcal {I} (\\boldsymbol {\\theta} ^ {\\star}) ^ {- 1}.",
+                                    "image_path": "950bd9e07022bef2405949302e6cda7129e685d1c346fdffe737d08fa709b1e5.jpg"
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+                {
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+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "type": "text",
+                                    "content": "From this it follows that the Fisher information serves as a lower bound on the mean-squared error:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
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+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\mathbb {E} _ {\\mathfrak {D} \\sim p _ {\\boldsymbol {\\theta} ^ {\\star}}} [ \\| \\widehat {\\boldsymbol {\\theta}} (\\mathfrak {D}) - \\boldsymbol {\\theta} ^ {\\star} \\| _ {2} ^ {2} ] = \\operatorname {t r} \\left(\\mathbb {E} _ {\\mathfrak {D} \\sim p _ {\\boldsymbol {\\theta} ^ {\\star}}} \\left[ \\left(\\widehat {\\boldsymbol {\\theta}} (\\mathfrak {D}) - \\boldsymbol {\\theta} ^ {\\star}\\right) \\left(\\widehat {\\boldsymbol {\\theta}} (\\mathfrak {D}) - \\boldsymbol {\\theta} ^ {\\star}\\right) ^ {\\top} \\right]\\right) \\geq T ^ {- 1} \\cdot \\operatorname {t r} \\left(\\mathcal {I} \\left(\\boldsymbol {\\theta} ^ {\\star}\\right) ^ {- 1}\\right). \\tag {1}",
+                                    "image_path": "cf38a8d4738f29767a6d5a1f3a5921ec6e4567240a30a1d1acf0152ce0372e0e.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
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+                    "angle": 0,
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "This is in general tight—for example, the maximum likelihood estimator satisfies (1) with equality as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    "type": "inline_equation",
+                                    "content": "T \\to \\infty"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " (Van der Vaart, 2000). The Fisher information thus serves as a fundamental lower bound on parameter estimation error, a key motivation for our exploration procedure."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        104,
+                        670,
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+                        697
+                    ],
+                    "type": "title",
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "4 ASID: TARGETED EXPLORATION FOR TEST-TIME SIMULATION CONSTRUCTION, IDENTIFICATION, AND POLICY OPTIMIZATION"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
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+                                    "type": "text",
+                                    "content": "In this section, we present our proposed approach, ASD, a three-stage pipeline illustrated in Figure 2. We describe each component of ASD in the following."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                }
+            ],
+            "discarded_blocks": [
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
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+                                    "type": "text",
+                                    "content": "4"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                }
+            ],
+            "page_size": [
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+            ],
+            "page_idx": 3
+        },
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+                                    "type": "text",
+                                    "content": "4.1 EXPLORATION VIA FISHER INFORMATION MAXIMIZATION"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
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+                {
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+                                    ],
+                                    "type": "text",
+                                    "content": "As motivated in Section 3, to learn a policy effectively accomplishing our task, it suffices to accurately identify "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta^{\\star}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
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+                                    ],
+                                    "type": "text",
+                                    "content": ". In the exploration phase, step 1 in our learning protocol, our goal is to then play an exploration policy "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        159
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi_{\\mathrm{exp}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        159
+                                    ],
+                                    "type": "text",
+                                    "content": " which generates a trajectory on the real environment that provides as much information on "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
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+                                        159
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta^{\\star}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        159
+                                    ],
+                                    "type": "text",
+                                    "content": " as possible. Following Section 3, the Fisher information gives a quantification of the usefulness of the data collected, which motivates our approach."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
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+                        504,
+                        198
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
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+                                        198
+                                    ],
+                                    "type": "text",
+                                    "content": "In our setting, the distribution over trajectories generated during exploration in real, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        163,
+                                        504,
+                                        198
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\tau}_{\\mathrm{real}} \\sim p_{\\pmb{\\theta}^*}(\\cdot \\mid \\pi_{\\mathrm{exp}})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": ", depends on the exploration policy, "
+                                },
+                                {
+                                    "bbox": [
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+                                        504,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi_{\\mathrm{exp}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": ", being played. As the Fisher information depends on the data distribution, it too scales with the choice of exploration policy:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
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+                    ],
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathcal {I} \\left(\\boldsymbol {\\theta} ^ {\\star}, \\pi_ {\\exp}\\right) := \\mathbb {E} _ {\\boldsymbol {\\tau} \\sim p _ {\\boldsymbol {\\theta} ^ {\\star}} (\\cdot | \\pi_ {\\exp})} \\left[ \\nabla_ {\\boldsymbol {\\theta}} \\log p _ {\\boldsymbol {\\theta} ^ {\\star}} (\\boldsymbol {\\tau} \\mid \\pi_ {\\exp}) \\cdot \\nabla_ {\\boldsymbol {\\theta}} \\log p _ {\\boldsymbol {\\theta} ^ {\\star}} (\\boldsymbol {\\tau} \\mid \\pi_ {\\exp}) ^ {\\top} \\right].",
+                                    "image_path": "ca96660fb4d69c15cedb5e4c69a1a3b4d21dd33ae78a4a8a3dd815a1bee79bea.jpg"
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Following (1), if we collect trajectories by playing "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        223,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi_{\\mathrm{exp}}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " and set "
+                                },
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\widehat{\\pmb{\\theta}}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " to any unbiased estimator of "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\theta}^{\\star}"
+                                },
+                                {
+                                    "bbox": [
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+                                        271
+                                    ],
+                                    "type": "text",
+                                    "content": " on these trajectories, the mean-squared error of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        223,
+                                        504,
+                                        271
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\widehat{\\pmb{\\theta}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        223,
+                                        504,
+                                        271
+                                    ],
+                                    "type": "text",
+                                    "content": " will be lower bounded by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        223,
+                                        504,
+                                        271
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{tr}(\\mathcal{I}(\\pmb{\\theta}^{\\star}, \\pi_{\\mathrm{exp}})^{-1})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        223,
+                                        504,
+                                        271
+                                    ],
+                                    "type": "text",
+                                    "content": ". The optimal exploration policy—the exploration policy which allows for the smallest estimation error—is, therefore, the policy which solves<sup>1</sup>"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        250,
+                                        275,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\arg \\min  _ {\\pi} \\operatorname {t r} \\left(\\mathcal {I} \\left(\\boldsymbol {\\theta} ^ {\\star}, \\pi\\right) ^ {- 1}\\right). \\tag {2}",
+                                    "image_path": "19d5b95497c07a960eb0e87f99d1b783f00e28ae6d7d3217de4b8777dfe54333.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        293,
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+                                    ],
+                                    "type": "text",
+                                    "content": "As an intuitive justification for this choice of exploration policy, note that the Fisher information is defined in terms of the gradient of the log-likelihood with respect to the unknown parameter. Thus, if playing some "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        293,
+                                        505,
+                                        392
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi_{\\mathrm{exp}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        293,
+                                        505,
+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": " makes the Fisher information \"large\", making "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        293,
+                                        505,
+                                        392
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{tr}(\\mathcal{I}(\\pmb{\\theta}^{\\star}, \\pi_{\\mathrm{exp}})^{-1})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        293,
+                                        505,
+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": " small, this suggests "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        293,
+                                        505,
+                                        392
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi_{\\mathrm{exp}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        293,
+                                        505,
+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": " induces trajectories that are very sensitive to the unknown parameters, i.e., trajectory that are significantly more likely under one set of parameters than another. By exploring to maximize the Fisher information, we, therefore, will collect trajectories that are maximally informative about the unknown parameters, since we will observe trajectories much more likely under one set of parameters than another. Motivated by this, we therefore seek to play a policy during exploration that solves (2)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Implementing Fisher Information Maximization: In practice, several issues arise in solving (2), which we address here. First, the form of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        404,
+                                        504,
+                                        449
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{I}(\\theta, \\pi)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        404,
+                                        504,
+                                        449
+                                    ],
+                                    "type": "text",
+                                    "content": " can be quite complicated, depending on the structure of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        404,
+                                        504,
+                                        449
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p_{\\theta}(\\cdot \\mid \\pi)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        404,
+                                        504,
+                                        449
+                                    ],
+                                    "type": "text",
+                                    "content": ", and it may not be possible to efficiently obtain a solution to (2). To address this, we make a simplifying assumption on the dynamics, that our next state, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        404,
+                                        504,
+                                        449
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s_{h+1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        404,
+                                        504,
+                                        449
+                                    ],
+                                    "type": "text",
+                                    "content": ", evolves as:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "s _ {h + 1} = f _ {\\boldsymbol {\\theta}} \\left(s _ {h}, a _ {h}\\right) + w _ {h}, \\tag {3}",
+                                    "image_path": "151540ebf56501c68175fe0c9c519126932e64c3de6ab527307527c907d70945.jpg"
+                                }
+                            ]
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+                    ],
+                    "index": 9
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+                {
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+                    ],
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+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        472,
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+                                        495
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s_h"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        472,
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+                                        495
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "a_h"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        472,
+                                        504,
+                                        495
+                                    ],
+                                    "type": "text",
+                                    "content": " are the current state and action, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        472,
+                                        504,
+                                        495
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_h \\sim \\mathcal{N}(0, \\sigma_w^2 \\cdot I)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        472,
+                                        504,
+                                        495
+                                    ],
+                                    "type": "text",
+                                    "content": " is Gaussian process noise, and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        472,
+                                        504,
+                                        495
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f_\\theta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        472,
+                                        504,
+                                        495
+                                    ],
+                                    "type": "text",
+                                    "content": " are the nominal dynamics. Under these dynamics, the Fisher information matrix reduces to"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
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+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathcal {I} (\\pmb {\\theta}, \\pi) = \\sigma_ {w} ^ {- 2} \\cdot \\mathbb {E} _ {p _ {\\pmb {\\theta}} (\\cdot | \\pi)} \\left[ \\sum_ {h = 1} ^ {H} \\nabla_ {\\pmb {\\theta}} f _ {\\pmb {\\theta}} (s _ {h}, a _ {h}) \\cdot \\nabla_ {\\pmb {\\theta}} f _ {\\pmb {\\theta}} (s _ {h}, a _ {h}) ^ {\\top} \\right].",
+                                    "image_path": "456a9a48f845532f23898e2ed11bcfc542418f4f4a2c4206ce9703201c2caec5.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "We argue that solving (2) with this form of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        524,
+                                        504,
+                                        582
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{I}(\\theta, \\pi)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        524,
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+                                        582
+                                    ],
+                                    "type": "text",
+                                    "content": " is a very intuitive objective, even in cases when the dynamics may not follow (3) exactly. Indeed, this suggests that during exploration, we should aim to reach states for which the dynamics "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        524,
+                                        504,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f_{\\theta}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        524,
+                                        504,
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+                                    ],
+                                    "type": "text",
+                                    "content": " have a large gradient with respect to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        524,
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+                                        582
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        524,
+                                        504,
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+                                    ],
+                                    "type": "text",
+                                    "content": "—states for which the next state predicted by the dynamics is very sensitive to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        524,
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+                                        582
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        524,
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+                                    ],
+                                    "type": "text",
+                                    "content": ". In such states, observing the next state gives us a significant amount of information on "
+                                },
+                                {
+                                    "bbox": [
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+                                        582
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        524,
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+                                    ],
+                                    "type": "text",
+                                    "content": ", allowing us to accurately identify "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        524,
+                                        504,
+                                        582
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta^{\\star}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                        504,
+                                        582
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "A second challenge in solving (2) is that we do not know the true parameter "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta^{\\star}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ", which the optimization (2) depends on. To circumvent this, we rely on domain randomization in choosing our exploration policy, solving instead:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
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+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\pi_ {\\exp} = \\arg \\min  _ {\\pi} \\mathbb {E} _ {\\boldsymbol {\\theta} \\sim q _ {0}} [ \\operatorname {t r} (\\mathcal {I} (\\boldsymbol {\\theta}, \\pi) ^ {- 1}) ] \\tag {4}",
+                                    "image_path": "e49f2fb2fae0d30cd7aedc527105346328a46313fa7e60fba916975f0be649ee.jpg"
+                                }
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+                {
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+                    ],
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+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "for some distribution over parameters "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "q_0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": ". While this is only an approximation of (2), in practice we find that this approximation yields effective exploration policies since, as described in Section 1, in many cases exploration policies require only a coarse model of the dynamics, and can therefore often be learned without precise knowledge of the unknown parameters."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
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+                                    "type": "text",
+                                    "content": "A final challenge is that, in general, we may not have access to a differentiable simulator, and our dynamics themselves may not be differentiable. In such cases, "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\nabla_{\\theta}f_{\\theta}(s_h,a_h)"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " is unknown or"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                }
+            ],
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
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+                                    "type": "text",
+                                    "content": "<sup>1</sup>In the experiment design literature, this is known as an A-optimal experiment design (Pukelsheim, 2006)."
+                                }
+                            ]
+                        }
+                    ],
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+                                    "content": "undefined, and the above approach cannot be applied. As a simple solution to this, we rely on a finite-differences approximation to the gradient, which still provides an effective measure of how sensitive the next state is to the unknown parameter. In practice, to solve (4) and obtain an exploration policy, we rely on standard policy optimization algorithms, such as PPO (Schulman et al., 2017b)."
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+                    ],
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+                                    "type": "text",
+                                    "content": "4.2 SYSTEM IDENTIFICATION"
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+                                    "content": "ASID runs the exploration policy "
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+                                    "type": "inline_equation",
+                                    "content": "\\pi_{\\mathrm{exp}}"
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+                                    "type": "text",
+                                    "content": " (Section 4.1) in the real environment to generate a single trajectory "
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+                                    "bbox": [
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+                                    "content": "\\pmb{\\tau}_{\\mathrm{real}} \\sim p_{\\pmb{\\theta}^{\\star}}(\\cdot \\mid \\pi_{\\mathrm{exp}})"
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ". In the system identification phase, ASID then updates the simulator parameters using the collected trajectory. The goal is to find a distribution over simulator parameters that yield trajectories that match "
+                                },
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+                                    "content": "\\pmb{\\tau}_{\\mathrm{real}}"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " as closely as possible. In particular, we wish to find some distribution over simulation parameters, "
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+                                    "type": "inline_equation",
+                                    "content": "q_{\\phi}"
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ", which minimizes:"
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+                            ]
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+                                    "type": "interline_equation",
+                                    "content": "\\mathbb {E} _ {\\boldsymbol {\\theta} \\sim q _ {\\phi}} \\big [ \\mathbb {E} _ {\\boldsymbol {\\tau} _ {\\mathrm {s i m}} \\sim p _ {\\boldsymbol {\\theta}} (\\cdot | \\mathcal {A} (\\boldsymbol {\\tau} _ {\\mathrm {r e a l}}))} \\big [ \\| \\boldsymbol {\\tau} _ {\\mathrm {r e a l}} - \\boldsymbol {\\tau} _ {\\mathrm {s i m}} \\| _ {2} ^ {2} \\big ] \\big ]",
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+                                    "type": "text",
+                                    "content": "where "
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+                                    "type": "inline_equation",
+                                    "content": "p_{\\theta}(\\cdot \\mid \\mathcal{A}(\\tau_{\\mathrm{real}}))"
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+                                    "type": "text",
+                                    "content": " denotes the distribution over trajectories generated by the simulator with parameter "
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+                                    "type": "text",
+                                    "content": ", and playing the same sequence of actions as were played in "
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": ". In practice, we apply REPS (Peters et al., 2010) for the simulation and Cross Entropy Method (CEM) for the real-world experiments. We stress that the ASD framework is generic, and other black-box optimization algorithms could be used instead."
+                                }
+                            ]
+                        }
+                    ],
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+                },
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+                                    "type": "text",
+                                    "content": "4.3 SOLVING THE DOWNSTREAM TASK"
+                                }
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+                        }
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+                                    "type": "text",
+                                    "content": "After exploration and identification, the simulator can include information about the kinematic tree, position, orientation, or size of the object of interest, and the physical parameters of the real environment. With such a high-fidelity simulator, we aim to solve the downstream tasks entirely in simulation and transfer the learned policies "
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": " to the real world in a zero-shot fashion. As with the system identification stage, ASD does not assume a particular method for solving the downstream task and any policy optimization algorithm can be used."
+                                }
+                            ]
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+                                    "type": "text",
+                                    "content": "5 EXPERIMENTAL EVALUATION"
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+                                    "content": "In our experimental evaluation, we aim to answer the following questions:"
+                                }
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+                                            "type": "text",
+                                            "content": "1. Does ASD's exploration strategy yield sufficient information to identify unknown parameters?"
+                                        }
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+                                            "type": "text",
+                                            "content": "2. Are downstream task policies learned via ASD successful when using the identified parameters?"
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+                                            "type": "text",
+                                            "content": "3. Does the paradigm suggested by ASD transfer to performing tasks in the real world?"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 12
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+                                    "content": "We conduct our experimental evaluation in two scenarios. First, we conduct empirical experiments entirely in simulation (Todorov et al., 2012) to validate the behavior of the exploration, system identification, and downstream task modules of ASD. This involves treating a true simulation environment as the real-world environment and then a reconstruction of this simulation environment as the approximate constructed simulation. Policies learned in this approximate simulation can then be evaluated back in the original simulation environment to judge efficacy. Second, we apply this to two real-world manipulation tasks that are inherently dependent on accurate parameter identification, showing the ability of ASD to function in the real world, using real-world data."
+                                }
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+                                    "content": "5.1 ABLATIONS AND BASELINE COMPARISONS"
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+                                    "content": "We compare ASD with several baselines and ablations for various portions of the pipeline."
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+                                    "content": "Exploration: To understand the importance of targeted exploration via Fisher information maximization we compare with two baselines. First, we compare with the naive approach of using data from a random policy for system identification that is not performing any targeted exploration. Secondly, we compare with Kumar et al. (2019), which aims to generate exploration trajectories that maximize mutual information with the parameters of interest. This method essentially learns a parameter inference network and then rewards exploration policies for minimizing its error."
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+                                            "type": "text",
+                                            "content": "(c) Multi-friction"
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+                                            "content": "Figure 3: Depiction of environments in simulation"
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+                                            "content": "(d) Articulation"
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+                                    "content": "System Identification: To evaluate the impact of system identification methods in this pipeline, we compare the effectiveness of using optimization-based system identification, e.g., (Peters et al., 2010; Memmel et al., 2022), with an end-to-end learned module (Kumar et al., 2019). This comparison shows the stability and extrapolation benefits of ASD over completely data-driven modeling techniques. In particular, we evaluate ASD with the optimization-based system identification replaced by the learned estimation module from Kumar et al. (2019) (ASID + estimator)."
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+                                    "content": "Downstream Policy Learning: Since the eventual goal is to solve the downstream task, we finally compare how effective it is to learn downstream policies in simulation as opposed to an uninformed Domain Randomization (DR) over the parameters, in contrast to the targeted and identified parameters that stem from the ASD pipeline."
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+                                    "content": "5.2 SIMULATED TASK DESCRIPTIONS"
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+                                    "content": "Sphere Manipulation: We consider two sphere manipulation tasks where physical parameters like rolling friction, object stiffness, and tangential surface friction are unknown and make a significant impact on policy performance: 1) striking a golf ball to reach a goal with unknown system friction (Figure 3b), 2) striking a golf ball in an environment with multiple patches that experience different surface friction (Figure 3c). In each scenario, the position of objects and the physical parameters are varied across evaluations. In this setting, we train downstream tasks with PPO."
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+                                    "content": "Rod Balancing: Balancing, or dynamic stacking of objects critically depends on an accurate estimate of the inertia parameters. We construct a rod balancing task where the agent can interact with a rod object (Figure 3a) to identify its physical parameters, in this case varying the distribution of mass along the rod. Once the parameters are identified, the rod must be balanced by placing it on a ledge (Figure 5 (left)). The error is measured by the tilting angle of the rod after placement, a task that requires accurate estimation of system parameters. In this case, we use the CEM to optimize downstream policies."
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+                                    "content": "Articulation: To stress generality, we consider environments that don't simply identify physical parameters like mass or friction but also the structure of the kinematic tree such as articulation and joint positioning. We consider an environment involving an articulated laptop model with a binary environment parameter representing whether articulation is present or not (Figure 3d)."
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+                                    "content": "5.3 DOES ASID LEARN EFFECTIVE EXPLORATION BEHAVIOR?"
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+                                    "content": "To evaluate the exploration behavior quantitatively, we consider the case of multiple unknown parameters, where learning each parameter requires exploration in a different region. In particular, we compare the exploration policy learned by our method with the exploration method of Kumar et al. (2019), on the multi-friction sphere manipulation environment illustrated in Figure 3c where the surface exhibits three different friction parameters (indicated by the grass, sand, and gravel textures). We initialize the golf ball in the leftmost region (grass)—to explore, the arm must strike the ball to other regions to identify their friction parameters. We plot a heat map of the regions visited by the ball during exploration in Figure 4. As can be seen, our approach achieves roughly uniform coverage over the entire space, learning to strike the ball into each region, and illustrating that our method is able to effectively handle complex exploration tasks that involve exciting multiple parameters. In contrast, the approach of Kumar et al. (2019) does not effectively move the sphere to regions different from the starting region."
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+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                            "content": "Table 1: Downstream task results in simulation: Random exploration fails in tasks where directed exploration is required, e.g., striking a sphere or finding the inertia of a rod. When placing the rod with a single action, domain randomization (DR) cannot solve the task without knowing the physical parameters. Learned system identification (Kumar et al. (2019) and "
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+                                            "content": " estimator) doesn't generalize to unseen trajectories and becomes far less effective than optimization-based system identification (cf. "
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+                                            "html": "<table><tr><td>Task</td><td colspan=\"3\">Rod Balancing</td><td>Sphere Striking</td></tr><tr><td>Metric</td><td colspan=\"3\">Tilt angle in degree°</td><td>Success Rate in % ↑</td></tr><tr><td>Parameter</td><td>Inertia (left)</td><td>Inertia (middle)</td><td>Inertia (right)</td><td>Friction ~ [1.1, 1.5]</td></tr><tr><td>Random exploration</td><td>12.44 ± 19.6</td><td>4.20 ± 6.5</td><td>15.34 ± 15.9</td><td>10.62 ± 4.3</td></tr><tr><td>Kumar et al. (2019)</td><td>13.70 ± 9.3</td><td>2.82 ± 2.7</td><td>15.26 ± 9.8</td><td>9.50 ± 2.4</td></tr><tr><td>DR</td><td>26.69 ± 7.0</td><td>13.05 ± 7.3</td><td>1.13 ± 1.3</td><td>8.75 ± 1.5</td></tr><tr><td>ASID + estimator</td><td>17.73 ± 13.1</td><td>4.65 ± 5.1</td><td>9.99 ± 6.8</td><td>11.00 ± 5.2</td></tr><tr><td>ASID + SysID (ours)</td><td>0.00 ± 0.0</td><td>0.72 ± 1.0</td><td>0.00 ± 0.0</td><td>28.00 ± 9.7</td></tr></table>",
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+                                    "content": "We further analyze the exploration behavior in the articulation environment (Figure 3d). Here our exploration policy interacts with the laptop "
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+                                    "content": " of the time as opposed to "
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+                                    "content": " for naïve baselines. Appendix A.2.2 shows that models trained to predict articulation from interaction data, e.g., Ditto (Jiang et al., 2022), can infer joint and part-level geometry from the collected data."
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+                                    "content": "QUANTITATIVELY IN SIMULATION ON DOWNSSTREAM TASKS?"
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+                                    "content": "To quantitatively assess the benefits of ASD over baselines and ablations, we evaluate the performance of ASD with respect to the baselines in simulation on downstream evaluation for the rod balancing (Figure 3a) and sphere manipulation (Figure 3c) task. Results are given in Table 1. Observing the performance on the rod environment, it becomes clear that previous methods struggle with one-shot tasks. When the inertia is not properly identified, CEM predicts the wrong action with high certainty, causing it to pick and place the rod at the wrong location. Domain randomization (DR) on the other hand tries to learn a policy over all possible mass distributions which in this case leads to a policy that always grasps the object at the center of the reference frame. The success here depends on \"getting lucky\" and sampling the correct action for the"
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+                                            "content": "(b) Kumar et al. (2019)"
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+                                            "content": "Figure 4: Visitation frequency of the sphere when explored by different exploration policies on multi-friction (Figure 3c). ASD activates the sphere over a much larger area, thereby identifying parameters more accurately"
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+                                    "content": "In the sphere environment, the distance rolled and bounce of the sphere are critically dependent on parameters such as friction and stiffness, and misidentifying these parameters can lead to significantly different trajectories and unsuccessful downstream behavior. This is reflected in the significantly higher success rate of ASD as compared to baselines that are uninformed of parameters and simply optimize for robust policies (DR), those that try to use learned estimators for exploration (Kumar et al., 2019) or using random exploration."
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+                                    "content": "5.5 DOES ASID ALLOW FOR REAL-WORLD CONTROLLER SYNTHESIS USING MINIMAL REAL-WORLD DATA?"
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+                                    "content": "We further evaluate ASD on real-world tasks, replicating the rod balancing task from simulation (Figure 5) and introducing a novel shuffleboard task (Figure 6). As in simulation, we use a Franka Emika Panda robot for exploration and task performance in the real world. We compute the object's position and orientation by color-threshold the pointclouds from two calibrated Intel RealSense D455 cameras—this approach could easily be replaced by a more sophisticated pose estimation system if desired."
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+                                    "content": "Rod Balancing: The goal of this task is to properly balance a rod with an unknown mass distribution by informatively exploring the environment in the real world and using the resulting data to identify the appropriate physics parameters in simulation. The optimized controller in simulation is then deployed to perform the task in the real world. In this case, the downstream task is picking the rod at a certain point along its axis and balancing it on a perch (Figure 1, Figure 5)."
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+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "content": "The policy executes the downstream task by a pick and place primitive parameterized by the exact pick point. We deploy ASD fully autonomously, executing exploration, system identification, downstream task training, and execution in an end-to-end fashion."
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+                                    "content": "Shuffleboard: This task is a scaled-down version of the popular bar game tabletop shuffle-"
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+                                            "content": "Figure 5: Real-world Rod Balancing: Simulation setup for training exploration and downstream task policies (left). Successful execution of autonomous real-world rod balancing with skewed mass (right)."
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+                                            "content": "Table 2: Downstream task results in real: ASD successfully balances the rod while domain randomization (DR) fails catastrophically."
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+                                            "html": "<table><tr><td>Task</td><td colspan=\"3\">Rod Balancing</td></tr><tr><td>Inertia</td><td>left</td><td>middle</td><td>right</td></tr><tr><td>DR</td><td>0/3</td><td>0/3</td><td>0/3</td></tr><tr><td>ASID (ours)</td><td>2/3</td><td>1/3</td><td>3/3</td></tr></table>",
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+                                    "type": "text",
+                                    "content": "board where a puck must be shot to a target area on a slippery board. We closely follow the original game and pour wax (sand) on the board to decrease the surface friction. This modification makes the task especially difficult as the surface friction on the board changes slightly after each shot since the puck displaces the wax. The goal is to strike the puck to one of the target regions—one closer (yellow) and one further (blue) away from the robot (Figure 2, Figure 6). After exploring the scene, ASD estimates the sliding and torsional friction of the puck to account for the changing surface friction of the board. For executing the downstream task, a primitive positions the endeffector at a fixed distance to the puck and a policy predicts a force value that parameterizes a shot attempt."
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+                                    "type": "text",
+                                    "content": "Due to the changing surface friction, the domain randomization baseline struggles to shoot the puck to the desired zone. While it succeeds 3/10 times—probably because the surface friction was part of its training distribution—the other attempts fall short or overshoot the target. With its dedicated exploration phase, ASD can accurately adapt the simulation to the current conditions and lands the puck in the desired zone 7/10 times (Table 3)."
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+                                    "content": "6 DISCUSSION"
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+                                    "content": "In this work, we introduced a novel pipeline for performing real-world control by autonomously exploring the environment, using the collected data for system identification, and the resulting identified system for downstream policy optimization. In essence, this sim-to-real-to-sim-to-real pipeline allows for targeted test-time construction of simulation environments in a way that enables the performance of downstream tasks. We demonstrate how this"
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+                                            "content": "Figure 6: Real-world Shuffleboard: Simulation setup for training exploration and downstream task policies (left). Successful strike to the yellow zone (right)."
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+                                            "content": "Table 3: Downstream task results in real: ASID outperforms domain randomization (DR) on shooting the puck to the desired zones."
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+                                    "content": "type of identification requires autonomous and carefully directed exploration, and introduce a novel algorithm based on Fisher information maximization that is able to accomplish this directed exploration. The autonomously collected trajectories can then be paired with downstream optimization-based system identification and reconstruction algorithms for accurate simulation construction and downstream policy learning. We show the efficacy of this paradigm on multiple environments in simulation, as well as on rod balancing and shuffleboard, two challenging real-world tasks."
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+                                            "type": "text",
+                                            "content": "Andrew Wagenmaker and Kevin Jamieson. Active learning for identification of linear dynamical systems. In Conference on Learning Theory, pp. 3487-3582. PMLR, 2020."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 16
+                        },
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+                                            "type": "text",
+                                            "content": "Andrew Wagenmaker, Guanya Shi, and Kevin Jamieson. Optimal exploration for model-based rl in nonlinear systems. arXiv preprint arXiv:2306.09210, 2023."
+                                        }
+                                    ]
+                                }
+                            ],
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+                                            "type": "text",
+                                            "content": "Andrew J. Wagenmaker, Max Simchowitz, and Kevin Jamieson. Task-optimal exploration in linear dynamical systems. In International Conference on Machine Learning, pp. 10641-10652. PMLR, 2021."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 18
+                        },
+                        {
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+                                            "type": "text",
+                                            "content": "Yian Wang, Ruihai Wu, Kaichun Mo, Jiaqi Ke, Qingnan Fan, Leonidas J Guibas, and Hao Dong. Adaafford: Learning to adapt manipulation affordance for 3d articulated objects via few-shot interactions. In ECCV, 2022."
+                                        }
+                                    ]
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+                                            "type": "text",
+                                            "content": "Grady Williams, Nolan Wagener, Brian Goldfain, Paul Drews, James M Rehg, Byron Boots, and Evangelos A Theodorou. Information theoretic mpc for model-based reinforcement learning. In 2017 IEEE international conference on robotics and automation (ICRA), pp. 1714-1721. IEEE, 2017."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 1
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+                                            "type": "text",
+                                            "content": "Zhenjia Xu, Jiajun Wu, Andy Zeng, Joshua B Tenenbaum, and Shuran Song. Densephysnet: Learning dense physical object representations via multi-step dynamic interactions. In RSS, 2019."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 2
+                        },
+                        {
+                            "bbox": [
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+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": "Tianhe Yu, Deirdre Quillen, Zhanpeng He, Ryan Julian, Karol Hausman, Chelsea Finn, and Sergey Levine. Meta-world: A benchmark and evaluation for multi-task and meta reinforcement learning. In Conference on robot learning, pp. 1094-1100. PMLR, 2020."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 3
+                        },
+                        {
+                            "bbox": [
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+                                        {
+                                            "bbox": [
+                                                105,
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+                                            "type": "text",
+                                            "content": "Chuning Zhu, Max Simchowitz, Siri Gadipudi, and Abhishek Gupta. Repo: Resilient model-based reinforcement learning by regularizing posterior predictability. In Thirty-seventh Conference on Neural Information Processing Systems, 2023. URL https://openreview.net/forum?id=OIJ3VXDy6s."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 4
+                        },
+                        {
+                            "bbox": [
+                                105,
+                                256,
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+                            ],
+                            "type": "ref_text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                        {
+                                            "bbox": [
+                                                105,
+                                                256,
+                                                504,
+                                                291
+                                            ],
+                                            "type": "text",
+                                            "content": "Henry Zhu, Abhishek Gupta, Aravind Rajeswaran, Sergey Levine, and Vikash Kumar. Dexterous manipulation with deep reinforcement learning: Efficient, general, and low-cost. In 2019 International Conference on Robotics and Automation (ICRA), pp. 3651-3657. IEEE, 2019."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 5
+                        },
+                        {
+                            "bbox": [
+                                105,
+                                297,
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+                            ],
+                            "type": "ref_text",
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+                                            "bbox": [
+                                                105,
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+                                            ],
+                                            "type": "text",
+                                            "content": "Shaojun Zhu, Andrew Kimmel, Kostas Bekris, and Abdeslam Boullarias. Fast model identification via physics engines for data-efficient policy search. In *IJCAI*, 2018."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 6
+                        }
+                    ],
+                    "sub_type": "ref_text"
+                }
+            ],
+            "discarded_blocks": [
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+                    ],
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+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
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+                    ],
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+                                    "content": "14"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                }
+            ],
+            "page_size": [
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+            ],
+            "page_idx": 13
+        },
+        {
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+                                    "type": "text",
+                                    "content": "A APPENDIX"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        105,
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+                                {
+                                    "bbox": [
+                                        105,
+                                        106,
+                                        198,
+                                        118
+                                    ],
+                                    "type": "text",
+                                    "content": "A.1 TASKDETAILS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        105,
+                        126,
+                        247,
+                        138
+                    ],
+                    "type": "title",
+                    "angle": 0,
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+                                {
+                                    "bbox": [
+                                        105,
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+                                        138
+                                    ],
+                                    "type": "text",
+                                    "content": "A.1.1 SPHERE MANIPULATION"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        145,
+                        506,
+                        224
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                145,
+                                506,
+                                224
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        145,
+                                        506,
+                                        224
+                                    ],
+                                    "type": "text",
+                                    "content": "In the sphere manipulation tasks, the observation space consists of endeffector position, sphere position, and robot joint angles. During the training of the exploration policy, we randomize the location of a sphere with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        145,
+                                        506,
+                                        224
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r = 0.03"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        145,
+                                        506,
+                                        224
+                                    ],
+                                    "type": "text",
+                                    "content": " to be between "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        145,
+                                        506,
+                                        224
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x \\in [0.430, 0.65]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        145,
+                                        506,
+                                        224
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        145,
+                                        506,
+                                        224
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "y \\in [-0.2, 0.23]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        145,
+                                        506,
+                                        224
+                                    ],
+                                    "type": "text",
+                                    "content": " for training and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        145,
+                                        506,
+                                        224
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x \\in [0.55, 0.65]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        145,
+                                        506,
+                                        224
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        145,
+                                        506,
+                                        224
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "y \\in [-0.2, 0.23]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        145,
+                                        506,
+                                        224
+                                    ],
+                                    "type": "text",
+                                    "content": " for evaluation. Parameter ranges for training are 1) friction "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        145,
+                                        506,
+                                        224
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta \\in [1e - 3, 5e - 3]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        145,
+                                        506,
+                                        224
+                                    ],
+                                    "type": "text",
+                                    "content": " and 2) patch friction "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        145,
+                                        506,
+                                        224
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta_0, \\theta_1, \\theta_2 \\in [1e - 5, 1e - 3]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        145,
+                                        506,
+                                        224
+                                    ],
+                                    "type": "text",
+                                    "content": " and set fixed sphere and goal locations as well as parameters for the downstream task. For 1) we attach a paddle and limit the endeffector position such that the robot has to strike the sphere to the goal."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        105,
+                        235,
+                        208,
+                        247
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                235,
+                                208,
+                                247
+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
+                                        235,
+                                        208,
+                                        247
+                                    ],
+                                    "type": "text",
+                                    "content": "A.1.2 ARTICULATION"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        255,
+                        504,
+                        310
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                255,
+                                504,
+                                310
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        310
+                                    ],
+                                    "type": "text",
+                                    "content": "In the case of the articulation environment, the parameter is binary and indicates whether articulation is present or not. During evaluation, articulation is always turned on and success is indicated by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        310
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\delta \\beta > 1e - 2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        310
+                                    ],
+                                    "type": "text",
+                                    "content": ". The initial laptop state is randomized over position "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        310
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x \\in [0.45, 0.65]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        310
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        310
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "y \\in [-0.1, 0.1]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        310
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        310
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "yaw \\in [0.00, 3.14]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        310
+                                    ],
+                                    "type": "text",
+                                    "content": " and lid angle "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        310
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta \\in [1.2, 2.5]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        310
+                                    ],
+                                    "type": "text",
+                                    "content": ". The observation space contains the endeffector position, joint angles, position, orientation, and joint angle of the laptop."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        105,
+                        322,
+                        217,
+                        333
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                322,
+                                217,
+                                333
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        322,
+                                        217,
+                                        333
+                                    ],
+                                    "type": "text",
+                                    "content": "A.1.3 ROD BALANCING"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
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+                                    "content": "Similar to the sphere manipulation, we restrict the exploration policy to endeffector control with "
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+                                            "content": "Figure 10: Exploration MAX vs. Fisher information: Trajectories collected (30) from policy checkpoints throughout training or from the last policy checkpoint. While Model-based Active Exploration (MAX) (Shyam et al., 2019) explores the state space over the course of training, getting distracted by the novel states induced by the blue spheres "
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+                                            "content": " (c.f. fig. 10a), Fisher information-based exploration (ours) shows directed exploration (c.f. fig. 10d), moving towards the sphere with changing parameters (red "
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+                                    "content": "We next illustrate the improvement using a simulator vs a fully learned dynamics model can give. We train a forward dynamics model (three layer MLP) on data generated both from MAX and our exploration procedure. When evaluated on out-of-distribution trajectories, i.e., trajectories not included in the training data, we find the model to be extremely inaccurate (see Figure 11). While the simulator extrapolates to unseen states and correctly predicts the movement of the sphere, the model hallucinates movement even when the end-effector does not interact with it at all! These findings make ASID preferable to a purely model-based approach."
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+                                            "type": "text",
+                                            "content": "Figure 11: Learned models vs. simulator: Evaluation of a forward model trained on trajectories (30) from MAX (Shyam et al., 2019) (checkpoints throughout training) and Fisher information (last checkpoint). In contrast to our simulator, the learned model fails to predict the red sphere's trajectory accurately even under no contact scenarios (c.f. fig. 11b, fig. 11d)."
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+                                    "content": "We evaluate the generalization capabilities of our exploration policy for different sphere locations and parameter combinations seen and unseen during training (Figure 13). Since we train the policies using an arena setup, we remove the arena during this evaluation to be able to query sphere locations outside of it (Figure 12)."
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+                                            "content": "Figure 12: Environment with free space. The red sphere is subject to changing parameter values and position is randomized."
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+                                            "content": "(a) Changing friction parameters (in distribution)"
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+                                            "content": "(b) Changing mass parameter (out of distribution)"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 8,
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+                                            "type": "text",
+                                            "content": "Figure 13: Generalization capabilities of an exploration policy trained on sphere friction for different sphere starting locations. The plot shows the success rate of pushing the sphere spawned at the corresponding x-y-location and evaluated over 5 different seeds, i.e., different physics parameter values. Initial endeffector position marked in red. Boxes denote sphere locations seen during training (purple box) and reachable area of the endeffector (orange box). Our policies experience surprising generalization capabilities to unseen sphere locations as long as they can be reached by the robot (cf.fig. 13a). Furthermore, the actions extrapolate to unseen physics parameters as long as they can be uncovered with similar interactions, e.g., pushing (cf.fig. 13b). This is the case because the policy does not have any information about the underlying physics parameters until it hits the sphere."
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diff --git a/2024/Accelerating Distributed Stochastic Optimization via Self-Repellent Random Walks/9e8fce71-9845-4483-82b8-2e8fb64c4467_content_list.json b/2024/Accelerating Distributed Stochastic Optimization via Self-Repellent Random Walks/9e8fce71-9845-4483-82b8-2e8fb64c4467_content_list.json
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+++ b/2024/Accelerating Distributed Stochastic Optimization via Self-Repellent Random Walks/9e8fce71-9845-4483-82b8-2e8fb64c4467_content_list.json	
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+[
+    {
+        "type": "text",
+        "text": "ACCELERATING DISTRIBUTED STOCHASTIC OPTIMIZATION VIA SELF-REPELLENT RANDOM WALKS",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Jie Hu\\*1, Vishwaraj Doshi\\*2, Do Young Eun",
+        "bbox": [
+            179,
+            169,
+            500,
+            184
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "$^{1}$ North Carolina State University,  $^{2}$ IQVIA Inc.",
+        "bbox": [
+            183,
+            185,
+            488,
+            199
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "{jhu29, dyeun}@ncsu.edu, vishwaraj.doshi@iqvia.com",
+        "bbox": [
+            183,
+            199,
+            661,
+            214
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT",
+        "text_level": 1,
+        "bbox": [
+            450,
+            250,
+            547,
+            263
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We study a family of distributed stochastic optimization algorithms where gradients are sampled by a token traversing a network of agents in random-walk fashion. Typically, these random-walks are chosen to be Markov chains that asymptotically sample from a desired target distribution, and play a critical role in the convergence of the optimization iterates. In this paper, we take a novel approach by replacing the standard linear Markovian token by one which follows a nonlinear Markov chain - namely the Self-Repellent Radom Walk (SRRW). Defined for any given 'base' Markov chain, the SRRW, parameterized by a positive scalar  $\\alpha$ , is less likely to transition to states that were highly visited in the past, thus the name. In the context of MCMC sampling on a graph, a recent breakthrough in Doshi et al. (2023) shows that the SRRW achieves  $O(1 / \\alpha)$  decrease in the asymptotic variance for sampling. We propose the use of a 'generalized' version of the SRRW to drive token algorithms for distributed stochastic optimization in the form of stochastic approximation, termed SA-SRRW. We prove that the optimization iterate errors of the resulting SA-SRRW converge to zero almost surely and prove a central limit theorem, deriving the explicit form of the resulting asymptotic covariance matrix corresponding to iterate errors. This asymptotic covariance is always smaller than that of an algorithm driven by the base Markov chain and decreases at rate  $O(1 / \\alpha^2)$  - the performance benefit of using SRRW thereby amplified in the stochastic optimization context. Empirical results support our theoretical findings.",
+        "bbox": [
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+            280,
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+            560
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION",
+        "text_level": 1,
+        "bbox": [
+            173,
+            585,
+            339,
+            599
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Stochastic optimization algorithms solve optimization problems of the form",
+        "bbox": [
+            169,
+            614,
+            671,
+            631
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {\\theta} ^ {*} \\in \\underset {\\boldsymbol {\\theta} \\in \\mathbb {R} ^ {d}} {\\arg \\min } f (\\boldsymbol {\\theta}), \\quad \\text {w h e r e} f (\\boldsymbol {\\theta}) \\triangleq \\mathbb {E} _ {X \\sim \\boldsymbol {\\mu}} [ F (\\boldsymbol {\\theta}, X) ] = \\sum_ {i \\in \\mathcal {N}} \\mu_ {i} F (\\boldsymbol {\\theta}, i), \\tag {1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            258,
+            633,
+            825,
+            666
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "with the objective function  $f: \\mathbb{R}^d \\to \\mathbb{R}$  and  $X$  taking values in a finite state space  $\\mathcal{N}$  with distribution  $\\mu \\triangleq [\\mu_i]_{i \\in \\mathcal{N}}$ . Leveraging partial gradient information per iteration, these algorithms have been recognized for their scalability and efficiency with large datasets (Bottou et al., 2018; Even, 2023). For any given noise sequence  $\\{X_n\\}_{n \\geq 0} \\subset \\mathcal{N}$ , and step size sequence  $\\{\\beta_n\\}_{n \\geq 0} \\subset \\mathbb{R}_+$ , most stochastic optimization algorithms can be classified as stochastic approximations (SA) of the form",
+        "bbox": [
+            169,
+            670,
+            826,
+            744
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} + \\beta_ {n + 1} H \\left(\\boldsymbol {\\theta} _ {n}, X _ {n + 1}\\right), \\quad \\forall n \\geq 0, \\tag {2}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            344,
+            750,
+            823,
+            767
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "where, roughly speaking,  $H(\\pmb{\\theta}, i)$  contains gradient information  $\\nabla_{\\pmb{\\theta}} F(\\pmb{\\theta}, i)$ , such that  $\\pmb{\\theta}^*$  solves  $\\mathbf{h}(\\pmb{\\theta}) \\triangleq \\mathbb{E}_{X \\sim \\pmb{\\mu}}[H(\\pmb{\\theta}, X)] = \\sum_{i \\in \\mathcal{N}} \\mu_i H(\\pmb{\\theta}, i) = \\mathbf{0}$ . Such SA iterations include the well-known stochastic gradient descent (SGD), stochastic heavy ball (SHB) (Gadat et al., 2018; Li et al., 2022), and some SGD-type algorithms employing additional auxiliary variables (Barakat et al., 2021). These algorithms typically have the stochastic noise term  $X_n$  generated by i.i.d. random variables with probability distribution  $\\pmb{\\mu}$  in each iteration. In this paper, we study a stochastic optimization algorithm where the noise sequence governing access to the gradient information is generated from general stochastic processes in place of i.i.d. random variables.",
+        "bbox": [
+            169,
+            772,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "page_footnote",
+        "text": "* Equal contributors.",
+        "bbox": [
+            192,
+            896,
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+            909
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "page_footnote",
+        "text": "Further illustrations of stochastic optimization algorithms of the form (2) are deferred to Appendix A.",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "page_number",
+        "text": "1",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This is commonly the case in distributed learning, where  $\\{X_{n}\\}$  is a (typically Markovian) random walk, and should asymptotically be able to sample the gradients from the desired probability distribution  $\\mu$ . This is equivalent to saying that the random walker's empirical distribution converges to  $\\mu$  almost surely (a.s.); that is,  $\\mathbf{x}_n \\triangleq \\frac{1}{n+1} \\sum_{k=0}^{n} \\delta_{X_k} \\xrightarrow[n\\to\\infty]{a.s.} \\mu$  for any initial  $X_0 \\in \\mathcal{N}$ , where  $\\delta_{X_k}$  is the delta measure whose  $X_k$ 'th entry is one, the rest being zero. Such convergence is most commonly achieved by employing the Metropolis Hastings random walk (MHRW) which can be designed to sample from any target measure  $\\mu$  and implemented in a scalable manner (Sun et al., 2018). Unsurprisingly, convergence characteristics of the employed Markov chain affect that of the SA sequence (2), and appear in both finite-time and asymptotic analyses. Finite-time bounds typically involve the second largest eigenvalue in modulus (SLEM) of the Markov chain's transition kernel  $\\mathbf{P}$ , which is critically connected to the mixing time of a Markov chain (Levin & Peres, 2017); whereas asymptotic results such as central limit theorems (CLT) involve asymptotic covariance matrices that embed information regarding the entire spectrum of  $\\mathbf{P}$ , i.e., all eigenvalues as well as eigenvectors (Bremaud, 2013), which are key to understanding the sampling efficiency of a Markov chain. Thus, the choice of random walker can significantly impact the performance of (2), and simply ensuring that it samples from  $\\mu$  asymptotically is not enough to achieve optimal algorithmic performance. In this paper, we take a closer look at the distributed stochastic optimization problem through the lens of a non-linear Markov chain, known as the Self Repellent Random Walk (SRRW), which was shown in Doshi et al. (2023) to achieve asymptotically minimal sampling variance for large values of  $\\alpha$ , a positive scalar controlling the strength of the random walker's self-repellence behaviour. Our proposed modification of (2) can be implemented within the settings of decentralized learning applications in a scalable manner, while also enjoying significant performance benefit over distributed stochastic optimization algorithms driven by vanilla Markov chains.",
+        "bbox": [
+            169,
+            103,
+            826,
+            431
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Token Algorithms for Decentralized Learning. In decentralized learning, agents like smartphones or IoT devices, each containing a subset of data, collaboratively train models on a graph  $\\mathcal{G}(\\mathcal{N},\\mathcal{E})$  by sharing information locally without a central server (McMahan et al., 2017). In this setup,  $N = |\\mathcal{N}|$  agents correspond to nodes  $i\\in \\mathcal{N}$ , and an edge  $(i,j)\\in \\mathcal{E}$  indicates direct communication between agents  $i$  and  $j$ . This decentralized approach offers several advantages compared to the traditional centralized learning setting, promoting data privacy and security by eliminating the need for raw data to be aggregated centrally and thus reducing the risk of data breach or misuse (Bottou et al., 2018; Nedic, 2020). Additionally, decentralized approaches are more scalable and can handle vast amounts of heterogeneous data from distributed agents without overwhelming a central server, alleviating concerns about single point of failure (Vogels et al., 2021).",
+        "bbox": [
+            169,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Among decentralized learning approaches, the class of 'Token' algorithms can be expressed as stochastic approximation iterations of the type (2), wherein the sequence  $\\{X_{n}\\}$  is realized as the sample path of a token that stochastically traverses the graph  $\\mathcal{G}$ , carrying with it the iterate  $\\theta_{n}$  for any time  $n\\geq 0$  and allowing each visited node (agent) to incrementally update  $\\theta_{n}$  using locally available gradient information. Token algorithms have gained popularity in recent years (Hu et al., 2022; Triastcyn et al., 2022; Hendrikx, 2023), and are provably more communication efficient (Even, 2023) when compared to consensus-based algorithms - another popular approach for solving distributed optimization problems (Boyd et al., 2006; Morral et al., 2017; Olshevsky, 2022). The construction of token algorithms means that they do not suffer from expensive costs of synchronization and communication that are typical of consensus-based approaches, where all agents (or a subset of agents selected by a coordinator (Boyd et al., 2006; Wang et al., 2019)) on the graph are required to take simultaneous actions, such as communicating on the graph at each iteration. While decentralized Federated learning has indeed helped mitigate the communication overhead by processing multiple SGD iterations prior to each aggregation (Lalitha et al., 2018; Ye et al., 2022; Chellapandi et al., 2023), they still cannot overcome challenges such as synchronization and straggler issues.",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Self Repellent Random Walk. As mentioned earlier, sample paths  $\\{X_{n}\\}$  of token algorithms are usually generated using Markov chains with  $\\mu \\in \\mathrm{Int}(\\Sigma)$  as their limiting distribution. Here,  $\\Sigma$  denotes the  $N$ -dimensional probability simplex, with  $\\mathrm{Int}(\\Sigma)$  representing its interior. A recent work by Doshi et al. (2023) pioneers the use of non-linear Markov chains to, in some sense, improve upon any given time-reversible Markov chain with transition kernel  $\\mathbf{P}$  whose stationary distribution is  $\\mu$ .",
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+        "page_idx": 1
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+    {
+        "type": "text",
+        "text": "They show that the non-linear transition kernel $^2$ $\\mathbf{K}[\\cdot ]:\\mathrm{Int}(\\Sigma)\\to [0,1]^{N\\times N}$ , given by",
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "text": "\n$$\nK _ {i j} [ \\mathbf {x} ] \\triangleq \\frac {P _ {i j} \\left(x _ {j} / \\mu_ {j}\\right) ^ {- \\alpha}}{\\sum_ {k \\in \\mathcal {N}} P _ {i k} \\left(x _ {k} / \\mu_ {k}\\right) ^ {- \\alpha}}, \\quad \\forall i, j \\in \\mathcal {N}, \\tag {3}\n$$\n",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "for any  $\\mathbf{x} \\in \\mathrm{Int}(\\Sigma)$ , when simulated as a self-interacting random walk (Del Moral & Miclo, 2006; Del Moral & Doucet, 2010), can achieve smaller asymptotic variance than the base Markov chain when sampling over a graph  $\\mathcal{G}$ , for all  $\\alpha > 0$ . The argument  $\\mathbf{x}$  for the kernel  $\\mathbf{K}[x]$  is taken to be the empirical distribution  $\\mathbf{x}_n$  at each time step  $n \\geq 0$ . For instance, if node  $j$  has been visited more often than other nodes so far, the entry  $x_j$  becomes larger (than target value  $\\mu_j$ ), resulting in a smaller transition probability from  $i$  to  $j$  under  $\\mathbf{K}[\\mathbf{x}]$  in (3) compared to  $P_{ij}$ . This ensures that a random walker prioritizes more seldom visited nodes in the process, and is thus 'self-repellent'. This effect is made more drastic by increasing  $\\alpha$ , and leads to asymptotically near-zero variance at a rate of  $O(1 / \\alpha)$ . Moreover, the polynomial function  $(x_i / \\mu_i)^{-\\alpha}$  chosen to encode self-repellent behaviour is shown in Doshi et al. (2023) to be the only one that allows the SRRW to inherit the so-called 'scale-invariance' property of the underlying Markov chain - a necessary component for the scalable implementation of a random walker over a large network without requiring knowledge of any graph-related global constants. Conclusively, such attributes render SRRW especially suitable for distributed optimization. $^3$",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Effect of Stochastic Noise - Finite time and Asymptotic Approaches. Most contemporary token algorithms driven by Markov chains are analyzed using the finite-time bounds approach for obtaining insights into their convergence rates (Sun et al., 2018; Doan et al., 2019; Triastcyn et al., 2022; Hendrikx, 2023). However, as also explained in Even (2023), in most cases these bounds are overly dependent on mixing time properties of the specific Markov chain employed therein. This makes them largely ineffective in capturing the exact contribution of the underlying random walk in a manner which is qualitative enough to be used for algorithm design; and performance enhancements are typically achieved via application of techniques such as variance reduction (Defazio et al., 2014; Schmidt et al., 2017), momentum/Nesterov's acceleration (Gadat et al., 2018; Li et al., 2022), adaptive step size (Kingma & Ba, 2015; Reddi et al., 2018), which work by modifying the algorithm iterations themselves, and never consider potential improvements to the stochastic input itself.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Complement to finite-time approaches, asymptotic analysis using CLT has proven to be an excellent tool to approach the design of stochastic algorithms (Hu et al., 2022; Devraj & Meyn, 2017; Morral et al., 2017; Chen et al., 2020a; Mou et al., 2020; Devraj & Meyn, 2021). Hu et al. (2022) shows how asymptotic analysis can be used to compare the performance of SGD algorithms for various stochastic inputs using their notion of efficiency ordering, and, as mentioned in Devraj & Meyn (2017), the asymptotic benefits from minimizing the limiting covariance matrix are known to be a good predictor of finite-time algorithmic performance, also observed empirically in Section 4.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "From the perspective of both finite-time analysis as well as asymptotic analysis of token algorithms, it is now well established that employing 'better' Markov chains can enhance the performance of stochastic optimization algorithm. For instance, Markov chains with smaller SLEMs yield tighter finite-time upper bounds (Sun et al., 2018; Ayache & El Rouayheb, 2021; Even, 2023). Similarly, Markov chains with smaller asymptotic variance for MCMC sampling tasks also provide better performance, resulting in smaller covariance matrix of SGD algorithms (Hu et al., 2022). Therefore, with these breakthrough results via SRRW achieving near-zero sampling variance, it is within reason to ask: Can we achieve near-zero variance in distributed stochastic optimization driven by SRRW-like token algorithms on any general graph?4 In this paper, we answer in the affirmative.",
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+        "type": "text",
+        "text": "SRRW Driven Algorithm and Analysis Approach. For any ergodic time-reversible Markov chain with transition probability matrix  $\\mathbf{P} \\triangleq [P_{ij}]_{i,j \\in \\mathcal{N}}$  and stationary distribution  $\\pmb{\\mu} \\in \\mathrm{Int}(\\Sigma)$ , we consider a general step size version of the SRRW stochastic process analysed in Doshi et al. (2023) and",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+    },
+    {
+        "type": "page_footnote",
+        "text": "2Here, non-linearity in the transition kernel implies that  $\\mathbf{K}[\\mathbf{x}]$  takes probability distribution  $\\mathbf{x}$  as the argument (Andrieu et al., 2007), as opposed to the kernel being a linear operator  $\\mathbf{K}[\\mathbf{x}] = \\mathbf{P}$  for a constant stochastic matrix  $\\mathbf{P}$  in a standard (linear) Markovian setting.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "page_footnote",
+        "text": "<sup>3</sup>Recently, Guo et al. (2020) introduce an optimization scheme, which designs self-repellence into the perturbation of the gradient descent iterates (Jin et al., 2017; 2018; 2021) with the goal of escaping saddle points. This notion of self-repellence is distinct from the SRRW, which is a probability kernel designed specifically for a token to sample from a target distribution  $\\mu$  over a set of nodes on an arbitrary graph.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "page_footnote",
+        "text": "This near-zero sampling variance implies a significantly smaller variance than even an i.i.d. sampling counterpart, while adhering to graph topological constraints of token algorithms.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "page_number",
+        "text": "3",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/97eb05def3588d5ce2425368f4f27c0629afcf80d345f83e6517c2c14928feea.jpg",
+        "image_caption": [
+            "Figure 1: Visualization of token algorithms using SRRW versus traditional MC in distributed learning. Our CLT analysis, extended from SRRW itself to distributed stochastic approximation, leads to near-zero variance for the SA iteration  $\\theta_{n}$ . Node numbers on the left denote visit counts."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "use it to drive the noise sequence in (2). Our SA-SRRW algorithm is as follows:",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Draw:  $X_{n + 1}\\sim \\mathbf{K}_{X_n,\\cdot}[\\mathbf{x}_n]$  (4a)",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Update:  $\\mathbf{x}_{n + 1} = \\mathbf{x}_n + \\gamma_{n + 1}(\\pmb{\\delta}_{X_{n + 1}} - \\mathbf{x}_n),$  (4b)",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} + \\beta_ {n + 1} H \\left(\\boldsymbol {\\theta} _ {n}, X _ {n + 1}\\right), \\tag {4c}\n$$\n",
+        "text_format": "latex",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where  $\\{\\beta_n\\}$  and  $\\{\\gamma_n\\}$  are step size sequences decreasing to zero, and  $\\mathbf{K}[\\mathbf{x}]$  is the SRRW kernel in (3). Current non-asymptotic analyses require globally Lipschitz mean field function (Chen et al., 2020b; Doan, 2021; Zeng et al., 2021; Even, 2023) and is thus inapplicable to SA-SRRW since the mean field function of the SRRW iterates (4b) is only locally Lipschitz (details deferred to Appendix B). Instead, we successfully obtain non-trivial results by taking an asymptotic CLT-based approach to analyze (4). This goes beyond just analyzing the asymptotic sampling covariance<sup>5</sup> as in Doshi et al. (2023), the result therein forming a special case of ours by setting  $\\gamma_{n} = 1 / (n + 1)$  and considering only (4a) and (4b), that is, in the absence of optimization iteration (4c). Specifically, we capture the effect of SRRW's hyper-parameter  $\\alpha$  on the asymptotic speed of convergence of the optimization error term  $\\theta_{n} - \\theta^{*}$  to zero via explicit deduction of its asymptotic covariance matrix. See Figure 1 for illustration.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Our Contributions.",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1. Given any time-reversible 'base' Markov chain with transition kernel  $\\mathbf{P}$  and stationary distribution  $\\mu$ , we generalize first and second order convergence results of  $\\mathbf{x}_n$  to target measure  $\\pmb{\\mu}$  (Theorems 4.1 and 4.2 in Doshi et al., 2023) to a class of weighted empirical measures, through the use of more general step sizes  $\\gamma_{n}$ . This includes showing that the asymptotic sampling covariance terms decrease to zero at rate  $O(1 / \\alpha)$ , thus quantifying the effect of self-repellent on  $\\mathbf{x}_n$ . Our generalization is not for the sake thereof and is shown in Section 3 to be crucial for the design of step sizes  $\\beta_{n},\\gamma_{n}$ .",
+            "2. Building upon the convergence results for iterates  $\\mathbf{x}_n$ , we analyze the algorithm (4) driven by the SRRW kernel in (3) with step sizes  $\\beta_{n}$  and  $\\gamma_{n}$  separated into three disjoint cases:"
+        ],
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+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "(i)  $\\beta_{n} = o(\\gamma_{n})$ , and we say that  $\\theta_{n}$  is on the slower timescale compared to  $\\mathbf{x}_n$ ;",
+            "(ii)  $\\beta_{n} = \\gamma_{n}$ , and we say that  $\\theta_{n}$  and  $\\mathbf{x}_n$  are on the same timescale;",
+            "(iii)  $\\gamma_{n} = o(\\beta_{n})$ , and we say that  $\\theta_{n}$  is on the faster timescale compared to  $\\mathbf{x}_n$ ."
+        ],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For any  $\\alpha \\geq 0$  and let  $k = 1,2$  and 3 refer to the corresponding cases (i), (ii) and (iii), we show that",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {\\theta} _ {n} \\xrightarrow [ n \\to \\infty ]{a . s .} \\boldsymbol {\\theta} ^ {*} \\quad \\text {a n d} \\quad (\\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*}) / \\sqrt {\\beta_ {n}} \\xrightarrow [ n \\to \\infty ]{d i s t.} N \\left(\\mathbf {0}, \\mathbf {V} _ {\\boldsymbol {\\theta}} ^ {(k)} (\\alpha)\\right),\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "featuring distinct asymptotic covariance matrices  $\\mathbf{V}_{\\theta}^{(1)}(\\alpha), \\mathbf{V}_{\\theta}^{(2)}(\\alpha)$  and  $\\mathbf{V}_{\\theta}^{(3)}(\\alpha)$ , respectively. The three matrices coincide when  $\\alpha = 0,6$ . Moreover, the derivation of the CLT for cases (i) and (iii), for which (4) corresponds to two-timescale SA with controlled Markov noise, is the first of its kind and thus a key technical contribution in this paper, as expanded upon in Section 3.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3. For case (i), we show that  $\\mathbf{V}_{\\theta}^{(1)}(\\alpha)$  decreases to zero (in the sense of Loewner ordering introduced in Section 2.1) as  $\\alpha$  increases, with rate  $O(1 / \\alpha^2)$ . This is especially surprising, since the asymptotic performance benefit from using the SRRW kernel with  $\\alpha$  in (3), to drive the noise terms  $X_{n}$ , is amplified in the context of distributed learning and estimating  $\\theta^{*}$ ; compared to the sampling case, for which the rate is  $O(1 / \\alpha)$  as mentioned earlier. For case (iii), we show that  $\\mathbf{V}_{\\theta}^{(3)}(\\alpha) = \\mathbf{V}_{\\theta}^{(3)}(0)$  for all  $\\alpha \\geq 0$ , implying that using the SRRW in this case provides no asymptotic benefit than the",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
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+            47
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "page_footnote",
+        "text": "${}^{5}$  Sampling covariance corresponds to only the empirical distribution  ${\\mathbf{x}}_{n}$  in (4b).",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "page_footnote",
+        "text": "The  $\\alpha = 0$  case is equivalent to simply running the base Markov chain, since from (3) we have  $\\mathbf{K}[\\cdot ] = \\mathbf{P}$ , thus bypassing the SRRW's effect and rendering all three cases nearly the same.",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "page_number",
+        "text": "4",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "original base Markov chain, and thus performs worse than case (i). In summary, we deduce that  $\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(\\alpha_2) < _L\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(\\alpha_1) < _L\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(0) = \\mathbf{V}_{\\pmb{\\theta}}^{(3)}(0) = \\mathbf{V}_{\\pmb{\\theta}}^{(3)}(\\alpha)$  for all  $\\alpha_{2} > \\alpha_{1} > 0$  and  $\\alpha >0$",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4. We numerically simulate our SA-SRRW algorithm on various real-world datasets, focusing on a binary classification task, to evaluate its performance across all three cases. By carefully choosing the function  $H$  in SA-SRRW, we test the SGD and algorithms driven by SRRW. Our findings consistently highlight the superiority of case (i) over cases (ii) and (iii) for diverse  $\\alpha$  values, even in their finite time performance. Notably, our tests validate the variance reduction at a rate of  $O(1 / \\alpha^2)$  for case (i), suggesting it as the best algorithmic choice among the three cases.",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2 PRELIMINARIES AND MODEL SETUP",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In Section 2.1, we first standardize the notations used throughout the paper, and define key mathematical terms and quantities used in our theoretical analyses. Then, in Section 2.2, we consolidate the model assumptions of our SA-SRRW algorithm (4). We then go on to discuss our assumptions, and provide additional interpretations of our use of generalized step-sizes.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.1 BASIC NOTATIONS AND DEFINITIONS",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Vectors are denoted by lower-case bold letters, e.g.,  $\\mathbf{v} \\triangleq [v_i] \\in \\mathbb{R}^D$ , and matrices by upper-case bold, e.g.,  $\\mathbf{M} \\triangleq [M_{ij}] \\in \\mathbb{R}^{D \\times D}$ .  $\\mathbf{M}^{-T}$  is the transpose of the matrix inverse  $\\mathbf{M}^{-1}$ . The diagonal matrix  $\\mathbf{D}_{\\mathbf{v}}$  is formed by vector  $\\mathbf{v}$  with  $v_i$  as the  $i$ 'th diagonal entry. Let  $\\mathbf{1}$  and  $\\mathbf{0}$  denote vectors of all ones and zeros, respectively. The identity matrix is represented by  $\\mathbf{I}$ , with subscripts indicating dimensions as needed. A matrix is Hurwitz if all its eigenvalues possess strictly negative real parts.  $\\mathbb{1}_{\\{\\cdot\\}}$  denotes an indicator function with condition in parentheses. We use  $\\|\\cdot\\|$  to denote both the Euclidean norm of vectors and the spectral norm of matrices. Two symmetric matrices  $\\mathbf{M}_1, \\mathbf{M}_2$  follow Loewner ordering  $\\mathbf{M}_1 <_L \\mathbf{M}_2$  if  $\\mathbf{M}_2 - \\mathbf{M}_1$  is positive semi-definite and  $\\mathbf{M}_1 \\neq \\mathbf{M}_2$ . This slightly differs from the conventional definition with  $\\leq_L$ , which allows  $\\mathbf{M}_1 = \\mathbf{M}_2$ .",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Throughout the paper, the matrix  $\\mathbf{P} \\triangleq [P_{i,j}]_{i,j \\in \\mathcal{N}}$  and vector  $\\pmb{\\mu} \\triangleq [\\mu_i]_{i \\in \\mathcal{N}}$  are used exclusively to denote an  $N \\times N$ -dimensional transition kernel of an ergodic Markov chain, and its stationary distribution, respectively. Without loss of generality, we assume  $P_{ij} > 0$  if and only if  $a_{ij} > 0$ . Markov chains satisfying the detailed balance equation, where  $\\mu_i P_{ij} = \\mu_j P_{ji}$  for all  $i, j \\in \\mathcal{N}$ , are termed time-reversible. For such chains, we use  $(\\lambda_i, \\mathbf{u}_i)$  (resp.  $(\\lambda_i, \\mathbf{v}_i)$ ) to denote the  $i$ 'th left (resp. right) eigenpair where the eigenvalues are ordered:  $-1 < \\lambda_1 \\leq \\dots \\leq \\lambda_{N-1} < \\lambda_N = 1$ , with  $\\mathbf{u}_N = \\pmb{\\mu}$  and  $\\mathbf{v}_N = \\mathbf{1}$  in  $\\mathbb{R}^N$ . We assume eigenvectors to be normalized such that  $\\mathbf{u}_i^T \\mathbf{v}_i = 1$  for all  $i$ , and we have  $\\mathbf{u}_i = \\mathbf{D}_{\\pmb{\\mu}} \\mathbf{v}_i$  and  $\\mathbf{u}_i^T \\mathbf{v}_j = 0$  for all  $i, j \\in \\mathcal{N}$ . We direct the reader to Aldous & Fill (2002, Chapter 3.4) for a detailed exposition on spectral properties of time-reversible Markov chains.",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "2.2 SA-SRRW: KEY ASSUMPTIONS AND DISCUSSIONS",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Assumptions: All results in our paper are proved under the following assumptions.",
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+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "(A1) The function  $H: \\mathbb{R}^D \\times \\mathcal{N} \\to \\mathbb{R}^D$ , is a continuous at every  $\\pmb{\\theta} \\in \\mathbb{R}^D$ , and there exists a positive constant  $L$  such that  $\\| H(\\pmb{\\theta}, i) \\| \\leq L (1 + \\| \\pmb{\\theta} \\|)$  for every  $\\pmb{\\theta} \\in \\mathbb{R}^D, i \\in \\mathcal{N}$ .",
+            "(A2) Step sizes  $\\beta_{n}$  and  $\\gamma_{n}$  follow  $\\beta_{n} = (n + 1)^{-b}$ , and  $\\gamma_{n} = (n + 1)^{-a}$ , where  $a, b \\in (0.5, 1]$ .",
+            "(A3) Roots of function  $\\mathbf{h}(\\cdot)$  are disjoint, which comprise the globally attracting set  $\\Theta \\triangleq \\left\\{\\pmb{\\theta}^{*}|\\mathbf{h}(\\pmb{\\theta}^{*}) = \\mathbf{0}, \\nabla \\mathbf{h}(\\pmb{\\theta}^{*}) + \\frac{\\mathbb{1}_{\\{b=1\\}}}{2}\\mathbf{I}\\text{ is Hurwitz}\\right\\} \\neq \\emptyset$  of the associated ordinary differential equation (ODE) for iteration (4c), given by  $d\\pmb{\\theta}(t)/dt = \\mathbf{h}(\\pmb{\\theta}(t))$ .",
+            "(A4) For any  $(\\pmb{\\theta}_0, \\mathbf{x}_0, X_0) \\in \\mathbb{R}^D \\times \\mathrm{Int}(\\Sigma) \\times \\mathcal{N}$ , the iterate sequence  $\\{\\pmb{\\theta}_n\\}_{n \\geq 0}$  (resp.  $\\{\\mathbf{x}_n\\}_{n \\geq 0}$ ) is  $\\mathbb{P}_{\\pmb{\\theta}_0, \\mathbf{x}_0, X_0}$ -almost surely contained within a compact subset of  $\\mathbb{R}^D$  (resp.  $\\mathrm{Int}(\\Sigma)$ )."
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+    },
+    {
+        "type": "text",
+        "text": "Discussions on Assumptions: Assumption A1 requires  $H$  to only be locally Lipschitz albeit with linear growth, and is less stringent than the globally Lipschitz assumption prevalent in optimization literature (Li & Wai, 2022; Hendrikx, 2023; Even, 2023).",
+        "bbox": [
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "page_footnote",
+        "text": "In particular, this is the reason why we advocate for a more general step size  $\\gamma_{n} = (n + 1)^{-a}$  in the SRRW iterates with  $a < 1$ , allowing us to choose  $\\beta_{n} = (n + 1)^{-b}$  with  $b \\in (a, 1]$  to satisfy  $\\beta_{n} = o(\\gamma_{n})$  for case (i).",
+        "bbox": [
+            169,
+            895,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "page_number",
+        "text": "5",
+        "bbox": [
+            493,
+            948,
+            504,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Assumption A2 is the general umbrella assumption under which cases (i), (ii) and (iii) mentioned in Section 1 are extracted by setting: (i)  $a < b$ , (ii)  $a = b$ , and (iii)  $a > b$ . Cases (i) and (iii) render  $\\theta_{n}, \\mathbf{x}_{n}$  on different timescales; the polynomial form of  $\\beta_{n}, \\gamma_{n}$  widely assumed in the two-timescale SA literature (Mokkadem & Pelletier, 2006; Zeng et al., 2021; Hong et al., 2023). Case (ii) characterizes the SA-SRRW algorithm (4) as a single-timescale SA with polynomially decreasing step size, and is among the most common assumptions in the SA literature (Borkar, 2022; Fort, 2015; Li et al., 2023). In all three cases, the form of  $\\gamma_{n}$  ensures  $\\gamma_{n} \\leq 1$  such that the SRRW iterates  $\\mathbf{x}_{n}$  in (4b) is within  $\\operatorname{Int}(\\Sigma)$ , ensuring that  $\\mathbf{K}[\\mathbf{x}_n]$  is well-defined for all  $n \\geq 0$ .",
+        "bbox": [
+            169,
+            103,
+            826,
+            217
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In Assumption A3, limiting dynamics of SA iterations  $\\{\\pmb{\\theta}_n\\}_{n\\geq 0}$  closely follow trajectories  $\\{\\pmb {\\theta}(t)\\}_{t\\geq 0}$  of their associated ODE, and assuming the existence of globally stable equilibria is standard (Borkar, 2022; Fort, 2015; Li et al., 2023). In optimization problems, this is equivalent to assuming the existence of at most countably many local minima.",
+        "bbox": [
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+            220,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Assumption A4 assumes almost sure boundedness of iterates  $\\theta_{n}$  and  $\\mathbf{x}_n$ , which is a common assumption in SA algorithms (Kushner & Yin, 2003; Chen, 2006; Borkar, 2022; Karmakar & Bhatnagar, 2018; Li et al., 2023) for the stability of the SA iterations by ensuring the well-definiteness of all quantities involved. Stability of the weighted empirical measure  $\\mathbf{x}_n$  of the SRRW process is practically ensured by studying (4b) with a truncation-based procedure (see Doshi et al., 2023, Remark 4.5 and Appendix E for a comprehensive explanation), while that for  $\\theta_{n}$  is usually ensured either as a by-product of the algorithm design, or via mechanisms such as projections onto a compact subset of  $\\mathbb{R}^D$ , depending on the application context. We now provide additional discussions regarding the step-size assumptions and their implications on the SRRW iteration (4b).",
+        "bbox": [
+            169,
+            285,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "SRRW with General Step Size: As shown in Benaim & Cloez (2015, Remark 1.1), albeit for a completely different non-linear Markov kernel driving the algorithm therein, iterates  $\\mathbf{x}_n$  of (4b) can also be expressed as weighted empirical measures of  $\\{X_n\\}_{n\\geq 0}$ , in the following form:",
+        "bbox": [
+            169,
+            416,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {x} _ {n} = \\frac {\\sum_ {i = 1} ^ {n} \\omega_ {i} \\delta_ {X _ {i}} + \\omega_ {0} \\mathbf {x} _ {0}}{\\sum_ {i = 0} ^ {n} \\omega_ {i}}, \\quad \\text {w h e r e} \\omega_ {0} = 1, \\text {a n d} \\omega_ {n} = \\frac {\\gamma_ {n}}{\\prod_ {i = 1} ^ {n} (1 - \\gamma_ {i})}, \\tag {5}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+            463,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "for all  $n > 0$ . For the special case when  $\\gamma_{n} = 1 / (n + 1)$  as in Doshi et al. (2023), we have  $\\omega_{n} = 1$  for all  $n \\geq 0$  and  $\\mathbf{x}_n$  is the typical, unweighted empirical measure. For the additional case considered in our paper, when  $a < 1$  for  $\\gamma_{n}$  as in assumption A2, we can approximate  $1 - \\gamma_{n} \\approx e^{-\\gamma_{n}}$  and  $\\omega_{n} \\approx n^{-a} e^{n^{(1 - a)} / (1 - a)}$ . This implies that  $\\omega_{n}$  will increase at sub-exponential rate, giving more weight to recent visit counts and allowing it to quickly 'forget' the poor initial measure  $\\mathbf{x}_0$  and shed the correlation with the initial choice of  $X_0$ . This 'speed up' effect by setting  $a < 1$  is guaranteed in case (i) irrespective of the choice of  $b$  in Assumption A2, and in Section 3 we show how this can lead to further reduction in covariance of optimization error  $\\theta_{n} = \\theta^{*}$  in the asymptotic regime.",
+        "bbox": [
+            169,
+            500,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Additional assumption for case (iii): Before moving on to Section 3, we take another look at the case when  $\\gamma_{n} = o(\\beta_{n})$ , and replace A3 with the following, stronger assumption only for case (iii).",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "(A3') For any  $\\mathbf{x} \\in \\mathrm{Int}(\\Sigma)$ , there exists a function  $\\rho: \\mathrm{Int}(\\Sigma) \\to \\mathbb{R}^D$  such that  $\\|\\rho(\\mathbf{x})\\| \\leq L_2(1 + \\| \\mathbf{x}\\|)$  for some  $L_2 > 0$ ,  $\\mathbb{E}_{i \\sim \\pi[\\mathbf{x}]}[H(\\rho(\\mathbf{x}), i)] = 0$  and  $\\mathbb{E}_{i \\sim \\pi[\\mathbf{x}]}[\\nabla H(\\rho(\\mathbf{x}), i)] + \\frac{\\mathbf{1}_{\\{b=1\\}}}{2} \\mathbf{I}$  is Hurwitz.",
+        "bbox": [
+            184,
+            657,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "While Assumption A3' for case (iii) is much stronger than A3, it is not detrimental to the overall results of our paper, since case (i) is of far greater interest as impressed upon in Section 1. This is discussed further in Appendix C.",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3 ASYMPTOTIC ANALYSIS OF THE SA-SRRW ALGORITHM",
+        "text_level": 1,
+        "bbox": [
+            171,
+            758,
+            681,
+            776
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section, we provide the main results for the SA-SRRW algorithm (4). We first present the a.s. convergence and the CLT result for SRRW with generalized step size, extending the results in Doshi et al. (2023). Building upon this, we present the a.s. convergence and the CLT result for the SA iterate  $\\theta_{n}$  under different settings of step sizes. We then shift our focus to the analysis of the different asymptotic covariance matrices emerging out of the CLT result, and capture the effect of  $\\alpha$  and the step sizes, particularly in cases (i) and (iii), on  $\\theta_{n} - \\theta^{*}$  via performance ordering.",
+        "bbox": [
+            169,
+            790,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Almost Sure convergence and CLT: The following result establishes first and second order convergence of the sequence  $\\{\\mathbf{x}_n\\}_{n\\geq 0}$ , which represents the weighted empirical measures of the SRRW process  $\\{X_{n}\\}_{n\\geq 0}$ , based on the update rule in (4b).",
+        "bbox": [
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+            881,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "page_number",
+        "text": "6",
+        "bbox": [
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+            948,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Lemma 3.1. Under Assumptions A1, A2 and A4, for the SRRW iterates (4b), we have",
+        "bbox": [
+            169,
+            103,
+            736,
+            118
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {x} _ {n} \\xrightarrow [ n \\to \\infty ]{a . s .} \\boldsymbol {\\mu}, \\quad a n d \\quad \\gamma_ {n} ^ {- 1 / 2} (\\mathbf {x} _ {n} - \\boldsymbol {\\mu}) \\xrightarrow [ n \\to \\infty ]{d i s t .} N (\\mathbf {0}, \\mathbf {V} _ {\\mathbf {x}} (\\alpha)),\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            294,
+            119,
+            699,
+            142
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nw h e r e \\quad \\mathbf {V} _ {\\mathbf {x}} (\\alpha) = \\sum_ {i = 1} ^ {N - 1} \\frac {1}{2 \\alpha \\left(1 + \\lambda_ {i}\\right) + 2 - \\mathbb {1} _ {\\{a = 1 \\}}} \\cdot \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T}. \\tag {6}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            281,
+            142,
+            823,
+            181
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Moreover, for all  $\\alpha_{2} > \\alpha_{1} > 0$ , we have  $\\mathbf{V}_{\\mathbf{x}}(\\alpha_{2}) < _{L}\\mathbf{V}_{\\mathbf{x}}(\\alpha_{1}) < _{L}\\mathbf{V}_{\\mathbf{x}}(0)$ .",
+        "bbox": [
+            174,
+            185,
+            663,
+            200
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Lemma 3.1 shows that the SRRW iterates  $\\mathbf{x}_n$  converges to the target distribution  $\\mu$  a.s. even under the general step size  $\\gamma_n = (n + 1)^{-a}$  for  $a\\in (0.5,1]$ . We also observe that the asymptotic covariance matrix  $\\mathbf{V}_{\\mathbf{x}}(\\alpha)$  decreases at rate  $O(1 / \\alpha)$ . Lemma 3.1 aligns with Doshi et al. (2023, Theorem 4.2 and Corollary 4.3) for the special case of  $a = 1$ , and is therefore more general. Critically, it helps us establish our next result regarding the first-order convergence for the optimization iterate sequence  $\\{\\theta_{n}\\}_{n\\geq 0}$  following update rule (4c), as well as its second-order convergence result, which follows shortly after. The proofs of Lemma 3.1 and our next result, Theorem 3.2, are deferred to Appendix D. In what follows,  $k = 1,2$ , and 3 refer to cases (i), (ii), and (iii) in Section 2.2, respectively. All subsequent results are proven under Assumptions A1 to A4, with  $\\mathrm{A}3^{\\prime}$  replacing A3 only when the step sizes  $\\beta_{n},\\gamma_{n}$  satisfy case (iii).",
+        "bbox": [
+            169,
+            204,
+            826,
+            344
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.2. For  $k \\in \\{1,2,3\\}$ , and any initial  $(\\pmb{\\theta}_0,\\mathbf{x}_0,X_0) \\in \\mathbb{R}^D \\times \\mathrm{Int}(\\Sigma) \\times \\mathcal{N}$ , we have  $\\pmb{\\theta}_n \\to \\pmb{\\theta}^*$  as  $n \\to \\infty$  for some  $\\pmb{\\theta}^* \\in \\Theta$ ,  $\\mathbb{P}_{\\pmb{\\theta}_0,\\mathbf{x}_0,X_0}$ -almost surely.",
+        "bbox": [
+            169,
+            345,
+            823,
+            378
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In the stochastic optimization context, the above result ensures convergence of iterates  $\\pmb{\\theta}_n$  to a local minimizer  $\\pmb{\\theta}^*$ . Loosely speaking, the first-order convergence of  $\\mathbf{x}_n$  in Lemma 3.1 as well as that of  $\\pmb{\\theta}_n$  are closely related to the convergence of trajectories  $\\{\\mathbf{z}(t) \\triangleq (\\pmb{\\theta}(t), \\mathbf{x}(t))\\}_{t \\geq 0}$  of the (coupled) mean-field ODE, written in a matrix-vector form as",
+        "bbox": [
+            169,
+            383,
+            823,
+            439
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\frac {d}{d t} \\mathbf {z} (t) = \\mathbf {g} (\\mathbf {z} (t)) \\triangleq \\left[ \\begin{array}{c} \\mathbf {H} (\\boldsymbol {\\theta} (t)) ^ {T} \\boldsymbol {\\pi} [ \\mathbf {x} (t) ] \\\\ \\boldsymbol {\\pi} [ \\mathbf {x} (t) ] - \\mathbf {x} (t) \\end{array} \\right] \\in \\mathbb {R} ^ {D + N}. \\tag {7}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            326,
+            439,
+            823,
+            472
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where matrix  $\\mathbf{H}(\\pmb{\\theta}) \\triangleq [H(\\pmb{\\theta}, 1), \\dots, H(\\pmb{\\theta}, N)]^T \\in \\mathbb{R}^{N \\times D}$  for any  $\\pmb{\\theta} \\in \\mathbb{R}^D$ . Here,  $\\pmb{\\pi}[\\mathbf{x}] \\in \\mathrm{Int}(\\Sigma)$  is the stationary distribution of the SRRW kernel  $\\mathbf{K}[\\mathbf{x}]$  and is shown in Doshi et al. (2023) to be given by  $\\pi_i[\\mathbf{x}] \\propto \\sum_{j \\in \\mathcal{N}} \\mu_i P_{ij}(x_i / \\mu_i)^{-\\alpha}(x_j / \\mu_j)^{-\\alpha}$ . The Jacobian matrix of (7) when evaluated at equilibria  $\\mathbf{z}^* = (\\pmb{\\theta}^*, \\pmb{\\mu})$  for  $\\pmb{\\theta}^* \\in \\Theta$  captures the behaviour of solutions of the mean-field in their vicinity, and plays an important role in the asymptotic covariance matrices arising out of our CLT results. We evaluate this Jacobian matrix  $\\mathbf{J}(\\alpha)$  as a function of  $\\alpha \\geq 0$  to be given by",
+        "bbox": [
+            169,
+            472,
+            823,
+            559
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {J} (\\alpha) \\triangleq \\nabla g \\left(\\mathbf {z} ^ {*}\\right) = \\left[ \\begin{array}{c c} \\nabla \\mathbf {h} \\left(\\boldsymbol {\\theta} ^ {*}\\right) & - \\alpha \\mathbf {H} \\left(\\boldsymbol {\\theta} ^ {*}\\right) ^ {T} \\left(\\mathbf {P} ^ {T} + \\mathbf {I}\\right) \\\\ \\mathbf {0} _ {N \\times D} & 2 \\alpha \\boldsymbol {\\mu} \\mathbf {1} ^ {T} - \\alpha \\mathbf {P} ^ {T} - (\\alpha + 1) \\mathbf {I} \\end{array} \\right] \\triangleq \\left[ \\begin{array}{l l} \\mathbf {J} _ {1 1} & \\mathbf {J} _ {1 2} (\\alpha) \\\\ \\mathbf {J} _ {2 1} & \\mathbf {J} _ {2 2} (\\alpha) \\end{array} \\right]. \\tag {8}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            251,
+            559,
+            823,
+            593
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The derivation of  $\\mathbf{J}(\\alpha)$  is referred to Appendix E.1.8 Here,  $\\mathbf{J}_{21}$  is a zero matrix since  $\\pi[\\mathbf{x}] - \\mathbf{x}$  is devoid of  $\\theta$ . While matrix  $\\mathbf{J}_{22}(\\alpha)$  is exactly of the form in Doshi et al. (2023, Lemma 3.4) to characterize the SRRW performance, our analysis includes an additional matrix  $\\mathbf{J}_{12}(\\alpha)$ , which captures the effect of  $\\mathbf{x}(t)$  on  $\\theta(t)$  in the ODE (7), which translates to the influence of our generalized SRRW empirical measure  $\\mathbf{x}_n$  on the SA iterates  $\\theta_n$  in (4).",
+        "bbox": [
+            169,
+            593,
+            823,
+            662
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For notational simplicity, and without loss of generality, all our remaining results are stated while conditioning on the event that  $\\{\\pmb {\\theta}_n\\to \\pmb {\\theta}^*\\}$ , for some  $\\pmb {\\theta}^{*}\\in \\Theta$ . We also adopt the shorthand notation  $\\mathbf{H}$  to represent  $\\mathbf{H}(\\pmb {\\theta}^{*})$ . Our main CLT result is as follows, with its proof deferred to Appendix E.",
+        "bbox": [
+            169,
+            669,
+            823,
+            713
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.3. For any  $\\alpha \\geq 0$ , we have: (a) There exists  $\\mathbf{V}^{(k)}(\\alpha)$  for all  $k \\in \\{1,2,3\\}$  such that",
+        "bbox": [
+            169,
+            715,
+            799,
+            732
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left[ \\begin{array}{c} \\beta_ {n} ^ {- 1 / 2} (\\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*}) \\\\ \\gamma_ {n} ^ {- 1 / 2} (\\mathbf {x} _ {n} - \\boldsymbol {\\mu}) \\end{array} \\right] \\xrightarrow [ n \\to \\infty ]{d i s t.} N \\left(\\mathbf {0}, \\mathbf {V} ^ {(k)} (\\alpha)\\right).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            349,
+            732,
+            648,
+            771
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "(b) For  $k = 2$ , matrix  $\\mathbf{V}^{(2)}(\\alpha)$  solves the Lyapunov equation  $\\mathbf{J}(\\alpha)\\mathbf{V}^{(2)}(\\alpha) + \\mathbf{V}^{(2)}(\\alpha)\\mathbf{J}(\\alpha)^T + \\mathbb{1}_{\\{b = 1\\}}\\mathbf{V}^{(2)}(\\alpha) = -\\mathbf{U}$ , where the Jacobian matrix  $\\mathbf{J}(\\alpha)$  is in (8), and",
+        "bbox": [
+            169,
+            771,
+            823,
+            804
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {U} \\triangleq \\sum_ {i = 1} ^ {N - 1} \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\cdot \\left[ \\begin{array}{c c} \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H} & \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\\\ \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H} & \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\end{array} \\right] \\triangleq \\left[ \\begin{array}{c c} \\mathbf {U} _ {1 1} & \\mathbf {U} _ {1 2} \\\\ \\mathbf {U} _ {2 1} & \\mathbf {U} _ {2 2} \\end{array} \\right]. \\tag {9}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            287,
+            805,
+            823,
+            844
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "(c) For  $k \\in \\{1, 3\\}$ ,  $\\mathbf{V}^{(k)}(\\alpha)$  becomes block diagonal, which is given by",
+        "bbox": [
+            169,
+            845,
+            640,
+            861
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {V} ^ {(k)} (\\alpha) = \\left[ \\begin{array}{l l} \\mathbf {V} _ {\\boldsymbol {\\theta}} ^ {(k)} (\\alpha) & \\mathbf {0} _ {D \\times N} \\\\ \\mathbf {0} _ {N \\times D} & \\mathbf {V} _ {\\mathbf {x}} (\\alpha) \\end{array} \\right], \\tag {10}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            383,
+            861,
+            823,
+            893
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "page_footnote",
+        "text": "The Jacobian  $\\mathbf{J}(\\alpha)$  is  $(D + N)\\times (D + N)$ -dimensional, with  $\\mathbf{J}_{11}\\in \\mathbb{R}^{D\\times D}$  and  $\\mathbf{J}_{22}(\\alpha)\\in \\mathbb{R}^{N\\times N}$ . Following this, all matrices written in a block form, such as matrix  $\\mathbf{U}$  in (9), will inherit the same dimensional structure.",
+        "bbox": [
+            169,
+            896,
+            823,
+            922
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "page_number",
+        "text": "7",
+        "bbox": [
+            493,
+            948,
+            504,
+            959
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where  $\\mathbf{V}_{\\mathbf{x}}(\\alpha)$  is as in (6), and  $\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(\\alpha)$  and  $\\mathbf{V}_{\\pmb{\\theta}}^{(3)}(\\alpha)$  can be written in the following explicit form:",
+        "bbox": [
+            169,
+            101,
+            815,
+            122
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {V} _ {\\boldsymbol {\\theta}} ^ {(1)} (\\alpha) = \\int_ {0} ^ {\\infty} e ^ {t (\\nabla_ {\\boldsymbol {\\theta}} \\mathbf {h} (\\boldsymbol {\\theta} ^ {*}) + \\frac {1}{2} \\{b = 1 \\} \\boldsymbol {I})} \\mathbf {U} _ {\\boldsymbol {\\theta}} (\\alpha) e ^ {t (\\nabla_ {\\boldsymbol {\\theta}} \\mathbf {h} (\\boldsymbol {\\theta} ^ {*}) + \\frac {1}{2} \\{b = 1 \\} \\boldsymbol {I}) ^ {T}} d t,\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            295,
+            127,
+            746,
+            151
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {V} _ {\\pmb {\\theta}} ^ {(3)} (\\alpha) = \\int_ {0} ^ {\\infty} e ^ {t \\nabla_ {\\pmb {\\theta}} \\mathbf {h} (\\pmb {\\theta} ^ {*})} \\mathbf {U} _ {1 1} e ^ {t \\nabla_ {\\pmb {\\theta}} \\mathbf {h} (\\pmb {\\theta} ^ {*})} d t,\n$$\n",
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+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nw h e r e \\mathbf {U} _ {\\boldsymbol {\\theta}} (\\alpha) = \\sum_ {i = 1} ^ {N - 1} \\frac {1}{\\left(\\alpha \\left(1 + \\lambda_ {i}\\right) + 1\\right) ^ {2}} \\cdot \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H}. \\tag {11}\n$$\n",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "For  $k \\in \\{1,3\\}$ , SA-SRRW in (4) is a two-timescale SA with controlled Markov noise. While a few works study the CLT of two-timescale SA with the stochastic input being a martingale-difference (i.i.d.) noise (Konda & Tsitsiklis, 2004; Mokkadem & Pelletier, 2006), a CLT result covering the case of controlled Markov noise (e.g.,  $k \\in \\{1,3\\}$ ), a far more general setting than martingale-difference noise, is still an open problem. Thus, we here prove our CLT for  $k \\in \\{1,3\\}$  from scratch by a series of careful decompositions of the Markovian noise, ultimately into a martingale-difference term and several non-vanishing noise terms through repeated application of the Poisson equation (Benveniste et al., 2012; Fort, 2015). Although the form of the resulting asymptotic covariance looks similar to that for the martingale-difference case in (Konda & Tsitsiklis, 2004; Mokkadem & Pelletier, 2006) at first glance, they are not equivalent. Specifically,  $\\mathbf{V}_{\\theta}^{(k)}(\\alpha)$  captures both the effect of SRRW hyper-parameter  $\\alpha$ , as well as that of the underlying base Markov chain via eigenpairs  $(\\lambda_i, \\mathbf{u}_i)$  of its transition probability matrix  $\\mathbf{P}$  in matrix  $\\mathbf{U}$ , whereas the latter only covers the martingale-difference noise terms as a special case.",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "When  $k = 2$ , that is,  $\\beta_{n} = \\gamma_{n}$ , algorithm (4) can be regarded as a single-timescale SA algorithm. In this case, we utilize the CLT in Fort (2015, Theorem 2.1) to obtain the implicit form of  $\\mathbf{V}^{(2)}(\\alpha)$  as shown in Theorem 3.3. However,  $\\mathbf{J}_{12}(\\alpha)$  being non-zero for  $\\alpha > 0$  restricts us from obtaining an explicit form for the covariance term corresponding to SA iterate errors  $\\theta_{n} - \\theta^{*}$ . On the other hand, for  $k \\in \\{1,3\\}$ , the nature of two-timescale structure causes  $\\theta_{n}$  and  $\\mathbf{x}_n$  to become asymptotically independent with zero correlation terms inside  $\\mathbf{V}^{(k)}(\\alpha)$  in (10), and we can explicitly deduce  $\\mathbf{V}_{\\pmb{\\theta}}^{(k)}(\\alpha)$ . We now take a deeper dive into  $\\alpha$  and study its effect on  $\\mathbf{V}_{\\pmb{\\theta}}^{(k)}(\\alpha)$ .",
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+    },
+    {
+        "type": "text",
+        "text": "Covariance Ordering of SA-SRRW: We refer the reader to Appendix F for proofs of all remaining results. We begin by focusing on case (i) and capturing the impact of  $\\alpha$  on  $\\mathbf{V}_{\\theta}^{(1)}(\\alpha)$ .",
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+    },
+    {
+        "type": "text",
+        "text": "Proposition 3.4. For all  $\\alpha_{2} > \\alpha_{1} > 0$ , we have  $\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(\\alpha_2) < _L\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(\\alpha_1) < _L\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(0)$ . Furthermore,  $\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(\\alpha)$  decreases to zero at a rate of  $O(1 / \\alpha^2)$ .",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Proposition 3.4 proves a monotonic reduction (in terms of Loewner ordering) of  $\\mathbf{V}_{\\theta}^{(1)}(\\alpha)$  as  $\\alpha$  increases. Moreover, the decrease rate  $O(1 / \\alpha^2)$  surpasses the  $O(1 / \\alpha)$  rate seen in  $\\mathbf{V}_{\\mathbf{x}}(\\alpha)$  and the sampling application in Doshi et al. (2023, Corollary 4.7), and is also empirically observed in our simulation in Section 4. Suppose we consider the same SA now driven by an i.i.d. sequence  $\\{X_{n}\\}$  with the same marginal distribution  $\\pmb{\\mu}$ . Then, our Proposition 3.4 asserts that a token algorithm employing SRRW (walk on a graph) with large enough  $\\alpha$  on a general graph can actually produce better SA iterates with its asymptotic covariance going down to zero, than a 'hypothetical situation' where the walker is able to access any node  $j$  with probability  $\\mu_{j}$  from anywhere in one step (more like a random jumper). This can be seen by noting that for large time  $n$ , the scaled MSE  $\\mathbb{E}[\\|\\pmb{\\theta}_n - \\pmb{\\theta}^*\\|^2] / \\beta_n$  is composed of the diagonal entries of the covariance matrix  $\\mathbf{V}_{\\theta}$ , which, as we discuss in detail in Appendix F.2, are decreasing in  $\\alpha$  as a consequence of the Loewner ordering in Proposition 3.4. For large enough  $\\alpha$ , the scaled MSE for SA-SRRW becomes smaller than its i.i.d. counterpart, which is always a constant. Although Doshi et al. (2023) alluded this for sampling applications with  $\\mathbf{V}_{\\mathbf{x}}(\\alpha)$ , we broaden its horizons to distributed optimization problems with  $\\mathbf{V}_{\\pmb{\\theta}}(\\alpha)$  using tokens on graphs. Our subsequent result concerns the performance comparison between cases (i) and (iii).",
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+    {
+        "type": "text",
+        "text": "Corollary 3.5. For any  $\\alpha >0$ , we have  $\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(\\alpha) < _L\\mathbf{V}_{\\pmb{\\theta}}^{(3)}(\\alpha) = \\mathbf{V}_{\\pmb{\\theta}}^{(3)}(0)$ .",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We show that case (i) is asymptotically better than case (iii) for  $\\alpha > 0$ . In view of Proposition 3.4 and Corollary 3.5, the advantages of case (i) become prominent.",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+    },
+    {
+        "type": "page_footnote",
+        "text": "Further insights of  $O(1 / \\alpha^2)$  are tied to the two-timescale structure, particularly  $\\beta_{n} = o(\\gamma_{n})$  in case (i), which places  $\\theta_{n}$  on the slow timescale so that the correlation terms  $\\mathbf{J}_{12}(\\alpha),\\mathbf{J}_{22}(\\alpha)$  in the Jacobian matrix  $\\mathbf{J}(\\alpha)$  in (8) come into play. Technical details are referred to Appendix E.2, where we show the form of  $\\mathbf{U}_{\\theta}(\\alpha)$ .",
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+    {
+        "type": "page_number",
+        "text": "8",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/1bc79c3f84b108613b42e4e1b6e41825e615535bc0bf287d96ae710edfe1bc20.jpg",
+        "image_caption": [
+            "(a) SGD-SRRW."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
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+        "type": "image",
+        "img_path": "images/849d61164ad08b70d0a82140ec0a774c9115b9287ff917667add7a60555a01b4.jpg",
+        "image_caption": [
+            "(b) SHB-SRRW."
+        ],
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+        "bbox": [
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+        ],
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+    {
+        "type": "image",
+        "img_path": "images/ef2728074860a48dcd3abbbae47d11f96aab393331e3abfb7f5f3bfe028bf323.jpg",
+        "image_caption": [
+            "(c) Curve fitting for MSE."
+        ],
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+    {
+        "type": "image",
+        "img_path": "images/99c1edae8a3c743d6e335debd63817aff3f5b8998330d174afdcf043685ea939.jpg",
+        "image_caption": [
+            "Figure 2: Simulation results under case (i): (a) and (b) show the performance of SGD-SRRW and SHB-SRRW for various  $\\alpha$  values. (c) shows that MSE decreases at  $O(1 / \\alpha^2)$  speed.",
+            "(a)  $\\alpha = 1$  ,SGD-SRRW",
+            "Figure 3: Comparison of the performance among cases (i) - (iii) for  $\\alpha \\in \\{1,5,10\\}$ ."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "image",
+        "img_path": "images/37d71bc66fd04ec9f787a87660d07ae255b7f1b8ddac4fc39b7b58bf790a6d78.jpg",
+        "image_caption": [
+            "(b)  $\\alpha = 5$  ,SGD-SRRW"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/127f341bd20cf5ab04a65c29d98d5d75f652e1582f6bcd73c062ddb4661b7783.jpg",
+        "image_caption": [
+            "(c)  $\\alpha = 10$  ,SGD-SRRW"
+        ],
+        "image_footnote": [],
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4 SIMULATION",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this section, we simulate our SA-SRRW algorithm on the wikiVote graph (Leskovec & Krevl, 2014), comprising 889 nodes and 2914 edges. We configure the SRRW's base Markov chain  $\\mathbf{P}$  as the MHRW with a uniform target distribution  $\\mu = \\frac{1}{N} \\mathbf{1}$ . For distributed optimization, we consider the following  $L_{2}$  regularized binary classification problem:",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left. \\min  _ {\\boldsymbol {\\theta} \\in \\mathbb {R} ^ {D}} \\left\\{f (\\boldsymbol {\\theta}) \\triangleq \\frac {1}{N} \\sum_ {i = 1} ^ {N} \\log \\left(1 + e ^ {\\boldsymbol {\\theta} ^ {T} \\mathbf {s} _ {i}}\\right) - y _ {i} \\left(\\boldsymbol {\\theta} ^ {T} \\mathbf {s} _ {i}\\right) + \\frac {\\kappa}{2} \\| \\boldsymbol {\\theta} \\| ^ {2} \\right\\}, \\right. \\tag {12}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            261,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "where  $\\{(\\mathbf{s}_i, y_i)\\}_{i=1}^N$  is the ijCNN1 dataset (with 22 features, i.e.,  $\\mathbf{s}_i \\in \\mathbb{R}^{22}$ ) from LIBSVM (Chang & Lin, 2011), and penalty parameter  $\\kappa = 1$ . Each node in the wikiVote graph is assigned one data point, thus 889 data points in total. We perform SRRW driven SGD (SGD-SRRW) and SRRW driven stochastic heavy ball (SHB-SRRW) algorithms (see (13) in Appendix A for its algorithm). We fix the step size  $\\beta_n = (n + 1)^{-0.9}$  for the SA iterates and adjust  $\\gamma_n = (n + 1)^{-a}$  in the SRRW iterates to cover all three cases discussed in this paper: (i)  $a = 0.8$ ; (ii)  $a = 0.9$ ; (iii)  $a = 1$ . We use mean square error (MSE), i.e.,  $\\mathbb{E}[\\|\\pmb{\\theta}_n - \\pmb{\\theta}^*\\|^2]$ , to measure the error on the SA iterates.",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Our results are presented in Figures 2 and 3, where each experiment is repeated 100 times. Figures 2a and 2b, based on wikiVote graph, highlight the consistent performance ordering across different  $\\alpha$  values for both algorithms over almost all time (not just asymptotically). Notably, curves for  $\\alpha \\geq 5$  outperform that of the i.i.d. sampling (in black) even under the graph constraints. Figure 2c on the smaller Dolphin graph (Rossi & Ahmed, 2015) - 62 nodes and 159 edges - illustrates that the points of  $(\\alpha, \\mathrm{MSE})$  pair arising from SGD-SRRW at time  $n = 10^7$  align with a curve in the form of  $g(x) = \\frac{c_1}{(x + c_2)^2} + c_3$  to showcase  $O(1 / \\alpha^2)$  rates. This smaller graph allows for longer simulations to observe the asymptotic behaviour. Additionally, among the three cases examined at identical  $\\alpha$  values, Figures 3a - 3c confirm that case (i) performs consistently better than the rest, underscoring its superiority in practice. Further results, including those from non-convex functions and additional datasets, are deferred to Appendix H due to space constraints.",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this paper, we show both theoretically and empirically that the SRRW as a drop-in replacement for Markov chains can provide significant performance improvements when used for token algorithms, where the acceleration comes purely from the careful analysis of the stochastic input of the algorithm, without changing the optimization iteration itself. Our paper is an instance where the asymptotic analysis approach allows the design of better algorithms despite the usage of unconventional noise sequences such as nonlinear Markov chains like the SRRW, for which traditional finite-time analytical approaches fall short, thus advocating their wider adoption.",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "page_number",
+        "text": "9",
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+    },
+    {
+        "type": "text",
+        "text": "6 ACKNOWLEDGMENTS AND DISCLOSURE OF FUNDING",
+        "text_level": 1,
+        "bbox": [
+            171,
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+        ],
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+    },
+    {
+        "type": "ref_text",
+        "text": "We thank the anonymous reviewers for their constructive comments, especially Reviewer DxLx for raising future directions. This work was supported in part by National Science Foundation under Grant Nos. CNS-2007423 and IIS-1910749.",
+        "bbox": [
+            171,
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+    },
+    {
+        "type": "text",
+        "text": "REFERENCES",
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+        "bbox": [
+            173,
+            196,
+            287,
+            210
+        ],
+        "page_idx": 9
+    },
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
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+    {
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+        "bbox": [
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+    },
+    {
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+        "sub_type": "ref_text",
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+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
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+        ],
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+    {
+        "type": "page_number",
+        "text": "13",
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+    },
+    {
+        "type": "text",
+        "text": "A EXAMPLES OF STOCHASTIC ALGORITHMS OF THE FORM (2).",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the literature of stochastic optimizations, many SGD variants have been proposed by introducing an auxiliary variable to improve convergence. In what follows, we present two SGD variants with decreasing step size that can be presented in the form of (2): SHB (Gadat et al., 2018; Li et al., 2022) and momentum-based algorithm (Barakat et al., 2021; Barakat & Bianchi, 2021).",
+        "bbox": [
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+    {
+        "type": "equation",
+        "text": "\n$$\n\\left\\{ \\begin{array}{l} \\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} - \\beta_ {n + 1} \\mathbf {m} _ {n} \\\\ \\mathbf {m} _ {n + 1} = \\mathbf {m} _ {n} + \\beta_ {n + 1} (\\nabla F (\\boldsymbol {\\theta} _ {n}, X _ {n + 1}) - \\mathbf {m} _ {n}), \\end{array} \\right. \\left\\{ \\begin{array}{l} \\mathbf {v} _ {n + 1} = \\mathbf {v} _ {n} + \\beta_ {n + 1} (\\nabla F (\\boldsymbol {\\theta} _ {n}, X _ {n + 1}) ^ {2} - \\mathbf {v} _ {n}), \\\\ \\mathbf {m} _ {n + 1} = \\mathbf {m} _ {n} + \\beta_ {n + 1} (\\nabla F (\\boldsymbol {\\theta} _ {n}, X _ {n + 1}) - \\mathbf {m} _ {n}), \\\\ \\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} - \\beta_ {n + 1} \\mathbf {m} _ {n} / \\sqrt {\\mathbf {v} _ {n} + \\epsilon}, \\end{array} \\right.\n$$\n",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "(a). SHB",
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+    {
+        "type": "text",
+        "text": "(b). Momentum-based Algorithm",
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+    {
+        "type": "text",
+        "text": "(13)",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where  $\\epsilon > 0$ ,  $\\pmb{\\theta}_n, \\mathbf{m}_n, \\mathbf{v}_n, \\nabla F(\\pmb{\\theta}, X) \\in \\mathbb{R}^d$ , and the square and square root in (13) (b) are element-wise operators. $^{10}$",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For SHB, we introduce an augmented variable  $\\mathbf{z}_n$  and function  $H(\\mathbf{z}_n,X_{n + 1})$  defined as follows:",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {z} _ {n} \\triangleq \\left[ \\begin{array}{c} \\boldsymbol {\\theta} _ {n} \\\\ \\mathbf {m} _ {n} \\end{array} \\right] \\in \\mathbb {R} ^ {2 d}, \\quad H (\\mathbf {z} _ {n}, X _ {n + 1}) \\triangleq \\left[ \\begin{array}{c} - \\mathbf {m} _ {n} \\\\ \\nabla F (\\boldsymbol {\\theta} _ {n}, X _ {n + 1}) - \\mathbf {m} _ {n} \\end{array} \\right] \\in \\mathbb {R} ^ {2 d}.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For the general momentum-based algorithm, we define",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {z} _ {n} \\triangleq \\left[ \\begin{array}{c} \\mathbf {v} _ {n} \\\\ \\mathbf {m} _ {n} \\\\ \\boldsymbol {\\theta} _ {n} \\end{array} \\right] \\in \\mathbb {R} ^ {3 d}, \\quad H (\\mathbf {z} _ {n}, X) \\triangleq \\left[ \\begin{array}{c} \\nabla F (\\boldsymbol {\\theta} _ {n}, X _ {n + 1}) ^ {2} - \\mathbf {v} _ {n} \\\\ \\nabla F (\\boldsymbol {\\theta} _ {n}, X _ {n + 1}) - \\mathbf {m} _ {n} \\\\ - \\mathbf {m} _ {n} / \\sqrt {\\mathbf {v} _ {n} + \\epsilon} \\end{array} \\right] \\in \\mathbb {R} ^ {3 d}.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Thus, we can reformulate both algorithms in (13) as  $\\mathbf{z}_{n + 1} = \\mathbf{z}_n + \\beta_{n + 1}H(\\mathbf{z}_n,X_{n + 1})$ . This augmentation approach was previously adopted in (Gadat et al., 2018; Barakat et al., 2021; Barakat & Bianchi, 2021; Li et al., 2022) to analyze the asymptotic performance of algorithms in (13) using an i.i.d. sequence  $\\{X_{n}\\}_{n\\geq 0}$ . Consequently, the general SA iteration (2) includes these SGD variants. However, we mainly focus on the CLT for the general SA driven by SRRW in this paper. Pursuing the explicit CLT results of these SGD variants with specific form of function  $H(\\theta ,X)$  driven by the SRRW sequence  $\\{X_{n}\\}$  is out of the scope of this paper.",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "When we numerically test the SHB algorithm in Section 4, we use the exact form of (13) (a) and the stochastic sequence  $\\{X_{n}\\}$  is now driven by the SRRW. Specifically, we consider MHRW with transition kernel  $\\mathbf{P}$  as the base Markov chain of the SRRW process, e.g.,",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nP _ {i j} = \\left\\{ \\begin{array}{l l} \\min  \\left\\{\\frac {1}{d _ {i}}, \\frac {1}{d _ {j}} \\right\\} & \\text {i f n o d e j i s t h e n e i g h b o r o f n o d e i ,} \\\\ 0 & \\text {o t h e r w i s e ,} \\end{array} \\right.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nP _ {i i} = 1 - \\sum_ {j \\in \\mathcal {N}} P _ {i j}.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Then, at each time step  $n$ ,",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Draw:  $X_{n + 1}\\sim \\mathbf{K}_{X_n,\\cdot}[\\mathbf{x}_n],$",
+        "bbox": [
+            341,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "where  $K_{ij}[\\mathbf{x}]\\triangleq \\frac{P_{ij}(x_j / \\mu_j)^{-\\alpha}}{\\sum_{k\\in\\mathcal{N}}P_{ik}(x_k / \\mu_k)^{-\\alpha}},\\quad \\forall i,j\\in \\mathcal{N},$",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Update:  $\\mathbf{x}_{n + 1} = \\mathbf{x}_n + \\gamma_{n + 1}(\\pmb{\\delta}_{X_{n + 1}} - \\mathbf{x}_n)$",
+        "bbox": [
+            344,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} - \\beta_ {n + 1} \\mathbf {m} _ {n},\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            419,
+            837,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {m} _ {n + 1} = \\mathbf {m} _ {n} + \\beta_ {n + 1} (\\nabla F (\\boldsymbol {\\theta} _ {n}, X _ {n + 1}) - \\mathbf {m} _ {n}).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            421,
+            859,
+            733,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
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+            47
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "page_footnote",
+        "text": "For ease of expression, we simplify the original SHB and momentum-based algorithms from Gadat et al. (2018); Li et al. (2022); Barakat et al. (2021); Barakat & Bianchi (2021), setting all tunable parameters to 1 and resulting in (13).",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "page_number",
+        "text": "14",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B DISCUSSION ON MEAN FIELD FUNCTION OF SRRW ITERATES (4b)",
+        "text_level": 1,
+        "bbox": [
+            171,
+            102,
+            767,
+            119
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Non-asymptotic analyses have seen extensive attention recently in both single-timescale SA literature (Sun et al., 2018; Karimi et al., 2019; Chen et al., 2020b; 2022) and two-timescale SA literature (Doan, 2021; Zeng et al., 2021). Specifically, single-timescale SA has the following form:",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {x} _ {n + 1} = \\mathbf {x} _ {n} + \\beta_ {n + 1} H (\\mathbf {x} _ {n}, X _ {n + 1}),\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "and function  $h(\\mathbf{x}) \\triangleq \\mathbb{E}_{X \\sim \\mu}[H(\\mathbf{x}, X)]$  is the mean field of function  $H(\\mathbf{x}, X)$ . Similarly, for two-timescale SA, we have the following recursions:",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {x} _ {n + 1} = \\mathbf {x} _ {n} + \\beta_ {n + 1} H _ {1} (\\mathbf {x} _ {n}, \\mathbf {y} _ {n}, X _ {n + 1}),\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            362,
+            248,
+            630,
+            263
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {y} _ {n + 1} = \\mathbf {y} _ {n} + \\gamma_ {n + 1} H _ {2} (\\mathbf {x} _ {n}, \\mathbf {y} _ {n}, X _ {n + 1}),\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            364,
+            267,
+            630,
+            282
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "where  $\\{\\beta_n\\}$  and  $\\{\\gamma_n\\}$  are on different timescales, and function  $h_i(\\mathbf{x},\\mathbf{y})\\triangleq \\mathbb{E}_{X\\sim \\mu}[H_i(\\mathbf{x},\\mathbf{y},X)]$  is the mean field of function  $H_{i}(\\mathbf{x},\\mathbf{y},X)$  for  $i = \\{1,2\\}$ . All the aforementioned works require the mean field function  $h(\\mathbf{x})$  in the single-timescale SA (or  $h_1(\\mathbf{x},\\mathbf{y}),h_2(\\mathbf{x},\\mathbf{y})$  in the two-timescale SA) to be globally Lipschitz with a Lipschitz constant  $L$  to proceed with the derivation of finite-time bounds including the constant  $L$ .",
+        "bbox": [
+            169,
+            292,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Here, we show that the mean field function  $\\pi[\\mathbf{x}] - \\mathbf{x}$  in the SRRW iterates (4b) is not globally Lipschitz, where  $\\pi[\\mathbf{x}]$  is the stationary distribution of the SRRW kernel  $\\mathbf{K}[\\mathbf{x}]$  defined in (3). To this end, we show that each entry of Jacobian matrix of  $\\pi[\\mathbf{x}] - \\mathbf{x}$  goes unbounded because a multivariate function is Lipschitz if and only if it has bounded partial derivatives. Note that from Doshi et al. (2023, Proposition 2.1), for the  $i$ -th entry of  $\\pi[\\mathbf{x}]$ , we have",
+        "bbox": [
+            169,
+            369,
+            825,
+            441
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {\\pi} _ {i} [ \\mathbf {x} ] = \\frac {\\sum_ {j \\in \\mathcal {N}} \\mu_ {i} P _ {i j} \\left(x _ {i} / \\mu_ {i}\\right) ^ {- \\alpha} \\left(x _ {j} / \\mu_ {j}\\right) ^ {- \\alpha}}{\\sum_ {i \\in \\mathcal {N}} \\sum_ {j \\in \\mathcal {N}} \\mu_ {i} P _ {i j} \\left(x _ {i} / \\mu_ {i}\\right) ^ {- \\alpha} \\left(x _ {j} / \\mu_ {j}\\right) ^ {- \\alpha}}. \\tag {16}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            320,
+            449,
+            823,
+            489
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Then, the Jacobian matrix of the mean field function  $\\pi[\\mathbf{x}] - \\mathbf{x}$ , which has been derived in Doshi et al. (2023, Proof of Lemma 3.4 in Appendix B), is given as follows:",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\frac {\\partial (\\boldsymbol {\\pi} _ {i} [ \\mathbf {x} ] - x _ {i})}{\\partial x _ {j}} \\\\ = \\frac {2 \\alpha}{x _ {j}} \\cdot \\frac {\\left(\\sum_ {k \\in \\mathcal {N}} \\mu_ {i} P _ {i k} \\left(x _ {i} / \\mu_ {i}\\right) ^ {- \\alpha} \\left(x _ {k} / \\mu_ {k}\\right) ^ {- \\alpha}\\right) \\left(\\sum_ {k \\in \\mathcal {N}} \\mu_ {j} P _ {j k} \\left(x _ {j} / \\mu_ {j}\\right) ^ {- \\alpha} \\left(x _ {k} / \\mu_ {k}\\right) ^ {- \\alpha}\\right)}{\\left(\\sum_ {l \\in \\mathcal {N}} \\sum_ {k \\in \\mathcal {N}} \\mu_ {l} P _ {l k} \\left(x _ {l} / \\mu_ {l}\\right) ^ {- \\alpha} \\left(x _ {k} / \\mu_ {k}\\right) ^ {- \\alpha}\\right) ^ {2}} \\tag {17} \\\\ - \\frac {\\alpha}{x _ {j}} \\cdot \\frac {\\mu_ {i} P _ {i j} (x _ {i} / \\mu_ {i}) ^ {- \\alpha} (x _ {j} / \\mu_ {j}) ^ {- \\alpha}}{\\sum_ {l \\in \\mathcal {N}} \\sum_ {k \\in \\mathcal {N}} \\mu_ {l} P _ {l k} (x _ {l} / \\mu_ {l}) ^ {- \\alpha} (x _ {k} / \\mu_ {k}) ^ {- \\alpha}} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            202,
+            534,
+            823,
+            648
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "for  $i,j\\in \\mathcal{N},i\\neq j$  , and",
+        "bbox": [
+            171,
+            655,
+            331,
+            671
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\frac {\\partial \\left(\\boldsymbol {\\pi} _ {i} [ \\mathbf {x} ] - x _ {i}\\right)}{\\partial x _ {i}} \\\\ = \\frac {2 \\alpha}{x _ {i}} \\cdot \\frac {\\left(\\sum_ {k \\in \\mathcal {N}} \\mu_ {i} P _ {i k} \\left(x _ {i} / \\mu_ {i}\\right) ^ {- \\alpha} \\left(x _ {k} / \\mu_ {k}\\right) ^ {- \\alpha}\\right) ^ {2}}{\\left(\\sum_ {l \\in \\mathcal {N}} \\sum_ {k \\in \\mathcal {N}} \\mu_ {l} P _ {l k} \\left(x _ {l} / \\mu_ {l}\\right) ^ {- \\alpha} \\left(x _ {k} / \\mu_ {k}\\right) ^ {- \\alpha}\\right) ^ {2}} \\tag {18} \\\\ - \\frac {\\alpha}{x _ {i}} \\cdot \\frac {\\sum_ {k \\in \\mathcal {N}} \\mu_ {i} P _ {i k} (x _ {i} / \\mu_ {i}) ^ {- \\alpha} (x _ {k} / \\mu_ {k}) ^ {- \\alpha} + \\mu_ {i} P _ {i i} (x _ {i} / \\mu_ {i}) ^ {- 2 \\alpha}}{\\sum_ {l \\in \\mathcal {N}} \\sum_ {k \\in \\mathcal {N}} \\mu_ {l} P _ {l k} (x _ {l} / \\mu_ {l}) ^ {- \\alpha} (x _ {k} / \\mu_ {k}) ^ {- \\alpha}} - 1 \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            264,
+            678,
+            825,
+            792
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "for  $i \\in \\mathcal{N}$ . Since the empirical distribution  $\\mathbf{x} \\in \\mathrm{Int}(\\Sigma)$ , we have  $x_i \\in (0,1)$  for all  $i \\in \\mathcal{N}$ . For fixed  $i$ , assume  $x_i = x_j$  and as they approach zero, the terms  $(x_i / \\mu_i)^{-\\alpha}, (x_j / \\mu_j)^{-\\alpha}$  dominate the fraction in (17) and both the numerator and the denominator of the fraction have the same order in terms of  $x_i, x_j$ . Thus, we have",
+        "bbox": [
+            169,
+            799,
+            823,
+            856
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\frac {\\partial (\\pmb {\\pi} _ {i} [ \\mathbf {x} ] - x _ {i})}{\\partial x _ {j}} = O \\left(\\frac {1}{x _ {j}}\\right)\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            406,
+            857,
+            588,
+            892
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "such that the  $(i,j)$ -th entry of the Jacobian matrix can go unbounded as  $x_{j} \\to 0$ . Consequently,  $\\pi[\\mathbf{x}] - \\mathbf{x}$  is not globally Lipschitz for  $\\mathbf{x} \\in \\operatorname{Int}(\\Sigma)$ .",
+        "bbox": [
+            169,
+            895,
+            823,
+            925
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "page_number",
+        "text": "15",
+        "bbox": [
+            490,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "C DISCUSSION ON ASSUMPTION A3'",
+        "text_level": 1,
+        "bbox": [
+            171,
+            101,
+            496,
+            118
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "When  $\\gamma_{n} = o(\\beta_{n})$ , iterates  $\\mathbf{x}_n$  has smaller step size compared to  $\\pmb{\\theta}_{n}$ , thus converges 'slower' than  $\\pmb{\\theta}_{n}$ . From Assumption A3',  $\\pmb{\\theta}_{n}$  will intuitively converge to some point  $\\rho (\\mathbf{x})$  with the current value  $\\mathbf{x}$  from the iteration  $\\mathbf{x}_n$ , i.e.,  $\\mathbb{E}_{X\\sim \\pi [\\mathbf{x}]}[H(\\rho (\\mathbf{x}),X)] = 0$ , while the Hurwitz condition is to ensure the stability around  $\\rho (\\mathbf{x})$ . We can see that Assumption A3 is less stringent than A3' in that it only assumes such condition when  $\\mathbf{x} = \\pmb{\\mu}$  such that  $\\rho (\\pmb {\\mu}) = \\pmb{\\theta}^{*}$  rather than for all  $\\mathbf{x}\\in \\mathrm{Int}(\\Sigma)$ .",
+        "bbox": [
+            169,
+            133,
+            823,
+            205
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "One special instance of Assumption A3' is by assuming the linear SA, e.g.,  $H(\\pmb{\\theta}, i) = A_i\\pmb{\\theta} + b_i$ . In this case,  $\\mathbb{E}_{X \\sim \\pi[\\mathbf{x}]}[H(\\rho(\\mathbf{x}), X)] = 0$  is equivalent to  $\\mathbb{E}_{i \\sim \\pi[\\mathbf{x}]}[A_i]\\rho(\\mathbf{x}) + \\mathbb{E}_{i \\sim \\pi[\\mathbf{x}]}[b_i] = 0$ . Under the condition that for every  $\\mathbf{x} \\in \\mathrm{Int}(\\Sigma)$ , matrix  $\\mathbb{E}_{i \\sim \\pi[\\mathbf{x}]}[A_i]$  is invertible, we then have",
+        "bbox": [
+            169,
+            210,
+            825,
+            258
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\rho (\\mathbf {x}) = - \\left(\\mathbb {E} _ {i \\sim \\boldsymbol {\\pi} [ \\mathbf {x} ]} [ A _ {i} ]\\right) ^ {- 1} \\cdot \\mathbb {E} _ {i \\sim \\boldsymbol {\\pi} [ \\mathbf {x} ]} [ b _ {i} ].\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            362,
+            265,
+            630,
+            282
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "However, this condition is quite strict. Loosely speaking,  $\\mathbb{E}_{i\\sim \\pi [\\mathbf{x}]}[A_i]$  being invertible for any  $\\mathbf{x}$  is similar to saying that any convex combination of  $\\{A_i\\}$  is invertible. For example, if we assume  $\\{A_i\\}_{i\\in \\mathcal{N}}$  are negative definite and they all share the same eigenbasis  $\\{\\mathbf{u}_i\\}$ , e.g.,  $A_{i} = \\sum_{j = 1}^{D}\\lambda_{j}^{i}\\mathbf{u}_{i}\\mathbf{u}_{i}^{T}$  and  $\\lambda_j^i < 0$  for all  $i\\in \\mathcal{N}, j\\in [D]$ . Then,  $\\mathbb{E}_{i\\sim \\pi [\\mathbf{x}]}[A_i]$  is invertible.",
+        "bbox": [
+            169,
+            290,
+            823,
+            358
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Another example for Assumption A3' is when  $H(\\pmb{\\theta}, i) = H(\\pmb{\\theta}, j)$  for all  $i, j \\in \\mathcal{N}$ , which implies that each agent in the distributed learning has the same local dataset to collaboratively train the model. In this example,  $\\rho(\\mathbf{x}) = \\pmb{\\theta}^*$  such that",
+        "bbox": [
+            169,
+            362,
+            823,
+            406
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\mathbb {E} _ {i \\sim \\pi [ \\mathbf {x} ]} [ H (\\rho (\\mathbf {x}), i) ] = h (\\boldsymbol {\\theta} ^ {*}) = 0, \\\\ \\mathbb {E} _ {i \\sim \\pi [ \\mathbf {x} ]} [ \\nabla H (\\rho (\\mathbf {x}), i) ] + \\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I} = \\nabla h (\\boldsymbol {\\theta} ^ {*}) + \\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I} \\quad \\text {b e i n g H u r w i t z}. \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            253,
+            412,
+            743,
+            468
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "D PROOF OF LEMMA 3.1 AND LEMMA 3.2",
+        "text_level": 1,
+        "bbox": [
+            171,
+            484,
+            542,
+            500
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "In this section, we demonstrate the almost sure convergence of both  $\\theta_{n}$  and  $\\mathbf{x}_n$  together. This proof naturally incorporates the almost certain convergence of the SRRW iteration in Lemma 3.1, since  $\\mathbf{x}_n$  is independent of  $\\theta_{n}$  (as indicated in (4)), allowing us to separate out its asymptotic results. The same reason applies to the CLT analysis of SRRW iterates and we refer the reader to Section E.1 for the CLT result of  $\\mathbf{x}_n$  in Lemma 3.1.",
+        "bbox": [
+            169,
+            516,
+            823,
+            585
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "We will use different techniques for different settings of step sizes in Assumption A2. Specifically, for step sizes  $\\gamma_{n} = (n + 1)^{-a},\\beta_{n} = (n + 1)^{-b}$ , we consider the following scenarios:",
+        "bbox": [
+            169,
+            592,
+            823,
+            622
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Scenario 1: We consider case(ii):  $1/2 < a = b \\leq 1$ , and will apply the almost sure convergence result of the single-timescale stochastic approximation in Theorem G.8 and verify all the conditions therein.",
+        "bbox": [
+            169,
+            632,
+            823,
+            675
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Scenario 2: We consider both case(i):  $1/2 < a < b \\leq 1$  and case (iii):  $1/2 < b < a \\leq 1$ . In these two cases, step sizes  $\\gamma_{n}, \\beta_{n}$  decrease at different rates, thereby putting iterates  $\\mathbf{x}_{n}, \\pmb{\\theta}_{n}$  on different timescales and resulting in a two-timescale structure. We will apply the existing almost sure convergence result of the two-timescale stochastic approximation with iteratedependent Markov chain in Yaji & Bhatnagar (2020, Theorem 4) where our SA-SRRW algorithm can be regarded as a special instance.[11]",
+        "bbox": [
+            169,
+            679,
+            823,
+            763
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "D.1 SCENARIO 1",
+        "text_level": 1,
+        "bbox": [
+            171,
+            780,
+            303,
+            794
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "In Scenario 1, we have  $\\beta_{n} = \\gamma_{n}$ . First, we rewrite (4) as",
+        "bbox": [
+            169,
+            806,
+            544,
+            821
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left[ \\begin{array}{l} \\boldsymbol {\\theta} _ {n + 1} \\\\ \\mathbf {x} _ {n + 1} \\end{array} \\right] = \\left[ \\begin{array}{l} \\boldsymbol {\\theta} _ {n} \\\\ \\mathbf {x} _ {n} \\end{array} \\right] + \\gamma_ {n + 1} \\left[ \\begin{array}{c} H \\left(\\boldsymbol {\\theta} _ {n}. X _ {n + 1}\\right) \\\\ \\boldsymbol {\\delta} _ {X _ {n + 1}} - \\mathbf {x} _ {n} \\end{array} \\right]. \\tag {19}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            357,
+            828,
+            823,
+            863
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "page_footnote",
+        "text": "<sup>11</sup>However, Yaji & Bhatnagar (2020) paper only analysed the almost sure convergence. The central limit theorem analysis remains unknown in the literature for the two-timescale stochastic approximation with iteratedependent Markov chains. Thus, our CLT analysis in Section E for this two-timescale structure with iteratedependent Markov chain is still novel and recognized as our contribution.",
+        "bbox": [
+            169,
+            871,
+            823,
+            925
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "page_number",
+        "text": "16",
+        "bbox": [
+            490,
+            948,
+            508,
+            959
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "By augmentations, we define the variable  $\\mathbf{z}_n \\triangleq \\left[ \\begin{array}{c} \\boldsymbol{\\theta}_n \\\\ \\mathbf{x}_n \\end{array} \\right] \\in \\mathbb{R}^{(N + D) \\times 1}$  and the function  $G(\\mathbf{z}_n, i) \\triangleq \\left[ \\begin{array}{c} H(\\boldsymbol{\\theta}_n, i) \\\\ \\delta_i - \\mathbf{x}_n \\end{array} \\right] \\in \\mathbb{R}^{(N + d) \\times 1}$ . In addition, we define a new Markov chain  $\\{Y_n\\}_{n \\geq 0}$  in the same state space  $\\mathcal{N}$  as SRRW sequence  $\\{X_n\\}_{n \\geq 0}$ . With slight abuse of notation, the transition kernel of  $\\{Y_n\\}$  is denoted by  $\\mathbf{K}'[\\mathbf{z}_n] \\equiv \\mathbf{K}[\\mathbf{x}_n]$  and its stationary distribution  $\\pi'[\\mathbf{z}_n] \\equiv \\pi[\\mathbf{x}_n]$ , where  $\\mathbf{K}[\\mathbf{x}_n]$  and  $\\pi(\\mathbf{x}_n)$  are the transition kernel and its corresponding stationary distribution of SRRW, with  $\\pi[\\mathbf{x}]$  of the form",
+        "bbox": [
+            169,
+            99,
+            826,
+            217
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\pi_ {i} [ \\mathbf {x} ] \\propto \\sum_ {j \\in \\mathcal {N}} \\mu_ {i} P _ {i j} \\left(x _ {i} / \\mu_ {i}\\right) ^ {- \\alpha} \\left(x _ {j} / \\mu_ {j}\\right) ^ {- \\alpha}. \\tag {20}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            215,
+            823,
+            247
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Recall that  $\\pmb{\\mu}$  is the fixed point, i.e.,  $\\pi[\\pmb{\\mu}] = \\pmb{\\mu}$ , and  $\\mathbf{P}$  is the base Markov chain inside SRRW (see (3)). Then, the mean field",
+        "bbox": [
+            169,
+            250,
+            823,
+            277
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\ng (\\mathbf {z}) = \\mathbb {E} _ {Y \\sim \\boldsymbol {\\pi^ {\\prime}} (\\mathbf {z})} [ G (\\mathbf {z}, Y) ] = \\left[ \\begin{array}{c} \\sum_ {i \\in \\mathcal {N}} \\pi_ {i} [ \\mathbf {x} ] H (\\boldsymbol {\\theta}, i) \\\\ \\boldsymbol {\\pi} [ \\mathbf {x} ] - \\mathbf {x} \\end{array} \\right],\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            279,
+            676,
+            313
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "and  $\\mathbf{z}^{*} = (\\pmb{\\theta}^{*},\\pmb{\\mu})$  for  $\\pmb{\\theta}^{*}\\in \\Theta$  in Assumption A3 is the root of  $g(\\mathbf{z})$ , i.e.,  $g(\\mathbf{z}^{*}) = 0$ . The augmented iteration (19) becomes",
+        "bbox": [
+            169,
+            314,
+            823,
+            340
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {z} _ {n + 1} = \\mathbf {z} _ {n} + \\gamma_ {n + 1} G \\left(\\mathbf {z} _ {n}, Y _ {n + 1}\\right) \\tag {21}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            388,
+            342,
+            821,
+            357
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "with the goal of solving  $g(\\mathbf{z}) = 0$ . Therefore, we can treat (21) as an SA algorithm driven by a Markov chain  $\\{Y_{n}\\}_{n\\geq 0}$  with its kernel  $\\mathbf{K}'[\\mathbf{z}]$  and stationary distribution  $\\pi^{\\prime}[\\mathbf{z}]$ , which has been widely studied in the literature (e.g., Delyon (2000); Benveniste et al. (2012); Fort (2015); Li et al. (2023)). In what follows, we demonstrate that for any initial point  $\\mathbf{z}_0 = (\\pmb {\\theta}_0,\\mathbf{x}_0)\\in \\mathbb{R}^D\\times \\mathrm{Int}(\\Sigma)$ , the SRRW iteration  $\\{\\mathbf{x}_n\\}_{n\\geq 0}$  will almost surely converge to the target distribution  $\\pmb{\\mu}$ , and the SA iteration  $\\{\\pmb {\\theta}_n\\}_{n\\geq 0}$  will almost surely converge to the set  $\\Theta$ .",
+        "bbox": [
+            169,
+            357,
+            823,
+            441
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Now we verify conditions C1 - C4 in Theorem G.8. Our assumption A4 is equivalent to condition C1 and assumption A2 corresponds to condition C2. For condition C3, we set  $\\nabla w(\\mathbf{z}) \\equiv -g(\\mathbf{z})$ , and the set  $S \\equiv \\{\\mathbf{z}^* | \\boldsymbol{\\theta}^* \\in \\Theta, \\mathbf{x}^* = \\boldsymbol{\\mu}\\}$ , including disjoint points. For condition C4, since  $\\mathbf{K}'[\\mathbf{z}]$ , or equivalently  $\\mathbf{K}[\\mathbf{x}]$ , is ergodic and time-reversible for a given  $\\mathbf{z}$ , as shown in the SRRW work Doshi et al. (2023), it automatically ensures a solution to the Poisson equation, which has been well discussed in Chen et al. (2020a, Section 2) and Benveniste et al. (2012); Meyn (2022). To show (97) and (98) in condition C4, for each given  $\\mathbf{z}$  and any  $i \\in \\mathcal{N}$ , we need to give the explicit solution  $m_{\\mathbf{z}}(i)$  to the Poisson equation  $m_{\\mathbf{z}}(i) - (\\mathbf{K}_{\\mathbf{z}}' m_{\\mathbf{z}})(i) = G(\\mathbf{z}, i) - g(\\mathbf{z})$  in (96). The notation  $(\\mathbf{K}_{\\mathbf{z}}' m_{\\mathbf{z}})(i)$  is defined as follows.",
+        "bbox": [
+            169,
+            446,
+            825,
+            571
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left(\\mathbf {K} _ {\\mathbf {z}} ^ {\\prime} m _ {\\mathbf {z}}\\right) (i) = \\sum_ {j \\in \\mathcal {N}} \\mathbf {K} _ {\\mathbf {z}} ^ {\\prime} (i, j) m (\\mathbf {z}, j).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            375,
+            569,
+            619,
+            601
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Let  $\\mathbf{G}(\\mathbf{z})\\triangleq [G(\\mathbf{z},1),\\dots ,G(\\mathbf{z},N)]^T\\in \\mathbb{R}^{N\\times D}$ . We use  $[\\mathbf{A}]_{:,i}$  to denote the  $i$ -th column of matrix  $\\mathbf{A}$ . Then, we let  $m_{\\mathbf{z}}(i)$  such that",
+        "bbox": [
+            169,
+            611,
+            823,
+            641
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nm _ {\\mathbf {z}} (i) = \\sum_ {k = 0} ^ {\\infty} \\left(\\left[ \\mathbf {G} (\\mathbf {z}) \\left(\\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] ^ {k}\\right) ^ {T} \\right] _ {[:, i ]} - g (\\mathbf {z})\\right) = \\sum_ {k = 0} ^ {\\infty} \\left[ \\mathbf {G} (\\mathbf {z}) \\left(\\left(\\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] ^ {k}\\right) ^ {T} - \\boldsymbol {\\pi} ^ {\\prime} [ \\mathbf {z} ] \\mathbf {1} ^ {T}\\right) \\right] _ {[:, i ]}. \\tag {22}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            204,
+            643,
+            823,
+            683
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "In addition,",
+        "bbox": [
+            171,
+            685,
+            253,
+            698
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left(\\mathbf {K} _ {\\mathbf {z}} ^ {\\prime} m _ {\\mathbf {z}}\\right) (i) = \\sum_ {k = 1} ^ {\\infty} \\left[ \\mathbf {G} (\\mathbf {z}) \\left(\\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] ^ {T} - \\boldsymbol {\\pi} ^ {\\prime} [ \\mathbf {z} ] \\mathbf {1} ^ {T}\\right) \\right] _ {[ i, i ]}. \\tag {23}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            334,
+            696,
+            823,
+            736
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "We can check that the  $m_{\\mathbf{z}}(i)$  form in (22) is indeed the solution of the above Poisson equation. Now, by induction, we get  $\\mathbf{K}'[\\mathbf{z}]^k - \\mathbf{1}\\pi'[\\mathbf{z}]^T = (\\mathbf{K}'[\\mathbf{z}] - \\mathbf{1}\\pi'[\\mathbf{z}]^T)^k$  for  $k \\geq 1$  and for  $k = 0$ ,  $\\mathbf{K}'[\\mathbf{z}]^0 - \\mathbf{1}\\pi'[\\mathbf{z}]^T = (\\mathbf{K}'[\\mathbf{z}] - \\mathbf{1}\\pi'[\\mathbf{z}]^T)^0 - \\mathbf{1}\\pi'[\\mathbf{z}]^T$ . Then,",
+        "bbox": [
+            169,
+            737,
+            823,
+            780
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} m _ {\\mathbf {z}} (i) = \\sum_ {k = 0} ^ {\\infty} [ \\mathbf {G} (\\mathbf {z}) (\\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] ^ {T} - \\boldsymbol {\\pi} ^ {\\prime} [ \\mathbf {z} ] \\mathbf {1} ^ {T}) ^ {k} ] _ {[; i ]} - g (\\mathbf {z}) \\\\ = \\left[ \\mathbf {G} (\\mathbf {z}) \\sum_ {k = 0} ^ {\\infty} \\left(\\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] ^ {T} - \\boldsymbol {\\pi} ^ {\\prime} [ \\mathbf {z} ] \\mathbf {1} ^ {T}\\right) ^ {k} \\right] _ {[;, i ]} - g (\\mathbf {z}) \\tag {24} \\\\ = \\left[ \\mathbf {G} (\\mathbf {z}) \\left(\\mathbf {I} - \\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] ^ {T} + \\boldsymbol {\\pi} ^ {\\prime} [ \\mathbf {z} ] \\mathbf {1} ^ {T}\\right) ^ {- 1} \\right] _ {[,: i ]} - g (\\mathbf {z}) \\\\ = \\sum_ {j \\in \\mathcal {N}} \\left(\\mathbf {I} - \\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] + \\mathbf {1} \\boldsymbol {\\pi} ^ {\\prime} [ \\mathbf {z} ] ^ {T}\\right) ^ {- 1} (i, j) G (\\mathbf {z}, j) - g (\\mathbf {z}). \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            299,
+            782,
+            823,
+            926
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "page_number",
+        "text": "17",
+        "bbox": [
+            490,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Here,  $(\\mathbf{I} - \\mathbf{K}'[\\mathbf{z}] + 1\\pi'[\\mathbf{z}]^T)^{-1}$  is well defined because  $\\mathbf{K}'[\\mathbf{z}]$  is ergodic and time-reversible for any given  $\\mathbf{z}$  (proved in Doshi et al. (2023, Appendix A)). Now that both functions  $H(\\theta, i)$  and  $\\delta_i - \\mathbf{x}$  are bounded for each compact subset of  $\\mathbb{R}^D \\times \\Sigma$  by our assumption A1, function  $G(\\mathbf{z}, i)$  is also bounded within the compact subset of its domain. Thus, function  $m_{\\mathbf{z}}(i)$  is bounded, and (97) is verified. Moreover, for a fixed  $i \\in \\mathcal{N}$ ,",
+        "bbox": [
+            169,
+            102,
+            823,
+            174
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\sum_ {j \\in \\mathcal {N}} (\\mathbf {I} - \\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] + \\mathbf {1} \\boldsymbol {\\pi} ^ {\\prime} [ \\mathbf {z} ] ^ {T}) ^ {- 1} (i, j) \\boldsymbol {\\delta} _ {j} = (\\mathbf {I} - \\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] + \\mathbf {1} \\boldsymbol {\\pi} ^ {\\prime} [ \\mathbf {z} ] ^ {T}) _ {[,: i ]} ^ {- 1} = (\\mathbf {I} - \\mathbf {K} [ \\mathbf {x} ] + \\mathbf {1} \\boldsymbol {\\pi} [ \\mathbf {x} ] ^ {T}) _ {[,: i ]} ^ {- 1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            184,
+            181,
+            810,
+            214
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "and this vector-valued function is continuous in  $\\mathbf{x}$  because  $\\mathbf{K}[\\mathbf{x}], \\pi[\\mathbf{x}]$  are continuous. We then rewrite (24) as",
+        "bbox": [
+            169,
+            222,
+            823,
+            251
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nm _ {\\mathbf {z}} (i) = \\left[ \\begin{array}{c} \\sum_ {j \\in \\mathcal {N}} (\\mathbf {I} - \\mathbf {K} [ \\mathbf {x} ] + \\mathbf {1} \\pi [ \\mathbf {x} ] ^ {T}) ^ {- 1} (i, j) H (\\mathbf {x}, j) \\\\ (\\mathbf {I} - \\mathbf {K} [ \\mathbf {x} ] ^ {T} + \\pi [ \\mathbf {x} ] \\mathbf {1} ^ {T}) _ {[,: i ]} ^ {- 1} \\end{array} \\right] - \\left[ \\begin{array}{c} \\sum_ {i \\in \\mathcal {N}} \\pi_ {i} [ \\mathbf {x} ] H (\\pmb {\\theta}, i) \\\\ \\pmb {\\pi} [ \\mathbf {x} ] - \\mathbf {x} \\end{array} \\right].\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            220,
+            258,
+            774,
+            301
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "With continuous functions  $H(\\theta, i)$ ,  $\\mathbf{K}[\\mathbf{x}]$ ,  $\\pi[\\mathbf{x}]$ , we have  $m_{\\mathbf{z}}(i)$  continuous with respect to  $\\mathbf{z}$ , so does  $(\\mathbf{K}_{\\mathbf{z}}^{\\prime} m_{\\mathbf{z}})(i)$ . This implies that functions  $m_{\\mathbf{z}}(i)$  and  $(\\mathbf{K}_{\\mathbf{z}}^{\\prime} m_{\\mathbf{z}})(i)$  are locally Lipschitz, which satisfies (98) with  $\\phi_{\\mathcal{C}}(x) = C_{\\mathcal{C}} x$  for some constant  $C_{\\mathcal{C}}$  that depends on the compact set  $\\mathcal{C}$ . Therefore, condition C4 is checked, and we can apply Theorem G.8 to show the almost convergence result of (19), i.e., almost surely,",
+        "bbox": [
+            169,
+            306,
+            825,
+            378
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\lim  _ {n \\to \\infty} \\mathbf {x} _ {n} = \\boldsymbol {\\mu}, \\quad \\text {a n d} \\quad \\lim  _ {n \\to \\infty} \\sup  _ {\\boldsymbol {\\theta} ^ {*} \\in \\Theta} \\inf  _ {\\boldsymbol {\\theta}} \\| \\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*} \\| = 0.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            318,
+            385,
+            676,
+            409
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Therefore, the almost sure convergence of  $\\mathbf{x}_n$  in Lemma 3.1 is also proved. This finishes the proof in Scenario 1.",
+        "bbox": [
+            169,
+            416,
+            826,
+            445
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "D.2 SCENARIO 2",
+        "text_level": 1,
+        "bbox": [
+            171,
+            462,
+            305,
+            476
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Now in this subsection, we consider the steps sizes  $\\gamma_{n},\\beta_{n}$  with  $1 / 2 < a < b\\leq 1$  and  $1 / 2 < b < a\\leq 1$ . We will frequently use assumptions (B1) - (B5) in Section G.3 and Theorem G.10 to prove the almost sure convergence.",
+        "bbox": [
+            169,
+            488,
+            823,
+            532
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "D.2.1 CASE (I):  $1/2 < a < b \\leq 1$",
+        "text_level": 1,
+        "bbox": [
+            171,
+            547,
+            416,
+            561
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "In case (i),  $\\theta_{n}$  is on the slow timescale and  $\\mathbf{x}_n$  is on the fast timescale because iteration  $\\theta_{n}$  has smaller step size than  $\\mathbf{x}_n$ , making  $\\theta_{n}$  converge slower than  $\\mathbf{x}_n$ . Here, we consider the two-timescale SA of the form:",
+        "bbox": [
+            169,
+            571,
+            823,
+            612
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} + \\beta_ {n + 1} H (\\boldsymbol {\\theta}, X _ {n + 1}),\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            385,
+            613,
+            609,
+            630
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {x} _ {n + 1} = \\mathbf {x} _ {n} + \\gamma_ {n + 1} (\\boldsymbol {\\delta} _ {X _ {n + 1}} - \\mathbf {x}).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            385,
+            633,
+            609,
+            652
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Now, we verify assumptions (B1) - (B5) listed in Section G.3.",
+        "bbox": [
+            169,
+            662,
+            578,
+            678
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- Assumptions (B1) and (B5) are satisfied by our assumptions A2 and A4.",
+            "- Our assumption A3 shows that the function  $H(\\pmb{\\theta}, X)$  is continuous and differentiable w.r.t  $\\pmb{\\theta}$  and grows linearly with  $\\| \\pmb{\\theta} \\|$ . In addition,  $\\delta_X - \\mathbf{x}$  also satisfies this property. Therefore, (B2) is satisfied.",
+            "- Now that the function  $\\pi[\\mathbf{x}] - \\mathbf{x}$  is independent of  $\\pmb{\\theta}$ , we can set  $\\rho(\\pmb{\\theta}) = \\pmb{\\mu}$  for any  $\\pmb{\\theta} \\in \\mathbb{R}^D$  such that  $\\pi[\\pmb{\\mu}] - \\pmb{\\mu} = 0$  from Doshi et al. (2023, Proposition 3.1), and"
+        ],
+        "bbox": [
+            215,
+            689,
+            823,
+            785
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\nabla_ {\\mathbf {x}} (\\boldsymbol {\\pi} (\\mathbf {x}) - \\mathbf {x}) | _ {\\mathbf {x} = \\boldsymbol {\\mu}} = 2 \\alpha \\mathbf {u} \\mathbf {1} ^ {T} - \\alpha \\mathbf {P} ^ {T} - (\\alpha + 1) \\mathbf {I}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            357,
+            792,
+            697,
+            811
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "from Doshi et al. (2023, Lemma 3.4), which is Hurwitz. Furthermore,  $\\rho(\\pmb{\\theta}) = \\pmb{\\mu}$  inherently satisfies the condition  $\\|\\rho(\\pmb{\\theta})\\| \\leq L_2(1 + \\|\\pmb{\\theta}\\|)$  for any  $L_2 \\geq \\|\\pmb{\\mu}\\|$ . Thus, conditions (i) - (iii) in (B3) are satisfied. Additionally,  $\\sum_{i \\in \\mathcal{N}} \\pi_i[\\rho(\\pmb{\\theta})] H(\\pmb{\\theta}, i) = \\sum_{i \\in \\mathcal{N}} \\pi_i[\\mathbf{x}] = \\mathbf{h}(\\pmb{\\theta})$  such that for  $\\pmb{\\theta}^* \\in \\Theta$  defined in assumption A3,  $\\sum_{i \\in \\mathcal{N}} \\pi_i[\\rho(\\pmb{\\theta}^*)] H(\\pmb{\\theta}^*, i) = \\mathbf{h}(\\pmb{\\theta}^*) = 0$ , and  $\\nabla_\\theta \\mathbf{h}(\\pmb{\\theta}^*)$  is Hurwitz. Therefore, (B3) is checked.",
+        "bbox": [
+            227,
+            816,
+            823,
+            891
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "- Assumption (B4) is verified by the nature of SRRW, i.e., its transition kernel  $\\mathbf{K}[\\mathbf{x}]$  and the corresponding stationary distribution  $\\pi[\\mathbf{x}]$  with  $\\pi[\\boldsymbol{\\mu}] = \\boldsymbol{\\mu}$ .",
+        "bbox": [
+            215,
+            895,
+            823,
+            925
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "page_number",
+        "text": "18",
+        "bbox": [
+            490,
+            946,
+            508,
+            960
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Consequently, assumptions (B1) - (B5) are satisfied by our assumeproins A1 - A4 and by Theorem G.10, we have  $\\lim_{n\\to \\infty}\\mathbf{x}_n = \\pmb{\\mu}$  and  $\\theta_{n}\\rightarrow \\Theta$  almost surely.",
+        "bbox": [
+            169,
+            103,
+            823,
+            133
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Next, we consider  $1/2 < b < a \\leq 1$ . As discussed before, (B1), (B2), (B4) and (B5) are satisfied by our assumptions A1 - A4 and the properties of SRRW. The only difference for this step size setting, compared to the previous one  $1/2 < a < b \\leq 1$ , is that the roles of  $\\theta_{n}, \\mathbf{x}_{n}$  are now flipped, that is,  $\\theta_{n}$  is now on the fast timescale while  $\\mathbf{x}_{n}$  is on the slow timescale. By a much stronger Assumption A3', for any  $\\mathbf{x} \\in \\mathrm{Int}(\\Sigma)$ , (i)  $\\mathbb{E}_{X \\sim \\pi[\\mathbf{x}]}[H(\\rho(\\mathbf{x}), X)] = 0$ ; (ii)  $\\mathbb{E}_{X \\sim \\pi[\\mathbf{x}]}[\\nabla H(\\rho(\\mathbf{x}), X)]$  is Hurwitz; (iii)  $\\|\\rho(\\mathbf{x})\\| \\leq L_{2}(1 + \\| \\mathbf{x}\\|)$ . Hence, conditions (i) - (iii) in (B3) are satisfied. Moreover, we have  $\\pi[\\pmb{\\mu}] - \\pmb{\\mu} = 0$ ,  $\\nabla(\\pi[\\mathbf{x}] - \\mathbf{x})|_{\\mathbf{x} = \\pmb{\\mu}}$  being Hurwitz, as mentioned in the previous part. Therefore, (B3) is verified. Accordingly, (B1) - (B5) are checked by our assumptions A1, A2, A3', A4. By Theorem G.10, we have  $\\lim_{n \\to \\infty} \\mathbf{x}_n = \\pmb{\\mu}$  and  $\\theta_{n} \\to \\Theta$  almost surely.",
+        "bbox": [
+            169,
+            138,
+            826,
+            267
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "E PROOF OF THEOREM 3.3",
+        "text_level": 1,
+        "bbox": [
+            171,
+            292,
+            415,
+            308
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "This section is devoted to the proof of Theorem 3.3, which also includes the proof of the CLT results for the SRRW iteration  $\\mathbf{x}_n$  in Lemma 3.1. We will use different techniques depending on the step sizes in Assumption A2. Specifically, for step sizes  $\\gamma_n = (n + 1)^{-a}$ ,  $\\beta_n = (n + 1)^{-b}$ , we will consider three cases: case (i):  $\\beta_n = o(\\gamma_n)$ ; case (ii):  $\\beta_n = \\gamma_n$ ; and case (iii):  $\\gamma_n = o(\\beta_n)$ . For case (ii), we will use the existing CLT result for single-timescale SA in Theorem G.9. For cases (i) and (iii), we will construct our own CLT analysis for the two-timescale structure. We start with case (ii).",
+        "bbox": [
+            169,
+            327,
+            823,
+            412
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "E.1 CASE (II):  $\\beta_{n} = \\gamma_{n}$",
+        "text_level": 1,
+        "bbox": [
+            171,
+            434,
+            351,
+            450
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "In this part, we stick to the notations for single-timescale SA studied in Section D.1. To utilize Theorem G.9, apart from Conditions C1 - C4 that have been checked in Section D.1, we still need to check conditions C5 and C6 listed in Section G.2.",
+        "bbox": [
+            169,
+            462,
+            823,
+            505
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Assumption A3 corresponds to condition C5. For condition C6, we need to obtain the explicit form of function  $Q_{\\mathbf{z}}$  to the Poisson equation defined in (96), that is,",
+        "bbox": [
+            169,
+            511,
+            823,
+            541
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nQ _ {\\mathbf {z}} (i) - \\left(\\mathbf {K} _ {\\mathbf {z}} ^ {\\prime} Q _ {\\mathbf {z}}\\right) (i) = \\psi (\\mathbf {z}, i) - \\mathbb {E} _ {j \\sim \\pi [ \\mathbf {z} ]} [ \\psi (\\mathbf {z}, j) ]\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            330,
+            551,
+            665,
+            570
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "where",
+        "bbox": [
+            171,
+            582,
+            217,
+            594
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\psi (\\mathbf {z}, i) \\triangleq \\sum_ {j \\in \\mathcal {N}} \\mathbf {K} _ {\\mathbf {z}} ^ {\\prime} (i, j) m _ {\\mathbf {z}} (j) m _ {\\mathbf {z}} (j) ^ {T} - (\\mathbf {K} _ {\\mathbf {z}} ^ {\\prime} m _ {\\mathbf {z}}) (i) (\\mathbf {K} _ {\\mathbf {z}} ^ {\\prime} m _ {\\mathbf {z}}) (i) ^ {T}.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            279,
+            599,
+            715,
+            633
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Following the similar steps in the derivation of  $m_{\\mathbf{z}}(i)$  from (22) to (24), we have",
+        "bbox": [
+            169,
+            645,
+            702,
+            661
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nQ _ {\\mathbf {z}} (i) = \\sum_ {j \\in \\mathcal {N}} (\\mathbf {I} - \\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] + \\mathbf {1} \\pmb {\\pi} ^ {\\prime} [ \\mathbf {z} ] ^ {T}) ^ {- 1} (i, j) m _ {\\mathbf {z}} (j) - \\pi_ {j} ^ {\\prime} [ \\mathbf {z} ] m _ {\\mathbf {z}} (j).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            279,
+            674,
+            715,
+            708
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "We also know that  $Q_{\\mathbf{z}}(i)$  and  $(\\mathbf{K}_{\\mathbf{z}}^{\\prime}Q_{\\mathbf{z}})(i)$  are continuous in  $\\mathbf{z}$  for any  $i \\in \\mathcal{N}$ . For any  $\\mathbf{z}$  in a compact set  $\\Omega$ ,  $Q_{\\mathbf{z}}(i)$  and  $(\\mathbf{K}_{\\mathbf{z}}^{\\prime}Q_{\\mathbf{z}})(i)$  are bounded because function  $m_{\\mathbf{z}}(i)$  is bounded. Therefore, C6 is checked. By Theorem G.9, assume  $\\mathbf{z}_n = \\begin{bmatrix} \\theta_n \\\\ \\mathbf{x}_n \\end{bmatrix}$  converges to a point  $\\mathbf{z}^* = \\begin{bmatrix} \\theta^* \\\\ \\mu \\end{bmatrix}$  for  $\\theta^* \\in \\Theta$ , we have",
+        "bbox": [
+            169,
+            720,
+            823,
+            792
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\gamma_ {n} ^ {- 1 / 2} \\left(\\mathbf {z} _ {n} - \\mathbf {z} ^ {*}\\right) \\xrightarrow [ n \\rightarrow \\infty ]{d i s t .} N (0, \\mathbf {V}), \\tag {26}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            383,
+            797,
+            825,
+            825
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "where  $\\mathbf{V}$  is the solution of the following Lyapunov equation",
+        "bbox": [
+            169,
+            834,
+            570,
+            851
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {V} \\left(\\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I} + \\nabla g \\left(\\boldsymbol {z} ^ {*}\\right) ^ {T}\\right) + \\left(\\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I} + \\nabla g \\left(\\boldsymbol {z} ^ {*}\\right)\\right) \\mathbf {V} + \\mathbf {U} = 0, \\tag {27}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            274,
+            861,
+            825,
+            897
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "and  $\\mathbf{U} = \\sum_{i\\in \\mathcal{N}}\\mu_i\\left(m_{\\mathbf{z}^*}(i)m_{\\mathbf{z}^*}(i)\\right)^T - (\\mathbf{K}_{\\mathbf{z}^*}m_{\\mathbf{z}^*})(i)(\\mathbf{K}_{\\mathbf{z}^*}m_{\\mathbf{z}^*})(i)^T.$",
+        "bbox": [
+            169,
+            907,
+            635,
+            928
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "page_number",
+        "text": "19",
+        "bbox": [
+            490,
+            946,
+            509,
+            960
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "By algebraic calculations of derivative of  $\\pi[\\mathbf{x}]$  with respect to  $\\mathbf{x}$  in (20), we can rewrite  $\\nabla g(\\mathbf{z}^*)$  in terms of  $\\mathbf{x}, \\pmb{\\theta}$ , i.e.,",
+        "bbox": [
+            169,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\mathbf {J} (\\alpha) \\triangleq \\nabla g (\\mathbf {z} ^ {*}) = \\left[ \\begin{array}{c c} \\frac {\\partial \\sum_ {i \\in \\mathcal {N}} \\pi_ {i} [ \\mathbf {x} ] H (\\boldsymbol {\\theta} , i)}{\\partial \\boldsymbol {\\theta}} & \\frac {\\partial \\sum_ {i \\in \\mathcal {N}} \\pi_ {i} [ \\mathbf {x} ] H (\\boldsymbol {\\theta} , i)}{\\partial \\mathbf {x}} \\\\ \\frac {\\partial (\\boldsymbol {\\pi} [ \\mathbf {x} ] - \\mathbf {x})}{\\partial \\boldsymbol {\\theta}} & \\frac {\\partial \\boldsymbol {\\pi} [ \\mathbf {x} ] - \\mathbf {x}}{\\partial \\mathbf {x}} \\end{array} \\right] _ {\\mathbf {z} = \\mathbf {z} ^ {*}} \\\\ = \\left[ \\begin{array}{c c} \\nabla \\mathbf {h} (\\boldsymbol {\\theta} ^ {*}) & - \\alpha \\mathbf {H} ^ {T} (\\mathbf {P} ^ {T} + \\mathbf {I}) \\\\ \\mathbf {0} & 2 \\alpha \\boldsymbol {\\mu} \\mathbf {1} ^ {T} - \\alpha \\mathbf {P} ^ {T} - (\\alpha + 1) \\mathbf {I} \\end{array} \\right] \\triangleq \\left[ \\begin{array}{c c} \\mathbf {J} _ {1 1} & \\mathbf {J} _ {1 2} (\\alpha) \\\\ \\mathbf {J} _ {2 1} & \\mathbf {J} _ {2 2} (\\alpha) \\end{array} \\right], \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            228,
+            138,
+            764,
+            219
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "where matrix  $\\mathbf{H} = [H(\\pmb{\\theta}^*, 1), \\dots, H(\\pmb{\\theta}, N)]^T$ . Then, we further clarify the matrix  $\\mathbf{U}$ . Note that",
+        "bbox": [
+            171,
+            228,
+            805,
+            244
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nm _ {\\mathbf {z} ^ {*}} (i) = \\sum_ {k = 0} ^ {\\infty} [ \\mathbf {G} (\\mathbf {z} ^ {*}) ((\\mathbf {P} ^ {k}) ^ {T} - \\boldsymbol {\\mu} \\mathbf {1} ^ {T}) ] _ {[:, i ]} = \\sum_ {k = 0} ^ {\\infty} [ \\mathbf {G} (\\mathbf {z} ^ {*}) (\\mathbf {P} ^ {k}) ^ {T} ] _ {[:, i ]} = \\mathbb {E} \\left[ \\sum_ {k = 0} ^ {\\infty} [ G (\\mathbf {z} ^ {*}, X _ {k}) ] \\right| X _ {0} = i ], \\tag {28}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            169,
+            253,
+            823,
+            306
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "where the first equality holds because  $\\mathbf{K}^{\\prime}[\\pmb {\\mu}] = \\mathbf{P}$  from the definition of SRRW kernel (3), the second equality stems from  $\\mathbf{G}(\\mathbf{z}^{*})\\pmb {\\mu} = g(\\mathbf{z}^{*}) = 0$  , and the last term is a conditional expectation over the base Markov chain  $\\{X_{k}\\}_{k\\geq 0}$  (with transition kernel  $\\mathbf{P}$ ) conditioned on  $X_0 = i$  . Similarly, with  $(\\mathbf{K}_{\\mathbf{z}}^{\\prime}m_{\\mathbf{z}})(i)$  in the form of (23), we have",
+        "bbox": [
+            169,
+            306,
+            823,
+            364
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n(\\mathbf {K} _ {\\mathbf {z}} ^ {\\prime} m _ {\\mathbf {z}}) (i) = \\mathbb {E} \\left[ \\sum_ {k = 1} ^ {\\infty} [ G (\\mathbf {z} ^ {*}, X _ {k}) ] \\Bigg | X _ {0} = i \\right].\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            346,
+            372,
+            650,
+            415
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "From the form  $\\sum_{i\\in \\mathcal{N}}\\mu_i$  inside the matrix  $\\mathbf{U}$ , the Markov chain  $\\{X_k\\}_{k\\geq 0}$  is in its stationary regime from the beginning, i.e.,  $X_{k}\\sim \\pmb{\\mu}$  for any  $k\\geq 0$ . Hence,",
+        "bbox": [
+            169,
+            431,
+            823,
+            460
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\mathbf {U} = \\mathbb {E} \\left[ \\left(\\sum_ {k = 0} ^ {\\infty} [ G (\\mathbf {z} ^ {*}, X _ {k}) ]\\right) \\left(\\sum_ {k = 0} ^ {\\infty} [ G (\\mathbf {z} ^ {*}, X _ {k}) ]\\right) ^ {T} \\right] \\\\ - \\mathbb {E} \\left[ \\left(\\sum_ {k = 1} ^ {\\infty} [ G (\\mathbf {z} ^ {*}, X _ {k}) ]\\right) \\left(\\sum_ {k = 1} ^ {\\infty} [ G (\\mathbf {z} ^ {*}, X _ {k}) ]\\right) ^ {T} \\right] \\\\ = \\mathbb {E} \\left[ G \\left(\\mathbf {z} ^ {*}, X _ {0}\\right) G \\left(\\mathbf {z} ^ {*}, X _ {0}\\right) ^ {T} \\right] + \\mathbb {E} \\left[ G \\left(\\mathbf {z} ^ {*}, X _ {0}\\right) \\left(\\sum_ {k = 1} ^ {\\infty} G \\left(\\mathbf {z} ^ {*}, X _ {k}\\right)\\right) ^ {T} \\right] \\tag {29} \\\\ + \\mathbb {E} \\left[ \\left(\\sum_ {k = 1} ^ {\\infty} G (\\mathbf {z} ^ {*}, X _ {k})\\right) G (\\mathbf {z} ^ {*}, X _ {0}) ^ {T} \\right] \\\\ = \\operatorname {C o v} \\left(G \\left(\\mathbf {z} ^ {*}, X _ {0}\\right), G \\left(\\mathbf {z} ^ {*}, X _ {0}\\right)\\right) \\\\ + \\sum_ {k = 1} ^ {\\infty} \\left[ \\operatorname {C o v} \\left(G \\left(\\mathbf {z} ^ {*}, X _ {0}\\right), G \\left(\\mathbf {z} ^ {*}, X _ {k}\\right)\\right) + \\operatorname {C o v} \\left(G \\left(\\mathbf {z} ^ {*}, X _ {k}\\right), G \\left(\\mathbf {z} ^ {*}, X _ {0}\\right)\\right) \\right], \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            179,
+            469,
+            823,
+            722
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "where the covariance between  $G(\\mathbf{z}^*, X_0)$  and  $G(\\mathbf{z}^*, X_k)$  for the Markov chain  $\\{X_n\\}$  in the stationary regime is  $\\mathrm{Cov}(G(\\mathbf{z}^*, X_0), G(\\mathbf{z}^*, X_k))$ . By Brémaud (2013, Theorem 6.3.7), it is demonstrated that  $\\mathbf{U}$  is the sampling covariance of the base Markov chain  $\\mathbf{P}$  for the test function  $G(\\mathbf{z}^*, \\cdot)$ . Moreover, Brémaud (2013, equation (6.34)) states that this sampling covariance  $\\mathbf{U}$  can be rewritten in the following form:",
+        "bbox": [
+            169,
+            733,
+            823,
+            804
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {U} = \\sum_ {i = 1} ^ {N - 1} \\mathbf {G} \\left(\\mathbf {z} ^ {*}\\right) ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} \\mathbf {G} \\left(\\mathbf {z} ^ {*}\\right) = \\sum_ {i = 1} ^ {N - 1} \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\left[ \\begin{array}{c c} \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H} & \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\\\ \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H} & \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\end{array} \\right] \\triangleq \\left[ \\begin{array}{l l} \\mathbf {U} _ {1 1} & \\mathbf {U} _ {1 2} \\\\ \\mathbf {U} _ {2 1} & \\mathbf {U} _ {2 2} \\end{array} \\right], \\tag {30}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            187,
+            813,
+            825,
+            856
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "where  $\\{(\\lambda_i, \\mathbf{u}_i)\\}_{i \\in \\mathcal{N}}$  is the eigenpair of the transition kernel  $\\mathbf{P}$  of the ergodic and time-reversible base Markov chain. This completes the proof of case 1.",
+        "bbox": [
+            169,
+            864,
+            823,
+            895
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "page_footnote",
+        "text": "12One may refer to Doshi et al. (2023, Appendix B, Proof of Lemma 3.4) for the computation of  $\\frac{\\partial\\pi|\\mathbf{x}| - \\mathbf{x}}{\\partial\\mathbf{x}}$",
+        "bbox": [
+            186,
+            907,
+            813,
+            926
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "page_number",
+        "text": "20",
+        "bbox": [
+            488,
+            946,
+            509,
+            959
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Remark E.1. For the CLT result (26), we can look further into the asymptotic covariance matrix  $\\mathbf{V}$  as in (27). For convenience, we denote  $\\mathbf{V} = \\begin{bmatrix} \\mathbf{V}_{11} & \\mathbf{V}_{12}\\\\ \\mathbf{V}_{21} & \\mathbf{V}_{22} \\end{bmatrix}$  and  $\\mathbf{U}$  in the form of (30) such that",
+        "bbox": [
+            169,
+            103,
+            823,
+            151
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left[ \\begin{array}{l l} \\mathbf {V} _ {1 1} & \\mathbf {V} _ {1 2} \\\\ \\mathbf {V} _ {2 1} & \\mathbf {V} _ {2 2} \\end{array} \\right] \\left(\\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I} + \\mathbf {J} (\\alpha) ^ {T}\\right) + \\left(\\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I} + \\mathbf {J} (\\alpha)\\right) \\left[ \\begin{array}{l l} \\mathbf {V} _ {1 1} & \\mathbf {V} _ {1 2} \\\\ \\mathbf {V} _ {2 1} & \\mathbf {V} _ {2 2} \\end{array} \\right] + \\mathbf {U} = 0. \\tag {31}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            207,
+            155,
+            825,
+            191
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "For the SRRW iteration  $\\mathbf{x}_n$ , from (26) we know that  $\\gamma_n^{-1/2}(\\mathbf{x}_n - \\boldsymbol{\\mu}) \\xrightarrow[n\\to\\infty]{dist.} N(\\mathbf{0}, \\mathbf{V}_4)$ . Thus, in this remark, we want to obtain the closed form of  $\\mathbf{V}_{22}$ . By algebraic computations of the bottom-right sub-block matrix, we have",
+        "bbox": [
+            169,
+            198,
+            826,
+            247
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\left(2 \\alpha \\boldsymbol {\\mu} \\mathbf {1} ^ {T} - \\alpha \\mathbf {P} ^ {T} - \\left(\\alpha + 1 - \\frac {\\mathbb {1} _ {\\{a = 1 \\}}}{2}\\right) \\mathbf {I}\\right) \\mathbf {V} _ {2 2} \\\\ + \\mathbf {V} _ {2 2} \\left(2 \\alpha \\boldsymbol {\\mu} \\mathbf {1} ^ {T} - \\alpha \\mathbf {P} ^ {T} - \\left(\\alpha + 1 - \\frac {\\mathbb {1} _ {\\{a = 1 \\}}}{2}\\right) \\mathbf {I}\\right) ^ {T} \\\\ + \\mathbf {U} _ {2 2} = 0. \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            330,
+            253,
+            668,
+            342
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "By using result of the closed form solution to the Lyapunov equation (e.g., Lemma G.1) and the eigendecomposition of  $\\mathbf{P}$ , we have",
+        "bbox": [
+            169,
+            349,
+            823,
+            378
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {V} _ {2 2} = \\sum_ {i = 1} ^ {N - 1} \\frac {1}{2 \\alpha \\left(1 + \\lambda_ {i}\\right) + 2 - \\mathbb {1} _ {\\{a = 1 \\}}} \\cdot \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T}. \\tag {32}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            385,
+            825,
+            425
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "E.2 CASE (I):  $\\beta_{n} = o(\\gamma_{n})$",
+        "text_level": 1,
+        "bbox": [
+            171,
+            441,
+            366,
+            458
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "In this part, we mainly focus on the CLT of the SA iteration  $\\theta_{n}$  because the SRRW iteration  $\\mathbf{x}_n$  is independent of  $\\theta_{n}$  and its CLT result has been shown in Remark E.1.",
+        "bbox": [
+            169,
+            467,
+            823,
+            498
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "E.2.1 DECOMPOSITION OF SA-SRRW ITERATION (4)",
+        "text_level": 1,
+        "bbox": [
+            171,
+            511,
+            558,
+            526
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "We slightly abuse the math notation and define the function",
+        "bbox": [
+            171,
+            536,
+            565,
+            550
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {h} (\\boldsymbol {\\theta}, \\mathbf {x}) \\triangleq \\mathbb {E} _ {i \\sim \\pi [ \\mathbf {x} ]} H (\\boldsymbol {\\theta}, i) = \\sum_ {i \\in \\mathcal {N}} \\pi_ {i} [ \\mathbf {x} ] H (\\boldsymbol {\\theta}, i)\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            339,
+            556,
+            653,
+            589
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "such that  $\\mathbf{h}(\\pmb {\\theta},\\pmb {\\mu})\\equiv \\mathbf{h}(\\pmb {\\theta})$  . Then, we reformulate (25) as",
+        "bbox": [
+            171,
+            595,
+            545,
+            611
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} + \\beta_ {n + 1} \\mathbf {h} \\left(\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}\\right) + \\beta_ {n + 1} \\left(H \\left(\\boldsymbol {\\theta} _ {n}, X _ {n + 1}\\right) - \\mathbf {h} \\left(\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}\\right)\\right). \\tag {33a}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            269,
+            616,
+            823,
+            633
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {x} _ {n + 1} = \\mathbf {x} _ {n} + \\gamma_ {n + 1} \\left(\\boldsymbol {\\pi} \\left[ \\mathbf {x} _ {n} \\right] - \\mathbf {x} _ {n}\\right) + \\gamma_ {n + 1} \\left(\\boldsymbol {\\delta} _ {X _ {n + 1}}\\right) - \\boldsymbol {\\pi} \\left[ \\mathbf {x} _ {n} \\right]). \\tag {33b}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            294,
+            638,
+            823,
+            656
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "There exist functions  $q_{\\mathbf{x}}:\\mathcal{N}\\to \\mathbb{R}^{N},\\tilde{H}_{\\boldsymbol{\\theta},\\mathbf{x}}:\\mathcal{N}\\to \\mathbb{R}^{D}$  satisfying the following Poisson equations",
+        "bbox": [
+            171,
+            660,
+            813,
+            676
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {\\delta} _ {i} - \\boldsymbol {\\pi} (\\mathbf {x}) = q _ {\\mathbf {x}} (i) - \\left(\\mathbf {K} _ {\\mathbf {x}} q _ {\\mathbf {x}}\\right) (i) \\tag {34a}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            387,
+            681,
+            823,
+            699
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nH (\\boldsymbol {\\theta}, i) - \\mathbf {h} (\\boldsymbol {\\theta}, \\mathbf {x}) = \\tilde {H} _ {\\boldsymbol {\\theta}, \\mathbf {x}} (i) - \\left(\\mathbf {K} _ {\\mathbf {x}} \\tilde {H} _ {\\boldsymbol {\\theta}, \\mathbf {x}}\\right) (i), \\tag {34b}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            341,
+            704,
+            823,
+            722
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "for any  $\\pmb{\\theta} \\in \\mathbb{R}^{D}$ ,  $\\mathbf{x} \\in \\mathrm{Int}(\\Sigma)$  and  $i \\in \\mathcal{N}$ , where  $(\\mathbf{K}_{\\mathbf{x}}q_{\\mathbf{x}})(i) \\triangleq \\sum_{j \\in \\mathcal{N}} K_{ij}[\\mathbf{x}] q_{\\mathbf{x}}(j)$ ,  $(\\mathbf{K}_{\\mathbf{x}}\\tilde{H}_{\\pmb{\\theta},\\mathbf{x}})(j) \\triangleq \\sum_{j \\in \\mathcal{N}} K_{ij}[\\mathbf{x}]\\tilde{H}_{\\pmb{\\theta},\\mathbf{x}}(j)$ . The existence and explicit form of the solutions  $q_{\\mathbf{x}}, \\tilde{H}_{\\pmb{\\theta},\\mathbf{x}}$ , which are continuous w.r.t  $\\mathbf{x}$ ,  $\\pmb{\\theta}$ , follow the similar steps that can be found in Section D.1 from (22) to (24). Thus, we can further decompose (33) into",
+        "bbox": [
+            169,
+            726,
+            826,
+            790
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} + \\beta_ {n + 1} \\mathbf {h} (\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}) + \\beta_ {n + 1} \\underbrace {(\\tilde {H} _ {\\boldsymbol {\\theta} _ {n} , \\mathbf {x} _ {n}} (X _ {n + 1}) - (\\mathbf {K} _ {\\mathbf {x} _ {n}} \\tilde {H} _ {\\boldsymbol {\\theta} _ {n} , \\mathbf {x} _ {n}}) (X _ {n}))} _ {M _ {n + 1} ^ {(\\boldsymbol {\\theta})}} \\\\ + \\beta_ {n + 1} \\underbrace {\\left(\\left(\\mathbf {K} _ {\\mathbf {x} _ {n + 1}} \\tilde {H} _ {\\boldsymbol {\\theta} _ {n + 1} , \\mathbf {x} _ {n + 1}}\\right) \\left(X _ {n + 1}\\right) - \\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} \\tilde {H} _ {\\boldsymbol {\\theta} _ {n} , \\mathbf {x} _ {n}}\\right) \\left(X _ {n + 1}\\right)\\right)} _ {r _ {n} ^ {(\\boldsymbol {\\theta}, 1)}} \\tag {35a} \\\\ + \\beta_ {n + 1} \\underbrace {((\\mathbf {K _ {x}} _ {n} \\tilde {H} ) (X _ {n}) - (\\mathbf {K _ {x}} _ {n + 1} \\tilde {H} ) (X _ {n + 1}))} _ {r _ {n} ^ {(\\pmb {\\theta}, 2)}}, \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            238,
+            796,
+            823,
+            928
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "page_number",
+        "text": "21",
+        "bbox": [
+            488,
+            946,
+            506,
+            960
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\mathbf {x} _ {n + 1} = \\mathbf {x} _ {n} + \\gamma_ {n + 1} (\\boldsymbol {\\pi} (\\mathbf {x} _ {n}) - \\mathbf {x} _ {n}) + \\gamma_ {n + 1} \\underbrace {(q _ {\\mathbf {x} _ {n}} (X _ {n + 1}) - (\\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}}) (X _ {n}))} _ {M _ {n + 1} ^ {(\\mathbf {x})}} \\\\ + \\gamma_ {n + 1} \\underbrace {\\left([ \\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}} ] \\left(X _ {n + 1}\\right) - [ \\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n + 1}} ] \\left(X _ {n + 1}\\right)\\right)} _ {r _ {n} ^ {(\\mathbf {x}, 1)}} \\tag {35b} \\\\ + \\gamma_ {n + 1} \\underbrace {\\left((\\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}}) (X _ {n}) - (\\mathbf {K} _ {\\mathbf {x} _ {n + 1}} q _ {\\mathbf {x} _ {n + 1}}) (X _ {n + 1})\\right)} _ {r _ {n} ^ {(\\mathbf {x}, 2)}}. \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            254,
+            95,
+            823,
+            220
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "such that",
+        "bbox": [
+            171,
+            223,
+            235,
+            234
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} + \\beta_ {n + 1} \\mathbf {h} \\left(\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}\\right) + \\beta_ {n + 1} M _ {n + 1} ^ {(\\boldsymbol {\\theta})} + \\beta_ {n + 1} r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} + \\beta_ {n + 1} r _ {n} ^ {(\\boldsymbol {\\theta}, 2)}, \\tag {36a}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            256,
+            238,
+            823,
+            258
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {x} _ {n + 1} = \\mathbf {x} _ {n} + \\gamma_ {n + 1} \\left(\\boldsymbol {\\pi} \\left(\\mathbf {x} _ {n}\\right) - \\mathbf {x} _ {n}\\right) + \\gamma_ {n + 1} M _ {n + 1} ^ {(\\mathbf {x})} + \\gamma_ {n + 1} r _ {n} ^ {(\\mathbf {x}, 1)} + \\gamma_ {n + 1} r _ {n} ^ {(\\mathbf {x}, 2)}. \\tag {36b}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            243,
+            263,
+            823,
+            282
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "We can observe that (36) differs from the expression in Konda & Tsitsiklis (2004); Mokkadem & Pelletier (2006), which studied the two-timescale SA with Martingale difference noise. Here, due to the presence of the iterate-dependent Markovian noise and the application of the Poisson equation technique, we have additional non-vanishing terms  $r_n^{(\\theta, 2)}$ ,  $r_n^{(\\mathbf{x}, 2)}$ , which will be further examined in Lemma E.2. Additionally, when we apply the Poisson equation to the Martingale difference terms  $M_{n+1}^{(\\theta)}$ ,  $M_{n+1}^{(\\mathbf{x})}$ , we find that there are some covariances that are also non-vanishing as in Lemma E.1. We will mention this again when we obtain those covariances. These extra non-zero noise terms make our analysis distinct from the previous ones since the key assumption (A4) in Mokkadem & Pelletier (2006) is not satisfied. We demonstrate that the long-term average performance of these terms can be managed so that they do not affect the final CLT result.",
+        "bbox": [
+            169,
+            282,
+            823,
+            429
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "Analysis of Terms  $M_{n + 1}^{(\\theta)},M_{n + 1}^{(\\mathbf{x})}$",
+        "bbox": [
+            171,
+            434,
+            393,
+            455
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "Consider the filtration  $\\mathcal{F}_n \\triangleq \\sigma(\\pmb{\\theta}_0, \\mathbf{x}_0, X_0, \\dots, \\pmb{\\theta}_n, \\mathbf{x}_n, X_n)$ , it is evident that  $M_{n+1}^{(\\pmb{\\theta})}, M_{n+1}^{(\\mathbf{x})}$  are Martingale difference sequences adapted to  $\\mathcal{F}_n$ . Then, we have",
+        "bbox": [
+            169,
+            460,
+            823,
+            494
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\mathbb {E} \\left[ M _ {n + 1} ^ {(\\mathbf {x})} (M _ {n + 1} ^ {(\\mathbf {x})}) ^ {T} \\Big | \\mathcal {F} _ {n} \\right] \\\\ = \\mathbb {E} \\left[ q _ {\\mathbf {x} _ {n}} \\left(X _ {n + 1}\\right) q _ {\\mathbf {x} _ {n}} \\left(X _ {n + 1}\\right) ^ {T} \\mid \\mathcal {F} _ {n} \\right] + \\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}}\\right) \\left(X _ {n}\\right) \\left(\\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}}\\right) \\left(X _ {n}\\right)\\right) ^ {T} \\tag {37} \\\\ - \\mathbb {E} \\left[ q _ {\\mathbf {x} _ {n}} \\left(X _ {n + 1}\\right) \\mid \\mathcal {F} _ {n} \\right] \\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}}\\right) \\left(X _ {n}\\right)) ^ {T} - \\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}}\\right) \\left(X _ {n}\\right) \\mathbb {E} \\left[ q _ {\\mathbf {x} _ {n}} \\left(X _ {n + 1}\\right) ^ {T} \\mid \\mathcal {F} _ {n} \\right] \\\\ = \\mathbb {E} \\left[ q _ {\\mathbf {x} _ {n}} \\left(X _ {n + 1}\\right) q _ {\\mathbf {x} _ {n}} \\left(X _ {n + 1}\\right) ^ {T} \\mid \\mathcal {F} _ {n} \\right] - \\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}}\\right) \\left(X _ {n}\\right) \\left(\\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}}\\right) \\left(X _ {n}\\right)\\right) ^ {T}. \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            233,
+            497,
+            823,
+            587
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "Similarly, we have",
+        "bbox": [
+            171,
+            590,
+            297,
+            604
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\mathbb {E} \\left[ M _ {n + 1} ^ {(\\boldsymbol {\\theta})} \\left(M _ {n + 1} ^ {(\\boldsymbol {\\theta})}\\right) ^ {T} \\mid \\mathcal {F} _ {n} \\right] \\tag {38} \\\\ = \\mathbb {E} \\left[ \\tilde {H} _ {\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}} \\left(X _ {n + 1}\\right) \\tilde {H} _ {\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}} \\left(X _ {n + 1}\\right) ^ {T} \\mid \\mathcal {F} _ {n} \\right] - \\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} \\tilde {H} _ {\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}}\\right) \\left(X _ {n}\\right) \\left(\\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} \\tilde {H} _ {\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}}\\right) \\left(X _ {n}\\right)\\right) ^ {T}, \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            191,
+            609,
+            823,
+            664
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "and",
+        "bbox": [
+            171,
+            667,
+            200,
+            680
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\mathbb {E} \\left[ M _ {n + 1} ^ {(\\mathbf {x})} (M _ {n + 1} ^ {(\\boldsymbol {\\theta})}) ^ {T} \\Big | \\mathcal {F} _ {n} \\right] \\\\ = \\mathbb {E} [ q _ {\\mathbf {x} _ {n}} (X _ {n + 1}) \\tilde {H} _ {\\pmb {\\theta} _ {n}, \\mathbf {x} _ {n}} (X _ {n + 1}) ^ {T} | \\mathcal {F} _ {n} ] - (\\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}}) (X _ {n}) ((\\mathbf {K} _ {\\mathbf {x} _ {n}} \\tilde {H} _ {\\pmb {\\theta} _ {n}, \\mathbf {x} _ {n}}) (X _ {n})) ^ {T}. \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            228,
+            684,
+            767,
+            739
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "We now focus on  $\\mathbb{E}\\left[M_{n + 1}^{(\\mathbf{x})}(M_{n + 1}^{(\\mathbf{x})})^T\\Big|\\mathcal{F}_n\\right]$ . Denote by",
+        "bbox": [
+            171,
+            750,
+            537,
+            777
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nV _ {1} (\\mathbf {x}, i) \\triangleq \\sum_ {j \\in \\mathcal {N}} \\mathbf {K} _ {i, j} [ \\mathbf {x} ] q _ {\\mathbf {x}} (j) q _ {\\mathbf {x}} (j) ^ {T} - \\left(\\mathbf {K} _ {\\mathbf {x}} q _ {\\mathbf {x}}\\right) (i) \\left(\\left(\\mathbf {K} _ {\\mathbf {x}} q _ {\\mathbf {x}}\\right) (i)\\right) ^ {T}, \\tag {39}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            281,
+            781,
+            823,
+            814
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "and let its expectation w.r.t the stationary distribution  $\\pi (\\mathbf{x})$  be  $v_{1}(\\mathbf{x})\\triangleq \\mathbb{E}_{i\\sim \\pi (\\mathbf{x})}[V_{1}(\\mathbf{x},i)]$ , we can construct another Poisson equation, i.e.,",
+        "bbox": [
+            169,
+            819,
+            823,
+            849
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\mathbb {E} \\left[ M _ {n + 1} ^ {(\\mathbf {x})} (M _ {n + 1} ^ {(\\mathbf {x})}) ^ {T} \\Big | \\mathcal {F} _ {n} \\right] - \\sum_ {X _ {n} \\in \\mathcal {N}} \\pi_ {X _ {n}} (\\mathbf {x} _ {n}) \\mathbb {E} \\left[ M _ {n + 1} ^ {(\\mathbf {x})} (M _ {n + 1} ^ {(\\mathbf {x})}) ^ {T} \\Big | \\mathcal {F} _ {n} \\right] \\\\ = V _ {1} \\left(\\mathbf {x} _ {n}, X _ {n + 1}\\right) - v _ {1} \\left(\\mathbf {x} _ {n}\\right) \\\\ = \\varphi_ {\\mathbf {x}} ^ {(1)} (X _ {n + 1}) - (\\mathbf {K} _ {\\mathbf {x} _ {n}} \\varphi_ {\\mathbf {x} _ {n}} ^ {(1)}) (X _ {n + 1}), \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            264,
+            852,
+            730,
+            928
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "page_number",
+        "text": "22",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "for some matrix-valued function  $\\varphi_{\\mathbf{x}}^{(1)}:\\mathcal{N}\\to \\mathbb{R}^{N\\times N}$ . Since  $q_{\\mathbf{x}}$  and  $\\mathbf{K}[\\mathbf{x}]$  are continuous in  $\\mathbf{x}$ , functions  $V_{1},v_{1}$  are also continuous in  $\\mathbf{x}$ . Then, we can decompose (39) into",
+        "bbox": [
+            169,
+            101,
+            823,
+            133
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} V _ {1} (\\mathbf {x} _ {n}, X _ {n + 1}) = \\underbrace {v _ {1} (\\boldsymbol {\\mu})} _ {\\mathbf {U} _ {2 2}} + \\underbrace {v _ {1} (\\mathbf {x} _ {n}) - v _ {1} (\\boldsymbol {\\mu})} _ {\\mathbf {D} _ {n} ^ {(1)}} + \\underbrace {\\varphi_ {\\mathbf {x} _ {n}} ^ {(1)} (X _ {n + 1}) - (\\mathbf {K} _ {\\mathbf {x} _ {n}} \\varphi_ {\\mathbf {x} _ {n}} ^ {(1)}) (X _ {n})} _ {\\mathbf {J} _ {n} ^ {(1, a)}} \\\\ + \\underbrace {\\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} \\varphi_ {\\mathbf {x} _ {n}} ^ {(1)}\\right) \\left(X _ {n}\\right) - \\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} \\varphi_ {\\mathbf {x} _ {n}} ^ {(1)}\\right) \\left(X _ {n + 1}\\right)} _ {\\mathbf {J} _ {n} ^ {(1, b)}}. \\tag {40} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            241,
+            138,
+            823,
+            227
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "Thus, we have",
+        "bbox": [
+            171,
+            234,
+            272,
+            248
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbb {E} \\left[ M _ {n + 1} ^ {(\\mathbf {x})} \\left(M _ {n + 1} ^ {(\\mathbf {x})}\\right) ^ {T} \\mid \\mathcal {F} _ {n} \\right] = \\mathbf {U} _ {2 2} + \\mathbf {D} _ {n} ^ {(1)} + \\mathbf {J} _ {n} ^ {(1)}, \\tag {41}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            341,
+            247,
+            823,
+            268
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "where  $\\mathbf{J}_n^{(1)} = \\mathbf{J}_n^{(1,a)} + \\mathbf{J}_n^{(1,b)}$ .",
+        "bbox": [
+            169,
+            272,
+            369,
+            292
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "Following the similar steps above, we can decompose  $\\mathbb{E}\\left[M_{n + 1}^{(\\mathbf{x})}(M_{n + 1}^{(\\boldsymbol{\\theta})})^T\\Big|\\mathcal{F}_n\\right]$  and  $\\mathbb{E}\\left[M_{n + 1}^{(\\boldsymbol{\\theta})}(M_{n + 1}^{(\\boldsymbol{\\theta})})^T\\Big|\\mathcal{F}_n\\right]$  as",
+        "bbox": [
+            169,
+            296,
+            823,
+            345
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbb {E} \\left[ M _ {n + 1} ^ {(\\mathbf {x})} \\left(M _ {n + 1} ^ {(\\boldsymbol {\\theta})}\\right) ^ {T} \\mid \\mathcal {F} _ {n} \\right] = \\mathbf {U} _ {2 1} + \\mathbf {D} _ {n} ^ {(2)} + \\mathbf {J} _ {n} ^ {(2)}, \\tag {42a}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            334,
+            352,
+            825,
+            380
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbb {E} \\left[ M _ {n + 1} ^ {(\\boldsymbol {\\theta})} \\left(M _ {n + 1} ^ {(\\boldsymbol {\\theta})}\\right) ^ {T} \\mid \\mathcal {F} _ {n} \\right] = \\mathbf {U} _ {1 1} + \\mathbf {D} _ {n} ^ {(3)} + \\mathbf {J} _ {n} ^ {(3)}. \\tag {42b}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            334,
+            387,
+            823,
+            414
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "where  $\\mathbf{J}_n^{(2)} = \\mathbf{J}_n^{(2,a)} + \\mathbf{J}_n^{(2,b)}$  and  $\\mathbf{J}_n^{(3)} = \\mathbf{J}_n^{(3,a)} + \\mathbf{J}_n^{(3,b)}$ . Here, we note that matrices  $\\mathbf{J}_n^i$  for  $i = 1,2,3$  are in presence of the current CLT analysis of the two-timescale SA with Martingale difference noise. In addition,  $\\mathbf{U}_{11}$ ,  $\\mathbf{U}_{12}$  and  $\\mathbf{U}_{22}$  inherently include the information of the underlying Markov chain (with its eigenpair  $(\\lambda_i,\\mathbf{u}_i)$ ), which is an extension of the previous works (Konda & Tsitsiklis, 2004; Mokkadem & Pelletier, 2006).",
+        "bbox": [
+            169,
+            417,
+            823,
+            492
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "Lemma E.1. For  $M_{n + 1}^{(\\theta)},M_{n + 1}^{(\\mathbf{x})}$  defined in (35) and their decomposition in (41) and (42), we have",
+        "bbox": [
+            169,
+            497,
+            818,
+            515
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {U} _ {1 1} = \\sum_ {i = 1} ^ {N - 1} \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T}, \\quad \\mathbf {U} _ {2 1} = \\sum_ {i = 1} ^ {N - 1} \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H}, \\quad \\mathbf {U} _ {1 1} = \\sum_ {i = 1} ^ {N - 1} \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H}, \\tag {43a}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            194,
+            523,
+            823,
+            577
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\lim  _ {n \\rightarrow \\infty} \\mathbf {D} _ {n} ^ {(i)} = 0 a. s. f o r i = 1, 2, 3, \\tag {43b}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            372,
+            577,
+            823,
+            601
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\lim  _ {n \\rightarrow \\infty} \\gamma_ {n} \\mathbb {E} \\left[\\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(i)} \\right\\|\\right] = 0, \\quad f o r \\quad i = 1, 2, 3. \\tag {43c}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            336,
+            604,
+            823,
+            648
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "Proof. We now provide the properties of the four terms inside (41) as an example. Note that",
+        "bbox": [
+            169,
+            662,
+            779,
+            679
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\mathbf {U} _ {1 1} = \\mathbb {E} _ {i \\sim \\boldsymbol {\\mu}} [ V _ {1} (\\boldsymbol {\\mu}, i) ] = \\sum_ {i \\in \\mathcal {N}} \\mu_ {i} \\left[ \\sum_ {j \\in \\mathcal {N}} \\mathbf {P} (i, j) q _ {\\boldsymbol {\\mu}} (j) q _ {\\boldsymbol {\\mu}} (j) ^ {T} - (\\mathbf {P} q _ {\\boldsymbol {\\mu}}) (i) ((\\mathbf {P} q _ {\\boldsymbol {\\mu}}) (i)) ^ {T} \\right] \\\\ = \\sum_ {j \\in \\mathcal {N}} \\mu_ {j} q _ {\\pmb {\\mu}} (j) q _ {\\pmb {\\mu}} (j) ^ {T} - (\\mathbf {P} q _ {\\pmb {\\mu}}) (j) ((\\mathbf {P} q _ {\\pmb {\\mu}}) (j)) ^ {T}. \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            210,
+            686,
+            779,
+            771
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "We can see that it has exactly the same structure as matrix  $\\mathbf{U}$  in (27). Following the similar steps in deducing the explicit form of  $\\mathbf{U}$  from (28) to (30), we get",
+        "bbox": [
+            169,
+            777,
+            823,
+            808
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {U} _ {1 1} = \\sum_ {i = 1} ^ {N - 1} \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T}. \\tag {44}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            408,
+            815,
+            823,
+            856
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "By the almost sure convergence result  $\\mathbf{x}_n\\to \\pmb{\\mu}$  in Lemma 3.1,  $v_{1}(\\mathbf{x}_{n})\\rightarrow v_{1}(\\pmb {\\mu})$  a.s. such that  $\\lim_{n\\to \\infty}\\mathbf{D}_n^{(1)} = 0$  a.s.",
+        "bbox": [
+            169,
+            864,
+            823,
+            897
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "We next prove that  $\\lim_{n\\to \\infty}\\gamma_n\\mathbb{E}\\left[\\left\\| \\sum_{k = 1}^n{\\bf J}_k^{(1,a)}\\right\\|\\right] = 0$  and  $\\lim_{n\\to \\infty}\\gamma_n\\mathbb{E}\\left[\\left\\| \\sum_{k = 1}^n{\\bf J}_k^{(1,b)}\\right\\|\\right] = 0.$",
+        "bbox": [
+            169,
+            902,
+            812,
+            929
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "page_number",
+        "text": "23",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "Since  $\\{\\mathbf{J}_n^{(1,a)}\\}$  is a Martingale difference sequence adapted to  $\\mathcal{F}_n$ , with the Burkholder inequality in Lemma G.2 and  $p = 1$ , we show that",
+        "bbox": [
+            169,
+            102,
+            823,
+            132
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbb {E} \\left[ \\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1, a)} \\right\\| \\right] \\leq C _ {1} \\mathbb {E} \\left[ \\sqrt {\\left(\\sum_ {k = 1} ^ {n} \\left\\| \\mathbf {J} _ {k} ^ {(1 , a)} \\right\\| ^ {2}\\right)} \\right]. \\tag {45}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            331,
+            137,
+            821,
+            184
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "By assumption A4,  $\\mathbf{x}_n$  is always within some compact set  $\\Omega$  such that  $\\sup_n\\| \\mathbf{J}_n^{(1,a)}\\| \\leq C_\\Omega < \\infty$  and for a given trajectory  $\\omega$  of  $\\mathbf{x}_n(\\omega)$",
+        "bbox": [
+            169,
+            188,
+            823,
+            219
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\gamma_ {n} C _ {p} \\sqrt {\\left(\\sum_ {k = 1} ^ {n} \\left\\| \\mathbf {J} _ {k} ^ {(1 , a)} \\right\\| ^ {2}\\right)} \\leq C _ {p} C _ {\\Omega} \\gamma_ {n} \\sqrt {n}, \\tag {46}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            357,
+            223,
+            821,
+            270
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "and the last term decreases to zero in  $n$  since  $a > 1 / 2$ .",
+        "bbox": [
+            169,
+            272,
+            529,
+            287
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "For  $\\mathbf{J}_n^{(1,b)}$ , we use Abel transformation and obtain",
+        "bbox": [
+            169,
+            294,
+            501,
+            311
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1, b)} = \\sum_ {k = 1} ^ {n} ((\\mathbf {K} _ {\\mathbf {x} _ {k}} \\varphi_ {\\mathbf {x} _ {k}} ^ {(1)}) (X _ {k - 1}) - (\\mathbf {K} _ {\\mathbf {x} _ {k - 1}} \\varphi_ {\\mathbf {x} _ {k - 1}} ^ {(1)}) (X _ {k - 1})) \\\\ + \\left(\\mathbf {K} _ {\\mathbf {x} _ {0}} \\varphi_ {\\mathbf {x} _ {0}} ^ {(1)}\\right) (X _ {0}) - \\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} \\varphi_ {\\mathbf {x} _ {n}} ^ {(1)}\\right) (X _ {n}). \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            292,
+            315,
+            704,
+            375
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "Since  $(\\mathbf{K}_{\\mathbf{x}}\\varphi_{\\mathbf{x}}^{(1)})(X)$  is continuous in  $\\mathbf{x}$ , for  $\\mathbf{x}_n$  within a compact set  $\\Omega$  (assumption A4), it is local Lipschitz with a constant  $L_{\\Omega}$  such that",
+        "bbox": [
+            169,
+            378,
+            823,
+            409
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left\\| \\right.\\left(\\mathbf {K} _ {\\mathbf {x} _ {k}} \\varphi_ {\\mathbf {x} _ {k}} ^ {(1)}\\right)\\left(X _ {k - 1}\\right) - \\mathbf {K} _ {\\mathbf {x} _ {k - 1}} \\varphi_ {\\mathbf {x} _ {k - 1}} ^ {(1)}\\left. \\right)\\left(X _ {k - 1}\\right)\\left. \\right\\| \\leq L _ {\\Omega} \\| \\mathbf {x} _ {k} - \\mathbf {x} _ {k - 1} \\| \\leq 2 L _ {\\Omega} \\gamma_ {k}.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            246,
+            412,
+            751,
+            430
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "where the last inequality arises from (4b), i.e.,  $\\mathbf{x}_k - \\mathbf{x}_{k-1} = \\gamma_k (\\delta_{X_k} - \\mathbf{x}_{k-1})$  and  $\\|\\delta_{X_k} - \\mathbf{x}_{k-1}\\| \\leq 2$  because  $\\mathbf{x}_n \\in \\mathrm{Int}(\\Sigma)$ . Also,  $\\|(K_{\\mathbf{x}_0} \\varphi_{\\mathbf{x}_0}^{(1)})(X_0)\\| + \\|(K_{\\mathbf{x}_n} \\varphi_{\\mathbf{x}_n}^{(1)})(X_n)\\|$  are upper-bounded by some positive constant  $C_\\Omega'$ . This implies that",
+        "bbox": [
+            169,
+            433,
+            823,
+            479
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1, b)} \\right\\| \\leq C _ {\\Omega} ^ {\\prime} + 2 L _ {\\Omega} \\sum_ {k = 1} ^ {n} \\gamma_ {k}.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            387,
+            483,
+            611,
+            523
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "Note that",
+        "bbox": [
+            171,
+            525,
+            236,
+            537
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\gamma_ {n} \\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1, b)} \\right\\| \\leq \\gamma_ {n} C _ {\\Omega} ^ {\\prime} + 2 L _ {\\Omega} \\gamma_ {n} \\sum_ {k = 1} ^ {n} \\gamma_ {k} \\leq \\gamma_ {n} C _ {\\Omega} ^ {\\prime} + \\frac {2 L _ {\\Omega}}{a} n ^ {1 - 2 a}, \\tag {47}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            281,
+            537,
+            821,
+            575
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "where the last inequality is from  $\\sum_{k=1}^{n} \\gamma_k < \\frac{1}{a} n^{1-a}$ . We observe that the last term in (47) is decreasing to zero in  $n$  because  $a > 1/2$ .",
+        "bbox": [
+            169,
+            578,
+            823,
+            608
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "Note that  $\\mathbf{J}_k^{(1)} = \\mathbf{J}_k^{(1,a)} + \\mathbf{J}_k^{(1,b)}$ , by triangular inequality we have",
+        "bbox": [
+            169,
+            614,
+            602,
+            633
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\gamma_ {n} \\mathbb {E} \\left[ \\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1 1)} \\right\\| \\right] \\leq \\gamma_ {n} \\mathbb {E} \\left[ \\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1 1, A)} \\right\\| \\right] + \\gamma_ {n} \\mathbb {E} \\left[ \\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1 1, B)} \\right\\| \\right] \\\\ \\leq \\gamma_ {n} C _ {1} \\mathbb {E} \\left[ \\sqrt {\\left(\\sum_ {k = 1} ^ {n} \\left\\| \\mathbf {J} _ {k} ^ {(1 1 , A)} \\right\\| ^ {2}\\right)} \\right] + \\gamma_ {n} \\mathbb {E} \\left[ \\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1 1, B)} \\right\\| \\right] \\tag {48} \\\\ = \\mathbb {E} \\left[ \\gamma_ {n} C _ {1} \\sqrt {\\left(\\sum_ {k = 1} ^ {n} \\left\\| \\mathbf {J} _ {k} ^ {(1 1 , A)} \\right\\| ^ {2}\\right)} + \\gamma_ {n} \\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1 1 , B)} \\right\\| \\right], \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            235,
+            637,
+            823,
+            780
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "where the second inequality comes from (45). By (46) and (47) we know that both terms in the last line of (48) are uniformly bounded by constants over time  $n$  that depend on the set  $\\Omega$ . Therefore, by dominated convergence theorem, taking the limit over the last line of (48) gives",
+        "bbox": [
+            169,
+            782,
+            823,
+            824
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\lim  _ {n \\rightarrow \\infty} \\mathbb {E} \\left[ \\gamma_ {n} C _ {1} \\sqrt {\\left(\\sum_ {k = 1} ^ {n} \\left\\| \\mathbf {J} _ {k} ^ {(1 1 , A)} \\right\\| ^ {2}\\right)} + \\gamma_ {n} \\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1 1, B)} \\right\\|\\right] \\\\ = \\mathbb {E} \\left[ \\lim  _ {n \\rightarrow \\infty} \\gamma_ {n} C _ {1} \\sqrt {\\left(\\sum_ {k = 1} ^ {n} \\left\\| \\mathbf {J} _ {k} ^ {(1 1 , A)} \\right\\| ^ {2}\\right)} + \\gamma_ {n} \\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1 1 , B)} \\right\\|\\right] = 0. \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            287,
+            828,
+            710,
+            928
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "page_number",
+        "text": "24",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "Therefore, we have",
+        "bbox": [
+            171,
+            104,
+            303,
+            117
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\lim  _ {n \\rightarrow \\infty} \\gamma_ {n} \\mathbb {E} \\left[\\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1)} \\right\\|\\right] = 0,\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            398,
+            114,
+            596,
+            157
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "In sum, in terms of  $\\mathbb{E}[M_{n + 1}^{(\\mathbf{x})}(M_{n + 1}^{(\\mathbf{x})})^T|\\mathcal{F}_n]$  in (41), we have  $\\mathbf{U}_{11}$  in (44),  $\\lim_{n\\to \\infty}\\mathbf{D}_n^{(1)} = 0$  a.s. and  $\\lim_{n\\to \\infty}\\gamma_n\\mathbb{E}\\left[\\left\\| \\sum_{k = 1}^n\\mathbf{J}_k^{(1)}\\right\\|\\right] = 0.$",
+        "bbox": [
+            169,
+            167,
+            823,
+            210
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "We can apply the same steps as above for the other two terms  $i = 2,3$  in (42) and obtain the results.",
+        "bbox": [
+            169,
+            215,
+            825,
+            244
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "Analysis of Terms  $r_n^{(\\pmb {\\theta},1)},r_n^{(\\pmb {\\theta},2)},r_n^{(\\mathbf{x},1)},r_n^{(\\mathbf{x},2)}$",
+        "bbox": [
+            171,
+            255,
+            473,
+            273
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "Lemma E.2. For  $r_n^{(\\theta, 1)}, r_n^{(\\theta, 2)}, r_n^{(\\mathbf{x}, 1)}, r_n^{(\\mathbf{x}, 2)}$  defined in (35), we have the following results:",
+        "bbox": [
+            169,
+            276,
+            766,
+            296
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left\\| r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} \\right\\| = O (\\gamma_ {n}) = o \\left(\\sqrt {\\beta_ {n}}\\right), \\quad \\sqrt {\\gamma_ {n}} \\left\\| \\sum_ {k = 1} ^ {n} r _ {k} ^ {(\\boldsymbol {\\theta}, 2)} \\right\\| = O \\left(\\sqrt {\\gamma_ {n}}\\right) = o (1). \\tag {49a}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            261,
+            299,
+            823,
+            339
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left\\| r _ {n} ^ {(\\mathbf {x}, 1)} \\right\\| = O \\left(\\gamma_ {n}\\right) = o \\left(\\sqrt {\\beta_ {n}}\\right), \\quad \\sqrt {\\gamma_ {n}} \\left\\| \\sum_ {k = 1} ^ {n} r _ {k} ^ {(\\mathbf {x}, 2)} \\right\\| = O \\left(\\sqrt {\\gamma_ {n}}\\right) = o (1). \\tag {49b}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            261,
+            344,
+            823,
+            385
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "Proof. For  $r_n^{(\\theta, 1)}$ , note that",
+        "bbox": [
+            169,
+            400,
+            354,
+            417
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} = \\left(\\mathbf {K} _ {\\mathbf {x} _ {n + 1}} \\tilde {H} _ {\\boldsymbol {\\theta} _ {n + 1}, \\mathbf {x} _ {n + 1}}\\right) \\left(X _ {n + 1}\\right) - \\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} \\tilde {H} _ {\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}}\\right) \\left(X _ {n + 1}\\right) \\\\ = \\sum_ {j \\in \\mathcal {N}} \\left(\\mathbf {K} _ {X _ {n}, j} [ \\mathbf {x} _ {n + 1} ] \\tilde {H} _ {\\boldsymbol {\\theta} _ {n + 1}, \\mathbf {x} _ {n + 1}} (j) - \\mathbf {K} _ {X _ {n}, j} [ \\mathbf {x} _ {n} ] \\tilde {H} _ {\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}} (j)\\right) \\tag {50} \\\\ \\leq \\sum_ {j \\in \\mathcal {N}} L _ {\\mathcal {C}} \\left(\\left\\| \\boldsymbol {\\theta} _ {n + 1} - \\boldsymbol {\\theta} _ {n} \\right\\| + \\left\\| \\mathbf {x} _ {n + 1} - \\mathbf {x} _ {n} \\right\\|\\right) \\\\ \\leq N L _ {\\mathcal {C}} \\left(C _ {\\mathcal {C}} \\beta_ {n + 1} + 2 \\gamma_ {n + 1}\\right) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            266,
+            421,
+            823,
+            532
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "where the second last inequality is because  $\\mathbf{K}_{i,j}[\\mathbf{x}]\\tilde{H}_{\\boldsymbol{\\theta},\\mathbf{x}}(j)$  is continuous in  $\\pmb {\\theta},\\mathbf{x}$ $\\mathbf{K}[\\mathbf{x}]$ , which stems from continuous functions  $\\pmb {K}[\\mathbf{x}]$  and  $\\tilde{H}_{\\pmb{\\theta},\\mathbf{x}}$ . The last inequality is from update rules (4) and  $(\\pmb {\\theta}_n,\\mathbf{x}_n)\\in \\Omega$  for some compact subset  $\\Omega$  by assumption A4. Then, we have  $\\| r_n^{(\\pmb {\\theta},1)}\\| = O(\\gamma_n) = o(\\sqrt{\\beta_n})$  because of  $a > 1 / 2\\geq b / 2$  by assumption A2.",
+        "bbox": [
+            169,
+            537,
+            823,
+            604
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "We let  $\\nu_{n} \\triangleq (\\mathbf{K}_{\\mathbf{x}_{n}}\\tilde{H}_{\\boldsymbol{\\theta}_{n},\\mathbf{x}_{n}})(X_{n})$  such that  $r_n^{(\\pmb{\\theta},2)} = \\nu_n - \\nu_{n + 1}$ . Note that  $\\sum_{k = 1}^{n}r_k^{(\\pmb{\\theta},2)} = \\nu_1 - \\nu_{n + 1}$  and by assumption A4,  $\\| \\nu_{n}\\|$  is upper bounded by a constant dependent on the compact set, which leads to",
+        "bbox": [
+            169,
+            609,
+            823,
+            652
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\sqrt {\\gamma_ {n}} \\left\\| \\sum_ {k = 1} ^ {n} r _ {k} ^ {(\\boldsymbol {\\theta}, 2)} \\right\\| = \\sqrt {\\gamma_ {n}} \\| \\nu_ {1} - \\nu_ {n + 1} \\| = O (\\sqrt {\\gamma_ {n}}) = o (1).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            302,
+            652,
+            692,
+            691
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "Similarly, we can also obtain  $\\| r_n^{(\\mathbf{x},1)}\\| = o(\\sqrt{\\beta_n})$  and  $\\sqrt{\\gamma_n}\\left\\| \\sum_{k = 1}^n r_k^{(\\mathbf{x},2)}\\right\\| = O(\\sqrt{\\gamma_n}) = o(1)$ .",
+        "bbox": [
+            169,
+            702,
+            823,
+            728
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "E.2.2 EFFECT OF SRRW ITERATION ON SA ITERATION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            739,
+            571,
+            753
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "In view of the almost sure convergence results in Lemma 3.1 and Lemma 3.2, for large enough  $n$  so that both iterations  $\\theta_{n},\\mathbf{x}_{n}$  are close to the equilibrium  $(\\theta^{*},\\mu)$ , we can apply the Taylor expansion to functions  $\\mathbf{h}(\\theta ,\\mathbf{x})$  and  $\\pi [\\mathbf{x}] - \\mathbf{x}$  in (36) at the point  $(\\theta^{*},\\mu)$ , which results in",
+        "bbox": [
+            169,
+            765,
+            823,
+            808
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {h} (\\boldsymbol {\\theta}, \\mathbf {x}) = \\mathbf {h} \\left(\\boldsymbol {\\theta} ^ {*}, \\boldsymbol {\\mu}\\right) + \\nabla_ {\\boldsymbol {\\theta}} \\mathbf {h} \\left(\\boldsymbol {\\theta} ^ {*}, \\boldsymbol {\\mu}\\right) \\left(\\boldsymbol {\\theta} - \\boldsymbol {\\theta} ^ {*}\\right) + \\nabla_ {\\mathbf {x}} \\mathbf {h} \\left(\\boldsymbol {\\theta} ^ {*}, \\boldsymbol {\\mu}\\right) (\\mathbf {x} - \\boldsymbol {\\mu}) + O \\left(\\left\\| \\boldsymbol {\\theta} - \\boldsymbol {\\theta} ^ {*} \\right\\| ^ {2} + \\| \\mathbf {x} - \\boldsymbol {\\mu} \\| ^ {2}\\right), \\tag {51a}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            179,
+            811,
+            823,
+            829
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\pi [ \\mathbf {x} ] - \\mathbf {x} = \\pi [ \\boldsymbol {\\mu} ] - \\boldsymbol {\\mu} + \\nabla_ {\\mathbf {x}} (\\pi (\\mathbf {x}) - \\mathbf {x}) | _ {\\mathbf {x} = \\boldsymbol {\\mu}} (\\mathbf {x} - \\boldsymbol {\\mu}) + O (\\| \\mathbf {x} - \\boldsymbol {\\mu} \\| ^ {2}). \\tag {51b}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            261,
+            833,
+            823,
+            849
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "With matrix  $\\mathbf{J}(\\alpha)$ , we have the following:",
+        "bbox": [
+            171,
+            851,
+            449,
+            866
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {J} _ {1 1} = \\nabla_ {\\boldsymbol {\\theta}} \\mathbf {h} \\left(\\boldsymbol {\\theta} ^ {*}, \\boldsymbol {\\mu}\\right) = \\nabla \\mathbf {h} \\left(\\boldsymbol {\\theta} ^ {*}\\right),\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            290,
+            869,
+            501,
+            886
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {J} _ {1 2} (\\alpha) = \\nabla_ {\\mathbf {x}} \\mathbf {h} \\left(\\boldsymbol {\\theta} ^ {*}, \\boldsymbol {\\mu}\\right) = - \\alpha \\mathbf {H} ^ {T} \\left(\\mathbf {P} ^ {T} + \\mathbf {I}\\right), \\tag {52}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            292,
+            888,
+            823,
+            906
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {J} _ {2 2} (\\alpha) = \\nabla_ {\\mathbf {x}} (\\boldsymbol {\\pi} (\\mathbf {x}) - \\mathbf {x}) | _ {\\mathbf {x} = \\boldsymbol {\\mu}} = 2 \\alpha \\boldsymbol {\\mu} \\mathbf {1} ^ {T} - \\alpha \\mathbf {P} ^ {T} - (\\alpha + 1) \\mathbf {I}.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            292,
+            910,
+            704,
+            928
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "page_number",
+        "text": "25",
+        "bbox": [
+            488,
+            946,
+            506,
+            959
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "Then, (36) becomes",
+        "bbox": [
+            171,
+            104,
+            307,
+            118
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} + \\beta_ {n + 1} \\left(\\mathbf {J} _ {1 1} \\left(\\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*}\\right) + \\mathbf {J} _ {1 2} (\\alpha) \\left(\\mathbf {x} _ {n} - \\boldsymbol {\\mu}\\right) + r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} + r _ {n} ^ {(\\boldsymbol {\\theta}, 2)} + M _ {n + 1} ^ {(\\boldsymbol {\\theta})} + \\eta_ {n} ^ {(\\boldsymbol {\\theta})}\\right), \\tag {53a}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            187,
+            123,
+            823,
+            142
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {x} _ {n + 1} = \\mathbf {x} _ {n} + \\gamma_ {n + 1} \\left(\\mathbf {J} _ {2 2} (\\alpha) \\left(\\mathbf {x} _ {n} - \\boldsymbol {\\mu}\\right) + r _ {n} ^ {(\\mathbf {x}, 1)} + r _ {n} ^ {(\\mathbf {x}, 2)} + M _ {n + 1} ^ {(\\mathbf {x})} + \\eta_ {n} ^ {(\\mathbf {x})}\\right), \\tag {53b}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            258,
+            148,
+            823,
+            167
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "where  $\\eta_n^{(\\pmb{\\theta})} = O(\\| \\mathbf{x}_n\\|^2 + \\| \\pmb{\\theta}_n\\|^2)$  and  $\\eta_n^{(\\mathbf{x})} = O(\\| \\mathbf{x}_n\\|^2)$ .",
+        "bbox": [
+            171,
+            169,
+            552,
+            189
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "Then, inspired by Mokkadem & Pelletier (2006), we decompose iterates  $\\{\\mathbf{x}_n\\}$  and  $\\{\\pmb {\\theta}_n\\}$  into  $\\mathbf{x}_n =$ $L_{n}^{(\\pmb {x})} + \\Delta_{n}^{(\\pmb {x})}$  and  $\\pmb {\\theta}_n = L_n^{(\\pmb {\\theta})} + R_n^{(\\pmb {\\theta})} + \\Delta_n^{(\\pmb {\\theta})}$ . Rewriting (53b) gives",
+        "bbox": [
+            171,
+            193,
+            823,
+            226
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {x} _ {n} - \\boldsymbol {\\mu} = \\gamma_ {n + 1} ^ {- 1} \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (\\mathbf {x} _ {n + 1} - \\mathbf {x} _ {n}) - \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (r _ {n} ^ {(\\mathbf {x}, 1)} + r _ {n} ^ {(\\mathbf {x}, 2)} + M _ {n + 1} ^ {(\\mathbf {x})} + \\eta_ {n} ^ {(\\mathbf {x})}),\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            215,
+            231,
+            779,
+            251
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "and substituting the above equation back in (53a) gives",
+        "bbox": [
+            171,
+            253,
+            535,
+            268
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\boldsymbol {\\theta} _ {n + 1} - \\boldsymbol {\\theta} ^ {*} = \\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*} + \\beta_ {n + 1} \\left(\\mathbf {J} _ {1 1} \\left(\\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*}\\right) + \\gamma_ {n + 1} ^ {- 1} \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(\\mathbf {x} _ {n + 1} - \\mathbf {x} _ {n}\\right) \\right. \\\\ \\left. - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(r _ {n} ^ {(\\mathbf {x}, 1)} + r _ {n} ^ {(\\mathbf {x}, 2)} + M _ {n + 1} ^ {(\\mathbf {x})} + \\eta_ {n} ^ {(\\mathbf {x})}\\right) + r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} + r _ {n} ^ {(\\boldsymbol {\\theta}, 2)} + M _ {n + 1} ^ {(\\boldsymbol {\\theta})} + \\eta_ {n} ^ {(\\boldsymbol {\\theta})}\\right) \\\\ = (\\mathbf {I} + \\beta_ {n + 1} \\mathbf {J} _ {1 1}) \\left(\\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*}\\right) + \\left[ \\beta_ {n + 1} \\gamma_ {n + 1} ^ {- 1} \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(\\mathbf {x} _ {n + 1} - \\mathbf {x} _ {n}\\right) \\right] \\\\ + \\beta_ {n + 1} (M _ {n + 1} ^ {(\\boldsymbol {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} M _ {n + 1} ^ {(\\mathbf {x})}) \\\\ + \\beta_ {n + 1} \\left(r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} + r _ {n} ^ {(\\boldsymbol {\\theta}, 2)} + \\eta_ {n} ^ {(\\boldsymbol {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(r _ {n} ^ {(\\mathbf {x}, 1)} + r _ {n} ^ {(\\mathbf {x}, 2)} + \\eta_ {n} ^ {(\\mathbf {x})}\\right)\\right), \\tag {54} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            191,
+            273,
+            823,
+            419
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "From (54) we can see the iteration  $\\{\\theta_n\\}$  implicitly embeds the recursions of three sequences",
+        "bbox": [
+            169,
+            424,
+            779,
+            440
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "-  $\\beta_{n+1} \\gamma_{n+1}^{-1} \\mathbf{J}_{12}(\\alpha) \\mathbf{J}_{22}(\\alpha)^{-1} (\\mathbf{x}_{n+1} - \\mathbf{x}_n)$ ;",
+            "-  $\\beta_{n+1}(M_{n+1}^{(\\boldsymbol{\\theta})} - \\mathbf{J}_{12}(\\alpha)\\mathbf{J}_{22}(\\alpha)^{-1}M_{n+1}^{(\\mathbf{x})})$ ;",
+            "-  $\\beta_{n+1}(r_n^{(\\pmb{\\theta}, 1)} + r_n^{(\\pmb{\\theta}, 2)} + \\eta_n^{(\\pmb{\\theta})} - \\mathbf{J}_{12}(\\alpha) \\mathbf{J}_{22}(\\alpha)^{-1}(r_n^{(\\mathbf{x}, 1)} + r_n^{(\\mathbf{x}, 2)} + \\eta_n^{(\\mathbf{x}))})$ ."
+        ],
+        "bbox": [
+            217,
+            450,
+            715,
+            513
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "Let  $u_{n} \\triangleq \\sum_{k=1}^{n} \\beta_{k}$  and  $s_{n} \\triangleq \\sum_{k=1}^{n} \\gamma_{k}$ . Below we define two iterations:",
+        "bbox": [
+            169,
+            523,
+            651,
+            541
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} L _ {n} ^ {(\\boldsymbol {\\theta})} = e ^ {\\beta_ {n} \\mathbf {J} _ {1 1}} L _ {n - 1} ^ {(\\boldsymbol {\\theta})} + \\beta_ {n} \\left(M _ {n} ^ {(\\boldsymbol {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} M _ {n} ^ {(\\mathbf {x})}\\right) \\\\ = \\sum_ {k = 1} ^ {n} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} \\beta_ {k} \\left(M _ {k} ^ {(\\boldsymbol {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} M _ {k} ^ {(\\mathbf {x})}\\right) \\tag {55a} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            269,
+            546,
+            821,
+            606
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} R _ {n} ^ {(\\boldsymbol {\\theta})} = e ^ {\\beta_ {n} \\mathbf {J} _ {1 1}} R _ {n - 1} ^ {(\\boldsymbol {\\theta})} + \\beta_ {n} \\gamma_ {n} ^ {- 1} \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (\\mathbf {x} _ {n} - \\mathbf {x} _ {n - 1}) \\\\ = \\sum_ {k = 1} ^ {n} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} \\beta_ {k} \\gamma_ {k} ^ {- 1} \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(\\mathbf {x} _ {k} - \\mathbf {x} _ {k - 1}\\right) \\tag {55b} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            267,
+            613,
+            821,
+            672
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "and a remaining term  $\\Delta_n^{(\\pmb{\\theta})} \\triangleq \\pmb{\\theta}_n - \\pmb{\\theta}^* - L_n^{(\\pmb{\\theta})} - R_n^{(\\pmb{\\theta})}$ .",
+        "bbox": [
+            171,
+            676,
+            532,
+            695
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "Similarly, for iteration  $\\mathbf{x}_n$ , define the sequence  $L_n^{(\\mathbf{x})}$  such that",
+        "bbox": [
+            171,
+            703,
+            576,
+            719
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nL _ {n} ^ {(\\mathbf {x})} = e ^ {\\gamma_ {n} \\mathbf {J} _ {2 2} (\\alpha)} L _ {n - 1} ^ {(\\mathbf {x})} + \\gamma_ {n} M _ {n} ^ {(\\mathbf {x})} = \\sum_ {k = 1} ^ {n} e ^ {(s _ {n} - s _ {k}) \\mathbf {J} _ {2 2} (\\alpha)} \\gamma_ {k} M _ {k} ^ {(\\mathbf {x})}, \\tag {56}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            287,
+            724,
+            823,
+            763
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "and a remaining term",
+        "bbox": [
+            171,
+            768,
+            315,
+            782
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\Delta_ {n} ^ {(\\mathbf {x})} \\triangleq \\mathbf {x} _ {n} - \\boldsymbol {\\mu} - L _ {n} ^ {(\\boldsymbol {\\theta})} \\tag {57}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            421,
+            781,
+            823,
+            799
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "The decomposition of  $\\theta_{n} - \\theta^{*}$  and  $\\mathbf{x}_n - \\pmb{\\mu}$  in the above form is also standard in the single-timescale SA literature (Delyon, 2000; Fort, 2015).",
+        "bbox": [
+            169,
+            800,
+            823,
+            829
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "Characterization of Sequences  $\\{L_n^{(\\theta)}\\}$  and  $\\{L_n^{(\\mathbf{x})}\\}$",
+        "bbox": [
+            169,
+            835,
+            517,
+            854
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "we set a Martingale  $Z^{(n)} = \\{Z_k^{(n)}\\}_{k\\geq 1}$  such that",
+        "bbox": [
+            171,
+            861,
+            496,
+            878
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nZ _ {k} ^ {(n)} = \\left( \\begin{array}{c c} \\beta_ {n} ^ {- 1 / 2} e ^ {u _ {n} \\mathbf {J} _ {1 1}} & 0 \\\\ 0 & \\gamma_ {n} ^ {- 1 / 2} e ^ {s _ {n} \\mathbf {J} _ {2 2} (\\alpha)} \\end{array} \\right) \\times \\sum_ {j = 1} ^ {k} \\left( \\begin{array}{c} e ^ {- u _ {k} \\mathbf {J} _ {1 1}} \\beta_ {k} (M _ {k} ^ {(\\boldsymbol {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} M _ {k} ^ {(\\mathbf {x})}) \\\\ e ^ {- s _ {k} \\mathbf {J} _ {2 2} (\\alpha)} \\gamma_ {k} M _ {k} ^ {(\\mathbf {x})} \\end{array} \\right).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            174,
+            883,
+            821,
+            926
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "page_number",
+        "text": "26",
+        "bbox": [
+            488,
+            948,
+            506,
+            959
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "Then, the Martingale difference array  $Z_{k}^{(n)} - Z_{k - 1}^{(n)}$  becomes",
+        "bbox": [
+            169,
+            101,
+            575,
+            121
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nZ _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)} = \\left( \\begin{array}{c} \\beta_ {n} ^ {- 1 / 2} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\beta_ {k} (M _ {k} ^ {(\\boldsymbol {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} M _ {k} ^ {(\\mathbf {x})}) \\\\ \\gamma_ {n} ^ {- 1 / 2} e ^ {(s _ {n} - s _ {k}) \\mathbf {J} _ {2 2} (\\alpha)} \\gamma_ {k} M _ {k} ^ {(\\mathbf {x})} \\end{array} \\right)\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            250,
+            125,
+            743,
+            165
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "and",
+        "bbox": [
+            171,
+            167,
+            200,
+            180
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\sum_ {k = 1} ^ {n} \\mathbb {E} \\left[ (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) ^ {T} | \\mathcal {F} _ {k - 1} \\right] = \\left( \\begin{array}{c c} A _ {1, n} & A _ {2, n} \\\\ A _ {2, n} ^ {T} & A _ {4, n} \\end{array} \\right),\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            274,
+            178,
+            718,
+            217
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "where, in view of decomposition of  $M_{n}^{(\\pmb{\\theta})}$  and  $M_{n}^{(\\mathbf{x})}$  in (41) and (42), respectively,",
+        "bbox": [
+            169,
+            219,
+            712,
+            238
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} A _ {1, n} = \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\beta_ {k} ^ {2} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\left(\\mathbf {U} _ {2 2} + \\mathbf {D} _ {k} ^ {(1)} + \\mathbf {J} _ {k} ^ {(1)} - \\left(\\mathbf {U} _ {2 1} + \\mathbf {D} _ {k} ^ {(2)} + \\mathbf {J} _ {k} ^ {(2)}\\right) \\left(\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}\\right) ^ {T} \\right. \\\\ + \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(\\mathbf {U} _ {1 1} + \\mathbf {D} _ {k} ^ {(3)} + \\mathbf {J} _ {k} ^ {(3)}\\right) \\left(\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}\\right) ^ {T} \\\\ \\left. - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(\\mathbf {U} _ {2 1} + \\mathbf {D} _ {k} ^ {(2)} + \\mathbf {J} _ {k} ^ {(2)}\\right) ^ {T}\\right) e ^ {\\left(u _ {n} - u _ {k}\\right) \\left(\\mathbf {J} _ {1 1}\\right) ^ {T}}, \\tag {58a} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            189,
+            241,
+            823,
+            351
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nA _ {2, n} = \\beta_ {n} ^ {- 1 / 2} \\gamma_ {n} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} \\beta_ {k} \\gamma_ {k} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} \\left(\\mathbf {U} _ {2 1} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\mathbf {U} _ {1 1}\\right) e ^ {\\left(s _ {n} - s _ {k}\\right) \\mathbf {J} _ {2 2} (\\alpha) ^ {T}}, \\tag {58b}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            186,
+            356,
+            823,
+            395
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nA _ {4, n} = \\gamma_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\gamma_ {k} ^ {2} e ^ {\\left(s _ {n} - s _ {k}\\right) \\mathbf {J} _ {2 2} (\\alpha)} \\left(\\mathbf {U} _ {1 1} + \\mathbf {D} _ {k} ^ {(3)} + \\mathbf {J} _ {k} ^ {(3)}\\right) e ^ {\\left(s _ {n} - s _ {k}\\right) \\mathbf {J} _ {2 2} (\\alpha) ^ {T}}. \\tag {58c}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            259,
+            398,
+            823,
+            436
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "We further decompose  $A_{1,n}$  into three parts:",
+        "bbox": [
+            169,
+            436,
+            465,
+            452
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} A _ {1, n} = \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\left(\\beta_ {k} ^ {2} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} \\left(\\mathbf {U} _ {2 2} - \\mathbf {U} _ {2 1} \\left(\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}\\right) ^ {T} \\right. \\right. \\\\ \\left. \\left. - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\mathbf {U} _ {1 2} + \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\mathbf {U} _ {1 1} \\left(\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}\\right) ^ {T}\\right) e ^ {\\left(u _ {n} - u _ {k}\\right) \\left(\\mathbf {J} _ {1 1}\\right) ^ {T}}\\right) \\\\ + \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\left(\\beta_ {k} ^ {2} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} \\left(\\mathbf {D} _ {k} ^ {(1)} + \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\mathbf {D} _ {k} ^ {(3)} \\left(\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}\\right) ^ {T} \\right. \\right. \\\\ \\left. - \\mathbf {D} _ {k} ^ {(2)} \\left(\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}\\right) ^ {T} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(\\mathbf {D} _ {k} ^ {(2)}\\right) ^ {T}\\right) e ^ {\\left(u _ {n} - u _ {k}\\right) \\left(\\mathbf {J} _ {1 1}\\right) ^ {T}} \\\\ + \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\left(\\beta_ {k} ^ {2} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} (\\mathbf {J} _ {k} ^ {(1)} + \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\mathbf {J} _ {k} ^ {(3)} (\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}) ^ {T} \\right. \\\\ \\left. - \\mathbf {J} _ {k} ^ {(2)} \\left(\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}\\right) ^ {T} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(\\mathbf {J} _ {k} ^ {(2)}\\right) ^ {T}\\right) e ^ {\\left(u _ {n} - u _ {k}\\right) \\left(\\mathbf {J} _ {1 1}\\right) ^ {T}} \\\\ \\triangleq A _ {1, n} ^ {(a)} + A _ {1, n} ^ {(b)} + A _ {1, n} ^ {(c)}. \\tag {59} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            176,
+            455,
+            823,
+            722
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "Here, we define  $\\mathbf{U}_{\\pmb{\\theta}}(\\alpha) \\triangleq \\mathbf{U}_{22} - \\mathbf{U}_{21}(\\mathbf{J}_{12}(\\alpha)\\mathbf{J}_{22}(\\alpha)^{-1})^T - \\mathbf{J}_{12}(\\alpha)\\mathbf{J}_{22}(\\alpha)^{-1}\\mathbf{U}_{12} + \\mathbf{J}_{12}(\\alpha)\\mathbf{J}_{22}(\\alpha)^{-1}\\mathbf{U}_{11} + \\mathbf{J}_{12}(\\alpha)\\mathbf{J}_{22}(\\alpha)^{-1}\\mathbf{J}_{12}$ . By (52) and (43a) in Lemma E.1, we have",
+        "bbox": [
+            169,
+            727,
+            823,
+            758
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {U} _ {\\boldsymbol {\\theta}} (\\alpha) = \\sum_ {i = 1} ^ {N - 1} \\frac {1}{(\\alpha (1 + \\lambda_ {i}) + 1) ^ {2}} \\cdot \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H}. \\tag {60}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            762,
+            823,
+            801
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "Then, we have the following lemma.",
+        "bbox": [
+            169,
+            805,
+            415,
+            819
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "Lemma E.3. For  $A_{1,n}^{(a)}, A_{1,n}^{(b)}, A_{1,n}^{(c)}$  defined in (59), we have",
+        "bbox": [
+            169,
+            821,
+            565,
+            842
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\lim  _ {n \\rightarrow \\infty} A _ {1, n} ^ {(a)} = \\mathbf {V} _ {\\boldsymbol {\\theta}} (\\alpha), \\quad \\lim  _ {n \\rightarrow \\infty} \\| A _ {1, n} ^ {(b)} \\| = 0, \\quad \\lim  _ {n \\rightarrow \\infty} \\| A _ {1, n} ^ {(c)} \\| = 0, \\tag {61}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            287,
+            845,
+            823,
+            869
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "where  $\\mathbf{V}_{\\theta}(\\alpha)$  is the solution to the Lyapunov equation",
+        "bbox": [
+            169,
+            873,
+            531,
+            888
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left(\\mathbf {J} _ {1 1} + \\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\boldsymbol {I}\\right) \\mathbf {V} _ {\\boldsymbol {\\theta}} (\\alpha) + \\mathbf {V} _ {\\boldsymbol {\\theta}} (\\alpha) \\left(\\mathbf {J} _ {1 1} + \\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\boldsymbol {I}\\right) ^ {T} + \\mathbf {U} _ {\\boldsymbol {\\theta}} (\\alpha) = 0.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            254,
+            892,
+            741,
+            929
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "page_number",
+        "text": "27",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "Proof. First, from Lemma G.4, we have for some  $c, T > 0$  such that",
+        "bbox": [
+            171,
+            103,
+            622,
+            119
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\| A _ {1, n} ^ {(b)} \\| \\leq \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\left\\| \\mathbf {D} _ {k} ^ {(1)} + \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\mathbf {D} _ {k} ^ {(3)} (\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}) ^ {T} - \\mathbf {D} _ {k} ^ {(2)} (\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}) ^ {T} \\right. \\\\ \\left. - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(\\mathbf {D} _ {k} ^ {(2)}\\right) ^ {T} \\right\\| \\cdot \\beta_ {k} ^ {2} c ^ {2} e ^ {- 2 T \\left(u _ {n} - u _ {k}\\right)}. \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            174,
+            125,
+            823,
+            200
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "Applying Lemma G.6, together with  $\\mathbf{D}_n^{(i)}\\to 0$  a.s. in Lemma E.1, gives",
+        "bbox": [
+            171,
+            205,
+            648,
+            224
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\lim \\sup \\| A _ {1, n} ^ {(b)} \\| \\\\ \\leq \\frac {1}{C (b , p)} \\lim  _ {n} \\sup  _ {n} \\| (\\mathbf {D} _ {n} ^ {(1)} + \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\mathbf {D} _ {n} ^ {(4)} (\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}) ^ {T} \\\\ \\left. - \\mathbf {D} _ {n} ^ {(2)} \\left(\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}\\right) ^ {T} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\mathbf {D} _ {n} ^ {(3)}\\right) \\| \\\\ = 0. \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            267,
+            229,
+            730,
+            323
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "We now consider  $\\| A_{1,n}^{(c)}\\|$ . Set",
+        "bbox": [
+            171,
+            334,
+            372,
+            353
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\Xi_ {n} \\triangleq \\sum_ {k = 1} ^ {n} \\left(\\mathbf {J} _ {n} ^ {(1)} + \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\mathbf {J} _ {k} ^ {(3)} (\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}) ^ {T} \\right. \\\\ - \\mathbf {J} _ {k} ^ {(2)} (\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}) ^ {T} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (\\mathbf {J} _ {k} ^ {(2)}) ^ {T}), \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            272,
+            361,
+            723,
+            422
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "we can rewrite  $A_{1,n}^{(c)}$  as",
+        "bbox": [
+            171,
+            428,
+            326,
+            446
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nA _ {1, n} ^ {(c)} = \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\beta_ {k} ^ {2} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} (\\Xi_ {k} - \\Xi_ {k - 1}) e ^ {(u _ {n} - u _ {k}) (\\mathbf {J} _ {1 1}) ^ {T}}.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            299,
+            454,
+            697,
+            493
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "By the Abel transformation, we have",
+        "bbox": [
+            171,
+            500,
+            418,
+            513
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} A _ {1, n} ^ {(c)} = \\beta_ {n} \\Xi_ {n} + \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n - 1} \\left[ \\beta_ {k} ^ {2} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} \\Xi_ {k} e ^ {\\left(u _ {n} - u _ {k}\\right) \\left(\\mathbf {J} _ {1 1}\\right) ^ {T}} \\right. \\tag {62} \\\\ \\left. - \\beta_ {k + 1} ^ {2} e ^ {(u _ {n} - u _ {k + 1}) \\mathbf {J} _ {1 1}} \\Xi_ {k} e ^ {(u _ {n} - u _ {k + 1}) (\\mathbf {J} _ {1 1}) ^ {T}} \\right]. \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            269,
+            521,
+            823,
+            588
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "We know from Lemma E.1 that  $\\beta_{n}\\Xi_{n}\\to 0$  a.s. because  $\\Xi_n = o(\\gamma_n^{-1})$ . Besides,",
+        "bbox": [
+            171,
+            594,
+            697,
+            609
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\left\\| \\beta_ {k} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} - \\beta_ {k + 1} e ^ {\\left(u _ {n} - u _ {k + 1}\\right) \\mathbf {J} _ {1 1}} \\right\\| \\\\ = \\| \\left(\\beta_ {k} - \\beta_ {k + 1}\\right) e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} + \\beta_ {k + 1} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} \\left(\\mathbf {I} - e ^ {- \\beta_ {k + 1} \\mathbf {J} _ {1 1}}\\right) \\| \\\\ \\leq C _ {1} \\beta_ {k} ^ {2} e ^ {- (u _ {n} - u _ {k}) T} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            236,
+            616,
+            759,
+            675
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "for some constant  $C_1 > 0$  because  $\\beta_{n} - \\beta_{n + 1}\\leq C_{2}\\beta_{n}^{2}$  and  $\\| \\mathbf{I} - e^{-\\beta_{k + 1}\\mathbf{J}_{11}}\\| \\leq C_3\\beta_{k + 1}$ . Moreover,",
+        "bbox": [
+            169,
+            681,
+            823,
+            698
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\left\\| \\beta_ {k} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} \\right\\| + \\left\\| \\beta_ {k + 1} e ^ {\\left(u _ {n} - u _ {k + 1}\\right) \\mathbf {J} _ {1 1}} \\right\\| \\\\ \\leq \\beta_ {k} \\| e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\| + \\beta_ {k} \\| e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\| \\cdot \\| e ^ {- \\beta_ {k + 1} \\mathbf {J} _ {1 1}} \\| \\\\ \\leq C _ {4} \\beta_ {k} e ^ {- (u _ {n} - u _ {k}) T}. \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            272,
+            704,
+            723,
+            762
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "Using Lemma G.7 on (62) gives",
+        "bbox": [
+            171,
+            768,
+            385,
+            784
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\| A _ {1, n} ^ {(c)} \\| \\leq C _ {1} C _ {4} \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n - 1} \\beta_ {k} ^ {2} e ^ {- 2 (u _ {n} - u _ {k}) T} \\| \\beta_ {k} \\Xi_ {k} \\| + \\| \\beta_ {n} \\Xi_ {n} \\|.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            297,
+            791,
+            699,
+            830
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "Applying Lemma G.6 again gives",
+        "bbox": [
+            171,
+            835,
+            397,
+            851
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\lim  _ {n} \\sup  _ {n} \\| A _ {1, n} ^ {(c)} \\| \\leq C _ {5} \\lim  _ {n} \\sup  _ {n} \\| \\beta_ {n} \\Xi_ {n} \\| = 0\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            357,
+            857,
+            640,
+            883
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "for some constant  $C_5 > 0$",
+        "bbox": [
+            171,
+            887,
+            349,
+            902
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "Finally, we provide an existing lemma below.",
+        "bbox": [
+            171,
+            909,
+            472,
+            924
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "page_number",
+        "text": "28",
+        "bbox": [
+            488,
+            946,
+            506,
+            959
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "Lemma E.4 (Mokkadem & Pelletier (2005) Lemma 4). For a sequence with decreasing step size  $\\beta_{n} = (n + 1)^{-b}$  for  $b\\in (1 / 2,1]$ ,  $u_{n} = \\sum_{k = 1}^{n}\\beta_{k}$ , a positive semi-definite matrix  $\\Gamma$  and a Hurwitz matrix  $\\mathbf{Q}$ , which is given by",
+        "bbox": [
+            169,
+            103,
+            826,
+            147
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\beta_ {n} ^ {2} e ^ {(u _ {n} - u _ {k}) \\mathbf {Q}} \\Gamma e ^ {(u _ {n} - u _ {k}) \\mathbf {Q} ^ {T}},\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            375,
+            152,
+            620,
+            191
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "we have",
+        "bbox": [
+            171,
+            199,
+            230,
+            212
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\lim  _ {n \\rightarrow \\infty} \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\beta_ {n} ^ {2} e ^ {(u _ {n} - u _ {k}) \\mathbf {Q}} \\Gamma e ^ {(u _ {n} - u _ {k}) \\mathbf {Q} ^ {T}} = \\mathbf {V}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            341,
+            210,
+            655,
+            250
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "where  $\\mathbf{V}$  is the solution of the Lyapunov equation",
+        "bbox": [
+            171,
+            252,
+            501,
+            267
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left(\\mathbf {Q} + \\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I}\\right) \\mathbf {V} + \\mathbf {V} \\left(\\mathbf {Q} ^ {T} + \\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I}\\right) + \\Gamma = 0.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            272,
+            679,
+            308
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "Then,  $\\lim_{n\\to \\infty}A_{1,n}^{(a)} = \\mathbf{V}_{\\pmb{\\theta}}(\\alpha)$  is a direct application of Lemma E.4.",
+        "bbox": [
+            171,
+            321,
+            624,
+            342
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "image",
+        "img_path": "images/7b45744ab21e80417e29886cf76609edd54e24c713dd773e4b043bc07136d5b3.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            325,
+            825,
+            337
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "We can follow the similar steps in Lemma E.3 to obtain",
+        "bbox": [
+            171,
+            354,
+            540,
+            369
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\lim  _ {n \\to \\infty} A _ {4, n} = \\mathbf {V} _ {\\mathbf {x}} (\\alpha),\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            424,
+            376,
+            571,
+            398
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "where  $\\mathbf{V}_{\\mathbf{x}}(\\alpha)$  is in the form of (32).",
+        "bbox": [
+            171,
+            404,
+            413,
+            420
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "The last step is to show  $\\lim_{n\\to \\infty}A_{2,n} = 0$  . Note that",
+        "bbox": [
+            171,
+            426,
+            527,
+            441
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\| A _ {2, n} \\| = O \\left(\\beta_ {n} ^ {- 1 / 2} \\gamma_ {n} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} \\beta_ {k} \\gamma_ {k} \\| e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\| \\| e ^ {(s _ {n} - s _ {k}) \\mathbf {J} _ {2 2} (\\alpha) ^ {T}} \\|\\right) \\\\ = O \\left(\\beta_ {n} ^ {- 1 / 2} \\gamma_ {n} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} \\beta_ {k} \\gamma_ {k} e ^ {- (u _ {n} - u _ {k}) T} e ^ {- (s _ {n} - s _ {k}) T ^ {\\prime}}\\right) \\\\ = O \\left(\\beta_ {n} ^ {- 1 / 2} \\gamma_ {n} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} \\beta_ {k} \\gamma_ {k} e ^ {- (s _ {n} - s _ {k}) T ^ {\\prime}}\\right), \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            259,
+            448,
+            730,
+            575
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "where the second equality is from Lemma G.4. Then, we use Lemma G.6 with  $p = 0$  to obtain",
+        "bbox": [
+            171,
+            583,
+            794,
+            598
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\sum_ {k = 1} ^ {n} \\beta_ {k} \\gamma_ {k} e ^ {- \\left(s _ {n} - s _ {k}\\right) T ^ {\\prime}} = O \\left(\\beta_ {n}\\right) \\tag {63}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            390,
+            604,
+            823,
+            643
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "Additionally, since  $\\beta_{n} = o(\\gamma_{n})$ , we have",
+        "bbox": [
+            171,
+            650,
+            444,
+            666
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\beta_ {n} ^ {- 1 / 2} \\gamma_ {n} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} \\beta_ {k} \\gamma_ {k} ^ {- 1 / 2} \\gamma_ {k} ^ {3 / 2} e ^ {- (s _ {n} - s _ {k}) T ^ {\\prime}} = O \\left(\\beta_ {n} ^ {1 / 2} \\gamma_ {n} ^ {- 1 / 2}\\right) = o (1).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            269,
+            672,
+            725,
+            712
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "Then, it follows that  $\\lim_{n\\to \\infty}A_{2,n} = 0$ . Therefore, we obtain",
+        "bbox": [
+            171,
+            719,
+            583,
+            734
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\lim  _ {n \\to \\infty} \\sum_ {k = 1} ^ {n} \\mathbb {E} \\left[ (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) ^ {T} | \\mathcal {F} _ {k - 1} \\right] = \\left( \\begin{array}{c c} \\mathbf {V} _ {\\boldsymbol {\\theta}} (\\alpha) & 0 \\\\ 0 & \\mathbf {V} _ {\\mathbf {x}} (\\alpha) \\end{array} \\right).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            241,
+            742,
+            751,
+            781
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "Now, we turn to verifying the conditions in Theorem G.3. For some  $\\tau > 0$ , we have",
+        "bbox": [
+            171,
+            795,
+            723,
+            810
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\sum_ {k = 1} ^ {n} \\mathbb {E} \\left[ \\| Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)} \\| ^ {2 + \\tau} | \\mathcal {F} _ {k - 1} \\right] \\\\ = O \\left(\\beta_ {n} ^ {- (1 + \\frac {\\tau}{2})} \\sum_ {k = 1} ^ {n} \\beta_ {k} ^ {2 + \\frac {\\tau}{2}} \\beta_ {k} ^ {\\frac {\\tau}{2}} e ^ {- (2 + \\tau) \\left(u _ {n} - u _ {k}\\right) T} + \\gamma_ {n} ^ {- (1 + \\frac {\\tau}{2})} \\sum_ {k = 1} ^ {n} \\gamma_ {k} ^ {2 + \\frac {\\tau}{2}} \\gamma_ {k} ^ {\\frac {\\tau}{2}} e ^ {- (2 + \\tau) \\left(s _ {n} - s _ {k}\\right) T ^ {\\prime}}\\right) \\tag {64} \\\\ = O \\left(\\beta_ {n} ^ {\\frac {\\tau}{2}} + \\gamma_ {n} ^ {\\frac {\\tau}{2}}\\right) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            184,
+            816,
+            823,
+            929
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "page_number",
+        "text": "29",
+        "bbox": [
+            488,
+            946,
+            509,
+            960
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "where the last equality comes from Lemma G.6. Since (64) also holds for  $\\tau = 0$ , we have",
+        "bbox": [
+            169,
+            103,
+            761,
+            119
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\sum_ {k = 1} ^ {n} \\mathbb {E} \\left[ \\| Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)} \\| ^ {2} | \\mathcal {F} _ {k - 1} \\right] = O (1) <   \\infty .\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            341,
+            126,
+            653,
+            164
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Therefore, all the conditions in Theorem G.3 are satisfied and its application then gives",
+        "bbox": [
+            169,
+            171,
+            745,
+            186
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nZ ^ {(n)} = \\left(\\begin{array}{c}\\sqrt {\\beta_ {n} ^ {- 1}} L _ {n} ^ {(\\boldsymbol {\\theta})}\\\\\\sqrt {\\gamma_ {n} ^ {- 1}} L _ {n} ^ {(\\mathbf {x})}\\end{array}\\right) \\xrightarrow [ d i s t . ]{n \\rightarrow \\infty} N \\left(0, \\left(\\begin{array}{c c}\\mathbf {V} _ {\\boldsymbol {\\theta}} (\\alpha)&0\\\\0&\\mathbf {V} _ {\\mathbf {x}} (\\alpha)\\end{array}\\right)\\right). \\tag {65}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            292,
+            193,
+            823,
+            234
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, we have the following lemma about the strong convergence rate of  $\\{L_n^{(\\pmb{\\theta})}\\}$  and  $\\{L_n^{(\\mathbf{x})}\\}$ .",
+        "bbox": [
+            169,
+            250,
+            823,
+            268
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Lemma E.5.",
+        "bbox": [
+            171,
+            270,
+            261,
+            284
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left\\| L _ {n} ^ {(\\boldsymbol {\\theta})} \\right\\| = O \\left(\\sqrt {\\beta_ {n} \\log \\left(u _ {n}\\right)}\\right) \\quad a. s. \\tag {66a}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            377,
+            282,
+            823,
+            306
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left\\| L _ {n} ^ {(\\mathbf {x})} \\right\\| = O \\left(\\sqrt {\\gamma_ {n} \\log \\left(s _ {n}\\right)}\\right) \\quad a. s. \\tag {66b}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            379,
+            310,
+            823,
+            335
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Proof. This proof follows Pelletier (1998, Lemma 1). We only need the special case of Pelletier (1998, Lemma 1) that fits our scenario; e.g., we let the two types of step sizes therein be the same. Specifically, we attach the following lemma.",
+        "bbox": [
+            169,
+            349,
+            823,
+            392
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Lemma E.6 (Pelletier (1998) Lemma 1). Consider a sequence",
+        "bbox": [
+            169,
+            398,
+            589,
+            414
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nL _ {n + 1} = e ^ {u _ {n} \\mathbf {H}} \\sum_ {k = 1} ^ {n} e ^ {- u _ {k} \\mathbf {H}} \\beta_ {k} M _ {k + 1},\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            380,
+            420,
+            616,
+            460
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "where  $\\beta_{n} = n^{-b}$ ,  $1/2 < b \\leq 1$ , and  $\\{M_n\\}$  is a Martingale difference sequence adapted to the filtration  $\\mathcal{F}$  such that, almost surely,  $\\lim \\sup_{n} \\mathbb{E}[\\| M_{n+1} \\|^2 | \\mathcal{F}_n] \\leq M^2$  and there exists  $\\tau \\in (0,2)$ ,  $b(2 + \\tau) > 2$ , such that  $\\sup_{n} \\mathbb{E}[\\| M_{n+1} \\|^2 + \\tau | \\mathcal{F}_n] < \\infty$ . Then, almost surely,",
+        "bbox": [
+            168,
+            468,
+            823,
+            513
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\lim  _ {n} \\sup  \\frac {\\left\\| L _ {n} \\right\\|}{\\sqrt {\\beta_ {n} \\log \\left(u _ {n}\\right)}} \\leq C _ {M}, \\tag {67}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            395,
+            518,
+            825,
+            554
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "where  $C_M$  is a constant dependent on  $M$ .",
+        "bbox": [
+            169,
+            559,
+            449,
+            574
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "By assumption A4, the iterates  $(\\pmb{\\theta}_n,\\mathbf{x}_n)$  are bounded within a compact subset  $\\Omega$ . Recall the form of  $M_{n + 1}^{(\\pmb{\\theta})},M_{n + 1}^{(\\mathbf{x})}$  defined in (35), it comprises the functions  $\\tilde{H}_{\\pmb{\\theta}_n,\\mathbf{x}_n}(i)$  and  $(\\mathbf{K}_{\\mathbf{x}_n}\\tilde{H}_{\\pmb{\\theta}_n,\\mathbf{x}_n})(i)$ , which in turn include the function  $H(\\pmb {\\theta},i)$ . We know that  $H(\\pmb {\\theta},i)$  is bounded for  $\\pmb{\\theta}$  in some compact set  $\\mathcal{C}$ . Thus, for any  $(\\pmb {\\theta}_n,\\mathbf{x}_n)\\in \\Omega$  for some compact set  $\\Omega$ ,  $M_{n + 1}^{(\\pmb{\\theta})},M_{n + 1}^{(\\mathbf{x})}$  are bounded and we denote by  $c_{\\pmb{\\theta}}$  and  $c_{\\mathbf{x}}$  as their upper bounds, i.e.,  $\\mathbb{E}[\\| M_{n + 1}^{(\\pmb{\\theta})}\\| ^2 |\\mathcal{F}_n]\\leq c_\\Omega^{(\\pmb{\\theta})}$  and  $\\mathbb{E}[\\| M_{n + 1}^{(\\mathbf{x})}\\| ^2 |\\mathcal{F}_n]\\leq c_\\Omega^{(\\mathbf{x})}$ . We only need to replace the upper bound  $c$  in Lemma E.6 by  $c_{\\Omega}^{(\\pmb{\\theta})}$  for the sequence  $\\{L_n^{(\\pmb{\\theta})}\\}$  (resp.  $c_{\\Omega}^{(\\mathbf{x})}$  for the sequence  $\\{L_n^{(\\mathbf{x})}\\}$ ), i.e.,",
+        "bbox": [
+            169,
+            588,
+            826,
+            708
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\lim  _ {n} \\sup  \\frac {\\left\\| L _ {n} ^ {(\\boldsymbol {\\theta})} \\right\\|}{\\sqrt {\\beta_ {n} \\log (u _ {n})}} \\leq C _ {\\Omega} ^ {(\\boldsymbol {\\theta})}, \\tag {68a}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            392,
+            715,
+            825,
+            755
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\lim  _ {n} \\sup  \\frac {\\| L _ {n} ^ {(\\mathbf {x})} \\|}{\\sqrt {\\gamma_ {n} \\log (s _ {n})}} \\leq C _ {\\Omega} ^ {(\\mathbf {x})}, \\tag {68b}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            393,
+            761,
+            823,
+            800
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "such that  $\\| L_n^{(\\pmb{\\theta})}\\| = O\\big(\\sqrt{\\beta_n\\log(u_n)}\\big)$  a.s. and  $\\| L_{n}^{(\\mathbf{x})}\\| = O\\big(\\sqrt{\\gamma_{n}\\log(s_{n})}\\big)$  a.s. which completes the proof.",
+        "bbox": [
+            169,
+            803,
+            826,
+            837
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Note that we have  $\\mathbf{x}_n - \\boldsymbol{\\mu}$  and  $L_{n}^{(\\mathbf{x})}$  weakly converge to the same Gaussian distribution from Remark E.1 and (65). Then,  $\\gamma_n^{-1 / 2}\\Delta_n^{(\\mathbf{x})}$  weakly converges to zero, implying that  $\\gamma_n^{-1 / 2}\\Delta_n^{(\\mathbf{x})}$  converges to zero with probability 1. Therefore, together with  $\\{\\gamma_n\\}$  being strictly positive, we have",
+        "bbox": [
+            169,
+            852,
+            825,
+            901
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\Delta_ {n} ^ {(\\mathbf {x})} = o \\left(\\sqrt {\\gamma_ {n}}\\right) \\quad a. s. \\tag {69}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            421,
+            907,
+            823,
+            926
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "page_number",
+        "text": "30",
+        "bbox": [
+            488,
+            946,
+            509,
+            959
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Characterization of Sequences  $\\{R_n^{(\\theta)}\\}$  and  $\\{\\Delta_n^{(\\theta)}\\}$",
+        "bbox": [
+            171,
+            101,
+            522,
+            119
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "text",
+        "text": "We first consider the sequence  $\\{R_n^{(\\theta)}\\}$ . We assume a positive real-valued bounded sequence  $\\{w_{n}\\}$  under the same conditions as in Mokkadem & Pelletier (2006, Definition 1), i.e.,",
+        "bbox": [
+            169,
+            128,
+            823,
+            157
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "text",
+        "text": "Definition E.1. In the case  $b < 1$ ,  $\\frac{w_n}{w_{n+1}} = 1 + o(\\beta_n)$ , which also implies  $\\frac{w_n}{w_{n+1}} = 1 + o(\\gamma_n)$ .",
+        "bbox": [
+            169,
+            159,
+            789,
+            179
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "text",
+        "text": "In the case  $b = 1$ , there exist  $\\epsilon \\geq 0$  and a nondecreasing slowly varying function  $l(n)$  such that  $w_{n} = n^{-\\epsilon}l(n)$ . When  $\\epsilon = 0$ , we require function  $l(n)$  to be bounded.",
+        "bbox": [
+            169,
+            184,
+            825,
+            214
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "text",
+        "text": "Since  $\\| \\mathbf{x}_n - \\boldsymbol{\\mu} \\| = o(1)$  by a.s. convergence result, we can assume that there exists  $\\{w_n\\}$  such that  $\\| \\mathbf{x}_n - \\boldsymbol{\\mu} \\| = O(w_n)$ . Then, from (55b), we can use the Abel transformation and obtain",
+        "bbox": [
+            169,
+            223,
+            823,
+            253
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} R _ {n} ^ {(\\boldsymbol {\\theta})} = \\beta_ {n} \\gamma_ {n} ^ {- 1} \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (\\mathbf {x} _ {n} - \\boldsymbol {\\mu}) - e ^ {(u _ {n} - u _ {1}) \\mathbf {J} _ {1 1}} \\beta_ {1} \\gamma_ {1} ^ {- 1} \\mathbf {U} _ {1 1} \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (\\mathbf {x} _ {1} - \\boldsymbol {\\mu}) \\\\ + e ^ {u _ {n} \\mathbf {J} _ {1 1}} \\sum_ {k = 1} ^ {n - 1} \\left(e ^ {- u _ {k} \\mathbf {J} _ {1 1}} \\beta_ {k} \\gamma_ {k} ^ {- 1} - e ^ {- u _ {k + 1} \\mathbf {J} _ {1 1}} \\beta_ {k + 1} \\gamma_ {k + 1} ^ {- 1}\\right) \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (\\mathbf {x} _ {k + 1} - \\boldsymbol {\\mu}), \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            192,
+            256,
+            805,
+            318
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "text",
+        "text": "where the last term on the RHS can be rewritten as",
+        "bbox": [
+            171,
+            320,
+            509,
+            334
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nW _ {n} = \\sum_ {k = 1} ^ {n - 1} e ^ {(u _ {n} - u _ {k + 1}) \\mathbf {J} _ {1 1}} \\beta_ {k + 1} \\gamma_ {k + 1} ^ {- 1} \\left(e ^ {\\beta_ {k + 1} \\mathbf {J} _ {1 1}} \\beta_ {k} \\beta_ {k + 1} ^ {- 1} \\gamma_ {k} ^ {- 1} \\gamma_ {k + 1} - \\mathbf {I}\\right) \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (\\mathbf {x} _ {k + 1} - \\boldsymbol {\\mu}).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            176,
+            335,
+            818,
+            377
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "text",
+        "text": "Using Lemma G.6 on  $W_{n}$  gives  $\\| W_n\\| = O(\\gamma_n^{-1}\\| e^{\\beta_n\\mathbf{J}_{11}} - \\mathbf{I}\\| \\| \\mathbf{x}_n - \\pmb {\\mu}\\|) = O(\\gamma_n^{-1}\\beta_n\\omega_n)$ . Then, it follows that for some  $T > 0$",
+        "bbox": [
+            169,
+            380,
+            823,
+            409
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left\\| R _ {n} ^ {(\\boldsymbol {\\theta})} \\right\\| = O \\left(\\beta_ {n} \\gamma_ {n} ^ {- 1} \\omega_ {n} + \\left\\| e ^ {u _ {n} \\mathbf {J} _ {1 1}} \\right\\|\\right) = O \\left(\\beta_ {n} \\gamma_ {n} ^ {- 1} \\omega_ {n} + e ^ {- u _ {n} T}\\right) \\tag {70}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            287,
+            411,
+            823,
+            430
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "text",
+        "text": "with the application of Lemma G.4 to the second equality.",
+        "bbox": [
+            171,
+            431,
+            555,
+            448
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "text",
+        "text": "Then, we shift our focus on  $\\{\\Delta_n^{(\\theta)}\\}$ . Specifically, we take (54), (55a), and (56) back to  $\\Delta_{n}^{(\\theta)} = \\pmb{\\theta}_{n} - \\pmb{\\theta}^{*} - L_{n}^{(\\theta)} - R_{n}^{(\\theta)}$ , and obtain",
+        "bbox": [
+            169,
+            453,
+            825,
+            488
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\Delta_ {n + 1} ^ {(\\boldsymbol {\\theta})} = (\\mathbf {I} + \\beta_ {n + 1} \\mathbf {J} _ {1 1}) (\\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*}) \\\\ + \\beta_ {n + 1} \\left(r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} + r _ {n} ^ {(\\boldsymbol {\\theta}, 2)} + \\eta_ {n} ^ {(\\boldsymbol {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(r _ {n} ^ {(\\mathbf {x}, 1)} + r _ {n} ^ {(\\mathbf {x}, 2)} + \\eta_ {n} ^ {(\\mathbf {x})}\\right)\\right) \\\\ - e ^ {\\beta_ {n + 1} \\mathbf {J} _ {1 1}} L _ {n} ^ {(\\boldsymbol {\\theta})} - e ^ {\\beta_ {n + 1} \\mathbf {J} _ {1 1}} R _ {n} ^ {(\\boldsymbol {\\theta})} \\\\ = \\left(\\mathbf {I} + \\beta_ {n + 1} \\mathbf {J} _ {1 1}\\right) \\left(\\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*}\\right) + \\beta_ {n + 1} \\left(r _ {x} ^ {(\\boldsymbol {\\theta}, 1)} + r _ {x} ^ {(\\boldsymbol {\\theta}, 2)} + \\eta_ {x} ^ {(\\boldsymbol {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(r _ {x} ^ {(\\mathbf {x}, 1)} + r _ {x} ^ {(\\mathbf {x}, 2)} + \\eta_ {x} ^ {(\\mathbf {x})}\\right)\\right) \\tag {71} \\\\ - \\left(\\mathbf {I} + \\beta_ {n + 1} \\mathbf {J} _ {1 1} + O \\left(\\beta_ {n + 1} ^ {2}\\right)\\right) L _ {n} ^ {(\\boldsymbol {\\theta})} - \\left(\\mathbf {I} + \\beta_ {n + 1} \\mathbf {J} _ {1 1} + O \\left(\\beta_ {n + 1} ^ {2}\\right)\\right) R _ {n} ^ {(\\boldsymbol {\\theta})} \\\\ = (\\mathbf {I} + \\beta_ {n + 1} \\mathbf {J} _ {1 1}) \\Delta_ {n} ^ {(\\pmb {\\theta})} + O (\\beta_ {n + 1} ^ {2}) (L _ {n} ^ {(\\pmb {\\theta})} + R _ {n} ^ {(\\pmb {\\theta}))}) \\\\ + \\beta_ {n + 1} (r _ {n} ^ {(\\pmb {\\theta}, 1)} + r _ {n} ^ {(\\pmb {\\theta}, 2)} + \\eta_ {n} ^ {(\\pmb {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (r _ {n} ^ {(\\mathbf {x}, 1)} + r _ {n} ^ {(\\mathbf {x}, 2)} + \\eta_ {n} ^ {(\\mathbf {x}))}), \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            207,
+            492,
+            823,
+            655
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "text",
+        "text": "where the second equality is by taking the Taylor expansion  $e^{\\beta_{n + 1}\\mathbf{J}_{11}} = \\mathbf{I} + \\beta_{n + 1}\\mathbf{J}_{11} + O(\\beta_{n + 1}^2)$ .",
+        "bbox": [
+            169,
+            657,
+            818,
+            676
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "text",
+        "text": "Define  $\\Phi_{k,n} \\triangleq \\prod_{j=k+1}^{n} (\\mathbf{I} + \\beta_j \\mathbf{J}_{11})$  and by convention  $\\Phi_{n,n} = \\mathbf{I}$ . Then, we rewrite (71) as",
+        "bbox": [
+            169,
+            681,
+            771,
+            700
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\Delta_ {n + 1} ^ {(\\boldsymbol {\\theta})} = \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(O (\\beta_ {k + 1}) L _ {k} ^ {(\\boldsymbol {\\theta})} + O (\\beta_ {k + 1}) R _ {k} ^ {(\\boldsymbol {\\theta})}\\right) \\\\ + \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} (r _ {k} ^ {(\\boldsymbol {\\theta}, 1)} + r _ {k} ^ {(\\boldsymbol {\\theta}, 2)} + \\eta_ {k} ^ {(\\boldsymbol {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (r _ {k} ^ {(\\mathbf {x}, 1)} + r _ {k} ^ {(\\mathbf {x}, 2)} + \\eta_ {k} ^ {(\\mathbf {x}))}) \\\\ = \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(O \\left(\\beta_ {k + 1}\\right) L _ {k} ^ {(\\boldsymbol {\\theta})} + O \\left(\\beta_ {k + 1}\\right) R _ {k} ^ {(\\boldsymbol {\\theta})}\\right) \\\\ + \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} (r _ {k} ^ {(\\pmb {\\theta}, 1)} + \\eta_ {k} ^ {(\\pmb {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (r _ {k} ^ {(\\mathbf {x}, 1)} + \\eta_ {k} ^ {(\\mathbf {x})})) \\\\ + \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(r _ {k} ^ {(\\boldsymbol {\\theta}, 2)} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} r _ {k} ^ {(\\mathbf {x}, 2)}\\right). \\tag {72} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            194,
+            705,
+            823,
+            926
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "page_number",
+        "text": "31",
+        "bbox": [
+            488,
+            946,
+            506,
+            959
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "text",
+        "text": "From (72), we can indeed decompose  $\\Delta_{n+1}^{(\\boldsymbol{x})}$  into two parts  $\\Delta_{n+1}^{(\\boldsymbol{\\theta})} = \\Delta_{n+1}^{(\\boldsymbol{\\theta}, 1)} + \\Delta_{n+1}^{(\\boldsymbol{\\theta}, 2)}$ , where",
+        "bbox": [
+            169,
+            99,
+            777,
+            121
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\Delta_ {n + 1} ^ {(\\boldsymbol {\\theta}, 1)} \\triangleq \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(O \\left(\\beta_ {k + 1}\\right) L _ {k} ^ {(\\boldsymbol {\\theta})} + O \\left(\\beta_ {k + 1}\\right) R _ {k} ^ {(\\boldsymbol {\\theta})}\\right) (73a) \\\\ + \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} (r _ {k} ^ {(\\pmb {\\theta}, 1)} + \\eta_ {k} ^ {(\\pmb {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (r _ {k} ^ {(\\mathbf {x}, 1)} + \\eta_ {k} ^ {(\\mathbf {x})})), \\\\ \\Delta_ {n + 1} ^ {(\\boldsymbol {\\theta}, 2)} \\triangleq \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(r _ {k} ^ {(\\boldsymbol {\\theta}, 2)} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} r _ {k} ^ {(\\mathbf {x}, 2)}\\right). (73b) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            248,
+            128,
+            823,
+            255
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "text",
+        "text": "This term  $\\Delta_{n+1}^{(\\theta,1)}$  shares the same recursive form as in the sequence defined in Mokkadem & Pelletier (2006, Lemma 6), which is given below.",
+        "bbox": [
+            169,
+            260,
+            823,
+            292
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "text",
+        "text": "Lemma E.7 (Mokkadem & Pelletier (2006) Lemma 6). For  $\\Delta_{n+1}^{(\\theta,1)}$  in the form of (73a), assume  $\\| \\mathbf{x}_n - \\boldsymbol{\\mu} \\| = O(\\omega_n)$  and  $\\| \\Delta_n^{(\\mathbf{x})} \\| = O(\\delta_n)$  for the sequences  $\\omega_n, \\delta_n$  defined in (E.1). Then, we have",
+        "bbox": [
+            169,
+            297,
+            825,
+            334
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\| \\Delta_ {n + 1} ^ {(\\boldsymbol {\\theta}, 1)} \\| = O \\left(\\beta_ {n} ^ {2} \\gamma_ {n} ^ {- 2} \\omega_ {n} ^ {2} + \\beta_ {n} \\gamma_ {n} ^ {- 1} \\delta_ {n}\\right) + o (\\sqrt {\\beta_ {n}}) \\quad a. s.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            313,
+            339,
+            681,
+            359
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "text",
+        "text": "Since we already have  $\\Delta_n^{(\\mathbf{x})} = o(\\sqrt{\\gamma_n})$  in (69), together with Lemma E.7, we have",
+        "bbox": [
+            169,
+            376,
+            717,
+            393
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\| \\Delta_ {n + 1} ^ {(\\boldsymbol {\\theta}, 1)} \\| = O (\\beta_ {n} ^ {2} \\gamma_ {n} ^ {- 2} \\omega_ {n} ^ {2}) + o (\\beta_ {n} \\gamma_ {n} ^ {- 1 / 2}) + o (\\sqrt {\\beta_ {n}}) = O (\\beta_ {n} ^ {2} \\gamma_ {n} ^ {- 2} \\omega_ {n} ^ {2}) + o (\\sqrt {\\beta_ {n}})\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            230,
+            400,
+            764,
+            421
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "text",
+        "text": "where the second equality comes from  $o(\\beta_n\\gamma_n^{-1 / 2}) = o(\\beta_n^{1 / 2}(\\beta_n\\gamma_n^{-1})^{1 / 2}) = o(\\beta_n^{1 / 2}).$",
+        "bbox": [
+            169,
+            428,
+            738,
+            448
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "text",
+        "text": "We now focus on  $\\Delta_{n + 1}^{(\\theta ,2)}$ . Define a sequence",
+        "bbox": [
+            171,
+            453,
+            465,
+            472
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\Psi_ {n} \\triangleq \\sum_ {k = 1} ^ {n} r _ {k} ^ {(\\boldsymbol {\\theta}, 2)} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} r _ {k} ^ {(\\mathbf {x}, 2)}, \\tag {74}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            356,
+            479,
+            823,
+            518
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "text",
+        "text": "and we have",
+        "bbox": [
+            171,
+            526,
+            258,
+            537
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\beta_ {n + 1} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(r _ {k} ^ {\\left(\\boldsymbol {\\theta}, 2\\right)} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} r _ {k} ^ {\\left(\\mathbf {x}, 2\\right)}\\right) \\\\ = \\beta_ {n + 1} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} (\\Psi_ {k} - \\Psi_ {k - 1}) \\\\ = \\beta_ {n + 1} ^ {1 / 2} \\Psi_ {n} + \\beta_ {n + 1} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n - 1} \\left(\\beta_ {k} \\Phi_ {k, n} - \\beta_ {k + 1} \\Phi_ {k + 1, n}\\right) \\Psi_ {k} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            546,
+            681,
+            670
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "text",
+        "text": "where the last equality comes from the Abel transformation. Note that",
+        "bbox": [
+            171,
+            676,
+            635,
+            691
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\left\\| \\beta_ {k} \\Phi_ {k, n} - \\beta_ {k + 1} \\Phi_ {k + 1, n} \\right\\| \\leq \\beta_ {k + 1} \\left\\| \\Phi_ {k, n} - \\Phi_ {k + 1, n} \\right\\| + \\left(\\beta_ {k} - \\beta_ {k + 1}\\right) \\left\\| \\Phi_ {k, n} \\right\\| \\\\ \\leq \\beta_ {k + 1} \\| \\Phi_ {k + 1, n} \\| \\beta_ {k} \\| \\mathbf {J} _ {1 1} \\| + C _ {7} \\beta_ {k} ^ {2} \\| \\Phi_ {k, n} \\| \\\\ \\leq C _ {8} \\beta_ {k} ^ {2} e ^ {- (u _ {n} - u _ {k}) T} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            254,
+            698,
+            741,
+            753
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "text",
+        "text": "for some constant  $C_7, C_8 > 0$ , where the last inequality is from Lemma G.4 and  $\\| \\Phi_{k+1,n} \\| \\leq C_9 \\| \\Phi_{k,n} \\|$  for some constant  $C_9 > 0$  that depends on  $e^{\\beta_0 T}$ . Then,",
+        "bbox": [
+            169,
+            762,
+            823,
+            792
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\beta_ {n + 1} ^ {- 1 / 2} \\left\\| \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(r _ {k} ^ {\\left(\\boldsymbol {\\theta}, 2\\right)} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} r _ {k} ^ {\\left(\\mathbf {x}, 2\\right)}\\right) \\right\\| \\\\ \\leq \\| \\beta_ {n + 1} ^ {1 / 2} \\Psi_ {n} \\| + \\left(\\frac {\\beta_ {n + 1}}{\\beta_ {n}}\\right) ^ {1 / 2} \\beta_ {n} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} \\| \\beta_ {k} \\Phi_ {k, n} - \\beta_ {k + 1} \\Phi_ {k + 1, n} \\| \\| \\Psi_ {k} \\| \\\\ \\leq \\| \\beta_ {n + 1} ^ {1 / 2} \\Psi_ {n} \\| + C _ {8} \\left(\\frac {\\beta_ {n + 1}}{\\beta_ {n}}\\right) ^ {1 / 2} \\beta_ {n} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} \\beta_ {k} ^ {3 / 2} e ^ {- (u _ {n} - u _ {k}) T} \\| \\beta_ {k} ^ {1 / 2} \\Psi_ {k} \\|. \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            261,
+            801,
+            733,
+            926
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "page_number",
+        "text": "32",
+        "bbox": [
+            488,
+            948,
+            508,
+            959
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "text",
+        "text": "By Lemma E.1, we have  $\\beta_n^{1/2} \\Psi_n \\to 0$  a.s. such that by Lemma G.6, it follows that",
+        "bbox": [
+            169,
+            102,
+            718,
+            119
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\lim _ {n} \\sup  _ {\\boldsymbol {n}} \\beta_ {n + 1} ^ {- 1 / 2} \\left\\| \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} (r _ {k} ^ {(\\boldsymbol {\\theta}, 2)} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} r _ {k} ^ {(\\mathbf {x}, 2)}) \\right\\| \\leq \\frac {\\lim  \\sup  _ {n} \\| \\beta_ {n} ^ {1 / 2} \\Psi_ {n} \\|}{C (T , 1 / 2)} = 0.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            187,
+            126,
+            808,
+            167
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "Therefore, we have",
+        "bbox": [
+            171,
+            175,
+            303,
+            189
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\Delta_ {n + 1} ^ {(\\boldsymbol {\\theta}, 2)} = \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(r _ {k} ^ {(\\boldsymbol {\\theta}, 2)} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} r _ {k} ^ {(\\mathbf {x}, 2)}\\right) = o \\left(\\sqrt {\\beta_ {n}}\\right). \\tag {75}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            267,
+            196,
+            823,
+            237
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "Consequently,  $\\Delta_{n + 1}^{(\\theta)} = O(\\beta_n^2\\gamma_n^{-2}\\omega_n^2) + o(\\sqrt{\\beta_n})$  almost surely.",
+        "bbox": [
+            169,
+            253,
+            591,
+            273
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "Now we are dealing with  $\\mathbf{x}_n - \\pmb{\\mu}$  and its related sequence  $\\omega_{n}$ . Note that by Lemma E.5 and (69), we have almost surely,",
+        "bbox": [
+            169,
+            277,
+            823,
+            306
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\left\\| \\mathbf {x} _ {n} - \\boldsymbol {\\mu} \\right\\| = O \\left(\\left\\| L _ {n} ^ {(\\mathbf {x})} \\right\\| + \\left\\| \\Delta_ {n} ^ {\\mathbf {x}} \\right\\|\\right) \\\\ = O \\left(\\sqrt {\\gamma_ {n} \\log \\left(s _ {n}\\right)} + o \\left(\\sqrt {\\gamma_ {n}}\\right)\\right) \\tag {76} \\\\ = O \\left(\\sqrt {\\gamma_ {n} \\log \\left(s _ {n}\\right)}\\right). \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            314,
+            823,
+            376
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "Thus, we can set  $\\omega_{n}\\equiv O(\\sqrt{\\gamma_{n}\\log(s_{n})})$  such that  $\\| R_n^{(\\pmb {\\theta})}\\|$  in (70) can be written as",
+        "bbox": [
+            169,
+            385,
+            718,
+            404
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left\\| R _ {n} ^ {(\\boldsymbol {\\theta})} \\right\\| = O \\left(n ^ {a / 2 - b} \\sqrt {\\log \\left(s _ {n}\\right)} + e ^ {- u _ {n} T}\\right),\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            357,
+            412,
+            638,
+            431
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "and",
+        "bbox": [
+            171,
+            438,
+            200,
+            450
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left\\| \\Delta_ {n + 1} ^ {(\\boldsymbol {\\theta})} \\right\\| = O \\left(n ^ {a - 2 b} \\log \\left(s _ {n}\\right)\\right) + o \\left(\\sqrt {\\beta_ {n}}\\right).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            449,
+            635,
+            469
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "In view of assumption A2 and  $\\beta_{n} = o(\\gamma_{n})$ ,  $a / 2 - b < -b / 2$  and  $a - 2b < -b$ , there exists a  $c > b / 2$  such that almost surely,",
+        "bbox": [
+            169,
+            472,
+            823,
+            502
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left\\| R _ {n} ^ {\\left(\\boldsymbol {\\theta}\\right)} \\right\\| = O \\left(n ^ {- s}\\right), \\quad \\left\\| \\Delta_ {n + 1} ^ {\\left(\\boldsymbol {\\theta}\\right)} \\right\\| = o (\\sqrt {\\beta_ {n}}).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            354,
+            508,
+            640,
+            530
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "Therefore,  $\\beta_{n}^{-1 / 2}(R_{n}^{(\\pmb{\\theta})} + \\Delta_{n + 1}^{(\\pmb{\\theta})})\\to 0$  almost surely. This completes the proof of Scenario 2.",
+        "bbox": [
+            169,
+            537,
+            782,
+            558
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "E.3 CASE (III):  $\\gamma_{n} = o(\\beta_{n})$",
+        "text_level": 1,
+        "bbox": [
+            171,
+            573,
+            375,
+            589
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "For  $\\gamma_{n} = o(\\beta_{n})$ , we can see that the roles of  $\\theta_{n}$  and  $\\mathbf{x}_n$  are flipped, i.e.,  $\\theta_{n}$  is now on fast timescale while  $\\mathbf{x}_n$  is on slow timescale.",
+        "bbox": [
+            169,
+            599,
+            823,
+            628
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "We still decompose  $\\mathbf{x}_n$  as  $\\mathbf{x}_n - \\boldsymbol{\\mu} = L_n^{(\\mathbf{x})} + \\Delta_n^{(\\mathbf{x})}$ , where  $L_{n}^{(\\mathbf{x})},\\Delta_{n}^{(\\mathbf{x})}$  are defined in (56) and (57), respectively. Since  $\\mathbf{x}_n$  is independent of  $\\theta_{n}$ , the results of  $L_{n}^{(\\mathbf{x})}$  and  $\\Delta_{n}^{(\\mathbf{x})}$  remain the same, i.e., almost surely,  $L_{n}^{(\\mathbf{x})} = O\\bigl (\\sqrt{\\gamma_{n}\\log(s_{n})}\\bigr)$  from Lemma E.5 and  $\\Delta_n^{(\\mathbf{x})} = o(\\sqrt{\\gamma_n})$  from (69). Then, we define sequences  $\\hat{L}_n^{(\\pmb {\\theta})}$  and  $\\hat{R}_n^{(\\pmb {\\theta})}$  as follows.",
+        "bbox": [
+            169,
+            635,
+            825,
+            705
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\hat {L} _ {n} ^ {(\\boldsymbol {\\theta})} \\triangleq e ^ {\\beta_ {n} \\mathbf {J} _ {1 1}} \\hat {L} _ {n - 1} ^ {(\\boldsymbol {\\theta})} + \\beta_ {n} M _ {n} ^ {(\\boldsymbol {\\theta})} = \\sum_ {k = 1} ^ {n} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\beta_ {k} M _ {k} ^ {(\\boldsymbol {\\theta})}, \\tag {77a}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            299,
+            714,
+            823,
+            753
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\hat {R} _ {n} ^ {(\\boldsymbol {\\theta})} \\triangleq e ^ {\\beta_ {n} \\mathbf {J} _ {1 1}} \\hat {R} _ {n - 1} ^ {(\\boldsymbol {\\theta})} + \\beta_ {n} \\mathbf {J} _ {1 2} (\\alpha) \\left(L _ {n - 1} ^ {(\\mathbf {x})} + R _ {n - 1} ^ {(\\mathbf {x})}\\right) = \\sum_ {k = 1} ^ {n} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} \\beta_ {k} \\mathbf {J} _ {1 2} (\\alpha) \\left(L _ {k - 1} ^ {(\\mathbf {x})} + R _ {k - 1} ^ {(\\mathbf {x})}\\right). \\tag {77b}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            763,
+            823,
+            814
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "Moreover, the remaining term  $\\hat{\\Delta}_n^{(\\pmb {\\theta})}\\triangleq \\pmb {\\theta}_n - \\pmb {\\theta}^* -\\hat{L}_n^{(\\pmb {\\theta})} - \\hat{R}_n^{(\\pmb {\\theta})}.$",
+        "bbox": [
+            169,
+            813,
+            589,
+            832
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "The proof outline is the same as in the previous scenario:",
+        "bbox": [
+            171,
+            838,
+            547,
+            853
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- We first show  $\\beta_n^{-1/2} \\hat{\\Delta}_n^{(\\theta)}$  weakly converges to the distribution  $N(0, \\mathbf{V}_{\\pmb{\\theta}}^{(3)})(\\alpha)$ ;",
+            "- We analyse  $\\hat{L}_n^{(\\theta)}$  and  $\\hat{R}_n^{(\\theta)}$  to ensure that these two terms decrease faster than the CLT scale  $\\beta_n^{-1/2}$ , i.e.,  $\\lim_{n\\to\\infty}\\beta_n^{-1/2}(\\hat{L}_n^{(\\theta)} - \\hat{R}_n^{(\\theta)}) = 0$ ;"
+        ],
+        "bbox": [
+            215,
+            864,
+            823,
+            925
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "page_number",
+        "text": "33",
+        "bbox": [
+            488,
+            946,
+            506,
+            959
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "- With above two steps, we can show that  $\\beta_{n}^{-1 / 2}(\\pmb{\\theta}_{n} - \\pmb{\\theta}^{*})$  weakly converges to the distribution  $N(0,\\mathbf{V}_{\\pmb{\\theta}}^{(3)})(\\alpha)$ .",
+        "bbox": [
+            215,
+            102,
+            823,
+            138
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "Analysis of  $\\hat{L}_n^{(\\theta)}$",
+        "bbox": [
+            171,
+            148,
+            287,
+            167
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "We first focus on  $\\hat{L}_n^{(\\theta)}$  and follow similar steps as we did when we analysed  $L_{n}^{(\\theta)}$  in the previous scenario. We set a Martingale  $Z^{(n)} = \\{Z_k^{(n)}\\}_{k\\geq 1}$  such that",
+        "bbox": [
+            169,
+            172,
+            823,
+            210
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nZ _ {k} ^ {(n)} = \\beta_ {n} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1} \\beta_ {k} M _ {k} ^ {(\\boldsymbol {\\theta})}}.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            369,
+            217,
+            625,
+            255
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "Then,",
+        "bbox": [
+            171,
+            262,
+            215,
+            275
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nA _ {n} \\triangleq \\sum_ {k = 1} ^ {n} \\mathbb {E} \\left[ (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) ^ {T} \\Big | \\mathcal {F} _ {k - 1} \\right].\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            313,
+            275,
+            679,
+            314
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "Following the similar steps in (59) to decompose  $M_{k}^{(\\pmb{\\theta})}$  with (42b), we have",
+        "bbox": [
+            169,
+            318,
+            671,
+            338
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} A _ {n} = \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\beta_ {k} ^ {2} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\left(\\mathbf {U} _ {1 1} + \\mathbf {D} _ {k} ^ {(3)} + \\mathbf {J} _ {k} ^ {(3)}\\right) e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1} ^ {T}} \\\\ = \\underbrace {\\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\beta_ {k} ^ {2} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\mathbf {U} _ {1 1} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1} ^ {T}}} _ {A _ {n} ^ {(a)}} + \\underbrace {\\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\beta_ {k} ^ {2} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\mathbf {D} _ {k} ^ {(3)} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1} ^ {T}}} _ {A _ {n} ^ {(b)}} \\tag {78} \\\\ + \\underbrace {\\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\beta_ {k} ^ {2} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\mathbf {J} _ {k} ^ {(3)} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1} ^ {T}}} _ {A _ {n} ^ {(c)}} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            187,
+            344,
+            823,
+            513
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "Since  $A_{n}^{(a)}, A_{n}^{(b)}, A_{n}^{(c)}$  share similar forms as in Lemma E.3, we follow the same steps as the proof therein, with the application of Lemma E.1. To avoid repetition, we omit the proof and directly give the following lemma.",
+        "bbox": [
+            169,
+            521,
+            826,
+            566
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "Lemma E.8. For  $A_{n}^{(a)}, A_{n}^{(b)}, A_{n}^{(c)}$  defined in (78), we have",
+        "bbox": [
+            169,
+            570,
+            563,
+            589
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\lim  _ {n \\rightarrow \\infty} A _ {n} ^ {(a)} = \\mathbf {V} _ {\\boldsymbol {\\theta}} ^ {(3)} (\\alpha), \\quad \\lim  _ {n \\rightarrow \\infty} \\| A _ {n} ^ {(b)} \\| = 0, \\quad \\lim  _ {n \\rightarrow \\infty} \\| A _ {n} ^ {(c)} \\| = 0, \\tag {79}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            285,
+            595,
+            823,
+            619
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "where  $\\mathbf{V}_{\\pmb{\\theta}}^{(3)}(\\alpha)$  is the solution to the Lyapunov equation",
+        "bbox": [
+            169,
+            627,
+            540,
+            646
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {J} _ {1 1} \\mathbf {V} + \\mathbf {V} \\mathbf {J} _ {1 1} ^ {T} + \\mathbf {U} _ {1 1} = 0.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            405,
+            652,
+            589,
+            670
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "Note that here we don't have the term  $\\frac{\\mathbb{1}_{\\{b = 1\\}}}{2}\\pmb{I}$  in above lemma, compared to Lemma E.3, because in the case of  $\\gamma_{n} = o(\\beta_{n})$ ,  $b < 1$  such that  $\\mathbb{1}_{\\{b = 1\\}} = 0$ . Then, applying Lemma G.1 to derive the closed form of  $\\mathbf{V}_{\\pmb{\\theta}}^{(3)}(\\alpha)$  gives",
+        "bbox": [
+            169,
+            685,
+            823,
+            736
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {V} _ {\\boldsymbol {\\theta}} ^ {(3)} (\\alpha) = \\int_ {0} ^ {\\infty} e ^ {t \\nabla_ {\\boldsymbol {\\theta}} \\mathbf {h} (\\boldsymbol {\\theta} ^ {*})} \\mathbf {U} _ {1 1} e ^ {t \\nabla_ {\\boldsymbol {\\theta}} \\mathbf {h} (\\boldsymbol {\\theta} ^ {*})} d t.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            354,
+            742,
+            638,
+            763
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "Thus, it follows that",
+        "bbox": [
+            171,
+            768,
+            307,
+            781
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\lim  _ {n \\rightarrow \\infty} \\sum_ {k = 1} ^ {n} \\mathbb {E} \\left[ (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) ^ {T} | \\mathcal {F} _ {k - 1} \\right] = \\mathbf {V} _ {\\boldsymbol {\\theta}} ^ {(3)} (\\alpha).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            282,
+            789,
+            712,
+            827
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "Again, we use the Martingale CLT result in Theorem G.3 and have the following result.",
+        "bbox": [
+            169,
+            833,
+            746,
+            849
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nZ ^ {n} = \\beta_ {n} ^ {- 1 / 2} \\hat {L} _ {n} ^ {(\\boldsymbol {\\theta})} \\xrightarrow [ d i s t . ]{n \\to \\infty} N \\left(0, \\mathbf {V} _ {\\boldsymbol {\\theta}} ^ {(3)} (\\alpha)\\right).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            854,
+            640,
+            882
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "Moreover, similar to the tighter upper bound of  $L_{n}^{(\\mathbf{x})}$  proved in Lemma E.5, we utilize the tighter upper bound Lemma E.6 in the proof thereof, and obtain  $\\hat{L}_n^{(\\pmb{\\theta})} = O(\\sqrt{\\beta_n\\log(u_n)})$ .",
+        "bbox": [
+            169,
+            891,
+            823,
+            926
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "page_number",
+        "text": "34",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "Analysis of  $\\hat{R}_n^{(\\theta)}$",
+        "bbox": [
+            171,
+            99,
+            287,
+            119
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "text",
+        "text": "Next, we turn to the term  $\\hat{R}_n^{(\\theta)}$  in (77b). Taking the norm gives the following inequality for some constant  $C, T > 0$  by applying Lemma G.4,",
+        "bbox": [
+            169,
+            126,
+            823,
+            159
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\| \\hat {R} _ {n} ^ {(\\boldsymbol {\\theta})} \\| \\leq C \\sum_ {k = 1} ^ {n} e ^ {- (u _ {n} - u _ {k}) T} \\beta_ {k} (\\| L _ {k - 1} ^ {(\\mathbf {x})} \\| + \\| R _ {k - 1} ^ {(\\mathbf {x})} \\)).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            318,
+            166,
+            676,
+            205
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "text",
+        "text": "Using Lemma G.6 gives",
+        "bbox": [
+            171,
+            212,
+            336,
+            227
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\sum_ {k = 1} ^ {n} e ^ {- (u _ {n} - u _ {k}) T} \\beta_ {k} (\\| L _ {k - 1} ^ {(\\mathbf {x})} \\| + \\| R _ {k - 1} ^ {(\\mathbf {x})} \\|) = O (\\| L _ {k - 1} ^ {(\\mathbf {x})} \\| + \\| R _ {n - 1} ^ {(\\mathbf {x})} \\)).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            274,
+            234,
+            720,
+            273
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "text",
+        "text": "Thus,  $\\beta_n^{-1/2} \\|\\hat{R}_n^{(\\pmb{\\theta})}\\| = o(\\sqrt{\\gamma_n \\beta_n^{-1}}) + O\\left(\\sqrt{\\gamma_n \\beta_n^{-1} \\log(s_n)}\\right)$ . Since  $\\gamma_n = o(\\beta_n), \\gamma_n \\beta_n^{-1} = (n + 1)^{b - a}$ , where  $b - a < 0$ . Then, there exists some  $s > 0$  such that  $b - a < -s < 0$ . Together with  $\\log(s_n) = O(\\log(n))$ , we have  $O\\left(\\sqrt{\\gamma_n \\beta_n^{-1} \\log(s_n)}\\right) = O(\\sqrt{n^{-s} \\log(n)}) = o(1)$ . Therefore, we have",
+        "bbox": [
+            169,
+            282,
+            823,
+            372
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\lim _ {n \\to \\infty} \\beta_ {n} ^ {- 1 / 2} \\hat {R} _ {n} ^ {(\\boldsymbol {\\theta})} = 0.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            423,
+            372,
+            571,
+            395
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "text",
+        "text": "Analysis of  $\\hat{\\Delta}_n^{(\\theta)}$",
+        "bbox": [
+            171,
+            407,
+            287,
+            425
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "text",
+        "text": "Lastly, let's focus on the term  $\\hat{\\Delta}_n^{(\\theta)}$ . We have",
+        "bbox": [
+            171,
+            431,
+            470,
+            450
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\hat {\\Delta} _ {n + 1} ^ {(\\boldsymbol {\\theta})} = \\boldsymbol {\\theta} _ {n + 1} - \\boldsymbol {\\theta} ^ {*} - \\hat {L} _ {n + 1} ^ {(\\boldsymbol {\\theta})} - \\hat {R} _ {n + 1} ^ {(\\boldsymbol {\\theta})} \\\\ = \\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*} + \\beta_ {n + 1} \\left(\\mathbf {J} _ {1 1} (\\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*}) + \\mathbf {J} _ {1 2} (\\alpha) (\\mathbf {x} _ {n} - \\boldsymbol {\\mu}) + M _ {n + 1} ^ {(\\boldsymbol {\\theta})} + r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} + r _ {n} ^ {(\\boldsymbol {\\theta}, 2)} + \\eta_ {n} ^ {(\\boldsymbol {\\theta})}\\right) \\\\ - e ^ {\\beta_ {n + 1} \\mathbf {J} _ {1 1}} \\hat {L} _ {n} ^ {(\\boldsymbol {\\theta})} - \\beta_ {n + 1} M _ {n + 1} ^ {(\\boldsymbol {\\theta})} - e ^ {\\beta_ {n + 1} \\mathbf {J} _ {1 1}} \\hat {R} _ {n} ^ {(\\boldsymbol {\\theta})} - \\beta_ {n + 1} \\mathbf {J} _ {1 2} (\\alpha) (L _ {n} ^ {(\\mathbf {x})} + R _ {n} ^ {(\\mathbf {x})}) \\\\ = (\\mathbf {I} + \\beta_ {n + 1} \\mathbf {J} _ {1 1}) \\left(\\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*}\\right) + \\beta_ {n + 1} \\mathbf {J} _ {1 2} (\\alpha) \\Delta_ {n} ^ {(\\mathbf {x})} + \\beta_ {n + 1} \\left(r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} + r _ {n} ^ {(\\boldsymbol {\\theta}, 2)} + \\eta_ {n} ^ {(\\boldsymbol {\\theta})}\\right) \\\\ - \\left(\\mathbf {I} + \\beta_ {n + 1} \\mathbf {J} _ {1 1} + O \\left(\\beta_ {n + 1} ^ {2}\\right)\\right) \\left(\\hat {L} _ {n} ^ {(\\boldsymbol {\\theta})} + \\hat {R} _ {n} ^ {(\\boldsymbol {\\theta})}\\right) \\\\ = \\left(\\mathbf {I} + \\beta_ {n + 1} \\mathbf {J} _ {1 1}\\right) \\hat {\\Delta} _ {n} ^ {(\\boldsymbol {\\theta})} + \\beta_ {n + 1} \\mathbf {J} _ {1 2} (\\alpha) \\Delta_ {n} ^ {(\\mathbf {x})} + \\beta_ {n + 1} \\left(r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} + r _ {n} ^ {(\\boldsymbol {\\theta}, 2)} + \\eta_ {n} ^ {(\\boldsymbol {\\theta})}\\right) \\\\ + O \\left(\\beta_ {n + 1} ^ {2}\\right) \\left(\\hat {L} _ {n} ^ {(\\boldsymbol {\\theta})} + \\hat {R} _ {n} ^ {(\\boldsymbol {\\theta})}\\right). \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            179,
+            458,
+            813,
+            612
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "text",
+        "text": "where the second equality is from (53a), the third equality stems from the approximation of  $e^{\\beta_{n+1}\\mathbf{J}_{11}}$ . Then, we again use the definition  $\\Phi_{k,n} \\triangleq \\prod_{j=k+1}^{n}(\\mathbf{I} + \\beta_{j}\\mathbf{J}_{11})$  and reiterate the above equation as",
+        "bbox": [
+            169,
+            621,
+            820,
+            655
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\hat {\\Delta} _ {n + 1} ^ {(\\boldsymbol {\\theta})} = \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(O (\\beta_ {k + 1}) L _ {k} ^ {(\\boldsymbol {\\theta})} + O (\\beta_ {k + 1}) R _ {k} ^ {(\\boldsymbol {\\theta})}\\right) \\\\ + \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\mathbf {J} _ {1 2} (\\alpha) \\Delta_ {n} ^ {(\\mathbf {x})} + \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(r _ {k} ^ {(\\boldsymbol {\\theta}, 1)} + \\eta_ {k} ^ {(\\boldsymbol {\\theta})}\\right) \\\\ + \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} r _ {k} ^ {(\\boldsymbol {\\theta}, 2)} \\\\ \\triangleq \\hat {\\Delta} _ {n + 1} ^ {(\\boldsymbol {\\theta}, 1)} + \\hat {\\Delta} _ {n + 1} ^ {(\\boldsymbol {\\theta}, 2)}, \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            269,
+            664,
+            725,
+            809
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "text",
+        "text": "where  $\\hat{\\Delta}_{n + 1}^{(\\pmb {\\theta},2)} = \\sum_{k = 1}^{n}\\Phi_{k,n}\\beta_{k + 1}r_{k}^{(\\pmb {\\theta},2)}$  and",
+        "bbox": [
+            171,
+            816,
+            460,
+            837
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\hat {\\Delta} _ {n + 1} ^ {(\\boldsymbol {\\theta}, 1)} = \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(O \\left(\\beta_ {k + 1}\\right) L _ {k} ^ {(\\boldsymbol {\\theta})} + O \\left(\\beta_ {k + 1}\\right) R _ {k} ^ {(\\boldsymbol {\\theta})}\\right) \\tag {80} \\\\ + \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(r _ {k} ^ {\\left(\\boldsymbol {\\theta}, 1\\right)} + \\eta_ {k} ^ {\\left(\\boldsymbol {\\theta}\\right)} + \\mathbf {J} _ {1 2} (\\alpha) \\Delta_ {n} ^ {\\left(\\mathbf {x}\\right)}\\right). \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            310,
+            847,
+            823,
+            926
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "page_number",
+        "text": "35",
+        "bbox": [
+            488,
+            946,
+            506,
+            959
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "text",
+        "text": "For  $\\hat{\\Delta}_{n+1}^{(\\theta,2)}$ , we follow the same steps from (74) to (75), and obtain  $\\hat{\\Delta}_{n+1}^{(\\theta,2)} = o\\left(\\sqrt{\\beta_n}\\right)$ .",
+        "bbox": [
+            169,
+            99,
+            733,
+            122
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "Next, we consider  $\\hat{\\Delta}_{n+1}^{(\\pmb{\\theta},1)}$  and want to show that  $\\hat{\\Delta}_{n+1}^{(\\pmb{\\theta},1)} = o(\\sqrt{\\beta_n})$ . Again, we utilize Mokkadem & Pelletier (2006, Lemma 6) for  $\\hat{\\Delta}_{n+1}^{(\\pmb{\\theta},1)}$  and adapt the notation here for the case  $\\gamma_n = o(\\beta_n)$ .",
+        "bbox": [
+            169,
+            127,
+            823,
+            167
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "Lemma E.9. For  $\\hat{\\Delta}_{n+1}^{(\\pmb{\\theta}, 1)}$  in the form of (80), assume  $\\|\\pmb{\\theta}_n - \\pmb{\\theta}^*\\| = O(\\omega_n)$  and  $\\|\\hat{\\Delta}_n^{(\\pmb{\\theta}, 1)}\\| = O(\\delta_n)$  for the sequences  $\\omega_n, \\delta_n$  defined in (E.1). Then, we have",
+        "bbox": [
+            168,
+            169,
+            823,
+            202
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left\\| \\hat {\\Delta} _ {n + 1} ^ {(\\boldsymbol {\\theta}, 1)} \\right\\| = O \\left(\\gamma_ {n} ^ {2} \\beta_ {n} ^ {- 2} \\omega_ {n} ^ {2} + \\gamma_ {n} \\beta_ {n} ^ {- 1} \\delta_ {n}\\right) + o (\\sqrt {\\gamma_ {n}}) \\quad a. s. \\tag {81}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            315,
+            207,
+            825,
+            229
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "Now we need to further analyse  $\\delta_{n}$  and tighten its big O form, starting from  $\\delta_{n} \\equiv 1$ , so that we can finally obtain the big O form of  $\\|\\hat{\\Delta}_{n+1}^{(\\theta,1)}\\|$ . The following steps are borrowed from the ideas in Mokkadem & Pelletier (2006, Section 2.3.2).",
+        "bbox": [
+            169,
+            239,
+            826,
+            287
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "By almost sure convergence result  $\\lim_{n\\to \\infty}\\pmb{\\theta}_n = \\pmb{\\theta}^*$ , we have  $\\lim_{n\\to \\infty}\\Delta_n^{(\\pmb{\\theta})} = 0$  a.s. such that we can first set  $\\delta_{n}\\equiv 1$ , and  $\\| \\hat{\\Delta}_{n + 1}^{(\\pmb {\\theta},1)}\\| = O(\\gamma_n^2\\beta_n^{-2}\\omega_n^2 +\\gamma_n\\beta_n^{-1}) + o(\\sqrt{\\gamma_n})$ . Then, we redefine",
+        "bbox": [
+            169,
+            294,
+            825,
+            330
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\delta_ {n} \\equiv O (\\gamma_ {n} ^ {2} \\beta_ {n} ^ {- 2} \\omega_ {n} ^ {2} + \\gamma_ {n} \\beta_ {n} ^ {- 1}) + o (\\sqrt {\\gamma_ {n}}),\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            335,
+            635,
+            354
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "and notice that it still satisfies definition E.1. Then, reapplying this  $\\delta_{n}$  form to (81) gives",
+        "bbox": [
+            169,
+            359,
+            753,
+            375
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\| \\hat {\\Delta} _ {n + 1} ^ {(\\pmb {\\theta}, 1)} \\| = O (\\gamma_ {n} ^ {2} \\beta_ {n} ^ {- 2} \\omega_ {n} ^ {2} + [ \\gamma_ {n} \\beta_ {n} ^ {- 1} ] ^ {2}) + o (\\sqrt {\\gamma_ {n}})\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            333,
+            380,
+            663,
+            401
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "and by induction we have for all integers  $k \\geq 1$ ,",
+        "bbox": [
+            169,
+            406,
+            491,
+            421
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\| \\hat {\\Delta} _ {n + 1} ^ {(\\pmb {\\theta}, 1)} \\| = O (\\gamma_ {n} ^ {2} \\beta_ {n} ^ {- 2} \\omega_ {n} ^ {2} + [ \\gamma_ {n} \\beta_ {n} ^ {- 1} ] ^ {k}) + o (\\sqrt {\\gamma_ {n}}).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            331,
+            426,
+            665,
+            448
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "Since  $[\\gamma_n\\beta_n^{-1}]^k = n^{(b - a)k}$ , there exists  $k_0 > a / 2(a - b)$  such that  $[\\gamma_n\\beta_n^{-1}]^{k_0} = o(\\sqrt{\\gamma_n})$ , and",
+        "bbox": [
+            169,
+            453,
+            789,
+            472
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left\\| \\hat {\\Delta} _ {n + 1} ^ {(\\boldsymbol {\\theta}, 1)} \\right\\| = O \\left(\\gamma_ {n} ^ {2} \\beta_ {n} ^ {- 2} \\omega_ {n} ^ {2}\\right) + o (\\sqrt {\\gamma_ {n}}). \\tag {82}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            372,
+            477,
+            823,
+            500
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "Then, as suggested in Mokkadem & Pelletier (2006, Section 2.3.2), we can choose  $\\omega_{n} = O(\\sqrt{\\beta_{n}\\log(u_{n})} +[\\gamma_{n}\\beta_{n}^{-1}]^{k})$ , which also satisfies definition E.1. Then,",
+        "bbox": [
+            169,
+            503,
+            823,
+            536
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\left\\| \\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*} \\right\\| = \\left\\| \\hat {L} _ {n} ^ {(\\boldsymbol {\\theta})} + \\hat {R} _ {n} ^ {(\\boldsymbol {\\theta})} + \\hat {\\Delta} _ {n} ^ {(\\boldsymbol {\\theta})} \\right\\| \\\\ = O \\left(\\sqrt {\\beta_ {n} \\log (u _ {n})} + \\sqrt {\\gamma_ {n} \\beta_ {n} ^ {- 1} \\log (s _ {n})} + \\left(\\left[ \\gamma_ {n} \\beta_ {n} ^ {- 1} \\right] ^ {k + 1} + \\gamma_ {n} \\beta_ {n} ^ {- 1} \\sqrt {\\beta_ {n} \\log (u _ {n})}\\right) ^ {2}\\right) \\\\ + o (\\sqrt {\\beta_ {n}} + \\sqrt {\\gamma_ {n}}) \\\\ = O (\\sqrt {\\beta_ {n} \\log (u _ {n})} + [ \\gamma_ {n} \\beta_ {n} ^ {- 1} ] ^ {k + 1}). \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            228,
+            541,
+            766,
+            640
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "By induction, this holds for all  $k \\geq 1$  such that there exists  $k_0$ ,  $[\\gamma_n\\beta_n^{-1}]^{k_0} = o(\\sqrt{\\beta_n})$  and  $\\| \\pmb{\\theta}_n - \\pmb{\\theta}^*\\| = O(\\sqrt{\\beta_n\\log(u_n)})$ . Equivalently,  $\\omega_{n} = \\sqrt{\\beta_{n}\\log(u_{n})}$ . Therefore, from (82) we have",
+        "bbox": [
+            169,
+            645,
+            823,
+            679
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\| \\hat {\\Delta} _ {n + 1} ^ {(\\boldsymbol {\\theta}, 1)} \\| = O (\\gamma_ {n} ^ {2} \\beta_ {n} ^ {- 1} \\log (u _ {n})) + o (\\sqrt {\\gamma_ {n}}) = o (\\sqrt {\\gamma_ {n}}).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            318,
+            685,
+            678,
+            705
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "Together with  $\\| \\hat{\\Delta}_{n + 1}^{(\\pmb {\\theta},2)}\\| = o(\\sqrt{\\beta_n})$  , we have  $\\beta_{n}^{-1 / 2}\\| \\hat{\\Delta}_{n + 1}^{(\\pmb {\\theta})}\\| = o(\\sqrt{\\gamma_{n}\\beta_{n}^{-1}}) + 1)$  such that",
+        "bbox": [
+            169,
+            710,
+            767,
+            733
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\lim  _ {n \\to \\infty} \\beta_ {n} ^ {- 1 / 2} \\hat {\\Delta} _ {n + 1} ^ {(\\boldsymbol {\\theta})} = 0.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            419,
+            737,
+            575,
+            763
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "Thus, we have finished the proof according to the proof outline mentioned at the beginning of this part.",
+        "bbox": [
+            169,
+            768,
+            823,
+            799
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "F DISCUSSION OF COVARIANCE ORDERING OF SA-SRRW",
+        "text_level": 1,
+        "bbox": [
+            169,
+            816,
+            676,
+            832
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "F.1 PROOF OF PROPOSITION 3.4",
+        "text_level": 1,
+        "bbox": [
+            171,
+            848,
+            413,
+            862
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "For any  $\\alpha > 0$  and any vector  $\\mathbf{x} \\in \\mathbb{R}^d$ , we have",
+        "bbox": [
+            169,
+            875,
+            486,
+            888
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {x} ^ {T} \\mathbf {V} _ {\\boldsymbol {\\theta}} ^ {(1)} (\\alpha) \\mathbf {x} = \\int_ {0} ^ {\\infty} \\mathbf {x} ^ {T} e ^ {t (\\nabla_ {\\boldsymbol {\\theta}} \\mathbf {h} (\\boldsymbol {\\theta} ^ {*}) + \\frac {1}{2} \\mathbf {\\Sigma} _ {\\{b = 1 \\}} ^ {*} \\boldsymbol {I})} \\mathbf {U} _ {\\boldsymbol {\\theta}} (\\alpha) e ^ {t (\\nabla_ {\\boldsymbol {\\theta}} \\mathbf {h} (\\boldsymbol {\\theta} ^ {*}) + \\frac {1}{2} \\mathbf {\\Sigma} _ {\\{b = 1 \\}} ^ {*} \\boldsymbol {I}) ^ {T}} \\mathbf {x} d t\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            238,
+            895,
+            754,
+            929
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "page_number",
+        "text": "36",
+        "bbox": [
+            488,
+            946,
+            509,
+            960
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "where the first equality is from the form of  $\\mathbf{V}_{\\theta}^{(1)}(\\alpha)$  in Theorem 3.3. Let  $\\mathbf{y} \\triangleq e^{t(\\nabla_{\\theta}\\mathbf{h}(\\boldsymbol{\\theta}^{*}) + \\frac{1}{2}\\left\\{b = 1\\right\\} I)}\\mathbf{x}$ , with the dependence on variable  $t$  left implicit. The matrix  $\\mathbf{U}_{\\theta}(\\alpha)$ , given explicitly in (11) positive semi definite, since  $\\lambda_i \\in (-1,1)$  for all  $i \\in \\{1,\\dots,N-1\\}$ . Thus, the terms  $\\mathbf{y}^T\\mathbf{U}_{\\theta}(\\alpha)\\mathbf{y}$  inside the integral are non-negative, and it is enough to provide an ordering on  $\\mathbf{y}^T\\mathbf{U}_{\\theta}(\\alpha)\\mathbf{y}$  with respect to  $\\alpha$ .",
+        "bbox": [
+            169,
+            101,
+            823,
+            164
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "For any  $\\alpha_{2} > \\alpha_{1} > 0$",
+        "bbox": [
+            171,
+            169,
+            321,
+            184
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\mathbf {y} ^ {T} \\mathbf {U} _ {\\boldsymbol {\\theta}} (\\alpha_ {2}) \\mathbf {y} = \\sum_ {i = 1} ^ {N - 1} \\frac {1}{(\\alpha_ {2} (1 + \\lambda_ {i}) + 1) ^ {2}} \\cdot \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {y} ^ {T} \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H} \\mathbf {y} \\\\ <   \\sum_ {i = 1} ^ {N - 1} \\frac {1}{\\left(\\alpha_ {1} \\left(1 + \\lambda_ {i}\\right) + 1\\right) ^ {2}} \\cdot \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {y} ^ {T} \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H} \\mathbf {y} = \\mathbf {y} ^ {T} \\mathbf {U} _ {\\boldsymbol {\\theta}} (\\alpha_ {1}) \\mathbf {y} \\\\ <   \\sum_ {i = 1} ^ {N - 1} \\cdot \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {y} ^ {T} \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H} \\mathbf {y} = \\mathbf {y} ^ {T} \\mathbf {U} _ {\\boldsymbol {\\theta}} (0) \\mathbf {y}, \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            225,
+            193,
+            767,
+            323
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "where the inequality $^{13}$  is because  $\\alpha(1 + \\lambda_i) > 0$  for all  $i \\in \\{1, \\dots, N\\}$  and any  $\\alpha > 0$ . In fact, the ordering is monotone in  $\\alpha$ , and  $\\mathbf{y}^T \\mathbf{U}_{\\theta}(\\alpha_2) \\mathbf{y}$  decreases at rate  $1 / \\alpha^2$  as seen from its form in the equation above. This completes the proof.",
+        "bbox": [
+            169,
+            340,
+            823,
+            385
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "F.2 DISCUSSION REGARDING PROPOSITION 3.4 AND MSE ORDERING",
+        "text_level": 1,
+        "bbox": [
+            171,
+            402,
+            668,
+            417
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "We can use Proposition 3.4 to show that the MSE of SA iterates of (4c) driven by SRRW eventually becomes smaller than that SA iterates when the stochastic noise is driven by an i.i.d. sequence of random variables. The diagonal entries of  $\\mathbf{V}_{\\theta}^{(1)}(\\alpha)$  are obtained by evaluating  $\\mathbf{e}_i^T\\mathbf{V}_{\\theta}^{(1)}(\\alpha)\\mathbf{e}_i$ , where  $\\mathbf{e}_i$  is the  $i$ 'th standard basis vector. $^{14}$  These diagonal entries are the asymptotic variance corresponding to the element-wise iterate errors, and for large enough  $n$ , we have  $\\mathbf{e}_i^T\\mathbf{V}_{\\theta}^{(1)}(\\alpha)\\mathbf{e}_i \\approx \\mathbb{E}[(\\boldsymbol{\\theta}_n - \\boldsymbol{\\theta}^*)_i^2] / \\beta_n$  for all  $i \\in \\{1, \\dots, D\\}$ . Thus, the trace of matrix  $\\mathbf{V}_{\\theta}^{(1)}(\\alpha)$  approximates the scaled MSE, that is  $\\mathrm{Tr}(\\mathbf{V}_{\\theta}^{(1)}(\\alpha)) = \\sum_i \\mathbf{e}_i^T\\mathbf{V}_{\\theta}^{(1)}(\\alpha)\\mathbf{e}_i \\approx \\sum_i \\mathbb{E}[(\\boldsymbol{\\theta}_n - \\boldsymbol{\\theta}^*)_i^2] / \\beta_n = \\mathbb{E}[||\\boldsymbol{\\theta}_n - \\boldsymbol{\\theta}^*||^2] / \\beta_n$  for large  $n$ . Since all entries of  $\\mathbf{V}_{\\theta}^{(1)}(\\alpha)$  go to zero as  $\\alpha$  increases, they get smaller than the corresponding term for the SA algorithm with i.i.d. input for large enough  $\\alpha$ , which achieves a constant MSE in the similarly scaled limit, since the asymptotic covariance is not a function of  $\\alpha$ . Moreover, the value of  $\\alpha$  only needs to be moderately large, since the asymptotic covariance terms decrease at rate  $O(1 / \\alpha^2)$  as shown in Proposition 3.4.",
+        "bbox": [
+            169,
+            429,
+            826,
+            619
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "F.3 PROOF OF COROLLARY 3.5",
+        "text_level": 1,
+        "bbox": [
+            171,
+            637,
+            403,
+            651
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "We see that  $\\mathbf{V}_{\\pmb{\\theta}}^{(3)}(\\alpha) = \\mathbf{V}_{\\pmb{\\theta}}^{(3)}(0)$  for all  $\\alpha > 0$ , because the form of  $\\mathbf{V}_{\\pmb{\\theta}}^{(3)}(\\alpha)$  in Theorem 3.3 is independent of  $\\alpha$ . To prove that  $\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(\\alpha) <_{L} \\mathbf{V}_{\\pmb{\\theta}}^{(3)}(0)$ , it is enough to show that  $\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(0) = \\mathbf{V}_{\\pmb{\\theta}}^{(3)}(0)$ , since  $\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(\\alpha) <_{L} \\mathbf{V}_{\\pmb{\\theta}}^{(1)}(0)$  from Proposition 3.4. This is easily checked by substituting  $\\alpha = 0$  in 11, for which  $\\mathbf{U}_{\\pmb{\\theta}}(0) = \\mathbf{U}_{11}$ . Substituting in the respective forms of  $\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(0)$  and  $\\mathbf{V}_{\\pmb{\\theta}}^{(3)}(0)$  in Theorem 3.3, we get equivalence. This completes the proof.",
+        "bbox": [
+            169,
+            660,
+            823,
+            748
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "G BACKGROUND THEORY",
+        "text_level": 1,
+        "bbox": [
+            171,
+            768,
+            408,
+            784
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "G.1 TECHNICAL LEMMAS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            801,
+            369,
+            816
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "Lemma G.1 (Solution to the Lyapunov Equation). If all the eigenvalues of matrix  $\\mathbf{M}$  have negative real part, then for every positive semi-definite matrix  $\\mathbf{U}$  there exists a unique positive semi-definite",
+        "bbox": [
+            169,
+            829,
+            823,
+            859
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "page_footnote",
+        "text": "<sup>13</sup>The inequality may not be strict when  $H$  is low rank, however it will always be true for some choice of  $\\mathbf{x}$ , since  $\\mathbf{H}$  is not a zero matrix. Thus, the ordering derived still follows our definition of  $<_{L}$  in Section 1, footnote 6.",
+        "bbox": [
+            169,
+            869,
+            823,
+            907
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "page_footnote",
+        "text": "$^{14}D$ -dimensional vector of all zeros except at the  $i$ 'th position which is 1.",
+        "bbox": [
+            187,
+            909,
+            622,
+            924
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "page_number",
+        "text": "37",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "matrix  $\\mathbf{V}$  satisfying the Lyapunov equation  $\\mathbf{U} + \\mathbf{M}\\mathbf{V} + \\mathbf{V}\\mathbf{M}^T = \\mathbf{0}$ . The explicit solution  $\\mathbf{V}$  is given as",
+        "bbox": [
+            169,
+            103,
+            823,
+            130
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {V} = \\int_ {0} ^ {\\infty} e ^ {\\mathbf {M} t} \\mathbf {U} e ^ {(\\mathbf {M} ^ {T}) t} d t. \\tag {83}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            406,
+            128,
+            825,
+            161
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "Chellaboina & Haddad (2008, Theorem 3.16) states that for a positive definite matrix  $\\mathbf{U}$ , there exists a positive definite matrix  $\\mathbf{V}$ . The reason they focus on the positive definite matrix  $\\mathbf{U}$  is that they require the related autonomous ODE system to be asymptotically stable. However, in this paper we don't need this requirement. The same steps therein can be used to prove Lemma G.1 and show that if  $\\mathbf{U}$  is positive semi-definite, then  $\\mathbf{V}$  in the form of (83) is unique and also positive semi-definite.",
+        "bbox": [
+            169,
+            169,
+            823,
+            239
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "Lemma G.2 (Burkholder Inequality, Davis (1970), Hall et al. (2014) Theorem 2.10). Given a Martingale difference sequence  $\\{M_{i,n}\\}_{i = 1}^{n}$ , for  $p\\geq 1$  and some positive constant  $C_p$ , we have",
+        "bbox": [
+            169,
+            241,
+            823,
+            272
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbb {E} \\left[ \\left\\| \\sum_ {i = 1} ^ {n} M _ {i, n} \\right\\| ^ {p} \\right] \\leq C _ {p} \\mathbb {E} \\left[ \\left(\\sum_ {i = 1} ^ {n} \\| M _ {i, n} \\| ^ {2}\\right) ^ {p / 2} \\right] \\tag {84}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            331,
+            275,
+            825,
+            325
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "Theorem G.3 (Martingale CLT, Delyon (2000) Theorem 30). If a Martingale difference array  $\\{X_{n,i}\\}$  satisfies the following condition: for some  $\\tau > 0$ ,",
+        "bbox": [
+            171,
+            327,
+            823,
+            357
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\sum_ {k = 1} ^ {n} \\mathbb {E} \\left[ \\| X _ {n, k} \\| ^ {2 + \\tau} \\mid \\mathcal {F} _ {k - 1} \\right] \\xrightarrow {\\mathbb {P}} 0, \\tag {85}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            390,
+            362,
+            825,
+            401
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\sup  _ {n} \\sum_ {k = 1} ^ {n} \\mathbb {E} \\left[ \\| X _ {n, k} \\| ^ {2} \\mid \\mathcal {F} _ {k - 1} \\right] <   \\infty , \\tag {86}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            383,
+            407,
+            825,
+            446
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "and",
+        "bbox": [
+            171,
+            446,
+            202,
+            460
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\sum_ {k = 1} ^ {n} \\mathbb {E} \\left[ X _ {n, k} X _ {n, k} ^ {T} \\mid \\mathcal {F} _ {k - 1} \\right] \\xrightarrow {\\mathbb {P}} \\boldsymbol {V}, \\tag {87}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            390,
+            458,
+            825,
+            497
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "then",
+        "bbox": [
+            171,
+            500,
+            205,
+            512
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\sum_ {i = 1} ^ {n} X _ {n, i} \\xrightarrow {\\text {d i s t .}} N (0, V). \\tag {88}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            411,
+            510,
+            823,
+            549
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "image",
+        "img_path": "images/d95b76f2eb15428d0dc289e8c29dfd2ff318ad2f40ca5095ea67f83225e15a5e.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            551,
+            823,
+            561
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "Lemma G.4 (Duflo (1996) Proposition 3.I.2). For a Hurwitz matrix  $\\mathbf{H}$ , there exist some positive constants  $C, b$  such that for any  $n$ ,",
+        "bbox": [
+            169,
+            566,
+            823,
+            595
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left\\| e ^ {\\mathbf {H} n} \\right\\| \\leq C e ^ {- b n}. \\tag {89}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            437,
+            594,
+            823,
+            613
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "Lemma G.5 (Fort (2015) Lemma 5.8). For a Hurwitz matrix  $\\mathbf{A}$ , denote by  $-r$ ,  $r > 0$ , the largest real part of its eigenvalues. Let a positive sequence  $\\{\\gamma_n\\}$  such that  $\\lim_{n} \\gamma_{n} = 0$ . Then for any  $0 < r' < r$ , there exists a positive constant  $C$  such that for any  $k < n$ ,",
+        "bbox": [
+            169,
+            614,
+            823,
+            657
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left\\| \\prod_ {j = k} ^ {n} (\\boldsymbol {I} + \\gamma_ {j} \\boldsymbol {A}) \\right\\| \\leq C e ^ {- r ^ {\\prime} \\sum_ {j = k} ^ {n} \\gamma_ {j}}. \\tag {90}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            377,
+            660,
+            823,
+            709
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "Lemma G.6 (Fort (2015) Lemma 5.9, Mokkadem & Pelletier (2006) Lemma 10). Let  $\\{\\gamma_n\\}$  be a positive sequence such that  $\\lim_{n} \\gamma_{n} = 0$  and  $\\sum_{n} \\gamma_{n} = \\infty$ . Let  $\\{\\epsilon_{n}, n \\geq 0\\}$  be a nonnegative sequence. Then, for  $b > 0$ ,  $p \\geq 0$ ,",
+        "bbox": [
+            169,
+            713,
+            823,
+            757
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\lim  _ {n} \\sup  _ {n} \\gamma_ {n} ^ {- p} \\sum_ {k = 1} ^ {n} \\gamma_ {k} ^ {p + 1} e ^ {- b \\sum_ {j = k + 1} ^ {n} \\gamma_ {j}} \\epsilon_ {k} \\leq \\frac {1}{C (b , p)} \\lim  _ {n} \\sup  _ {n} \\epsilon_ {n} \\tag {91}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            297,
+            761,
+            823,
+            800
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "for some constant  $C(b,p) > 0$",
+        "bbox": [
+            168,
+            804,
+            380,
+            820
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "When  $p = 0$  and define a positive sequence  $\\{w_{n}\\}$  satisfying  $w_{n - 1} / w_{n} = 1 + o(\\gamma_{n})$ , we have",
+        "bbox": [
+            171,
+            825,
+            784,
+            842
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\sum_ {k = 1} ^ {n} \\gamma_ {k} e ^ {- b \\sum_ {j = k + 1} ^ {n} \\gamma_ {j}} \\epsilon_ {k} = \\left\\{ \\begin{array}{l l} O (w _ {n}), & \\text {i f} \\epsilon_ {n} = O (w _ {n}), \\\\ o (w _ {n}), & \\text {i f} \\epsilon_ {n} = o (w _ {n}). \\end{array} \\right. \\tag {92}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            308,
+            845,
+            823,
+            885
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "Lemma G.7 (Fort (2015) Lemma 5.10). For any matrices  $A, B, C$ ,",
+        "bbox": [
+            169,
+            888,
+            617,
+            904
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left\\| A B A ^ {T} - C B C ^ {T} \\right\\| \\leq \\| A - C \\| \\| B \\| \\left(\\| A \\| + \\| C \\|\\right). \\tag {93}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            907,
+            823,
+            926
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "page_number",
+        "text": "38",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "G.2 ASYMPTOTIC RESULTS OF SINGLE-TIMESCALE SA",
+        "text_level": 1,
+        "bbox": [
+            171,
+            103,
+            573,
+            118
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "Consider the stochastic approximation in the form of",
+        "bbox": [
+            171,
+            128,
+            524,
+            143
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {z} _ {n + 1} = \\boldsymbol {z} _ {n} + \\gamma_ {n + 1} G \\left(\\boldsymbol {z} _ {n}, X _ {n + 1}\\right). \\tag {94}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            383,
+            148,
+            823,
+            164
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "Let  $\\mathbf{K}_z$  be the transition kernel of the underlying Markov chain  $\\{X_n\\}_{n\\geq 0}$  with stationary distribution  $\\pi (z)$  such that  $g(z)\\triangleq \\mathbb{E}_{X\\sim \\pi (z)}[G(z,X)]$  with domain  $\\mathcal{O}\\subseteq \\mathbb{R}^d$ . Define an operator  $\\mathbf{K}_zf$  for any function  $f:\\mathcal{N}\\to \\mathbb{R}^{D}$  such that",
+        "bbox": [
+            169,
+            167,
+            825,
+            215
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left(\\mathbf {K} _ {\\boldsymbol {z}} f\\right) (i) = \\sum_ {j \\in \\mathcal {N}} f (j) \\mathbf {K} _ {\\boldsymbol {z}} (i, j). \\tag {95}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            392,
+            219,
+            825,
+            252
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "Assume that",
+        "bbox": [
+            171,
+            256,
+            259,
+            268
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "C1. W.p.1, the closure of  $\\{z_{n}\\}_{n\\geq 0}$  is a compact subset of  $\\mathcal{O}$ .",
+            "C2.  $\\gamma_{n} = \\gamma_{0} / n^{a}, a\\in (1 / 2,1]$",
+            "C3. Function  $g$  is continuous on  $\\mathcal{O}$  and there exists a non-negative  $C^1$  function  $w$  and a compact set  $\\mathcal{K} \\subset \\mathcal{O}$  such that"
+        ],
+        "bbox": [
+            197,
+            280,
+            823,
+            344
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "-  $\\nabla w(\\boldsymbol{z})^T g(\\boldsymbol{z}) \\leq 0$  for all  $\\boldsymbol{z} \\in \\mathcal{O}$  and  $\\nabla w(\\boldsymbol{z})^T g(\\boldsymbol{z}) < 0$  if  $\\boldsymbol{z} \\notin \\mathcal{K}$ ;",
+            "- the set  $S \\triangleq \\{z \\mid \\nabla w(z)^T g(z) = 0\\}$  is such that  $w(S)$  has an empty interior;"
+        ],
+        "bbox": [
+            246,
+            348,
+            767,
+            382
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "C4. For every  $z$ , there exists a solution  $m_z: \\mathcal{N} \\to \\mathbb{R}^d$  for the following Poisson equation",
+        "bbox": [
+            197,
+            385,
+            790,
+            400
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nm _ {z} (i) - \\left(\\mathbf {K} _ {z} m _ {z}\\right) (i) = G (z, i) - g (z) \\tag {96}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            395,
+            402,
+            823,
+            420
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "for any  $i\\in \\mathcal{N}$  ; for any compact set  $\\mathcal{C}\\subset \\mathcal{O}$",
+        "bbox": [
+            228,
+            422,
+            517,
+            438
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\sup  _ {\\boldsymbol {z} \\in \\mathcal {C}, i \\in \\mathcal {N}} \\| \\left(\\mathbf {K} _ {\\boldsymbol {z}} m _ {\\boldsymbol {z}}\\right) (i) \\| + \\| m _ {\\boldsymbol {z}} (i) \\| <   \\infty \\tag {97}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            390,
+            441,
+            825,
+            467
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "and there exist a continuous function  $\\phi_{\\mathcal{C}},\\phi_{\\mathcal{C}}(0) = 0$  , such that for any  $\\pmb {z},\\pmb{z}^{\\prime}\\in \\mathcal{C}$",
+        "bbox": [
+            228,
+            472,
+            759,
+            487
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\sup  _ {i \\in \\mathcal {N}} \\| \\left(\\mathbf {K} _ {\\boldsymbol {z}} m _ {\\boldsymbol {z}}\\right) (i) - \\left(\\mathbf {K} _ {\\boldsymbol {z} ^ {\\prime}} m _ {\\boldsymbol {z} ^ {\\prime}}\\right) (i) \\| \\leq \\phi_ {\\mathcal {C}} (\\| \\boldsymbol {z} - \\boldsymbol {z} ^ {\\prime} \\|). \\tag {98}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            349,
+            489,
+            823,
+            515
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "C5. Denote by  $-r$  the largest real part of the eigenvalues of the Jacobian matrix  $\\nabla g(z^{*})$  and assume  $r > \\frac{\\mathbb{1}_{\\{a = 1\\}}}{2}$ .",
+            "C6. For every  $z$ , there exists a solution  $Q_{z}:\\mathcal{N}\\to \\mathbb{R}^{d\\times d}$  for the following Poisson equation"
+        ],
+        "bbox": [
+            197,
+            521,
+            823,
+            575
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nQ _ {z} (i) - \\left(\\mathbf {K} _ {z} Q _ {z}\\right) (i) = F (z, i) - \\mathbb {E} _ {j \\sim \\pi (z)} [ F (z, j) ] \\tag {99}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            354,
+            578,
+            823,
+            595
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "for any  $i\\in \\mathcal{N}$  , where",
+        "bbox": [
+            228,
+            598,
+            375,
+            613
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nF (\\boldsymbol {z}, i) \\triangleq \\sum_ {j \\in \\mathcal {N}} m _ {\\boldsymbol {z}} (j) m _ {\\boldsymbol {z}} (j) ^ {T} \\mathbf {K} _ {\\boldsymbol {z}} (i, j) - \\left(\\mathbf {K} _ {\\boldsymbol {z}} m _ {\\boldsymbol {z}}\\right) (i) \\left(\\mathbf {K} _ {\\boldsymbol {z}} m _ {\\boldsymbol {z}}\\right) (i) ^ {T}. \\tag {100}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            303,
+            617,
+            825,
+            648
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "For any compact set  $\\mathcal{C} \\subset \\mathcal{O}$ ,",
+        "bbox": [
+            228,
+            652,
+            419,
+            667
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\sup  _ {\\boldsymbol {z} \\in \\mathcal {C}, i \\in \\mathcal {N}} \\| Q _ {\\boldsymbol {z}} (i) \\| + \\| \\left(\\mathbf {K} _ {\\boldsymbol {z}} Q _ {\\boldsymbol {z}}\\right) (i) \\| <   \\infty \\tag {101}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            393,
+            672,
+            823,
+            696
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "and there exist  $p, C_{\\mathcal{C}} > 0$ , such that for any  $\\mathbf{z}, \\mathbf{z}' \\in \\mathcal{C}$",
+        "bbox": [
+            228,
+            702,
+            586,
+            717
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\sup  _ {i \\in \\mathcal {N}} \\| \\left(\\mathbf {K} _ {\\boldsymbol {z}} Q _ {\\boldsymbol {z}}\\right) (i) - \\left(\\mathbf {K} _ {\\boldsymbol {z} ^ {\\prime}} Q _ {\\boldsymbol {z} ^ {\\prime}}\\right) (i) \\| \\leq C _ {\\mathcal {C}} \\| \\boldsymbol {z} - \\boldsymbol {z} ^ {\\prime} \\| ^ {p}. \\tag {102}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            354,
+            720,
+            823,
+            744
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "Theorem G.8 (Delyon et al. (1999) Theorem 2). Consider (94) and assume C1 - C4. Then, w.p.1,  $\\lim \\sup_{n}d(\\pmb{z}_n,S) = 0$ .",
+        "bbox": [
+            169,
+            747,
+            823,
+            776
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "Theorem G.9 (Fort (2015) Theorem 2.1 & Proposition 4.1). Consider (94) and assume  $C1 - C6$ . Then, given the condition that  $z_{n}$  converges to one point  $z^{*} \\in S$ , we have",
+        "bbox": [
+            169,
+            777,
+            823,
+            808
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\gamma_ {n} ^ {- 1 / 2} \\left(\\boldsymbol {z} _ {n} - \\boldsymbol {z} ^ {*}\\right) \\xrightarrow [ n \\rightarrow \\infty ]{\\text {d i s t .}} N (0, \\mathbf {V}), \\tag {103}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            382,
+            811,
+            823,
+            837
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "where",
+        "bbox": [
+            171,
+            840,
+            217,
+            854
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {V} \\left(\\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I} + \\nabla g \\left(\\boldsymbol {z} ^ {*}\\right) ^ {T}\\right) + \\left(\\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I} + \\nabla g \\left(\\boldsymbol {z} ^ {*}\\right)\\right) \\mathbf {V} + \\mathbf {U} = 0, \\tag {104}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            274,
+            849,
+            825,
+            883
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "and",
+        "bbox": [
+            171,
+            885,
+            202,
+            898
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {U} \\triangleq \\sum_ {i \\in \\mathcal {N}} \\mu_ {i} \\left(m _ {\\mathbf {z} ^ {*}} (i) m _ {\\mathbf {z} ^ {*}} (i) ^ {T} - \\left(\\mathbf {K} _ {\\mathbf {z} ^ {*}} m _ {\\mathbf {z} ^ {*}}\\right) (i) \\left(\\mathbf {K} _ {\\mathbf {z} ^ {*}} m _ {\\mathbf {z} ^ {*}}\\right) (i) ^ {T}\\right). \\tag {105}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            287,
+            895,
+            825,
+            926
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "page_number",
+        "text": "39",
+        "bbox": [
+            488,
+            946,
+            509,
+            960
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "G.3 ASYMPTOTIC RESULTS OF TWO-TIMESCALE SA",
+        "text_level": 1,
+        "bbox": [
+            171,
+            103,
+            555,
+            118
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "text",
+        "text": "For the two-timescale SA with iterate-dependent Markov chain, we have the following iterations:",
+        "bbox": [
+            169,
+            128,
+            810,
+            145
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {z} _ {n + 1} = \\mathbf {z} _ {n} + \\beta_ {n + 1} G _ {1} \\left(\\mathbf {z} _ {n}, \\mathbf {y} _ {n} X _ {n + 1}\\right), \\tag {106a}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            370,
+            151,
+            823,
+            167
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {y} _ {n + 1} = \\mathbf {y} _ {n} + \\gamma_ {n + 1} G _ {2} \\left(\\mathbf {z} _ {n}, \\mathbf {y} _ {n}, X _ {n + 1}\\right), \\tag {106b}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            366,
+            172,
+            823,
+            189
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "text",
+        "text": "with the goal of finding the root  $(\\mathbf{z}^{*},\\mathbf{y}^{*})$  such that",
+        "bbox": [
+            169,
+            191,
+            506,
+            205
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\ng _ {1} \\left(\\mathbf {z} ^ {*}, \\mathbf {y} ^ {*}\\right) = \\mathbb {E} _ {X \\sim \\boldsymbol {\\mu}} \\left[ G _ {1} \\left(\\mathbf {z} ^ {*}, \\mathbf {y} ^ {*}, X\\right) \\right] = 0, \\quad g _ {2} \\left(\\mathbf {z} ^ {*}, \\mathbf {y} ^ {*}\\right) = \\mathbb {E} _ {X \\sim \\boldsymbol {\\mu}} \\left[ G _ {2} \\left(\\mathbf {z} ^ {*}, \\mathbf {y} ^ {*}, X\\right) \\right] = 0. \\tag {107}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            194,
+            210,
+            825,
+            229
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "text",
+        "text": "We present here a simplified version of the assumptions for single-valued functions  $G_{1}, G_{2}$  that are necessary for the almost sure convergence result in Yaji & Bhatnagar (2020, Theorem 4). The original assumptions are intended for more general set-valued functions  $G_{1}, G_{2}$ .",
+        "bbox": [
+            169,
+            241,
+            826,
+            285
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "(B1) The step sizes  $\\beta_{n} \\triangleq n^{-b}$  and  $\\gamma_{n} \\triangleq n^{-a}$ , where  $0.5 < a < b \\leq 1$ .",
+            "(B2) Assume the function  $G_{1}(\\mathbf{z},\\mathbf{y},X)$  is continuous and differentiable with respect to  $\\mathbf{z},\\mathbf{y}$ . There exists a positive constant  $L_{1}$  such that  $\\| G_{1}(\\mathbf{z},\\mathbf{y},X)\\| \\leq L_{1}(1 + \\| \\mathbf{z}\\| +\\| \\mathbf{y}\\|)$  for every  $\\mathbf{z}\\in \\mathbb{R}^{d_1},\\mathbf{y}\\in \\mathbb{R}^{d_2},X\\in \\mathcal{N}$ . The same condition holds for the function  $G_{2}$  as well.",
+            "(B3) Assume there exists a function  $\\rho : \\mathbb{R}^{d_1} \\to \\mathbb{R}^{d_2}$  such that the following three properties hold: (i)  $\\| \\rho(\\mathbf{z}) \\| \\leq L_2(1 + \\| \\mathbf{z} \\|)$  for some positive constant  $L_2$ ; (ii) the ODE  $\\dot{\\mathbf{y}} = g_2(\\mathbf{z}, \\mathbf{y})$  has a globally asymptotically stable equilibrium  $\\lambda(\\mathbf{z})$  such that  $g_2(\\mathbf{z}, \\rho(\\mathbf{z})) = 0$ . Additionally, let  $\\hat{g}_1(\\mathbf{z}) \\triangleq g_1(\\mathbf{z}, \\rho(\\mathbf{z}))$ , there exists a set of disjoint roots  $\\Lambda \\triangleq \\{\\mathbf{z}^* : \\hat{g}_1(\\mathbf{z}^*) = 0\\}$ , which is the set of globally asymptotically stable equilibria of the ODE  $\\dot{\\mathbf{z}} = \\hat{g}_1(\\mathbf{z})$ .",
+            "(B4)  $\\{X_{n}\\}_{n\\geq 0}$  is an iterate-dependent Markov process in finite state space  $\\mathcal{N}$ . For every  $n\\geq 0$ ,  $P(X_{n + 1} = j|\\mathbf{z}_m,\\mathbf{y}_m,X_m,0\\leq m\\leq n) = P(X_{n + 1} = j|\\mathbf{z}_n,\\mathbf{y}_n,X_n = i) = \\mathbf{P}_{i,j}[\\mathbf{z}_n,\\mathbf{y}_n]$ , where the transition kernel  $\\mathbf{P}[\\mathbf{z},\\mathbf{y}]$  is continuous in  $\\mathbf{z},\\mathbf{y}$ , and the Markov chain generated by  $\\mathbf{P}[\\mathbf{z},\\mathbf{y}]$  is ergodic so that it admits a stationary distribution  $\\pi (\\mathbf{z},\\mathbf{y})$ , and  $\\pi (\\mathbf{z}^{*},\\rho (\\mathbf{z}^{*})) = \\pmb{\\mu}$ .",
+            "(B5)  $\\sup_{n\\geq 0}\\bigl (\\| \\mathbf{z}_n\\| +\\| \\mathbf{y}_n\\| \\bigr) <   \\infty$  a.s."
+        ],
+        "bbox": [
+            191,
+            295,
+            823,
+            529
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "text",
+        "text": "Yaji & Bhatnagar (2020) included assumptions A1 - A9 and A11 for the following Theorem G.10. We briefly show the correspondence of our assumptions (B1) - (B5) and theirs: (B1) with A5, (B2) with A1 and A2, (B3) with A9 and A11, (B4) with A3 and A4, and (B5) with A8. Given that our two-timescale SA framework (106) excludes additional noises (setting them to zero), A6 and A7 therein are inherently met.",
+        "bbox": [
+            169,
+            537,
+            826,
+            609
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "text",
+        "text": "Theorem G.10 (Yaji & Bhatnagar (2020) Theorem 4). Under Assumptions (B1) - (B5), iterations  $(\\mathbf{z}_n, \\mathbf{y}_n)$  in (106) almost surely converge to a set of roots, i.e.,  $(\\mathbf{z}_n, \\mathbf{y}_n) \\to \\bigcup_{\\mathbf{z}^* \\in \\Lambda} (\\mathbf{z}^*, \\rho(\\mathbf{z}^*))$  a.s.",
+        "bbox": [
+            169,
+            612,
+            823,
+            643
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "text",
+        "text": "H ADDITIONAL SIMULATION RESULTS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            661,
+            513,
+            676
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "text",
+        "text": "H.1 BINARY CLASSIFICATION ON ADDITIONAL DATSETS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            691,
+            591,
+            705
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "text",
+        "text": "In this part, we perform the binary classification task as in Section 4 on additional datasets, i.e.,  $a9a$  (with 123 features) and splice (with 60 features) from LIBSVM (Chang & Lin, 2011). Figure 4 provides the performance ordering of different  $\\alpha$  values, and we empirically demonstrate that the curves with  $\\alpha \\geq 5$  still outperform the i.i.d. counterpart. Additionally, Figure 5 compare cases (i) - (iii) under both  $a9a$  and splice datasets, and case (i) consistently perform the best.",
+        "bbox": [
+            169,
+            718,
+            823,
+            789
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "text",
+        "text": "H.2 NON-CONVEX LINEAR REGRESSION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            805,
+            470,
+            819
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "text",
+        "text": "We further test SGD-SRRW and SHB-SRRW algorithms with a non-convex function to demonstrate the efficiency of our SA-SRRW algorithm beyond the convex setting. In this task, we simulate the following linear regression problem in Khaled & Richtárik (2023) with non-convex regularization",
+        "bbox": [
+            169,
+            830,
+            823,
+            873
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\min  _ {\\boldsymbol {\\theta} \\in \\mathbb {R} ^ {d}} \\left\\{f (\\boldsymbol {\\theta}) \\triangleq \\frac {1}{N} \\sum_ {i = 1} ^ {N} l _ {i} (\\boldsymbol {\\theta}) + \\kappa \\sum_ {j = 1} ^ {d} \\frac {\\boldsymbol {\\theta} _ {j} ^ {2}}{\\boldsymbol {\\theta} _ {j} ^ {2} + 1} \\right\\} \\tag {108}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            341,
+            878,
+            823,
+            928
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "page_number",
+        "text": "40",
+        "bbox": [
+            488,
+            946,
+            509,
+            959
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "image",
+        "img_path": "images/2d1e8f8f246d13e6aa251844ff8a067dbf0170e28fea236f3258cbba8058ba8c.jpg",
+        "image_caption": [
+            "(a) SGD-SRRW,  $a9a$"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            171,
+            99,
+            488,
+            273
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "image",
+        "img_path": "images/8f2c6102f9a6088003a51d97deaa629169541c9c71233b2cc0570ed68c7d2053.jpg",
+        "image_caption": [
+            "(b) SGD-SRRW, splice"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            506,
+            99,
+            826,
+            273
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "image",
+        "img_path": "images/b8866562c6e07806e4f4c55c24f8e8e309d3bf7ec91a0ddef4f6ad41baa82448.jpg",
+        "image_caption": [
+            "(a)  $\\alpha = 5$  ,SGD-SRRW,  $a9a$"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            171,
+            332,
+            385,
+            441
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "image",
+        "img_path": "images/264b693785fa52b8afe5e2db5f54679f226463c06f8cfe1225ab7f1d6d6e6e1c.jpg",
+        "image_caption": [
+            "Figure 4: Simulation results with various  $\\alpha$  values in  $a9a$  and splice datasets."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            392,
+            330,
+            606,
+            440
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "image",
+        "img_path": "images/1a0d5f354d77a5b47896bcd75d825c3586cd1b099c55f28b0ee7c7f21c729e01.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            612,
+            330,
+            825,
+            439
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "image",
+        "img_path": "images/1577c5522b60a09974502a32cefeb168ec93c72b2edbe4bcdf77210e4c4f7713.jpg",
+        "image_caption": [
+            "(d)  $\\alpha = 5$  ,SGD-SRRW, splice"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            171,
+            459,
+            385,
+            565
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "image",
+        "img_path": "images/8c84d1d6c94b4e3086c8b7bf6884cb09a72b02849cfc7f603b16018cd8ee3b18.jpg",
+        "image_caption": [
+            "(b)  $\\alpha = 10$  ,SGD-SRRW,  $a9a$",
+            "(e)  $\\alpha = 10$  ,SGD-SRRW,splice"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            392,
+            459,
+            604,
+            566
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "image",
+        "img_path": "images/9c36ee1f937b69d21db470aff9b2d69cb83fc14bcd7e80899e2e16ddb03e694c.jpg",
+        "image_caption": [
+            "(c)  $\\alpha = 20$  ,SGD-SRRW,  $a9a$",
+            "(f)  $\\alpha = 20$  ,SGD-SRRW,splice",
+            "Figure 5: Performance comparison among cases (i) - (iii) for  $\\alpha \\in \\{5,10,20\\}$  in  $a9a$  and splice datasets."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            612,
+            459,
+            825,
+            565
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "text",
+        "text": "where the loss function  $l_{i}(\\pmb{\\theta}) = \\|\\mathbf{s}_{i}^{T}\\pmb{\\theta} - y_{i}\\|^{2}$  and  $\\kappa = 1$ , with the data points  $\\{(\\mathbf{s}_i,y_i)\\}_{i\\in \\mathcal{N}}$  from the ijCNN1 dataset of LIBVIM (Chang & Lin, 2011). We still perform the optimization over the wikiVote graph, as done in Section 4.",
+        "bbox": [
+            169,
+            650,
+            823,
+            694
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "text",
+        "text": "The numerical results for the non-convex linear regression taks are presented in Figures 6 and 7, where each experiment is repeated 100 times. Figures 6a and 6b show that the performance ordering across different  $\\alpha$  values is still preserved for both algorithms over almost all time, and curves for  $\\alpha \\geq 5$  outperform that of the i.i.d. sampling (in black) under the graph topological constraints. Additionally, among the three cases examined at identical  $\\alpha$  values, Figures 7a - 7c confirm that case (i) performs consistently better than the other two cases, implying that case (i) can even become the best choice for non-convex distributed optimization tasks.",
+        "bbox": [
+            169,
+            700,
+            826,
+            799
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "page_number",
+        "text": "41",
+        "bbox": [
+            488,
+            946,
+            506,
+            959
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "image",
+        "img_path": "images/94f57fdbc14f2a8f7f300bdc7855b74e0bc099ed55eed335cf7c26a74ed1b3d7.jpg",
+        "image_caption": [
+            "(a) SGD-SRRW"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            171,
+            209,
+            488,
+            381
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "image",
+        "img_path": "images/d29eb500eb3ae4ad0a5543bcd74368c38296c152e0e34ac58bbf76d0c4291400.jpg",
+        "image_caption": [
+            "(b) SHB-SRRW",
+            "Figure 6: Simulation results for non-convex linear regression under case (i) with various  $\\alpha$  values."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            506,
+            210,
+            823,
+            383
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "image",
+        "img_path": "images/ad925edb5647cb031d4b4cc76c2d11ec1c343d050407da3fa9f661246eec1754.jpg",
+        "image_caption": [
+            "(a)  $\\alpha = 5$  ,SGD-SRRW",
+            "Figure 7: Performance comparison among cases (i) - (iii) for non-convex regression."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            171,
+            657,
+            385,
+            763
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "image",
+        "img_path": "images/6a8587f7d29fcf98434402e7aae5e6b9d8ef46bfadc5b1dd5e7d86ebbc2a8788.jpg",
+        "image_caption": [
+            "(b)  $\\alpha = 10$  ,SGD-SRRW"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            393,
+            657,
+            604,
+            763
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "image",
+        "img_path": "images/a78add41258ce9f7def976cf23892a9d4df59d33a6f273c56037fad8cff23d92.jpg",
+        "image_caption": [
+            "(c)  $\\alpha = 20$  ,SGD-SRRW"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            612,
+            657,
+            823,
+            763
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "page_number",
+        "text": "42",
+        "bbox": [
+            488,
+            946,
+            509,
+            959
+        ],
+        "page_idx": 41
+    }
+]
\ No newline at end of file
diff --git a/2024/Accelerating Distributed Stochastic Optimization via Self-Repellent Random Walks/9e8fce71-9845-4483-82b8-2e8fb64c4467_model.json b/2024/Accelerating Distributed Stochastic Optimization via Self-Repellent Random Walks/9e8fce71-9845-4483-82b8-2e8fb64c4467_model.json
new file mode 100644
index 0000000000000000000000000000000000000000..af01574e96291a15aec32f8eacc6e0b9742dccb3
--- /dev/null
+++ b/2024/Accelerating Distributed Stochastic Optimization via Self-Repellent Random Walks/9e8fce71-9845-4483-82b8-2e8fb64c4467_model.json	
@@ -0,0 +1,9469 @@
+[
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.1,
+                0.825,
+                0.149
+            ],
+            "angle": 0,
+            "content": "ACCELERATING DISTRIBUTED STOCHASTIC OPTIMIZATION VIA SELF-REPELLENT RANDOM WALKS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.181,
+                0.17,
+                0.501,
+                0.185
+            ],
+            "angle": 0,
+            "content": "Jie Hu\\*1, Vishwaraj Doshi\\*2, Do Young Eun"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.184,
+                0.186,
+                0.49,
+                0.2
+            ],
+            "angle": 0,
+            "content": "\\(^{1}\\)North Carolina State University, \\(^{2}\\)IQVIA Inc."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.184,
+                0.2,
+                0.663,
+                0.215
+            ],
+            "angle": 0,
+            "content": "{jhu29, dyeun}@ncsu.edu, vishwaraj.doshi@iqvia.com"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.451,
+                0.25,
+                0.548,
+                0.265
+            ],
+            "angle": 0,
+            "content": "ABSTRACT"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.23,
+                0.281,
+                0.77,
+                0.561
+            ],
+            "angle": 0,
+            "content": "We study a family of distributed stochastic optimization algorithms where gradients are sampled by a token traversing a network of agents in random-walk fashion. Typically, these random-walks are chosen to be Markov chains that asymptotically sample from a desired target distribution, and play a critical role in the convergence of the optimization iterates. In this paper, we take a novel approach by replacing the standard linear Markovian token by one which follows a nonlinear Markov chain - namely the Self-Repellent Radom Walk (SRRW). Defined for any given 'base' Markov chain, the SRRW, parameterized by a positive scalar \\(\\alpha\\), is less likely to transition to states that were highly visited in the past, thus the name. In the context of MCMC sampling on a graph, a recent breakthrough in Doshi et al. (2023) shows that the SRRW achieves \\(O(1 / \\alpha)\\) decrease in the asymptotic variance for sampling. We propose the use of a 'generalized' version of the SRRW to drive token algorithms for distributed stochastic optimization in the form of stochastic approximation, termed SA-SRRW. We prove that the optimization iterate errors of the resulting SA-SRRW converge to zero almost surely and prove a central limit theorem, deriving the explicit form of the resulting asymptotic covariance matrix corresponding to iterate errors. This asymptotic covariance is always smaller than that of an algorithm driven by the base Markov chain and decreases at rate \\(O(1 / \\alpha^2)\\) - the performance benefit of using SRRW thereby amplified in the stochastic optimization context. Empirical results support our theoretical findings."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.174,
+                0.586,
+                0.34,
+                0.601
+            ],
+            "angle": 0,
+            "content": "1 INTRODUCTION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.616,
+                0.673,
+                0.632
+            ],
+            "angle": 0,
+            "content": "Stochastic optimization algorithms solve optimization problems of the form"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.259,
+                0.635,
+                0.826,
+                0.667
+            ],
+            "angle": 0,
+            "content": "\\[\n\\boldsymbol {\\theta} ^ {*} \\in \\underset {\\boldsymbol {\\theta} \\in \\mathbb {R} ^ {d}} {\\arg \\min } f (\\boldsymbol {\\theta}), \\quad \\text {w h e r e} f (\\boldsymbol {\\theta}) \\triangleq \\mathbb {E} _ {X \\sim \\boldsymbol {\\mu}} [ F (\\boldsymbol {\\theta}, X) ] = \\sum_ {i \\in \\mathcal {N}} \\mu_ {i} F (\\boldsymbol {\\theta}, i), \\tag {1}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.671,
+                0.827,
+                0.745
+            ],
+            "angle": 0,
+            "content": "with the objective function \\( f: \\mathbb{R}^d \\to \\mathbb{R} \\) and \\( X \\) taking values in a finite state space \\( \\mathcal{N} \\) with distribution \\( \\mu \\triangleq [\\mu_i]_{i \\in \\mathcal{N}} \\). Leveraging partial gradient information per iteration, these algorithms have been recognized for their scalability and efficiency with large datasets (Bottou et al., 2018; Even, 2023). For any given noise sequence \\( \\{X_n\\}_{n \\geq 0} \\subset \\mathcal{N} \\), and step size sequence \\( \\{\\beta_n\\}_{n \\geq 0} \\subset \\mathbb{R}_+ \\), most stochastic optimization algorithms can be classified as stochastic approximations (SA) of the form"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.346,
+                0.75,
+                0.825,
+                0.768
+            ],
+            "angle": 0,
+            "content": "\\[\n\\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} + \\beta_ {n + 1} H \\left(\\boldsymbol {\\theta} _ {n}, X _ {n + 1}\\right), \\quad \\forall n \\geq 0, \\tag {2}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.773,
+                0.827,
+                0.888
+            ],
+            "angle": 0,
+            "content": "where, roughly speaking, \\(H(\\pmb{\\theta}, i)\\) contains gradient information \\(\\nabla_{\\pmb{\\theta}} F(\\pmb{\\theta}, i)\\), such that \\(\\pmb{\\theta}^*\\) solves \\(\\mathbf{h}(\\pmb{\\theta}) \\triangleq \\mathbb{E}_{X \\sim \\pmb{\\mu}}[H(\\pmb{\\theta}, X)] = \\sum_{i \\in \\mathcal{N}} \\mu_i H(\\pmb{\\theta}, i) = \\mathbf{0}\\). Such SA iterations include the well-known stochastic gradient descent (SGD), stochastic heavy ball (SHB) (Gadat et al., 2018; Li et al., 2022), and some SGD-type algorithms employing additional auxiliary variables (Barakat et al., 2021). These algorithms typically have the stochastic noise term \\(X_n\\) generated by i.i.d. random variables with probability distribution \\(\\pmb{\\mu}\\) in each iteration. In this paper, we study a stochastic optimization algorithm where the noise sequence governing access to the gradient information is generated from general stochastic processes in place of i.i.d. random variables."
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.193,
+                0.897,
+                0.316,
+                0.91
+            ],
+            "angle": 0,
+            "content": "* Equal contributors."
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.193,
+                0.91,
+                0.8,
+                0.925
+            ],
+            "angle": 0,
+            "content": "Further illustrations of stochastic optimization algorithms of the form (2) are deferred to Appendix A."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.193,
+                0.897,
+                0.8,
+                0.925
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.495,
+                0.949,
+                0.505,
+                0.96
+            ],
+            "angle": 0,
+            "content": "1"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.827,
+                0.432
+            ],
+            "angle": 0,
+            "content": "This is commonly the case in distributed learning, where \\(\\{X_{n}\\}\\) is a (typically Markovian) random walk, and should asymptotically be able to sample the gradients from the desired probability distribution \\(\\mu\\). This is equivalent to saying that the random walker's empirical distribution converges to \\(\\mu\\) almost surely (a.s.); that is, \\(\\mathbf{x}_n \\triangleq \\frac{1}{n+1} \\sum_{k=0}^{n} \\delta_{X_k} \\xrightarrow[n\\to\\infty]{a.s.} \\mu\\) for any initial \\(X_0 \\in \\mathcal{N}\\), where \\(\\delta_{X_k}\\) is the delta measure whose \\(X_k\\)'th entry is one, the rest being zero. Such convergence is most commonly achieved by employing the Metropolis Hastings random walk (MHRW) which can be designed to sample from any target measure \\(\\mu\\) and implemented in a scalable manner (Sun et al., 2018). Unsurprisingly, convergence characteristics of the employed Markov chain affect that of the SA sequence (2), and appear in both finite-time and asymptotic analyses. Finite-time bounds typically involve the second largest eigenvalue in modulus (SLEM) of the Markov chain's transition kernel \\(\\mathbf{P}\\), which is critically connected to the mixing time of a Markov chain (Levin & Peres, 2017); whereas asymptotic results such as central limit theorems (CLT) involve asymptotic covariance matrices that embed information regarding the entire spectrum of \\(\\mathbf{P}\\), i.e., all eigenvalues as well as eigenvectors (Bremaud, 2013), which are key to understanding the sampling efficiency of a Markov chain. Thus, the choice of random walker can significantly impact the performance of (2), and simply ensuring that it samples from \\(\\mu\\) asymptotically is not enough to achieve optimal algorithmic performance. In this paper, we take a closer look at the distributed stochastic optimization problem through the lens of a non-linear Markov chain, known as the Self Repellent Random Walk (SRRW), which was shown in Doshi et al. (2023) to achieve asymptotically minimal sampling variance for large values of \\(\\alpha\\), a positive scalar controlling the strength of the random walker's self-repellence behaviour. Our proposed modification of (2) can be implemented within the settings of decentralized learning applications in a scalable manner, while also enjoying significant performance benefit over distributed stochastic optimization algorithms driven by vanilla Markov chains."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.438,
+                0.825,
+                0.577
+            ],
+            "angle": 0,
+            "content": "Token Algorithms for Decentralized Learning. In decentralized learning, agents like smartphones or IoT devices, each containing a subset of data, collaboratively train models on a graph \\(\\mathcal{G}(\\mathcal{N},\\mathcal{E})\\) by sharing information locally without a central server (McMahan et al., 2017). In this setup, \\(N = |\\mathcal{N}|\\) agents correspond to nodes \\(i\\in \\mathcal{N}\\), and an edge \\((i,j)\\in \\mathcal{E}\\) indicates direct communication between agents \\(i\\) and \\(j\\). This decentralized approach offers several advantages compared to the traditional centralized learning setting, promoting data privacy and security by eliminating the need for raw data to be aggregated centrally and thus reducing the risk of data breach or misuse (Bottou et al., 2018; Nedic, 2020). Additionally, decentralized approaches are more scalable and can handle vast amounts of heterogeneous data from distributed agents without overwhelming a central server, alleviating concerns about single point of failure (Vogels et al., 2021)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.584,
+                0.825,
+                0.793
+            ],
+            "angle": 0,
+            "content": "Among decentralized learning approaches, the class of 'Token' algorithms can be expressed as stochastic approximation iterations of the type (2), wherein the sequence \\(\\{X_{n}\\}\\) is realized as the sample path of a token that stochastically traverses the graph \\(\\mathcal{G}\\), carrying with it the iterate \\(\\theta_{n}\\) for any time \\(n\\geq 0\\) and allowing each visited node (agent) to incrementally update \\(\\theta_{n}\\) using locally available gradient information. Token algorithms have gained popularity in recent years (Hu et al., 2022; Triastcyn et al., 2022; Hendrikx, 2023), and are provably more communication efficient (Even, 2023) when compared to consensus-based algorithms - another popular approach for solving distributed optimization problems (Boyd et al., 2006; Morral et al., 2017; Olshevsky, 2022). The construction of token algorithms means that they do not suffer from expensive costs of synchronization and communication that are typical of consensus-based approaches, where all agents (or a subset of agents selected by a coordinator (Boyd et al., 2006; Wang et al., 2019)) on the graph are required to take simultaneous actions, such as communicating on the graph at each iteration. While decentralized Federated learning has indeed helped mitigate the communication overhead by processing multiple SGD iterations prior to each aggregation (Lalitha et al., 2018; Ye et al., 2022; Chellapandi et al., 2023), they still cannot overcome challenges such as synchronization and straggler issues."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.798,
+                0.825,
+                0.87
+            ],
+            "angle": 0,
+            "content": "Self Repellent Random Walk. As mentioned earlier, sample paths \\(\\{X_{n}\\}\\) of token algorithms are usually generated using Markov chains with \\(\\mu \\in \\mathrm{Int}(\\Sigma)\\) as their limiting distribution. Here, \\(\\Sigma\\) denotes the \\(N\\)-dimensional probability simplex, with \\(\\mathrm{Int}(\\Sigma)\\) representing its interior. A recent work by Doshi et al. (2023) pioneers the use of non-linear Markov chains to, in some sense, improve upon any given time-reversible Markov chain with transition kernel \\(\\mathbf{P}\\) whose stationary distribution is \\(\\mu\\)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.948,
+                0.506,
+                0.96
+            ],
+            "angle": 0,
+            "content": "2"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.103,
+                0.735,
+                0.12
+            ],
+            "angle": 0,
+            "content": "They show that the non-linear transition kernel\\(^2\\) \\(\\mathbf{K}[\\cdot ]:\\mathrm{Int}(\\Sigma)\\to [0,1]^{N\\times N}\\), given by"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.334,
+                0.121,
+                0.825,
+                0.155
+            ],
+            "angle": 0,
+            "content": "\\[\nK _ {i j} [ \\mathbf {x} ] \\triangleq \\frac {P _ {i j} \\left(x _ {j} / \\mu_ {j}\\right) ^ {- \\alpha}}{\\sum_ {k \\in \\mathcal {N}} P _ {i k} \\left(x _ {k} / \\mu_ {k}\\right) ^ {- \\alpha}}, \\quad \\forall i, j \\in \\mathcal {N}, \\tag {3}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.155,
+                0.827,
+                0.35
+            ],
+            "angle": 0,
+            "content": "for any \\(\\mathbf{x} \\in \\mathrm{Int}(\\Sigma)\\), when simulated as a self-interacting random walk (Del Moral & Miclo, 2006; Del Moral & Doucet, 2010), can achieve smaller asymptotic variance than the base Markov chain when sampling over a graph \\(\\mathcal{G}\\), for all \\(\\alpha > 0\\). The argument \\(\\mathbf{x}\\) for the kernel \\(\\mathbf{K}[x]\\) is taken to be the empirical distribution \\(\\mathbf{x}_n\\) at each time step \\(n \\geq 0\\). For instance, if node \\(j\\) has been visited more often than other nodes so far, the entry \\(x_j\\) becomes larger (than target value \\(\\mu_j\\)), resulting in a smaller transition probability from \\(i\\) to \\(j\\) under \\(\\mathbf{K}[\\mathbf{x}]\\) in (3) compared to \\(P_{ij}\\). This ensures that a random walker prioritizes more seldom visited nodes in the process, and is thus 'self-repellent'. This effect is made more drastic by increasing \\(\\alpha\\), and leads to asymptotically near-zero variance at a rate of \\(O(1 / \\alpha)\\). Moreover, the polynomial function \\((x_i / \\mu_i)^{-\\alpha}\\) chosen to encode self-repellent behaviour is shown in Doshi et al. (2023) to be the only one that allows the SRRW to inherit the so-called 'scale-invariance' property of the underlying Markov chain - a necessary component for the scalable implementation of a random walker over a large network without requiring knowledge of any graph-related global constants. Conclusively, such attributes render SRRW especially suitable for distributed optimization.\\(^3\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.357,
+                0.825,
+                0.51
+            ],
+            "angle": 0,
+            "content": "Effect of Stochastic Noise - Finite time and Asymptotic Approaches. Most contemporary token algorithms driven by Markov chains are analyzed using the finite-time bounds approach for obtaining insights into their convergence rates (Sun et al., 2018; Doan et al., 2019; Triastcyn et al., 2022; Hendrikx, 2023). However, as also explained in Even (2023), in most cases these bounds are overly dependent on mixing time properties of the specific Markov chain employed therein. This makes them largely ineffective in capturing the exact contribution of the underlying random walk in a manner which is qualitative enough to be used for algorithm design; and performance enhancements are typically achieved via application of techniques such as variance reduction (Defazio et al., 2014; Schmidt et al., 2017), momentum/Nesterov's acceleration (Gadat et al., 2018; Li et al., 2022), adaptive step size (Kingma & Ba, 2015; Reddi et al., 2018), which work by modifying the algorithm iterations themselves, and never consider potential improvements to the stochastic input itself."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.516,
+                0.825,
+                0.614
+            ],
+            "angle": 0,
+            "content": "Complement to finite-time approaches, asymptotic analysis using CLT has proven to be an excellent tool to approach the design of stochastic algorithms (Hu et al., 2022; Devraj & Meyn, 2017; Morral et al., 2017; Chen et al., 2020a; Mou et al., 2020; Devraj & Meyn, 2021). Hu et al. (2022) shows how asymptotic analysis can be used to compare the performance of SGD algorithms for various stochastic inputs using their notion of efficiency ordering, and, as mentioned in Devraj & Meyn (2017), the asymptotic benefits from minimizing the limiting covariance matrix are known to be a good predictor of finite-time algorithmic performance, also observed empirically in Section 4."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.621,
+                0.825,
+                0.747
+            ],
+            "angle": 0,
+            "content": "From the perspective of both finite-time analysis as well as asymptotic analysis of token algorithms, it is now well established that employing 'better' Markov chains can enhance the performance of stochastic optimization algorithm. For instance, Markov chains with smaller SLEMs yield tighter finite-time upper bounds (Sun et al., 2018; Ayache & El Rouayheb, 2021; Even, 2023). Similarly, Markov chains with smaller asymptotic variance for MCMC sampling tasks also provide better performance, resulting in smaller covariance matrix of SGD algorithms (Hu et al., 2022). Therefore, with these breakthrough results via SRRW achieving near-zero sampling variance, it is within reason to ask: Can we achieve near-zero variance in distributed stochastic optimization driven by SRRW-like token algorithms on any general graph?4 In this paper, we answer in the affirmative."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.753,
+                0.825,
+                0.798
+            ],
+            "angle": 0,
+            "content": "SRRW Driven Algorithm and Analysis Approach. For any ergodic time-reversible Markov chain with transition probability matrix \\(\\mathbf{P} \\triangleq [P_{ij}]_{i,j \\in \\mathcal{N}}\\) and stationary distribution \\(\\pmb{\\mu} \\in \\mathrm{Int}(\\Sigma)\\), we consider a general step size version of the SRRW stochastic process analysed in Doshi et al. (2023) and"
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.171,
+                0.807,
+                0.825,
+                0.846
+            ],
+            "angle": 0,
+            "content": "2Here, non-linearity in the transition kernel implies that \\(\\mathbf{K}[\\mathbf{x}]\\) takes probability distribution \\(\\mathbf{x}\\) as the argument (Andrieu et al., 2007), as opposed to the kernel being a linear operator \\(\\mathbf{K}[\\mathbf{x}] = \\mathbf{P}\\) for a constant stochastic matrix \\(\\mathbf{P}\\) in a standard (linear) Markovian setting."
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.171,
+                0.847,
+                0.825,
+                0.898
+            ],
+            "angle": 0,
+            "content": "<sup>3</sup>Recently, Guo et al. (2020) introduce an optimization scheme, which designs self-repellence into the perturbation of the gradient descent iterates (Jin et al., 2017; 2018; 2021) with the goal of escaping saddle points. This notion of self-repellence is distinct from the SRRW, which is a probability kernel designed specifically for a token to sample from a target distribution \\(\\mu\\) over a set of nodes on an arbitrary graph."
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.171,
+                0.898,
+                0.825,
+                0.925
+            ],
+            "angle": 0,
+            "content": "This near-zero sampling variance implies a significantly smaller variance than even an i.i.d. sampling counterpart, while adhering to graph topological constraints of token algorithms."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.171,
+                0.807,
+                0.825,
+                0.925
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.504,
+                0.96
+            ],
+            "angle": 0,
+            "content": "3"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.237,
+                0.102,
+                0.76,
+                0.193
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.171,
+                0.193,
+                0.825,
+                0.237
+            ],
+            "angle": 0,
+            "content": "Figure 1: Visualization of token algorithms using SRRW versus traditional MC in distributed learning. Our CLT analysis, extended from SRRW itself to distributed stochastic approximation, leads to near-zero variance for the SA iteration \\(\\theta_{n}\\). Node numbers on the left denote visit counts."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.244,
+                0.7,
+                0.259
+            ],
+            "angle": 0,
+            "content": "use it to drive the noise sequence in (2). Our SA-SRRW algorithm is as follows:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.344,
+                0.259,
+                0.825,
+                0.274
+            ],
+            "angle": 0,
+            "content": "Draw: \\(X_{n + 1}\\sim \\mathbf{K}_{X_n,\\cdot}[\\mathbf{x}_n]\\) (4a)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.344,
+                0.276,
+                0.825,
+                0.293
+            ],
+            "angle": 0,
+            "content": "Update: \\(\\mathbf{x}_{n + 1} = \\mathbf{x}_n + \\gamma_{n + 1}(\\pmb{\\delta}_{X_{n + 1}} - \\mathbf{x}_n),\\) (4b)"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.422,
+                0.295,
+                0.825,
+                0.311
+            ],
+            "angle": 0,
+            "content": "\\[\n\\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} + \\beta_ {n + 1} H \\left(\\boldsymbol {\\theta} _ {n}, X _ {n + 1}\\right), \\tag {4c}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.313,
+                0.827,
+                0.466
+            ],
+            "angle": 0,
+            "content": "where \\(\\{\\beta_n\\}\\) and \\(\\{\\gamma_n\\}\\) are step size sequences decreasing to zero, and \\(\\mathbf{K}[\\mathbf{x}]\\) is the SRRW kernel in (3). Current non-asymptotic analyses require globally Lipschitz mean field function (Chen et al., 2020b; Doan, 2021; Zeng et al., 2021; Even, 2023) and is thus inapplicable to SA-SRRW since the mean field function of the SRRW iterates (4b) is only locally Lipschitz (details deferred to Appendix B). Instead, we successfully obtain non-trivial results by taking an asymptotic CLT-based approach to analyze (4). This goes beyond just analyzing the asymptotic sampling covariance<sup>5</sup> as in Doshi et al. (2023), the result therein forming a special case of ours by setting \\(\\gamma_{n} = 1 / (n + 1)\\) and considering only (4a) and (4b), that is, in the absence of optimization iteration (4c). Specifically, we capture the effect of SRRW's hyper-parameter \\(\\alpha\\) on the asymptotic speed of convergence of the optimization error term \\(\\theta_{n} - \\theta^{*}\\) to zero via explicit deduction of its asymptotic covariance matrix. See Figure 1 for illustration."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.473,
+                0.313,
+                0.487
+            ],
+            "angle": 0,
+            "content": "Our Contributions."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.494,
+                0.825,
+                0.581
+            ],
+            "angle": 0,
+            "content": "1. Given any time-reversible 'base' Markov chain with transition kernel \\(\\mathbf{P}\\) and stationary distribution \\(\\mu\\), we generalize first and second order convergence results of \\(\\mathbf{x}_n\\) to target measure \\(\\pmb{\\mu}\\) (Theorems 4.1 and 4.2 in Doshi et al., 2023) to a class of weighted empirical measures, through the use of more general step sizes \\(\\gamma_{n}\\). This includes showing that the asymptotic sampling covariance terms decrease to zero at rate \\(O(1 / \\alpha)\\), thus quantifying the effect of self-repellent on \\(\\mathbf{x}_n\\). Our generalization is not for the sake thereof and is shown in Section 3 to be crucial for the design of step sizes \\(\\beta_{n},\\gamma_{n}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.585,
+                0.825,
+                0.614
+            ],
+            "angle": 0,
+            "content": "2. Building upon the convergence results for iterates \\(\\mathbf{x}_n\\), we analyze the algorithm (4) driven by the SRRW kernel in (3) with step sizes \\(\\beta_{n}\\) and \\(\\gamma_{n}\\) separated into three disjoint cases:"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.171,
+                0.494,
+                0.825,
+                0.614
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.205,
+                0.618,
+                0.73,
+                0.633
+            ],
+            "angle": 0,
+            "content": "(i) \\(\\beta_{n} = o(\\gamma_{n})\\), and we say that \\(\\theta_{n}\\) is on the slower timescale compared to \\(\\mathbf{x}_n\\);"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.202,
+                0.633,
+                0.649,
+                0.648
+            ],
+            "angle": 0,
+            "content": "(ii) \\(\\beta_{n} = \\gamma_{n}\\), and we say that \\(\\theta_{n}\\) and \\(\\mathbf{x}_n\\) are on the same timescale;"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.648,
+                0.721,
+                0.664
+            ],
+            "angle": 0,
+            "content": "(iii) \\(\\gamma_{n} = o(\\beta_{n})\\), and we say that \\(\\theta_{n}\\) is on the faster timescale compared to \\(\\mathbf{x}_n\\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.198,
+                0.618,
+                0.73,
+                0.664
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.667,
+                0.82,
+                0.682
+            ],
+            "angle": 0,
+            "content": "For any \\(\\alpha \\geq 0\\) and let \\(k = 1,2\\) and 3 refer to the corresponding cases (i), (ii) and (iii), we show that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.282,
+                0.685,
+                0.714,
+                0.711
+            ],
+            "angle": 0,
+            "content": "\\[\n\\boldsymbol {\\theta} _ {n} \\xrightarrow [ n \\to \\infty ]{a . s .} \\boldsymbol {\\theta} ^ {*} \\quad \\text {a n d} \\quad (\\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*}) / \\sqrt {\\beta_ {n}} \\xrightarrow [ n \\to \\infty ]{d i s t.} N \\left(\\mathbf {0}, \\mathbf {V} _ {\\boldsymbol {\\theta}} ^ {(k)} (\\alpha)\\right),\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.716,
+                0.825,
+                0.777
+            ],
+            "angle": 0,
+            "content": "featuring distinct asymptotic covariance matrices \\(\\mathbf{V}_{\\theta}^{(1)}(\\alpha), \\mathbf{V}_{\\theta}^{(2)}(\\alpha)\\) and \\(\\mathbf{V}_{\\theta}^{(3)}(\\alpha)\\), respectively. The three matrices coincide when \\(\\alpha = 0,6\\). Moreover, the derivation of the CLT for cases (i) and (iii), for which (4) corresponds to two-timescale SA with controlled Markov noise, is the first of its kind and thus a key technical contribution in this paper, as expanded upon in Section 3."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.785,
+                0.826,
+                0.876
+            ],
+            "angle": 0,
+            "content": "3. For case (i), we show that \\(\\mathbf{V}_{\\theta}^{(1)}(\\alpha)\\) decreases to zero (in the sense of Loewner ordering introduced in Section 2.1) as \\(\\alpha\\) increases, with rate \\(O(1 / \\alpha^2)\\). This is especially surprising, since the asymptotic performance benefit from using the SRRW kernel with \\(\\alpha\\) in (3), to drive the noise terms \\(X_{n}\\), is amplified in the context of distributed learning and estimating \\(\\theta^{*}\\); compared to the sampling case, for which the rate is \\(O(1 / \\alpha)\\) as mentioned earlier. For case (iii), we show that \\(\\mathbf{V}_{\\theta}^{(3)}(\\alpha) = \\mathbf{V}_{\\theta}^{(3)}(0)\\) for all \\(\\alpha \\geq 0\\), implying that using the SRRW in this case provides no asymptotic benefit than the"
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.191,
+                0.884,
+                0.666,
+                0.898
+            ],
+            "angle": 0,
+            "content": "\\( {}^{5} \\) Sampling covariance corresponds to only the empirical distribution \\( {\\mathbf{x}}_{n} \\) in (4b)."
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.171,
+                0.897,
+                0.825,
+                0.925
+            ],
+            "angle": 0,
+            "content": "The \\(\\alpha = 0\\) case is equivalent to simply running the base Markov chain, since from (3) we have \\(\\mathbf{K}[\\cdot ] = \\mathbf{P}\\), thus bypassing the SRRW's effect and rendering all three cases nearly the same."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.171,
+                0.884,
+                0.825,
+                0.925
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.505,
+                0.96
+            ],
+            "angle": 0,
+            "content": "4"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.825,
+                0.138
+            ],
+            "angle": 0,
+            "content": "original base Markov chain, and thus performs worse than case (i). In summary, we deduce that \\(\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(\\alpha_2) < _L\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(\\alpha_1) < _L\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(0) = \\mathbf{V}_{\\pmb{\\theta}}^{(3)}(0) = \\mathbf{V}_{\\pmb{\\theta}}^{(3)}(\\alpha)\\) for all \\(\\alpha_{2} > \\alpha_{1} > 0\\) and \\(\\alpha >0\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.143,
+                0.825,
+                0.228
+            ],
+            "angle": 0,
+            "content": "4. We numerically simulate our SA-SRRW algorithm on various real-world datasets, focusing on a binary classification task, to evaluate its performance across all three cases. By carefully choosing the function \\( H \\) in SA-SRRW, we test the SGD and algorithms driven by SRRW. Our findings consistently highlight the superiority of case (i) over cases (ii) and (iii) for diverse \\( \\alpha \\) values, even in their finite time performance. Notably, our tests validate the variance reduction at a rate of \\( O(1 / \\alpha^2) \\) for case (i), suggesting it as the best algorithmic choice among the three cases."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.247,
+                0.513,
+                0.264
+            ],
+            "angle": 0,
+            "content": "2 PRELIMINARIES AND MODEL SETUP"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.278,
+                0.825,
+                0.336
+            ],
+            "angle": 0,
+            "content": "In Section 2.1, we first standardize the notations used throughout the paper, and define key mathematical terms and quantities used in our theoretical analyses. Then, in Section 2.2, we consolidate the model assumptions of our SA-SRRW algorithm (4). We then go on to discuss our assumptions, and provide additional interpretations of our use of generalized step-sizes."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.351,
+                0.478,
+                0.365
+            ],
+            "angle": 0,
+            "content": "2.1 BASIC NOTATIONS AND DEFINITIONS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.376,
+                0.827,
+                0.505
+            ],
+            "angle": 0,
+            "content": "Vectors are denoted by lower-case bold letters, e.g., \\(\\mathbf{v} \\triangleq [v_i] \\in \\mathbb{R}^D\\), and matrices by upper-case bold, e.g., \\(\\mathbf{M} \\triangleq [M_{ij}] \\in \\mathbb{R}^{D \\times D}\\). \\(\\mathbf{M}^{-T}\\) is the transpose of the matrix inverse \\(\\mathbf{M}^{-1}\\). The diagonal matrix \\(\\mathbf{D}_{\\mathbf{v}}\\) is formed by vector \\(\\mathbf{v}\\) with \\(v_i\\) as the \\(i\\)'th diagonal entry. Let \\(\\mathbf{1}\\) and \\(\\mathbf{0}\\) denote vectors of all ones and zeros, respectively. The identity matrix is represented by \\(\\mathbf{I}\\), with subscripts indicating dimensions as needed. A matrix is Hurwitz if all its eigenvalues possess strictly negative real parts. \\(\\mathbb{1}_{\\{\\cdot\\}}\\) denotes an indicator function with condition in parentheses. We use \\(\\|\\cdot\\|\\) to denote both the Euclidean norm of vectors and the spectral norm of matrices. Two symmetric matrices \\(\\mathbf{M}_1, \\mathbf{M}_2\\) follow Loewner ordering \\(\\mathbf{M}_1 <_L \\mathbf{M}_2\\) if \\(\\mathbf{M}_2 - \\mathbf{M}_1\\) is positive semi-definite and \\(\\mathbf{M}_1 \\neq \\mathbf{M}_2\\). This slightly differs from the conventional definition with \\(\\leq_L\\), which allows \\(\\mathbf{M}_1 = \\mathbf{M}_2\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.512,
+                0.827,
+                0.641
+            ],
+            "angle": 0,
+            "content": "Throughout the paper, the matrix \\(\\mathbf{P} \\triangleq [P_{i,j}]_{i,j \\in \\mathcal{N}}\\) and vector \\(\\pmb{\\mu} \\triangleq [\\mu_i]_{i \\in \\mathcal{N}}\\) are used exclusively to denote an \\(N \\times N\\)-dimensional transition kernel of an ergodic Markov chain, and its stationary distribution, respectively. Without loss of generality, we assume \\(P_{ij} > 0\\) if and only if \\(a_{ij} > 0\\). Markov chains satisfying the detailed balance equation, where \\(\\mu_i P_{ij} = \\mu_j P_{ji}\\) for all \\(i, j \\in \\mathcal{N}\\), are termed time-reversible. For such chains, we use \\((\\lambda_i, \\mathbf{u}_i)\\) (resp. \\((\\lambda_i, \\mathbf{v}_i)\\)) to denote the \\(i\\)'th left (resp. right) eigenpair where the eigenvalues are ordered: \\(-1 < \\lambda_1 \\leq \\dots \\leq \\lambda_{N-1} < \\lambda_N = 1\\), with \\(\\mathbf{u}_N = \\pmb{\\mu}\\) and \\(\\mathbf{v}_N = \\mathbf{1}\\) in \\(\\mathbb{R}^N\\). We assume eigenvectors to be normalized such that \\(\\mathbf{u}_i^T \\mathbf{v}_i = 1\\) for all \\(i\\), and we have \\(\\mathbf{u}_i = \\mathbf{D}_{\\pmb{\\mu}} \\mathbf{v}_i\\) and \\(\\mathbf{u}_i^T \\mathbf{v}_j = 0\\) for all \\(i, j \\in \\mathcal{N}\\). We direct the reader to Aldous & Fill (2002, Chapter 3.4) for a detailed exposition on spectral properties of time-reversible Markov chains."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.656,
+                0.574,
+                0.67
+            ],
+            "angle": 0,
+            "content": "2.2 SA-SRRW: KEY ASSUMPTIONS AND DISCUSSIONS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.682,
+                0.722,
+                0.697
+            ],
+            "angle": 0,
+            "content": "Assumptions: All results in our paper are proved under the following assumptions."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.191,
+                0.7,
+                0.825,
+                0.73
+            ],
+            "angle": 0,
+            "content": "(A1) The function \\( H: \\mathbb{R}^D \\times \\mathcal{N} \\to \\mathbb{R}^D \\), is a continuous at every \\( \\pmb{\\theta} \\in \\mathbb{R}^D \\), and there exists a positive constant \\( L \\) such that \\( \\| H(\\pmb{\\theta}, i) \\| \\leq L (1 + \\| \\pmb{\\theta} \\|) \\) for every \\( \\pmb{\\theta} \\in \\mathbb{R}^D, i \\in \\mathcal{N} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.191,
+                0.733,
+                0.797,
+                0.749
+            ],
+            "angle": 0,
+            "content": "(A2) Step sizes \\(\\beta_{n}\\) and \\(\\gamma_{n}\\) follow \\(\\beta_{n} = (n + 1)^{-b}\\), and \\(\\gamma_{n} = (n + 1)^{-a}\\), where \\(a, b \\in (0.5, 1]\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.191,
+                0.753,
+                0.825,
+                0.807
+            ],
+            "angle": 0,
+            "content": "(A3) Roots of function \\(\\mathbf{h}(\\cdot)\\) are disjoint, which comprise the globally attracting set \\(\\Theta \\triangleq \\left\\{\\pmb{\\theta}^{*}|\\mathbf{h}(\\pmb{\\theta}^{*}) = \\mathbf{0}, \\nabla \\mathbf{h}(\\pmb{\\theta}^{*}) + \\frac{\\mathbb{1}_{\\{b=1\\}}}{2}\\mathbf{I}\\text{ is Hurwitz}\\right\\} \\neq \\emptyset\\) of the associated ordinary differential equation (ODE) for iteration (4c), given by \\(d\\pmb{\\theta}(t)/dt = \\mathbf{h}(\\pmb{\\theta}(t))\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.191,
+                0.81,
+                0.825,
+                0.842
+            ],
+            "angle": 0,
+            "content": "(A4) For any \\((\\pmb{\\theta}_0, \\mathbf{x}_0, X_0) \\in \\mathbb{R}^D \\times \\mathrm{Int}(\\Sigma) \\times \\mathcal{N}\\), the iterate sequence \\(\\{\\pmb{\\theta}_n\\}_{n \\geq 0}\\) (resp. \\(\\{\\mathbf{x}_n\\}_{n \\geq 0}\\)) is \\(\\mathbb{P}_{\\pmb{\\theta}_0, \\mathbf{x}_0, X_0}\\)-almost surely contained within a compact subset of \\(\\mathbb{R}^D\\) (resp. \\(\\mathrm{Int}(\\Sigma)\\))."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.191,
+                0.7,
+                0.825,
+                0.842
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.845,
+                0.825,
+                0.888
+            ],
+            "angle": 0,
+            "content": "Discussions on Assumptions: Assumption A1 requires \\( H \\) to only be locally Lipschitz albeit with linear growth, and is less stringent than the globally Lipschitz assumption prevalent in optimization literature (Li & Wai, 2022; Hendrikx, 2023; Even, 2023)."
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.171,
+                0.896,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "In particular, this is the reason why we advocate for a more general step size \\(\\gamma_{n} = (n + 1)^{-a}\\) in the SRRW iterates with \\(a < 1\\), allowing us to choose \\(\\beta_{n} = (n + 1)^{-b}\\) with \\(b \\in (a, 1]\\) to satisfy \\(\\beta_{n} = o(\\gamma_{n})\\) for case (i)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.505,
+                0.96
+            ],
+            "angle": 0,
+            "content": "5"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.104,
+                0.827,
+                0.218
+            ],
+            "angle": 0,
+            "content": "Assumption A2 is the general umbrella assumption under which cases (i), (ii) and (iii) mentioned in Section 1 are extracted by setting: (i) \\(a < b\\), (ii) \\(a = b\\), and (iii) \\(a > b\\). Cases (i) and (iii) render \\(\\theta_{n}, \\mathbf{x}_{n}\\) on different timescales; the polynomial form of \\(\\beta_{n}, \\gamma_{n}\\) widely assumed in the two-timescale SA literature (Mokkadem & Pelletier, 2006; Zeng et al., 2021; Hong et al., 2023). Case (ii) characterizes the SA-SRRW algorithm (4) as a single-timescale SA with polynomially decreasing step size, and is among the most common assumptions in the SA literature (Borkar, 2022; Fort, 2015; Li et al., 2023). In all three cases, the form of \\(\\gamma_{n}\\) ensures \\(\\gamma_{n} \\leq 1\\) such that the SRRW iterates \\(\\mathbf{x}_{n}\\) in (4b) is within \\(\\operatorname{Int}(\\Sigma)\\), ensuring that \\(\\mathbf{K}[\\mathbf{x}_n]\\) is well-defined for all \\(n \\geq 0\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.222,
+                0.825,
+                0.28
+            ],
+            "angle": 0,
+            "content": "In Assumption A3, limiting dynamics of SA iterations \\(\\{\\pmb{\\theta}_n\\}_{n\\geq 0}\\) closely follow trajectories \\(\\{\\pmb {\\theta}(t)\\}_{t\\geq 0}\\) of their associated ODE, and assuming the existence of globally stable equilibria is standard (Borkar, 2022; Fort, 2015; Li et al., 2023). In optimization problems, this is equivalent to assuming the existence of at most countably many local minima."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.286,
+                0.825,
+                0.412
+            ],
+            "angle": 0,
+            "content": "Assumption A4 assumes almost sure boundedness of iterates \\(\\theta_{n}\\) and \\(\\mathbf{x}_n\\), which is a common assumption in SA algorithms (Kushner & Yin, 2003; Chen, 2006; Borkar, 2022; Karmakar & Bhatnagar, 2018; Li et al., 2023) for the stability of the SA iterations by ensuring the well-definiteness of all quantities involved. Stability of the weighted empirical measure \\(\\mathbf{x}_n\\) of the SRRW process is practically ensured by studying (4b) with a truncation-based procedure (see Doshi et al., 2023, Remark 4.5 and Appendix E for a comprehensive explanation), while that for \\(\\theta_{n}\\) is usually ensured either as a by-product of the algorithm design, or via mechanisms such as projections onto a compact subset of \\(\\mathbb{R}^D\\), depending on the application context. We now provide additional discussions regarding the step-size assumptions and their implications on the SRRW iteration (4b)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.417,
+                0.825,
+                0.461
+            ],
+            "angle": 0,
+            "content": "SRRW with General Step Size: As shown in Benaim & Cloez (2015, Remark 1.1), albeit for a completely different non-linear Markov kernel driving the algorithm therein, iterates \\(\\mathbf{x}_n\\) of (4b) can also be expressed as weighted empirical measures of \\(\\{X_n\\}_{n\\geq 0}\\), in the following form:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.251,
+                0.464,
+                0.825,
+                0.498
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {x} _ {n} = \\frac {\\sum_ {i = 1} ^ {n} \\omega_ {i} \\delta_ {X _ {i}} + \\omega_ {0} \\mathbf {x} _ {0}}{\\sum_ {i = 0} ^ {n} \\omega_ {i}}, \\quad \\text {w h e r e} \\omega_ {0} = 1, \\text {a n d} \\omega_ {n} = \\frac {\\gamma_ {n}}{\\prod_ {i = 1} ^ {n} (1 - \\gamma_ {i})}, \\tag {5}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.501,
+                0.825,
+                0.616
+            ],
+            "angle": 0,
+            "content": "for all \\(n > 0\\). For the special case when \\(\\gamma_{n} = 1 / (n + 1)\\) as in Doshi et al. (2023), we have \\(\\omega_{n} = 1\\) for all \\(n \\geq 0\\) and \\(\\mathbf{x}_n\\) is the typical, unweighted empirical measure. For the additional case considered in our paper, when \\(a < 1\\) for \\(\\gamma_{n}\\) as in assumption A2, we can approximate \\(1 - \\gamma_{n} \\approx e^{-\\gamma_{n}}\\) and \\(\\omega_{n} \\approx n^{-a} e^{n^{(1 - a)} / (1 - a)}\\). This implies that \\(\\omega_{n}\\) will increase at sub-exponential rate, giving more weight to recent visit counts and allowing it to quickly 'forget' the poor initial measure \\(\\mathbf{x}_0\\) and shed the correlation with the initial choice of \\(X_0\\). This 'speed up' effect by setting \\(a < 1\\) is guaranteed in case (i) irrespective of the choice of \\(b\\) in Assumption A2, and in Section 3 we show how this can lead to further reduction in covariance of optimization error \\(\\theta_{n} = \\theta^{*}\\) in the asymptotic regime."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.622,
+                0.825,
+                0.652
+            ],
+            "angle": 0,
+            "content": "Additional assumption for case (iii): Before moving on to Section 3, we take another look at the case when \\(\\gamma_{n} = o(\\beta_{n})\\), and replace A3 with the following, stronger assumption only for case (iii)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.186,
+                0.658,
+                0.825,
+                0.693
+            ],
+            "angle": 0,
+            "content": "(A3') For any \\(\\mathbf{x} \\in \\mathrm{Int}(\\Sigma)\\), there exists a function \\(\\rho: \\mathrm{Int}(\\Sigma) \\to \\mathbb{R}^D\\) such that \\(\\|\\rho(\\mathbf{x})\\| \\leq L_2(1 + \\| \\mathbf{x}\\|)\\) for some \\(L_2 > 0\\), \\(\\mathbb{E}_{i \\sim \\pi[\\mathbf{x}]}[H(\\rho(\\mathbf{x}), i)] = 0\\) and \\(\\mathbb{E}_{i \\sim \\pi[\\mathbf{x}]}[\\nabla H(\\rho(\\mathbf{x}), i)] + \\frac{\\mathbf{1}_{\\{b=1\\}}}{2} \\mathbf{I}\\) is Hurwitz."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.698,
+                0.825,
+                0.741
+            ],
+            "angle": 0,
+            "content": "While Assumption A3' for case (iii) is much stronger than A3, it is not detrimental to the overall results of our paper, since case (i) is of far greater interest as impressed upon in Section 1. This is discussed further in Appendix C."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.76,
+                0.682,
+                0.777
+            ],
+            "angle": 0,
+            "content": "3 ASYMPTOTIC ANALYSIS OF THE SA-SRRW ALGORITHM"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.791,
+                0.825,
+                0.877
+            ],
+            "angle": 0,
+            "content": "In this section, we provide the main results for the SA-SRRW algorithm (4). We first present the a.s. convergence and the CLT result for SRRW with generalized step size, extending the results in Doshi et al. (2023). Building upon this, we present the a.s. convergence and the CLT result for the SA iterate \\(\\theta_{n}\\) under different settings of step sizes. We then shift our focus to the analysis of the different asymptotic covariance matrices emerging out of the CLT result, and capture the effect of \\(\\alpha\\) and the step sizes, particularly in cases (i) and (iii), on \\(\\theta_{n} - \\theta^{*}\\) via performance ordering."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.882,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "Almost Sure convergence and CLT: The following result establishes first and second order convergence of the sequence \\(\\{\\mathbf{x}_n\\}_{n\\geq 0}\\), which represents the weighted empirical measures of the SRRW process \\(\\{X_{n}\\}_{n\\geq 0}\\), based on the update rule in (4b)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.505,
+                0.96
+            ],
+            "angle": 0,
+            "content": "6"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.738,
+                0.119
+            ],
+            "angle": 0,
+            "content": "Lemma 3.1. Under Assumptions A1, A2 and A4, for the SRRW iterates (4b), we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.295,
+                0.12,
+                0.7,
+                0.143
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {x} _ {n} \\xrightarrow [ n \\to \\infty ]{a . s .} \\boldsymbol {\\mu}, \\quad a n d \\quad \\gamma_ {n} ^ {- 1 / 2} (\\mathbf {x} _ {n} - \\boldsymbol {\\mu}) \\xrightarrow [ n \\to \\infty ]{d i s t .} N (\\mathbf {0}, \\mathbf {V} _ {\\mathbf {x}} (\\alpha)),\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.282,
+                0.143,
+                0.825,
+                0.183
+            ],
+            "angle": 0,
+            "content": "\\[\nw h e r e \\quad \\mathbf {V} _ {\\mathbf {x}} (\\alpha) = \\sum_ {i = 1} ^ {N - 1} \\frac {1}{2 \\alpha \\left(1 + \\lambda_ {i}\\right) + 2 - \\mathbb {1} _ {\\{a = 1 \\}}} \\cdot \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T}. \\tag {6}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.175,
+                0.186,
+                0.664,
+                0.202
+            ],
+            "angle": 0,
+            "content": "Moreover, for all \\(\\alpha_{2} > \\alpha_{1} > 0\\), we have \\(\\mathbf{V}_{\\mathbf{x}}(\\alpha_{2}) < _{L}\\mathbf{V}_{\\mathbf{x}}(\\alpha_{1}) < _{L}\\mathbf{V}_{\\mathbf{x}}(0)\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.205,
+                0.827,
+                0.345
+            ],
+            "angle": 0,
+            "content": "Lemma 3.1 shows that the SRRW iterates \\(\\mathbf{x}_n\\) converges to the target distribution \\(\\mu\\) a.s. even under the general step size \\(\\gamma_n = (n + 1)^{-a}\\) for \\(a\\in (0.5,1]\\). We also observe that the asymptotic covariance matrix \\(\\mathbf{V}_{\\mathbf{x}}(\\alpha)\\) decreases at rate \\(O(1 / \\alpha)\\). Lemma 3.1 aligns with Doshi et al. (2023, Theorem 4.2 and Corollary 4.3) for the special case of \\(a = 1\\), and is therefore more general. Critically, it helps us establish our next result regarding the first-order convergence for the optimization iterate sequence \\(\\{\\theta_{n}\\}_{n\\geq 0}\\) following update rule (4c), as well as its second-order convergence result, which follows shortly after. The proofs of Lemma 3.1 and our next result, Theorem 3.2, are deferred to Appendix D. In what follows, \\(k = 1,2\\), and 3 refer to cases (i), (ii), and (iii) in Section 2.2, respectively. All subsequent results are proven under Assumptions A1 to A4, with \\(\\mathrm{A}3^{\\prime}\\) replacing A3 only when the step sizes \\(\\beta_{n},\\gamma_{n}\\) satisfy case (iii)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.347,
+                0.825,
+                0.379
+            ],
+            "angle": 0,
+            "content": "Theorem 3.2. For \\(k \\in \\{1,2,3\\}\\), and any initial \\((\\pmb{\\theta}_0,\\mathbf{x}_0,X_0) \\in \\mathbb{R}^D \\times \\mathrm{Int}(\\Sigma) \\times \\mathcal{N}\\), we have \\(\\pmb{\\theta}_n \\to \\pmb{\\theta}^*\\) as \\(n \\to \\infty\\) for some \\(\\pmb{\\theta}^* \\in \\Theta\\), \\(\\mathbb{P}_{\\pmb{\\theta}_0,\\mathbf{x}_0,X_0}\\)-almost surely."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.384,
+                0.825,
+                0.44
+            ],
+            "angle": 0,
+            "content": "In the stochastic optimization context, the above result ensures convergence of iterates \\(\\pmb{\\theta}_n\\) to a local minimizer \\(\\pmb{\\theta}^*\\). Loosely speaking, the first-order convergence of \\(\\mathbf{x}_n\\) in Lemma 3.1 as well as that of \\(\\pmb{\\theta}_n\\) are closely related to the convergence of trajectories \\(\\{\\mathbf{z}(t) \\triangleq (\\pmb{\\theta}(t), \\mathbf{x}(t))\\}_{t \\geq 0}\\) of the (coupled) mean-field ODE, written in a matrix-vector form as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.328,
+                0.44,
+                0.825,
+                0.473
+            ],
+            "angle": 0,
+            "content": "\\[\n\\frac {d}{d t} \\mathbf {z} (t) = \\mathbf {g} (\\mathbf {z} (t)) \\triangleq \\left[ \\begin{array}{c} \\mathbf {H} (\\boldsymbol {\\theta} (t)) ^ {T} \\boldsymbol {\\pi} [ \\mathbf {x} (t) ] \\\\ \\boldsymbol {\\pi} [ \\mathbf {x} (t) ] - \\mathbf {x} (t) \\end{array} \\right] \\in \\mathbb {R} ^ {D + N}. \\tag {7}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.473,
+                0.825,
+                0.56
+            ],
+            "angle": 0,
+            "content": "where matrix \\(\\mathbf{H}(\\pmb{\\theta}) \\triangleq [H(\\pmb{\\theta}, 1), \\dots, H(\\pmb{\\theta}, N)]^T \\in \\mathbb{R}^{N \\times D}\\) for any \\(\\pmb{\\theta} \\in \\mathbb{R}^D\\). Here, \\(\\pmb{\\pi}[\\mathbf{x}] \\in \\mathrm{Int}(\\Sigma)\\) is the stationary distribution of the SRRW kernel \\(\\mathbf{K}[\\mathbf{x}]\\) and is shown in Doshi et al. (2023) to be given by \\(\\pi_i[\\mathbf{x}] \\propto \\sum_{j \\in \\mathcal{N}} \\mu_i P_{ij}(x_i / \\mu_i)^{-\\alpha}(x_j / \\mu_j)^{-\\alpha}\\). The Jacobian matrix of (7) when evaluated at equilibria \\(\\mathbf{z}^* = (\\pmb{\\theta}^*, \\pmb{\\mu})\\) for \\(\\pmb{\\theta}^* \\in \\Theta\\) captures the behaviour of solutions of the mean-field in their vicinity, and plays an important role in the asymptotic covariance matrices arising out of our CLT results. We evaluate this Jacobian matrix \\(\\mathbf{J}(\\alpha)\\) as a function of \\(\\alpha \\geq 0\\) to be given by"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.252,
+                0.56,
+                0.825,
+                0.594
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {J} (\\alpha) \\triangleq \\nabla g \\left(\\mathbf {z} ^ {*}\\right) = \\left[ \\begin{array}{c c} \\nabla \\mathbf {h} \\left(\\boldsymbol {\\theta} ^ {*}\\right) & - \\alpha \\mathbf {H} \\left(\\boldsymbol {\\theta} ^ {*}\\right) ^ {T} \\left(\\mathbf {P} ^ {T} + \\mathbf {I}\\right) \\\\ \\mathbf {0} _ {N \\times D} & 2 \\alpha \\boldsymbol {\\mu} \\mathbf {1} ^ {T} - \\alpha \\mathbf {P} ^ {T} - (\\alpha + 1) \\mathbf {I} \\end{array} \\right] \\triangleq \\left[ \\begin{array}{l l} \\mathbf {J} _ {1 1} & \\mathbf {J} _ {1 2} (\\alpha) \\\\ \\mathbf {J} _ {2 1} & \\mathbf {J} _ {2 2} (\\alpha) \\end{array} \\right]. \\tag {8}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.594,
+                0.825,
+                0.664
+            ],
+            "angle": 0,
+            "content": "The derivation of \\(\\mathbf{J}(\\alpha)\\) is referred to Appendix E.1.8 Here, \\(\\mathbf{J}_{21}\\) is a zero matrix since \\(\\pi[\\mathbf{x}] - \\mathbf{x}\\) is devoid of \\(\\theta\\). While matrix \\(\\mathbf{J}_{22}(\\alpha)\\) is exactly of the form in Doshi et al. (2023, Lemma 3.4) to characterize the SRRW performance, our analysis includes an additional matrix \\(\\mathbf{J}_{12}(\\alpha)\\), which captures the effect of \\(\\mathbf{x}(t)\\) on \\(\\theta(t)\\) in the ODE (7), which translates to the influence of our generalized SRRW empirical measure \\(\\mathbf{x}_n\\) on the SA iterates \\(\\theta_n\\) in (4)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.67,
+                0.825,
+                0.714
+            ],
+            "angle": 0,
+            "content": "For notational simplicity, and without loss of generality, all our remaining results are stated while conditioning on the event that \\(\\{\\pmb {\\theta}_n\\to \\pmb {\\theta}^*\\}\\), for some \\(\\pmb {\\theta}^{*}\\in \\Theta\\). We also adopt the shorthand notation \\(\\mathbf{H}\\) to represent \\(\\mathbf{H}(\\pmb {\\theta}^{*})\\). Our main CLT result is as follows, with its proof deferred to Appendix E."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.717,
+                0.8,
+                0.733
+            ],
+            "angle": 0,
+            "content": "Theorem 3.3. For any \\(\\alpha \\geq 0\\), we have: (a) There exists \\(\\mathbf{V}^{(k)}(\\alpha)\\) for all \\(k \\in \\{1,2,3\\}\\) such that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.35,
+                0.733,
+                0.649,
+                0.772
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left[ \\begin{array}{c} \\beta_ {n} ^ {- 1 / 2} (\\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*}) \\\\ \\gamma_ {n} ^ {- 1 / 2} (\\mathbf {x} _ {n} - \\boldsymbol {\\mu}) \\end{array} \\right] \\xrightarrow [ n \\to \\infty ]{d i s t.} N \\left(\\mathbf {0}, \\mathbf {V} ^ {(k)} (\\alpha)\\right).\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.772,
+                0.825,
+                0.805
+            ],
+            "angle": 0,
+            "content": "(b) For \\(k = 2\\), matrix \\(\\mathbf{V}^{(2)}(\\alpha)\\) solves the Lyapunov equation \\(\\mathbf{J}(\\alpha)\\mathbf{V}^{(2)}(\\alpha) + \\mathbf{V}^{(2)}(\\alpha)\\mathbf{J}(\\alpha)^T + \\mathbb{1}_{\\{b = 1\\}}\\mathbf{V}^{(2)}(\\alpha) = -\\mathbf{U}\\), where the Jacobian matrix \\(\\mathbf{J}(\\alpha)\\) is in (8), and"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.288,
+                0.806,
+                0.825,
+                0.845
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {U} \\triangleq \\sum_ {i = 1} ^ {N - 1} \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\cdot \\left[ \\begin{array}{c c} \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H} & \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\\\ \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H} & \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\end{array} \\right] \\triangleq \\left[ \\begin{array}{c c} \\mathbf {U} _ {1 1} & \\mathbf {U} _ {1 2} \\\\ \\mathbf {U} _ {2 1} & \\mathbf {U} _ {2 2} \\end{array} \\right]. \\tag {9}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.847,
+                0.642,
+                0.862
+            ],
+            "angle": 0,
+            "content": "(c) For \\( k \\in \\{1, 3\\} \\), \\( \\mathbf{V}^{(k)}(\\alpha) \\) becomes block diagonal, which is given by"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.385,
+                0.862,
+                0.825,
+                0.895
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {V} ^ {(k)} (\\alpha) = \\left[ \\begin{array}{l l} \\mathbf {V} _ {\\boldsymbol {\\theta}} ^ {(k)} (\\alpha) & \\mathbf {0} _ {D \\times N} \\\\ \\mathbf {0} _ {N \\times D} & \\mathbf {V} _ {\\mathbf {x}} (\\alpha) \\end{array} \\right], \\tag {10}\n\\]"
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.171,
+                0.897,
+                0.825,
+                0.924
+            ],
+            "angle": 0,
+            "content": "The Jacobian \\(\\mathbf{J}(\\alpha)\\) is \\((D + N)\\times (D + N)\\)-dimensional, with \\(\\mathbf{J}_{11}\\in \\mathbb{R}^{D\\times D}\\) and \\(\\mathbf{J}_{22}(\\alpha)\\in \\mathbb{R}^{N\\times N}\\). Following this, all matrices written in a block form, such as matrix \\(\\mathbf{U}\\) in (9), will inherit the same dimensional structure."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.505,
+                0.96
+            ],
+            "angle": 0,
+            "content": "7"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.102,
+                0.816,
+                0.123
+            ],
+            "angle": 0,
+            "content": "where \\(\\mathbf{V}_{\\mathbf{x}}(\\alpha)\\) is as in (6), and \\(\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(\\alpha)\\) and \\(\\mathbf{V}_{\\pmb{\\theta}}^{(3)}(\\alpha)\\) can be written in the following explicit form:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.296,
+                0.128,
+                0.747,
+                0.152
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {V} _ {\\boldsymbol {\\theta}} ^ {(1)} (\\alpha) = \\int_ {0} ^ {\\infty} e ^ {t (\\nabla_ {\\boldsymbol {\\theta}} \\mathbf {h} (\\boldsymbol {\\theta} ^ {*}) + \\frac {1}{2} \\{b = 1 \\} \\boldsymbol {I})} \\mathbf {U} _ {\\boldsymbol {\\theta}} (\\alpha) e ^ {t (\\nabla_ {\\boldsymbol {\\theta}} \\mathbf {h} (\\boldsymbol {\\theta} ^ {*}) + \\frac {1}{2} \\{b = 1 \\} \\boldsymbol {I}) ^ {T}} d t,\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.296,
+                0.153,
+                0.584,
+                0.174
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {V} _ {\\pmb {\\theta}} ^ {(3)} (\\alpha) = \\int_ {0} ^ {\\infty} e ^ {t \\nabla_ {\\pmb {\\theta}} \\mathbf {h} (\\pmb {\\theta} ^ {*})} \\mathbf {U} _ {1 1} e ^ {t \\nabla_ {\\pmb {\\theta}} \\mathbf {h} (\\pmb {\\theta} ^ {*})} d t,\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.25,
+                0.177,
+                0.826,
+                0.217
+            ],
+            "angle": 0,
+            "content": "\\[\nw h e r e \\mathbf {U} _ {\\boldsymbol {\\theta}} (\\alpha) = \\sum_ {i = 1} ^ {N - 1} \\frac {1}{\\left(\\alpha \\left(1 + \\lambda_ {i}\\right) + 1\\right) ^ {2}} \\cdot \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H}. \\tag {11}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.217,
+                0.827,
+                0.402
+            ],
+            "angle": 0,
+            "content": "For \\( k \\in \\{1,3\\} \\), SA-SRRW in (4) is a two-timescale SA with controlled Markov noise. While a few works study the CLT of two-timescale SA with the stochastic input being a martingale-difference (i.i.d.) noise (Konda & Tsitsiklis, 2004; Mokkadem & Pelletier, 2006), a CLT result covering the case of controlled Markov noise (e.g., \\( k \\in \\{1,3\\} \\)), a far more general setting than martingale-difference noise, is still an open problem. Thus, we here prove our CLT for \\( k \\in \\{1,3\\} \\) from scratch by a series of careful decompositions of the Markovian noise, ultimately into a martingale-difference term and several non-vanishing noise terms through repeated application of the Poisson equation (Benveniste et al., 2012; Fort, 2015). Although the form of the resulting asymptotic covariance looks similar to that for the martingale-difference case in (Konda & Tsitsiklis, 2004; Mokkadem & Pelletier, 2006) at first glance, they are not equivalent. Specifically, \\( \\mathbf{V}_{\\theta}^{(k)}(\\alpha) \\) captures both the effect of SRRW hyper-parameter \\( \\alpha \\), as well as that of the underlying base Markov chain via eigenpairs \\( (\\lambda_i, \\mathbf{u}_i) \\) of its transition probability matrix \\( \\mathbf{P} \\) in matrix \\( \\mathbf{U} \\), whereas the latter only covers the martingale-difference noise terms as a special case."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.408,
+                0.825,
+                0.515
+            ],
+            "angle": 0,
+            "content": "When \\(k = 2\\), that is, \\(\\beta_{n} = \\gamma_{n}\\), algorithm (4) can be regarded as a single-timescale SA algorithm. In this case, we utilize the CLT in Fort (2015, Theorem 2.1) to obtain the implicit form of \\(\\mathbf{V}^{(2)}(\\alpha)\\) as shown in Theorem 3.3. However, \\(\\mathbf{J}_{12}(\\alpha)\\) being non-zero for \\(\\alpha > 0\\) restricts us from obtaining an explicit form for the covariance term corresponding to SA iterate errors \\(\\theta_{n} - \\theta^{*}\\). On the other hand, for \\(k \\in \\{1,3\\}\\), the nature of two-timescale structure causes \\(\\theta_{n}\\) and \\(\\mathbf{x}_n\\) to become asymptotically independent with zero correlation terms inside \\(\\mathbf{V}^{(k)}(\\alpha)\\) in (10), and we can explicitly deduce \\(\\mathbf{V}_{\\pmb{\\theta}}^{(k)}(\\alpha)\\). We now take a deeper dive into \\(\\alpha\\) and study its effect on \\(\\mathbf{V}_{\\pmb{\\theta}}^{(k)}(\\alpha)\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.519,
+                0.825,
+                0.553
+            ],
+            "angle": 0,
+            "content": "Covariance Ordering of SA-SRRW: We refer the reader to Appendix F for proofs of all remaining results. We begin by focusing on case (i) and capturing the impact of \\(\\alpha\\) on \\(\\mathbf{V}_{\\theta}^{(1)}(\\alpha)\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.557,
+                0.825,
+                0.595
+            ],
+            "angle": 0,
+            "content": "Proposition 3.4. For all \\(\\alpha_{2} > \\alpha_{1} > 0\\), we have \\(\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(\\alpha_2) < _L\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(\\alpha_1) < _L\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(0)\\). Furthermore, \\(\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(\\alpha)\\) decreases to zero at a rate of \\(O(1 / \\alpha^2)\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.6,
+                0.827,
+                0.814
+            ],
+            "angle": 0,
+            "content": "Proposition 3.4 proves a monotonic reduction (in terms of Loewner ordering) of \\(\\mathbf{V}_{\\theta}^{(1)}(\\alpha)\\) as \\(\\alpha\\) increases. Moreover, the decrease rate \\(O(1 / \\alpha^2)\\) surpasses the \\(O(1 / \\alpha)\\) rate seen in \\(\\mathbf{V}_{\\mathbf{x}}(\\alpha)\\) and the sampling application in Doshi et al. (2023, Corollary 4.7), and is also empirically observed in our simulation in Section 4. Suppose we consider the same SA now driven by an i.i.d. sequence \\(\\{X_{n}\\}\\) with the same marginal distribution \\(\\pmb{\\mu}\\). Then, our Proposition 3.4 asserts that a token algorithm employing SRRW (walk on a graph) with large enough \\(\\alpha\\) on a general graph can actually produce better SA iterates with its asymptotic covariance going down to zero, than a 'hypothetical situation' where the walker is able to access any node \\(j\\) with probability \\(\\mu_{j}\\) from anywhere in one step (more like a random jumper). This can be seen by noting that for large time \\(n\\), the scaled MSE \\(\\mathbb{E}[\\|\\pmb{\\theta}_n - \\pmb{\\theta}^*\\|^2] / \\beta_n\\) is composed of the diagonal entries of the covariance matrix \\(\\mathbf{V}_{\\theta}\\), which, as we discuss in detail in Appendix F.2, are decreasing in \\(\\alpha\\) as a consequence of the Loewner ordering in Proposition 3.4. For large enough \\(\\alpha\\), the scaled MSE for SA-SRRW becomes smaller than its i.i.d. counterpart, which is always a constant. Although Doshi et al. (2023) alluded this for sampling applications with \\(\\mathbf{V}_{\\mathbf{x}}(\\alpha)\\), we broaden its horizons to distributed optimization problems with \\(\\mathbf{V}_{\\pmb{\\theta}}(\\alpha)\\) using tokens on graphs. Our subsequent result concerns the performance comparison between cases (i) and (iii)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.817,
+                0.663,
+                0.838
+            ],
+            "angle": 0,
+            "content": "Corollary 3.5. For any \\(\\alpha >0\\), we have \\(\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(\\alpha) < _L\\mathbf{V}_{\\pmb{\\theta}}^{(3)}(\\alpha) = \\mathbf{V}_{\\pmb{\\theta}}^{(3)}(0)\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.846,
+                0.826,
+                0.876
+            ],
+            "angle": 0,
+            "content": "We show that case (i) is asymptotically better than case (iii) for \\(\\alpha > 0\\). In view of Proposition 3.4 and Corollary 3.5, the advantages of case (i) become prominent."
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.171,
+                0.884,
+                0.826,
+                0.926
+            ],
+            "angle": 0,
+            "content": "Further insights of \\(O(1 / \\alpha^2)\\) are tied to the two-timescale structure, particularly \\(\\beta_{n} = o(\\gamma_{n})\\) in case (i), which places \\(\\theta_{n}\\) on the slow timescale so that the correlation terms \\(\\mathbf{J}_{12}(\\alpha),\\mathbf{J}_{22}(\\alpha)\\) in the Jacobian matrix \\(\\mathbf{J}(\\alpha)\\) in (8) come into play. Technical details are referred to Appendix E.2, where we show the form of \\(\\mathbf{U}_{\\theta}(\\alpha)\\)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.505,
+                0.96
+            ],
+            "angle": 0,
+            "content": "8"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.175,
+                0.101,
+                0.378,
+                0.21
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.223,
+                0.214,
+                0.323,
+                0.227
+            ],
+            "angle": 0,
+            "content": "(a) SGD-SRRW."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.398,
+                0.101,
+                0.599,
+                0.21
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.446,
+                0.214,
+                0.547,
+                0.227
+            ],
+            "angle": 0,
+            "content": "(b) SHB-SRRW."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.622,
+                0.11,
+                0.822,
+                0.21
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.643,
+                0.214,
+                0.799,
+                0.227
+            ],
+            "angle": 0,
+            "content": "(c) Curve fitting for MSE."
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.171,
+                0.228,
+                0.825,
+                0.257
+            ],
+            "angle": 0,
+            "content": "Figure 2: Simulation results under case (i): (a) and (b) show the performance of SGD-SRRW and SHB-SRRW for various \\(\\alpha\\) values. (c) shows that MSE decreases at \\(O(1 / \\alpha^2)\\) speed."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.174,
+                0.26,
+                0.373,
+                0.359
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.201,
+                0.364,
+                0.343,
+                0.377
+            ],
+            "angle": 0,
+            "content": "(a) \\(\\alpha = 1\\) ,SGD-SRRW"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.398,
+                0.26,
+                0.597,
+                0.359
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.425,
+                0.364,
+                0.568,
+                0.377
+            ],
+            "angle": 0,
+            "content": "(b) \\(\\alpha = 5\\) ,SGD-SRRW"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.623,
+                0.26,
+                0.822,
+                0.359
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.647,
+                0.364,
+                0.796,
+                0.377
+            ],
+            "angle": 0,
+            "content": "(c) \\(\\alpha = 10\\) ,SGD-SRRW"
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.223,
+                0.379,
+                0.772,
+                0.394
+            ],
+            "angle": 0,
+            "content": "Figure 3: Comparison of the performance among cases (i) - (iii) for \\(\\alpha \\in \\{1,5,10\\}\\)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.399,
+                0.315,
+                0.414
+            ],
+            "angle": 0,
+            "content": "4 SIMULATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.43,
+                0.825,
+                0.486
+            ],
+            "angle": 0,
+            "content": "In this section, we simulate our SA-SRRW algorithm on the wikiVote graph (Leskovec & Krevl, 2014), comprising 889 nodes and 2914 edges. We configure the SRRW's base Markov chain \\(\\mathbf{P}\\) as the MHRW with a uniform target distribution \\(\\mu = \\frac{1}{N} \\mathbf{1}\\). For distributed optimization, we consider the following \\(L_{2}\\) regularized binary classification problem:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.263,
+                0.488,
+                0.825,
+                0.514
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left. \\min  _ {\\boldsymbol {\\theta} \\in \\mathbb {R} ^ {D}} \\left\\{f (\\boldsymbol {\\theta}) \\triangleq \\frac {1}{N} \\sum_ {i = 1} ^ {N} \\log \\left(1 + e ^ {\\boldsymbol {\\theta} ^ {T} \\mathbf {s} _ {i}}\\right) - y _ {i} \\left(\\boldsymbol {\\theta} ^ {T} \\mathbf {s} _ {i}\\right) + \\frac {\\kappa}{2} \\| \\boldsymbol {\\theta} \\| ^ {2} \\right\\}, \\right. \\tag {12}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.515,
+                0.825,
+                0.614
+            ],
+            "angle": 0,
+            "content": "where \\(\\{(\\mathbf{s}_i, y_i)\\}_{i=1}^N\\) is the ijCNN1 dataset (with 22 features, i.e., \\(\\mathbf{s}_i \\in \\mathbb{R}^{22}\\)) from LIBSVM (Chang & Lin, 2011), and penalty parameter \\(\\kappa = 1\\). Each node in the wikiVote graph is assigned one data point, thus 889 data points in total. We perform SRRW driven SGD (SGD-SRRW) and SRRW driven stochastic heavy ball (SHB-SRRW) algorithms (see (13) in Appendix A for its algorithm). We fix the step size \\(\\beta_n = (n + 1)^{-0.9}\\) for the SA iterates and adjust \\(\\gamma_n = (n + 1)^{-a}\\) in the SRRW iterates to cover all three cases discussed in this paper: (i) \\(a = 0.8\\); (ii) \\(a = 0.9\\); (iii) \\(a = 1\\). We use mean square error (MSE), i.e., \\(\\mathbb{E}[\\|\\pmb{\\theta}_n - \\pmb{\\theta}^*\\|^2]\\), to measure the error on the SA iterates."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.62,
+                0.827,
+                0.777
+            ],
+            "angle": 0,
+            "content": "Our results are presented in Figures 2 and 3, where each experiment is repeated 100 times. Figures 2a and 2b, based on wikiVote graph, highlight the consistent performance ordering across different \\(\\alpha\\) values for both algorithms over almost all time (not just asymptotically). Notably, curves for \\(\\alpha \\geq 5\\) outperform that of the i.i.d. sampling (in black) even under the graph constraints. Figure 2c on the smaller Dolphin graph (Rossi & Ahmed, 2015) - 62 nodes and 159 edges - illustrates that the points of \\((\\alpha, \\mathrm{MSE})\\) pair arising from SGD-SRRW at time \\(n = 10^7\\) align with a curve in the form of \\(g(x) = \\frac{c_1}{(x + c_2)^2} + c_3\\) to showcase \\(O(1 / \\alpha^2)\\) rates. This smaller graph allows for longer simulations to observe the asymptotic behaviour. Additionally, among the three cases examined at identical \\(\\alpha\\) values, Figures 3a - 3c confirm that case (i) performs consistently better than the rest, underscoring its superiority in practice. Further results, including those from non-convex functions and additional datasets, are deferred to Appendix H due to space constraints."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.796,
+                0.32,
+                0.811
+            ],
+            "angle": 0,
+            "content": "5 CONCLUSION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.827,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "In this paper, we show both theoretically and empirically that the SRRW as a drop-in replacement for Markov chains can provide significant performance improvements when used for token algorithms, where the acceleration comes purely from the careful analysis of the stochastic input of the algorithm, without changing the optimization iteration itself. Our paper is an instance where the asymptotic analysis approach allows the design of better algorithms despite the usage of unconventional noise sequences such as nonlinear Markov chains like the SRRW, for which traditional finite-time analytical approaches fall short, thus advocating their wider adoption."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.504,
+                0.96
+            ],
+            "angle": 0,
+            "content": "9"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.103,
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+            ],
+            "angle": 0,
+            "content": "6 ACKNOWLEDGMENTS AND DISCLOSURE OF FUNDING"
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
+                0.172,
+                0.134,
+                0.827,
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+            ],
+            "angle": 0,
+            "content": "We thank the anonymous reviewers for their constructive comments, especially Reviewer DxLx for raising future directions. This work was supported in part by National Science Foundation under Grant Nos. CNS-2007423 and IIS-1910749."
+        },
+        {
+            "type": "title",
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+        {
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+            "angle": 0,
+            "content": null
+        },
+        {
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+            "angle": 0,
+            "content": "13"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
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+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.103,
+                0.714,
+                0.119
+            ],
+            "angle": 0,
+            "content": "A EXAMPLES OF STOCHASTIC ALGORITHMS OF THE FORM (2)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.135,
+                0.825,
+                0.193
+            ],
+            "angle": 0,
+            "content": "In the literature of stochastic optimizations, many SGD variants have been proposed by introducing an auxiliary variable to improve convergence. In what follows, we present two SGD variants with decreasing step size that can be presented in the form of (2): SHB (Gadat et al., 2018; Li et al., 2022) and momentum-based algorithm (Barakat et al., 2021; Barakat & Bianchi, 2021)."
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.191,
+                0.201,
+                0.792,
+                0.252
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left\\{ \\begin{array}{l} \\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} - \\beta_ {n + 1} \\mathbf {m} _ {n} \\\\ \\mathbf {m} _ {n + 1} = \\mathbf {m} _ {n} + \\beta_ {n + 1} (\\nabla F (\\boldsymbol {\\theta} _ {n}, X _ {n + 1}) - \\mathbf {m} _ {n}), \\end{array} \\right. \\left\\{ \\begin{array}{l} \\mathbf {v} _ {n + 1} = \\mathbf {v} _ {n} + \\beta_ {n + 1} (\\nabla F (\\boldsymbol {\\theta} _ {n}, X _ {n + 1}) ^ {2} - \\mathbf {v} _ {n}), \\\\ \\mathbf {m} _ {n + 1} = \\mathbf {m} _ {n} + \\beta_ {n + 1} (\\nabla F (\\boldsymbol {\\theta} _ {n}, X _ {n + 1}) - \\mathbf {m} _ {n}), \\\\ \\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} - \\beta_ {n + 1} \\mathbf {m} _ {n} / \\sqrt {\\mathbf {v} _ {n} + \\epsilon}, \\end{array} \\right.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.297,
+                0.255,
+                0.361,
+                0.27
+            ],
+            "angle": 0,
+            "content": "(a). SHB"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.504,
+                0.256,
+                0.726,
+                0.27
+            ],
+            "angle": 0,
+            "content": "(b). Momentum-based Algorithm"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.794,
+                0.271,
+                0.824,
+                0.285
+            ],
+            "angle": 0,
+            "content": "(13)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.296,
+                0.825,
+                0.327
+            ],
+            "angle": 0,
+            "content": "where \\(\\epsilon > 0\\), \\(\\pmb{\\theta}_n, \\mathbf{m}_n, \\mathbf{v}_n, \\nabla F(\\pmb{\\theta}, X) \\in \\mathbb{R}^d\\), and the square and square root in (13) (b) are element-wise operators.\\(^{10}\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.331,
+                0.808,
+                0.347
+            ],
+            "angle": 0,
+            "content": "For SHB, we introduce an augmented variable \\(\\mathbf{z}_n\\) and function \\(H(\\mathbf{z}_n,X_{n + 1})\\) defined as follows:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.255,
+                0.354,
+                0.741,
+                0.39
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {z} _ {n} \\triangleq \\left[ \\begin{array}{c} \\boldsymbol {\\theta} _ {n} \\\\ \\mathbf {m} _ {n} \\end{array} \\right] \\in \\mathbb {R} ^ {2 d}, \\quad H (\\mathbf {z} _ {n}, X _ {n + 1}) \\triangleq \\left[ \\begin{array}{c} - \\mathbf {m} _ {n} \\\\ \\nabla F (\\boldsymbol {\\theta} _ {n}, X _ {n + 1}) - \\mathbf {m} _ {n} \\end{array} \\right] \\in \\mathbb {R} ^ {2 d}.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.397,
+                0.535,
+                0.412
+            ],
+            "angle": 0,
+            "content": "For the general momentum-based algorithm, we define"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.264,
+                0.42,
+                0.731,
+                0.468
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {z} _ {n} \\triangleq \\left[ \\begin{array}{c} \\mathbf {v} _ {n} \\\\ \\mathbf {m} _ {n} \\\\ \\boldsymbol {\\theta} _ {n} \\end{array} \\right] \\in \\mathbb {R} ^ {3 d}, \\quad H (\\mathbf {z} _ {n}, X) \\triangleq \\left[ \\begin{array}{c} \\nabla F (\\boldsymbol {\\theta} _ {n}, X _ {n + 1}) ^ {2} - \\mathbf {v} _ {n} \\\\ \\nabla F (\\boldsymbol {\\theta} _ {n}, X _ {n + 1}) - \\mathbf {m} _ {n} \\\\ - \\mathbf {m} _ {n} / \\sqrt {\\mathbf {v} _ {n} + \\epsilon} \\end{array} \\right] \\in \\mathbb {R} ^ {3 d}.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.476,
+                0.825,
+                0.576
+            ],
+            "angle": 0,
+            "content": "Thus, we can reformulate both algorithms in (13) as \\(\\mathbf{z}_{n + 1} = \\mathbf{z}_n + \\beta_{n + 1}H(\\mathbf{z}_n,X_{n + 1})\\). This augmentation approach was previously adopted in (Gadat et al., 2018; Barakat et al., 2021; Barakat & Bianchi, 2021; Li et al., 2022) to analyze the asymptotic performance of algorithms in (13) using an i.i.d. sequence \\(\\{X_{n}\\}_{n\\geq 0}\\). Consequently, the general SA iteration (2) includes these SGD variants. However, we mainly focus on the CLT for the general SA driven by SRRW in this paper. Pursuing the explicit CLT results of these SGD variants with specific form of function \\(H(\\theta ,X)\\) driven by the SRRW sequence \\(\\{X_{n}\\}\\) is out of the scope of this paper."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.581,
+                0.825,
+                0.625
+            ],
+            "angle": 0,
+            "content": "When we numerically test the SHB algorithm in Section 4, we use the exact form of (13) (a) and the stochastic sequence \\(\\{X_{n}\\}\\) is now driven by the SRRW. Specifically, we consider MHRW with transition kernel \\(\\mathbf{P}\\) as the base Markov chain of the SRRW process, e.g.,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.299,
+                0.632,
+                0.697,
+                0.674
+            ],
+            "angle": 0,
+            "content": "\\[\nP _ {i j} = \\left\\{ \\begin{array}{l l} \\min  \\left\\{\\frac {1}{d _ {i}}, \\frac {1}{d _ {j}} \\right\\} & \\text {i f n o d e j i s t h e n e i g h b o r o f n o d e i ,} \\\\ 0 & \\text {o t h e r w i s e ,} \\end{array} \\right.\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.337,
+                0.685,
+                0.469,
+                0.719
+            ],
+            "angle": 0,
+            "content": "\\[\nP _ {i i} = 1 - \\sum_ {j \\in \\mathcal {N}} P _ {i j}.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.724,
+                0.347,
+                0.739
+            ],
+            "angle": 0,
+            "content": "Then, at each time step \\(n\\),"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.342,
+                0.746,
+                0.555,
+                0.765
+            ],
+            "angle": 0,
+            "content": "Draw: \\(X_{n + 1}\\sim \\mathbf{K}_{X_n,\\cdot}[\\mathbf{x}_n],\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.346,
+                0.774,
+                0.745,
+                0.809
+            ],
+            "angle": 0,
+            "content": "where \\(K_{ij}[\\mathbf{x}]\\triangleq \\frac{P_{ij}(x_j / \\mu_j)^{-\\alpha}}{\\sum_{k\\in\\mathcal{N}}P_{ik}(x_k / \\mu_k)^{-\\alpha}},\\quad \\forall i,j\\in \\mathcal{N},\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.346,
+                0.815,
+                0.652,
+                0.832
+            ],
+            "angle": 0,
+            "content": "Update: \\(\\mathbf{x}_{n + 1} = \\mathbf{x}_n + \\gamma_{n + 1}(\\pmb{\\delta}_{X_{n + 1}} - \\mathbf{x}_n)\\)"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.421,
+                0.838,
+                0.584,
+                0.854
+            ],
+            "angle": 0,
+            "content": "\\[\n\\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} - \\beta_ {n + 1} \\mathbf {m} _ {n},\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.422,
+                0.86,
+                0.734,
+                0.877
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {m} _ {n + 1} = \\mathbf {m} _ {n} + \\beta_ {n + 1} (\\nabla F (\\boldsymbol {\\theta} _ {n}, X _ {n + 1}) - \\mathbf {m} _ {n}).\n\\]"
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.171,
+                0.885,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "For ease of expression, we simplify the original SHB and momentum-based algorithms from Gadat et al. (2018); Li et al. (2022); Barakat et al. (2021); Barakat & Bianchi (2021), setting all tunable parameters to 1 and resulting in (13)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.491,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "14"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.103,
+                0.768,
+                0.12
+            ],
+            "angle": 0,
+            "content": "B DISCUSSION ON MEAN FIELD FUNCTION OF SRRW ITERATES (4b)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.135,
+                0.825,
+                0.178
+            ],
+            "angle": 0,
+            "content": "Non-asymptotic analyses have seen extensive attention recently in both single-timescale SA literature (Sun et al., 2018; Karimi et al., 2019; Chen et al., 2020b; 2022) and two-timescale SA literature (Doan, 2021; Zeng et al., 2021). Specifically, single-timescale SA has the following form:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.381,
+                0.186,
+                0.615,
+                0.202
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {x} _ {n + 1} = \\mathbf {x} _ {n} + \\beta_ {n + 1} H (\\mathbf {x} _ {n}, X _ {n + 1}),\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.211,
+                0.825,
+                0.24
+            ],
+            "angle": 0,
+            "content": "and function \\( h(\\mathbf{x}) \\triangleq \\mathbb{E}_{X \\sim \\mu}[H(\\mathbf{x}, X)] \\) is the mean field of function \\( H(\\mathbf{x}, X) \\). Similarly, for two-timescale SA, we have the following recursions:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.364,
+                0.249,
+                0.632,
+                0.265
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {x} _ {n + 1} = \\mathbf {x} _ {n} + \\beta_ {n + 1} H _ {1} (\\mathbf {x} _ {n}, \\mathbf {y} _ {n}, X _ {n + 1}),\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.366,
+                0.268,
+                0.631,
+                0.283
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {y} _ {n + 1} = \\mathbf {y} _ {n} + \\gamma_ {n + 1} H _ {2} (\\mathbf {x} _ {n}, \\mathbf {y} _ {n}, X _ {n + 1}),\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.293,
+                0.825,
+                0.364
+            ],
+            "angle": 0,
+            "content": "where \\(\\{\\beta_n\\}\\) and \\(\\{\\gamma_n\\}\\) are on different timescales, and function \\(h_i(\\mathbf{x},\\mathbf{y})\\triangleq \\mathbb{E}_{X\\sim \\mu}[H_i(\\mathbf{x},\\mathbf{y},X)]\\) is the mean field of function \\(H_{i}(\\mathbf{x},\\mathbf{y},X)\\) for \\(i = \\{1,2\\}\\). All the aforementioned works require the mean field function \\(h(\\mathbf{x})\\) in the single-timescale SA (or \\(h_1(\\mathbf{x},\\mathbf{y}),h_2(\\mathbf{x},\\mathbf{y})\\) in the two-timescale SA) to be globally Lipschitz with a Lipschitz constant \\(L\\) to proceed with the derivation of finite-time bounds including the constant \\(L\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.37,
+                0.826,
+                0.442
+            ],
+            "angle": 0,
+            "content": "Here, we show that the mean field function \\(\\pi[\\mathbf{x}] - \\mathbf{x}\\) in the SRRW iterates (4b) is not globally Lipschitz, where \\(\\pi[\\mathbf{x}]\\) is the stationary distribution of the SRRW kernel \\(\\mathbf{K}[\\mathbf{x}]\\) defined in (3). To this end, we show that each entry of Jacobian matrix of \\(\\pi[\\mathbf{x}] - \\mathbf{x}\\) goes unbounded because a multivariate function is Lipschitz if and only if it has bounded partial derivatives. Note that from Doshi et al. (2023, Proposition 2.1), for the \\(i\\)-th entry of \\(\\pi[\\mathbf{x}]\\), we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.321,
+                0.45,
+                0.825,
+                0.49
+            ],
+            "angle": 0,
+            "content": "\\[\n\\boldsymbol {\\pi} _ {i} [ \\mathbf {x} ] = \\frac {\\sum_ {j \\in \\mathcal {N}} \\mu_ {i} P _ {i j} \\left(x _ {i} / \\mu_ {i}\\right) ^ {- \\alpha} \\left(x _ {j} / \\mu_ {j}\\right) ^ {- \\alpha}}{\\sum_ {i \\in \\mathcal {N}} \\sum_ {j \\in \\mathcal {N}} \\mu_ {i} P _ {i j} \\left(x _ {i} / \\mu_ {i}\\right) ^ {- \\alpha} \\left(x _ {j} / \\mu_ {j}\\right) ^ {- \\alpha}}. \\tag {16}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.498,
+                0.825,
+                0.527
+            ],
+            "angle": 0,
+            "content": "Then, the Jacobian matrix of the mean field function \\(\\pi[\\mathbf{x}] - \\mathbf{x}\\), which has been derived in Doshi et al. (2023, Proof of Lemma 3.4 in Appendix B), is given as follows:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.204,
+                0.535,
+                0.825,
+                0.65
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\frac {\\partial (\\boldsymbol {\\pi} _ {i} [ \\mathbf {x} ] - x _ {i})}{\\partial x _ {j}} \\\\ = \\frac {2 \\alpha}{x _ {j}} \\cdot \\frac {\\left(\\sum_ {k \\in \\mathcal {N}} \\mu_ {i} P _ {i k} \\left(x _ {i} / \\mu_ {i}\\right) ^ {- \\alpha} \\left(x _ {k} / \\mu_ {k}\\right) ^ {- \\alpha}\\right) \\left(\\sum_ {k \\in \\mathcal {N}} \\mu_ {j} P _ {j k} \\left(x _ {j} / \\mu_ {j}\\right) ^ {- \\alpha} \\left(x _ {k} / \\mu_ {k}\\right) ^ {- \\alpha}\\right)}{\\left(\\sum_ {l \\in \\mathcal {N}} \\sum_ {k \\in \\mathcal {N}} \\mu_ {l} P _ {l k} \\left(x _ {l} / \\mu_ {l}\\right) ^ {- \\alpha} \\left(x _ {k} / \\mu_ {k}\\right) ^ {- \\alpha}\\right) ^ {2}} \\tag {17} \\\\ - \\frac {\\alpha}{x _ {j}} \\cdot \\frac {\\mu_ {i} P _ {i j} (x _ {i} / \\mu_ {i}) ^ {- \\alpha} (x _ {j} / \\mu_ {j}) ^ {- \\alpha}}{\\sum_ {l \\in \\mathcal {N}} \\sum_ {k \\in \\mathcal {N}} \\mu_ {l} P _ {l k} (x _ {l} / \\mu_ {l}) ^ {- \\alpha} (x _ {k} / \\mu_ {k}) ^ {- \\alpha}} \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.656,
+                0.332,
+                0.672
+            ],
+            "angle": 0,
+            "content": "for \\(i,j\\in \\mathcal{N},i\\neq j\\) , and"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.266,
+                0.679,
+                0.826,
+                0.793
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\frac {\\partial \\left(\\boldsymbol {\\pi} _ {i} [ \\mathbf {x} ] - x _ {i}\\right)}{\\partial x _ {i}} \\\\ = \\frac {2 \\alpha}{x _ {i}} \\cdot \\frac {\\left(\\sum_ {k \\in \\mathcal {N}} \\mu_ {i} P _ {i k} \\left(x _ {i} / \\mu_ {i}\\right) ^ {- \\alpha} \\left(x _ {k} / \\mu_ {k}\\right) ^ {- \\alpha}\\right) ^ {2}}{\\left(\\sum_ {l \\in \\mathcal {N}} \\sum_ {k \\in \\mathcal {N}} \\mu_ {l} P _ {l k} \\left(x _ {l} / \\mu_ {l}\\right) ^ {- \\alpha} \\left(x _ {k} / \\mu_ {k}\\right) ^ {- \\alpha}\\right) ^ {2}} \\tag {18} \\\\ - \\frac {\\alpha}{x _ {i}} \\cdot \\frac {\\sum_ {k \\in \\mathcal {N}} \\mu_ {i} P _ {i k} (x _ {i} / \\mu_ {i}) ^ {- \\alpha} (x _ {k} / \\mu_ {k}) ^ {- \\alpha} + \\mu_ {i} P _ {i i} (x _ {i} / \\mu_ {i}) ^ {- 2 \\alpha}}{\\sum_ {l \\in \\mathcal {N}} \\sum_ {k \\in \\mathcal {N}} \\mu_ {l} P _ {l k} (x _ {l} / \\mu_ {l}) ^ {- \\alpha} (x _ {k} / \\mu_ {k}) ^ {- \\alpha}} - 1 \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.8,
+                0.825,
+                0.857
+            ],
+            "angle": 0,
+            "content": "for \\(i \\in \\mathcal{N}\\). Since the empirical distribution \\(\\mathbf{x} \\in \\mathrm{Int}(\\Sigma)\\), we have \\(x_i \\in (0,1)\\) for all \\(i \\in \\mathcal{N}\\). For fixed \\(i\\), assume \\(x_i = x_j\\) and as they approach zero, the terms \\((x_i / \\mu_i)^{-\\alpha}, (x_j / \\mu_j)^{-\\alpha}\\) dominate the fraction in (17) and both the numerator and the denominator of the fraction have the same order in terms of \\(x_i, x_j\\). Thus, we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.408,
+                0.858,
+                0.589,
+                0.893
+            ],
+            "angle": 0,
+            "content": "\\[\n\\frac {\\partial (\\pmb {\\pi} _ {i} [ \\mathbf {x} ] - x _ {i})}{\\partial x _ {j}} = O \\left(\\frac {1}{x _ {j}}\\right)\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.896,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "such that the \\((i,j)\\)-th entry of the Jacobian matrix can go unbounded as \\(x_{j} \\to 0\\). Consequently, \\(\\pi[\\mathbf{x}] - \\mathbf{x}\\) is not globally Lipschitz for \\(\\mathbf{x} \\in \\operatorname{Int}(\\Sigma)\\)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.491,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "15"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.102,
+                0.498,
+                0.119
+            ],
+            "angle": 0,
+            "content": "C DISCUSSION ON ASSUMPTION A3'"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.134,
+                0.825,
+                0.207
+            ],
+            "angle": 0,
+            "content": "When \\(\\gamma_{n} = o(\\beta_{n})\\), iterates \\(\\mathbf{x}_n\\) has smaller step size compared to \\(\\pmb{\\theta}_{n}\\), thus converges 'slower' than \\(\\pmb{\\theta}_{n}\\). From Assumption A3', \\(\\pmb{\\theta}_{n}\\) will intuitively converge to some point \\(\\rho (\\mathbf{x})\\) with the current value \\(\\mathbf{x}\\) from the iteration \\(\\mathbf{x}_n\\), i.e., \\(\\mathbb{E}_{X\\sim \\pi [\\mathbf{x}]}[H(\\rho (\\mathbf{x}),X)] = 0\\), while the Hurwitz condition is to ensure the stability around \\(\\rho (\\mathbf{x})\\). We can see that Assumption A3 is less stringent than A3' in that it only assumes such condition when \\(\\mathbf{x} = \\pmb{\\mu}\\) such that \\(\\rho (\\pmb {\\mu}) = \\pmb{\\theta}^{*}\\) rather than for all \\(\\mathbf{x}\\in \\mathrm{Int}(\\Sigma)\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.212,
+                0.826,
+                0.259
+            ],
+            "angle": 0,
+            "content": "One special instance of Assumption A3' is by assuming the linear SA, e.g., \\( H(\\pmb{\\theta}, i) = A_i\\pmb{\\theta} + b_i \\). In this case, \\( \\mathbb{E}_{X \\sim \\pi[\\mathbf{x}]}[H(\\rho(\\mathbf{x}), X)] = 0 \\) is equivalent to \\( \\mathbb{E}_{i \\sim \\pi[\\mathbf{x}]}[A_i]\\rho(\\mathbf{x}) + \\mathbb{E}_{i \\sim \\pi[\\mathbf{x}]}[b_i] = 0 \\). Under the condition that for every \\( \\mathbf{x} \\in \\mathrm{Int}(\\Sigma) \\), matrix \\( \\mathbb{E}_{i \\sim \\pi[\\mathbf{x}]}[A_i] \\) is invertible, we then have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.364,
+                0.266,
+                0.632,
+                0.284
+            ],
+            "angle": 0,
+            "content": "\\[\n\\rho (\\mathbf {x}) = - \\left(\\mathbb {E} _ {i \\sim \\boldsymbol {\\pi} [ \\mathbf {x} ]} [ A _ {i} ]\\right) ^ {- 1} \\cdot \\mathbb {E} _ {i \\sim \\boldsymbol {\\pi} [ \\mathbf {x} ]} [ b _ {i} ].\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.291,
+                0.825,
+                0.359
+            ],
+            "angle": 0,
+            "content": "However, this condition is quite strict. Loosely speaking, \\(\\mathbb{E}_{i\\sim \\pi [\\mathbf{x}]}[A_i]\\) being invertible for any \\(\\mathbf{x}\\) is similar to saying that any convex combination of \\(\\{A_i\\}\\) is invertible. For example, if we assume \\(\\{A_i\\}_{i\\in \\mathcal{N}}\\) are negative definite and they all share the same eigenbasis \\(\\{\\mathbf{u}_i\\}\\), e.g., \\(A_{i} = \\sum_{j = 1}^{D}\\lambda_{j}^{i}\\mathbf{u}_{i}\\mathbf{u}_{i}^{T}\\) and \\(\\lambda_j^i < 0\\) for all \\(i\\in \\mathcal{N}, j\\in [D]\\). Then, \\(\\mathbb{E}_{i\\sim \\pi [\\mathbf{x}]}[A_i]\\) is invertible."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.363,
+                0.825,
+                0.407
+            ],
+            "angle": 0,
+            "content": "Another example for Assumption A3' is when \\( H(\\pmb{\\theta}, i) = H(\\pmb{\\theta}, j) \\) for all \\( i, j \\in \\mathcal{N} \\), which implies that each agent in the distributed learning has the same local dataset to collaboratively train the model. In this example, \\( \\rho(\\mathbf{x}) = \\pmb{\\theta}^* \\) such that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.254,
+                0.414,
+                0.745,
+                0.469
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\mathbb {E} _ {i \\sim \\pi [ \\mathbf {x} ]} [ H (\\rho (\\mathbf {x}), i) ] = h (\\boldsymbol {\\theta} ^ {*}) = 0, \\\\ \\mathbb {E} _ {i \\sim \\pi [ \\mathbf {x} ]} [ \\nabla H (\\rho (\\mathbf {x}), i) ] + \\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I} = \\nabla h (\\boldsymbol {\\theta} ^ {*}) + \\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I} \\quad \\text {b e i n g H u r w i t z}. \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.485,
+                0.543,
+                0.5
+            ],
+            "angle": 0,
+            "content": "D PROOF OF LEMMA 3.1 AND LEMMA 3.2"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.517,
+                0.825,
+                0.587
+            ],
+            "angle": 0,
+            "content": "In this section, we demonstrate the almost sure convergence of both \\(\\theta_{n}\\) and \\(\\mathbf{x}_n\\) together. This proof naturally incorporates the almost certain convergence of the SRRW iteration in Lemma 3.1, since \\(\\mathbf{x}_n\\) is independent of \\(\\theta_{n}\\) (as indicated in (4)), allowing us to separate out its asymptotic results. The same reason applies to the CLT analysis of SRRW iterates and we refer the reader to Section E.1 for the CLT result of \\(\\mathbf{x}_n\\) in Lemma 3.1."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.593,
+                0.825,
+                0.623
+            ],
+            "angle": 0,
+            "content": "We will use different techniques for different settings of step sizes in Assumption A2. Specifically, for step sizes \\(\\gamma_{n} = (n + 1)^{-a},\\beta_{n} = (n + 1)^{-b}\\), we consider the following scenarios:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.633,
+                0.825,
+                0.676
+            ],
+            "angle": 0,
+            "content": "Scenario 1: We consider case(ii): \\( 1/2 < a = b \\leq 1 \\), and will apply the almost sure convergence result of the single-timescale stochastic approximation in Theorem G.8 and verify all the conditions therein."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.68,
+                0.825,
+                0.765
+            ],
+            "angle": 0,
+            "content": "Scenario 2: We consider both case(i): \\(1/2 < a < b \\leq 1\\) and case (iii): \\(1/2 < b < a \\leq 1\\). In these two cases, step sizes \\(\\gamma_{n}, \\beta_{n}\\) decrease at different rates, thereby putting iterates \\(\\mathbf{x}_{n}, \\pmb{\\theta}_{n}\\) on different timescales and resulting in a two-timescale structure. We will apply the existing almost sure convergence result of the two-timescale stochastic approximation with iteratedependent Markov chain in Yaji & Bhatnagar (2020, Theorem 4) where our SA-SRRW algorithm can be regarded as a special instance.[11]"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.781,
+                0.305,
+                0.795
+            ],
+            "angle": 0,
+            "content": "D.1 SCENARIO 1"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.807,
+                0.545,
+                0.823
+            ],
+            "angle": 0,
+            "content": "In Scenario 1, we have \\(\\beta_{n} = \\gamma_{n}\\). First, we rewrite (4) as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.358,
+                0.829,
+                0.825,
+                0.864
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left[ \\begin{array}{l} \\boldsymbol {\\theta} _ {n + 1} \\\\ \\mathbf {x} _ {n + 1} \\end{array} \\right] = \\left[ \\begin{array}{l} \\boldsymbol {\\theta} _ {n} \\\\ \\mathbf {x} _ {n} \\end{array} \\right] + \\gamma_ {n + 1} \\left[ \\begin{array}{c} H \\left(\\boldsymbol {\\theta} _ {n}. X _ {n + 1}\\right) \\\\ \\boldsymbol {\\delta} _ {X _ {n + 1}} - \\mathbf {x} _ {n} \\end{array} \\right]. \\tag {19}\n\\]"
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.171,
+                0.872,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "<sup>11</sup>However, Yaji & Bhatnagar (2020) paper only analysed the almost sure convergence. The central limit theorem analysis remains unknown in the literature for the two-timescale stochastic approximation with iteratedependent Markov chains. Thus, our CLT analysis in Section E for this two-timescale structure with iteratedependent Markov chain is still novel and recognized as our contribution."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.491,
+                0.949,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "16"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.101,
+                0.827,
+                0.218
+            ],
+            "angle": 0,
+            "content": "By augmentations, we define the variable \\(\\mathbf{z}_n \\triangleq \\left[ \\begin{array}{c} \\boldsymbol{\\theta}_n \\\\ \\mathbf{x}_n \\end{array} \\right] \\in \\mathbb{R}^{(N + D) \\times 1}\\) and the function \\(G(\\mathbf{z}_n, i) \\triangleq \\left[ \\begin{array}{c} H(\\boldsymbol{\\theta}_n, i) \\\\ \\delta_i - \\mathbf{x}_n \\end{array} \\right] \\in \\mathbb{R}^{(N + d) \\times 1}\\). In addition, we define a new Markov chain \\(\\{Y_n\\}_{n \\geq 0}\\) in the same state space \\(\\mathcal{N}\\) as SRRW sequence \\(\\{X_n\\}_{n \\geq 0}\\). With slight abuse of notation, the transition kernel of \\(\\{Y_n\\}\\) is denoted by \\(\\mathbf{K}'[\\mathbf{z}_n] \\equiv \\mathbf{K}[\\mathbf{x}_n]\\) and its stationary distribution \\(\\pi'[\\mathbf{z}_n] \\equiv \\pi[\\mathbf{x}_n]\\), where \\(\\mathbf{K}[\\mathbf{x}_n]\\) and \\(\\pi(\\mathbf{x}_n)\\) are the transition kernel and its corresponding stationary distribution of SRRW, with \\(\\pi[\\mathbf{x}]\\) of the form"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.36,
+                0.217,
+                0.825,
+                0.248
+            ],
+            "angle": 0,
+            "content": "\\[\n\\pi_ {i} [ \\mathbf {x} ] \\propto \\sum_ {j \\in \\mathcal {N}} \\mu_ {i} P _ {i j} \\left(x _ {i} / \\mu_ {i}\\right) ^ {- \\alpha} \\left(x _ {j} / \\mu_ {j}\\right) ^ {- \\alpha}. \\tag {20}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.25,
+                0.825,
+                0.278
+            ],
+            "angle": 0,
+            "content": "Recall that \\(\\pmb{\\mu}\\) is the fixed point, i.e., \\(\\pi[\\pmb{\\mu}] = \\pmb{\\mu}\\), and \\(\\mathbf{P}\\) is the base Markov chain inside SRRW (see (3)). Then, the mean field"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.318,
+                0.28,
+                0.677,
+                0.314
+            ],
+            "angle": 0,
+            "content": "\\[\ng (\\mathbf {z}) = \\mathbb {E} _ {Y \\sim \\boldsymbol {\\pi^ {\\prime}} (\\mathbf {z})} [ G (\\mathbf {z}, Y) ] = \\left[ \\begin{array}{c} \\sum_ {i \\in \\mathcal {N}} \\pi_ {i} [ \\mathbf {x} ] H (\\boldsymbol {\\theta}, i) \\\\ \\boldsymbol {\\pi} [ \\mathbf {x} ] - \\mathbf {x} \\end{array} \\right],\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.315,
+                0.825,
+                0.342
+            ],
+            "angle": 0,
+            "content": "and \\(\\mathbf{z}^{*} = (\\pmb{\\theta}^{*},\\pmb{\\mu})\\) for \\(\\pmb{\\theta}^{*}\\in \\Theta\\) in Assumption A3 is the root of \\(g(\\mathbf{z})\\), i.e., \\(g(\\mathbf{z}^{*}) = 0\\). The augmented iteration (19) becomes"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.389,
+                0.343,
+                0.823,
+                0.358
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {z} _ {n + 1} = \\mathbf {z} _ {n} + \\gamma_ {n + 1} G \\left(\\mathbf {z} _ {n}, Y _ {n + 1}\\right) \\tag {21}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.358,
+                0.825,
+                0.443
+            ],
+            "angle": 0,
+            "content": "with the goal of solving \\( g(\\mathbf{z}) = 0 \\). Therefore, we can treat (21) as an SA algorithm driven by a Markov chain \\( \\{Y_{n}\\}_{n\\geq 0} \\) with its kernel \\( \\mathbf{K}'[\\mathbf{z}] \\) and stationary distribution \\( \\pi^{\\prime}[\\mathbf{z}] \\), which has been widely studied in the literature (e.g., Delyon (2000); Benveniste et al. (2012); Fort (2015); Li et al. (2023)). In what follows, we demonstrate that for any initial point \\( \\mathbf{z}_0 = (\\pmb {\\theta}_0,\\mathbf{x}_0)\\in \\mathbb{R}^D\\times \\mathrm{Int}(\\Sigma) \\), the SRRW iteration \\( \\{\\mathbf{x}_n\\}_{n\\geq 0} \\) will almost surely converge to the target distribution \\( \\pmb{\\mu} \\), and the SA iteration \\( \\{\\pmb {\\theta}_n\\}_{n\\geq 0} \\) will almost surely converge to the set \\( \\Theta \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.448,
+                0.826,
+                0.572
+            ],
+            "angle": 0,
+            "content": "Now we verify conditions C1 - C4 in Theorem G.8. Our assumption A4 is equivalent to condition C1 and assumption A2 corresponds to condition C2. For condition C3, we set \\(\\nabla w(\\mathbf{z}) \\equiv -g(\\mathbf{z})\\), and the set \\(S \\equiv \\{\\mathbf{z}^* | \\boldsymbol{\\theta}^* \\in \\Theta, \\mathbf{x}^* = \\boldsymbol{\\mu}\\}\\), including disjoint points. For condition C4, since \\(\\mathbf{K}'[\\mathbf{z}]\\), or equivalently \\(\\mathbf{K}[\\mathbf{x}]\\), is ergodic and time-reversible for a given \\(\\mathbf{z}\\), as shown in the SRRW work Doshi et al. (2023), it automatically ensures a solution to the Poisson equation, which has been well discussed in Chen et al. (2020a, Section 2) and Benveniste et al. (2012); Meyn (2022). To show (97) and (98) in condition C4, for each given \\(\\mathbf{z}\\) and any \\(i \\in \\mathcal{N}\\), we need to give the explicit solution \\(m_{\\mathbf{z}}(i)\\) to the Poisson equation \\(m_{\\mathbf{z}}(i) - (\\mathbf{K}_{\\mathbf{z}}' m_{\\mathbf{z}})(i) = G(\\mathbf{z}, i) - g(\\mathbf{z})\\) in (96). The notation \\((\\mathbf{K}_{\\mathbf{z}}' m_{\\mathbf{z}})(i)\\) is defined as follows."
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.377,
+                0.57,
+                0.62,
+                0.602
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left(\\mathbf {K} _ {\\mathbf {z}} ^ {\\prime} m _ {\\mathbf {z}}\\right) (i) = \\sum_ {j \\in \\mathcal {N}} \\mathbf {K} _ {\\mathbf {z}} ^ {\\prime} (i, j) m (\\mathbf {z}, j).\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.612,
+                0.825,
+                0.642
+            ],
+            "angle": 0,
+            "content": "Let \\(\\mathbf{G}(\\mathbf{z})\\triangleq [G(\\mathbf{z},1),\\dots ,G(\\mathbf{z},N)]^T\\in \\mathbb{R}^{N\\times D}\\). We use \\([\\mathbf{A}]_{:,i}\\) to denote the \\(i\\)-th column of matrix \\(\\mathbf{A}\\). Then, we let \\(m_{\\mathbf{z}}(i)\\) such that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.205,
+                0.644,
+                0.825,
+                0.684
+            ],
+            "angle": 0,
+            "content": "\\[\nm _ {\\mathbf {z}} (i) = \\sum_ {k = 0} ^ {\\infty} \\left(\\left[ \\mathbf {G} (\\mathbf {z}) \\left(\\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] ^ {k}\\right) ^ {T} \\right] _ {[:, i ]} - g (\\mathbf {z})\\right) = \\sum_ {k = 0} ^ {\\infty} \\left[ \\mathbf {G} (\\mathbf {z}) \\left(\\left(\\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] ^ {k}\\right) ^ {T} - \\boldsymbol {\\pi} ^ {\\prime} [ \\mathbf {z} ] \\mathbf {1} ^ {T}\\right) \\right] _ {[:, i ]}. \\tag {22}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.686,
+                0.254,
+                0.699
+            ],
+            "angle": 0,
+            "content": "In addition,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.335,
+                0.698,
+                0.825,
+                0.737
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left(\\mathbf {K} _ {\\mathbf {z}} ^ {\\prime} m _ {\\mathbf {z}}\\right) (i) = \\sum_ {k = 1} ^ {\\infty} \\left[ \\mathbf {G} (\\mathbf {z}) \\left(\\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] ^ {T} - \\boldsymbol {\\pi} ^ {\\prime} [ \\mathbf {z} ] \\mathbf {1} ^ {T}\\right) \\right] _ {[ i, i ]}. \\tag {23}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.738,
+                0.825,
+                0.781
+            ],
+            "angle": 0,
+            "content": "We can check that the \\(m_{\\mathbf{z}}(i)\\) form in (22) is indeed the solution of the above Poisson equation. Now, by induction, we get \\(\\mathbf{K}'[\\mathbf{z}]^k - \\mathbf{1}\\pi'[\\mathbf{z}]^T = (\\mathbf{K}'[\\mathbf{z}] - \\mathbf{1}\\pi'[\\mathbf{z}]^T)^k\\) for \\(k \\geq 1\\) and for \\(k = 0\\), \\(\\mathbf{K}'[\\mathbf{z}]^0 - \\mathbf{1}\\pi'[\\mathbf{z}]^T = (\\mathbf{K}'[\\mathbf{z}] - \\mathbf{1}\\pi'[\\mathbf{z}]^T)^0 - \\mathbf{1}\\pi'[\\mathbf{z}]^T\\). Then,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.3,
+                0.783,
+                0.825,
+                0.928
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} m _ {\\mathbf {z}} (i) = \\sum_ {k = 0} ^ {\\infty} [ \\mathbf {G} (\\mathbf {z}) (\\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] ^ {T} - \\boldsymbol {\\pi} ^ {\\prime} [ \\mathbf {z} ] \\mathbf {1} ^ {T}) ^ {k} ] _ {[; i ]} - g (\\mathbf {z}) \\\\ = \\left[ \\mathbf {G} (\\mathbf {z}) \\sum_ {k = 0} ^ {\\infty} \\left(\\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] ^ {T} - \\boldsymbol {\\pi} ^ {\\prime} [ \\mathbf {z} ] \\mathbf {1} ^ {T}\\right) ^ {k} \\right] _ {[;, i ]} - g (\\mathbf {z}) \\tag {24} \\\\ = \\left[ \\mathbf {G} (\\mathbf {z}) \\left(\\mathbf {I} - \\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] ^ {T} + \\boldsymbol {\\pi} ^ {\\prime} [ \\mathbf {z} ] \\mathbf {1} ^ {T}\\right) ^ {- 1} \\right] _ {[,: i ]} - g (\\mathbf {z}) \\\\ = \\sum_ {j \\in \\mathcal {N}} \\left(\\mathbf {I} - \\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] + \\mathbf {1} \\boldsymbol {\\pi} ^ {\\prime} [ \\mathbf {z} ] ^ {T}\\right) ^ {- 1} (i, j) G (\\mathbf {z}, j) - g (\\mathbf {z}). \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.491,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "17"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.103,
+                0.825,
+                0.175
+            ],
+            "angle": 0,
+            "content": "Here, \\((\\mathbf{I} - \\mathbf{K}'[\\mathbf{z}] + 1\\pi'[\\mathbf{z}]^T)^{-1}\\) is well defined because \\(\\mathbf{K}'[\\mathbf{z}]\\) is ergodic and time-reversible for any given \\(\\mathbf{z}\\) (proved in Doshi et al. (2023, Appendix A)). Now that both functions \\(H(\\theta, i)\\) and \\(\\delta_i - \\mathbf{x}\\) are bounded for each compact subset of \\(\\mathbb{R}^D \\times \\Sigma\\) by our assumption A1, function \\(G(\\mathbf{z}, i)\\) is also bounded within the compact subset of its domain. Thus, function \\(m_{\\mathbf{z}}(i)\\) is bounded, and (97) is verified. Moreover, for a fixed \\(i \\in \\mathcal{N}\\),"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.185,
+                0.182,
+                0.812,
+                0.215
+            ],
+            "angle": 0,
+            "content": "\\[\n\\sum_ {j \\in \\mathcal {N}} (\\mathbf {I} - \\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] + \\mathbf {1} \\boldsymbol {\\pi} ^ {\\prime} [ \\mathbf {z} ] ^ {T}) ^ {- 1} (i, j) \\boldsymbol {\\delta} _ {j} = (\\mathbf {I} - \\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] + \\mathbf {1} \\boldsymbol {\\pi} ^ {\\prime} [ \\mathbf {z} ] ^ {T}) _ {[,: i ]} ^ {- 1} = (\\mathbf {I} - \\mathbf {K} [ \\mathbf {x} ] + \\mathbf {1} \\boldsymbol {\\pi} [ \\mathbf {x} ] ^ {T}) _ {[,: i ]} ^ {- 1}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.223,
+                0.825,
+                0.252
+            ],
+            "angle": 0,
+            "content": "and this vector-valued function is continuous in \\( \\mathbf{x} \\) because \\( \\mathbf{K}[\\mathbf{x}], \\pi[\\mathbf{x}] \\) are continuous. We then rewrite (24) as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.221,
+                0.259,
+                0.775,
+                0.302
+            ],
+            "angle": 0,
+            "content": "\\[\nm _ {\\mathbf {z}} (i) = \\left[ \\begin{array}{c} \\sum_ {j \\in \\mathcal {N}} (\\mathbf {I} - \\mathbf {K} [ \\mathbf {x} ] + \\mathbf {1} \\pi [ \\mathbf {x} ] ^ {T}) ^ {- 1} (i, j) H (\\mathbf {x}, j) \\\\ (\\mathbf {I} - \\mathbf {K} [ \\mathbf {x} ] ^ {T} + \\pi [ \\mathbf {x} ] \\mathbf {1} ^ {T}) _ {[,: i ]} ^ {- 1} \\end{array} \\right] - \\left[ \\begin{array}{c} \\sum_ {i \\in \\mathcal {N}} \\pi_ {i} [ \\mathbf {x} ] H (\\pmb {\\theta}, i) \\\\ \\pmb {\\pi} [ \\mathbf {x} ] - \\mathbf {x} \\end{array} \\right].\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.307,
+                0.826,
+                0.379
+            ],
+            "angle": 0,
+            "content": "With continuous functions \\(H(\\theta, i)\\), \\(\\mathbf{K}[\\mathbf{x}]\\), \\(\\pi[\\mathbf{x}]\\), we have \\(m_{\\mathbf{z}}(i)\\) continuous with respect to \\(\\mathbf{z}\\), so does \\((\\mathbf{K}_{\\mathbf{z}}^{\\prime} m_{\\mathbf{z}})(i)\\). This implies that functions \\(m_{\\mathbf{z}}(i)\\) and \\((\\mathbf{K}_{\\mathbf{z}}^{\\prime} m_{\\mathbf{z}})(i)\\) are locally Lipschitz, which satisfies (98) with \\(\\phi_{\\mathcal{C}}(x) = C_{\\mathcal{C}} x\\) for some constant \\(C_{\\mathcal{C}}\\) that depends on the compact set \\(\\mathcal{C}\\). Therefore, condition C4 is checked, and we can apply Theorem G.8 to show the almost convergence result of (19), i.e., almost surely,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.32,
+                0.386,
+                0.677,
+                0.41
+            ],
+            "angle": 0,
+            "content": "\\[\n\\lim  _ {n \\to \\infty} \\mathbf {x} _ {n} = \\boldsymbol {\\mu}, \\quad \\text {a n d} \\quad \\lim  _ {n \\to \\infty} \\sup  _ {\\boldsymbol {\\theta} ^ {*} \\in \\Theta} \\inf  _ {\\boldsymbol {\\theta}} \\| \\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*} \\| = 0.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.417,
+                0.827,
+                0.446
+            ],
+            "angle": 0,
+            "content": "Therefore, the almost sure convergence of \\(\\mathbf{x}_n\\) in Lemma 3.1 is also proved. This finishes the proof in Scenario 1."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.463,
+                0.307,
+                0.477
+            ],
+            "angle": 0,
+            "content": "D.2 SCENARIO 2"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.489,
+                0.825,
+                0.533
+            ],
+            "angle": 0,
+            "content": "Now in this subsection, we consider the steps sizes \\(\\gamma_{n},\\beta_{n}\\) with \\(1 / 2 < a < b\\leq 1\\) and \\(1 / 2 < b < a\\leq 1\\). We will frequently use assumptions (B1) - (B5) in Section G.3 and Theorem G.10 to prove the almost sure convergence."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.548,
+                0.418,
+                0.563
+            ],
+            "angle": 0,
+            "content": "D.2.1 CASE (I): \\( 1/2 < a < b \\leq 1 \\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.572,
+                0.825,
+                0.613
+            ],
+            "angle": 0,
+            "content": "In case (i), \\(\\theta_{n}\\) is on the slow timescale and \\(\\mathbf{x}_n\\) is on the fast timescale because iteration \\(\\theta_{n}\\) has smaller step size than \\(\\mathbf{x}_n\\), making \\(\\theta_{n}\\) converge slower than \\(\\mathbf{x}_n\\). Here, we consider the two-timescale SA of the form:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.386,
+                0.614,
+                0.61,
+                0.631
+            ],
+            "angle": 0,
+            "content": "\\[\n\\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} + \\beta_ {n + 1} H (\\boldsymbol {\\theta}, X _ {n + 1}),\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.387,
+                0.635,
+                0.61,
+                0.653
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {x} _ {n + 1} = \\mathbf {x} _ {n} + \\gamma_ {n + 1} (\\boldsymbol {\\delta} _ {X _ {n + 1}} - \\mathbf {x}).\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.663,
+                0.58,
+                0.679
+            ],
+            "angle": 0,
+            "content": "Now, we verify assumptions (B1) - (B5) listed in Section G.3."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.216,
+                0.69,
+                0.707,
+                0.705
+            ],
+            "angle": 0,
+            "content": "- Assumptions (B1) and (B5) are satisfied by our assumptions A2 and A4."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.216,
+                0.709,
+                0.825,
+                0.751
+            ],
+            "angle": 0,
+            "content": "- Our assumption A3 shows that the function \\( H(\\pmb{\\theta}, X) \\) is continuous and differentiable w.r.t \\( \\pmb{\\theta} \\) and grows linearly with \\( \\| \\pmb{\\theta} \\| \\). In addition, \\( \\delta_X - \\mathbf{x} \\) also satisfies this property. Therefore, (B2) is satisfied."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.216,
+                0.756,
+                0.825,
+                0.786
+            ],
+            "angle": 0,
+            "content": "- Now that the function \\(\\pi[\\mathbf{x}] - \\mathbf{x}\\) is independent of \\(\\pmb{\\theta}\\), we can set \\(\\rho(\\pmb{\\theta}) = \\pmb{\\mu}\\) for any \\(\\pmb{\\theta} \\in \\mathbb{R}^D\\) such that \\(\\pi[\\pmb{\\mu}] - \\pmb{\\mu} = 0\\) from Doshi et al. (2023, Proposition 3.1), and"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.216,
+                0.69,
+                0.825,
+                0.786
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.358,
+                0.793,
+                0.698,
+                0.812
+            ],
+            "angle": 0,
+            "content": "\\[\n\\nabla_ {\\mathbf {x}} (\\boldsymbol {\\pi} (\\mathbf {x}) - \\mathbf {x}) | _ {\\mathbf {x} = \\boldsymbol {\\mu}} = 2 \\alpha \\mathbf {u} \\mathbf {1} ^ {T} - \\alpha \\mathbf {P} ^ {T} - (\\alpha + 1) \\mathbf {I}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.228,
+                0.818,
+                0.825,
+                0.892
+            ],
+            "angle": 0,
+            "content": "from Doshi et al. (2023, Lemma 3.4), which is Hurwitz. Furthermore, \\(\\rho(\\pmb{\\theta}) = \\pmb{\\mu}\\) inherently satisfies the condition \\(\\|\\rho(\\pmb{\\theta})\\| \\leq L_2(1 + \\|\\pmb{\\theta}\\|)\\) for any \\(L_2 \\geq \\|\\pmb{\\mu}\\|\\). Thus, conditions (i) - (iii) in (B3) are satisfied. Additionally, \\(\\sum_{i \\in \\mathcal{N}} \\pi_i[\\rho(\\pmb{\\theta})] H(\\pmb{\\theta}, i) = \\sum_{i \\in \\mathcal{N}} \\pi_i[\\mathbf{x}] = \\mathbf{h}(\\pmb{\\theta})\\) such that for \\(\\pmb{\\theta}^* \\in \\Theta\\) defined in assumption A3, \\(\\sum_{i \\in \\mathcal{N}} \\pi_i[\\rho(\\pmb{\\theta}^*)] H(\\pmb{\\theta}^*, i) = \\mathbf{h}(\\pmb{\\theta}^*) = 0\\), and \\(\\nabla_\\theta \\mathbf{h}(\\pmb{\\theta}^*)\\) is Hurwitz. Therefore, (B3) is checked."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.216,
+                0.896,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "- Assumption (B4) is verified by the nature of SRRW, i.e., its transition kernel \\(\\mathbf{K}[\\mathbf{x}]\\) and the corresponding stationary distribution \\(\\pi[\\mathbf{x}]\\) with \\(\\pi[\\boldsymbol{\\mu}] = \\boldsymbol{\\mu}\\)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.491,
+                0.948,
+                0.509,
+                0.961
+            ],
+            "angle": 0,
+            "content": "18"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.104,
+                0.825,
+                0.135
+            ],
+            "angle": 0,
+            "content": "Consequently, assumptions (B1) - (B5) are satisfied by our assumeproins A1 - A4 and by Theorem G.10, we have \\(\\lim_{n\\to \\infty}\\mathbf{x}_n = \\pmb{\\mu}\\) and \\(\\theta_{n}\\rightarrow \\Theta\\) almost surely."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.139,
+                0.827,
+                0.268
+            ],
+            "angle": 0,
+            "content": "Next, we consider \\(1/2 < b < a \\leq 1\\). As discussed before, (B1), (B2), (B4) and (B5) are satisfied by our assumptions A1 - A4 and the properties of SRRW. The only difference for this step size setting, compared to the previous one \\(1/2 < a < b \\leq 1\\), is that the roles of \\(\\theta_{n}, \\mathbf{x}_{n}\\) are now flipped, that is, \\(\\theta_{n}\\) is now on the fast timescale while \\(\\mathbf{x}_{n}\\) is on the slow timescale. By a much stronger Assumption A3', for any \\(\\mathbf{x} \\in \\mathrm{Int}(\\Sigma)\\), (i) \\(\\mathbb{E}_{X \\sim \\pi[\\mathbf{x}]}[H(\\rho(\\mathbf{x}), X)] = 0\\); (ii) \\(\\mathbb{E}_{X \\sim \\pi[\\mathbf{x}]}[\\nabla H(\\rho(\\mathbf{x}), X)]\\) is Hurwitz; (iii) \\(\\|\\rho(\\mathbf{x})\\| \\leq L_{2}(1 + \\| \\mathbf{x}\\|)\\). Hence, conditions (i) - (iii) in (B3) are satisfied. Moreover, we have \\(\\pi[\\pmb{\\mu}] - \\pmb{\\mu} = 0\\), \\(\\nabla(\\pi[\\mathbf{x}] - \\mathbf{x})|_{\\mathbf{x} = \\pmb{\\mu}}\\) being Hurwitz, as mentioned in the previous part. Therefore, (B3) is verified. Accordingly, (B1) - (B5) are checked by our assumptions A1, A2, A3', A4. By Theorem G.10, we have \\(\\lim_{n \\to \\infty} \\mathbf{x}_n = \\pmb{\\mu}\\) and \\(\\theta_{n} \\to \\Theta\\) almost surely."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.293,
+                0.416,
+                0.309
+            ],
+            "angle": 0,
+            "content": "E PROOF OF THEOREM 3.3"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.328,
+                0.825,
+                0.414
+            ],
+            "angle": 0,
+            "content": "This section is devoted to the proof of Theorem 3.3, which also includes the proof of the CLT results for the SRRW iteration \\(\\mathbf{x}_n\\) in Lemma 3.1. We will use different techniques depending on the step sizes in Assumption A2. Specifically, for step sizes \\(\\gamma_n = (n + 1)^{-a}\\), \\(\\beta_n = (n + 1)^{-b}\\), we will consider three cases: case (i): \\(\\beta_n = o(\\gamma_n)\\); case (ii): \\(\\beta_n = \\gamma_n\\); and case (iii): \\(\\gamma_n = o(\\beta_n)\\). For case (ii), we will use the existing CLT result for single-timescale SA in Theorem G.9. For cases (i) and (iii), we will construct our own CLT analysis for the two-timescale structure. We start with case (ii)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.435,
+                0.352,
+                0.451
+            ],
+            "angle": 0,
+            "content": "E.1 CASE (II): \\(\\beta_{n} = \\gamma_{n}\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.463,
+                0.825,
+                0.506
+            ],
+            "angle": 0,
+            "content": "In this part, we stick to the notations for single-timescale SA studied in Section D.1. To utilize Theorem G.9, apart from Conditions C1 - C4 that have been checked in Section D.1, we still need to check conditions C5 and C6 listed in Section G.2."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.512,
+                0.825,
+                0.542
+            ],
+            "angle": 0,
+            "content": "Assumption A3 corresponds to condition C5. For condition C6, we need to obtain the explicit form of function \\( Q_{\\mathbf{z}} \\) to the Poisson equation defined in (96), that is,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.331,
+                0.553,
+                0.666,
+                0.571
+            ],
+            "angle": 0,
+            "content": "\\[\nQ _ {\\mathbf {z}} (i) - \\left(\\mathbf {K} _ {\\mathbf {z}} ^ {\\prime} Q _ {\\mathbf {z}}\\right) (i) = \\psi (\\mathbf {z}, i) - \\mathbb {E} _ {j \\sim \\pi [ \\mathbf {z} ]} [ \\psi (\\mathbf {z}, j) ]\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.583,
+                0.218,
+                0.595
+            ],
+            "angle": 0,
+            "content": "where"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.28,
+                0.6,
+                0.717,
+                0.635
+            ],
+            "angle": 0,
+            "content": "\\[\n\\psi (\\mathbf {z}, i) \\triangleq \\sum_ {j \\in \\mathcal {N}} \\mathbf {K} _ {\\mathbf {z}} ^ {\\prime} (i, j) m _ {\\mathbf {z}} (j) m _ {\\mathbf {z}} (j) ^ {T} - (\\mathbf {K} _ {\\mathbf {z}} ^ {\\prime} m _ {\\mathbf {z}}) (i) (\\mathbf {K} _ {\\mathbf {z}} ^ {\\prime} m _ {\\mathbf {z}}) (i) ^ {T}.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.646,
+                0.704,
+                0.662
+            ],
+            "angle": 0,
+            "content": "Following the similar steps in the derivation of \\(m_{\\mathbf{z}}(i)\\) from (22) to (24), we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.28,
+                0.675,
+                0.717,
+                0.709
+            ],
+            "angle": 0,
+            "content": "\\[\nQ _ {\\mathbf {z}} (i) = \\sum_ {j \\in \\mathcal {N}} (\\mathbf {I} - \\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] + \\mathbf {1} \\pmb {\\pi} ^ {\\prime} [ \\mathbf {z} ] ^ {T}) ^ {- 1} (i, j) m _ {\\mathbf {z}} (j) - \\pi_ {j} ^ {\\prime} [ \\mathbf {z} ] m _ {\\mathbf {z}} (j).\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.721,
+                0.825,
+                0.793
+            ],
+            "angle": 0,
+            "content": "We also know that \\( Q_{\\mathbf{z}}(i) \\) and \\( (\\mathbf{K}_{\\mathbf{z}}^{\\prime}Q_{\\mathbf{z}})(i) \\) are continuous in \\( \\mathbf{z} \\) for any \\( i \\in \\mathcal{N} \\). For any \\( \\mathbf{z} \\) in a compact set \\( \\Omega \\), \\( Q_{\\mathbf{z}}(i) \\) and \\( (\\mathbf{K}_{\\mathbf{z}}^{\\prime}Q_{\\mathbf{z}})(i) \\) are bounded because function \\( m_{\\mathbf{z}}(i) \\) is bounded. Therefore, C6 is checked. By Theorem G.9, assume \\( \\mathbf{z}_n = \\begin{bmatrix} \\theta_n \\\\ \\mathbf{x}_n \\end{bmatrix} \\) converges to a point \\( \\mathbf{z}^* = \\begin{bmatrix} \\theta^* \\\\ \\mu \\end{bmatrix} \\) for \\( \\theta^* \\in \\Theta \\), we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.384,
+                0.799,
+                0.826,
+                0.826
+            ],
+            "angle": 0,
+            "content": "\\[\n\\gamma_ {n} ^ {- 1 / 2} \\left(\\mathbf {z} _ {n} - \\mathbf {z} ^ {*}\\right) \\xrightarrow [ n \\rightarrow \\infty ]{d i s t .} N (0, \\mathbf {V}), \\tag {26}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.835,
+                0.571,
+                0.852
+            ],
+            "angle": 0,
+            "content": "where \\(\\mathbf{V}\\) is the solution of the following Lyapunov equation"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.275,
+                0.862,
+                0.826,
+                0.898
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {V} \\left(\\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I} + \\nabla g \\left(\\boldsymbol {z} ^ {*}\\right) ^ {T}\\right) + \\left(\\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I} + \\nabla g \\left(\\boldsymbol {z} ^ {*}\\right)\\right) \\mathbf {V} + \\mathbf {U} = 0, \\tag {27}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.908,
+                0.636,
+                0.929
+            ],
+            "angle": 0,
+            "content": "and \\(\\mathbf{U} = \\sum_{i\\in \\mathcal{N}}\\mu_i\\left(m_{\\mathbf{z}^*}(i)m_{\\mathbf{z}^*}(i)\\right)^T - (\\mathbf{K}_{\\mathbf{z}^*}m_{\\mathbf{z}^*})(i)(\\mathbf{K}_{\\mathbf{z}^*}m_{\\mathbf{z}^*})(i)^T.\\)"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.491,
+                0.948,
+                0.51,
+                0.961
+            ],
+            "angle": 0,
+            "content": "19"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.825,
+                0.133
+            ],
+            "angle": 0,
+            "content": "By algebraic calculations of derivative of \\(\\pi[\\mathbf{x}]\\) with respect to \\(\\mathbf{x}\\) in (20), we can rewrite \\(\\nabla g(\\mathbf{z}^*)\\) in terms of \\(\\mathbf{x}, \\pmb{\\theta}\\), i.e.,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.23,
+                0.14,
+                0.766,
+                0.22
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\mathbf {J} (\\alpha) \\triangleq \\nabla g (\\mathbf {z} ^ {*}) = \\left[ \\begin{array}{c c} \\frac {\\partial \\sum_ {i \\in \\mathcal {N}} \\pi_ {i} [ \\mathbf {x} ] H (\\boldsymbol {\\theta} , i)}{\\partial \\boldsymbol {\\theta}} & \\frac {\\partial \\sum_ {i \\in \\mathcal {N}} \\pi_ {i} [ \\mathbf {x} ] H (\\boldsymbol {\\theta} , i)}{\\partial \\mathbf {x}} \\\\ \\frac {\\partial (\\boldsymbol {\\pi} [ \\mathbf {x} ] - \\mathbf {x})}{\\partial \\boldsymbol {\\theta}} & \\frac {\\partial \\boldsymbol {\\pi} [ \\mathbf {x} ] - \\mathbf {x}}{\\partial \\mathbf {x}} \\end{array} \\right] _ {\\mathbf {z} = \\mathbf {z} ^ {*}} \\\\ = \\left[ \\begin{array}{c c} \\nabla \\mathbf {h} (\\boldsymbol {\\theta} ^ {*}) & - \\alpha \\mathbf {H} ^ {T} (\\mathbf {P} ^ {T} + \\mathbf {I}) \\\\ \\mathbf {0} & 2 \\alpha \\boldsymbol {\\mu} \\mathbf {1} ^ {T} - \\alpha \\mathbf {P} ^ {T} - (\\alpha + 1) \\mathbf {I} \\end{array} \\right] \\triangleq \\left[ \\begin{array}{c c} \\mathbf {J} _ {1 1} & \\mathbf {J} _ {1 2} (\\alpha) \\\\ \\mathbf {J} _ {2 1} & \\mathbf {J} _ {2 2} (\\alpha) \\end{array} \\right], \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.229,
+                0.806,
+                0.246
+            ],
+            "angle": 0,
+            "content": "where matrix \\(\\mathbf{H} = [H(\\pmb{\\theta}^*, 1), \\dots, H(\\pmb{\\theta}, N)]^T\\). Then, we further clarify the matrix \\(\\mathbf{U}\\). Note that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.171,
+                0.255,
+                0.825,
+                0.308
+            ],
+            "angle": 0,
+            "content": "\\[\nm _ {\\mathbf {z} ^ {*}} (i) = \\sum_ {k = 0} ^ {\\infty} [ \\mathbf {G} (\\mathbf {z} ^ {*}) ((\\mathbf {P} ^ {k}) ^ {T} - \\boldsymbol {\\mu} \\mathbf {1} ^ {T}) ] _ {[:, i ]} = \\sum_ {k = 0} ^ {\\infty} [ \\mathbf {G} (\\mathbf {z} ^ {*}) (\\mathbf {P} ^ {k}) ^ {T} ] _ {[:, i ]} = \\mathbb {E} \\left[ \\sum_ {k = 0} ^ {\\infty} [ G (\\mathbf {z} ^ {*}, X _ {k}) ] \\right| X _ {0} = i ], \\tag {28}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.308,
+                0.825,
+                0.366
+            ],
+            "angle": 0,
+            "content": "where the first equality holds because \\(\\mathbf{K}^{\\prime}[\\pmb {\\mu}] = \\mathbf{P}\\) from the definition of SRRW kernel (3), the second equality stems from \\(\\mathbf{G}(\\mathbf{z}^{*})\\pmb {\\mu} = g(\\mathbf{z}^{*}) = 0\\) , and the last term is a conditional expectation over the base Markov chain \\(\\{X_{k}\\}_{k\\geq 0}\\) (with transition kernel \\(\\mathbf{P}\\)) conditioned on \\(X_0 = i\\) . Similarly, with \\((\\mathbf{K}_{\\mathbf{z}}^{\\prime}m_{\\mathbf{z}})(i)\\) in the form of (23), we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.347,
+                0.373,
+                0.651,
+                0.416
+            ],
+            "angle": 0,
+            "content": "\\[\n(\\mathbf {K} _ {\\mathbf {z}} ^ {\\prime} m _ {\\mathbf {z}}) (i) = \\mathbb {E} \\left[ \\sum_ {k = 1} ^ {\\infty} [ G (\\mathbf {z} ^ {*}, X _ {k}) ] \\Bigg | X _ {0} = i \\right].\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.432,
+                0.825,
+                0.462
+            ],
+            "angle": 0,
+            "content": "From the form \\(\\sum_{i\\in \\mathcal{N}}\\mu_i\\) inside the matrix \\(\\mathbf{U}\\), the Markov chain \\(\\{X_k\\}_{k\\geq 0}\\) is in its stationary regime from the beginning, i.e., \\(X_{k}\\sim \\pmb{\\mu}\\) for any \\(k\\geq 0\\). Hence,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.181,
+                0.47,
+                0.825,
+                0.723
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\mathbf {U} = \\mathbb {E} \\left[ \\left(\\sum_ {k = 0} ^ {\\infty} [ G (\\mathbf {z} ^ {*}, X _ {k}) ]\\right) \\left(\\sum_ {k = 0} ^ {\\infty} [ G (\\mathbf {z} ^ {*}, X _ {k}) ]\\right) ^ {T} \\right] \\\\ - \\mathbb {E} \\left[ \\left(\\sum_ {k = 1} ^ {\\infty} [ G (\\mathbf {z} ^ {*}, X _ {k}) ]\\right) \\left(\\sum_ {k = 1} ^ {\\infty} [ G (\\mathbf {z} ^ {*}, X _ {k}) ]\\right) ^ {T} \\right] \\\\ = \\mathbb {E} \\left[ G \\left(\\mathbf {z} ^ {*}, X _ {0}\\right) G \\left(\\mathbf {z} ^ {*}, X _ {0}\\right) ^ {T} \\right] + \\mathbb {E} \\left[ G \\left(\\mathbf {z} ^ {*}, X _ {0}\\right) \\left(\\sum_ {k = 1} ^ {\\infty} G \\left(\\mathbf {z} ^ {*}, X _ {k}\\right)\\right) ^ {T} \\right] \\tag {29} \\\\ + \\mathbb {E} \\left[ \\left(\\sum_ {k = 1} ^ {\\infty} G (\\mathbf {z} ^ {*}, X _ {k})\\right) G (\\mathbf {z} ^ {*}, X _ {0}) ^ {T} \\right] \\\\ = \\operatorname {C o v} \\left(G \\left(\\mathbf {z} ^ {*}, X _ {0}\\right), G \\left(\\mathbf {z} ^ {*}, X _ {0}\\right)\\right) \\\\ + \\sum_ {k = 1} ^ {\\infty} \\left[ \\operatorname {C o v} \\left(G \\left(\\mathbf {z} ^ {*}, X _ {0}\\right), G \\left(\\mathbf {z} ^ {*}, X _ {k}\\right)\\right) + \\operatorname {C o v} \\left(G \\left(\\mathbf {z} ^ {*}, X _ {k}\\right), G \\left(\\mathbf {z} ^ {*}, X _ {0}\\right)\\right) \\right], \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.734,
+                0.825,
+                0.805
+            ],
+            "angle": 0,
+            "content": "where the covariance between \\( G(\\mathbf{z}^*, X_0) \\) and \\( G(\\mathbf{z}^*, X_k) \\) for the Markov chain \\( \\{X_n\\} \\) in the stationary regime is \\( \\mathrm{Cov}(G(\\mathbf{z}^*, X_0), G(\\mathbf{z}^*, X_k)) \\). By Brémaud (2013, Theorem 6.3.7), it is demonstrated that \\( \\mathbf{U} \\) is the sampling covariance of the base Markov chain \\( \\mathbf{P} \\) for the test function \\( G(\\mathbf{z}^*, \\cdot) \\). Moreover, Brémaud (2013, equation (6.34)) states that this sampling covariance \\( \\mathbf{U} \\) can be rewritten in the following form:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.188,
+                0.814,
+                0.826,
+                0.857
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {U} = \\sum_ {i = 1} ^ {N - 1} \\mathbf {G} \\left(\\mathbf {z} ^ {*}\\right) ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} \\mathbf {G} \\left(\\mathbf {z} ^ {*}\\right) = \\sum_ {i = 1} ^ {N - 1} \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\left[ \\begin{array}{c c} \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H} & \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\\\ \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H} & \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\end{array} \\right] \\triangleq \\left[ \\begin{array}{l l} \\mathbf {U} _ {1 1} & \\mathbf {U} _ {1 2} \\\\ \\mathbf {U} _ {2 1} & \\mathbf {U} _ {2 2} \\end{array} \\right], \\tag {30}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.866,
+                0.825,
+                0.896
+            ],
+            "angle": 0,
+            "content": "where \\(\\{(\\lambda_i, \\mathbf{u}_i)\\}_{i \\in \\mathcal{N}}\\) is the eigenpair of the transition kernel \\(\\mathbf{P}\\) of the ergodic and time-reversible base Markov chain. This completes the proof of case 1."
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.187,
+                0.908,
+                0.814,
+                0.927
+            ],
+            "angle": 0,
+            "content": "12One may refer to Doshi et al. (2023, Appendix B, Proof of Lemma 3.4) for the computation of \\(\\frac{\\partial\\pi|\\mathbf{x}| - \\mathbf{x}}{\\partial\\mathbf{x}}\\)"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.51,
+                0.96
+            ],
+            "angle": 0,
+            "content": "20"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.825,
+                0.152
+            ],
+            "angle": 0,
+            "content": "Remark E.1. For the CLT result (26), we can look further into the asymptotic covariance matrix \\(\\mathbf{V}\\) as in (27). For convenience, we denote \\(\\mathbf{V} = \\begin{bmatrix} \\mathbf{V}_{11} & \\mathbf{V}_{12}\\\\ \\mathbf{V}_{21} & \\mathbf{V}_{22} \\end{bmatrix}\\) and \\(\\mathbf{U}\\) in the form of (30) such that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.208,
+                0.156,
+                0.826,
+                0.192
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left[ \\begin{array}{l l} \\mathbf {V} _ {1 1} & \\mathbf {V} _ {1 2} \\\\ \\mathbf {V} _ {2 1} & \\mathbf {V} _ {2 2} \\end{array} \\right] \\left(\\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I} + \\mathbf {J} (\\alpha) ^ {T}\\right) + \\left(\\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I} + \\mathbf {J} (\\alpha)\\right) \\left[ \\begin{array}{l l} \\mathbf {V} _ {1 1} & \\mathbf {V} _ {1 2} \\\\ \\mathbf {V} _ {2 1} & \\mathbf {V} _ {2 2} \\end{array} \\right] + \\mathbf {U} = 0. \\tag {31}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.199,
+                0.827,
+                0.248
+            ],
+            "angle": 0,
+            "content": "For the SRRW iteration \\(\\mathbf{x}_n\\), from (26) we know that \\(\\gamma_n^{-1/2}(\\mathbf{x}_n - \\boldsymbol{\\mu}) \\xrightarrow[n\\to\\infty]{dist.} N(\\mathbf{0}, \\mathbf{V}_4)\\). Thus, in this remark, we want to obtain the closed form of \\(\\mathbf{V}_{22}\\). By algebraic computations of the bottom-right sub-block matrix, we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.331,
+                0.254,
+                0.669,
+                0.343
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\left(2 \\alpha \\boldsymbol {\\mu} \\mathbf {1} ^ {T} - \\alpha \\mathbf {P} ^ {T} - \\left(\\alpha + 1 - \\frac {\\mathbb {1} _ {\\{a = 1 \\}}}{2}\\right) \\mathbf {I}\\right) \\mathbf {V} _ {2 2} \\\\ + \\mathbf {V} _ {2 2} \\left(2 \\alpha \\boldsymbol {\\mu} \\mathbf {1} ^ {T} - \\alpha \\mathbf {P} ^ {T} - \\left(\\alpha + 1 - \\frac {\\mathbb {1} _ {\\{a = 1 \\}}}{2}\\right) \\mathbf {I}\\right) ^ {T} \\\\ + \\mathbf {U} _ {2 2} = 0. \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.35,
+                0.825,
+                0.379
+            ],
+            "angle": 0,
+            "content": "By using result of the closed form solution to the Lyapunov equation (e.g., Lemma G.1) and the eigendecomposition of \\(\\mathbf{P}\\), we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.318,
+                0.386,
+                0.826,
+                0.426
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {V} _ {2 2} = \\sum_ {i = 1} ^ {N - 1} \\frac {1}{2 \\alpha \\left(1 + \\lambda_ {i}\\right) + 2 - \\mathbb {1} _ {\\{a = 1 \\}}} \\cdot \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T}. \\tag {32}\n\\]"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.442,
+                0.367,
+                0.459
+            ],
+            "angle": 0,
+            "content": "E.2 CASE (I): \\(\\beta_{n} = o(\\gamma_{n})\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.468,
+                0.825,
+                0.499
+            ],
+            "angle": 0,
+            "content": "In this part, we mainly focus on the CLT of the SA iteration \\(\\theta_{n}\\) because the SRRW iteration \\(\\mathbf{x}_n\\) is independent of \\(\\theta_{n}\\) and its CLT result has been shown in Remark E.1."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.512,
+                0.559,
+                0.527
+            ],
+            "angle": 0,
+            "content": "E.2.1 DECOMPOSITION OF SA-SRRW ITERATION (4)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.537,
+                0.566,
+                0.551
+            ],
+            "angle": 0,
+            "content": "We slightly abuse the math notation and define the function"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.341,
+                0.558,
+                0.655,
+                0.59
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {h} (\\boldsymbol {\\theta}, \\mathbf {x}) \\triangleq \\mathbb {E} _ {i \\sim \\pi [ \\mathbf {x} ]} H (\\boldsymbol {\\theta}, i) = \\sum_ {i \\in \\mathcal {N}} \\pi_ {i} [ \\mathbf {x} ] H (\\boldsymbol {\\theta}, i)\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.596,
+                0.547,
+                0.612
+            ],
+            "angle": 0,
+            "content": "such that \\(\\mathbf{h}(\\pmb {\\theta},\\pmb {\\mu})\\equiv \\mathbf{h}(\\pmb {\\theta})\\) . Then, we reformulate (25) as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.271,
+                0.617,
+                0.825,
+                0.635
+            ],
+            "angle": 0,
+            "content": "\\[\n\\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} + \\beta_ {n + 1} \\mathbf {h} \\left(\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}\\right) + \\beta_ {n + 1} \\left(H \\left(\\boldsymbol {\\theta} _ {n}, X _ {n + 1}\\right) - \\mathbf {h} \\left(\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}\\right)\\right). \\tag {33a}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.295,
+                0.639,
+                0.825,
+                0.657
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {x} _ {n + 1} = \\mathbf {x} _ {n} + \\gamma_ {n + 1} \\left(\\boldsymbol {\\pi} \\left[ \\mathbf {x} _ {n} \\right] - \\mathbf {x} _ {n}\\right) + \\gamma_ {n + 1} \\left(\\boldsymbol {\\delta} _ {X _ {n + 1}}\\right) - \\boldsymbol {\\pi} \\left[ \\mathbf {x} _ {n} \\right]). \\tag {33b}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.661,
+                0.814,
+                0.677
+            ],
+            "angle": 0,
+            "content": "There exist functions \\(q_{\\mathbf{x}}:\\mathcal{N}\\to \\mathbb{R}^{N},\\tilde{H}_{\\boldsymbol{\\theta},\\mathbf{x}}:\\mathcal{N}\\to \\mathbb{R}^{D}\\) satisfying the following Poisson equations"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.388,
+                0.683,
+                0.825,
+                0.7
+            ],
+            "angle": 0,
+            "content": "\\[\n\\boldsymbol {\\delta} _ {i} - \\boldsymbol {\\pi} (\\mathbf {x}) = q _ {\\mathbf {x}} (i) - \\left(\\mathbf {K} _ {\\mathbf {x}} q _ {\\mathbf {x}}\\right) (i) \\tag {34a}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.343,
+                0.705,
+                0.825,
+                0.723
+            ],
+            "angle": 0,
+            "content": "\\[\nH (\\boldsymbol {\\theta}, i) - \\mathbf {h} (\\boldsymbol {\\theta}, \\mathbf {x}) = \\tilde {H} _ {\\boldsymbol {\\theta}, \\mathbf {x}} (i) - \\left(\\mathbf {K} _ {\\mathbf {x}} \\tilde {H} _ {\\boldsymbol {\\theta}, \\mathbf {x}}\\right) (i), \\tag {34b}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.727,
+                0.827,
+                0.791
+            ],
+            "angle": 0,
+            "content": "for any \\(\\pmb{\\theta} \\in \\mathbb{R}^{D}\\), \\(\\mathbf{x} \\in \\mathrm{Int}(\\Sigma)\\) and \\(i \\in \\mathcal{N}\\), where \\((\\mathbf{K}_{\\mathbf{x}}q_{\\mathbf{x}})(i) \\triangleq \\sum_{j \\in \\mathcal{N}} K_{ij}[\\mathbf{x}] q_{\\mathbf{x}}(j)\\), \\((\\mathbf{K}_{\\mathbf{x}}\\tilde{H}_{\\pmb{\\theta},\\mathbf{x}})(j) \\triangleq \\sum_{j \\in \\mathcal{N}} K_{ij}[\\mathbf{x}]\\tilde{H}_{\\pmb{\\theta},\\mathbf{x}}(j)\\). The existence and explicit form of the solutions \\(q_{\\mathbf{x}}, \\tilde{H}_{\\pmb{\\theta},\\mathbf{x}}\\), which are continuous w.r.t \\(\\mathbf{x}\\), \\(\\pmb{\\theta}\\), follow the similar steps that can be found in Section D.1 from (22) to (24). Thus, we can further decompose (33) into"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.24,
+                0.797,
+                0.825,
+                0.929
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} + \\beta_ {n + 1} \\mathbf {h} (\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}) + \\beta_ {n + 1} \\underbrace {(\\tilde {H} _ {\\boldsymbol {\\theta} _ {n} , \\mathbf {x} _ {n}} (X _ {n + 1}) - (\\mathbf {K} _ {\\mathbf {x} _ {n}} \\tilde {H} _ {\\boldsymbol {\\theta} _ {n} , \\mathbf {x} _ {n}}) (X _ {n}))} _ {M _ {n + 1} ^ {(\\boldsymbol {\\theta})}} \\\\ + \\beta_ {n + 1} \\underbrace {\\left(\\left(\\mathbf {K} _ {\\mathbf {x} _ {n + 1}} \\tilde {H} _ {\\boldsymbol {\\theta} _ {n + 1} , \\mathbf {x} _ {n + 1}}\\right) \\left(X _ {n + 1}\\right) - \\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} \\tilde {H} _ {\\boldsymbol {\\theta} _ {n} , \\mathbf {x} _ {n}}\\right) \\left(X _ {n + 1}\\right)\\right)} _ {r _ {n} ^ {(\\boldsymbol {\\theta}, 1)}} \\tag {35a} \\\\ + \\beta_ {n + 1} \\underbrace {((\\mathbf {K _ {x}} _ {n} \\tilde {H} ) (X _ {n}) - (\\mathbf {K _ {x}} _ {n + 1} \\tilde {H} ) (X _ {n + 1}))} _ {r _ {n} ^ {(\\pmb {\\theta}, 2)}}, \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.961
+            ],
+            "angle": 0,
+            "content": "21"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.255,
+                0.097,
+                0.824,
+                0.222
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\mathbf {x} _ {n + 1} = \\mathbf {x} _ {n} + \\gamma_ {n + 1} (\\boldsymbol {\\pi} (\\mathbf {x} _ {n}) - \\mathbf {x} _ {n}) + \\gamma_ {n + 1} \\underbrace {(q _ {\\mathbf {x} _ {n}} (X _ {n + 1}) - (\\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}}) (X _ {n}))} _ {M _ {n + 1} ^ {(\\mathbf {x})}} \\\\ + \\gamma_ {n + 1} \\underbrace {\\left([ \\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}} ] \\left(X _ {n + 1}\\right) - [ \\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n + 1}} ] \\left(X _ {n + 1}\\right)\\right)} _ {r _ {n} ^ {(\\mathbf {x}, 1)}} \\tag {35b} \\\\ + \\gamma_ {n + 1} \\underbrace {\\left((\\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}}) (X _ {n}) - (\\mathbf {K} _ {\\mathbf {x} _ {n + 1}} q _ {\\mathbf {x} _ {n + 1}}) (X _ {n + 1})\\right)} _ {r _ {n} ^ {(\\mathbf {x}, 2)}}. \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.224,
+                0.236,
+                0.236
+            ],
+            "angle": 0,
+            "content": "such that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.258,
+                0.239,
+                0.825,
+                0.259
+            ],
+            "angle": 0,
+            "content": "\\[\n\\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} + \\beta_ {n + 1} \\mathbf {h} \\left(\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}\\right) + \\beta_ {n + 1} M _ {n + 1} ^ {(\\boldsymbol {\\theta})} + \\beta_ {n + 1} r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} + \\beta_ {n + 1} r _ {n} ^ {(\\boldsymbol {\\theta}, 2)}, \\tag {36a}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.244,
+                0.264,
+                0.824,
+                0.283
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {x} _ {n + 1} = \\mathbf {x} _ {n} + \\gamma_ {n + 1} \\left(\\boldsymbol {\\pi} \\left(\\mathbf {x} _ {n}\\right) - \\mathbf {x} _ {n}\\right) + \\gamma_ {n + 1} M _ {n + 1} ^ {(\\mathbf {x})} + \\gamma_ {n + 1} r _ {n} ^ {(\\mathbf {x}, 1)} + \\gamma_ {n + 1} r _ {n} ^ {(\\mathbf {x}, 2)}. \\tag {36b}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.284,
+                0.825,
+                0.43
+            ],
+            "angle": 0,
+            "content": "We can observe that (36) differs from the expression in Konda & Tsitsiklis (2004); Mokkadem & Pelletier (2006), which studied the two-timescale SA with Martingale difference noise. Here, due to the presence of the iterate-dependent Markovian noise and the application of the Poisson equation technique, we have additional non-vanishing terms \\( r_n^{(\\theta, 2)} \\), \\( r_n^{(\\mathbf{x}, 2)} \\), which will be further examined in Lemma E.2. Additionally, when we apply the Poisson equation to the Martingale difference terms \\( M_{n+1}^{(\\theta)} \\), \\( M_{n+1}^{(\\mathbf{x})} \\), we find that there are some covariances that are also non-vanishing as in Lemma E.1. We will mention this again when we obtain those covariances. These extra non-zero noise terms make our analysis distinct from the previous ones since the key assumption (A4) in Mokkadem & Pelletier (2006) is not satisfied. We demonstrate that the long-term average performance of these terms can be managed so that they do not affect the final CLT result."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.435,
+                0.395,
+                0.456
+            ],
+            "angle": 0,
+            "content": "Analysis of Terms \\(M_{n + 1}^{(\\theta)},M_{n + 1}^{(\\mathbf{x})}\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.462,
+                0.825,
+                0.495
+            ],
+            "angle": 0,
+            "content": "Consider the filtration \\(\\mathcal{F}_n \\triangleq \\sigma(\\pmb{\\theta}_0, \\mathbf{x}_0, X_0, \\dots, \\pmb{\\theta}_n, \\mathbf{x}_n, X_n)\\), it is evident that \\(M_{n+1}^{(\\pmb{\\theta})}, M_{n+1}^{(\\mathbf{x})}\\) are Martingale difference sequences adapted to \\(\\mathcal{F}_n\\). Then, we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.235,
+                0.498,
+                0.824,
+                0.588
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\mathbb {E} \\left[ M _ {n + 1} ^ {(\\mathbf {x})} (M _ {n + 1} ^ {(\\mathbf {x})}) ^ {T} \\Big | \\mathcal {F} _ {n} \\right] \\\\ = \\mathbb {E} \\left[ q _ {\\mathbf {x} _ {n}} \\left(X _ {n + 1}\\right) q _ {\\mathbf {x} _ {n}} \\left(X _ {n + 1}\\right) ^ {T} \\mid \\mathcal {F} _ {n} \\right] + \\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}}\\right) \\left(X _ {n}\\right) \\left(\\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}}\\right) \\left(X _ {n}\\right)\\right) ^ {T} \\tag {37} \\\\ - \\mathbb {E} \\left[ q _ {\\mathbf {x} _ {n}} \\left(X _ {n + 1}\\right) \\mid \\mathcal {F} _ {n} \\right] \\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}}\\right) \\left(X _ {n}\\right)) ^ {T} - \\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}}\\right) \\left(X _ {n}\\right) \\mathbb {E} \\left[ q _ {\\mathbf {x} _ {n}} \\left(X _ {n + 1}\\right) ^ {T} \\mid \\mathcal {F} _ {n} \\right] \\\\ = \\mathbb {E} \\left[ q _ {\\mathbf {x} _ {n}} \\left(X _ {n + 1}\\right) q _ {\\mathbf {x} _ {n}} \\left(X _ {n + 1}\\right) ^ {T} \\mid \\mathcal {F} _ {n} \\right] - \\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}}\\right) \\left(X _ {n}\\right) \\left(\\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}}\\right) \\left(X _ {n}\\right)\\right) ^ {T}. \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.591,
+                0.299,
+                0.606
+            ],
+            "angle": 0,
+            "content": "Similarly, we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.192,
+                0.61,
+                0.824,
+                0.665
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\mathbb {E} \\left[ M _ {n + 1} ^ {(\\boldsymbol {\\theta})} \\left(M _ {n + 1} ^ {(\\boldsymbol {\\theta})}\\right) ^ {T} \\mid \\mathcal {F} _ {n} \\right] \\tag {38} \\\\ = \\mathbb {E} \\left[ \\tilde {H} _ {\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}} \\left(X _ {n + 1}\\right) \\tilde {H} _ {\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}} \\left(X _ {n + 1}\\right) ^ {T} \\mid \\mathcal {F} _ {n} \\right] - \\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} \\tilde {H} _ {\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}}\\right) \\left(X _ {n}\\right) \\left(\\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} \\tilde {H} _ {\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}}\\right) \\left(X _ {n}\\right)\\right) ^ {T}, \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.669,
+                0.202,
+                0.681
+            ],
+            "angle": 0,
+            "content": "and"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.229,
+                0.685,
+                0.769,
+                0.741
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\mathbb {E} \\left[ M _ {n + 1} ^ {(\\mathbf {x})} (M _ {n + 1} ^ {(\\boldsymbol {\\theta})}) ^ {T} \\Big | \\mathcal {F} _ {n} \\right] \\\\ = \\mathbb {E} [ q _ {\\mathbf {x} _ {n}} (X _ {n + 1}) \\tilde {H} _ {\\pmb {\\theta} _ {n}, \\mathbf {x} _ {n}} (X _ {n + 1}) ^ {T} | \\mathcal {F} _ {n} ] - (\\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}}) (X _ {n}) ((\\mathbf {K} _ {\\mathbf {x} _ {n}} \\tilde {H} _ {\\pmb {\\theta} _ {n}, \\mathbf {x} _ {n}}) (X _ {n})) ^ {T}. \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.751,
+                0.538,
+                0.778
+            ],
+            "angle": 0,
+            "content": "We now focus on \\(\\mathbb{E}\\left[M_{n + 1}^{(\\mathbf{x})}(M_{n + 1}^{(\\mathbf{x})})^T\\Big|\\mathcal{F}_n\\right]\\). Denote by"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.282,
+                0.782,
+                0.825,
+                0.815
+            ],
+            "angle": 0,
+            "content": "\\[\nV _ {1} (\\mathbf {x}, i) \\triangleq \\sum_ {j \\in \\mathcal {N}} \\mathbf {K} _ {i, j} [ \\mathbf {x} ] q _ {\\mathbf {x}} (j) q _ {\\mathbf {x}} (j) ^ {T} - \\left(\\mathbf {K} _ {\\mathbf {x}} q _ {\\mathbf {x}}\\right) (i) \\left(\\left(\\mathbf {K} _ {\\mathbf {x}} q _ {\\mathbf {x}}\\right) (i)\\right) ^ {T}, \\tag {39}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.82,
+                0.825,
+                0.85
+            ],
+            "angle": 0,
+            "content": "and let its expectation w.r.t the stationary distribution \\(\\pi (\\mathbf{x})\\) be \\(v_{1}(\\mathbf{x})\\triangleq \\mathbb{E}_{i\\sim \\pi (\\mathbf{x})}[V_{1}(\\mathbf{x},i)]\\), we can construct another Poisson equation, i.e.,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.266,
+                0.853,
+                0.731,
+                0.929
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\mathbb {E} \\left[ M _ {n + 1} ^ {(\\mathbf {x})} (M _ {n + 1} ^ {(\\mathbf {x})}) ^ {T} \\Big | \\mathcal {F} _ {n} \\right] - \\sum_ {X _ {n} \\in \\mathcal {N}} \\pi_ {X _ {n}} (\\mathbf {x} _ {n}) \\mathbb {E} \\left[ M _ {n + 1} ^ {(\\mathbf {x})} (M _ {n + 1} ^ {(\\mathbf {x})}) ^ {T} \\Big | \\mathcal {F} _ {n} \\right] \\\\ = V _ {1} \\left(\\mathbf {x} _ {n}, X _ {n + 1}\\right) - v _ {1} \\left(\\mathbf {x} _ {n}\\right) \\\\ = \\varphi_ {\\mathbf {x}} ^ {(1)} (X _ {n + 1}) - (\\mathbf {K} _ {\\mathbf {x} _ {n}} \\varphi_ {\\mathbf {x} _ {n}} ^ {(1)}) (X _ {n + 1}), \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "22"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.102,
+                0.825,
+                0.134
+            ],
+            "angle": 0,
+            "content": "for some matrix-valued function \\(\\varphi_{\\mathbf{x}}^{(1)}:\\mathcal{N}\\to \\mathbb{R}^{N\\times N}\\). Since \\(q_{\\mathbf{x}}\\) and \\(\\mathbf{K}[\\mathbf{x}]\\) are continuous in \\(\\mathbf{x}\\), functions \\(V_{1},v_{1}\\) are also continuous in \\(\\mathbf{x}\\). Then, we can decompose (39) into"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.243,
+                0.14,
+                0.824,
+                0.228
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} V _ {1} (\\mathbf {x} _ {n}, X _ {n + 1}) = \\underbrace {v _ {1} (\\boldsymbol {\\mu})} _ {\\mathbf {U} _ {2 2}} + \\underbrace {v _ {1} (\\mathbf {x} _ {n}) - v _ {1} (\\boldsymbol {\\mu})} _ {\\mathbf {D} _ {n} ^ {(1)}} + \\underbrace {\\varphi_ {\\mathbf {x} _ {n}} ^ {(1)} (X _ {n + 1}) - (\\mathbf {K} _ {\\mathbf {x} _ {n}} \\varphi_ {\\mathbf {x} _ {n}} ^ {(1)}) (X _ {n})} _ {\\mathbf {J} _ {n} ^ {(1, a)}} \\\\ + \\underbrace {\\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} \\varphi_ {\\mathbf {x} _ {n}} ^ {(1)}\\right) \\left(X _ {n}\\right) - \\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} \\varphi_ {\\mathbf {x} _ {n}} ^ {(1)}\\right) \\left(X _ {n + 1}\\right)} _ {\\mathbf {J} _ {n} ^ {(1, b)}}. \\tag {40} \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.235,
+                0.274,
+                0.249
+            ],
+            "angle": 0,
+            "content": "Thus, we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.343,
+                0.248,
+                0.825,
+                0.269
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbb {E} \\left[ M _ {n + 1} ^ {(\\mathbf {x})} \\left(M _ {n + 1} ^ {(\\mathbf {x})}\\right) ^ {T} \\mid \\mathcal {F} _ {n} \\right] = \\mathbf {U} _ {2 2} + \\mathbf {D} _ {n} ^ {(1)} + \\mathbf {J} _ {n} ^ {(1)}, \\tag {41}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.273,
+                0.37,
+                0.293
+            ],
+            "angle": 0,
+            "content": "where \\(\\mathbf{J}_n^{(1)} = \\mathbf{J}_n^{(1,a)} + \\mathbf{J}_n^{(1,b)}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.297,
+                0.825,
+                0.346
+            ],
+            "angle": 0,
+            "content": "Following the similar steps above, we can decompose \\(\\mathbb{E}\\left[M_{n + 1}^{(\\mathbf{x})}(M_{n + 1}^{(\\boldsymbol{\\theta})})^T\\Big|\\mathcal{F}_n\\right]\\) and \\(\\mathbb{E}\\left[M_{n + 1}^{(\\boldsymbol{\\theta})}(M_{n + 1}^{(\\boldsymbol{\\theta})})^T\\Big|\\mathcal{F}_n\\right]\\) as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.335,
+                0.353,
+                0.826,
+                0.381
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbb {E} \\left[ M _ {n + 1} ^ {(\\mathbf {x})} \\left(M _ {n + 1} ^ {(\\boldsymbol {\\theta})}\\right) ^ {T} \\mid \\mathcal {F} _ {n} \\right] = \\mathbf {U} _ {2 1} + \\mathbf {D} _ {n} ^ {(2)} + \\mathbf {J} _ {n} ^ {(2)}, \\tag {42a}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.336,
+                0.388,
+                0.825,
+                0.415
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbb {E} \\left[ M _ {n + 1} ^ {(\\boldsymbol {\\theta})} \\left(M _ {n + 1} ^ {(\\boldsymbol {\\theta})}\\right) ^ {T} \\mid \\mathcal {F} _ {n} \\right] = \\mathbf {U} _ {1 1} + \\mathbf {D} _ {n} ^ {(3)} + \\mathbf {J} _ {n} ^ {(3)}. \\tag {42b}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.419,
+                0.825,
+                0.493
+            ],
+            "angle": 0,
+            "content": "where \\(\\mathbf{J}_n^{(2)} = \\mathbf{J}_n^{(2,a)} + \\mathbf{J}_n^{(2,b)}\\) and \\(\\mathbf{J}_n^{(3)} = \\mathbf{J}_n^{(3,a)} + \\mathbf{J}_n^{(3,b)}\\). Here, we note that matrices \\(\\mathbf{J}_n^i\\) for \\(i = 1,2,3\\) are in presence of the current CLT analysis of the two-timescale SA with Martingale difference noise. In addition, \\(\\mathbf{U}_{11}\\), \\(\\mathbf{U}_{12}\\) and \\(\\mathbf{U}_{22}\\) inherently include the information of the underlying Markov chain (with its eigenpair \\((\\lambda_i,\\mathbf{u}_i)\\)), which is an extension of the previous works (Konda & Tsitsiklis, 2004; Mokkadem & Pelletier, 2006)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.498,
+                0.819,
+                0.516
+            ],
+            "angle": 0,
+            "content": "Lemma E.1. For \\(M_{n + 1}^{(\\theta)},M_{n + 1}^{(\\mathbf{x})}\\) defined in (35) and their decomposition in (41) and (42), we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.195,
+                0.524,
+                0.825,
+                0.578
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {U} _ {1 1} = \\sum_ {i = 1} ^ {N - 1} \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T}, \\quad \\mathbf {U} _ {2 1} = \\sum_ {i = 1} ^ {N - 1} \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H}, \\quad \\mathbf {U} _ {1 1} = \\sum_ {i = 1} ^ {N - 1} \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H}, \\tag {43a}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.373,
+                0.578,
+                0.825,
+                0.602
+            ],
+            "angle": 0,
+            "content": "\\[\n\\lim  _ {n \\rightarrow \\infty} \\mathbf {D} _ {n} ^ {(i)} = 0 a. s. f o r i = 1, 2, 3, \\tag {43b}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.338,
+                0.606,
+                0.825,
+                0.649
+            ],
+            "angle": 0,
+            "content": "\\[\n\\lim  _ {n \\rightarrow \\infty} \\gamma_ {n} \\mathbb {E} \\left[\\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(i)} \\right\\|\\right] = 0, \\quad f o r \\quad i = 1, 2, 3. \\tag {43c}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.664,
+                0.78,
+                0.68
+            ],
+            "angle": 0,
+            "content": "Proof. We now provide the properties of the four terms inside (41) as an example. Note that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.212,
+                0.687,
+                0.78,
+                0.772
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\mathbf {U} _ {1 1} = \\mathbb {E} _ {i \\sim \\boldsymbol {\\mu}} [ V _ {1} (\\boldsymbol {\\mu}, i) ] = \\sum_ {i \\in \\mathcal {N}} \\mu_ {i} \\left[ \\sum_ {j \\in \\mathcal {N}} \\mathbf {P} (i, j) q _ {\\boldsymbol {\\mu}} (j) q _ {\\boldsymbol {\\mu}} (j) ^ {T} - (\\mathbf {P} q _ {\\boldsymbol {\\mu}}) (i) ((\\mathbf {P} q _ {\\boldsymbol {\\mu}}) (i)) ^ {T} \\right] \\\\ = \\sum_ {j \\in \\mathcal {N}} \\mu_ {j} q _ {\\pmb {\\mu}} (j) q _ {\\pmb {\\mu}} (j) ^ {T} - (\\mathbf {P} q _ {\\pmb {\\mu}}) (j) ((\\mathbf {P} q _ {\\pmb {\\mu}}) (j)) ^ {T}. \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.779,
+                0.825,
+                0.809
+            ],
+            "angle": 0,
+            "content": "We can see that it has exactly the same structure as matrix \\( \\mathbf{U} \\) in (27). Following the similar steps in deducing the explicit form of \\( \\mathbf{U} \\) from (28) to (30), we get"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.41,
+                0.816,
+                0.825,
+                0.857
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {U} _ {1 1} = \\sum_ {i = 1} ^ {N - 1} \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T}. \\tag {44}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.865,
+                0.825,
+                0.898
+            ],
+            "angle": 0,
+            "content": "By the almost sure convergence result \\(\\mathbf{x}_n\\to \\pmb{\\mu}\\) in Lemma 3.1, \\(v_{1}(\\mathbf{x}_{n})\\rightarrow v_{1}(\\pmb {\\mu})\\) a.s. such that \\(\\lim_{n\\to \\infty}\\mathbf{D}_n^{(1)} = 0\\) a.s."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.903,
+                0.813,
+                0.93
+            ],
+            "angle": 0,
+            "content": "We next prove that \\(\\lim_{n\\to \\infty}\\gamma_n\\mathbb{E}\\left[\\left\\| \\sum_{k = 1}^n{\\bf J}_k^{(1,a)}\\right\\|\\right] = 0\\) and \\(\\lim_{n\\to \\infty}\\gamma_n\\mathbb{E}\\left[\\left\\| \\sum_{k = 1}^n{\\bf J}_k^{(1,b)}\\right\\|\\right] = 0.\\)"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "23"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.103,
+                0.825,
+                0.133
+            ],
+            "angle": 0,
+            "content": "Since \\(\\{\\mathbf{J}_n^{(1,a)}\\}\\) is a Martingale difference sequence adapted to \\(\\mathcal{F}_n\\), with the Burkholder inequality in Lemma G.2 and \\(p = 1\\), we show that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.333,
+                0.138,
+                0.823,
+                0.185
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbb {E} \\left[ \\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1, a)} \\right\\| \\right] \\leq C _ {1} \\mathbb {E} \\left[ \\sqrt {\\left(\\sum_ {k = 1} ^ {n} \\left\\| \\mathbf {J} _ {k} ^ {(1 , a)} \\right\\| ^ {2}\\right)} \\right]. \\tag {45}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.189,
+                0.825,
+                0.22
+            ],
+            "angle": 0,
+            "content": "By assumption A4, \\(\\mathbf{x}_n\\) is always within some compact set \\(\\Omega\\) such that \\(\\sup_n\\| \\mathbf{J}_n^{(1,a)}\\| \\leq C_\\Omega < \\infty\\) and for a given trajectory \\(\\omega\\) of \\(\\mathbf{x}_n(\\omega)\\)"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.359,
+                0.224,
+                0.823,
+                0.271
+            ],
+            "angle": 0,
+            "content": "\\[\n\\gamma_ {n} C _ {p} \\sqrt {\\left(\\sum_ {k = 1} ^ {n} \\left\\| \\mathbf {J} _ {k} ^ {(1 , a)} \\right\\| ^ {2}\\right)} \\leq C _ {p} C _ {\\Omega} \\gamma_ {n} \\sqrt {n}, \\tag {46}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.273,
+                0.531,
+                0.289
+            ],
+            "angle": 0,
+            "content": "and the last term decreases to zero in \\(n\\) since \\(a > 1 / 2\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.295,
+                0.503,
+                0.312
+            ],
+            "angle": 0,
+            "content": "For \\(\\mathbf{J}_n^{(1,b)}\\), we use Abel transformation and obtain"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.293,
+                0.316,
+                0.705,
+                0.376
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1, b)} = \\sum_ {k = 1} ^ {n} ((\\mathbf {K} _ {\\mathbf {x} _ {k}} \\varphi_ {\\mathbf {x} _ {k}} ^ {(1)}) (X _ {k - 1}) - (\\mathbf {K} _ {\\mathbf {x} _ {k - 1}} \\varphi_ {\\mathbf {x} _ {k - 1}} ^ {(1)}) (X _ {k - 1})) \\\\ + \\left(\\mathbf {K} _ {\\mathbf {x} _ {0}} \\varphi_ {\\mathbf {x} _ {0}} ^ {(1)}\\right) (X _ {0}) - \\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} \\varphi_ {\\mathbf {x} _ {n}} ^ {(1)}\\right) (X _ {n}). \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.379,
+                0.825,
+                0.41
+            ],
+            "angle": 0,
+            "content": "Since \\((\\mathbf{K}_{\\mathbf{x}}\\varphi_{\\mathbf{x}}^{(1)})(X)\\) is continuous in \\(\\mathbf{x}\\), for \\(\\mathbf{x}_n\\) within a compact set \\(\\Omega\\) (assumption A4), it is local Lipschitz with a constant \\(L_{\\Omega}\\) such that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.248,
+                0.413,
+                0.752,
+                0.431
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left\\| \\right.\\left(\\mathbf {K} _ {\\mathbf {x} _ {k}} \\varphi_ {\\mathbf {x} _ {k}} ^ {(1)}\\right)\\left(X _ {k - 1}\\right) - \\mathbf {K} _ {\\mathbf {x} _ {k - 1}} \\varphi_ {\\mathbf {x} _ {k - 1}} ^ {(1)}\\left. \\right)\\left(X _ {k - 1}\\right)\\left. \\right\\| \\leq L _ {\\Omega} \\| \\mathbf {x} _ {k} - \\mathbf {x} _ {k - 1} \\| \\leq 2 L _ {\\Omega} \\gamma_ {k}.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.434,
+                0.825,
+                0.481
+            ],
+            "angle": 0,
+            "content": "where the last inequality arises from (4b), i.e., \\(\\mathbf{x}_k - \\mathbf{x}_{k-1} = \\gamma_k (\\delta_{X_k} - \\mathbf{x}_{k-1})\\) and \\(\\|\\delta_{X_k} - \\mathbf{x}_{k-1}\\| \\leq 2\\) because \\(\\mathbf{x}_n \\in \\mathrm{Int}(\\Sigma)\\). Also, \\(\\|(K_{\\mathbf{x}_0} \\varphi_{\\mathbf{x}_0}^{(1)})(X_0)\\| + \\|(K_{\\mathbf{x}_n} \\varphi_{\\mathbf{x}_n}^{(1)})(X_n)\\|\\) are upper-bounded by some positive constant \\(C_\\Omega'\\). This implies that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.388,
+                0.484,
+                0.612,
+                0.525
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1, b)} \\right\\| \\leq C _ {\\Omega} ^ {\\prime} + 2 L _ {\\Omega} \\sum_ {k = 1} ^ {n} \\gamma_ {k}.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.526,
+                0.238,
+                0.539
+            ],
+            "angle": 0,
+            "content": "Note that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.282,
+                0.538,
+                0.823,
+                0.577
+            ],
+            "angle": 0,
+            "content": "\\[\n\\gamma_ {n} \\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1, b)} \\right\\| \\leq \\gamma_ {n} C _ {\\Omega} ^ {\\prime} + 2 L _ {\\Omega} \\gamma_ {n} \\sum_ {k = 1} ^ {n} \\gamma_ {k} \\leq \\gamma_ {n} C _ {\\Omega} ^ {\\prime} + \\frac {2 L _ {\\Omega}}{a} n ^ {1 - 2 a}, \\tag {47}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.579,
+                0.825,
+                0.609
+            ],
+            "angle": 0,
+            "content": "where the last inequality is from \\(\\sum_{k=1}^{n} \\gamma_k < \\frac{1}{a} n^{1-a}\\). We observe that the last term in (47) is decreasing to zero in \\(n\\) because \\(a > 1/2\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.615,
+                0.604,
+                0.635
+            ],
+            "angle": 0,
+            "content": "Note that \\(\\mathbf{J}_k^{(1)} = \\mathbf{J}_k^{(1,a)} + \\mathbf{J}_k^{(1,b)}\\), by triangular inequality we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.236,
+                0.638,
+                0.824,
+                0.781
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\gamma_ {n} \\mathbb {E} \\left[ \\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1 1)} \\right\\| \\right] \\leq \\gamma_ {n} \\mathbb {E} \\left[ \\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1 1, A)} \\right\\| \\right] + \\gamma_ {n} \\mathbb {E} \\left[ \\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1 1, B)} \\right\\| \\right] \\\\ \\leq \\gamma_ {n} C _ {1} \\mathbb {E} \\left[ \\sqrt {\\left(\\sum_ {k = 1} ^ {n} \\left\\| \\mathbf {J} _ {k} ^ {(1 1 , A)} \\right\\| ^ {2}\\right)} \\right] + \\gamma_ {n} \\mathbb {E} \\left[ \\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1 1, B)} \\right\\| \\right] \\tag {48} \\\\ = \\mathbb {E} \\left[ \\gamma_ {n} C _ {1} \\sqrt {\\left(\\sum_ {k = 1} ^ {n} \\left\\| \\mathbf {J} _ {k} ^ {(1 1 , A)} \\right\\| ^ {2}\\right)} + \\gamma_ {n} \\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1 1 , B)} \\right\\| \\right], \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.783,
+                0.825,
+                0.825
+            ],
+            "angle": 0,
+            "content": "where the second inequality comes from (45). By (46) and (47) we know that both terms in the last line of (48) are uniformly bounded by constants over time \\( n \\) that depend on the set \\( \\Omega \\). Therefore, by dominated convergence theorem, taking the limit over the last line of (48) gives"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.289,
+                0.829,
+                0.711,
+                0.929
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\lim  _ {n \\rightarrow \\infty} \\mathbb {E} \\left[ \\gamma_ {n} C _ {1} \\sqrt {\\left(\\sum_ {k = 1} ^ {n} \\left\\| \\mathbf {J} _ {k} ^ {(1 1 , A)} \\right\\| ^ {2}\\right)} + \\gamma_ {n} \\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1 1, B)} \\right\\|\\right] \\\\ = \\mathbb {E} \\left[ \\lim  _ {n \\rightarrow \\infty} \\gamma_ {n} C _ {1} \\sqrt {\\left(\\sum_ {k = 1} ^ {n} \\left\\| \\mathbf {J} _ {k} ^ {(1 1 , A)} \\right\\| ^ {2}\\right)} + \\gamma_ {n} \\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1 1 , B)} \\right\\|\\right] = 0. \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "24"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.105,
+                0.304,
+                0.118
+            ],
+            "angle": 0,
+            "content": "Therefore, we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.399,
+                0.116,
+                0.597,
+                0.158
+            ],
+            "angle": 0,
+            "content": "\\[\n\\lim  _ {n \\rightarrow \\infty} \\gamma_ {n} \\mathbb {E} \\left[\\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1)} \\right\\|\\right] = 0,\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.168,
+                0.825,
+                0.211
+            ],
+            "angle": 0,
+            "content": "In sum, in terms of \\(\\mathbb{E}[M_{n + 1}^{(\\mathbf{x})}(M_{n + 1}^{(\\mathbf{x})})^T|\\mathcal{F}_n]\\) in (41), we have \\(\\mathbf{U}_{11}\\) in (44), \\(\\lim_{n\\to \\infty}\\mathbf{D}_n^{(1)} = 0\\) a.s. and \\(\\lim_{n\\to \\infty}\\gamma_n\\mathbb{E}\\left[\\left\\| \\sum_{k = 1}^n\\mathbf{J}_k^{(1)}\\right\\|\\right] = 0.\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.216,
+                0.826,
+                0.245
+            ],
+            "angle": 0,
+            "content": "We can apply the same steps as above for the other two terms \\( i = 2,3 \\) in (42) and obtain the results."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.256,
+                0.474,
+                0.274
+            ],
+            "angle": 0,
+            "content": "Analysis of Terms \\(r_n^{(\\pmb {\\theta},1)},r_n^{(\\pmb {\\theta},2)},r_n^{(\\mathbf{x},1)},r_n^{(\\mathbf{x},2)}\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.277,
+                0.767,
+                0.297
+            ],
+            "angle": 0,
+            "content": "Lemma E.2. For \\(r_n^{(\\theta, 1)}, r_n^{(\\theta, 2)}, r_n^{(\\mathbf{x}, 1)}, r_n^{(\\mathbf{x}, 2)}\\) defined in (35), we have the following results:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.263,
+                0.3,
+                0.825,
+                0.34
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left\\| r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} \\right\\| = O (\\gamma_ {n}) = o \\left(\\sqrt {\\beta_ {n}}\\right), \\quad \\sqrt {\\gamma_ {n}} \\left\\| \\sum_ {k = 1} ^ {n} r _ {k} ^ {(\\boldsymbol {\\theta}, 2)} \\right\\| = O \\left(\\sqrt {\\gamma_ {n}}\\right) = o (1). \\tag {49a}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.263,
+                0.345,
+                0.825,
+                0.386
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left\\| r _ {n} ^ {(\\mathbf {x}, 1)} \\right\\| = O \\left(\\gamma_ {n}\\right) = o \\left(\\sqrt {\\beta_ {n}}\\right), \\quad \\sqrt {\\gamma_ {n}} \\left\\| \\sum_ {k = 1} ^ {n} r _ {k} ^ {(\\mathbf {x}, 2)} \\right\\| = O \\left(\\sqrt {\\gamma_ {n}}\\right) = o (1). \\tag {49b}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.401,
+                0.356,
+                0.419
+            ],
+            "angle": 0,
+            "content": "Proof. For \\(r_n^{(\\theta, 1)}\\), note that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.267,
+                0.422,
+                0.824,
+                0.534
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} = \\left(\\mathbf {K} _ {\\mathbf {x} _ {n + 1}} \\tilde {H} _ {\\boldsymbol {\\theta} _ {n + 1}, \\mathbf {x} _ {n + 1}}\\right) \\left(X _ {n + 1}\\right) - \\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} \\tilde {H} _ {\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}}\\right) \\left(X _ {n + 1}\\right) \\\\ = \\sum_ {j \\in \\mathcal {N}} \\left(\\mathbf {K} _ {X _ {n}, j} [ \\mathbf {x} _ {n + 1} ] \\tilde {H} _ {\\boldsymbol {\\theta} _ {n + 1}, \\mathbf {x} _ {n + 1}} (j) - \\mathbf {K} _ {X _ {n}, j} [ \\mathbf {x} _ {n} ] \\tilde {H} _ {\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}} (j)\\right) \\tag {50} \\\\ \\leq \\sum_ {j \\in \\mathcal {N}} L _ {\\mathcal {C}} \\left(\\left\\| \\boldsymbol {\\theta} _ {n + 1} - \\boldsymbol {\\theta} _ {n} \\right\\| + \\left\\| \\mathbf {x} _ {n + 1} - \\mathbf {x} _ {n} \\right\\|\\right) \\\\ \\leq N L _ {\\mathcal {C}} \\left(C _ {\\mathcal {C}} \\beta_ {n + 1} + 2 \\gamma_ {n + 1}\\right) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.539,
+                0.825,
+                0.605
+            ],
+            "angle": 0,
+            "content": "where the second last inequality is because \\(\\mathbf{K}_{i,j}[\\mathbf{x}]\\tilde{H}_{\\boldsymbol{\\theta},\\mathbf{x}}(j)\\) is continuous in \\(\\pmb {\\theta},\\mathbf{x}\\) \\(\\mathbf{K}[\\mathbf{x}]\\), which stems from continuous functions \\(\\pmb {K}[\\mathbf{x}]\\) and \\(\\tilde{H}_{\\pmb{\\theta},\\mathbf{x}}\\). The last inequality is from update rules (4) and \\((\\pmb {\\theta}_n,\\mathbf{x}_n)\\in \\Omega\\) for some compact subset \\(\\Omega\\) by assumption A4. Then, we have \\(\\| r_n^{(\\pmb {\\theta},1)}\\| = O(\\gamma_n) = o(\\sqrt{\\beta_n})\\) because of \\(a > 1 / 2\\geq b / 2\\) by assumption A2."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.61,
+                0.825,
+                0.653
+            ],
+            "angle": 0,
+            "content": "We let \\(\\nu_{n} \\triangleq (\\mathbf{K}_{\\mathbf{x}_{n}}\\tilde{H}_{\\boldsymbol{\\theta}_{n},\\mathbf{x}_{n}})(X_{n})\\) such that \\(r_n^{(\\pmb{\\theta},2)} = \\nu_n - \\nu_{n + 1}\\). Note that \\(\\sum_{k = 1}^{n}r_k^{(\\pmb{\\theta},2)} = \\nu_1 - \\nu_{n + 1}\\) and by assumption A4, \\(\\| \\nu_{n}\\|\\) is upper bounded by a constant dependent on the compact set, which leads to"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.303,
+                0.653,
+                0.693,
+                0.693
+            ],
+            "angle": 0,
+            "content": "\\[\n\\sqrt {\\gamma_ {n}} \\left\\| \\sum_ {k = 1} ^ {n} r _ {k} ^ {(\\boldsymbol {\\theta}, 2)} \\right\\| = \\sqrt {\\gamma_ {n}} \\| \\nu_ {1} - \\nu_ {n + 1} \\| = O (\\sqrt {\\gamma_ {n}}) = o (1).\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.703,
+                0.825,
+                0.729
+            ],
+            "angle": 0,
+            "content": "Similarly, we can also obtain \\(\\| r_n^{(\\mathbf{x},1)}\\| = o(\\sqrt{\\beta_n})\\) and \\(\\sqrt{\\gamma_n}\\left\\| \\sum_{k = 1}^n r_k^{(\\mathbf{x},2)}\\right\\| = O(\\sqrt{\\gamma_n}) = o(1)\\)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.741,
+                0.573,
+                0.755
+            ],
+            "angle": 0,
+            "content": "E.2.2 EFFECT OF SRRW ITERATION ON SA ITERATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.766,
+                0.825,
+                0.809
+            ],
+            "angle": 0,
+            "content": "In view of the almost sure convergence results in Lemma 3.1 and Lemma 3.2, for large enough \\(n\\) so that both iterations \\(\\theta_{n},\\mathbf{x}_{n}\\) are close to the equilibrium \\((\\theta^{*},\\mu)\\), we can apply the Taylor expansion to functions \\(\\mathbf{h}(\\theta ,\\mathbf{x})\\) and \\(\\pi [\\mathbf{x}] - \\mathbf{x}\\) in (36) at the point \\((\\theta^{*},\\mu)\\), which results in"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.18,
+                0.812,
+                0.825,
+                0.83
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {h} (\\boldsymbol {\\theta}, \\mathbf {x}) = \\mathbf {h} \\left(\\boldsymbol {\\theta} ^ {*}, \\boldsymbol {\\mu}\\right) + \\nabla_ {\\boldsymbol {\\theta}} \\mathbf {h} \\left(\\boldsymbol {\\theta} ^ {*}, \\boldsymbol {\\mu}\\right) \\left(\\boldsymbol {\\theta} - \\boldsymbol {\\theta} ^ {*}\\right) + \\nabla_ {\\mathbf {x}} \\mathbf {h} \\left(\\boldsymbol {\\theta} ^ {*}, \\boldsymbol {\\mu}\\right) (\\mathbf {x} - \\boldsymbol {\\mu}) + O \\left(\\left\\| \\boldsymbol {\\theta} - \\boldsymbol {\\theta} ^ {*} \\right\\| ^ {2} + \\| \\mathbf {x} - \\boldsymbol {\\mu} \\| ^ {2}\\right), \\tag {51a}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.263,
+                0.834,
+                0.825,
+                0.851
+            ],
+            "angle": 0,
+            "content": "\\[\n\\pi [ \\mathbf {x} ] - \\mathbf {x} = \\pi [ \\boldsymbol {\\mu} ] - \\boldsymbol {\\mu} + \\nabla_ {\\mathbf {x}} (\\pi (\\mathbf {x}) - \\mathbf {x}) | _ {\\mathbf {x} = \\boldsymbol {\\mu}} (\\mathbf {x} - \\boldsymbol {\\mu}) + O (\\| \\mathbf {x} - \\boldsymbol {\\mu} \\| ^ {2}). \\tag {51b}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.852,
+                0.45,
+                0.867
+            ],
+            "angle": 0,
+            "content": "With matrix \\(\\mathbf{J}(\\alpha)\\), we have the following:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.291,
+                0.871,
+                0.503,
+                0.887
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {J} _ {1 1} = \\nabla_ {\\boldsymbol {\\theta}} \\mathbf {h} \\left(\\boldsymbol {\\theta} ^ {*}, \\boldsymbol {\\mu}\\right) = \\nabla \\mathbf {h} \\left(\\boldsymbol {\\theta} ^ {*}\\right),\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.293,
+                0.89,
+                0.825,
+                0.907
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {J} _ {1 2} (\\alpha) = \\nabla_ {\\mathbf {x}} \\mathbf {h} \\left(\\boldsymbol {\\theta} ^ {*}, \\boldsymbol {\\mu}\\right) = - \\alpha \\mathbf {H} ^ {T} \\left(\\mathbf {P} ^ {T} + \\mathbf {I}\\right), \\tag {52}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.293,
+                0.911,
+                0.705,
+                0.929
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {J} _ {2 2} (\\alpha) = \\nabla_ {\\mathbf {x}} (\\boldsymbol {\\pi} (\\mathbf {x}) - \\mathbf {x}) | _ {\\mathbf {x} = \\boldsymbol {\\mu}} = 2 \\alpha \\boldsymbol {\\mu} \\mathbf {1} ^ {T} - \\alpha \\mathbf {P} ^ {T} - (\\alpha + 1) \\mathbf {I}.\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "25"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.105,
+                0.308,
+                0.119
+            ],
+            "angle": 0,
+            "content": "Then, (36) becomes"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.188,
+                0.124,
+                0.825,
+                0.143
+            ],
+            "angle": 0,
+            "content": "\\[\n\\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} + \\beta_ {n + 1} \\left(\\mathbf {J} _ {1 1} \\left(\\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*}\\right) + \\mathbf {J} _ {1 2} (\\alpha) \\left(\\mathbf {x} _ {n} - \\boldsymbol {\\mu}\\right) + r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} + r _ {n} ^ {(\\boldsymbol {\\theta}, 2)} + M _ {n + 1} ^ {(\\boldsymbol {\\theta})} + \\eta_ {n} ^ {(\\boldsymbol {\\theta})}\\right), \\tag {53a}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.259,
+                0.149,
+                0.824,
+                0.168
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {x} _ {n + 1} = \\mathbf {x} _ {n} + \\gamma_ {n + 1} \\left(\\mathbf {J} _ {2 2} (\\alpha) \\left(\\mathbf {x} _ {n} - \\boldsymbol {\\mu}\\right) + r _ {n} ^ {(\\mathbf {x}, 1)} + r _ {n} ^ {(\\mathbf {x}, 2)} + M _ {n + 1} ^ {(\\mathbf {x})} + \\eta_ {n} ^ {(\\mathbf {x})}\\right), \\tag {53b}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.17,
+                0.553,
+                0.19
+            ],
+            "angle": 0,
+            "content": "where \\(\\eta_n^{(\\pmb{\\theta})} = O(\\| \\mathbf{x}_n\\|^2 + \\| \\pmb{\\theta}_n\\|^2)\\) and \\(\\eta_n^{(\\mathbf{x})} = O(\\| \\mathbf{x}_n\\|^2)\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.194,
+                0.825,
+                0.227
+            ],
+            "angle": 0,
+            "content": "Then, inspired by Mokkadem & Pelletier (2006), we decompose iterates \\(\\{\\mathbf{x}_n\\}\\) and \\(\\{\\pmb {\\theta}_n\\}\\) into \\(\\mathbf{x}_n =\\) \\(L_{n}^{(\\pmb {x})} + \\Delta_{n}^{(\\pmb {x})}\\) and \\(\\pmb {\\theta}_n = L_n^{(\\pmb {\\theta})} + R_n^{(\\pmb {\\theta})} + \\Delta_n^{(\\pmb {\\theta})}\\). Rewriting (53b) gives"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.217,
+                0.232,
+                0.78,
+                0.252
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {x} _ {n} - \\boldsymbol {\\mu} = \\gamma_ {n + 1} ^ {- 1} \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (\\mathbf {x} _ {n + 1} - \\mathbf {x} _ {n}) - \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (r _ {n} ^ {(\\mathbf {x}, 1)} + r _ {n} ^ {(\\mathbf {x}, 2)} + M _ {n + 1} ^ {(\\mathbf {x})} + \\eta_ {n} ^ {(\\mathbf {x})}),\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.255,
+                0.536,
+                0.27
+            ],
+            "angle": 0,
+            "content": "and substituting the above equation back in (53a) gives"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.192,
+                0.275,
+                0.824,
+                0.42
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\boldsymbol {\\theta} _ {n + 1} - \\boldsymbol {\\theta} ^ {*} = \\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*} + \\beta_ {n + 1} \\left(\\mathbf {J} _ {1 1} \\left(\\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*}\\right) + \\gamma_ {n + 1} ^ {- 1} \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(\\mathbf {x} _ {n + 1} - \\mathbf {x} _ {n}\\right) \\right. \\\\ \\left. - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(r _ {n} ^ {(\\mathbf {x}, 1)} + r _ {n} ^ {(\\mathbf {x}, 2)} + M _ {n + 1} ^ {(\\mathbf {x})} + \\eta_ {n} ^ {(\\mathbf {x})}\\right) + r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} + r _ {n} ^ {(\\boldsymbol {\\theta}, 2)} + M _ {n + 1} ^ {(\\boldsymbol {\\theta})} + \\eta_ {n} ^ {(\\boldsymbol {\\theta})}\\right) \\\\ = (\\mathbf {I} + \\beta_ {n + 1} \\mathbf {J} _ {1 1}) \\left(\\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*}\\right) + \\left[ \\beta_ {n + 1} \\gamma_ {n + 1} ^ {- 1} \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(\\mathbf {x} _ {n + 1} - \\mathbf {x} _ {n}\\right) \\right] \\\\ + \\beta_ {n + 1} (M _ {n + 1} ^ {(\\boldsymbol {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} M _ {n + 1} ^ {(\\mathbf {x})}) \\\\ + \\beta_ {n + 1} \\left(r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} + r _ {n} ^ {(\\boldsymbol {\\theta}, 2)} + \\eta_ {n} ^ {(\\boldsymbol {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(r _ {n} ^ {(\\mathbf {x}, 1)} + r _ {n} ^ {(\\mathbf {x}, 2)} + \\eta_ {n} ^ {(\\mathbf {x})}\\right)\\right), \\tag {54} \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.425,
+                0.78,
+                0.441
+            ],
+            "angle": 0,
+            "content": "From (54) we can see the iteration \\(\\{\\theta_n\\}\\) implicitly embeds the recursions of three sequences"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.218,
+                0.451,
+                0.506,
+                0.468
+            ],
+            "angle": 0,
+            "content": "- \\(\\beta_{n+1} \\gamma_{n+1}^{-1} \\mathbf{J}_{12}(\\alpha) \\mathbf{J}_{22}(\\alpha)^{-1} (\\mathbf{x}_{n+1} - \\mathbf{x}_n)\\);"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.218,
+                0.472,
+                0.501,
+                0.492
+            ],
+            "angle": 0,
+            "content": "- \\(\\beta_{n+1}(M_{n+1}^{(\\boldsymbol{\\theta})} - \\mathbf{J}_{12}(\\alpha)\\mathbf{J}_{22}(\\alpha)^{-1}M_{n+1}^{(\\mathbf{x})})\\);"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.218,
+                0.495,
+                0.717,
+                0.514
+            ],
+            "angle": 0,
+            "content": "- \\(\\beta_{n+1}(r_n^{(\\pmb{\\theta}, 1)} + r_n^{(\\pmb{\\theta}, 2)} + \\eta_n^{(\\pmb{\\theta})} - \\mathbf{J}_{12}(\\alpha) \\mathbf{J}_{22}(\\alpha)^{-1}(r_n^{(\\mathbf{x}, 1)} + r_n^{(\\mathbf{x}, 2)} + \\eta_n^{(\\mathbf{x}))})\\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.218,
+                0.451,
+                0.717,
+                0.514
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.524,
+                0.652,
+                0.542
+            ],
+            "angle": 0,
+            "content": "Let \\( u_{n} \\triangleq \\sum_{k=1}^{n} \\beta_{k} \\) and \\( s_{n} \\triangleq \\sum_{k=1}^{n} \\gamma_{k} \\). Below we define two iterations:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.271,
+                0.547,
+                0.823,
+                0.607
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} L _ {n} ^ {(\\boldsymbol {\\theta})} = e ^ {\\beta_ {n} \\mathbf {J} _ {1 1}} L _ {n - 1} ^ {(\\boldsymbol {\\theta})} + \\beta_ {n} \\left(M _ {n} ^ {(\\boldsymbol {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} M _ {n} ^ {(\\mathbf {x})}\\right) \\\\ = \\sum_ {k = 1} ^ {n} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} \\beta_ {k} \\left(M _ {k} ^ {(\\boldsymbol {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} M _ {k} ^ {(\\mathbf {x})}\\right) \\tag {55a} \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.268,
+                0.614,
+                0.823,
+                0.674
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} R _ {n} ^ {(\\boldsymbol {\\theta})} = e ^ {\\beta_ {n} \\mathbf {J} _ {1 1}} R _ {n - 1} ^ {(\\boldsymbol {\\theta})} + \\beta_ {n} \\gamma_ {n} ^ {- 1} \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (\\mathbf {x} _ {n} - \\mathbf {x} _ {n - 1}) \\\\ = \\sum_ {k = 1} ^ {n} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} \\beta_ {k} \\gamma_ {k} ^ {- 1} \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(\\mathbf {x} _ {k} - \\mathbf {x} _ {k - 1}\\right) \\tag {55b} \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.677,
+                0.534,
+                0.696
+            ],
+            "angle": 0,
+            "content": "and a remaining term \\(\\Delta_n^{(\\pmb{\\theta})} \\triangleq \\pmb{\\theta}_n - \\pmb{\\theta}^* - L_n^{(\\pmb{\\theta})} - R_n^{(\\pmb{\\theta})}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.704,
+                0.578,
+                0.72
+            ],
+            "angle": 0,
+            "content": "Similarly, for iteration \\(\\mathbf{x}_n\\), define the sequence \\(L_n^{(\\mathbf{x})}\\) such that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.288,
+                0.726,
+                0.824,
+                0.764
+            ],
+            "angle": 0,
+            "content": "\\[\nL _ {n} ^ {(\\mathbf {x})} = e ^ {\\gamma_ {n} \\mathbf {J} _ {2 2} (\\alpha)} L _ {n - 1} ^ {(\\mathbf {x})} + \\gamma_ {n} M _ {n} ^ {(\\mathbf {x})} = \\sum_ {k = 1} ^ {n} e ^ {(s _ {n} - s _ {k}) \\mathbf {J} _ {2 2} (\\alpha)} \\gamma_ {k} M _ {k} ^ {(\\mathbf {x})}, \\tag {56}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.769,
+                0.316,
+                0.783
+            ],
+            "angle": 0,
+            "content": "and a remaining term"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.422,
+                0.782,
+                0.824,
+                0.8
+            ],
+            "angle": 0,
+            "content": "\\[\n\\Delta_ {n} ^ {(\\mathbf {x})} \\triangleq \\mathbf {x} _ {n} - \\boldsymbol {\\mu} - L _ {n} ^ {(\\boldsymbol {\\theta})} \\tag {57}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.801,
+                0.825,
+                0.83
+            ],
+            "angle": 0,
+            "content": "The decomposition of \\(\\theta_{n} - \\theta^{*}\\) and \\(\\mathbf{x}_n - \\pmb{\\mu}\\) in the above form is also standard in the single-timescale SA literature (Delyon, 2000; Fort, 2015)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.836,
+                0.519,
+                0.855
+            ],
+            "angle": 0,
+            "content": "Characterization of Sequences \\(\\{L_n^{(\\theta)}\\}\\) and \\(\\{L_n^{(\\mathbf{x})}\\}\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.862,
+                0.498,
+                0.88
+            ],
+            "angle": 0,
+            "content": "we set a Martingale \\(Z^{(n)} = \\{Z_k^{(n)}\\}_{k\\geq 1}\\) such that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.175,
+                0.885,
+                0.822,
+                0.928
+            ],
+            "angle": 0,
+            "content": "\\[\nZ _ {k} ^ {(n)} = \\left( \\begin{array}{c c} \\beta_ {n} ^ {- 1 / 2} e ^ {u _ {n} \\mathbf {J} _ {1 1}} & 0 \\\\ 0 & \\gamma_ {n} ^ {- 1 / 2} e ^ {s _ {n} \\mathbf {J} _ {2 2} (\\alpha)} \\end{array} \\right) \\times \\sum_ {j = 1} ^ {k} \\left( \\begin{array}{c} e ^ {- u _ {k} \\mathbf {J} _ {1 1}} \\beta_ {k} (M _ {k} ^ {(\\boldsymbol {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} M _ {k} ^ {(\\mathbf {x})}) \\\\ e ^ {- s _ {k} \\mathbf {J} _ {2 2} (\\alpha)} \\gamma_ {k} M _ {k} ^ {(\\mathbf {x})} \\end{array} \\right).\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.949,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "26"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.102,
+                0.576,
+                0.122
+            ],
+            "angle": 0,
+            "content": "Then, the Martingale difference array \\( Z_{k}^{(n)} - Z_{k - 1}^{(n)} \\) becomes"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.251,
+                0.125,
+                0.744,
+                0.166
+            ],
+            "angle": 0,
+            "content": "\\[\nZ _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)} = \\left( \\begin{array}{c} \\beta_ {n} ^ {- 1 / 2} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\beta_ {k} (M _ {k} ^ {(\\boldsymbol {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} M _ {k} ^ {(\\mathbf {x})}) \\\\ \\gamma_ {n} ^ {- 1 / 2} e ^ {(s _ {n} - s _ {k}) \\mathbf {J} _ {2 2} (\\alpha)} \\gamma_ {k} M _ {k} ^ {(\\mathbf {x})} \\end{array} \\right)\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.168,
+                0.202,
+                0.181
+            ],
+            "angle": 0,
+            "content": "and"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.275,
+                0.179,
+                0.72,
+                0.218
+            ],
+            "angle": 0,
+            "content": "\\[\n\\sum_ {k = 1} ^ {n} \\mathbb {E} \\left[ (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) ^ {T} | \\mathcal {F} _ {k - 1} \\right] = \\left( \\begin{array}{c c} A _ {1, n} & A _ {2, n} \\\\ A _ {2, n} ^ {T} & A _ {4, n} \\end{array} \\right),\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.22,
+                0.714,
+                0.239
+            ],
+            "angle": 0,
+            "content": "where, in view of decomposition of \\(M_{n}^{(\\pmb{\\theta})}\\) and \\(M_{n}^{(\\mathbf{x})}\\) in (41) and (42), respectively,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.19,
+                0.242,
+                0.824,
+                0.352
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} A _ {1, n} = \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\beta_ {k} ^ {2} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\left(\\mathbf {U} _ {2 2} + \\mathbf {D} _ {k} ^ {(1)} + \\mathbf {J} _ {k} ^ {(1)} - \\left(\\mathbf {U} _ {2 1} + \\mathbf {D} _ {k} ^ {(2)} + \\mathbf {J} _ {k} ^ {(2)}\\right) \\left(\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}\\right) ^ {T} \\right. \\\\ + \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(\\mathbf {U} _ {1 1} + \\mathbf {D} _ {k} ^ {(3)} + \\mathbf {J} _ {k} ^ {(3)}\\right) \\left(\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}\\right) ^ {T} \\\\ \\left. - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(\\mathbf {U} _ {2 1} + \\mathbf {D} _ {k} ^ {(2)} + \\mathbf {J} _ {k} ^ {(2)}\\right) ^ {T}\\right) e ^ {\\left(u _ {n} - u _ {k}\\right) \\left(\\mathbf {J} _ {1 1}\\right) ^ {T}}, \\tag {58a} \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.187,
+                0.357,
+                0.825,
+                0.396
+            ],
+            "angle": 0,
+            "content": "\\[\nA _ {2, n} = \\beta_ {n} ^ {- 1 / 2} \\gamma_ {n} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} \\beta_ {k} \\gamma_ {k} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} \\left(\\mathbf {U} _ {2 1} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\mathbf {U} _ {1 1}\\right) e ^ {\\left(s _ {n} - s _ {k}\\right) \\mathbf {J} _ {2 2} (\\alpha) ^ {T}}, \\tag {58b}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.261,
+                0.399,
+                0.825,
+                0.437
+            ],
+            "angle": 0,
+            "content": "\\[\nA _ {4, n} = \\gamma_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\gamma_ {k} ^ {2} e ^ {\\left(s _ {n} - s _ {k}\\right) \\mathbf {J} _ {2 2} (\\alpha)} \\left(\\mathbf {U} _ {1 1} + \\mathbf {D} _ {k} ^ {(3)} + \\mathbf {J} _ {k} ^ {(3)}\\right) e ^ {\\left(s _ {n} - s _ {k}\\right) \\mathbf {J} _ {2 2} (\\alpha) ^ {T}}. \\tag {58c}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.438,
+                0.467,
+                0.453
+            ],
+            "angle": 0,
+            "content": "We further decompose \\(A_{1,n}\\) into three parts:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.177,
+                0.457,
+                0.824,
+                0.723
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} A _ {1, n} = \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\left(\\beta_ {k} ^ {2} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} \\left(\\mathbf {U} _ {2 2} - \\mathbf {U} _ {2 1} \\left(\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}\\right) ^ {T} \\right. \\right. \\\\ \\left. \\left. - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\mathbf {U} _ {1 2} + \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\mathbf {U} _ {1 1} \\left(\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}\\right) ^ {T}\\right) e ^ {\\left(u _ {n} - u _ {k}\\right) \\left(\\mathbf {J} _ {1 1}\\right) ^ {T}}\\right) \\\\ + \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\left(\\beta_ {k} ^ {2} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} \\left(\\mathbf {D} _ {k} ^ {(1)} + \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\mathbf {D} _ {k} ^ {(3)} \\left(\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}\\right) ^ {T} \\right. \\right. \\\\ \\left. - \\mathbf {D} _ {k} ^ {(2)} \\left(\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}\\right) ^ {T} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(\\mathbf {D} _ {k} ^ {(2)}\\right) ^ {T}\\right) e ^ {\\left(u _ {n} - u _ {k}\\right) \\left(\\mathbf {J} _ {1 1}\\right) ^ {T}} \\\\ + \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\left(\\beta_ {k} ^ {2} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} (\\mathbf {J} _ {k} ^ {(1)} + \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\mathbf {J} _ {k} ^ {(3)} (\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}) ^ {T} \\right. \\\\ \\left. - \\mathbf {J} _ {k} ^ {(2)} \\left(\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}\\right) ^ {T} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(\\mathbf {J} _ {k} ^ {(2)}\\right) ^ {T}\\right) e ^ {\\left(u _ {n} - u _ {k}\\right) \\left(\\mathbf {J} _ {1 1}\\right) ^ {T}} \\\\ \\triangleq A _ {1, n} ^ {(a)} + A _ {1, n} ^ {(b)} + A _ {1, n} ^ {(c)}. \\tag {59} \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.728,
+                0.825,
+                0.759
+            ],
+            "angle": 0,
+            "content": "Here, we define \\(\\mathbf{U}_{\\pmb{\\theta}}(\\alpha) \\triangleq \\mathbf{U}_{22} - \\mathbf{U}_{21}(\\mathbf{J}_{12}(\\alpha)\\mathbf{J}_{22}(\\alpha)^{-1})^T - \\mathbf{J}_{12}(\\alpha)\\mathbf{J}_{22}(\\alpha)^{-1}\\mathbf{U}_{12} + \\mathbf{J}_{12}(\\alpha)\\mathbf{J}_{22}(\\alpha)^{-1}\\mathbf{U}_{11} + \\mathbf{J}_{12}(\\alpha)\\mathbf{J}_{22}(\\alpha)^{-1}\\mathbf{J}_{12}\\). By (52) and (43a) in Lemma E.1, we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.317,
+                0.763,
+                0.825,
+                0.803
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {U} _ {\\boldsymbol {\\theta}} (\\alpha) = \\sum_ {i = 1} ^ {N - 1} \\frac {1}{(\\alpha (1 + \\lambda_ {i}) + 1) ^ {2}} \\cdot \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H}. \\tag {60}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.806,
+                0.416,
+                0.82
+            ],
+            "angle": 0,
+            "content": "Then, we have the following lemma."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.823,
+                0.566,
+                0.843
+            ],
+            "angle": 0,
+            "content": "Lemma E.3. For \\(A_{1,n}^{(a)}, A_{1,n}^{(b)}, A_{1,n}^{(c)}\\) defined in (59), we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.288,
+                0.846,
+                0.825,
+                0.871
+            ],
+            "angle": 0,
+            "content": "\\[\n\\lim  _ {n \\rightarrow \\infty} A _ {1, n} ^ {(a)} = \\mathbf {V} _ {\\boldsymbol {\\theta}} (\\alpha), \\quad \\lim  _ {n \\rightarrow \\infty} \\| A _ {1, n} ^ {(b)} \\| = 0, \\quad \\lim  _ {n \\rightarrow \\infty} \\| A _ {1, n} ^ {(c)} \\| = 0, \\tag {61}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.874,
+                0.532,
+                0.89
+            ],
+            "angle": 0,
+            "content": "where \\(\\mathbf{V}_{\\theta}(\\alpha)\\) is the solution to the Lyapunov equation"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.256,
+                0.893,
+                0.742,
+                0.93
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left(\\mathbf {J} _ {1 1} + \\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\boldsymbol {I}\\right) \\mathbf {V} _ {\\boldsymbol {\\theta}} (\\alpha) + \\mathbf {V} _ {\\boldsymbol {\\theta}} (\\alpha) \\left(\\mathbf {J} _ {1 1} + \\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\boldsymbol {I}\\right) ^ {T} + \\mathbf {U} _ {\\boldsymbol {\\theta}} (\\alpha) = 0.\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "27"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.104,
+                0.624,
+                0.12
+            ],
+            "angle": 0,
+            "content": "Proof. First, from Lemma G.4, we have for some \\(c, T > 0\\) such that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.176,
+                0.126,
+                0.825,
+                0.201
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\| A _ {1, n} ^ {(b)} \\| \\leq \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\left\\| \\mathbf {D} _ {k} ^ {(1)} + \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\mathbf {D} _ {k} ^ {(3)} (\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}) ^ {T} - \\mathbf {D} _ {k} ^ {(2)} (\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}) ^ {T} \\right. \\\\ \\left. - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(\\mathbf {D} _ {k} ^ {(2)}\\right) ^ {T} \\right\\| \\cdot \\beta_ {k} ^ {2} c ^ {2} e ^ {- 2 T \\left(u _ {n} - u _ {k}\\right)}. \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.206,
+                0.65,
+                0.225
+            ],
+            "angle": 0,
+            "content": "Applying Lemma G.6, together with \\(\\mathbf{D}_n^{(i)}\\to 0\\) a.s. in Lemma E.1, gives"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.269,
+                0.23,
+                0.731,
+                0.324
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\lim \\sup \\| A _ {1, n} ^ {(b)} \\| \\\\ \\leq \\frac {1}{C (b , p)} \\lim  _ {n} \\sup  _ {n} \\| (\\mathbf {D} _ {n} ^ {(1)} + \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\mathbf {D} _ {n} ^ {(4)} (\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}) ^ {T} \\\\ \\left. - \\mathbf {D} _ {n} ^ {(2)} \\left(\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}\\right) ^ {T} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\mathbf {D} _ {n} ^ {(3)}\\right) \\| \\\\ = 0. \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.335,
+                0.373,
+                0.354
+            ],
+            "angle": 0,
+            "content": "We now consider \\(\\| A_{1,n}^{(c)}\\|\\). Set"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.274,
+                0.362,
+                0.724,
+                0.423
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\Xi_ {n} \\triangleq \\sum_ {k = 1} ^ {n} \\left(\\mathbf {J} _ {n} ^ {(1)} + \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\mathbf {J} _ {k} ^ {(3)} (\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}) ^ {T} \\right. \\\\ - \\mathbf {J} _ {k} ^ {(2)} (\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}) ^ {T} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (\\mathbf {J} _ {k} ^ {(2)}) ^ {T}), \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.429,
+                0.327,
+                0.448
+            ],
+            "angle": 0,
+            "content": "we can rewrite \\(A_{1,n}^{(c)}\\) as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.3,
+                0.455,
+                0.699,
+                0.494
+            ],
+            "angle": 0,
+            "content": "\\[\nA _ {1, n} ^ {(c)} = \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\beta_ {k} ^ {2} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} (\\Xi_ {k} - \\Xi_ {k - 1}) e ^ {(u _ {n} - u _ {k}) (\\mathbf {J} _ {1 1}) ^ {T}}.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.5,
+                0.419,
+                0.514
+            ],
+            "angle": 0,
+            "content": "By the Abel transformation, we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.271,
+                0.522,
+                0.824,
+                0.589
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} A _ {1, n} ^ {(c)} = \\beta_ {n} \\Xi_ {n} + \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n - 1} \\left[ \\beta_ {k} ^ {2} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} \\Xi_ {k} e ^ {\\left(u _ {n} - u _ {k}\\right) \\left(\\mathbf {J} _ {1 1}\\right) ^ {T}} \\right. \\tag {62} \\\\ \\left. - \\beta_ {k + 1} ^ {2} e ^ {(u _ {n} - u _ {k + 1}) \\mathbf {J} _ {1 1}} \\Xi_ {k} e ^ {(u _ {n} - u _ {k + 1}) (\\mathbf {J} _ {1 1}) ^ {T}} \\right]. \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.595,
+                0.698,
+                0.611
+            ],
+            "angle": 0,
+            "content": "We know from Lemma E.1 that \\(\\beta_{n}\\Xi_{n}\\to 0\\) a.s. because \\(\\Xi_n = o(\\gamma_n^{-1})\\). Besides,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.238,
+                0.617,
+                0.761,
+                0.676
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\left\\| \\beta_ {k} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} - \\beta_ {k + 1} e ^ {\\left(u _ {n} - u _ {k + 1}\\right) \\mathbf {J} _ {1 1}} \\right\\| \\\\ = \\| \\left(\\beta_ {k} - \\beta_ {k + 1}\\right) e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} + \\beta_ {k + 1} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} \\left(\\mathbf {I} - e ^ {- \\beta_ {k + 1} \\mathbf {J} _ {1 1}}\\right) \\| \\\\ \\leq C _ {1} \\beta_ {k} ^ {2} e ^ {- (u _ {n} - u _ {k}) T} \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.683,
+                0.825,
+                0.699
+            ],
+            "angle": 0,
+            "content": "for some constant \\(C_1 > 0\\) because \\(\\beta_{n} - \\beta_{n + 1}\\leq C_{2}\\beta_{n}^{2}\\) and \\(\\| \\mathbf{I} - e^{-\\beta_{k + 1}\\mathbf{J}_{11}}\\| \\leq C_3\\beta_{k + 1}\\). Moreover,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.274,
+                0.705,
+                0.725,
+                0.763
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\left\\| \\beta_ {k} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} \\right\\| + \\left\\| \\beta_ {k + 1} e ^ {\\left(u _ {n} - u _ {k + 1}\\right) \\mathbf {J} _ {1 1}} \\right\\| \\\\ \\leq \\beta_ {k} \\| e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\| + \\beta_ {k} \\| e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\| \\cdot \\| e ^ {- \\beta_ {k + 1} \\mathbf {J} _ {1 1}} \\| \\\\ \\leq C _ {4} \\beta_ {k} e ^ {- (u _ {n} - u _ {k}) T}. \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.77,
+                0.387,
+                0.785
+            ],
+            "angle": 0,
+            "content": "Using Lemma G.7 on (62) gives"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.299,
+                0.792,
+                0.7,
+                0.832
+            ],
+            "angle": 0,
+            "content": "\\[\n\\| A _ {1, n} ^ {(c)} \\| \\leq C _ {1} C _ {4} \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n - 1} \\beta_ {k} ^ {2} e ^ {- 2 (u _ {n} - u _ {k}) T} \\| \\beta_ {k} \\Xi_ {k} \\| + \\| \\beta_ {n} \\Xi_ {n} \\|.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.837,
+                0.398,
+                0.852
+            ],
+            "angle": 0,
+            "content": "Applying Lemma G.6 again gives"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.358,
+                0.858,
+                0.642,
+                0.884
+            ],
+            "angle": 0,
+            "content": "\\[\n\\lim  _ {n} \\sup  _ {n} \\| A _ {1, n} ^ {(c)} \\| \\leq C _ {5} \\lim  _ {n} \\sup  _ {n} \\| \\beta_ {n} \\Xi_ {n} \\| = 0\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.888,
+                0.351,
+                0.903
+            ],
+            "angle": 0,
+            "content": "for some constant \\(C_5 > 0\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.91,
+                0.473,
+                0.925
+            ],
+            "angle": 0,
+            "content": "Finally, we provide an existing lemma below."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "28"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.827,
+                0.148
+            ],
+            "angle": 0,
+            "content": "Lemma E.4 (Mokkadem & Pelletier (2005) Lemma 4). For a sequence with decreasing step size \\(\\beta_{n} = (n + 1)^{-b}\\) for \\(b\\in (1 / 2,1]\\), \\(u_{n} = \\sum_{k = 1}^{n}\\beta_{k}\\), a positive semi-definite matrix \\(\\Gamma\\) and a Hurwitz matrix \\(\\mathbf{Q}\\), which is given by"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.376,
+                0.154,
+                0.622,
+                0.193
+            ],
+            "angle": 0,
+            "content": "\\[\n\\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\beta_ {n} ^ {2} e ^ {(u _ {n} - u _ {k}) \\mathbf {Q}} \\Gamma e ^ {(u _ {n} - u _ {k}) \\mathbf {Q} ^ {T}},\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.2,
+                0.232,
+                0.213
+            ],
+            "angle": 0,
+            "content": "we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.342,
+                0.211,
+                0.656,
+                0.25
+            ],
+            "angle": 0,
+            "content": "\\[\n\\lim  _ {n \\rightarrow \\infty} \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\beta_ {n} ^ {2} e ^ {(u _ {n} - u _ {k}) \\mathbf {Q}} \\Gamma e ^ {(u _ {n} - u _ {k}) \\mathbf {Q} ^ {T}} = \\mathbf {V}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.253,
+                0.502,
+                0.268
+            ],
+            "angle": 0,
+            "content": "where \\(\\mathbf{V}\\) is the solution of the Lyapunov equation"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.318,
+                0.273,
+                0.681,
+                0.309
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left(\\mathbf {Q} + \\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I}\\right) \\mathbf {V} + \\mathbf {V} \\left(\\mathbf {Q} ^ {T} + \\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I}\\right) + \\Gamma = 0.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.323,
+                0.625,
+                0.343
+            ],
+            "angle": 0,
+            "content": "Then, \\(\\lim_{n\\to \\infty}A_{1,n}^{(a)} = \\mathbf{V}_{\\pmb{\\theta}}(\\alpha)\\) is a direct application of Lemma E.4."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.326,
+                0.826,
+                0.338
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.355,
+                0.542,
+                0.371
+            ],
+            "angle": 0,
+            "content": "We can follow the similar steps in Lemma E.3 to obtain"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.425,
+                0.377,
+                0.572,
+                0.399
+            ],
+            "angle": 0,
+            "content": "\\[\n\\lim  _ {n \\to \\infty} A _ {4, n} = \\mathbf {V} _ {\\mathbf {x}} (\\alpha),\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.405,
+                0.414,
+                0.421
+            ],
+            "angle": 0,
+            "content": "where \\(\\mathbf{V}_{\\mathbf{x}}(\\alpha)\\) is in the form of (32)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.427,
+                0.528,
+                0.442
+            ],
+            "angle": 0,
+            "content": "The last step is to show \\(\\lim_{n\\to \\infty}A_{2,n} = 0\\) . Note that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.261,
+                0.449,
+                0.732,
+                0.577
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\| A _ {2, n} \\| = O \\left(\\beta_ {n} ^ {- 1 / 2} \\gamma_ {n} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} \\beta_ {k} \\gamma_ {k} \\| e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\| \\| e ^ {(s _ {n} - s _ {k}) \\mathbf {J} _ {2 2} (\\alpha) ^ {T}} \\|\\right) \\\\ = O \\left(\\beta_ {n} ^ {- 1 / 2} \\gamma_ {n} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} \\beta_ {k} \\gamma_ {k} e ^ {- (u _ {n} - u _ {k}) T} e ^ {- (s _ {n} - s _ {k}) T ^ {\\prime}}\\right) \\\\ = O \\left(\\beta_ {n} ^ {- 1 / 2} \\gamma_ {n} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} \\beta_ {k} \\gamma_ {k} e ^ {- (s _ {n} - s _ {k}) T ^ {\\prime}}\\right), \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.584,
+                0.795,
+                0.599
+            ],
+            "angle": 0,
+            "content": "where the second equality is from Lemma G.4. Then, we use Lemma G.6 with \\( p = 0 \\) to obtain"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.392,
+                0.605,
+                0.825,
+                0.645
+            ],
+            "angle": 0,
+            "content": "\\[\n\\sum_ {k = 1} ^ {n} \\beta_ {k} \\gamma_ {k} e ^ {- \\left(s _ {n} - s _ {k}\\right) T ^ {\\prime}} = O \\left(\\beta_ {n}\\right) \\tag {63}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.651,
+                0.446,
+                0.667
+            ],
+            "angle": 0,
+            "content": "Additionally, since \\(\\beta_{n} = o(\\gamma_{n})\\), we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.27,
+                0.674,
+                0.726,
+                0.713
+            ],
+            "angle": 0,
+            "content": "\\[\n\\beta_ {n} ^ {- 1 / 2} \\gamma_ {n} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} \\beta_ {k} \\gamma_ {k} ^ {- 1 / 2} \\gamma_ {k} ^ {3 / 2} e ^ {- (s _ {n} - s _ {k}) T ^ {\\prime}} = O \\left(\\beta_ {n} ^ {1 / 2} \\gamma_ {n} ^ {- 1 / 2}\\right) = o (1).\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.72,
+                0.584,
+                0.735
+            ],
+            "angle": 0,
+            "content": "Then, it follows that \\(\\lim_{n\\to \\infty}A_{2,n} = 0\\). Therefore, we obtain"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.243,
+                0.743,
+                0.753,
+                0.782
+            ],
+            "angle": 0,
+            "content": "\\[\n\\lim  _ {n \\to \\infty} \\sum_ {k = 1} ^ {n} \\mathbb {E} \\left[ (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) ^ {T} | \\mathcal {F} _ {k - 1} \\right] = \\left( \\begin{array}{c c} \\mathbf {V} _ {\\boldsymbol {\\theta}} (\\alpha) & 0 \\\\ 0 & \\mathbf {V} _ {\\mathbf {x}} (\\alpha) \\end{array} \\right).\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.796,
+                0.724,
+                0.811
+            ],
+            "angle": 0,
+            "content": "Now, we turn to verifying the conditions in Theorem G.3. For some \\(\\tau > 0\\), we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.185,
+                0.818,
+                0.825,
+                0.93
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\sum_ {k = 1} ^ {n} \\mathbb {E} \\left[ \\| Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)} \\| ^ {2 + \\tau} | \\mathcal {F} _ {k - 1} \\right] \\\\ = O \\left(\\beta_ {n} ^ {- (1 + \\frac {\\tau}{2})} \\sum_ {k = 1} ^ {n} \\beta_ {k} ^ {2 + \\frac {\\tau}{2}} \\beta_ {k} ^ {\\frac {\\tau}{2}} e ^ {- (2 + \\tau) \\left(u _ {n} - u _ {k}\\right) T} + \\gamma_ {n} ^ {- (1 + \\frac {\\tau}{2})} \\sum_ {k = 1} ^ {n} \\gamma_ {k} ^ {2 + \\frac {\\tau}{2}} \\gamma_ {k} ^ {\\frac {\\tau}{2}} e ^ {- (2 + \\tau) \\left(s _ {n} - s _ {k}\\right) T ^ {\\prime}}\\right) \\tag {64} \\\\ = O \\left(\\beta_ {n} ^ {\\frac {\\tau}{2}} + \\gamma_ {n} ^ {\\frac {\\tau}{2}}\\right) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.51,
+                0.961
+            ],
+            "angle": 0,
+            "content": "29"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.763,
+                0.12
+            ],
+            "angle": 0,
+            "content": "where the last equality comes from Lemma G.6. Since (64) also holds for \\(\\tau = 0\\), we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.342,
+                0.127,
+                0.655,
+                0.165
+            ],
+            "angle": 0,
+            "content": "\\[\n\\sum_ {k = 1} ^ {n} \\mathbb {E} \\left[ \\| Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)} \\| ^ {2} | \\mathcal {F} _ {k - 1} \\right] = O (1) <   \\infty .\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.172,
+                0.746,
+                0.187
+            ],
+            "angle": 0,
+            "content": "Therefore, all the conditions in Theorem G.3 are satisfied and its application then gives"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.294,
+                0.194,
+                0.825,
+                0.236
+            ],
+            "angle": 0,
+            "content": "\\[\nZ ^ {(n)} = \\left(\\begin{array}{c}\\sqrt {\\beta_ {n} ^ {- 1}} L _ {n} ^ {(\\boldsymbol {\\theta})}\\\\\\sqrt {\\gamma_ {n} ^ {- 1}} L _ {n} ^ {(\\mathbf {x})}\\end{array}\\right) \\xrightarrow [ d i s t . ]{n \\rightarrow \\infty} N \\left(0, \\left(\\begin{array}{c c}\\mathbf {V} _ {\\boldsymbol {\\theta}} (\\alpha)&0\\\\0&\\mathbf {V} _ {\\mathbf {x}} (\\alpha)\\end{array}\\right)\\right). \\tag {65}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.25,
+                0.825,
+                0.269
+            ],
+            "angle": 0,
+            "content": "Furthermore, we have the following lemma about the strong convergence rate of \\(\\{L_n^{(\\pmb{\\theta})}\\}\\) and \\(\\{L_n^{(\\mathbf{x})}\\}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.271,
+                0.262,
+                0.285
+            ],
+            "angle": 0,
+            "content": "Lemma E.5."
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.378,
+                0.283,
+                0.825,
+                0.308
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left\\| L _ {n} ^ {(\\boldsymbol {\\theta})} \\right\\| = O \\left(\\sqrt {\\beta_ {n} \\log \\left(u _ {n}\\right)}\\right) \\quad a. s. \\tag {66a}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.38,
+                0.311,
+                0.825,
+                0.337
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left\\| L _ {n} ^ {(\\mathbf {x})} \\right\\| = O \\left(\\sqrt {\\gamma_ {n} \\log \\left(s _ {n}\\right)}\\right) \\quad a. s. \\tag {66b}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.35,
+                0.825,
+                0.393
+            ],
+            "angle": 0,
+            "content": "Proof. This proof follows Pelletier (1998, Lemma 1). We only need the special case of Pelletier (1998, Lemma 1) that fits our scenario; e.g., we let the two types of step sizes therein be the same. Specifically, we attach the following lemma."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.399,
+                0.591,
+                0.415
+            ],
+            "angle": 0,
+            "content": "Lemma E.6 (Pelletier (1998) Lemma 1). Consider a sequence"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.381,
+                0.421,
+                0.617,
+                0.461
+            ],
+            "angle": 0,
+            "content": "\\[\nL _ {n + 1} = e ^ {u _ {n} \\mathbf {H}} \\sum_ {k = 1} ^ {n} e ^ {- u _ {k} \\mathbf {H}} \\beta_ {k} M _ {k + 1},\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.169,
+                0.469,
+                0.825,
+                0.514
+            ],
+            "angle": 0,
+            "content": "where \\(\\beta_{n} = n^{-b}\\), \\(1/2 < b \\leq 1\\), and \\(\\{M_n\\}\\) is a Martingale difference sequence adapted to the filtration \\(\\mathcal{F}\\) such that, almost surely, \\(\\lim \\sup_{n} \\mathbb{E}[\\| M_{n+1} \\|^2 | \\mathcal{F}_n] \\leq M^2\\) and there exists \\(\\tau \\in (0,2)\\), \\(b(2 + \\tau) > 2\\), such that \\(\\sup_{n} \\mathbb{E}[\\| M_{n+1} \\|^2 + \\tau | \\mathcal{F}_n] < \\infty\\). Then, almost surely,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.396,
+                0.519,
+                0.826,
+                0.555
+            ],
+            "angle": 0,
+            "content": "\\[\n\\lim  _ {n} \\sup  \\frac {\\left\\| L _ {n} \\right\\|}{\\sqrt {\\beta_ {n} \\log \\left(u _ {n}\\right)}} \\leq C _ {M}, \\tag {67}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.56,
+                0.45,
+                0.575
+            ],
+            "angle": 0,
+            "content": "where \\(C_M\\) is a constant dependent on \\(M\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.589,
+                0.827,
+                0.709
+            ],
+            "angle": 0,
+            "content": "By assumption A4, the iterates \\((\\pmb{\\theta}_n,\\mathbf{x}_n)\\) are bounded within a compact subset \\(\\Omega\\). Recall the form of \\(M_{n + 1}^{(\\pmb{\\theta})},M_{n + 1}^{(\\mathbf{x})}\\) defined in (35), it comprises the functions \\(\\tilde{H}_{\\pmb{\\theta}_n,\\mathbf{x}_n}(i)\\) and \\((\\mathbf{K}_{\\mathbf{x}_n}\\tilde{H}_{\\pmb{\\theta}_n,\\mathbf{x}_n})(i)\\), which in turn include the function \\(H(\\pmb {\\theta},i)\\). We know that \\(H(\\pmb {\\theta},i)\\) is bounded for \\(\\pmb{\\theta}\\) in some compact set \\(\\mathcal{C}\\). Thus, for any \\((\\pmb {\\theta}_n,\\mathbf{x}_n)\\in \\Omega\\) for some compact set \\(\\Omega\\), \\(M_{n + 1}^{(\\pmb{\\theta})},M_{n + 1}^{(\\mathbf{x})}\\) are bounded and we denote by \\(c_{\\pmb{\\theta}}\\) and \\(c_{\\mathbf{x}}\\) as their upper bounds, i.e., \\(\\mathbb{E}[\\| M_{n + 1}^{(\\pmb{\\theta})}\\| ^2 |\\mathcal{F}_n]\\leq c_\\Omega^{(\\pmb{\\theta})}\\) and \\(\\mathbb{E}[\\| M_{n + 1}^{(\\mathbf{x})}\\| ^2 |\\mathcal{F}_n]\\leq c_\\Omega^{(\\mathbf{x})}\\). We only need to replace the upper bound \\(c\\) in Lemma E.6 by \\(c_{\\Omega}^{(\\pmb{\\theta})}\\) for the sequence \\(\\{L_n^{(\\pmb{\\theta})}\\}\\) (resp. \\(c_{\\Omega}^{(\\mathbf{x})}\\) for the sequence \\(\\{L_n^{(\\mathbf{x})}\\}\\)), i.e.,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.393,
+                0.716,
+                0.826,
+                0.756
+            ],
+            "angle": 0,
+            "content": "\\[\n\\lim  _ {n} \\sup  \\frac {\\left\\| L _ {n} ^ {(\\boldsymbol {\\theta})} \\right\\|}{\\sqrt {\\beta_ {n} \\log (u _ {n})}} \\leq C _ {\\Omega} ^ {(\\boldsymbol {\\theta})}, \\tag {68a}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.395,
+                0.762,
+                0.825,
+                0.801
+            ],
+            "angle": 0,
+            "content": "\\[\n\\lim  _ {n} \\sup  \\frac {\\| L _ {n} ^ {(\\mathbf {x})} \\|}{\\sqrt {\\gamma_ {n} \\log (s _ {n})}} \\leq C _ {\\Omega} ^ {(\\mathbf {x})}, \\tag {68b}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.804,
+                0.827,
+                0.838
+            ],
+            "angle": 0,
+            "content": "such that \\(\\| L_n^{(\\pmb{\\theta})}\\| = O\\big(\\sqrt{\\beta_n\\log(u_n)}\\big)\\) a.s. and \\(\\| L_{n}^{(\\mathbf{x})}\\| = O\\big(\\sqrt{\\gamma_{n}\\log(s_{n})}\\big)\\) a.s. which completes the proof."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.853,
+                0.826,
+                0.902
+            ],
+            "angle": 0,
+            "content": "Note that we have \\(\\mathbf{x}_n - \\boldsymbol{\\mu}\\) and \\(L_{n}^{(\\mathbf{x})}\\) weakly converge to the same Gaussian distribution from Remark E.1 and (65). Then, \\(\\gamma_n^{-1 / 2}\\Delta_n^{(\\mathbf{x})}\\) weakly converges to zero, implying that \\(\\gamma_n^{-1 / 2}\\Delta_n^{(\\mathbf{x})}\\) converges to zero with probability 1. Therefore, together with \\(\\{\\gamma_n\\}\\) being strictly positive, we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.423,
+                0.908,
+                0.825,
+                0.927
+            ],
+            "angle": 0,
+            "content": "\\[\n\\Delta_ {n} ^ {(\\mathbf {x})} = o \\left(\\sqrt {\\gamma_ {n}}\\right) \\quad a. s. \\tag {69}\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.51,
+                0.96
+            ],
+            "angle": 0,
+            "content": "30"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.102,
+                0.523,
+                0.121
+            ],
+            "angle": 0,
+            "content": "Characterization of Sequences \\(\\{R_n^{(\\theta)}\\}\\) and \\(\\{\\Delta_n^{(\\theta)}\\}\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.129,
+                0.825,
+                0.159
+            ],
+            "angle": 0,
+            "content": "We first consider the sequence \\(\\{R_n^{(\\theta)}\\}\\). We assume a positive real-valued bounded sequence \\(\\{w_{n}\\}\\) under the same conditions as in Mokkadem & Pelletier (2006, Definition 1), i.e.,"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.16,
+                0.79,
+                0.18
+            ],
+            "angle": 0,
+            "content": "Definition E.1. In the case \\( b < 1 \\), \\( \\frac{w_n}{w_{n+1}} = 1 + o(\\beta_n) \\), which also implies \\( \\frac{w_n}{w_{n+1}} = 1 + o(\\gamma_n) \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.185,
+                0.826,
+                0.215
+            ],
+            "angle": 0,
+            "content": "In the case \\( b = 1 \\), there exist \\( \\epsilon \\geq 0 \\) and a nondecreasing slowly varying function \\( l(n) \\) such that \\( w_{n} = n^{-\\epsilon}l(n) \\). When \\( \\epsilon = 0 \\), we require function \\( l(n) \\) to be bounded."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.224,
+                0.825,
+                0.254
+            ],
+            "angle": 0,
+            "content": "Since \\( \\| \\mathbf{x}_n - \\boldsymbol{\\mu} \\| = o(1) \\) by a.s. convergence result, we can assume that there exists \\( \\{w_n\\} \\) such that \\( \\| \\mathbf{x}_n - \\boldsymbol{\\mu} \\| = O(w_n) \\). Then, from (55b), we can use the Abel transformation and obtain"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.194,
+                0.257,
+                0.806,
+                0.319
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} R _ {n} ^ {(\\boldsymbol {\\theta})} = \\beta_ {n} \\gamma_ {n} ^ {- 1} \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (\\mathbf {x} _ {n} - \\boldsymbol {\\mu}) - e ^ {(u _ {n} - u _ {1}) \\mathbf {J} _ {1 1}} \\beta_ {1} \\gamma_ {1} ^ {- 1} \\mathbf {U} _ {1 1} \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (\\mathbf {x} _ {1} - \\boldsymbol {\\mu}) \\\\ + e ^ {u _ {n} \\mathbf {J} _ {1 1}} \\sum_ {k = 1} ^ {n - 1} \\left(e ^ {- u _ {k} \\mathbf {J} _ {1 1}} \\beta_ {k} \\gamma_ {k} ^ {- 1} - e ^ {- u _ {k + 1} \\mathbf {J} _ {1 1}} \\beta_ {k + 1} \\gamma_ {k + 1} ^ {- 1}\\right) \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (\\mathbf {x} _ {k + 1} - \\boldsymbol {\\mu}), \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.321,
+                0.51,
+                0.335
+            ],
+            "angle": 0,
+            "content": "where the last term on the RHS can be rewritten as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.177,
+                0.337,
+                0.82,
+                0.378
+            ],
+            "angle": 0,
+            "content": "\\[\nW _ {n} = \\sum_ {k = 1} ^ {n - 1} e ^ {(u _ {n} - u _ {k + 1}) \\mathbf {J} _ {1 1}} \\beta_ {k + 1} \\gamma_ {k + 1} ^ {- 1} \\left(e ^ {\\beta_ {k + 1} \\mathbf {J} _ {1 1}} \\beta_ {k} \\beta_ {k + 1} ^ {- 1} \\gamma_ {k} ^ {- 1} \\gamma_ {k + 1} - \\mathbf {I}\\right) \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (\\mathbf {x} _ {k + 1} - \\boldsymbol {\\mu}).\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.381,
+                0.825,
+                0.41
+            ],
+            "angle": 0,
+            "content": "Using Lemma G.6 on \\(W_{n}\\) gives \\(\\| W_n\\| = O(\\gamma_n^{-1}\\| e^{\\beta_n\\mathbf{J}_{11}} - \\mathbf{I}\\| \\| \\mathbf{x}_n - \\pmb {\\mu}\\|) = O(\\gamma_n^{-1}\\beta_n\\omega_n)\\). Then, it follows that for some \\(T > 0\\)"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.288,
+                0.412,
+                0.825,
+                0.431
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left\\| R _ {n} ^ {(\\boldsymbol {\\theta})} \\right\\| = O \\left(\\beta_ {n} \\gamma_ {n} ^ {- 1} \\omega_ {n} + \\left\\| e ^ {u _ {n} \\mathbf {J} _ {1 1}} \\right\\|\\right) = O \\left(\\beta_ {n} \\gamma_ {n} ^ {- 1} \\omega_ {n} + e ^ {- u _ {n} T}\\right) \\tag {70}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.433,
+                0.557,
+                0.449
+            ],
+            "angle": 0,
+            "content": "with the application of Lemma G.4 to the second equality."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.454,
+                0.826,
+                0.489
+            ],
+            "angle": 0,
+            "content": "Then, we shift our focus on \\(\\{\\Delta_n^{(\\theta)}\\}\\). Specifically, we take (54), (55a), and (56) back to \\(\\Delta_{n}^{(\\theta)} = \\pmb{\\theta}_{n} - \\pmb{\\theta}^{*} - L_{n}^{(\\theta)} - R_{n}^{(\\theta)}\\), and obtain"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.209,
+                0.493,
+                0.824,
+                0.656
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\Delta_ {n + 1} ^ {(\\boldsymbol {\\theta})} = (\\mathbf {I} + \\beta_ {n + 1} \\mathbf {J} _ {1 1}) (\\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*}) \\\\ + \\beta_ {n + 1} \\left(r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} + r _ {n} ^ {(\\boldsymbol {\\theta}, 2)} + \\eta_ {n} ^ {(\\boldsymbol {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(r _ {n} ^ {(\\mathbf {x}, 1)} + r _ {n} ^ {(\\mathbf {x}, 2)} + \\eta_ {n} ^ {(\\mathbf {x})}\\right)\\right) \\\\ - e ^ {\\beta_ {n + 1} \\mathbf {J} _ {1 1}} L _ {n} ^ {(\\boldsymbol {\\theta})} - e ^ {\\beta_ {n + 1} \\mathbf {J} _ {1 1}} R _ {n} ^ {(\\boldsymbol {\\theta})} \\\\ = \\left(\\mathbf {I} + \\beta_ {n + 1} \\mathbf {J} _ {1 1}\\right) \\left(\\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*}\\right) + \\beta_ {n + 1} \\left(r _ {x} ^ {(\\boldsymbol {\\theta}, 1)} + r _ {x} ^ {(\\boldsymbol {\\theta}, 2)} + \\eta_ {x} ^ {(\\boldsymbol {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(r _ {x} ^ {(\\mathbf {x}, 1)} + r _ {x} ^ {(\\mathbf {x}, 2)} + \\eta_ {x} ^ {(\\mathbf {x})}\\right)\\right) \\tag {71} \\\\ - \\left(\\mathbf {I} + \\beta_ {n + 1} \\mathbf {J} _ {1 1} + O \\left(\\beta_ {n + 1} ^ {2}\\right)\\right) L _ {n} ^ {(\\boldsymbol {\\theta})} - \\left(\\mathbf {I} + \\beta_ {n + 1} \\mathbf {J} _ {1 1} + O \\left(\\beta_ {n + 1} ^ {2}\\right)\\right) R _ {n} ^ {(\\boldsymbol {\\theta})} \\\\ = (\\mathbf {I} + \\beta_ {n + 1} \\mathbf {J} _ {1 1}) \\Delta_ {n} ^ {(\\pmb {\\theta})} + O (\\beta_ {n + 1} ^ {2}) (L _ {n} ^ {(\\pmb {\\theta})} + R _ {n} ^ {(\\pmb {\\theta}))}) \\\\ + \\beta_ {n + 1} (r _ {n} ^ {(\\pmb {\\theta}, 1)} + r _ {n} ^ {(\\pmb {\\theta}, 2)} + \\eta_ {n} ^ {(\\pmb {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (r _ {n} ^ {(\\mathbf {x}, 1)} + r _ {n} ^ {(\\mathbf {x}, 2)} + \\eta_ {n} ^ {(\\mathbf {x}))}), \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.659,
+                0.819,
+                0.677
+            ],
+            "angle": 0,
+            "content": "where the second equality is by taking the Taylor expansion \\(e^{\\beta_{n + 1}\\mathbf{J}_{11}} = \\mathbf{I} + \\beta_{n + 1}\\mathbf{J}_{11} + O(\\beta_{n + 1}^2)\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.683,
+                0.772,
+                0.701
+            ],
+            "angle": 0,
+            "content": "Define \\(\\Phi_{k,n} \\triangleq \\prod_{j=k+1}^{n} (\\mathbf{I} + \\beta_j \\mathbf{J}_{11})\\) and by convention \\(\\Phi_{n,n} = \\mathbf{I}\\). Then, we rewrite (71) as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.196,
+                0.706,
+                0.824,
+                0.927
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\Delta_ {n + 1} ^ {(\\boldsymbol {\\theta})} = \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(O (\\beta_ {k + 1}) L _ {k} ^ {(\\boldsymbol {\\theta})} + O (\\beta_ {k + 1}) R _ {k} ^ {(\\boldsymbol {\\theta})}\\right) \\\\ + \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} (r _ {k} ^ {(\\boldsymbol {\\theta}, 1)} + r _ {k} ^ {(\\boldsymbol {\\theta}, 2)} + \\eta_ {k} ^ {(\\boldsymbol {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (r _ {k} ^ {(\\mathbf {x}, 1)} + r _ {k} ^ {(\\mathbf {x}, 2)} + \\eta_ {k} ^ {(\\mathbf {x}))}) \\\\ = \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(O \\left(\\beta_ {k + 1}\\right) L _ {k} ^ {(\\boldsymbol {\\theta})} + O \\left(\\beta_ {k + 1}\\right) R _ {k} ^ {(\\boldsymbol {\\theta})}\\right) \\\\ + \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} (r _ {k} ^ {(\\pmb {\\theta}, 1)} + \\eta_ {k} ^ {(\\pmb {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (r _ {k} ^ {(\\mathbf {x}, 1)} + \\eta_ {k} ^ {(\\mathbf {x})})) \\\\ + \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(r _ {k} ^ {(\\boldsymbol {\\theta}, 2)} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} r _ {k} ^ {(\\mathbf {x}, 2)}\\right). \\tag {72} \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "31"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.101,
+                0.778,
+                0.122
+            ],
+            "angle": 0,
+            "content": "From (72), we can indeed decompose \\(\\Delta_{n+1}^{(\\boldsymbol{x})}\\) into two parts \\(\\Delta_{n+1}^{(\\boldsymbol{\\theta})} = \\Delta_{n+1}^{(\\boldsymbol{\\theta}, 1)} + \\Delta_{n+1}^{(\\boldsymbol{\\theta}, 2)}\\), where"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.249,
+                0.13,
+                0.825,
+                0.256
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\Delta_ {n + 1} ^ {(\\boldsymbol {\\theta}, 1)} \\triangleq \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(O \\left(\\beta_ {k + 1}\\right) L _ {k} ^ {(\\boldsymbol {\\theta})} + O \\left(\\beta_ {k + 1}\\right) R _ {k} ^ {(\\boldsymbol {\\theta})}\\right) (73a) \\\\ + \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} (r _ {k} ^ {(\\pmb {\\theta}, 1)} + \\eta_ {k} ^ {(\\pmb {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (r _ {k} ^ {(\\mathbf {x}, 1)} + \\eta_ {k} ^ {(\\mathbf {x})})), \\\\ \\Delta_ {n + 1} ^ {(\\boldsymbol {\\theta}, 2)} \\triangleq \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(r _ {k} ^ {(\\boldsymbol {\\theta}, 2)} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} r _ {k} ^ {(\\mathbf {x}, 2)}\\right). (73b) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.261,
+                0.825,
+                0.293
+            ],
+            "angle": 0,
+            "content": "This term \\(\\Delta_{n+1}^{(\\theta,1)}\\) shares the same recursive form as in the sequence defined in Mokkadem & Pelletier (2006, Lemma 6), which is given below."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.298,
+                0.826,
+                0.335
+            ],
+            "angle": 0,
+            "content": "Lemma E.7 (Mokkadem & Pelletier (2006) Lemma 6). For \\(\\Delta_{n+1}^{(\\theta,1)}\\) in the form of (73a), assume \\(\\| \\mathbf{x}_n - \\boldsymbol{\\mu} \\| = O(\\omega_n)\\) and \\(\\| \\Delta_n^{(\\mathbf{x})} \\| = O(\\delta_n)\\) for the sequences \\(\\omega_n, \\delta_n\\) defined in (E.1). Then, we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.315,
+                0.34,
+                0.682,
+                0.361
+            ],
+            "angle": 0,
+            "content": "\\[\n\\| \\Delta_ {n + 1} ^ {(\\boldsymbol {\\theta}, 1)} \\| = O \\left(\\beta_ {n} ^ {2} \\gamma_ {n} ^ {- 2} \\omega_ {n} ^ {2} + \\beta_ {n} \\gamma_ {n} ^ {- 1} \\delta_ {n}\\right) + o (\\sqrt {\\beta_ {n}}) \\quad a. s.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.377,
+                0.718,
+                0.395
+            ],
+            "angle": 0,
+            "content": "Since we already have \\(\\Delta_n^{(\\mathbf{x})} = o(\\sqrt{\\gamma_n})\\) in (69), together with Lemma E.7, we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.232,
+                0.401,
+                0.766,
+                0.422
+            ],
+            "angle": 0,
+            "content": "\\[\n\\| \\Delta_ {n + 1} ^ {(\\boldsymbol {\\theta}, 1)} \\| = O (\\beta_ {n} ^ {2} \\gamma_ {n} ^ {- 2} \\omega_ {n} ^ {2}) + o (\\beta_ {n} \\gamma_ {n} ^ {- 1 / 2}) + o (\\sqrt {\\beta_ {n}}) = O (\\beta_ {n} ^ {2} \\gamma_ {n} ^ {- 2} \\omega_ {n} ^ {2}) + o (\\sqrt {\\beta_ {n}})\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.429,
+                0.739,
+                0.449
+            ],
+            "angle": 0,
+            "content": "where the second equality comes from \\(o(\\beta_n\\gamma_n^{-1 / 2}) = o(\\beta_n^{1 / 2}(\\beta_n\\gamma_n^{-1})^{1 / 2}) = o(\\beta_n^{1 / 2}).\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.454,
+                0.466,
+                0.473
+            ],
+            "angle": 0,
+            "content": "We now focus on \\(\\Delta_{n + 1}^{(\\theta ,2)}\\). Define a sequence"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.357,
+                0.481,
+                0.825,
+                0.52
+            ],
+            "angle": 0,
+            "content": "\\[\n\\Psi_ {n} \\triangleq \\sum_ {k = 1} ^ {n} r _ {k} ^ {(\\boldsymbol {\\theta}, 2)} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} r _ {k} ^ {(\\mathbf {x}, 2)}, \\tag {74}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.527,
+                0.259,
+                0.539
+            ],
+            "angle": 0,
+            "content": "and we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.317,
+                0.547,
+                0.682,
+                0.671
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\beta_ {n + 1} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(r _ {k} ^ {\\left(\\boldsymbol {\\theta}, 2\\right)} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} r _ {k} ^ {\\left(\\mathbf {x}, 2\\right)}\\right) \\\\ = \\beta_ {n + 1} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} (\\Psi_ {k} - \\Psi_ {k - 1}) \\\\ = \\beta_ {n + 1} ^ {1 / 2} \\Psi_ {n} + \\beta_ {n + 1} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n - 1} \\left(\\beta_ {k} \\Phi_ {k, n} - \\beta_ {k + 1} \\Phi_ {k + 1, n}\\right) \\Psi_ {k} \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.678,
+                0.636,
+                0.692
+            ],
+            "angle": 0,
+            "content": "where the last equality comes from the Abel transformation. Note that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.255,
+                0.699,
+                0.743,
+                0.755
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\left\\| \\beta_ {k} \\Phi_ {k, n} - \\beta_ {k + 1} \\Phi_ {k + 1, n} \\right\\| \\leq \\beta_ {k + 1} \\left\\| \\Phi_ {k, n} - \\Phi_ {k + 1, n} \\right\\| + \\left(\\beta_ {k} - \\beta_ {k + 1}\\right) \\left\\| \\Phi_ {k, n} \\right\\| \\\\ \\leq \\beta_ {k + 1} \\| \\Phi_ {k + 1, n} \\| \\beta_ {k} \\| \\mathbf {J} _ {1 1} \\| + C _ {7} \\beta_ {k} ^ {2} \\| \\Phi_ {k, n} \\| \\\\ \\leq C _ {8} \\beta_ {k} ^ {2} e ^ {- (u _ {n} - u _ {k}) T} \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.763,
+                0.825,
+                0.794
+            ],
+            "angle": 0,
+            "content": "for some constant \\( C_7, C_8 > 0 \\), where the last inequality is from Lemma G.4 and \\( \\| \\Phi_{k+1,n} \\| \\leq C_9 \\| \\Phi_{k,n} \\| \\) for some constant \\( C_9 > 0 \\) that depends on \\( e^{\\beta_0 T} \\). Then,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.262,
+                0.802,
+                0.735,
+                0.927
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\beta_ {n + 1} ^ {- 1 / 2} \\left\\| \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(r _ {k} ^ {\\left(\\boldsymbol {\\theta}, 2\\right)} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} r _ {k} ^ {\\left(\\mathbf {x}, 2\\right)}\\right) \\right\\| \\\\ \\leq \\| \\beta_ {n + 1} ^ {1 / 2} \\Psi_ {n} \\| + \\left(\\frac {\\beta_ {n + 1}}{\\beta_ {n}}\\right) ^ {1 / 2} \\beta_ {n} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} \\| \\beta_ {k} \\Phi_ {k, n} - \\beta_ {k + 1} \\Phi_ {k + 1, n} \\| \\| \\Psi_ {k} \\| \\\\ \\leq \\| \\beta_ {n + 1} ^ {1 / 2} \\Psi_ {n} \\| + C _ {8} \\left(\\frac {\\beta_ {n + 1}}{\\beta_ {n}}\\right) ^ {1 / 2} \\beta_ {n} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} \\beta_ {k} ^ {3 / 2} e ^ {- (u _ {n} - u _ {k}) T} \\| \\beta_ {k} ^ {1 / 2} \\Psi_ {k} \\|. \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.949,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "32"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.103,
+                0.719,
+                0.12
+            ],
+            "angle": 0,
+            "content": "By Lemma E.1, we have \\(\\beta_n^{1/2} \\Psi_n \\to 0\\) a.s. such that by Lemma G.6, it follows that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.188,
+                0.127,
+                0.809,
+                0.169
+            ],
+            "angle": 0,
+            "content": "\\[\n\\lim _ {n} \\sup  _ {\\boldsymbol {n}} \\beta_ {n + 1} ^ {- 1 / 2} \\left\\| \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} (r _ {k} ^ {(\\boldsymbol {\\theta}, 2)} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} r _ {k} ^ {(\\mathbf {x}, 2)}) \\right\\| \\leq \\frac {\\lim  \\sup  _ {n} \\| \\beta_ {n} ^ {1 / 2} \\Psi_ {n} \\|}{C (T , 1 / 2)} = 0.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.176,
+                0.304,
+                0.19
+            ],
+            "angle": 0,
+            "content": "Therefore, we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.268,
+                0.198,
+                0.825,
+                0.238
+            ],
+            "angle": 0,
+            "content": "\\[\n\\Delta_ {n + 1} ^ {(\\boldsymbol {\\theta}, 2)} = \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(r _ {k} ^ {(\\boldsymbol {\\theta}, 2)} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} r _ {k} ^ {(\\mathbf {x}, 2)}\\right) = o \\left(\\sqrt {\\beta_ {n}}\\right). \\tag {75}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.254,
+                0.592,
+                0.274
+            ],
+            "angle": 0,
+            "content": "Consequently, \\(\\Delta_{n + 1}^{(\\theta)} = O(\\beta_n^2\\gamma_n^{-2}\\omega_n^2) + o(\\sqrt{\\beta_n})\\) almost surely."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.278,
+                0.825,
+                0.307
+            ],
+            "angle": 0,
+            "content": "Now we are dealing with \\(\\mathbf{x}_n - \\pmb{\\mu}\\) and its related sequence \\(\\omega_{n}\\). Note that by Lemma E.5 and (69), we have almost surely,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.36,
+                0.315,
+                0.825,
+                0.377
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\left\\| \\mathbf {x} _ {n} - \\boldsymbol {\\mu} \\right\\| = O \\left(\\left\\| L _ {n} ^ {(\\mathbf {x})} \\right\\| + \\left\\| \\Delta_ {n} ^ {\\mathbf {x}} \\right\\|\\right) \\\\ = O \\left(\\sqrt {\\gamma_ {n} \\log \\left(s _ {n}\\right)} + o \\left(\\sqrt {\\gamma_ {n}}\\right)\\right) \\tag {76} \\\\ = O \\left(\\sqrt {\\gamma_ {n} \\log \\left(s _ {n}\\right)}\\right). \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.386,
+                0.72,
+                0.405
+            ],
+            "angle": 0,
+            "content": "Thus, we can set \\(\\omega_{n}\\equiv O(\\sqrt{\\gamma_{n}\\log(s_{n})})\\) such that \\(\\| R_n^{(\\pmb {\\theta})}\\|\\) in (70) can be written as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.358,
+                0.413,
+                0.64,
+                0.432
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left\\| R _ {n} ^ {(\\boldsymbol {\\theta})} \\right\\| = O \\left(n ^ {a / 2 - b} \\sqrt {\\log \\left(s _ {n}\\right)} + e ^ {- u _ {n} T}\\right),\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.439,
+                0.202,
+                0.452
+            ],
+            "angle": 0,
+            "content": "and"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.36,
+                0.45,
+                0.637,
+                0.47
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left\\| \\Delta_ {n + 1} ^ {(\\boldsymbol {\\theta})} \\right\\| = O \\left(n ^ {a - 2 b} \\log \\left(s _ {n}\\right)\\right) + o \\left(\\sqrt {\\beta_ {n}}\\right).\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.473,
+                0.825,
+                0.503
+            ],
+            "angle": 0,
+            "content": "In view of assumption A2 and \\(\\beta_{n} = o(\\gamma_{n})\\), \\(a / 2 - b < -b / 2\\) and \\(a - 2b < -b\\), there exists a \\(c > b / 2\\) such that almost surely,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.355,
+                0.51,
+                0.641,
+                0.531
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left\\| R _ {n} ^ {\\left(\\boldsymbol {\\theta}\\right)} \\right\\| = O \\left(n ^ {- s}\\right), \\quad \\left\\| \\Delta_ {n + 1} ^ {\\left(\\boldsymbol {\\theta}\\right)} \\right\\| = o (\\sqrt {\\beta_ {n}}).\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.539,
+                0.784,
+                0.559
+            ],
+            "angle": 0,
+            "content": "Therefore, \\(\\beta_{n}^{-1 / 2}(R_{n}^{(\\pmb{\\theta})} + \\Delta_{n + 1}^{(\\pmb{\\theta})})\\to 0\\) almost surely. This completes the proof of Scenario 2."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.574,
+                0.377,
+                0.59
+            ],
+            "angle": 0,
+            "content": "E.3 CASE (III): \\(\\gamma_{n} = o(\\beta_{n})\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.601,
+                0.825,
+                0.629
+            ],
+            "angle": 0,
+            "content": "For \\(\\gamma_{n} = o(\\beta_{n})\\), we can see that the roles of \\(\\theta_{n}\\) and \\(\\mathbf{x}_n\\) are flipped, i.e., \\(\\theta_{n}\\) is now on fast timescale while \\(\\mathbf{x}_n\\) is on slow timescale."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.636,
+                0.826,
+                0.707
+            ],
+            "angle": 0,
+            "content": "We still decompose \\(\\mathbf{x}_n\\) as \\(\\mathbf{x}_n - \\boldsymbol{\\mu} = L_n^{(\\mathbf{x})} + \\Delta_n^{(\\mathbf{x})}\\), where \\(L_{n}^{(\\mathbf{x})},\\Delta_{n}^{(\\mathbf{x})}\\) are defined in (56) and (57), respectively. Since \\(\\mathbf{x}_n\\) is independent of \\(\\theta_{n}\\), the results of \\(L_{n}^{(\\mathbf{x})}\\) and \\(\\Delta_{n}^{(\\mathbf{x})}\\) remain the same, i.e., almost surely, \\(L_{n}^{(\\mathbf{x})} = O\\bigl (\\sqrt{\\gamma_{n}\\log(s_{n})}\\bigr)\\) from Lemma E.5 and \\(\\Delta_n^{(\\mathbf{x})} = o(\\sqrt{\\gamma_n})\\) from (69). Then, we define sequences \\(\\hat{L}_n^{(\\pmb {\\theta})}\\) and \\(\\hat{R}_n^{(\\pmb {\\theta})}\\) as follows."
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.3,
+                0.715,
+                0.825,
+                0.755
+            ],
+            "angle": 0,
+            "content": "\\[\n\\hat {L} _ {n} ^ {(\\boldsymbol {\\theta})} \\triangleq e ^ {\\beta_ {n} \\mathbf {J} _ {1 1}} \\hat {L} _ {n - 1} ^ {(\\boldsymbol {\\theta})} + \\beta_ {n} M _ {n} ^ {(\\boldsymbol {\\theta})} = \\sum_ {k = 1} ^ {n} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\beta_ {k} M _ {k} ^ {(\\boldsymbol {\\theta})}, \\tag {77a}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.182,
+                0.764,
+                0.825,
+                0.815
+            ],
+            "angle": 0,
+            "content": "\\[\n\\hat {R} _ {n} ^ {(\\boldsymbol {\\theta})} \\triangleq e ^ {\\beta_ {n} \\mathbf {J} _ {1 1}} \\hat {R} _ {n - 1} ^ {(\\boldsymbol {\\theta})} + \\beta_ {n} \\mathbf {J} _ {1 2} (\\alpha) \\left(L _ {n - 1} ^ {(\\mathbf {x})} + R _ {n - 1} ^ {(\\mathbf {x})}\\right) = \\sum_ {k = 1} ^ {n} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} \\beta_ {k} \\mathbf {J} _ {1 2} (\\alpha) \\left(L _ {k - 1} ^ {(\\mathbf {x})} + R _ {k - 1} ^ {(\\mathbf {x})}\\right). \\tag {77b}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.814,
+                0.591,
+                0.833
+            ],
+            "angle": 0,
+            "content": "Moreover, the remaining term \\(\\hat{\\Delta}_n^{(\\pmb {\\theta})}\\triangleq \\pmb {\\theta}_n - \\pmb {\\theta}^* -\\hat{L}_n^{(\\pmb {\\theta})} - \\hat{R}_n^{(\\pmb {\\theta})}.\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.839,
+                0.549,
+                0.854
+            ],
+            "angle": 0,
+            "content": "The proof outline is the same as in the previous scenario:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.216,
+                0.865,
+                0.748,
+                0.885
+            ],
+            "angle": 0,
+            "content": "- We first show \\(\\beta_n^{-1/2} \\hat{\\Delta}_n^{(\\theta)}\\) weakly converges to the distribution \\(N(0, \\mathbf{V}_{\\pmb{\\theta}}^{(3)})(\\alpha)\\);"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.216,
+                0.89,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "- We analyse \\(\\hat{L}_n^{(\\theta)}\\) and \\(\\hat{R}_n^{(\\theta)}\\) to ensure that these two terms decrease faster than the CLT scale \\(\\beta_n^{-1/2}\\), i.e., \\(\\lim_{n\\to\\infty}\\beta_n^{-1/2}(\\hat{L}_n^{(\\theta)} - \\hat{R}_n^{(\\theta)}) = 0\\);"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.216,
+                0.865,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "33"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.216,
+                0.103,
+                0.825,
+                0.139
+            ],
+            "angle": 0,
+            "content": "- With above two steps, we can show that \\(\\beta_{n}^{-1 / 2}(\\pmb{\\theta}_{n} - \\pmb{\\theta}^{*})\\) weakly converges to the distribution \\(N(0,\\mathbf{V}_{\\pmb{\\theta}}^{(3)})(\\alpha)\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.15,
+                0.289,
+                0.168
+            ],
+            "angle": 0,
+            "content": "Analysis of \\(\\hat{L}_n^{(\\theta)}\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.174,
+                0.825,
+                0.211
+            ],
+            "angle": 0,
+            "content": "We first focus on \\(\\hat{L}_n^{(\\theta)}\\) and follow similar steps as we did when we analysed \\(L_{n}^{(\\theta)}\\) in the previous scenario. We set a Martingale \\(Z^{(n)} = \\{Z_k^{(n)}\\}_{k\\geq 1}\\) such that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.37,
+                0.218,
+                0.626,
+                0.256
+            ],
+            "angle": 0,
+            "content": "\\[\nZ _ {k} ^ {(n)} = \\beta_ {n} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1} \\beta_ {k} M _ {k} ^ {(\\boldsymbol {\\theta})}}.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.263,
+                0.216,
+                0.276
+            ],
+            "angle": 0,
+            "content": "Then,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.315,
+                0.276,
+                0.681,
+                0.315
+            ],
+            "angle": 0,
+            "content": "\\[\nA _ {n} \\triangleq \\sum_ {k = 1} ^ {n} \\mathbb {E} \\left[ (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) ^ {T} \\Big | \\mathcal {F} _ {k - 1} \\right].\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.319,
+                0.673,
+                0.339
+            ],
+            "angle": 0,
+            "content": "Following the similar steps in (59) to decompose \\( M_{k}^{(\\pmb{\\theta})} \\) with (42b), we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.189,
+                0.345,
+                0.824,
+                0.514
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} A _ {n} = \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\beta_ {k} ^ {2} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\left(\\mathbf {U} _ {1 1} + \\mathbf {D} _ {k} ^ {(3)} + \\mathbf {J} _ {k} ^ {(3)}\\right) e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1} ^ {T}} \\\\ = \\underbrace {\\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\beta_ {k} ^ {2} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\mathbf {U} _ {1 1} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1} ^ {T}}} _ {A _ {n} ^ {(a)}} + \\underbrace {\\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\beta_ {k} ^ {2} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\mathbf {D} _ {k} ^ {(3)} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1} ^ {T}}} _ {A _ {n} ^ {(b)}} \\tag {78} \\\\ + \\underbrace {\\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\beta_ {k} ^ {2} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\mathbf {J} _ {k} ^ {(3)} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1} ^ {T}}} _ {A _ {n} ^ {(c)}} \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.522,
+                0.827,
+                0.568
+            ],
+            "angle": 0,
+            "content": "Since \\( A_{n}^{(a)}, A_{n}^{(b)}, A_{n}^{(c)} \\) share similar forms as in Lemma E.3, we follow the same steps as the proof therein, with the application of Lemma E.1. To avoid repetition, we omit the proof and directly give the following lemma."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.571,
+                0.564,
+                0.59
+            ],
+            "angle": 0,
+            "content": "Lemma E.8. For \\(A_{n}^{(a)}, A_{n}^{(b)}, A_{n}^{(c)}\\) defined in (78), we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.286,
+                0.596,
+                0.825,
+                0.621
+            ],
+            "angle": 0,
+            "content": "\\[\n\\lim  _ {n \\rightarrow \\infty} A _ {n} ^ {(a)} = \\mathbf {V} _ {\\boldsymbol {\\theta}} ^ {(3)} (\\alpha), \\quad \\lim  _ {n \\rightarrow \\infty} \\| A _ {n} ^ {(b)} \\| = 0, \\quad \\lim  _ {n \\rightarrow \\infty} \\| A _ {n} ^ {(c)} \\| = 0, \\tag {79}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.628,
+                0.542,
+                0.647
+            ],
+            "angle": 0,
+            "content": "where \\(\\mathbf{V}_{\\pmb{\\theta}}^{(3)}(\\alpha)\\) is the solution to the Lyapunov equation"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.406,
+                0.653,
+                0.591,
+                0.671
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {J} _ {1 1} \\mathbf {V} + \\mathbf {V} \\mathbf {J} _ {1 1} ^ {T} + \\mathbf {U} _ {1 1} = 0.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.686,
+                0.825,
+                0.737
+            ],
+            "angle": 0,
+            "content": "Note that here we don't have the term \\(\\frac{\\mathbb{1}_{\\{b = 1\\}}}{2}\\pmb{I}\\) in above lemma, compared to Lemma E.3, because in the case of \\(\\gamma_{n} = o(\\beta_{n})\\), \\(b < 1\\) such that \\(\\mathbb{1}_{\\{b = 1\\}} = 0\\). Then, applying Lemma G.1 to derive the closed form of \\(\\mathbf{V}_{\\pmb{\\theta}}^{(3)}(\\alpha)\\) gives"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.355,
+                0.743,
+                0.64,
+                0.765
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {V} _ {\\boldsymbol {\\theta}} ^ {(3)} (\\alpha) = \\int_ {0} ^ {\\infty} e ^ {t \\nabla_ {\\boldsymbol {\\theta}} \\mathbf {h} (\\boldsymbol {\\theta} ^ {*})} \\mathbf {U} _ {1 1} e ^ {t \\nabla_ {\\boldsymbol {\\theta}} \\mathbf {h} (\\boldsymbol {\\theta} ^ {*})} d t.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.769,
+                0.308,
+                0.782
+            ],
+            "angle": 0,
+            "content": "Thus, it follows that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.283,
+                0.79,
+                0.713,
+                0.828
+            ],
+            "angle": 0,
+            "content": "\\[\n\\lim  _ {n \\rightarrow \\infty} \\sum_ {k = 1} ^ {n} \\mathbb {E} \\left[ (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) ^ {T} | \\mathcal {F} _ {k - 1} \\right] = \\mathbf {V} _ {\\boldsymbol {\\theta}} ^ {(3)} (\\alpha).\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.834,
+                0.747,
+                0.85
+            ],
+            "angle": 0,
+            "content": "Again, we use the Martingale CLT result in Theorem G.3 and have the following result."
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.354,
+                0.856,
+                0.641,
+                0.883
+            ],
+            "angle": 0,
+            "content": "\\[\nZ ^ {n} = \\beta_ {n} ^ {- 1 / 2} \\hat {L} _ {n} ^ {(\\boldsymbol {\\theta})} \\xrightarrow [ d i s t . ]{n \\to \\infty} N \\left(0, \\mathbf {V} _ {\\boldsymbol {\\theta}} ^ {(3)} (\\alpha)\\right).\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.892,
+                0.825,
+                0.927
+            ],
+            "angle": 0,
+            "content": "Moreover, similar to the tighter upper bound of \\( L_{n}^{(\\mathbf{x})} \\) proved in Lemma E.5, we utilize the tighter upper bound Lemma E.6 in the proof thereof, and obtain \\( \\hat{L}_n^{(\\pmb{\\theta})} = O(\\sqrt{\\beta_n\\log(u_n)}) \\)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "34"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.101,
+                0.289,
+                0.12
+            ],
+            "angle": 0,
+            "content": "Analysis of \\(\\hat{R}_n^{(\\theta)}\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.127,
+                0.825,
+                0.16
+            ],
+            "angle": 0,
+            "content": "Next, we turn to the term \\(\\hat{R}_n^{(\\theta)}\\) in (77b). Taking the norm gives the following inequality for some constant \\(C, T > 0\\) by applying Lemma G.4,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.32,
+                0.167,
+                0.677,
+                0.207
+            ],
+            "angle": 0,
+            "content": "\\[\n\\| \\hat {R} _ {n} ^ {(\\boldsymbol {\\theta})} \\| \\leq C \\sum_ {k = 1} ^ {n} e ^ {- (u _ {n} - u _ {k}) T} \\beta_ {k} (\\| L _ {k - 1} ^ {(\\mathbf {x})} \\| + \\| R _ {k - 1} ^ {(\\mathbf {x})} \\)).\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.213,
+                0.338,
+                0.228
+            ],
+            "angle": 0,
+            "content": "Using Lemma G.6 gives"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.275,
+                0.236,
+                0.721,
+                0.275
+            ],
+            "angle": 0,
+            "content": "\\[\n\\sum_ {k = 1} ^ {n} e ^ {- (u _ {n} - u _ {k}) T} \\beta_ {k} (\\| L _ {k - 1} ^ {(\\mathbf {x})} \\| + \\| R _ {k - 1} ^ {(\\mathbf {x})} \\|) = O (\\| L _ {k - 1} ^ {(\\mathbf {x})} \\| + \\| R _ {n - 1} ^ {(\\mathbf {x})} \\)).\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.284,
+                0.825,
+                0.373
+            ],
+            "angle": 0,
+            "content": "Thus, \\(\\beta_n^{-1/2} \\|\\hat{R}_n^{(\\pmb{\\theta})}\\| = o(\\sqrt{\\gamma_n \\beta_n^{-1}}) + O\\left(\\sqrt{\\gamma_n \\beta_n^{-1} \\log(s_n)}\\right)\\). Since \\(\\gamma_n = o(\\beta_n), \\gamma_n \\beta_n^{-1} = (n + 1)^{b - a}\\), where \\(b - a < 0\\). Then, there exists some \\(s > 0\\) such that \\(b - a < -s < 0\\). Together with \\(\\log(s_n) = O(\\log(n))\\), we have \\(O\\left(\\sqrt{\\gamma_n \\beta_n^{-1} \\log(s_n)}\\right) = O(\\sqrt{n^{-s} \\log(n)}) = o(1)\\). Therefore, we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.424,
+                0.373,
+                0.572,
+                0.396
+            ],
+            "angle": 0,
+            "content": "\\[\n\\lim _ {n \\to \\infty} \\beta_ {n} ^ {- 1 / 2} \\hat {R} _ {n} ^ {(\\boldsymbol {\\theta})} = 0.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.408,
+                0.289,
+                0.426
+            ],
+            "angle": 0,
+            "content": "Analysis of \\(\\hat{\\Delta}_n^{(\\theta)}\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.433,
+                0.472,
+                0.452
+            ],
+            "angle": 0,
+            "content": "Lastly, let's focus on the term \\(\\hat{\\Delta}_n^{(\\theta)}\\). We have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.181,
+                0.459,
+                0.815,
+                0.613
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\hat {\\Delta} _ {n + 1} ^ {(\\boldsymbol {\\theta})} = \\boldsymbol {\\theta} _ {n + 1} - \\boldsymbol {\\theta} ^ {*} - \\hat {L} _ {n + 1} ^ {(\\boldsymbol {\\theta})} - \\hat {R} _ {n + 1} ^ {(\\boldsymbol {\\theta})} \\\\ = \\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*} + \\beta_ {n + 1} \\left(\\mathbf {J} _ {1 1} (\\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*}) + \\mathbf {J} _ {1 2} (\\alpha) (\\mathbf {x} _ {n} - \\boldsymbol {\\mu}) + M _ {n + 1} ^ {(\\boldsymbol {\\theta})} + r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} + r _ {n} ^ {(\\boldsymbol {\\theta}, 2)} + \\eta_ {n} ^ {(\\boldsymbol {\\theta})}\\right) \\\\ - e ^ {\\beta_ {n + 1} \\mathbf {J} _ {1 1}} \\hat {L} _ {n} ^ {(\\boldsymbol {\\theta})} - \\beta_ {n + 1} M _ {n + 1} ^ {(\\boldsymbol {\\theta})} - e ^ {\\beta_ {n + 1} \\mathbf {J} _ {1 1}} \\hat {R} _ {n} ^ {(\\boldsymbol {\\theta})} - \\beta_ {n + 1} \\mathbf {J} _ {1 2} (\\alpha) (L _ {n} ^ {(\\mathbf {x})} + R _ {n} ^ {(\\mathbf {x})}) \\\\ = (\\mathbf {I} + \\beta_ {n + 1} \\mathbf {J} _ {1 1}) \\left(\\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*}\\right) + \\beta_ {n + 1} \\mathbf {J} _ {1 2} (\\alpha) \\Delta_ {n} ^ {(\\mathbf {x})} + \\beta_ {n + 1} \\left(r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} + r _ {n} ^ {(\\boldsymbol {\\theta}, 2)} + \\eta_ {n} ^ {(\\boldsymbol {\\theta})}\\right) \\\\ - \\left(\\mathbf {I} + \\beta_ {n + 1} \\mathbf {J} _ {1 1} + O \\left(\\beta_ {n + 1} ^ {2}\\right)\\right) \\left(\\hat {L} _ {n} ^ {(\\boldsymbol {\\theta})} + \\hat {R} _ {n} ^ {(\\boldsymbol {\\theta})}\\right) \\\\ = \\left(\\mathbf {I} + \\beta_ {n + 1} \\mathbf {J} _ {1 1}\\right) \\hat {\\Delta} _ {n} ^ {(\\boldsymbol {\\theta})} + \\beta_ {n + 1} \\mathbf {J} _ {1 2} (\\alpha) \\Delta_ {n} ^ {(\\mathbf {x})} + \\beta_ {n + 1} \\left(r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} + r _ {n} ^ {(\\boldsymbol {\\theta}, 2)} + \\eta_ {n} ^ {(\\boldsymbol {\\theta})}\\right) \\\\ + O \\left(\\beta_ {n + 1} ^ {2}\\right) \\left(\\hat {L} _ {n} ^ {(\\boldsymbol {\\theta})} + \\hat {R} _ {n} ^ {(\\boldsymbol {\\theta})}\\right). \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.622,
+                0.821,
+                0.656
+            ],
+            "angle": 0,
+            "content": "where the second equality is from (53a), the third equality stems from the approximation of \\(e^{\\beta_{n+1}\\mathbf{J}_{11}}\\). Then, we again use the definition \\(\\Phi_{k,n} \\triangleq \\prod_{j=k+1}^{n}(\\mathbf{I} + \\beta_{j}\\mathbf{J}_{11})\\) and reiterate the above equation as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.271,
+                0.665,
+                0.727,
+                0.81
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\hat {\\Delta} _ {n + 1} ^ {(\\boldsymbol {\\theta})} = \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(O (\\beta_ {k + 1}) L _ {k} ^ {(\\boldsymbol {\\theta})} + O (\\beta_ {k + 1}) R _ {k} ^ {(\\boldsymbol {\\theta})}\\right) \\\\ + \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\mathbf {J} _ {1 2} (\\alpha) \\Delta_ {n} ^ {(\\mathbf {x})} + \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(r _ {k} ^ {(\\boldsymbol {\\theta}, 1)} + \\eta_ {k} ^ {(\\boldsymbol {\\theta})}\\right) \\\\ + \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} r _ {k} ^ {(\\boldsymbol {\\theta}, 2)} \\\\ \\triangleq \\hat {\\Delta} _ {n + 1} ^ {(\\boldsymbol {\\theta}, 1)} + \\hat {\\Delta} _ {n + 1} ^ {(\\boldsymbol {\\theta}, 2)}, \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.818,
+                0.462,
+                0.838
+            ],
+            "angle": 0,
+            "content": "where \\(\\hat{\\Delta}_{n + 1}^{(\\pmb {\\theta},2)} = \\sum_{k = 1}^{n}\\Phi_{k,n}\\beta_{k + 1}r_{k}^{(\\pmb {\\theta},2)}\\) and"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.311,
+                0.848,
+                0.824,
+                0.927
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\hat {\\Delta} _ {n + 1} ^ {(\\boldsymbol {\\theta}, 1)} = \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(O \\left(\\beta_ {k + 1}\\right) L _ {k} ^ {(\\boldsymbol {\\theta})} + O \\left(\\beta_ {k + 1}\\right) R _ {k} ^ {(\\boldsymbol {\\theta})}\\right) \\tag {80} \\\\ + \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(r _ {k} ^ {\\left(\\boldsymbol {\\theta}, 1\\right)} + \\eta_ {k} ^ {\\left(\\boldsymbol {\\theta}\\right)} + \\mathbf {J} _ {1 2} (\\alpha) \\Delta_ {n} ^ {\\left(\\mathbf {x}\\right)}\\right). \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "35"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.101,
+                0.735,
+                0.123
+            ],
+            "angle": 0,
+            "content": "For \\(\\hat{\\Delta}_{n+1}^{(\\theta,2)}\\), we follow the same steps from (74) to (75), and obtain \\(\\hat{\\Delta}_{n+1}^{(\\theta,2)} = o\\left(\\sqrt{\\beta_n}\\right)\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.128,
+                0.825,
+                0.168
+            ],
+            "angle": 0,
+            "content": "Next, we consider \\(\\hat{\\Delta}_{n+1}^{(\\pmb{\\theta},1)}\\) and want to show that \\(\\hat{\\Delta}_{n+1}^{(\\pmb{\\theta},1)} = o(\\sqrt{\\beta_n})\\). Again, we utilize Mokkadem & Pelletier (2006, Lemma 6) for \\(\\hat{\\Delta}_{n+1}^{(\\pmb{\\theta},1)}\\) and adapt the notation here for the case \\(\\gamma_n = o(\\beta_n)\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.169,
+                0.17,
+                0.825,
+                0.203
+            ],
+            "angle": 0,
+            "content": "Lemma E.9. For \\(\\hat{\\Delta}_{n+1}^{(\\pmb{\\theta}, 1)}\\) in the form of (80), assume \\(\\|\\pmb{\\theta}_n - \\pmb{\\theta}^*\\| = O(\\omega_n)\\) and \\(\\|\\hat{\\Delta}_n^{(\\pmb{\\theta}, 1)}\\| = O(\\delta_n)\\) for the sequences \\(\\omega_n, \\delta_n\\) defined in (E.1). Then, we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.316,
+                0.208,
+                0.826,
+                0.23
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left\\| \\hat {\\Delta} _ {n + 1} ^ {(\\boldsymbol {\\theta}, 1)} \\right\\| = O \\left(\\gamma_ {n} ^ {2} \\beta_ {n} ^ {- 2} \\omega_ {n} ^ {2} + \\gamma_ {n} \\beta_ {n} ^ {- 1} \\delta_ {n}\\right) + o (\\sqrt {\\gamma_ {n}}) \\quad a. s. \\tag {81}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.241,
+                0.827,
+                0.288
+            ],
+            "angle": 0,
+            "content": "Now we need to further analyse \\(\\delta_{n}\\) and tighten its big O form, starting from \\(\\delta_{n} \\equiv 1\\), so that we can finally obtain the big O form of \\(\\|\\hat{\\Delta}_{n+1}^{(\\theta,1)}\\|\\). The following steps are borrowed from the ideas in Mokkadem & Pelletier (2006, Section 2.3.2)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.295,
+                0.826,
+                0.332
+            ],
+            "angle": 0,
+            "content": "By almost sure convergence result \\(\\lim_{n\\to \\infty}\\pmb{\\theta}_n = \\pmb{\\theta}^*\\), we have \\(\\lim_{n\\to \\infty}\\Delta_n^{(\\pmb{\\theta})} = 0\\) a.s. such that we can first set \\(\\delta_{n}\\equiv 1\\), and \\(\\| \\hat{\\Delta}_{n + 1}^{(\\pmb {\\theta},1)}\\| = O(\\gamma_n^2\\beta_n^{-2}\\omega_n^2 +\\gamma_n\\beta_n^{-1}) + o(\\sqrt{\\gamma_n})\\). Then, we redefine"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.36,
+                0.336,
+                0.637,
+                0.355
+            ],
+            "angle": 0,
+            "content": "\\[\n\\delta_ {n} \\equiv O (\\gamma_ {n} ^ {2} \\beta_ {n} ^ {- 2} \\omega_ {n} ^ {2} + \\gamma_ {n} \\beta_ {n} ^ {- 1}) + o (\\sqrt {\\gamma_ {n}}),\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.36,
+                0.754,
+                0.376
+            ],
+            "angle": 0,
+            "content": "and notice that it still satisfies definition E.1. Then, reapplying this \\(\\delta_{n}\\) form to (81) gives"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.334,
+                0.381,
+                0.664,
+                0.402
+            ],
+            "angle": 0,
+            "content": "\\[\n\\| \\hat {\\Delta} _ {n + 1} ^ {(\\pmb {\\theta}, 1)} \\| = O (\\gamma_ {n} ^ {2} \\beta_ {n} ^ {- 2} \\omega_ {n} ^ {2} + [ \\gamma_ {n} \\beta_ {n} ^ {- 1} ] ^ {2}) + o (\\sqrt {\\gamma_ {n}})\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.407,
+                0.492,
+                0.422
+            ],
+            "angle": 0,
+            "content": "and by induction we have for all integers \\(k \\geq 1\\),"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.332,
+                0.428,
+                0.666,
+                0.449
+            ],
+            "angle": 0,
+            "content": "\\[\n\\| \\hat {\\Delta} _ {n + 1} ^ {(\\pmb {\\theta}, 1)} \\| = O (\\gamma_ {n} ^ {2} \\beta_ {n} ^ {- 2} \\omega_ {n} ^ {2} + [ \\gamma_ {n} \\beta_ {n} ^ {- 1} ] ^ {k}) + o (\\sqrt {\\gamma_ {n}}).\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.454,
+                0.79,
+                0.473
+            ],
+            "angle": 0,
+            "content": "Since \\([\\gamma_n\\beta_n^{-1}]^k = n^{(b - a)k}\\), there exists \\(k_0 > a / 2(a - b)\\) such that \\([\\gamma_n\\beta_n^{-1}]^{k_0} = o(\\sqrt{\\gamma_n})\\), and"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.373,
+                0.478,
+                0.825,
+                0.5
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left\\| \\hat {\\Delta} _ {n + 1} ^ {(\\boldsymbol {\\theta}, 1)} \\right\\| = O \\left(\\gamma_ {n} ^ {2} \\beta_ {n} ^ {- 2} \\omega_ {n} ^ {2}\\right) + o (\\sqrt {\\gamma_ {n}}). \\tag {82}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.504,
+                0.825,
+                0.537
+            ],
+            "angle": 0,
+            "content": "Then, as suggested in Mokkadem & Pelletier (2006, Section 2.3.2), we can choose \\(\\omega_{n} = O(\\sqrt{\\beta_{n}\\log(u_{n})} +[\\gamma_{n}\\beta_{n}^{-1}]^{k})\\), which also satisfies definition E.1. Then,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.229,
+                0.542,
+                0.767,
+                0.641
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\left\\| \\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*} \\right\\| = \\left\\| \\hat {L} _ {n} ^ {(\\boldsymbol {\\theta})} + \\hat {R} _ {n} ^ {(\\boldsymbol {\\theta})} + \\hat {\\Delta} _ {n} ^ {(\\boldsymbol {\\theta})} \\right\\| \\\\ = O \\left(\\sqrt {\\beta_ {n} \\log (u _ {n})} + \\sqrt {\\gamma_ {n} \\beta_ {n} ^ {- 1} \\log (s _ {n})} + \\left(\\left[ \\gamma_ {n} \\beta_ {n} ^ {- 1} \\right] ^ {k + 1} + \\gamma_ {n} \\beta_ {n} ^ {- 1} \\sqrt {\\beta_ {n} \\log (u _ {n})}\\right) ^ {2}\\right) \\\\ + o (\\sqrt {\\beta_ {n}} + \\sqrt {\\gamma_ {n}}) \\\\ = O (\\sqrt {\\beta_ {n} \\log (u _ {n})} + [ \\gamma_ {n} \\beta_ {n} ^ {- 1} ] ^ {k + 1}). \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.646,
+                0.825,
+                0.68
+            ],
+            "angle": 0,
+            "content": "By induction, this holds for all \\( k \\geq 1 \\) such that there exists \\( k_0 \\), \\( [\\gamma_n\\beta_n^{-1}]^{k_0} = o(\\sqrt{\\beta_n}) \\) and \\( \\| \\pmb{\\theta}_n - \\pmb{\\theta}^*\\| = O(\\sqrt{\\beta_n\\log(u_n)}) \\). Equivalently, \\( \\omega_{n} = \\sqrt{\\beta_{n}\\log(u_{n})} \\). Therefore, from (82) we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.319,
+                0.686,
+                0.679,
+                0.707
+            ],
+            "angle": 0,
+            "content": "\\[\n\\| \\hat {\\Delta} _ {n + 1} ^ {(\\boldsymbol {\\theta}, 1)} \\| = O (\\gamma_ {n} ^ {2} \\beta_ {n} ^ {- 1} \\log (u _ {n})) + o (\\sqrt {\\gamma_ {n}}) = o (\\sqrt {\\gamma_ {n}}).\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.712,
+                0.768,
+                0.734
+            ],
+            "angle": 0,
+            "content": "Together with \\(\\| \\hat{\\Delta}_{n + 1}^{(\\pmb {\\theta},2)}\\| = o(\\sqrt{\\beta_n})\\) , we have \\(\\beta_{n}^{-1 / 2}\\| \\hat{\\Delta}_{n + 1}^{(\\pmb {\\theta})}\\| = o(\\sqrt{\\gamma_{n}\\beta_{n}^{-1}}) + 1)\\) such that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.42,
+                0.738,
+                0.576,
+                0.764
+            ],
+            "angle": 0,
+            "content": "\\[\n\\lim  _ {n \\to \\infty} \\beta_ {n} ^ {- 1 / 2} \\hat {\\Delta} _ {n + 1} ^ {(\\boldsymbol {\\theta})} = 0.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.77,
+                0.825,
+                0.8
+            ],
+            "angle": 0,
+            "content": "Thus, we have finished the proof according to the proof outline mentioned at the beginning of this part."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.171,
+                0.817,
+                0.677,
+                0.833
+            ],
+            "angle": 0,
+            "content": "F DISCUSSION OF COVARIANCE ORDERING OF SA-SRRW"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.849,
+                0.414,
+                0.863
+            ],
+            "angle": 0,
+            "content": "F.1 PROOF OF PROPOSITION 3.4"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.875,
+                0.487,
+                0.89
+            ],
+            "angle": 0,
+            "content": "For any \\(\\alpha > 0\\) and any vector \\(\\mathbf{x} \\in \\mathbb{R}^d\\), we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.24,
+                0.896,
+                0.756,
+                0.93
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {x} ^ {T} \\mathbf {V} _ {\\boldsymbol {\\theta}} ^ {(1)} (\\alpha) \\mathbf {x} = \\int_ {0} ^ {\\infty} \\mathbf {x} ^ {T} e ^ {t (\\nabla_ {\\boldsymbol {\\theta}} \\mathbf {h} (\\boldsymbol {\\theta} ^ {*}) + \\frac {1}{2} \\mathbf {\\Sigma} _ {\\{b = 1 \\}} ^ {*} \\boldsymbol {I})} \\mathbf {U} _ {\\boldsymbol {\\theta}} (\\alpha) e ^ {t (\\nabla_ {\\boldsymbol {\\theta}} \\mathbf {h} (\\boldsymbol {\\theta} ^ {*}) + \\frac {1}{2} \\mathbf {\\Sigma} _ {\\{b = 1 \\}} ^ {*} \\boldsymbol {I}) ^ {T}} \\mathbf {x} d t\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.51,
+                0.961
+            ],
+            "angle": 0,
+            "content": "36"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.102,
+                0.825,
+                0.165
+            ],
+            "angle": 0,
+            "content": "where the first equality is from the form of \\(\\mathbf{V}_{\\theta}^{(1)}(\\alpha)\\) in Theorem 3.3. Let \\(\\mathbf{y} \\triangleq e^{t(\\nabla_{\\theta}\\mathbf{h}(\\boldsymbol{\\theta}^{*}) + \\frac{1}{2}\\left\\{b = 1\\right\\} I)}\\mathbf{x}\\), with the dependence on variable \\(t\\) left implicit. The matrix \\(\\mathbf{U}_{\\theta}(\\alpha)\\), given explicitly in (11) positive semi definite, since \\(\\lambda_i \\in (-1,1)\\) for all \\(i \\in \\{1,\\dots,N-1\\}\\). Thus, the terms \\(\\mathbf{y}^T\\mathbf{U}_{\\theta}(\\alpha)\\mathbf{y}\\) inside the integral are non-negative, and it is enough to provide an ordering on \\(\\mathbf{y}^T\\mathbf{U}_{\\theta}(\\alpha)\\mathbf{y}\\) with respect to \\(\\alpha\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.17,
+                0.323,
+                0.185
+            ],
+            "angle": 0,
+            "content": "For any \\(\\alpha_{2} > \\alpha_{1} > 0\\)"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.227,
+                0.194,
+                0.768,
+                0.324
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\mathbf {y} ^ {T} \\mathbf {U} _ {\\boldsymbol {\\theta}} (\\alpha_ {2}) \\mathbf {y} = \\sum_ {i = 1} ^ {N - 1} \\frac {1}{(\\alpha_ {2} (1 + \\lambda_ {i}) + 1) ^ {2}} \\cdot \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {y} ^ {T} \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H} \\mathbf {y} \\\\ <   \\sum_ {i = 1} ^ {N - 1} \\frac {1}{\\left(\\alpha_ {1} \\left(1 + \\lambda_ {i}\\right) + 1\\right) ^ {2}} \\cdot \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {y} ^ {T} \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H} \\mathbf {y} = \\mathbf {y} ^ {T} \\mathbf {U} _ {\\boldsymbol {\\theta}} (\\alpha_ {1}) \\mathbf {y} \\\\ <   \\sum_ {i = 1} ^ {N - 1} \\cdot \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {y} ^ {T} \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H} \\mathbf {y} = \\mathbf {y} ^ {T} \\mathbf {U} _ {\\boldsymbol {\\theta}} (0) \\mathbf {y}, \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.342,
+                0.825,
+                0.386
+            ],
+            "angle": 0,
+            "content": "where the inequality\\(^{13}\\) is because \\(\\alpha(1 + \\lambda_i) > 0\\) for all \\(i \\in \\{1, \\dots, N\\}\\) and any \\(\\alpha > 0\\). In fact, the ordering is monotone in \\(\\alpha\\), and \\(\\mathbf{y}^T \\mathbf{U}_{\\theta}(\\alpha_2) \\mathbf{y}\\) decreases at rate \\(1 / \\alpha^2\\) as seen from its form in the equation above. This completes the proof."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.404,
+                0.669,
+                0.418
+            ],
+            "angle": 0,
+            "content": "F.2 DISCUSSION REGARDING PROPOSITION 3.4 AND MSE ORDERING"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.43,
+                0.827,
+                0.62
+            ],
+            "angle": 0,
+            "content": "We can use Proposition 3.4 to show that the MSE of SA iterates of (4c) driven by SRRW eventually becomes smaller than that SA iterates when the stochastic noise is driven by an i.i.d. sequence of random variables. The diagonal entries of \\(\\mathbf{V}_{\\theta}^{(1)}(\\alpha)\\) are obtained by evaluating \\(\\mathbf{e}_i^T\\mathbf{V}_{\\theta}^{(1)}(\\alpha)\\mathbf{e}_i\\), where \\(\\mathbf{e}_i\\) is the \\(i\\)'th standard basis vector.\\(^{14}\\) These diagonal entries are the asymptotic variance corresponding to the element-wise iterate errors, and for large enough \\(n\\), we have \\(\\mathbf{e}_i^T\\mathbf{V}_{\\theta}^{(1)}(\\alpha)\\mathbf{e}_i \\approx \\mathbb{E}[(\\boldsymbol{\\theta}_n - \\boldsymbol{\\theta}^*)_i^2] / \\beta_n\\) for all \\(i \\in \\{1, \\dots, D\\}\\). Thus, the trace of matrix \\(\\mathbf{V}_{\\theta}^{(1)}(\\alpha)\\) approximates the scaled MSE, that is \\(\\mathrm{Tr}(\\mathbf{V}_{\\theta}^{(1)}(\\alpha)) = \\sum_i \\mathbf{e}_i^T\\mathbf{V}_{\\theta}^{(1)}(\\alpha)\\mathbf{e}_i \\approx \\sum_i \\mathbb{E}[(\\boldsymbol{\\theta}_n - \\boldsymbol{\\theta}^*)_i^2] / \\beta_n = \\mathbb{E}[||\\boldsymbol{\\theta}_n - \\boldsymbol{\\theta}^*||^2] / \\beta_n\\) for large \\(n\\). Since all entries of \\(\\mathbf{V}_{\\theta}^{(1)}(\\alpha)\\) go to zero as \\(\\alpha\\) increases, they get smaller than the corresponding term for the SA algorithm with i.i.d. input for large enough \\(\\alpha\\), which achieves a constant MSE in the similarly scaled limit, since the asymptotic covariance is not a function of \\(\\alpha\\). Moreover, the value of \\(\\alpha\\) only needs to be moderately large, since the asymptotic covariance terms decrease at rate \\(O(1 / \\alpha^2)\\) as shown in Proposition 3.4."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.638,
+                0.405,
+                0.652
+            ],
+            "angle": 0,
+            "content": "F.3 PROOF OF COROLLARY 3.5"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.661,
+                0.825,
+                0.749
+            ],
+            "angle": 0,
+            "content": "We see that \\(\\mathbf{V}_{\\pmb{\\theta}}^{(3)}(\\alpha) = \\mathbf{V}_{\\pmb{\\theta}}^{(3)}(0)\\) for all \\(\\alpha > 0\\), because the form of \\(\\mathbf{V}_{\\pmb{\\theta}}^{(3)}(\\alpha)\\) in Theorem 3.3 is independent of \\(\\alpha\\). To prove that \\(\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(\\alpha) <_{L} \\mathbf{V}_{\\pmb{\\theta}}^{(3)}(0)\\), it is enough to show that \\(\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(0) = \\mathbf{V}_{\\pmb{\\theta}}^{(3)}(0)\\), since \\(\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(\\alpha) <_{L} \\mathbf{V}_{\\pmb{\\theta}}^{(1)}(0)\\) from Proposition 3.4. This is easily checked by substituting \\(\\alpha = 0\\) in 11, for which \\(\\mathbf{U}_{\\pmb{\\theta}}(0) = \\mathbf{U}_{11}\\). Substituting in the respective forms of \\(\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(0)\\) and \\(\\mathbf{V}_{\\pmb{\\theta}}^{(3)}(0)\\) in Theorem 3.3, we get equivalence. This completes the proof."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.77,
+                0.409,
+                0.785
+            ],
+            "angle": 0,
+            "content": "G BACKGROUND THEORY"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.803,
+                0.37,
+                0.817
+            ],
+            "angle": 0,
+            "content": "G.1 TECHNICAL LEMMAS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.83,
+                0.825,
+                0.86
+            ],
+            "angle": 0,
+            "content": "Lemma G.1 (Solution to the Lyapunov Equation). If all the eigenvalues of matrix \\(\\mathbf{M}\\) have negative real part, then for every positive semi-definite matrix \\(\\mathbf{U}\\) there exists a unique positive semi-definite"
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.171,
+                0.871,
+                0.825,
+                0.908
+            ],
+            "angle": 0,
+            "content": "<sup>13</sup>The inequality may not be strict when \\( H \\) is low rank, however it will always be true for some choice of \\( \\mathbf{x} \\), since \\( \\mathbf{H} \\) is not a zero matrix. Thus, the ordering derived still follows our definition of \\( <_{L} \\) in Section 1, footnote 6."
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.188,
+                0.91,
+                0.624,
+                0.925
+            ],
+            "angle": 0,
+            "content": "\\(^{14}D\\)-dimensional vector of all zeros except at the \\(i\\)'th position which is 1."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.171,
+                0.871,
+                0.825,
+                0.925
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "37"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.104,
+                0.825,
+                0.131
+            ],
+            "angle": 0,
+            "content": "matrix \\(\\mathbf{V}\\) satisfying the Lyapunov equation \\(\\mathbf{U} + \\mathbf{M}\\mathbf{V} + \\mathbf{V}\\mathbf{M}^T = \\mathbf{0}\\). The explicit solution \\(\\mathbf{V}\\) is given as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.408,
+                0.13,
+                0.826,
+                0.162
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {V} = \\int_ {0} ^ {\\infty} e ^ {\\mathbf {M} t} \\mathbf {U} e ^ {(\\mathbf {M} ^ {T}) t} d t. \\tag {83}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.17,
+                0.825,
+                0.24
+            ],
+            "angle": 0,
+            "content": "Chellaboina & Haddad (2008, Theorem 3.16) states that for a positive definite matrix \\(\\mathbf{U}\\), there exists a positive definite matrix \\(\\mathbf{V}\\). The reason they focus on the positive definite matrix \\(\\mathbf{U}\\) is that they require the related autonomous ODE system to be asymptotically stable. However, in this paper we don't need this requirement. The same steps therein can be used to prove Lemma G.1 and show that if \\(\\mathbf{U}\\) is positive semi-definite, then \\(\\mathbf{V}\\) in the form of (83) is unique and also positive semi-definite."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.242,
+                0.825,
+                0.273
+            ],
+            "angle": 0,
+            "content": "Lemma G.2 (Burkholder Inequality, Davis (1970), Hall et al. (2014) Theorem 2.10). Given a Martingale difference sequence \\(\\{M_{i,n}\\}_{i = 1}^{n}\\), for \\(p\\geq 1\\) and some positive constant \\(C_p\\), we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.333,
+                0.276,
+                0.826,
+                0.326
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbb {E} \\left[ \\left\\| \\sum_ {i = 1} ^ {n} M _ {i, n} \\right\\| ^ {p} \\right] \\leq C _ {p} \\mathbb {E} \\left[ \\left(\\sum_ {i = 1} ^ {n} \\| M _ {i, n} \\| ^ {2}\\right) ^ {p / 2} \\right] \\tag {84}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.328,
+                0.825,
+                0.358
+            ],
+            "angle": 0,
+            "content": "Theorem G.3 (Martingale CLT, Delyon (2000) Theorem 30). If a Martingale difference array \\(\\{X_{n,i}\\}\\) satisfies the following condition: for some \\(\\tau > 0\\),"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.391,
+                0.363,
+                0.826,
+                0.402
+            ],
+            "angle": 0,
+            "content": "\\[\n\\sum_ {k = 1} ^ {n} \\mathbb {E} \\left[ \\| X _ {n, k} \\| ^ {2 + \\tau} \\mid \\mathcal {F} _ {k - 1} \\right] \\xrightarrow {\\mathbb {P}} 0, \\tag {85}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.385,
+                0.408,
+                0.826,
+                0.448
+            ],
+            "angle": 0,
+            "content": "\\[\n\\sup  _ {n} \\sum_ {k = 1} ^ {n} \\mathbb {E} \\left[ \\| X _ {n, k} \\| ^ {2} \\mid \\mathcal {F} _ {k - 1} \\right] <   \\infty , \\tag {86}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.448,
+                0.204,
+                0.461
+            ],
+            "angle": 0,
+            "content": "and"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.391,
+                0.459,
+                0.826,
+                0.498
+            ],
+            "angle": 0,
+            "content": "\\[\n\\sum_ {k = 1} ^ {n} \\mathbb {E} \\left[ X _ {n, k} X _ {n, k} ^ {T} \\mid \\mathcal {F} _ {k - 1} \\right] \\xrightarrow {\\mathbb {P}} \\boldsymbol {V}, \\tag {87}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.501,
+                0.206,
+                0.513
+            ],
+            "angle": 0,
+            "content": "then"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.413,
+                0.511,
+                0.825,
+                0.55
+            ],
+            "angle": 0,
+            "content": "\\[\n\\sum_ {i = 1} ^ {n} X _ {n, i} \\xrightarrow {\\text {d i s t .}} N (0, V). \\tag {88}\n\\]"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.552,
+                0.825,
+                0.563
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.568,
+                0.825,
+                0.597
+            ],
+            "angle": 0,
+            "content": "Lemma G.4 (Duflo (1996) Proposition 3.I.2). For a Hurwitz matrix \\( \\mathbf{H} \\), there exist some positive constants \\( C, b \\) such that for any \\( n \\),"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.438,
+                0.595,
+                0.825,
+                0.614
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left\\| e ^ {\\mathbf {H} n} \\right\\| \\leq C e ^ {- b n}. \\tag {89}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.615,
+                0.825,
+                0.658
+            ],
+            "angle": 0,
+            "content": "Lemma G.5 (Fort (2015) Lemma 5.8). For a Hurwitz matrix \\( \\mathbf{A} \\), denote by \\( -r \\), \\( r > 0 \\), the largest real part of its eigenvalues. Let a positive sequence \\( \\{\\gamma_n\\} \\) such that \\( \\lim_{n} \\gamma_{n} = 0 \\). Then for any \\( 0 < r' < r \\), there exists a positive constant \\( C \\) such that for any \\( k < n \\),"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.379,
+                0.661,
+                0.825,
+                0.71
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left\\| \\prod_ {j = k} ^ {n} (\\boldsymbol {I} + \\gamma_ {j} \\boldsymbol {A}) \\right\\| \\leq C e ^ {- r ^ {\\prime} \\sum_ {j = k} ^ {n} \\gamma_ {j}}. \\tag {90}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.714,
+                0.825,
+                0.758
+            ],
+            "angle": 0,
+            "content": "Lemma G.6 (Fort (2015) Lemma 5.9, Mokkadem & Pelletier (2006) Lemma 10). Let \\(\\{\\gamma_n\\}\\) be a positive sequence such that \\(\\lim_{n} \\gamma_{n} = 0\\) and \\(\\sum_{n} \\gamma_{n} = \\infty\\). Let \\(\\{\\epsilon_{n}, n \\geq 0\\}\\) be a nonnegative sequence. Then, for \\(b > 0\\), \\(p \\geq 0\\),"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.299,
+                0.762,
+                0.825,
+                0.801
+            ],
+            "angle": 0,
+            "content": "\\[\n\\lim  _ {n} \\sup  _ {n} \\gamma_ {n} ^ {- p} \\sum_ {k = 1} ^ {n} \\gamma_ {k} ^ {p + 1} e ^ {- b \\sum_ {j = k + 1} ^ {n} \\gamma_ {j}} \\epsilon_ {k} \\leq \\frac {1}{C (b , p)} \\lim  _ {n} \\sup  _ {n} \\epsilon_ {n} \\tag {91}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.169,
+                0.805,
+                0.381,
+                0.821
+            ],
+            "angle": 0,
+            "content": "for some constant \\(C(b,p) > 0\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.826,
+                0.785,
+                0.843
+            ],
+            "angle": 0,
+            "content": "When \\( p = 0 \\) and define a positive sequence \\( \\{w_{n}\\} \\) satisfying \\( w_{n - 1} / w_{n} = 1 + o(\\gamma_{n}) \\), we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.31,
+                0.847,
+                0.825,
+                0.886
+            ],
+            "angle": 0,
+            "content": "\\[\n\\sum_ {k = 1} ^ {n} \\gamma_ {k} e ^ {- b \\sum_ {j = k + 1} ^ {n} \\gamma_ {j}} \\epsilon_ {k} = \\left\\{ \\begin{array}{l l} O (w _ {n}), & \\text {i f} \\epsilon_ {n} = O (w _ {n}), \\\\ o (w _ {n}), & \\text {i f} \\epsilon_ {n} = o (w _ {n}). \\end{array} \\right. \\tag {92}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.89,
+                0.619,
+                0.905
+            ],
+            "angle": 0,
+            "content": "Lemma G.7 (Fort (2015) Lemma 5.10). For any matrices \\(A, B, C\\),"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.326,
+                0.908,
+                0.825,
+                0.927
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left\\| A B A ^ {T} - C B C ^ {T} \\right\\| \\leq \\| A - C \\| \\| B \\| \\left(\\| A \\| + \\| C \\|\\right). \\tag {93}\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "38"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.104,
+                0.574,
+                0.119
+            ],
+            "angle": 0,
+            "content": "G.2 ASYMPTOTIC RESULTS OF SINGLE-TIMESCALE SA"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.13,
+                0.525,
+                0.145
+            ],
+            "angle": 0,
+            "content": "Consider the stochastic approximation in the form of"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.384,
+                0.149,
+                0.825,
+                0.165
+            ],
+            "angle": 0,
+            "content": "\\[\n\\boldsymbol {z} _ {n + 1} = \\boldsymbol {z} _ {n} + \\gamma_ {n + 1} G \\left(\\boldsymbol {z} _ {n}, X _ {n + 1}\\right). \\tag {94}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.168,
+                0.826,
+                0.216
+            ],
+            "angle": 0,
+            "content": "Let \\(\\mathbf{K}_z\\) be the transition kernel of the underlying Markov chain \\(\\{X_n\\}_{n\\geq 0}\\) with stationary distribution \\(\\pi (z)\\) such that \\(g(z)\\triangleq \\mathbb{E}_{X\\sim \\pi (z)}[G(z,X)]\\) with domain \\(\\mathcal{O}\\subseteq \\mathbb{R}^d\\). Define an operator \\(\\mathbf{K}_zf\\) for any function \\(f:\\mathcal{N}\\to \\mathbb{R}^{D}\\) such that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.393,
+                0.22,
+                0.826,
+                0.253
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left(\\mathbf {K} _ {\\boldsymbol {z}} f\\right) (i) = \\sum_ {j \\in \\mathcal {N}} f (j) \\mathbf {K} _ {\\boldsymbol {z}} (i, j). \\tag {95}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.257,
+                0.261,
+                0.27
+            ],
+            "angle": 0,
+            "content": "Assume that"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.281,
+                0.605,
+                0.298
+            ],
+            "angle": 0,
+            "content": "C1. W.p.1, the closure of \\(\\{z_{n}\\}_{n\\geq 0}\\) is a compact subset of \\(\\mathcal{O}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.199,
+                0.299,
+                0.407,
+                0.315
+            ],
+            "angle": 0,
+            "content": "C2. \\(\\gamma_{n} = \\gamma_{0} / n^{a}, a\\in (1 / 2,1]\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.317,
+                0.825,
+                0.345
+            ],
+            "angle": 0,
+            "content": "C3. Function \\( g \\) is continuous on \\( \\mathcal{O} \\) and there exists a non-negative \\( C^1 \\) function \\( w \\) and a compact set \\( \\mathcal{K} \\subset \\mathcal{O} \\) such that"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.198,
+                0.281,
+                0.825,
+                0.345
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.248,
+                0.349,
+                0.694,
+                0.365
+            ],
+            "angle": 0,
+            "content": "- \\(\\nabla w(\\boldsymbol{z})^T g(\\boldsymbol{z}) \\leq 0\\) for all \\(\\boldsymbol{z} \\in \\mathcal{O}\\) and \\(\\nabla w(\\boldsymbol{z})^T g(\\boldsymbol{z}) < 0\\) if \\(\\boldsymbol{z} \\notin \\mathcal{K}\\);"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.248,
+                0.367,
+                0.769,
+                0.383
+            ],
+            "angle": 0,
+            "content": "- the set \\( S \\triangleq \\{z \\mid \\nabla w(z)^T g(z) = 0\\} \\) is such that \\( w(S) \\) has an empty interior;"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.248,
+                0.349,
+                0.769,
+                0.383
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.386,
+                0.792,
+                0.401
+            ],
+            "angle": 0,
+            "content": "C4. For every \\(z\\), there exists a solution \\(m_z: \\mathcal{N} \\to \\mathbb{R}^d\\) for the following Poisson equation"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.396,
+                0.404,
+                0.825,
+                0.421
+            ],
+            "angle": 0,
+            "content": "\\[\nm _ {z} (i) - \\left(\\mathbf {K} _ {z} m _ {z}\\right) (i) = G (z, i) - g (z) \\tag {96}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.23,
+                0.424,
+                0.518,
+                0.439
+            ],
+            "angle": 0,
+            "content": "for any \\(i\\in \\mathcal{N}\\) ; for any compact set \\(\\mathcal{C}\\subset \\mathcal{O}\\)"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.392,
+                0.442,
+                0.826,
+                0.468
+            ],
+            "angle": 0,
+            "content": "\\[\n\\sup  _ {\\boldsymbol {z} \\in \\mathcal {C}, i \\in \\mathcal {N}} \\| \\left(\\mathbf {K} _ {\\boldsymbol {z}} m _ {\\boldsymbol {z}}\\right) (i) \\| + \\| m _ {\\boldsymbol {z}} (i) \\| <   \\infty \\tag {97}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.23,
+                0.473,
+                0.761,
+                0.488
+            ],
+            "angle": 0,
+            "content": "and there exist a continuous function \\(\\phi_{\\mathcal{C}},\\phi_{\\mathcal{C}}(0) = 0\\) , such that for any \\(\\pmb {z},\\pmb{z}^{\\prime}\\in \\mathcal{C}\\)"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.351,
+                0.491,
+                0.825,
+                0.516
+            ],
+            "angle": 0,
+            "content": "\\[\n\\sup  _ {i \\in \\mathcal {N}} \\| \\left(\\mathbf {K} _ {\\boldsymbol {z}} m _ {\\boldsymbol {z}}\\right) (i) - \\left(\\mathbf {K} _ {\\boldsymbol {z} ^ {\\prime}} m _ {\\boldsymbol {z} ^ {\\prime}}\\right) (i) \\| \\leq \\phi_ {\\mathcal {C}} (\\| \\boldsymbol {z} - \\boldsymbol {z} ^ {\\prime} \\|). \\tag {98}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.522,
+                0.825,
+                0.556
+            ],
+            "angle": 0,
+            "content": "C5. Denote by \\(-r\\) the largest real part of the eigenvalues of the Jacobian matrix \\(\\nabla g(z^{*})\\) and assume \\(r > \\frac{\\mathbb{1}_{\\{a = 1\\}}}{2}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.56,
+                0.807,
+                0.576
+            ],
+            "angle": 0,
+            "content": "C6. For every \\(z\\), there exists a solution \\(Q_{z}:\\mathcal{N}\\to \\mathbb{R}^{d\\times d}\\) for the following Poisson equation"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.198,
+                0.522,
+                0.825,
+                0.576
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.356,
+                0.579,
+                0.825,
+                0.596
+            ],
+            "angle": 0,
+            "content": "\\[\nQ _ {z} (i) - \\left(\\mathbf {K} _ {z} Q _ {z}\\right) (i) = F (z, i) - \\mathbb {E} _ {j \\sim \\pi (z)} [ F (z, j) ] \\tag {99}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.23,
+                0.599,
+                0.376,
+                0.614
+            ],
+            "angle": 0,
+            "content": "for any \\(i\\in \\mathcal{N}\\) , where"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.304,
+                0.618,
+                0.826,
+                0.65
+            ],
+            "angle": 0,
+            "content": "\\[\nF (\\boldsymbol {z}, i) \\triangleq \\sum_ {j \\in \\mathcal {N}} m _ {\\boldsymbol {z}} (j) m _ {\\boldsymbol {z}} (j) ^ {T} \\mathbf {K} _ {\\boldsymbol {z}} (i, j) - \\left(\\mathbf {K} _ {\\boldsymbol {z}} m _ {\\boldsymbol {z}}\\right) (i) \\left(\\mathbf {K} _ {\\boldsymbol {z}} m _ {\\boldsymbol {z}}\\right) (i) ^ {T}. \\tag {100}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.23,
+                0.654,
+                0.421,
+                0.669
+            ],
+            "angle": 0,
+            "content": "For any compact set \\(\\mathcal{C} \\subset \\mathcal{O}\\),"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.394,
+                0.673,
+                0.825,
+                0.698
+            ],
+            "angle": 0,
+            "content": "\\[\n\\sup  _ {\\boldsymbol {z} \\in \\mathcal {C}, i \\in \\mathcal {N}} \\| Q _ {\\boldsymbol {z}} (i) \\| + \\| \\left(\\mathbf {K} _ {\\boldsymbol {z}} Q _ {\\boldsymbol {z}}\\right) (i) \\| <   \\infty \\tag {101}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.23,
+                0.703,
+                0.587,
+                0.718
+            ],
+            "angle": 0,
+            "content": "and there exist \\(p, C_{\\mathcal{C}} > 0\\), such that for any \\(\\mathbf{z}, \\mathbf{z}' \\in \\mathcal{C}\\)"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.355,
+                0.721,
+                0.825,
+                0.745
+            ],
+            "angle": 0,
+            "content": "\\[\n\\sup  _ {i \\in \\mathcal {N}} \\| \\left(\\mathbf {K} _ {\\boldsymbol {z}} Q _ {\\boldsymbol {z}}\\right) (i) - \\left(\\mathbf {K} _ {\\boldsymbol {z} ^ {\\prime}} Q _ {\\boldsymbol {z} ^ {\\prime}}\\right) (i) \\| \\leq C _ {\\mathcal {C}} \\| \\boldsymbol {z} - \\boldsymbol {z} ^ {\\prime} \\| ^ {p}. \\tag {102}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.748,
+                0.825,
+                0.777
+            ],
+            "angle": 0,
+            "content": "Theorem G.8 (Delyon et al. (1999) Theorem 2). Consider (94) and assume C1 - C4. Then, w.p.1, \\(\\lim \\sup_{n}d(\\pmb{z}_n,S) = 0\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.779,
+                0.825,
+                0.809
+            ],
+            "angle": 0,
+            "content": "Theorem G.9 (Fort (2015) Theorem 2.1 & Proposition 4.1). Consider (94) and assume \\( C1 - C6 \\). Then, given the condition that \\( z_{n} \\) converges to one point \\( z^{*} \\in S \\), we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.383,
+                0.812,
+                0.825,
+                0.838
+            ],
+            "angle": 0,
+            "content": "\\[\n\\gamma_ {n} ^ {- 1 / 2} \\left(\\boldsymbol {z} _ {n} - \\boldsymbol {z} ^ {*}\\right) \\xrightarrow [ n \\rightarrow \\infty ]{\\text {d i s t .}} N (0, \\mathbf {V}), \\tag {103}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.842,
+                0.218,
+                0.855
+            ],
+            "angle": 0,
+            "content": "where"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.276,
+                0.851,
+                0.826,
+                0.885
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {V} \\left(\\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I} + \\nabla g \\left(\\boldsymbol {z} ^ {*}\\right) ^ {T}\\right) + \\left(\\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I} + \\nabla g \\left(\\boldsymbol {z} ^ {*}\\right)\\right) \\mathbf {V} + \\mathbf {U} = 0, \\tag {104}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.886,
+                0.204,
+                0.899
+            ],
+            "angle": 0,
+            "content": "and"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.288,
+                0.896,
+                0.826,
+                0.928
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {U} \\triangleq \\sum_ {i \\in \\mathcal {N}} \\mu_ {i} \\left(m _ {\\mathbf {z} ^ {*}} (i) m _ {\\mathbf {z} ^ {*}} (i) ^ {T} - \\left(\\mathbf {K} _ {\\mathbf {z} ^ {*}} m _ {\\mathbf {z} ^ {*}}\\right) (i) \\left(\\mathbf {K} _ {\\mathbf {z} ^ {*}} m _ {\\mathbf {z} ^ {*}}\\right) (i) ^ {T}\\right). \\tag {105}\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.51,
+                0.961
+            ],
+            "angle": 0,
+            "content": "39"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.104,
+                0.557,
+                0.119
+            ],
+            "angle": 0,
+            "content": "G.3 ASYMPTOTIC RESULTS OF TWO-TIMESCALE SA"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.13,
+                0.812,
+                0.146
+            ],
+            "angle": 0,
+            "content": "For the two-timescale SA with iterate-dependent Markov chain, we have the following iterations:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.372,
+                0.152,
+                0.825,
+                0.168
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {z} _ {n + 1} = \\mathbf {z} _ {n} + \\beta_ {n + 1} G _ {1} \\left(\\mathbf {z} _ {n}, \\mathbf {y} _ {n} X _ {n + 1}\\right), \\tag {106a}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.367,
+                0.174,
+                0.825,
+                0.19
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {y} _ {n + 1} = \\mathbf {y} _ {n} + \\gamma_ {n + 1} G _ {2} \\left(\\mathbf {z} _ {n}, \\mathbf {y} _ {n}, X _ {n + 1}\\right), \\tag {106b}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.192,
+                0.507,
+                0.207
+            ],
+            "angle": 0,
+            "content": "with the goal of finding the root \\((\\mathbf{z}^{*},\\mathbf{y}^{*})\\) such that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.196,
+                0.212,
+                0.826,
+                0.231
+            ],
+            "angle": 0,
+            "content": "\\[\ng _ {1} \\left(\\mathbf {z} ^ {*}, \\mathbf {y} ^ {*}\\right) = \\mathbb {E} _ {X \\sim \\boldsymbol {\\mu}} \\left[ G _ {1} \\left(\\mathbf {z} ^ {*}, \\mathbf {y} ^ {*}, X\\right) \\right] = 0, \\quad g _ {2} \\left(\\mathbf {z} ^ {*}, \\mathbf {y} ^ {*}\\right) = \\mathbb {E} _ {X \\sim \\boldsymbol {\\mu}} \\left[ G _ {2} \\left(\\mathbf {z} ^ {*}, \\mathbf {y} ^ {*}, X\\right) \\right] = 0. \\tag {107}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.242,
+                0.827,
+                0.286
+            ],
+            "angle": 0,
+            "content": "We present here a simplified version of the assumptions for single-valued functions \\( G_{1}, G_{2} \\) that are necessary for the almost sure convergence result in Yaji & Bhatnagar (2020, Theorem 4). The original assumptions are intended for more general set-valued functions \\( G_{1}, G_{2} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.296,
+                0.662,
+                0.312
+            ],
+            "angle": 0,
+            "content": "(B1) The step sizes \\(\\beta_{n} \\triangleq n^{-b}\\) and \\(\\gamma_{n} \\triangleq n^{-a}\\), where \\(0.5 < a < b \\leq 1\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.316,
+                0.825,
+                0.358
+            ],
+            "angle": 0,
+            "content": "(B2) Assume the function \\( G_{1}(\\mathbf{z},\\mathbf{y},X) \\) is continuous and differentiable with respect to \\( \\mathbf{z},\\mathbf{y} \\). There exists a positive constant \\( L_{1} \\) such that \\( \\| G_{1}(\\mathbf{z},\\mathbf{y},X)\\| \\leq L_{1}(1 + \\| \\mathbf{z}\\| +\\| \\mathbf{y}\\|) \\) for every \\( \\mathbf{z}\\in \\mathbb{R}^{d_1},\\mathbf{y}\\in \\mathbb{R}^{d_2},X\\in \\mathcal{N} \\). The same condition holds for the function \\( G_{2} \\) as well."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.362,
+                0.825,
+                0.436
+            ],
+            "angle": 0,
+            "content": "(B3) Assume there exists a function \\(\\rho : \\mathbb{R}^{d_1} \\to \\mathbb{R}^{d_2}\\) such that the following three properties hold: (i) \\(\\| \\rho(\\mathbf{z}) \\| \\leq L_2(1 + \\| \\mathbf{z} \\|)\\) for some positive constant \\(L_2\\); (ii) the ODE \\(\\dot{\\mathbf{y}} = g_2(\\mathbf{z}, \\mathbf{y})\\) has a globally asymptotically stable equilibrium \\(\\lambda(\\mathbf{z})\\) such that \\(g_2(\\mathbf{z}, \\rho(\\mathbf{z})) = 0\\). Additionally, let \\(\\hat{g}_1(\\mathbf{z}) \\triangleq g_1(\\mathbf{z}, \\rho(\\mathbf{z}))\\), there exists a set of disjoint roots \\(\\Lambda \\triangleq \\{\\mathbf{z}^* : \\hat{g}_1(\\mathbf{z}^*) = 0\\}\\), which is the set of globally asymptotically stable equilibria of the ODE \\(\\dot{\\mathbf{z}} = \\hat{g}_1(\\mathbf{z})\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.439,
+                0.825,
+                0.51
+            ],
+            "angle": 0,
+            "content": "(B4) \\(\\{X_{n}\\}_{n\\geq 0}\\) is an iterate-dependent Markov process in finite state space \\(\\mathcal{N}\\). For every \\(n\\geq 0\\), \\(P(X_{n + 1} = j|\\mathbf{z}_m,\\mathbf{y}_m,X_m,0\\leq m\\leq n) = P(X_{n + 1} = j|\\mathbf{z}_n,\\mathbf{y}_n,X_n = i) = \\mathbf{P}_{i,j}[\\mathbf{z}_n,\\mathbf{y}_n]\\), where the transition kernel \\(\\mathbf{P}[\\mathbf{z},\\mathbf{y}]\\) is continuous in \\(\\mathbf{z},\\mathbf{y}\\), and the Markov chain generated by \\(\\mathbf{P}[\\mathbf{z},\\mathbf{y}]\\) is ergodic so that it admits a stationary distribution \\(\\pi (\\mathbf{z},\\mathbf{y})\\), and \\(\\pi (\\mathbf{z}^{*},\\rho (\\mathbf{z}^{*})) = \\pmb{\\mu}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.193,
+                0.513,
+                0.451,
+                0.53
+            ],
+            "angle": 0,
+            "content": "(B5) \\(\\sup_{n\\geq 0}\\bigl (\\| \\mathbf{z}_n\\| +\\| \\mathbf{y}_n\\| \\bigr) <   \\infty\\) a.s."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.192,
+                0.296,
+                0.825,
+                0.53
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.539,
+                0.827,
+                0.61
+            ],
+            "angle": 0,
+            "content": "Yaji & Bhatnagar (2020) included assumptions A1 - A9 and A11 for the following Theorem G.10. We briefly show the correspondence of our assumptions (B1) - (B5) and theirs: (B1) with A5, (B2) with A1 and A2, (B3) with A9 and A11, (B4) with A3 and A4, and (B5) with A8. Given that our two-timescale SA framework (106) excludes additional noises (setting them to zero), A6 and A7 therein are inherently met."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.613,
+                0.825,
+                0.644
+            ],
+            "angle": 0,
+            "content": "Theorem G.10 (Yaji & Bhatnagar (2020) Theorem 4). Under Assumptions (B1) - (B5), iterations \\((\\mathbf{z}_n, \\mathbf{y}_n)\\) in (106) almost surely converge to a set of roots, i.e., \\((\\mathbf{z}_n, \\mathbf{y}_n) \\to \\bigcup_{\\mathbf{z}^* \\in \\Lambda} (\\mathbf{z}^*, \\rho(\\mathbf{z}^*))\\) a.s."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.662,
+                0.514,
+                0.677
+            ],
+            "angle": 0,
+            "content": "H ADDITIONAL SIMULATION RESULTS"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.693,
+                0.592,
+                0.707
+            ],
+            "angle": 0,
+            "content": "H.1 BINARY CLASSIFICATION ON ADDITIONAL DATSETS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.719,
+                0.825,
+                0.79
+            ],
+            "angle": 0,
+            "content": "In this part, we perform the binary classification task as in Section 4 on additional datasets, i.e., \\( a9a \\) (with 123 features) and splice (with 60 features) from LIBSVM (Chang & Lin, 2011). Figure 4 provides the performance ordering of different \\( \\alpha \\) values, and we empirically demonstrate that the curves with \\( \\alpha \\geq 5 \\) still outperform the i.i.d. counterpart. Additionally, Figure 5 compare cases (i) - (iii) under both \\( a9a \\) and splice datasets, and case (i) consistently perform the best."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.806,
+                0.472,
+                0.82
+            ],
+            "angle": 0,
+            "content": "H.2 NON-CONVEX LINEAR REGRESSION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.832,
+                0.825,
+                0.874
+            ],
+            "angle": 0,
+            "content": "We further test SGD-SRRW and SHB-SRRW algorithms with a non-convex function to demonstrate the efficiency of our SA-SRRW algorithm beyond the convex setting. In this task, we simulate the following linear regression problem in Khaled & Richtárik (2023) with non-convex regularization"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.342,
+                0.88,
+                0.825,
+                0.929
+            ],
+            "angle": 0,
+            "content": "\\[\n\\min  _ {\\boldsymbol {\\theta} \\in \\mathbb {R} ^ {d}} \\left\\{f (\\boldsymbol {\\theta}) \\triangleq \\frac {1}{N} \\sum_ {i = 1} ^ {N} l _ {i} (\\boldsymbol {\\theta}) + \\kappa \\sum_ {j = 1} ^ {d} \\frac {\\boldsymbol {\\theta} _ {j} ^ {2}}{\\boldsymbol {\\theta} _ {j} ^ {2} + 1} \\right\\} \\tag {108}\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.51,
+                0.96
+            ],
+            "angle": 0,
+            "content": "40"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.172,
+                0.101,
+                0.49,
+                0.274
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.268,
+                0.278,
+                0.396,
+                0.292
+            ],
+            "angle": 0,
+            "content": "(a) SGD-SRRW, \\(a9a\\)"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.507,
+                0.101,
+                0.828,
+                0.274
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.597,
+                0.278,
+                0.738,
+                0.293
+            ],
+            "angle": 0,
+            "content": "(b) SGD-SRRW, splice"
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.244,
+                0.303,
+                0.751,
+                0.319
+            ],
+            "angle": 0,
+            "content": "Figure 4: Simulation results with various \\(\\alpha\\) values in \\(a9a\\) and splice datasets."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.172,
+                0.333,
+                0.387,
+                0.442
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.194,
+                0.445,
+                0.366,
+                0.459
+            ],
+            "angle": 0,
+            "content": "(a) \\(\\alpha = 5\\) ,SGD-SRRW, \\(a9a\\)"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.393,
+                0.332,
+                0.607,
+                0.441
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.409,
+                0.445,
+                0.591,
+                0.459
+            ],
+            "angle": 0,
+            "content": "(b) \\(\\alpha = 10\\) ,SGD-SRRW, \\(a9a\\)"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.613,
+                0.332,
+                0.826,
+                0.44
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.629,
+                0.445,
+                0.808,
+                0.459
+            ],
+            "angle": 0,
+            "content": "(c) \\(\\alpha = 20\\) ,SGD-SRRW, \\(a9a\\)"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.172,
+                0.46,
+                0.387,
+                0.566
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.187,
+                0.571,
+                0.372,
+                0.586
+            ],
+            "angle": 0,
+            "content": "(d) \\(\\alpha = 5\\) ,SGD-SRRW, splice"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.393,
+                0.46,
+                0.606,
+                0.567
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.404,
+                0.571,
+                0.595,
+                0.586
+            ],
+            "angle": 0,
+            "content": "(e) \\(\\alpha = 10\\) ,SGD-SRRW,splice"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.613,
+                0.46,
+                0.826,
+                0.566
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.625,
+                0.571,
+                0.814,
+                0.586
+            ],
+            "angle": 0,
+            "content": "(f) \\(\\alpha = 20\\) ,SGD-SRRW,splice"
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.171,
+                0.596,
+                0.825,
+                0.625
+            ],
+            "angle": 0,
+            "content": "Figure 5: Performance comparison among cases (i) - (iii) for \\(\\alpha \\in \\{5,10,20\\}\\) in \\(a9a\\) and splice datasets."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.651,
+                0.825,
+                0.695
+            ],
+            "angle": 0,
+            "content": "where the loss function \\( l_{i}(\\pmb{\\theta}) = \\|\\mathbf{s}_{i}^{T}\\pmb{\\theta} - y_{i}\\|^{2} \\) and \\( \\kappa = 1 \\), with the data points \\( \\{(\\mathbf{s}_i,y_i)\\}_{i\\in \\mathcal{N}} \\) from the ijCNN1 dataset of LIBVIM (Chang & Lin, 2011). We still perform the optimization over the wikiVote graph, as done in Section 4."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.701,
+                0.827,
+                0.8
+            ],
+            "angle": 0,
+            "content": "The numerical results for the non-convex linear regression taks are presented in Figures 6 and 7, where each experiment is repeated 100 times. Figures 6a and 6b show that the performance ordering across different \\(\\alpha\\) values is still preserved for both algorithms over almost all time, and curves for \\(\\alpha \\geq 5\\) outperform that of the i.i.d. sampling (in black) under the graph topological constraints. Additionally, among the three cases examined at identical \\(\\alpha\\) values, Figures 7a - 7c confirm that case (i) performs consistently better than the other two cases, implying that case (i) can even become the best choice for non-convex distributed optimization tasks."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.489,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "41"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.172,
+                0.21,
+                0.49,
+                0.382
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.282,
+                0.388,
+                0.38,
+                0.401
+            ],
+            "angle": 0,
+            "content": "(a) SGD-SRRW"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.508,
+                0.212,
+                0.825,
+                0.384
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.619,
+                0.388,
+                0.715,
+                0.401
+            ],
+            "angle": 0,
+            "content": "(b) SHB-SRRW"
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.176,
+                0.413,
+                0.82,
+                0.429
+            ],
+            "angle": 0,
+            "content": "Figure 6: Simulation results for non-convex linear regression under case (i) with various \\(\\alpha\\) values."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.173,
+                0.658,
+                0.387,
+                0.765
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.208,
+                0.77,
+                0.35,
+                0.784
+            ],
+            "angle": 0,
+            "content": "(a) \\(\\alpha = 5\\) ,SGD-SRRW"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.395,
+                0.659,
+                0.605,
+                0.765
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.424,
+                0.77,
+                0.575,
+                0.784
+            ],
+            "angle": 0,
+            "content": "(b) \\(\\alpha = 10\\) ,SGD-SRRW"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.614,
+                0.658,
+                0.825,
+                0.765
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.645,
+                0.77,
+                0.794,
+                0.784
+            ],
+            "angle": 0,
+            "content": "(c) \\(\\alpha = 20\\) ,SGD-SRRW"
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.22,
+                0.795,
+                0.776,
+                0.812
+            ],
+            "angle": 0,
+            "content": "Figure 7: Performance comparison among cases (i) - (iii) for non-convex regression."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.489,
+                0.948,
+                0.51,
+                0.96
+            ],
+            "angle": 0,
+            "content": "42"
+        }
+    ]
+]
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+# ACCELERATING DISTRIBUTED STOCHASTIC OPTIMIZATION VIA SELF-REPELLENT RANDOM WALKS
+
+Jie Hu\*1, Vishwaraj Doshi\*2, Do Young Eun
+
+$^{1}$ North Carolina State University,  $^{2}$ IQVIA Inc.
+
+{jhu29, dyeun}@ncsu.edu, vishwaraj.doshi@iqvia.com
+
+# ABSTRACT
+
+We study a family of distributed stochastic optimization algorithms where gradients are sampled by a token traversing a network of agents in random-walk fashion. Typically, these random-walks are chosen to be Markov chains that asymptotically sample from a desired target distribution, and play a critical role in the convergence of the optimization iterates. In this paper, we take a novel approach by replacing the standard linear Markovian token by one which follows a nonlinear Markov chain - namely the Self-Repellent Radom Walk (SRRW). Defined for any given 'base' Markov chain, the SRRW, parameterized by a positive scalar  $\alpha$ , is less likely to transition to states that were highly visited in the past, thus the name. In the context of MCMC sampling on a graph, a recent breakthrough in Doshi et al. (2023) shows that the SRRW achieves  $O(1 / \alpha)$  decrease in the asymptotic variance for sampling. We propose the use of a 'generalized' version of the SRRW to drive token algorithms for distributed stochastic optimization in the form of stochastic approximation, termed SA-SRRW. We prove that the optimization iterate errors of the resulting SA-SRRW converge to zero almost surely and prove a central limit theorem, deriving the explicit form of the resulting asymptotic covariance matrix corresponding to iterate errors. This asymptotic covariance is always smaller than that of an algorithm driven by the base Markov chain and decreases at rate  $O(1 / \alpha^2)$  - the performance benefit of using SRRW thereby amplified in the stochastic optimization context. Empirical results support our theoretical findings.
+
+# 1 INTRODUCTION
+
+Stochastic optimization algorithms solve optimization problems of the form
+
+$$
+\boldsymbol {\theta} ^ {*} \in \underset {\boldsymbol {\theta} \in \mathbb {R} ^ {d}} {\arg \min } f (\boldsymbol {\theta}), \quad \text {w h e r e} f (\boldsymbol {\theta}) \triangleq \mathbb {E} _ {X \sim \boldsymbol {\mu}} [ F (\boldsymbol {\theta}, X) ] = \sum_ {i \in \mathcal {N}} \mu_ {i} F (\boldsymbol {\theta}, i), \tag {1}
+$$
+
+with the objective function  $f: \mathbb{R}^d \to \mathbb{R}$  and  $X$  taking values in a finite state space  $\mathcal{N}$  with distribution  $\mu \triangleq [\mu_i]_{i \in \mathcal{N}}$ . Leveraging partial gradient information per iteration, these algorithms have been recognized for their scalability and efficiency with large datasets (Bottou et al., 2018; Even, 2023). For any given noise sequence  $\{X_n\}_{n \geq 0} \subset \mathcal{N}$ , and step size sequence  $\{\beta_n\}_{n \geq 0} \subset \mathbb{R}_+$ , most stochastic optimization algorithms can be classified as stochastic approximations (SA) of the form
+
+$$
+\boldsymbol {\theta} _ {n + 1} = \boldsymbol {\theta} _ {n} + \beta_ {n + 1} H \left(\boldsymbol {\theta} _ {n}, X _ {n + 1}\right), \quad \forall n \geq 0, \tag {2}
+$$
+
+where, roughly speaking,  $H(\pmb{\theta}, i)$  contains gradient information  $\nabla_{\pmb{\theta}} F(\pmb{\theta}, i)$ , such that  $\pmb{\theta}^*$  solves  $\mathbf{h}(\pmb{\theta}) \triangleq \mathbb{E}_{X \sim \pmb{\mu}}[H(\pmb{\theta}, X)] = \sum_{i \in \mathcal{N}} \mu_i H(\pmb{\theta}, i) = \mathbf{0}$ . Such SA iterations include the well-known stochastic gradient descent (SGD), stochastic heavy ball (SHB) (Gadat et al., 2018; Li et al., 2022), and some SGD-type algorithms employing additional auxiliary variables (Barakat et al., 2021). These algorithms typically have the stochastic noise term  $X_n$  generated by i.i.d. random variables with probability distribution  $\pmb{\mu}$  in each iteration. In this paper, we study a stochastic optimization algorithm where the noise sequence governing access to the gradient information is generated from general stochastic processes in place of i.i.d. random variables.
+
+This is commonly the case in distributed learning, where  $\{X_{n}\}$  is a (typically Markovian) random walk, and should asymptotically be able to sample the gradients from the desired probability distribution  $\mu$ . This is equivalent to saying that the random walker's empirical distribution converges to  $\mu$  almost surely (a.s.); that is,  $\mathbf{x}_n \triangleq \frac{1}{n+1} \sum_{k=0}^{n} \delta_{X_k} \xrightarrow[n\to\infty]{a.s.} \mu$  for any initial  $X_0 \in \mathcal{N}$ , where  $\delta_{X_k}$  is the delta measure whose  $X_k$ 'th entry is one, the rest being zero. Such convergence is most commonly achieved by employing the Metropolis Hastings random walk (MHRW) which can be designed to sample from any target measure  $\mu$  and implemented in a scalable manner (Sun et al., 2018). Unsurprisingly, convergence characteristics of the employed Markov chain affect that of the SA sequence (2), and appear in both finite-time and asymptotic analyses. Finite-time bounds typically involve the second largest eigenvalue in modulus (SLEM) of the Markov chain's transition kernel  $\mathbf{P}$ , which is critically connected to the mixing time of a Markov chain (Levin & Peres, 2017); whereas asymptotic results such as central limit theorems (CLT) involve asymptotic covariance matrices that embed information regarding the entire spectrum of  $\mathbf{P}$ , i.e., all eigenvalues as well as eigenvectors (Bremaud, 2013), which are key to understanding the sampling efficiency of a Markov chain. Thus, the choice of random walker can significantly impact the performance of (2), and simply ensuring that it samples from  $\mu$  asymptotically is not enough to achieve optimal algorithmic performance. In this paper, we take a closer look at the distributed stochastic optimization problem through the lens of a non-linear Markov chain, known as the Self Repellent Random Walk (SRRW), which was shown in Doshi et al. (2023) to achieve asymptotically minimal sampling variance for large values of  $\alpha$ , a positive scalar controlling the strength of the random walker's self-repellence behaviour. Our proposed modification of (2) can be implemented within the settings of decentralized learning applications in a scalable manner, while also enjoying significant performance benefit over distributed stochastic optimization algorithms driven by vanilla Markov chains.
+
+Token Algorithms for Decentralized Learning. In decentralized learning, agents like smartphones or IoT devices, each containing a subset of data, collaboratively train models on a graph  $\mathcal{G}(\mathcal{N},\mathcal{E})$  by sharing information locally without a central server (McMahan et al., 2017). In this setup,  $N = |\mathcal{N}|$  agents correspond to nodes  $i\in \mathcal{N}$ , and an edge  $(i,j)\in \mathcal{E}$  indicates direct communication between agents  $i$  and  $j$ . This decentralized approach offers several advantages compared to the traditional centralized learning setting, promoting data privacy and security by eliminating the need for raw data to be aggregated centrally and thus reducing the risk of data breach or misuse (Bottou et al., 2018; Nedic, 2020). Additionally, decentralized approaches are more scalable and can handle vast amounts of heterogeneous data from distributed agents without overwhelming a central server, alleviating concerns about single point of failure (Vogels et al., 2021).
+
+Among decentralized learning approaches, the class of 'Token' algorithms can be expressed as stochastic approximation iterations of the type (2), wherein the sequence  $\{X_{n}\}$  is realized as the sample path of a token that stochastically traverses the graph  $\mathcal{G}$ , carrying with it the iterate  $\theta_{n}$  for any time  $n\geq 0$  and allowing each visited node (agent) to incrementally update  $\theta_{n}$  using locally available gradient information. Token algorithms have gained popularity in recent years (Hu et al., 2022; Triastcyn et al., 2022; Hendrikx, 2023), and are provably more communication efficient (Even, 2023) when compared to consensus-based algorithms - another popular approach for solving distributed optimization problems (Boyd et al., 2006; Morral et al., 2017; Olshevsky, 2022). The construction of token algorithms means that they do not suffer from expensive costs of synchronization and communication that are typical of consensus-based approaches, where all agents (or a subset of agents selected by a coordinator (Boyd et al., 2006; Wang et al., 2019)) on the graph are required to take simultaneous actions, such as communicating on the graph at each iteration. While decentralized Federated learning has indeed helped mitigate the communication overhead by processing multiple SGD iterations prior to each aggregation (Lalitha et al., 2018; Ye et al., 2022; Chellapandi et al., 2023), they still cannot overcome challenges such as synchronization and straggler issues.
+
+Self Repellent Random Walk. As mentioned earlier, sample paths  $\{X_{n}\}$  of token algorithms are usually generated using Markov chains with  $\mu \in \mathrm{Int}(\Sigma)$  as their limiting distribution. Here,  $\Sigma$  denotes the  $N$ -dimensional probability simplex, with  $\mathrm{Int}(\Sigma)$  representing its interior. A recent work by Doshi et al. (2023) pioneers the use of non-linear Markov chains to, in some sense, improve upon any given time-reversible Markov chain with transition kernel  $\mathbf{P}$  whose stationary distribution is  $\mu$ .
+
+They show that the non-linear transition kernel $^2$ $\mathbf{K}[\cdot ]:\mathrm{Int}(\Sigma)\to [0,1]^{N\times N}$ , given by
+
+$$
+K _ {i j} [ \mathbf {x} ] \triangleq \frac {P _ {i j} \left(x _ {j} / \mu_ {j}\right) ^ {- \alpha}}{\sum_ {k \in \mathcal {N}} P _ {i k} \left(x _ {k} / \mu_ {k}\right) ^ {- \alpha}}, \quad \forall i, j \in \mathcal {N}, \tag {3}
+$$
+
+for any  $\mathbf{x} \in \mathrm{Int}(\Sigma)$ , when simulated as a self-interacting random walk (Del Moral & Miclo, 2006; Del Moral & Doucet, 2010), can achieve smaller asymptotic variance than the base Markov chain when sampling over a graph  $\mathcal{G}$ , for all  $\alpha > 0$ . The argument  $\mathbf{x}$  for the kernel  $\mathbf{K}[x]$  is taken to be the empirical distribution  $\mathbf{x}_n$  at each time step  $n \geq 0$ . For instance, if node  $j$  has been visited more often than other nodes so far, the entry  $x_j$  becomes larger (than target value  $\mu_j$ ), resulting in a smaller transition probability from  $i$  to  $j$  under  $\mathbf{K}[\mathbf{x}]$  in (3) compared to  $P_{ij}$ . This ensures that a random walker prioritizes more seldom visited nodes in the process, and is thus 'self-repellent'. This effect is made more drastic by increasing  $\alpha$ , and leads to asymptotically near-zero variance at a rate of  $O(1 / \alpha)$ . Moreover, the polynomial function  $(x_i / \mu_i)^{-\alpha}$  chosen to encode self-repellent behaviour is shown in Doshi et al. (2023) to be the only one that allows the SRRW to inherit the so-called 'scale-invariance' property of the underlying Markov chain - a necessary component for the scalable implementation of a random walker over a large network without requiring knowledge of any graph-related global constants. Conclusively, such attributes render SRRW especially suitable for distributed optimization. $^3$
+
+Effect of Stochastic Noise - Finite time and Asymptotic Approaches. Most contemporary token algorithms driven by Markov chains are analyzed using the finite-time bounds approach for obtaining insights into their convergence rates (Sun et al., 2018; Doan et al., 2019; Triastcyn et al., 2022; Hendrikx, 2023). However, as also explained in Even (2023), in most cases these bounds are overly dependent on mixing time properties of the specific Markov chain employed therein. This makes them largely ineffective in capturing the exact contribution of the underlying random walk in a manner which is qualitative enough to be used for algorithm design; and performance enhancements are typically achieved via application of techniques such as variance reduction (Defazio et al., 2014; Schmidt et al., 2017), momentum/Nesterov's acceleration (Gadat et al., 2018; Li et al., 2022), adaptive step size (Kingma & Ba, 2015; Reddi et al., 2018), which work by modifying the algorithm iterations themselves, and never consider potential improvements to the stochastic input itself.
+
+Complement to finite-time approaches, asymptotic analysis using CLT has proven to be an excellent tool to approach the design of stochastic algorithms (Hu et al., 2022; Devraj & Meyn, 2017; Morral et al., 2017; Chen et al., 2020a; Mou et al., 2020; Devraj & Meyn, 2021). Hu et al. (2022) shows how asymptotic analysis can be used to compare the performance of SGD algorithms for various stochastic inputs using their notion of efficiency ordering, and, as mentioned in Devraj & Meyn (2017), the asymptotic benefits from minimizing the limiting covariance matrix are known to be a good predictor of finite-time algorithmic performance, also observed empirically in Section 4.
+
+From the perspective of both finite-time analysis as well as asymptotic analysis of token algorithms, it is now well established that employing 'better' Markov chains can enhance the performance of stochastic optimization algorithm. For instance, Markov chains with smaller SLEMs yield tighter finite-time upper bounds (Sun et al., 2018; Ayache & El Rouayheb, 2021; Even, 2023). Similarly, Markov chains with smaller asymptotic variance for MCMC sampling tasks also provide better performance, resulting in smaller covariance matrix of SGD algorithms (Hu et al., 2022). Therefore, with these breakthrough results via SRRW achieving near-zero sampling variance, it is within reason to ask: Can we achieve near-zero variance in distributed stochastic optimization driven by SRRW-like token algorithms on any general graph?4 In this paper, we answer in the affirmative.
+
+SRRW Driven Algorithm and Analysis Approach. For any ergodic time-reversible Markov chain with transition probability matrix  $\mathbf{P} \triangleq [P_{ij}]_{i,j \in \mathcal{N}}$  and stationary distribution  $\pmb{\mu} \in \mathrm{Int}(\Sigma)$ , we consider a general step size version of the SRRW stochastic process analysed in Doshi et al. (2023) and
+
+![](images/97eb05def3588d5ce2425368f4f27c0629afcf80d345f83e6517c2c14928feea.jpg)  
+Figure 1: Visualization of token algorithms using SRRW versus traditional MC in distributed learning. Our CLT analysis, extended from SRRW itself to distributed stochastic approximation, leads to near-zero variance for the SA iteration  $\theta_{n}$ . Node numbers on the left denote visit counts.
+
+use it to drive the noise sequence in (2). Our SA-SRRW algorithm is as follows:
+
+Draw:  $X_{n + 1}\sim \mathbf{K}_{X_n,\cdot}[\mathbf{x}_n]$  (4a)
+
+Update:  $\mathbf{x}_{n + 1} = \mathbf{x}_n + \gamma_{n + 1}(\pmb{\delta}_{X_{n + 1}} - \mathbf{x}_n),$  (4b)
+
+$$
+\boldsymbol {\theta} _ {n + 1} = \boldsymbol {\theta} _ {n} + \beta_ {n + 1} H \left(\boldsymbol {\theta} _ {n}, X _ {n + 1}\right), \tag {4c}
+$$
+
+where  $\{\beta_n\}$  and  $\{\gamma_n\}$  are step size sequences decreasing to zero, and  $\mathbf{K}[\mathbf{x}]$  is the SRRW kernel in (3). Current non-asymptotic analyses require globally Lipschitz mean field function (Chen et al., 2020b; Doan, 2021; Zeng et al., 2021; Even, 2023) and is thus inapplicable to SA-SRRW since the mean field function of the SRRW iterates (4b) is only locally Lipschitz (details deferred to Appendix B). Instead, we successfully obtain non-trivial results by taking an asymptotic CLT-based approach to analyze (4). This goes beyond just analyzing the asymptotic sampling covariance<sup>5</sup> as in Doshi et al. (2023), the result therein forming a special case of ours by setting  $\gamma_{n} = 1 / (n + 1)$  and considering only (4a) and (4b), that is, in the absence of optimization iteration (4c). Specifically, we capture the effect of SRRW's hyper-parameter  $\alpha$  on the asymptotic speed of convergence of the optimization error term  $\theta_{n} - \theta^{*}$  to zero via explicit deduction of its asymptotic covariance matrix. See Figure 1 for illustration.
+
+# Our Contributions.
+
+1. Given any time-reversible 'base' Markov chain with transition kernel  $\mathbf{P}$  and stationary distribution  $\mu$ , we generalize first and second order convergence results of  $\mathbf{x}_n$  to target measure  $\pmb{\mu}$  (Theorems 4.1 and 4.2 in Doshi et al., 2023) to a class of weighted empirical measures, through the use of more general step sizes  $\gamma_{n}$ . This includes showing that the asymptotic sampling covariance terms decrease to zero at rate  $O(1 / \alpha)$ , thus quantifying the effect of self-repellent on  $\mathbf{x}_n$ . Our generalization is not for the sake thereof and is shown in Section 3 to be crucial for the design of step sizes  $\beta_{n},\gamma_{n}$ .  
+2. Building upon the convergence results for iterates  $\mathbf{x}_n$ , we analyze the algorithm (4) driven by the SRRW kernel in (3) with step sizes  $\beta_{n}$  and  $\gamma_{n}$  separated into three disjoint cases:
+
+(i)  $\beta_{n} = o(\gamma_{n})$ , and we say that  $\theta_{n}$  is on the slower timescale compared to  $\mathbf{x}_n$ ;  
+(ii)  $\beta_{n} = \gamma_{n}$ , and we say that  $\theta_{n}$  and  $\mathbf{x}_n$  are on the same timescale;  
+(iii)  $\gamma_{n} = o(\beta_{n})$ , and we say that  $\theta_{n}$  is on the faster timescale compared to  $\mathbf{x}_n$ .
+
+For any  $\alpha \geq 0$  and let  $k = 1,2$  and 3 refer to the corresponding cases (i), (ii) and (iii), we show that
+
+$$
+\boldsymbol {\theta} _ {n} \xrightarrow [ n \to \infty ]{a . s .} \boldsymbol {\theta} ^ {*} \quad \text {a n d} \quad (\boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*}) / \sqrt {\beta_ {n}} \xrightarrow [ n \to \infty ]{d i s t.} N \left(\mathbf {0}, \mathbf {V} _ {\boldsymbol {\theta}} ^ {(k)} (\alpha)\right),
+$$
+
+featuring distinct asymptotic covariance matrices  $\mathbf{V}_{\theta}^{(1)}(\alpha), \mathbf{V}_{\theta}^{(2)}(\alpha)$  and  $\mathbf{V}_{\theta}^{(3)}(\alpha)$ , respectively. The three matrices coincide when  $\alpha = 0,6$ . Moreover, the derivation of the CLT for cases (i) and (iii), for which (4) corresponds to two-timescale SA with controlled Markov noise, is the first of its kind and thus a key technical contribution in this paper, as expanded upon in Section 3.
+
+3. For case (i), we show that  $\mathbf{V}_{\theta}^{(1)}(\alpha)$  decreases to zero (in the sense of Loewner ordering introduced in Section 2.1) as  $\alpha$  increases, with rate  $O(1 / \alpha^2)$ . This is especially surprising, since the asymptotic performance benefit from using the SRRW kernel with  $\alpha$  in (3), to drive the noise terms  $X_{n}$ , is amplified in the context of distributed learning and estimating  $\theta^{*}$ ; compared to the sampling case, for which the rate is  $O(1 / \alpha)$  as mentioned earlier. For case (iii), we show that  $\mathbf{V}_{\theta}^{(3)}(\alpha) = \mathbf{V}_{\theta}^{(3)}(0)$  for all  $\alpha \geq 0$ , implying that using the SRRW in this case provides no asymptotic benefit than the
+
+original base Markov chain, and thus performs worse than case (i). In summary, we deduce that  $\mathbf{V}_{\pmb{\theta}}^{(1)}(\alpha_2) < _L\mathbf{V}_{\pmb{\theta}}^{(1)}(\alpha_1) < _L\mathbf{V}_{\pmb{\theta}}^{(1)}(0) = \mathbf{V}_{\pmb{\theta}}^{(3)}(0) = \mathbf{V}_{\pmb{\theta}}^{(3)}(\alpha)$  for all  $\alpha_{2} > \alpha_{1} > 0$  and  $\alpha >0$
+
+4. We numerically simulate our SA-SRRW algorithm on various real-world datasets, focusing on a binary classification task, to evaluate its performance across all three cases. By carefully choosing the function  $H$  in SA-SRRW, we test the SGD and algorithms driven by SRRW. Our findings consistently highlight the superiority of case (i) over cases (ii) and (iii) for diverse  $\alpha$  values, even in their finite time performance. Notably, our tests validate the variance reduction at a rate of  $O(1 / \alpha^2)$  for case (i), suggesting it as the best algorithmic choice among the three cases.
+
+# 2 PRELIMINARIES AND MODEL SETUP
+
+In Section 2.1, we first standardize the notations used throughout the paper, and define key mathematical terms and quantities used in our theoretical analyses. Then, in Section 2.2, we consolidate the model assumptions of our SA-SRRW algorithm (4). We then go on to discuss our assumptions, and provide additional interpretations of our use of generalized step-sizes.
+
+# 2.1 BASIC NOTATIONS AND DEFINITIONS
+
+Vectors are denoted by lower-case bold letters, e.g.,  $\mathbf{v} \triangleq [v_i] \in \mathbb{R}^D$ , and matrices by upper-case bold, e.g.,  $\mathbf{M} \triangleq [M_{ij}] \in \mathbb{R}^{D \times D}$ .  $\mathbf{M}^{-T}$  is the transpose of the matrix inverse  $\mathbf{M}^{-1}$ . The diagonal matrix  $\mathbf{D}_{\mathbf{v}}$  is formed by vector  $\mathbf{v}$  with  $v_i$  as the  $i$ 'th diagonal entry. Let  $\mathbf{1}$  and  $\mathbf{0}$  denote vectors of all ones and zeros, respectively. The identity matrix is represented by  $\mathbf{I}$ , with subscripts indicating dimensions as needed. A matrix is Hurwitz if all its eigenvalues possess strictly negative real parts.  $\mathbb{1}_{\{\cdot\}}$  denotes an indicator function with condition in parentheses. We use  $\|\cdot\|$  to denote both the Euclidean norm of vectors and the spectral norm of matrices. Two symmetric matrices  $\mathbf{M}_1, \mathbf{M}_2$  follow Loewner ordering  $\mathbf{M}_1 <_L \mathbf{M}_2$  if  $\mathbf{M}_2 - \mathbf{M}_1$  is positive semi-definite and  $\mathbf{M}_1 \neq \mathbf{M}_2$ . This slightly differs from the conventional definition with  $\leq_L$ , which allows  $\mathbf{M}_1 = \mathbf{M}_2$ .
+
+Throughout the paper, the matrix  $\mathbf{P} \triangleq [P_{i,j}]_{i,j \in \mathcal{N}}$  and vector  $\pmb{\mu} \triangleq [\mu_i]_{i \in \mathcal{N}}$  are used exclusively to denote an  $N \times N$ -dimensional transition kernel of an ergodic Markov chain, and its stationary distribution, respectively. Without loss of generality, we assume  $P_{ij} > 0$  if and only if  $a_{ij} > 0$ . Markov chains satisfying the detailed balance equation, where  $\mu_i P_{ij} = \mu_j P_{ji}$  for all  $i, j \in \mathcal{N}$ , are termed time-reversible. For such chains, we use  $(\lambda_i, \mathbf{u}_i)$  (resp.  $(\lambda_i, \mathbf{v}_i)$ ) to denote the  $i$ 'th left (resp. right) eigenpair where the eigenvalues are ordered:  $-1 < \lambda_1 \leq \dots \leq \lambda_{N-1} < \lambda_N = 1$ , with  $\mathbf{u}_N = \pmb{\mu}$  and  $\mathbf{v}_N = \mathbf{1}$  in  $\mathbb{R}^N$ . We assume eigenvectors to be normalized such that  $\mathbf{u}_i^T \mathbf{v}_i = 1$  for all  $i$ , and we have  $\mathbf{u}_i = \mathbf{D}_{\pmb{\mu}} \mathbf{v}_i$  and  $\mathbf{u}_i^T \mathbf{v}_j = 0$  for all  $i, j \in \mathcal{N}$ . We direct the reader to Aldous & Fill (2002, Chapter 3.4) for a detailed exposition on spectral properties of time-reversible Markov chains.
+
+# 2.2 SA-SRRW: KEY ASSUMPTIONS AND DISCUSSIONS
+
+Assumptions: All results in our paper are proved under the following assumptions.
+
+(A1) The function  $H: \mathbb{R}^D \times \mathcal{N} \to \mathbb{R}^D$ , is a continuous at every  $\pmb{\theta} \in \mathbb{R}^D$ , and there exists a positive constant  $L$  such that  $\| H(\pmb{\theta}, i) \| \leq L (1 + \| \pmb{\theta} \|)$  for every  $\pmb{\theta} \in \mathbb{R}^D, i \in \mathcal{N}$ .  
+(A2) Step sizes  $\beta_{n}$  and  $\gamma_{n}$  follow  $\beta_{n} = (n + 1)^{-b}$ , and  $\gamma_{n} = (n + 1)^{-a}$ , where  $a, b \in (0.5, 1]$ .  
+(A3) Roots of function  $\mathbf{h}(\cdot)$  are disjoint, which comprise the globally attracting set  $\Theta \triangleq \left\{\pmb{\theta}^{*}|\mathbf{h}(\pmb{\theta}^{*}) = \mathbf{0}, \nabla \mathbf{h}(\pmb{\theta}^{*}) + \frac{\mathbb{1}_{\{b=1\}}}{2}\mathbf{I}\text{ is Hurwitz}\right\} \neq \emptyset$  of the associated ordinary differential equation (ODE) for iteration (4c), given by  $d\pmb{\theta}(t)/dt = \mathbf{h}(\pmb{\theta}(t))$ .  
+(A4) For any  $(\pmb{\theta}_0, \mathbf{x}_0, X_0) \in \mathbb{R}^D \times \mathrm{Int}(\Sigma) \times \mathcal{N}$ , the iterate sequence  $\{\pmb{\theta}_n\}_{n \geq 0}$  (resp.  $\{\mathbf{x}_n\}_{n \geq 0}$ ) is  $\mathbb{P}_{\pmb{\theta}_0, \mathbf{x}_0, X_0}$ -almost surely contained within a compact subset of  $\mathbb{R}^D$  (resp.  $\mathrm{Int}(\Sigma)$ ).
+
+Discussions on Assumptions: Assumption A1 requires  $H$  to only be locally Lipschitz albeit with linear growth, and is less stringent than the globally Lipschitz assumption prevalent in optimization literature (Li & Wai, 2022; Hendrikx, 2023; Even, 2023).
+
+Assumption A2 is the general umbrella assumption under which cases (i), (ii) and (iii) mentioned in Section 1 are extracted by setting: (i)  $a < b$ , (ii)  $a = b$ , and (iii)  $a > b$ . Cases (i) and (iii) render  $\theta_{n}, \mathbf{x}_{n}$  on different timescales; the polynomial form of  $\beta_{n}, \gamma_{n}$  widely assumed in the two-timescale SA literature (Mokkadem & Pelletier, 2006; Zeng et al., 2021; Hong et al., 2023). Case (ii) characterizes the SA-SRRW algorithm (4) as a single-timescale SA with polynomially decreasing step size, and is among the most common assumptions in the SA literature (Borkar, 2022; Fort, 2015; Li et al., 2023). In all three cases, the form of  $\gamma_{n}$  ensures  $\gamma_{n} \leq 1$  such that the SRRW iterates  $\mathbf{x}_{n}$  in (4b) is within  $\operatorname{Int}(\Sigma)$ , ensuring that  $\mathbf{K}[\mathbf{x}_n]$  is well-defined for all  $n \geq 0$ .
+
+In Assumption A3, limiting dynamics of SA iterations  $\{\pmb{\theta}_n\}_{n\geq 0}$  closely follow trajectories  $\{\pmb {\theta}(t)\}_{t\geq 0}$  of their associated ODE, and assuming the existence of globally stable equilibria is standard (Borkar, 2022; Fort, 2015; Li et al., 2023). In optimization problems, this is equivalent to assuming the existence of at most countably many local minima.
+
+Assumption A4 assumes almost sure boundedness of iterates  $\theta_{n}$  and  $\mathbf{x}_n$ , which is a common assumption in SA algorithms (Kushner & Yin, 2003; Chen, 2006; Borkar, 2022; Karmakar & Bhatnagar, 2018; Li et al., 2023) for the stability of the SA iterations by ensuring the well-definiteness of all quantities involved. Stability of the weighted empirical measure  $\mathbf{x}_n$  of the SRRW process is practically ensured by studying (4b) with a truncation-based procedure (see Doshi et al., 2023, Remark 4.5 and Appendix E for a comprehensive explanation), while that for  $\theta_{n}$  is usually ensured either as a by-product of the algorithm design, or via mechanisms such as projections onto a compact subset of  $\mathbb{R}^D$ , depending on the application context. We now provide additional discussions regarding the step-size assumptions and their implications on the SRRW iteration (4b).
+
+SRRW with General Step Size: As shown in Benaim & Cloez (2015, Remark 1.1), albeit for a completely different non-linear Markov kernel driving the algorithm therein, iterates  $\mathbf{x}_n$  of (4b) can also be expressed as weighted empirical measures of  $\{X_n\}_{n\geq 0}$ , in the following form:
+
+$$
+\mathbf {x} _ {n} = \frac {\sum_ {i = 1} ^ {n} \omega_ {i} \delta_ {X _ {i}} + \omega_ {0} \mathbf {x} _ {0}}{\sum_ {i = 0} ^ {n} \omega_ {i}}, \quad \text {w h e r e} \omega_ {0} = 1, \text {a n d} \omega_ {n} = \frac {\gamma_ {n}}{\prod_ {i = 1} ^ {n} (1 - \gamma_ {i})}, \tag {5}
+$$
+
+for all  $n > 0$ . For the special case when  $\gamma_{n} = 1 / (n + 1)$  as in Doshi et al. (2023), we have  $\omega_{n} = 1$  for all  $n \geq 0$  and  $\mathbf{x}_n$  is the typical, unweighted empirical measure. For the additional case considered in our paper, when  $a < 1$  for  $\gamma_{n}$  as in assumption A2, we can approximate  $1 - \gamma_{n} \approx e^{-\gamma_{n}}$  and  $\omega_{n} \approx n^{-a} e^{n^{(1 - a)} / (1 - a)}$ . This implies that  $\omega_{n}$  will increase at sub-exponential rate, giving more weight to recent visit counts and allowing it to quickly 'forget' the poor initial measure  $\mathbf{x}_0$  and shed the correlation with the initial choice of  $X_0$ . This 'speed up' effect by setting  $a < 1$  is guaranteed in case (i) irrespective of the choice of  $b$  in Assumption A2, and in Section 3 we show how this can lead to further reduction in covariance of optimization error  $\theta_{n} = \theta^{*}$  in the asymptotic regime.
+
+Additional assumption for case (iii): Before moving on to Section 3, we take another look at the case when  $\gamma_{n} = o(\beta_{n})$ , and replace A3 with the following, stronger assumption only for case (iii).
+
+(A3') For any  $\mathbf{x} \in \mathrm{Int}(\Sigma)$ , there exists a function  $\rho: \mathrm{Int}(\Sigma) \to \mathbb{R}^D$  such that  $\|\rho(\mathbf{x})\| \leq L_2(1 + \| \mathbf{x}\|)$  for some  $L_2 > 0$ ,  $\mathbb{E}_{i \sim \pi[\mathbf{x}]}[H(\rho(\mathbf{x}), i)] = 0$  and  $\mathbb{E}_{i \sim \pi[\mathbf{x}]}[\nabla H(\rho(\mathbf{x}), i)] + \frac{\mathbf{1}_{\{b=1\}}}{2} \mathbf{I}$  is Hurwitz.
+
+While Assumption A3' for case (iii) is much stronger than A3, it is not detrimental to the overall results of our paper, since case (i) is of far greater interest as impressed upon in Section 1. This is discussed further in Appendix C.
+
+# 3 ASYMPTOTIC ANALYSIS OF THE SA-SRRW ALGORITHM
+
+In this section, we provide the main results for the SA-SRRW algorithm (4). We first present the a.s. convergence and the CLT result for SRRW with generalized step size, extending the results in Doshi et al. (2023). Building upon this, we present the a.s. convergence and the CLT result for the SA iterate  $\theta_{n}$  under different settings of step sizes. We then shift our focus to the analysis of the different asymptotic covariance matrices emerging out of the CLT result, and capture the effect of  $\alpha$  and the step sizes, particularly in cases (i) and (iii), on  $\theta_{n} - \theta^{*}$  via performance ordering.
+
+Almost Sure convergence and CLT: The following result establishes first and second order convergence of the sequence  $\{\mathbf{x}_n\}_{n\geq 0}$ , which represents the weighted empirical measures of the SRRW process  $\{X_{n}\}_{n\geq 0}$ , based on the update rule in (4b).
+
+Lemma 3.1. Under Assumptions A1, A2 and A4, for the SRRW iterates (4b), we have
+
+$$
+\mathbf {x} _ {n} \xrightarrow [ n \to \infty ]{a . s .} \boldsymbol {\mu}, \quad a n d \quad \gamma_ {n} ^ {- 1 / 2} (\mathbf {x} _ {n} - \boldsymbol {\mu}) \xrightarrow [ n \to \infty ]{d i s t .} N (\mathbf {0}, \mathbf {V} _ {\mathbf {x}} (\alpha)),
+$$
+
+$$
+w h e r e \quad \mathbf {V} _ {\mathbf {x}} (\alpha) = \sum_ {i = 1} ^ {N - 1} \frac {1}{2 \alpha \left(1 + \lambda_ {i}\right) + 2 - \mathbb {1} _ {\{a = 1 \}}} \cdot \frac {1 + \lambda_ {i}}{1 - \lambda_ {i}} \mathbf {u} _ {i} \mathbf {u} _ {i} ^ {T}. \tag {6}
+$$
+
+Moreover, for all  $\alpha_{2} > \alpha_{1} > 0$ , we have  $\mathbf{V}_{\mathbf{x}}(\alpha_{2}) < _{L}\mathbf{V}_{\mathbf{x}}(\alpha_{1}) < _{L}\mathbf{V}_{\mathbf{x}}(0)$ .
+
+Lemma 3.1 shows that the SRRW iterates  $\mathbf{x}_n$  converges to the target distribution  $\mu$  a.s. even under the general step size  $\gamma_n = (n + 1)^{-a}$  for  $a\in (0.5,1]$ . We also observe that the asymptotic covariance matrix  $\mathbf{V}_{\mathbf{x}}(\alpha)$  decreases at rate  $O(1 / \alpha)$ . Lemma 3.1 aligns with Doshi et al. (2023, Theorem 4.2 and Corollary 4.3) for the special case of  $a = 1$ , and is therefore more general. Critically, it helps us establish our next result regarding the first-order convergence for the optimization iterate sequence  $\{\theta_{n}\}_{n\geq 0}$  following update rule (4c), as well as its second-order convergence result, which follows shortly after. The proofs of Lemma 3.1 and our next result, Theorem 3.2, are deferred to Appendix D. In what follows,  $k = 1,2$ , and 3 refer to cases (i), (ii), and (iii) in Section 2.2, respectively. All subsequent results are proven under Assumptions A1 to A4, with  $\mathrm{A}3^{\prime}$  replacing A3 only when the step sizes  $\beta_{n},\gamma_{n}$  satisfy case (iii).
+
+Theorem 3.2. For  $k \in \{1,2,3\}$ , and any initial  $(\pmb{\theta}_0,\mathbf{x}_0,X_0) \in \mathbb{R}^D \times \mathrm{Int}(\Sigma) \times \mathcal{N}$ , we have  $\pmb{\theta}_n \to \pmb{\theta}^*$  as  $n \to \infty$  for some  $\pmb{\theta}^* \in \Theta$ ,  $\mathbb{P}_{\pmb{\theta}_0,\mathbf{x}_0,X_0}$ -almost surely.
+
+In the stochastic optimization context, the above result ensures convergence of iterates  $\pmb{\theta}_n$  to a local minimizer  $\pmb{\theta}^*$ . Loosely speaking, the first-order convergence of  $\mathbf{x}_n$  in Lemma 3.1 as well as that of  $\pmb{\theta}_n$  are closely related to the convergence of trajectories  $\{\mathbf{z}(t) \triangleq (\pmb{\theta}(t), \mathbf{x}(t))\}_{t \geq 0}$  of the (coupled) mean-field ODE, written in a matrix-vector form as
+
+$$
+\frac {d}{d t} \mathbf {z} (t) = \mathbf {g} (\mathbf {z} (t)) \triangleq \left[ \begin{array}{c} \mathbf {H} (\boldsymbol {\theta} (t)) ^ {T} \boldsymbol {\pi} [ \mathbf {x} (t) ] \\ \boldsymbol {\pi} [ \mathbf {x} (t) ] - \mathbf {x} (t) \end{array} \right] \in \mathbb {R} ^ {D + N}. \tag {7}
+$$
+
+where matrix  $\mathbf{H}(\pmb{\theta}) \triangleq [H(\pmb{\theta}, 1), \dots, H(\pmb{\theta}, N)]^T \in \mathbb{R}^{N \times D}$  for any  $\pmb{\theta} \in \mathbb{R}^D$ . Here,  $\pmb{\pi}[\mathbf{x}] \in \mathrm{Int}(\Sigma)$  is the stationary distribution of the SRRW kernel  $\mathbf{K}[\mathbf{x}]$  and is shown in Doshi et al. (2023) to be given by  $\pi_i[\mathbf{x}] \propto \sum_{j \in \mathcal{N}} \mu_i P_{ij}(x_i / \mu_i)^{-\alpha}(x_j / \mu_j)^{-\alpha}$ . The Jacobian matrix of (7) when evaluated at equilibria  $\mathbf{z}^* = (\pmb{\theta}^*, \pmb{\mu})$  for  $\pmb{\theta}^* \in \Theta$  captures the behaviour of solutions of the mean-field in their vicinity, and plays an important role in the asymptotic covariance matrices arising out of our CLT results. We evaluate this Jacobian matrix  $\mathbf{J}(\alpha)$  as a function of  $\alpha \geq 0$  to be given by
+
+$$
+\mathbf {J} (\alpha) \triangleq \nabla g \left(\mathbf {z} ^ {*}\right) = \left[ \begin{array}{c c} \nabla \mathbf {h} \left(\boldsymbol {\theta} ^ {*}\right) & - \alpha \mathbf {H} \left(\boldsymbol {\theta} ^ {*}\right) ^ {T} \left(\mathbf {P} ^ {T} + \mathbf {I}\right) \\ \mathbf {0} _ {N \times D} & 2 \alpha \boldsymbol {\mu} \mathbf {1} ^ {T} - \alpha \mathbf {P} ^ {T} - (\alpha + 1) \mathbf {I} \end{array} \right] \triangleq \left[ \begin{array}{l l} \mathbf {J} _ {1 1} & \mathbf {J} _ {1 2} (\alpha) \\ \mathbf {J} _ {2 1} & \mathbf {J} _ {2 2} (\alpha) \end{array} \right]. \tag {8}
+$$
+
+The derivation of  $\mathbf{J}(\alpha)$  is referred to Appendix E.1.8 Here,  $\mathbf{J}_{21}$  is a zero matrix since  $\pi[\mathbf{x}] - \mathbf{x}$  is devoid of  $\theta$ . While matrix  $\mathbf{J}_{22}(\alpha)$  is exactly of the form in Doshi et al. (2023, Lemma 3.4) to characterize the SRRW performance, our analysis includes an additional matrix  $\mathbf{J}_{12}(\alpha)$ , which captures the effect of  $\mathbf{x}(t)$  on  $\theta(t)$  in the ODE (7), which translates to the influence of our generalized SRRW empirical measure  $\mathbf{x}_n$  on the SA iterates  $\theta_n$  in (4).
+
+For notational simplicity, and without loss of generality, all our remaining results are stated while conditioning on the event that  $\{\pmb {\theta}_n\to \pmb {\theta}^*\}$ , for some  $\pmb {\theta}^{*}\in \Theta$ . We also adopt the shorthand notation  $\mathbf{H}$  to represent  $\mathbf{H}(\pmb {\theta}^{*})$ . Our main CLT result is as follows, with its proof deferred to Appendix E.
+
+Theorem 3.3. For any  $\alpha \geq 0$ , we have: (a) There exists  $\mathbf{V}^{(k)}(\alpha)$  for all  $k \in \{1,2,3\}$  such that
+
+$$
+\left[ \begin{array}{c} \beta_ {n} ^ {- 1 / 2} (\boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*}) \\ \gamma_ {n} ^ {- 1 / 2} (\mathbf {x} _ {n} - \boldsymbol {\mu}) \end{array} \right] \xrightarrow [ n \to \infty ]{d i s t.} N \left(\mathbf {0}, \mathbf {V} ^ {(k)} (\alpha)\right).
+$$
+
+(b) For  $k = 2$ , matrix  $\mathbf{V}^{(2)}(\alpha)$  solves the Lyapunov equation  $\mathbf{J}(\alpha)\mathbf{V}^{(2)}(\alpha) + \mathbf{V}^{(2)}(\alpha)\mathbf{J}(\alpha)^T + \mathbb{1}_{\{b = 1\}}\mathbf{V}^{(2)}(\alpha) = -\mathbf{U}$ , where the Jacobian matrix  $\mathbf{J}(\alpha)$  is in (8), and
+
+$$
+\mathbf {U} \triangleq \sum_ {i = 1} ^ {N - 1} \frac {1 + \lambda_ {i}}{1 - \lambda_ {i}} \cdot \left[ \begin{array}{c c} \mathbf {H} ^ {T} \mathbf {u} _ {i} \mathbf {u} _ {i} ^ {T} \mathbf {H} & \mathbf {H} ^ {T} \mathbf {u} _ {i} \mathbf {u} _ {i} ^ {T} \\ \mathbf {u} _ {i} \mathbf {u} _ {i} ^ {T} \mathbf {H} & \mathbf {u} _ {i} \mathbf {u} _ {i} ^ {T} \end{array} \right] \triangleq \left[ \begin{array}{c c} \mathbf {U} _ {1 1} & \mathbf {U} _ {1 2} \\ \mathbf {U} _ {2 1} & \mathbf {U} _ {2 2} \end{array} \right]. \tag {9}
+$$
+
+(c) For  $k \in \{1, 3\}$ ,  $\mathbf{V}^{(k)}(\alpha)$  becomes block diagonal, which is given by
+
+$$
+\mathbf {V} ^ {(k)} (\alpha) = \left[ \begin{array}{l l} \mathbf {V} _ {\boldsymbol {\theta}} ^ {(k)} (\alpha) & \mathbf {0} _ {D \times N} \\ \mathbf {0} _ {N \times D} & \mathbf {V} _ {\mathbf {x}} (\alpha) \end{array} \right], \tag {10}
+$$
+
+where  $\mathbf{V}_{\mathbf{x}}(\alpha)$  is as in (6), and  $\mathbf{V}_{\pmb{\theta}}^{(1)}(\alpha)$  and  $\mathbf{V}_{\pmb{\theta}}^{(3)}(\alpha)$  can be written in the following explicit form:
+
+$$
+\mathbf {V} _ {\boldsymbol {\theta}} ^ {(1)} (\alpha) = \int_ {0} ^ {\infty} e ^ {t (\nabla_ {\boldsymbol {\theta}} \mathbf {h} (\boldsymbol {\theta} ^ {*}) + \frac {1}{2} \{b = 1 \} \boldsymbol {I})} \mathbf {U} _ {\boldsymbol {\theta}} (\alpha) e ^ {t (\nabla_ {\boldsymbol {\theta}} \mathbf {h} (\boldsymbol {\theta} ^ {*}) + \frac {1}{2} \{b = 1 \} \boldsymbol {I}) ^ {T}} d t,
+$$
+
+$$
+\mathbf {V} _ {\pmb {\theta}} ^ {(3)} (\alpha) = \int_ {0} ^ {\infty} e ^ {t \nabla_ {\pmb {\theta}} \mathbf {h} (\pmb {\theta} ^ {*})} \mathbf {U} _ {1 1} e ^ {t \nabla_ {\pmb {\theta}} \mathbf {h} (\pmb {\theta} ^ {*})} d t,
+$$
+
+$$
+w h e r e \mathbf {U} _ {\boldsymbol {\theta}} (\alpha) = \sum_ {i = 1} ^ {N - 1} \frac {1}{\left(\alpha \left(1 + \lambda_ {i}\right) + 1\right) ^ {2}} \cdot \frac {1 + \lambda_ {i}}{1 - \lambda_ {i}} \mathbf {H} ^ {T} \mathbf {u} _ {i} \mathbf {u} _ {i} ^ {T} \mathbf {H}. \tag {11}
+$$
+
+For  $k \in \{1,3\}$ , SA-SRRW in (4) is a two-timescale SA with controlled Markov noise. While a few works study the CLT of two-timescale SA with the stochastic input being a martingale-difference (i.i.d.) noise (Konda & Tsitsiklis, 2004; Mokkadem & Pelletier, 2006), a CLT result covering the case of controlled Markov noise (e.g.,  $k \in \{1,3\}$ ), a far more general setting than martingale-difference noise, is still an open problem. Thus, we here prove our CLT for  $k \in \{1,3\}$  from scratch by a series of careful decompositions of the Markovian noise, ultimately into a martingale-difference term and several non-vanishing noise terms through repeated application of the Poisson equation (Benveniste et al., 2012; Fort, 2015). Although the form of the resulting asymptotic covariance looks similar to that for the martingale-difference case in (Konda & Tsitsiklis, 2004; Mokkadem & Pelletier, 2006) at first glance, they are not equivalent. Specifically,  $\mathbf{V}_{\theta}^{(k)}(\alpha)$  captures both the effect of SRRW hyper-parameter  $\alpha$ , as well as that of the underlying base Markov chain via eigenpairs  $(\lambda_i, \mathbf{u}_i)$  of its transition probability matrix  $\mathbf{P}$  in matrix  $\mathbf{U}$ , whereas the latter only covers the martingale-difference noise terms as a special case.
+
+When  $k = 2$ , that is,  $\beta_{n} = \gamma_{n}$ , algorithm (4) can be regarded as a single-timescale SA algorithm. In this case, we utilize the CLT in Fort (2015, Theorem 2.1) to obtain the implicit form of  $\mathbf{V}^{(2)}(\alpha)$  as shown in Theorem 3.3. However,  $\mathbf{J}_{12}(\alpha)$  being non-zero for  $\alpha > 0$  restricts us from obtaining an explicit form for the covariance term corresponding to SA iterate errors  $\theta_{n} - \theta^{*}$ . On the other hand, for  $k \in \{1,3\}$ , the nature of two-timescale structure causes  $\theta_{n}$  and  $\mathbf{x}_n$  to become asymptotically independent with zero correlation terms inside  $\mathbf{V}^{(k)}(\alpha)$  in (10), and we can explicitly deduce  $\mathbf{V}_{\pmb{\theta}}^{(k)}(\alpha)$ . We now take a deeper dive into  $\alpha$  and study its effect on  $\mathbf{V}_{\pmb{\theta}}^{(k)}(\alpha)$ .
+
+Covariance Ordering of SA-SRRW: We refer the reader to Appendix F for proofs of all remaining results. We begin by focusing on case (i) and capturing the impact of  $\alpha$  on  $\mathbf{V}_{\theta}^{(1)}(\alpha)$ .
+
+Proposition 3.4. For all  $\alpha_{2} > \alpha_{1} > 0$ , we have  $\mathbf{V}_{\pmb{\theta}}^{(1)}(\alpha_2) < _L\mathbf{V}_{\pmb{\theta}}^{(1)}(\alpha_1) < _L\mathbf{V}_{\pmb{\theta}}^{(1)}(0)$ . Furthermore,  $\mathbf{V}_{\pmb{\theta}}^{(1)}(\alpha)$  decreases to zero at a rate of  $O(1 / \alpha^2)$ .
+
+Proposition 3.4 proves a monotonic reduction (in terms of Loewner ordering) of  $\mathbf{V}_{\theta}^{(1)}(\alpha)$  as  $\alpha$  increases. Moreover, the decrease rate  $O(1 / \alpha^2)$  surpasses the  $O(1 / \alpha)$  rate seen in  $\mathbf{V}_{\mathbf{x}}(\alpha)$  and the sampling application in Doshi et al. (2023, Corollary 4.7), and is also empirically observed in our simulation in Section 4. Suppose we consider the same SA now driven by an i.i.d. sequence  $\{X_{n}\}$  with the same marginal distribution  $\pmb{\mu}$ . Then, our Proposition 3.4 asserts that a token algorithm employing SRRW (walk on a graph) with large enough  $\alpha$  on a general graph can actually produce better SA iterates with its asymptotic covariance going down to zero, than a 'hypothetical situation' where the walker is able to access any node  $j$  with probability  $\mu_{j}$  from anywhere in one step (more like a random jumper). This can be seen by noting that for large time  $n$ , the scaled MSE  $\mathbb{E}[\|\pmb{\theta}_n - \pmb{\theta}^*\|^2] / \beta_n$  is composed of the diagonal entries of the covariance matrix  $\mathbf{V}_{\theta}$ , which, as we discuss in detail in Appendix F.2, are decreasing in  $\alpha$  as a consequence of the Loewner ordering in Proposition 3.4. For large enough  $\alpha$ , the scaled MSE for SA-SRRW becomes smaller than its i.i.d. counterpart, which is always a constant. Although Doshi et al. (2023) alluded this for sampling applications with  $\mathbf{V}_{\mathbf{x}}(\alpha)$ , we broaden its horizons to distributed optimization problems with  $\mathbf{V}_{\pmb{\theta}}(\alpha)$  using tokens on graphs. Our subsequent result concerns the performance comparison between cases (i) and (iii).
+
+Corollary 3.5. For any  $\alpha >0$ , we have  $\mathbf{V}_{\pmb{\theta}}^{(1)}(\alpha) < _L\mathbf{V}_{\pmb{\theta}}^{(3)}(\alpha) = \mathbf{V}_{\pmb{\theta}}^{(3)}(0)$ .
+
+We show that case (i) is asymptotically better than case (iii) for  $\alpha > 0$ . In view of Proposition 3.4 and Corollary 3.5, the advantages of case (i) become prominent.
+
+![](images/1bc79c3f84b108613b42e4e1b6e41825e615535bc0bf287d96ae710edfe1bc20.jpg)  
+(a) SGD-SRRW.
+
+![](images/849d61164ad08b70d0a82140ec0a774c9115b9287ff917667add7a60555a01b4.jpg)  
+(b) SHB-SRRW.
+
+![](images/ef2728074860a48dcd3abbbae47d11f96aab393331e3abfb7f5f3bfe028bf323.jpg)  
+(c) Curve fitting for MSE.
+
+![](images/99c1edae8a3c743d6e335debd63817aff3f5b8998330d174afdcf043685ea939.jpg)  
+Figure 2: Simulation results under case (i): (a) and (b) show the performance of SGD-SRRW and SHB-SRRW for various  $\alpha$  values. (c) shows that MSE decreases at  $O(1 / \alpha^2)$  speed.  
+(a)  $\alpha = 1$  ,SGD-SRRW  
+Figure 3: Comparison of the performance among cases (i) - (iii) for  $\alpha \in \{1,5,10\}$ .
+
+![](images/37d71bc66fd04ec9f787a87660d07ae255b7f1b8ddac4fc39b7b58bf790a6d78.jpg)  
+(b)  $\alpha = 5$  ,SGD-SRRW
+
+![](images/127f341bd20cf5ab04a65c29d98d5d75f652e1582f6bcd73c062ddb4661b7783.jpg)  
+(c)  $\alpha = 10$  ,SGD-SRRW
+
+# 4 SIMULATION
+
+In this section, we simulate our SA-SRRW algorithm on the wikiVote graph (Leskovec & Krevl, 2014), comprising 889 nodes and 2914 edges. We configure the SRRW's base Markov chain  $\mathbf{P}$  as the MHRW with a uniform target distribution  $\mu = \frac{1}{N} \mathbf{1}$ . For distributed optimization, we consider the following  $L_{2}$  regularized binary classification problem:
+
+$$
+\left. \min  _ {\boldsymbol {\theta} \in \mathbb {R} ^ {D}} \left\{f (\boldsymbol {\theta}) \triangleq \frac {1}{N} \sum_ {i = 1} ^ {N} \log \left(1 + e ^ {\boldsymbol {\theta} ^ {T} \mathbf {s} _ {i}}\right) - y _ {i} \left(\boldsymbol {\theta} ^ {T} \mathbf {s} _ {i}\right) + \frac {\kappa}{2} \| \boldsymbol {\theta} \| ^ {2} \right\}, \right. \tag {12}
+$$
+
+where  $\{(\mathbf{s}_i, y_i)\}_{i=1}^N$  is the ijCNN1 dataset (with 22 features, i.e.,  $\mathbf{s}_i \in \mathbb{R}^{22}$ ) from LIBSVM (Chang & Lin, 2011), and penalty parameter  $\kappa = 1$ . Each node in the wikiVote graph is assigned one data point, thus 889 data points in total. We perform SRRW driven SGD (SGD-SRRW) and SRRW driven stochastic heavy ball (SHB-SRRW) algorithms (see (13) in Appendix A for its algorithm). We fix the step size  $\beta_n = (n + 1)^{-0.9}$  for the SA iterates and adjust  $\gamma_n = (n + 1)^{-a}$  in the SRRW iterates to cover all three cases discussed in this paper: (i)  $a = 0.8$ ; (ii)  $a = 0.9$ ; (iii)  $a = 1$ . We use mean square error (MSE), i.e.,  $\mathbb{E}[\|\pmb{\theta}_n - \pmb{\theta}^*\|^2]$ , to measure the error on the SA iterates.
+
+Our results are presented in Figures 2 and 3, where each experiment is repeated 100 times. Figures 2a and 2b, based on wikiVote graph, highlight the consistent performance ordering across different  $\alpha$  values for both algorithms over almost all time (not just asymptotically). Notably, curves for  $\alpha \geq 5$  outperform that of the i.i.d. sampling (in black) even under the graph constraints. Figure 2c on the smaller Dolphin graph (Rossi & Ahmed, 2015) - 62 nodes and 159 edges - illustrates that the points of  $(\alpha, \mathrm{MSE})$  pair arising from SGD-SRRW at time  $n = 10^7$  align with a curve in the form of  $g(x) = \frac{c_1}{(x + c_2)^2} + c_3$  to showcase  $O(1 / \alpha^2)$  rates. This smaller graph allows for longer simulations to observe the asymptotic behaviour. Additionally, among the three cases examined at identical  $\alpha$  values, Figures 3a - 3c confirm that case (i) performs consistently better than the rest, underscoring its superiority in practice. Further results, including those from non-convex functions and additional datasets, are deferred to Appendix H due to space constraints.
+
+# 5 CONCLUSION
+
+In this paper, we show both theoretically and empirically that the SRRW as a drop-in replacement for Markov chains can provide significant performance improvements when used for token algorithms, where the acceleration comes purely from the careful analysis of the stochastic input of the algorithm, without changing the optimization iteration itself. Our paper is an instance where the asymptotic analysis approach allows the design of better algorithms despite the usage of unconventional noise sequences such as nonlinear Markov chains like the SRRW, for which traditional finite-time analytical approaches fall short, thus advocating their wider adoption.
+
+# 6 ACKNOWLEDGMENTS AND DISCLOSURE OF FUNDING
+
+We thank the anonymous reviewers for their constructive comments, especially Reviewer DxLx for raising future directions. This work was supported in part by National Science Foundation under Grant Nos. CNS-2007423 and IIS-1910749.
+
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+
+# A EXAMPLES OF STOCHASTIC ALGORITHMS OF THE FORM (2).
+
+In the literature of stochastic optimizations, many SGD variants have been proposed by introducing an auxiliary variable to improve convergence. In what follows, we present two SGD variants with decreasing step size that can be presented in the form of (2): SHB (Gadat et al., 2018; Li et al., 2022) and momentum-based algorithm (Barakat et al., 2021; Barakat & Bianchi, 2021).
+
+$$
+\left\{ \begin{array}{l} \boldsymbol {\theta} _ {n + 1} = \boldsymbol {\theta} _ {n} - \beta_ {n + 1} \mathbf {m} _ {n} \\ \mathbf {m} _ {n + 1} = \mathbf {m} _ {n} + \beta_ {n + 1} (\nabla F (\boldsymbol {\theta} _ {n}, X _ {n + 1}) - \mathbf {m} _ {n}), \end{array} \right. \left\{ \begin{array}{l} \mathbf {v} _ {n + 1} = \mathbf {v} _ {n} + \beta_ {n + 1} (\nabla F (\boldsymbol {\theta} _ {n}, X _ {n + 1}) ^ {2} - \mathbf {v} _ {n}), \\ \mathbf {m} _ {n + 1} = \mathbf {m} _ {n} + \beta_ {n + 1} (\nabla F (\boldsymbol {\theta} _ {n}, X _ {n + 1}) - \mathbf {m} _ {n}), \\ \boldsymbol {\theta} _ {n + 1} = \boldsymbol {\theta} _ {n} - \beta_ {n + 1} \mathbf {m} _ {n} / \sqrt {\mathbf {v} _ {n} + \epsilon}, \end{array} \right.
+$$
+
+(a). SHB
+
+(b). Momentum-based Algorithm
+
+(13)
+
+where  $\epsilon > 0$ ,  $\pmb{\theta}_n, \mathbf{m}_n, \mathbf{v}_n, \nabla F(\pmb{\theta}, X) \in \mathbb{R}^d$ , and the square and square root in (13) (b) are element-wise operators. $^{10}$
+
+For SHB, we introduce an augmented variable  $\mathbf{z}_n$  and function  $H(\mathbf{z}_n,X_{n + 1})$  defined as follows:
+
+$$
+\mathbf {z} _ {n} \triangleq \left[ \begin{array}{c} \boldsymbol {\theta} _ {n} \\ \mathbf {m} _ {n} \end{array} \right] \in \mathbb {R} ^ {2 d}, \quad H (\mathbf {z} _ {n}, X _ {n + 1}) \triangleq \left[ \begin{array}{c} - \mathbf {m} _ {n} \\ \nabla F (\boldsymbol {\theta} _ {n}, X _ {n + 1}) - \mathbf {m} _ {n} \end{array} \right] \in \mathbb {R} ^ {2 d}.
+$$
+
+For the general momentum-based algorithm, we define
+
+$$
+\mathbf {z} _ {n} \triangleq \left[ \begin{array}{c} \mathbf {v} _ {n} \\ \mathbf {m} _ {n} \\ \boldsymbol {\theta} _ {n} \end{array} \right] \in \mathbb {R} ^ {3 d}, \quad H (\mathbf {z} _ {n}, X) \triangleq \left[ \begin{array}{c} \nabla F (\boldsymbol {\theta} _ {n}, X _ {n + 1}) ^ {2} - \mathbf {v} _ {n} \\ \nabla F (\boldsymbol {\theta} _ {n}, X _ {n + 1}) - \mathbf {m} _ {n} \\ - \mathbf {m} _ {n} / \sqrt {\mathbf {v} _ {n} + \epsilon} \end{array} \right] \in \mathbb {R} ^ {3 d}.
+$$
+
+Thus, we can reformulate both algorithms in (13) as  $\mathbf{z}_{n + 1} = \mathbf{z}_n + \beta_{n + 1}H(\mathbf{z}_n,X_{n + 1})$ . This augmentation approach was previously adopted in (Gadat et al., 2018; Barakat et al., 2021; Barakat & Bianchi, 2021; Li et al., 2022) to analyze the asymptotic performance of algorithms in (13) using an i.i.d. sequence  $\{X_{n}\}_{n\geq 0}$ . Consequently, the general SA iteration (2) includes these SGD variants. However, we mainly focus on the CLT for the general SA driven by SRRW in this paper. Pursuing the explicit CLT results of these SGD variants with specific form of function  $H(\theta ,X)$  driven by the SRRW sequence  $\{X_{n}\}$  is out of the scope of this paper.
+
+When we numerically test the SHB algorithm in Section 4, we use the exact form of (13) (a) and the stochastic sequence  $\{X_{n}\}$  is now driven by the SRRW. Specifically, we consider MHRW with transition kernel  $\mathbf{P}$  as the base Markov chain of the SRRW process, e.g.,
+
+$$
+P _ {i j} = \left\{ \begin{array}{l l} \min  \left\{\frac {1}{d _ {i}}, \frac {1}{d _ {j}} \right\} & \text {i f n o d e j i s t h e n e i g h b o r o f n o d e i ,} \\ 0 & \text {o t h e r w i s e ,} \end{array} \right.
+$$
+
+$$
+P _ {i i} = 1 - \sum_ {j \in \mathcal {N}} P _ {i j}.
+$$
+
+Then, at each time step  $n$ ,
+
+Draw:  $X_{n + 1}\sim \mathbf{K}_{X_n,\cdot}[\mathbf{x}_n],$
+
+where  $K_{ij}[\mathbf{x}]\triangleq \frac{P_{ij}(x_j / \mu_j)^{-\alpha}}{\sum_{k\in\mathcal{N}}P_{ik}(x_k / \mu_k)^{-\alpha}},\quad \forall i,j\in \mathcal{N},$
+
+Update:  $\mathbf{x}_{n + 1} = \mathbf{x}_n + \gamma_{n + 1}(\pmb{\delta}_{X_{n + 1}} - \mathbf{x}_n)$
+
+$$
+\boldsymbol {\theta} _ {n + 1} = \boldsymbol {\theta} _ {n} - \beta_ {n + 1} \mathbf {m} _ {n},
+$$
+
+$$
+\mathbf {m} _ {n + 1} = \mathbf {m} _ {n} + \beta_ {n + 1} (\nabla F (\boldsymbol {\theta} _ {n}, X _ {n + 1}) - \mathbf {m} _ {n}).
+$$
+
+# B DISCUSSION ON MEAN FIELD FUNCTION OF SRRW ITERATES (4b)
+
+Non-asymptotic analyses have seen extensive attention recently in both single-timescale SA literature (Sun et al., 2018; Karimi et al., 2019; Chen et al., 2020b; 2022) and two-timescale SA literature (Doan, 2021; Zeng et al., 2021). Specifically, single-timescale SA has the following form:
+
+$$
+\mathbf {x} _ {n + 1} = \mathbf {x} _ {n} + \beta_ {n + 1} H (\mathbf {x} _ {n}, X _ {n + 1}),
+$$
+
+and function  $h(\mathbf{x}) \triangleq \mathbb{E}_{X \sim \mu}[H(\mathbf{x}, X)]$  is the mean field of function  $H(\mathbf{x}, X)$ . Similarly, for two-timescale SA, we have the following recursions:
+
+$$
+\mathbf {x} _ {n + 1} = \mathbf {x} _ {n} + \beta_ {n + 1} H _ {1} (\mathbf {x} _ {n}, \mathbf {y} _ {n}, X _ {n + 1}),
+$$
+
+$$
+\mathbf {y} _ {n + 1} = \mathbf {y} _ {n} + \gamma_ {n + 1} H _ {2} (\mathbf {x} _ {n}, \mathbf {y} _ {n}, X _ {n + 1}),
+$$
+
+where  $\{\beta_n\}$  and  $\{\gamma_n\}$  are on different timescales, and function  $h_i(\mathbf{x},\mathbf{y})\triangleq \mathbb{E}_{X\sim \mu}[H_i(\mathbf{x},\mathbf{y},X)]$  is the mean field of function  $H_{i}(\mathbf{x},\mathbf{y},X)$  for  $i = \{1,2\}$ . All the aforementioned works require the mean field function  $h(\mathbf{x})$  in the single-timescale SA (or  $h_1(\mathbf{x},\mathbf{y}),h_2(\mathbf{x},\mathbf{y})$  in the two-timescale SA) to be globally Lipschitz with a Lipschitz constant  $L$  to proceed with the derivation of finite-time bounds including the constant  $L$ .
+
+Here, we show that the mean field function  $\pi[\mathbf{x}] - \mathbf{x}$  in the SRRW iterates (4b) is not globally Lipschitz, where  $\pi[\mathbf{x}]$  is the stationary distribution of the SRRW kernel  $\mathbf{K}[\mathbf{x}]$  defined in (3). To this end, we show that each entry of Jacobian matrix of  $\pi[\mathbf{x}] - \mathbf{x}$  goes unbounded because a multivariate function is Lipschitz if and only if it has bounded partial derivatives. Note that from Doshi et al. (2023, Proposition 2.1), for the  $i$ -th entry of  $\pi[\mathbf{x}]$ , we have
+
+$$
+\boldsymbol {\pi} _ {i} [ \mathbf {x} ] = \frac {\sum_ {j \in \mathcal {N}} \mu_ {i} P _ {i j} \left(x _ {i} / \mu_ {i}\right) ^ {- \alpha} \left(x _ {j} / \mu_ {j}\right) ^ {- \alpha}}{\sum_ {i \in \mathcal {N}} \sum_ {j \in \mathcal {N}} \mu_ {i} P _ {i j} \left(x _ {i} / \mu_ {i}\right) ^ {- \alpha} \left(x _ {j} / \mu_ {j}\right) ^ {- \alpha}}. \tag {16}
+$$
+
+Then, the Jacobian matrix of the mean field function  $\pi[\mathbf{x}] - \mathbf{x}$ , which has been derived in Doshi et al. (2023, Proof of Lemma 3.4 in Appendix B), is given as follows:
+
+$$
+\begin{array}{l} \frac {\partial (\boldsymbol {\pi} _ {i} [ \mathbf {x} ] - x _ {i})}{\partial x _ {j}} \\ = \frac {2 \alpha}{x _ {j}} \cdot \frac {\left(\sum_ {k \in \mathcal {N}} \mu_ {i} P _ {i k} \left(x _ {i} / \mu_ {i}\right) ^ {- \alpha} \left(x _ {k} / \mu_ {k}\right) ^ {- \alpha}\right) \left(\sum_ {k \in \mathcal {N}} \mu_ {j} P _ {j k} \left(x _ {j} / \mu_ {j}\right) ^ {- \alpha} \left(x _ {k} / \mu_ {k}\right) ^ {- \alpha}\right)}{\left(\sum_ {l \in \mathcal {N}} \sum_ {k \in \mathcal {N}} \mu_ {l} P _ {l k} \left(x _ {l} / \mu_ {l}\right) ^ {- \alpha} \left(x _ {k} / \mu_ {k}\right) ^ {- \alpha}\right) ^ {2}} \tag {17} \\ - \frac {\alpha}{x _ {j}} \cdot \frac {\mu_ {i} P _ {i j} (x _ {i} / \mu_ {i}) ^ {- \alpha} (x _ {j} / \mu_ {j}) ^ {- \alpha}}{\sum_ {l \in \mathcal {N}} \sum_ {k \in \mathcal {N}} \mu_ {l} P _ {l k} (x _ {l} / \mu_ {l}) ^ {- \alpha} (x _ {k} / \mu_ {k}) ^ {- \alpha}} \\ \end{array}
+$$
+
+for  $i,j\in \mathcal{N},i\neq j$  , and
+
+$$
+\begin{array}{l} \frac {\partial \left(\boldsymbol {\pi} _ {i} [ \mathbf {x} ] - x _ {i}\right)}{\partial x _ {i}} \\ = \frac {2 \alpha}{x _ {i}} \cdot \frac {\left(\sum_ {k \in \mathcal {N}} \mu_ {i} P _ {i k} \left(x _ {i} / \mu_ {i}\right) ^ {- \alpha} \left(x _ {k} / \mu_ {k}\right) ^ {- \alpha}\right) ^ {2}}{\left(\sum_ {l \in \mathcal {N}} \sum_ {k \in \mathcal {N}} \mu_ {l} P _ {l k} \left(x _ {l} / \mu_ {l}\right) ^ {- \alpha} \left(x _ {k} / \mu_ {k}\right) ^ {- \alpha}\right) ^ {2}} \tag {18} \\ - \frac {\alpha}{x _ {i}} \cdot \frac {\sum_ {k \in \mathcal {N}} \mu_ {i} P _ {i k} (x _ {i} / \mu_ {i}) ^ {- \alpha} (x _ {k} / \mu_ {k}) ^ {- \alpha} + \mu_ {i} P _ {i i} (x _ {i} / \mu_ {i}) ^ {- 2 \alpha}}{\sum_ {l \in \mathcal {N}} \sum_ {k \in \mathcal {N}} \mu_ {l} P _ {l k} (x _ {l} / \mu_ {l}) ^ {- \alpha} (x _ {k} / \mu_ {k}) ^ {- \alpha}} - 1 \\ \end{array}
+$$
+
+for  $i \in \mathcal{N}$ . Since the empirical distribution  $\mathbf{x} \in \mathrm{Int}(\Sigma)$ , we have  $x_i \in (0,1)$  for all  $i \in \mathcal{N}$ . For fixed  $i$ , assume  $x_i = x_j$  and as they approach zero, the terms  $(x_i / \mu_i)^{-\alpha}, (x_j / \mu_j)^{-\alpha}$  dominate the fraction in (17) and both the numerator and the denominator of the fraction have the same order in terms of  $x_i, x_j$ . Thus, we have
+
+$$
+\frac {\partial (\pmb {\pi} _ {i} [ \mathbf {x} ] - x _ {i})}{\partial x _ {j}} = O \left(\frac {1}{x _ {j}}\right)
+$$
+
+such that the  $(i,j)$ -th entry of the Jacobian matrix can go unbounded as  $x_{j} \to 0$ . Consequently,  $\pi[\mathbf{x}] - \mathbf{x}$  is not globally Lipschitz for  $\mathbf{x} \in \operatorname{Int}(\Sigma)$ .
+
+# C DISCUSSION ON ASSUMPTION A3'
+
+When  $\gamma_{n} = o(\beta_{n})$ , iterates  $\mathbf{x}_n$  has smaller step size compared to  $\pmb{\theta}_{n}$ , thus converges 'slower' than  $\pmb{\theta}_{n}$ . From Assumption A3',  $\pmb{\theta}_{n}$  will intuitively converge to some point  $\rho (\mathbf{x})$  with the current value  $\mathbf{x}$  from the iteration  $\mathbf{x}_n$ , i.e.,  $\mathbb{E}_{X\sim \pi [\mathbf{x}]}[H(\rho (\mathbf{x}),X)] = 0$ , while the Hurwitz condition is to ensure the stability around  $\rho (\mathbf{x})$ . We can see that Assumption A3 is less stringent than A3' in that it only assumes such condition when  $\mathbf{x} = \pmb{\mu}$  such that  $\rho (\pmb {\mu}) = \pmb{\theta}^{*}$  rather than for all  $\mathbf{x}\in \mathrm{Int}(\Sigma)$ .
+
+One special instance of Assumption A3' is by assuming the linear SA, e.g.,  $H(\pmb{\theta}, i) = A_i\pmb{\theta} + b_i$ . In this case,  $\mathbb{E}_{X \sim \pi[\mathbf{x}]}[H(\rho(\mathbf{x}), X)] = 0$  is equivalent to  $\mathbb{E}_{i \sim \pi[\mathbf{x}]}[A_i]\rho(\mathbf{x}) + \mathbb{E}_{i \sim \pi[\mathbf{x}]}[b_i] = 0$ . Under the condition that for every  $\mathbf{x} \in \mathrm{Int}(\Sigma)$ , matrix  $\mathbb{E}_{i \sim \pi[\mathbf{x}]}[A_i]$  is invertible, we then have
+
+$$
+\rho (\mathbf {x}) = - \left(\mathbb {E} _ {i \sim \boldsymbol {\pi} [ \mathbf {x} ]} [ A _ {i} ]\right) ^ {- 1} \cdot \mathbb {E} _ {i \sim \boldsymbol {\pi} [ \mathbf {x} ]} [ b _ {i} ].
+$$
+
+However, this condition is quite strict. Loosely speaking,  $\mathbb{E}_{i\sim \pi [\mathbf{x}]}[A_i]$  being invertible for any  $\mathbf{x}$  is similar to saying that any convex combination of  $\{A_i\}$  is invertible. For example, if we assume  $\{A_i\}_{i\in \mathcal{N}}$  are negative definite and they all share the same eigenbasis  $\{\mathbf{u}_i\}$ , e.g.,  $A_{i} = \sum_{j = 1}^{D}\lambda_{j}^{i}\mathbf{u}_{i}\mathbf{u}_{i}^{T}$  and  $\lambda_j^i < 0$  for all  $i\in \mathcal{N}, j\in [D]$ . Then,  $\mathbb{E}_{i\sim \pi [\mathbf{x}]}[A_i]$  is invertible.
+
+Another example for Assumption A3' is when  $H(\pmb{\theta}, i) = H(\pmb{\theta}, j)$  for all  $i, j \in \mathcal{N}$ , which implies that each agent in the distributed learning has the same local dataset to collaboratively train the model. In this example,  $\rho(\mathbf{x}) = \pmb{\theta}^*$  such that
+
+$$
+\begin{array}{l} \mathbb {E} _ {i \sim \pi [ \mathbf {x} ]} [ H (\rho (\mathbf {x}), i) ] = h (\boldsymbol {\theta} ^ {*}) = 0, \\ \mathbb {E} _ {i \sim \pi [ \mathbf {x} ]} [ \nabla H (\rho (\mathbf {x}), i) ] + \frac {\mathbb {1} _ {\{b = 1 \}}}{2} \mathbf {I} = \nabla h (\boldsymbol {\theta} ^ {*}) + \frac {\mathbb {1} _ {\{b = 1 \}}}{2} \mathbf {I} \quad \text {b e i n g H u r w i t z}. \\ \end{array}
+$$
+
+# D PROOF OF LEMMA 3.1 AND LEMMA 3.2
+
+In this section, we demonstrate the almost sure convergence of both  $\theta_{n}$  and  $\mathbf{x}_n$  together. This proof naturally incorporates the almost certain convergence of the SRRW iteration in Lemma 3.1, since  $\mathbf{x}_n$  is independent of  $\theta_{n}$  (as indicated in (4)), allowing us to separate out its asymptotic results. The same reason applies to the CLT analysis of SRRW iterates and we refer the reader to Section E.1 for the CLT result of  $\mathbf{x}_n$  in Lemma 3.1.
+
+We will use different techniques for different settings of step sizes in Assumption A2. Specifically, for step sizes  $\gamma_{n} = (n + 1)^{-a},\beta_{n} = (n + 1)^{-b}$ , we consider the following scenarios:
+
+Scenario 1: We consider case(ii):  $1/2 < a = b \leq 1$ , and will apply the almost sure convergence result of the single-timescale stochastic approximation in Theorem G.8 and verify all the conditions therein.
+
+Scenario 2: We consider both case(i):  $1/2 < a < b \leq 1$  and case (iii):  $1/2 < b < a \leq 1$ . In these two cases, step sizes  $\gamma_{n}, \beta_{n}$  decrease at different rates, thereby putting iterates  $\mathbf{x}_{n}, \pmb{\theta}_{n}$  on different timescales and resulting in a two-timescale structure. We will apply the existing almost sure convergence result of the two-timescale stochastic approximation with iteratedependent Markov chain in Yaji & Bhatnagar (2020, Theorem 4) where our SA-SRRW algorithm can be regarded as a special instance.[11]
+
+# D.1 SCENARIO 1
+
+In Scenario 1, we have  $\beta_{n} = \gamma_{n}$ . First, we rewrite (4) as
+
+$$
+\left[ \begin{array}{l} \boldsymbol {\theta} _ {n + 1} \\ \mathbf {x} _ {n + 1} \end{array} \right] = \left[ \begin{array}{l} \boldsymbol {\theta} _ {n} \\ \mathbf {x} _ {n} \end{array} \right] + \gamma_ {n + 1} \left[ \begin{array}{c} H \left(\boldsymbol {\theta} _ {n}. X _ {n + 1}\right) \\ \boldsymbol {\delta} _ {X _ {n + 1}} - \mathbf {x} _ {n} \end{array} \right]. \tag {19}
+$$
+
+By augmentations, we define the variable  $\mathbf{z}_n \triangleq \left[ \begin{array}{c} \boldsymbol{\theta}_n \\ \mathbf{x}_n \end{array} \right] \in \mathbb{R}^{(N + D) \times 1}$  and the function  $G(\mathbf{z}_n, i) \triangleq \left[ \begin{array}{c} H(\boldsymbol{\theta}_n, i) \\ \delta_i - \mathbf{x}_n \end{array} \right] \in \mathbb{R}^{(N + d) \times 1}$ . In addition, we define a new Markov chain  $\{Y_n\}_{n \geq 0}$  in the same state space  $\mathcal{N}$  as SRRW sequence  $\{X_n\}_{n \geq 0}$ . With slight abuse of notation, the transition kernel of  $\{Y_n\}$  is denoted by  $\mathbf{K}'[\mathbf{z}_n] \equiv \mathbf{K}[\mathbf{x}_n]$  and its stationary distribution  $\pi'[\mathbf{z}_n] \equiv \pi[\mathbf{x}_n]$ , where  $\mathbf{K}[\mathbf{x}_n]$  and  $\pi(\mathbf{x}_n)$  are the transition kernel and its corresponding stationary distribution of SRRW, with  $\pi[\mathbf{x}]$  of the form
+
+$$
+\pi_ {i} [ \mathbf {x} ] \propto \sum_ {j \in \mathcal {N}} \mu_ {i} P _ {i j} \left(x _ {i} / \mu_ {i}\right) ^ {- \alpha} \left(x _ {j} / \mu_ {j}\right) ^ {- \alpha}. \tag {20}
+$$
+
+Recall that  $\pmb{\mu}$  is the fixed point, i.e.,  $\pi[\pmb{\mu}] = \pmb{\mu}$ , and  $\mathbf{P}$  is the base Markov chain inside SRRW (see (3)). Then, the mean field
+
+$$
+g (\mathbf {z}) = \mathbb {E} _ {Y \sim \boldsymbol {\pi^ {\prime}} (\mathbf {z})} [ G (\mathbf {z}, Y) ] = \left[ \begin{array}{c} \sum_ {i \in \mathcal {N}} \pi_ {i} [ \mathbf {x} ] H (\boldsymbol {\theta}, i) \\ \boldsymbol {\pi} [ \mathbf {x} ] - \mathbf {x} \end{array} \right],
+$$
+
+and  $\mathbf{z}^{*} = (\pmb{\theta}^{*},\pmb{\mu})$  for  $\pmb{\theta}^{*}\in \Theta$  in Assumption A3 is the root of  $g(\mathbf{z})$ , i.e.,  $g(\mathbf{z}^{*}) = 0$ . The augmented iteration (19) becomes
+
+$$
+\mathbf {z} _ {n + 1} = \mathbf {z} _ {n} + \gamma_ {n + 1} G \left(\mathbf {z} _ {n}, Y _ {n + 1}\right) \tag {21}
+$$
+
+with the goal of solving  $g(\mathbf{z}) = 0$ . Therefore, we can treat (21) as an SA algorithm driven by a Markov chain  $\{Y_{n}\}_{n\geq 0}$  with its kernel  $\mathbf{K}'[\mathbf{z}]$  and stationary distribution  $\pi^{\prime}[\mathbf{z}]$ , which has been widely studied in the literature (e.g., Delyon (2000); Benveniste et al. (2012); Fort (2015); Li et al. (2023)). In what follows, we demonstrate that for any initial point  $\mathbf{z}_0 = (\pmb {\theta}_0,\mathbf{x}_0)\in \mathbb{R}^D\times \mathrm{Int}(\Sigma)$ , the SRRW iteration  $\{\mathbf{x}_n\}_{n\geq 0}$  will almost surely converge to the target distribution  $\pmb{\mu}$ , and the SA iteration  $\{\pmb {\theta}_n\}_{n\geq 0}$  will almost surely converge to the set  $\Theta$ .
+
+Now we verify conditions C1 - C4 in Theorem G.8. Our assumption A4 is equivalent to condition C1 and assumption A2 corresponds to condition C2. For condition C3, we set  $\nabla w(\mathbf{z}) \equiv -g(\mathbf{z})$ , and the set  $S \equiv \{\mathbf{z}^* | \boldsymbol{\theta}^* \in \Theta, \mathbf{x}^* = \boldsymbol{\mu}\}$ , including disjoint points. For condition C4, since  $\mathbf{K}'[\mathbf{z}]$ , or equivalently  $\mathbf{K}[\mathbf{x}]$ , is ergodic and time-reversible for a given  $\mathbf{z}$ , as shown in the SRRW work Doshi et al. (2023), it automatically ensures a solution to the Poisson equation, which has been well discussed in Chen et al. (2020a, Section 2) and Benveniste et al. (2012); Meyn (2022). To show (97) and (98) in condition C4, for each given  $\mathbf{z}$  and any  $i \in \mathcal{N}$ , we need to give the explicit solution  $m_{\mathbf{z}}(i)$  to the Poisson equation  $m_{\mathbf{z}}(i) - (\mathbf{K}_{\mathbf{z}}' m_{\mathbf{z}})(i) = G(\mathbf{z}, i) - g(\mathbf{z})$  in (96). The notation  $(\mathbf{K}_{\mathbf{z}}' m_{\mathbf{z}})(i)$  is defined as follows.
+
+$$
+\left(\mathbf {K} _ {\mathbf {z}} ^ {\prime} m _ {\mathbf {z}}\right) (i) = \sum_ {j \in \mathcal {N}} \mathbf {K} _ {\mathbf {z}} ^ {\prime} (i, j) m (\mathbf {z}, j).
+$$
+
+Let  $\mathbf{G}(\mathbf{z})\triangleq [G(\mathbf{z},1),\dots ,G(\mathbf{z},N)]^T\in \mathbb{R}^{N\times D}$ . We use  $[\mathbf{A}]_{:,i}$  to denote the  $i$ -th column of matrix  $\mathbf{A}$ . Then, we let  $m_{\mathbf{z}}(i)$  such that
+
+$$
+m _ {\mathbf {z}} (i) = \sum_ {k = 0} ^ {\infty} \left(\left[ \mathbf {G} (\mathbf {z}) \left(\mathbf {K} ^ {\prime} [ \mathbf {z} ] ^ {k}\right) ^ {T} \right] _ {[:, i ]} - g (\mathbf {z})\right) = \sum_ {k = 0} ^ {\infty} \left[ \mathbf {G} (\mathbf {z}) \left(\left(\mathbf {K} ^ {\prime} [ \mathbf {z} ] ^ {k}\right) ^ {T} - \boldsymbol {\pi} ^ {\prime} [ \mathbf {z} ] \mathbf {1} ^ {T}\right) \right] _ {[:, i ]}. \tag {22}
+$$
+
+In addition,
+
+$$
+\left(\mathbf {K} _ {\mathbf {z}} ^ {\prime} m _ {\mathbf {z}}\right) (i) = \sum_ {k = 1} ^ {\infty} \left[ \mathbf {G} (\mathbf {z}) \left(\mathbf {K} ^ {\prime} [ \mathbf {z} ] ^ {T} - \boldsymbol {\pi} ^ {\prime} [ \mathbf {z} ] \mathbf {1} ^ {T}\right) \right] _ {[ i, i ]}. \tag {23}
+$$
+
+We can check that the  $m_{\mathbf{z}}(i)$  form in (22) is indeed the solution of the above Poisson equation. Now, by induction, we get  $\mathbf{K}'[\mathbf{z}]^k - \mathbf{1}\pi'[\mathbf{z}]^T = (\mathbf{K}'[\mathbf{z}] - \mathbf{1}\pi'[\mathbf{z}]^T)^k$  for  $k \geq 1$  and for  $k = 0$ ,  $\mathbf{K}'[\mathbf{z}]^0 - \mathbf{1}\pi'[\mathbf{z}]^T = (\mathbf{K}'[\mathbf{z}] - \mathbf{1}\pi'[\mathbf{z}]^T)^0 - \mathbf{1}\pi'[\mathbf{z}]^T$ . Then,
+
+$$
+\begin{array}{l} m _ {\mathbf {z}} (i) = \sum_ {k = 0} ^ {\infty} [ \mathbf {G} (\mathbf {z}) (\mathbf {K} ^ {\prime} [ \mathbf {z} ] ^ {T} - \boldsymbol {\pi} ^ {\prime} [ \mathbf {z} ] \mathbf {1} ^ {T}) ^ {k} ] _ {[; i ]} - g (\mathbf {z}) \\ = \left[ \mathbf {G} (\mathbf {z}) \sum_ {k = 0} ^ {\infty} \left(\mathbf {K} ^ {\prime} [ \mathbf {z} ] ^ {T} - \boldsymbol {\pi} ^ {\prime} [ \mathbf {z} ] \mathbf {1} ^ {T}\right) ^ {k} \right] _ {[;, i ]} - g (\mathbf {z}) \tag {24} \\ = \left[ \mathbf {G} (\mathbf {z}) \left(\mathbf {I} - \mathbf {K} ^ {\prime} [ \mathbf {z} ] ^ {T} + \boldsymbol {\pi} ^ {\prime} [ \mathbf {z} ] \mathbf {1} ^ {T}\right) ^ {- 1} \right] _ {[,: i ]} - g (\mathbf {z}) \\ = \sum_ {j \in \mathcal {N}} \left(\mathbf {I} - \mathbf {K} ^ {\prime} [ \mathbf {z} ] + \mathbf {1} \boldsymbol {\pi} ^ {\prime} [ \mathbf {z} ] ^ {T}\right) ^ {- 1} (i, j) G (\mathbf {z}, j) - g (\mathbf {z}). \\ \end{array}
+$$
+
+Here,  $(\mathbf{I} - \mathbf{K}'[\mathbf{z}] + 1\pi'[\mathbf{z}]^T)^{-1}$  is well defined because  $\mathbf{K}'[\mathbf{z}]$  is ergodic and time-reversible for any given  $\mathbf{z}$  (proved in Doshi et al. (2023, Appendix A)). Now that both functions  $H(\theta, i)$  and  $\delta_i - \mathbf{x}$  are bounded for each compact subset of  $\mathbb{R}^D \times \Sigma$  by our assumption A1, function  $G(\mathbf{z}, i)$  is also bounded within the compact subset of its domain. Thus, function  $m_{\mathbf{z}}(i)$  is bounded, and (97) is verified. Moreover, for a fixed  $i \in \mathcal{N}$ ,
+
+$$
+\sum_ {j \in \mathcal {N}} (\mathbf {I} - \mathbf {K} ^ {\prime} [ \mathbf {z} ] + \mathbf {1} \boldsymbol {\pi} ^ {\prime} [ \mathbf {z} ] ^ {T}) ^ {- 1} (i, j) \boldsymbol {\delta} _ {j} = (\mathbf {I} - \mathbf {K} ^ {\prime} [ \mathbf {z} ] + \mathbf {1} \boldsymbol {\pi} ^ {\prime} [ \mathbf {z} ] ^ {T}) _ {[,: i ]} ^ {- 1} = (\mathbf {I} - \mathbf {K} [ \mathbf {x} ] + \mathbf {1} \boldsymbol {\pi} [ \mathbf {x} ] ^ {T}) _ {[,: i ]} ^ {- 1}
+$$
+
+and this vector-valued function is continuous in  $\mathbf{x}$  because  $\mathbf{K}[\mathbf{x}], \pi[\mathbf{x}]$  are continuous. We then rewrite (24) as
+
+$$
+m _ {\mathbf {z}} (i) = \left[ \begin{array}{c} \sum_ {j \in \mathcal {N}} (\mathbf {I} - \mathbf {K} [ \mathbf {x} ] + \mathbf {1} \pi [ \mathbf {x} ] ^ {T}) ^ {- 1} (i, j) H (\mathbf {x}, j) \\ (\mathbf {I} - \mathbf {K} [ \mathbf {x} ] ^ {T} + \pi [ \mathbf {x} ] \mathbf {1} ^ {T}) _ {[,: i ]} ^ {- 1} \end{array} \right] - \left[ \begin{array}{c} \sum_ {i \in \mathcal {N}} \pi_ {i} [ \mathbf {x} ] H (\pmb {\theta}, i) \\ \pmb {\pi} [ \mathbf {x} ] - \mathbf {x} \end{array} \right].
+$$
+
+With continuous functions  $H(\theta, i)$ ,  $\mathbf{K}[\mathbf{x}]$ ,  $\pi[\mathbf{x}]$ , we have  $m_{\mathbf{z}}(i)$  continuous with respect to  $\mathbf{z}$ , so does  $(\mathbf{K}_{\mathbf{z}}^{\prime} m_{\mathbf{z}})(i)$ . This implies that functions  $m_{\mathbf{z}}(i)$  and  $(\mathbf{K}_{\mathbf{z}}^{\prime} m_{\mathbf{z}})(i)$  are locally Lipschitz, which satisfies (98) with  $\phi_{\mathcal{C}}(x) = C_{\mathcal{C}} x$  for some constant  $C_{\mathcal{C}}$  that depends on the compact set  $\mathcal{C}$ . Therefore, condition C4 is checked, and we can apply Theorem G.8 to show the almost convergence result of (19), i.e., almost surely,
+
+$$
+\lim  _ {n \to \infty} \mathbf {x} _ {n} = \boldsymbol {\mu}, \quad \text {a n d} \quad \lim  _ {n \to \infty} \sup  _ {\boldsymbol {\theta} ^ {*} \in \Theta} \inf  _ {\boldsymbol {\theta}} \| \boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*} \| = 0.
+$$
+
+Therefore, the almost sure convergence of  $\mathbf{x}_n$  in Lemma 3.1 is also proved. This finishes the proof in Scenario 1.
+
+# D.2 SCENARIO 2
+
+Now in this subsection, we consider the steps sizes  $\gamma_{n},\beta_{n}$  with  $1 / 2 < a < b\leq 1$  and  $1 / 2 < b < a\leq 1$ . We will frequently use assumptions (B1) - (B5) in Section G.3 and Theorem G.10 to prove the almost sure convergence.
+
+# D.2.1 CASE (I):  $1/2 < a < b \leq 1$
+
+In case (i),  $\theta_{n}$  is on the slow timescale and  $\mathbf{x}_n$  is on the fast timescale because iteration  $\theta_{n}$  has smaller step size than  $\mathbf{x}_n$ , making  $\theta_{n}$  converge slower than  $\mathbf{x}_n$ . Here, we consider the two-timescale SA of the form:
+
+$$
+\boldsymbol {\theta} _ {n + 1} = \boldsymbol {\theta} _ {n} + \beta_ {n + 1} H (\boldsymbol {\theta}, X _ {n + 1}),
+$$
+
+$$
+\mathbf {x} _ {n + 1} = \mathbf {x} _ {n} + \gamma_ {n + 1} (\boldsymbol {\delta} _ {X _ {n + 1}} - \mathbf {x}).
+$$
+
+Now, we verify assumptions (B1) - (B5) listed in Section G.3.
+
+- Assumptions (B1) and (B5) are satisfied by our assumptions A2 and A4.  
+- Our assumption A3 shows that the function  $H(\pmb{\theta}, X)$  is continuous and differentiable w.r.t  $\pmb{\theta}$  and grows linearly with  $\| \pmb{\theta} \|$ . In addition,  $\delta_X - \mathbf{x}$  also satisfies this property. Therefore, (B2) is satisfied.  
+- Now that the function  $\pi[\mathbf{x}] - \mathbf{x}$  is independent of  $\pmb{\theta}$ , we can set  $\rho(\pmb{\theta}) = \pmb{\mu}$  for any  $\pmb{\theta} \in \mathbb{R}^D$  such that  $\pi[\pmb{\mu}] - \pmb{\mu} = 0$  from Doshi et al. (2023, Proposition 3.1), and
+
+$$
+\nabla_ {\mathbf {x}} (\boldsymbol {\pi} (\mathbf {x}) - \mathbf {x}) | _ {\mathbf {x} = \boldsymbol {\mu}} = 2 \alpha \mathbf {u} \mathbf {1} ^ {T} - \alpha \mathbf {P} ^ {T} - (\alpha + 1) \mathbf {I}
+$$
+
+from Doshi et al. (2023, Lemma 3.4), which is Hurwitz. Furthermore,  $\rho(\pmb{\theta}) = \pmb{\mu}$  inherently satisfies the condition  $\|\rho(\pmb{\theta})\| \leq L_2(1 + \|\pmb{\theta}\|)$  for any  $L_2 \geq \|\pmb{\mu}\|$ . Thus, conditions (i) - (iii) in (B3) are satisfied. Additionally,  $\sum_{i \in \mathcal{N}} \pi_i[\rho(\pmb{\theta})] H(\pmb{\theta}, i) = \sum_{i \in \mathcal{N}} \pi_i[\mathbf{x}] = \mathbf{h}(\pmb{\theta})$  such that for  $\pmb{\theta}^* \in \Theta$  defined in assumption A3,  $\sum_{i \in \mathcal{N}} \pi_i[\rho(\pmb{\theta}^*)] H(\pmb{\theta}^*, i) = \mathbf{h}(\pmb{\theta}^*) = 0$ , and  $\nabla_\theta \mathbf{h}(\pmb{\theta}^*)$  is Hurwitz. Therefore, (B3) is checked.
+
+- Assumption (B4) is verified by the nature of SRRW, i.e., its transition kernel  $\mathbf{K}[\mathbf{x}]$  and the corresponding stationary distribution  $\pi[\mathbf{x}]$  with  $\pi[\boldsymbol{\mu}] = \boldsymbol{\mu}$ .
+
+Consequently, assumptions (B1) - (B5) are satisfied by our assumeproins A1 - A4 and by Theorem G.10, we have  $\lim_{n\to \infty}\mathbf{x}_n = \pmb{\mu}$  and  $\theta_{n}\rightarrow \Theta$  almost surely.
+
+Next, we consider  $1/2 < b < a \leq 1$ . As discussed before, (B1), (B2), (B4) and (B5) are satisfied by our assumptions A1 - A4 and the properties of SRRW. The only difference for this step size setting, compared to the previous one  $1/2 < a < b \leq 1$ , is that the roles of  $\theta_{n}, \mathbf{x}_{n}$  are now flipped, that is,  $\theta_{n}$  is now on the fast timescale while  $\mathbf{x}_{n}$  is on the slow timescale. By a much stronger Assumption A3', for any  $\mathbf{x} \in \mathrm{Int}(\Sigma)$ , (i)  $\mathbb{E}_{X \sim \pi[\mathbf{x}]}[H(\rho(\mathbf{x}), X)] = 0$ ; (ii)  $\mathbb{E}_{X \sim \pi[\mathbf{x}]}[\nabla H(\rho(\mathbf{x}), X)]$  is Hurwitz; (iii)  $\|\rho(\mathbf{x})\| \leq L_{2}(1 + \| \mathbf{x}\|)$ . Hence, conditions (i) - (iii) in (B3) are satisfied. Moreover, we have  $\pi[\pmb{\mu}] - \pmb{\mu} = 0$ ,  $\nabla(\pi[\mathbf{x}] - \mathbf{x})|_{\mathbf{x} = \pmb{\mu}}$  being Hurwitz, as mentioned in the previous part. Therefore, (B3) is verified. Accordingly, (B1) - (B5) are checked by our assumptions A1, A2, A3', A4. By Theorem G.10, we have  $\lim_{n \to \infty} \mathbf{x}_n = \pmb{\mu}$  and  $\theta_{n} \to \Theta$  almost surely.
+
+# E PROOF OF THEOREM 3.3
+
+This section is devoted to the proof of Theorem 3.3, which also includes the proof of the CLT results for the SRRW iteration  $\mathbf{x}_n$  in Lemma 3.1. We will use different techniques depending on the step sizes in Assumption A2. Specifically, for step sizes  $\gamma_n = (n + 1)^{-a}$ ,  $\beta_n = (n + 1)^{-b}$ , we will consider three cases: case (i):  $\beta_n = o(\gamma_n)$ ; case (ii):  $\beta_n = \gamma_n$ ; and case (iii):  $\gamma_n = o(\beta_n)$ . For case (ii), we will use the existing CLT result for single-timescale SA in Theorem G.9. For cases (i) and (iii), we will construct our own CLT analysis for the two-timescale structure. We start with case (ii).
+
+# E.1 CASE (II):  $\beta_{n} = \gamma_{n}$
+
+In this part, we stick to the notations for single-timescale SA studied in Section D.1. To utilize Theorem G.9, apart from Conditions C1 - C4 that have been checked in Section D.1, we still need to check conditions C5 and C6 listed in Section G.2.
+
+Assumption A3 corresponds to condition C5. For condition C6, we need to obtain the explicit form of function  $Q_{\mathbf{z}}$  to the Poisson equation defined in (96), that is,
+
+$$
+Q _ {\mathbf {z}} (i) - \left(\mathbf {K} _ {\mathbf {z}} ^ {\prime} Q _ {\mathbf {z}}\right) (i) = \psi (\mathbf {z}, i) - \mathbb {E} _ {j \sim \pi [ \mathbf {z} ]} [ \psi (\mathbf {z}, j) ]
+$$
+
+where
+
+$$
+\psi (\mathbf {z}, i) \triangleq \sum_ {j \in \mathcal {N}} \mathbf {K} _ {\mathbf {z}} ^ {\prime} (i, j) m _ {\mathbf {z}} (j) m _ {\mathbf {z}} (j) ^ {T} - (\mathbf {K} _ {\mathbf {z}} ^ {\prime} m _ {\mathbf {z}}) (i) (\mathbf {K} _ {\mathbf {z}} ^ {\prime} m _ {\mathbf {z}}) (i) ^ {T}.
+$$
+
+Following the similar steps in the derivation of  $m_{\mathbf{z}}(i)$  from (22) to (24), we have
+
+$$
+Q _ {\mathbf {z}} (i) = \sum_ {j \in \mathcal {N}} (\mathbf {I} - \mathbf {K} ^ {\prime} [ \mathbf {z} ] + \mathbf {1} \pmb {\pi} ^ {\prime} [ \mathbf {z} ] ^ {T}) ^ {- 1} (i, j) m _ {\mathbf {z}} (j) - \pi_ {j} ^ {\prime} [ \mathbf {z} ] m _ {\mathbf {z}} (j).
+$$
+
+We also know that  $Q_{\mathbf{z}}(i)$  and  $(\mathbf{K}_{\mathbf{z}}^{\prime}Q_{\mathbf{z}})(i)$  are continuous in  $\mathbf{z}$  for any  $i \in \mathcal{N}$ . For any  $\mathbf{z}$  in a compact set  $\Omega$ ,  $Q_{\mathbf{z}}(i)$  and  $(\mathbf{K}_{\mathbf{z}}^{\prime}Q_{\mathbf{z}})(i)$  are bounded because function  $m_{\mathbf{z}}(i)$  is bounded. Therefore, C6 is checked. By Theorem G.9, assume  $\mathbf{z}_n = \begin{bmatrix} \theta_n \\ \mathbf{x}_n \end{bmatrix}$  converges to a point  $\mathbf{z}^* = \begin{bmatrix} \theta^* \\ \mu \end{bmatrix}$  for  $\theta^* \in \Theta$ , we have
+
+$$
+\gamma_ {n} ^ {- 1 / 2} \left(\mathbf {z} _ {n} - \mathbf {z} ^ {*}\right) \xrightarrow [ n \rightarrow \infty ]{d i s t .} N (0, \mathbf {V}), \tag {26}
+$$
+
+where  $\mathbf{V}$  is the solution of the following Lyapunov equation
+
+$$
+\mathbf {V} \left(\frac {\mathbb {1} _ {\{b = 1 \}}}{2} \mathbf {I} + \nabla g \left(\boldsymbol {z} ^ {*}\right) ^ {T}\right) + \left(\frac {\mathbb {1} _ {\{b = 1 \}}}{2} \mathbf {I} + \nabla g \left(\boldsymbol {z} ^ {*}\right)\right) \mathbf {V} + \mathbf {U} = 0, \tag {27}
+$$
+
+and  $\mathbf{U} = \sum_{i\in \mathcal{N}}\mu_i\left(m_{\mathbf{z}^*}(i)m_{\mathbf{z}^*}(i)\right)^T - (\mathbf{K}_{\mathbf{z}^*}m_{\mathbf{z}^*})(i)(\mathbf{K}_{\mathbf{z}^*}m_{\mathbf{z}^*})(i)^T.$
+
+By algebraic calculations of derivative of  $\pi[\mathbf{x}]$  with respect to  $\mathbf{x}$  in (20), we can rewrite  $\nabla g(\mathbf{z}^*)$  in terms of  $\mathbf{x}, \pmb{\theta}$ , i.e.,
+
+$$
+\begin{array}{l} \mathbf {J} (\alpha) \triangleq \nabla g (\mathbf {z} ^ {*}) = \left[ \begin{array}{c c} \frac {\partial \sum_ {i \in \mathcal {N}} \pi_ {i} [ \mathbf {x} ] H (\boldsymbol {\theta} , i)}{\partial \boldsymbol {\theta}} & \frac {\partial \sum_ {i \in \mathcal {N}} \pi_ {i} [ \mathbf {x} ] H (\boldsymbol {\theta} , i)}{\partial \mathbf {x}} \\ \frac {\partial (\boldsymbol {\pi} [ \mathbf {x} ] - \mathbf {x})}{\partial \boldsymbol {\theta}} & \frac {\partial \boldsymbol {\pi} [ \mathbf {x} ] - \mathbf {x}}{\partial \mathbf {x}} \end{array} \right] _ {\mathbf {z} = \mathbf {z} ^ {*}} \\ = \left[ \begin{array}{c c} \nabla \mathbf {h} (\boldsymbol {\theta} ^ {*}) & - \alpha \mathbf {H} ^ {T} (\mathbf {P} ^ {T} + \mathbf {I}) \\ \mathbf {0} & 2 \alpha \boldsymbol {\mu} \mathbf {1} ^ {T} - \alpha \mathbf {P} ^ {T} - (\alpha + 1) \mathbf {I} \end{array} \right] \triangleq \left[ \begin{array}{c c} \mathbf {J} _ {1 1} & \mathbf {J} _ {1 2} (\alpha) \\ \mathbf {J} _ {2 1} & \mathbf {J} _ {2 2} (\alpha) \end{array} \right], \\ \end{array}
+$$
+
+where matrix  $\mathbf{H} = [H(\pmb{\theta}^*, 1), \dots, H(\pmb{\theta}, N)]^T$ . Then, we further clarify the matrix  $\mathbf{U}$ . Note that
+
+$$
+m _ {\mathbf {z} ^ {*}} (i) = \sum_ {k = 0} ^ {\infty} [ \mathbf {G} (\mathbf {z} ^ {*}) ((\mathbf {P} ^ {k}) ^ {T} - \boldsymbol {\mu} \mathbf {1} ^ {T}) ] _ {[:, i ]} = \sum_ {k = 0} ^ {\infty} [ \mathbf {G} (\mathbf {z} ^ {*}) (\mathbf {P} ^ {k}) ^ {T} ] _ {[:, i ]} = \mathbb {E} \left[ \sum_ {k = 0} ^ {\infty} [ G (\mathbf {z} ^ {*}, X _ {k}) ] \right| X _ {0} = i ], \tag {28}
+$$
+
+where the first equality holds because  $\mathbf{K}^{\prime}[\pmb {\mu}] = \mathbf{P}$  from the definition of SRRW kernel (3), the second equality stems from  $\mathbf{G}(\mathbf{z}^{*})\pmb {\mu} = g(\mathbf{z}^{*}) = 0$  , and the last term is a conditional expectation over the base Markov chain  $\{X_{k}\}_{k\geq 0}$  (with transition kernel  $\mathbf{P}$ ) conditioned on  $X_0 = i$  . Similarly, with  $(\mathbf{K}_{\mathbf{z}}^{\prime}m_{\mathbf{z}})(i)$  in the form of (23), we have
+
+$$
+(\mathbf {K} _ {\mathbf {z}} ^ {\prime} m _ {\mathbf {z}}) (i) = \mathbb {E} \left[ \sum_ {k = 1} ^ {\infty} [ G (\mathbf {z} ^ {*}, X _ {k}) ] \Bigg | X _ {0} = i \right].
+$$
+
+From the form  $\sum_{i\in \mathcal{N}}\mu_i$  inside the matrix  $\mathbf{U}$ , the Markov chain  $\{X_k\}_{k\geq 0}$  is in its stationary regime from the beginning, i.e.,  $X_{k}\sim \pmb{\mu}$  for any  $k\geq 0$ . Hence,
+
+$$
+\begin{array}{l} \mathbf {U} = \mathbb {E} \left[ \left(\sum_ {k = 0} ^ {\infty} [ G (\mathbf {z} ^ {*}, X _ {k}) ]\right) \left(\sum_ {k = 0} ^ {\infty} [ G (\mathbf {z} ^ {*}, X _ {k}) ]\right) ^ {T} \right] \\ - \mathbb {E} \left[ \left(\sum_ {k = 1} ^ {\infty} [ G (\mathbf {z} ^ {*}, X _ {k}) ]\right) \left(\sum_ {k = 1} ^ {\infty} [ G (\mathbf {z} ^ {*}, X _ {k}) ]\right) ^ {T} \right] \\ = \mathbb {E} \left[ G \left(\mathbf {z} ^ {*}, X _ {0}\right) G \left(\mathbf {z} ^ {*}, X _ {0}\right) ^ {T} \right] + \mathbb {E} \left[ G \left(\mathbf {z} ^ {*}, X _ {0}\right) \left(\sum_ {k = 1} ^ {\infty} G \left(\mathbf {z} ^ {*}, X _ {k}\right)\right) ^ {T} \right] \tag {29} \\ + \mathbb {E} \left[ \left(\sum_ {k = 1} ^ {\infty} G (\mathbf {z} ^ {*}, X _ {k})\right) G (\mathbf {z} ^ {*}, X _ {0}) ^ {T} \right] \\ = \operatorname {C o v} \left(G \left(\mathbf {z} ^ {*}, X _ {0}\right), G \left(\mathbf {z} ^ {*}, X _ {0}\right)\right) \\ + \sum_ {k = 1} ^ {\infty} \left[ \operatorname {C o v} \left(G \left(\mathbf {z} ^ {*}, X _ {0}\right), G \left(\mathbf {z} ^ {*}, X _ {k}\right)\right) + \operatorname {C o v} \left(G \left(\mathbf {z} ^ {*}, X _ {k}\right), G \left(\mathbf {z} ^ {*}, X _ {0}\right)\right) \right], \\ \end{array}
+$$
+
+where the covariance between  $G(\mathbf{z}^*, X_0)$  and  $G(\mathbf{z}^*, X_k)$  for the Markov chain  $\{X_n\}$  in the stationary regime is  $\mathrm{Cov}(G(\mathbf{z}^*, X_0), G(\mathbf{z}^*, X_k))$ . By Brémaud (2013, Theorem 6.3.7), it is demonstrated that  $\mathbf{U}$  is the sampling covariance of the base Markov chain  $\mathbf{P}$  for the test function  $G(\mathbf{z}^*, \cdot)$ . Moreover, Brémaud (2013, equation (6.34)) states that this sampling covariance  $\mathbf{U}$  can be rewritten in the following form:
+
+$$
+\mathbf {U} = \sum_ {i = 1} ^ {N - 1} \mathbf {G} \left(\mathbf {z} ^ {*}\right) ^ {T} \mathbf {u} _ {i} \mathbf {u} _ {i} \mathbf {G} \left(\mathbf {z} ^ {*}\right) = \sum_ {i = 1} ^ {N - 1} \frac {1 + \lambda_ {i}}{1 - \lambda_ {i}} \left[ \begin{array}{c c} \mathbf {H} ^ {T} \mathbf {u} _ {i} \mathbf {u} _ {i} ^ {T} \mathbf {H} & \mathbf {H} ^ {T} \mathbf {u} _ {i} \mathbf {u} _ {i} ^ {T} \\ \mathbf {u} _ {i} \mathbf {u} _ {i} ^ {T} \mathbf {H} & \mathbf {u} _ {i} \mathbf {u} _ {i} ^ {T} \end{array} \right] \triangleq \left[ \begin{array}{l l} \mathbf {U} _ {1 1} & \mathbf {U} _ {1 2} \\ \mathbf {U} _ {2 1} & \mathbf {U} _ {2 2} \end{array} \right], \tag {30}
+$$
+
+where  $\{(\lambda_i, \mathbf{u}_i)\}_{i \in \mathcal{N}}$  is the eigenpair of the transition kernel  $\mathbf{P}$  of the ergodic and time-reversible base Markov chain. This completes the proof of case 1.
+
+Remark E.1. For the CLT result (26), we can look further into the asymptotic covariance matrix  $\mathbf{V}$  as in (27). For convenience, we denote  $\mathbf{V} = \begin{bmatrix} \mathbf{V}_{11} & \mathbf{V}_{12}\\ \mathbf{V}_{21} & \mathbf{V}_{22} \end{bmatrix}$  and  $\mathbf{U}$  in the form of (30) such that
+
+$$
+\left[ \begin{array}{l l} \mathbf {V} _ {1 1} & \mathbf {V} _ {1 2} \\ \mathbf {V} _ {2 1} & \mathbf {V} _ {2 2} \end{array} \right] \left(\frac {\mathbb {1} _ {\{b = 1 \}}}{2} \mathbf {I} + \mathbf {J} (\alpha) ^ {T}\right) + \left(\frac {\mathbb {1} _ {\{b = 1 \}}}{2} \mathbf {I} + \mathbf {J} (\alpha)\right) \left[ \begin{array}{l l} \mathbf {V} _ {1 1} & \mathbf {V} _ {1 2} \\ \mathbf {V} _ {2 1} & \mathbf {V} _ {2 2} \end{array} \right] + \mathbf {U} = 0. \tag {31}
+$$
+
+For the SRRW iteration  $\mathbf{x}_n$ , from (26) we know that  $\gamma_n^{-1/2}(\mathbf{x}_n - \boldsymbol{\mu}) \xrightarrow[n\to\infty]{dist.} N(\mathbf{0}, \mathbf{V}_4)$ . Thus, in this remark, we want to obtain the closed form of  $\mathbf{V}_{22}$ . By algebraic computations of the bottom-right sub-block matrix, we have
+
+$$
+\begin{array}{l} \left(2 \alpha \boldsymbol {\mu} \mathbf {1} ^ {T} - \alpha \mathbf {P} ^ {T} - \left(\alpha + 1 - \frac {\mathbb {1} _ {\{a = 1 \}}}{2}\right) \mathbf {I}\right) \mathbf {V} _ {2 2} \\ + \mathbf {V} _ {2 2} \left(2 \alpha \boldsymbol {\mu} \mathbf {1} ^ {T} - \alpha \mathbf {P} ^ {T} - \left(\alpha + 1 - \frac {\mathbb {1} _ {\{a = 1 \}}}{2}\right) \mathbf {I}\right) ^ {T} \\ + \mathbf {U} _ {2 2} = 0. \\ \end{array}
+$$
+
+By using result of the closed form solution to the Lyapunov equation (e.g., Lemma G.1) and the eigendecomposition of  $\mathbf{P}$ , we have
+
+$$
+\mathbf {V} _ {2 2} = \sum_ {i = 1} ^ {N - 1} \frac {1}{2 \alpha \left(1 + \lambda_ {i}\right) + 2 - \mathbb {1} _ {\{a = 1 \}}} \cdot \frac {1 + \lambda_ {i}}{1 - \lambda_ {i}} \mathbf {u} _ {i} \mathbf {u} _ {i} ^ {T}. \tag {32}
+$$
+
+# E.2 CASE (I):  $\beta_{n} = o(\gamma_{n})$
+
+In this part, we mainly focus on the CLT of the SA iteration  $\theta_{n}$  because the SRRW iteration  $\mathbf{x}_n$  is independent of  $\theta_{n}$  and its CLT result has been shown in Remark E.1.
+
+# E.2.1 DECOMPOSITION OF SA-SRRW ITERATION (4)
+
+We slightly abuse the math notation and define the function
+
+$$
+\mathbf {h} (\boldsymbol {\theta}, \mathbf {x}) \triangleq \mathbb {E} _ {i \sim \pi [ \mathbf {x} ]} H (\boldsymbol {\theta}, i) = \sum_ {i \in \mathcal {N}} \pi_ {i} [ \mathbf {x} ] H (\boldsymbol {\theta}, i)
+$$
+
+such that  $\mathbf{h}(\pmb {\theta},\pmb {\mu})\equiv \mathbf{h}(\pmb {\theta})$  . Then, we reformulate (25) as
+
+$$
+\boldsymbol {\theta} _ {n + 1} = \boldsymbol {\theta} _ {n} + \beta_ {n + 1} \mathbf {h} \left(\boldsymbol {\theta} _ {n}, \mathbf {x} _ {n}\right) + \beta_ {n + 1} \left(H \left(\boldsymbol {\theta} _ {n}, X _ {n + 1}\right) - \mathbf {h} \left(\boldsymbol {\theta} _ {n}, \mathbf {x} _ {n}\right)\right). \tag {33a}
+$$
+
+$$
+\mathbf {x} _ {n + 1} = \mathbf {x} _ {n} + \gamma_ {n + 1} \left(\boldsymbol {\pi} \left[ \mathbf {x} _ {n} \right] - \mathbf {x} _ {n}\right) + \gamma_ {n + 1} \left(\boldsymbol {\delta} _ {X _ {n + 1}}\right) - \boldsymbol {\pi} \left[ \mathbf {x} _ {n} \right]). \tag {33b}
+$$
+
+There exist functions  $q_{\mathbf{x}}:\mathcal{N}\to \mathbb{R}^{N},\tilde{H}_{\boldsymbol{\theta},\mathbf{x}}:\mathcal{N}\to \mathbb{R}^{D}$  satisfying the following Poisson equations
+
+$$
+\boldsymbol {\delta} _ {i} - \boldsymbol {\pi} (\mathbf {x}) = q _ {\mathbf {x}} (i) - \left(\mathbf {K} _ {\mathbf {x}} q _ {\mathbf {x}}\right) (i) \tag {34a}
+$$
+
+$$
+H (\boldsymbol {\theta}, i) - \mathbf {h} (\boldsymbol {\theta}, \mathbf {x}) = \tilde {H} _ {\boldsymbol {\theta}, \mathbf {x}} (i) - \left(\mathbf {K} _ {\mathbf {x}} \tilde {H} _ {\boldsymbol {\theta}, \mathbf {x}}\right) (i), \tag {34b}
+$$
+
+for any  $\pmb{\theta} \in \mathbb{R}^{D}$ ,  $\mathbf{x} \in \mathrm{Int}(\Sigma)$  and  $i \in \mathcal{N}$ , where  $(\mathbf{K}_{\mathbf{x}}q_{\mathbf{x}})(i) \triangleq \sum_{j \in \mathcal{N}} K_{ij}[\mathbf{x}] q_{\mathbf{x}}(j)$ ,  $(\mathbf{K}_{\mathbf{x}}\tilde{H}_{\pmb{\theta},\mathbf{x}})(j) \triangleq \sum_{j \in \mathcal{N}} K_{ij}[\mathbf{x}]\tilde{H}_{\pmb{\theta},\mathbf{x}}(j)$ . The existence and explicit form of the solutions  $q_{\mathbf{x}}, \tilde{H}_{\pmb{\theta},\mathbf{x}}$ , which are continuous w.r.t  $\mathbf{x}$ ,  $\pmb{\theta}$ , follow the similar steps that can be found in Section D.1 from (22) to (24). Thus, we can further decompose (33) into
+
+$$
+\begin{array}{l} \boldsymbol {\theta} _ {n + 1} = \boldsymbol {\theta} _ {n} + \beta_ {n + 1} \mathbf {h} (\boldsymbol {\theta} _ {n}, \mathbf {x} _ {n}) + \beta_ {n + 1} \underbrace {(\tilde {H} _ {\boldsymbol {\theta} _ {n} , \mathbf {x} _ {n}} (X _ {n + 1}) - (\mathbf {K} _ {\mathbf {x} _ {n}} \tilde {H} _ {\boldsymbol {\theta} _ {n} , \mathbf {x} _ {n}}) (X _ {n}))} _ {M _ {n + 1} ^ {(\boldsymbol {\theta})}} \\ + \beta_ {n + 1} \underbrace {\left(\left(\mathbf {K} _ {\mathbf {x} _ {n + 1}} \tilde {H} _ {\boldsymbol {\theta} _ {n + 1} , \mathbf {x} _ {n + 1}}\right) \left(X _ {n + 1}\right) - \left(\mathbf {K} _ {\mathbf {x} _ {n}} \tilde {H} _ {\boldsymbol {\theta} _ {n} , \mathbf {x} _ {n}}\right) \left(X _ {n + 1}\right)\right)} _ {r _ {n} ^ {(\boldsymbol {\theta}, 1)}} \tag {35a} \\ + \beta_ {n + 1} \underbrace {((\mathbf {K _ {x}} _ {n} \tilde {H} ) (X _ {n}) - (\mathbf {K _ {x}} _ {n + 1} \tilde {H} ) (X _ {n + 1}))} _ {r _ {n} ^ {(\pmb {\theta}, 2)}}, \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \mathbf {x} _ {n + 1} = \mathbf {x} _ {n} + \gamma_ {n + 1} (\boldsymbol {\pi} (\mathbf {x} _ {n}) - \mathbf {x} _ {n}) + \gamma_ {n + 1} \underbrace {(q _ {\mathbf {x} _ {n}} (X _ {n + 1}) - (\mathbf {K} _ {\mathbf {x} _ {n}} q _ {\mathbf {x} _ {n}}) (X _ {n}))} _ {M _ {n + 1} ^ {(\mathbf {x})}} \\ + \gamma_ {n + 1} \underbrace {\left([ \mathbf {K} _ {\mathbf {x} _ {n}} q _ {\mathbf {x} _ {n}} ] \left(X _ {n + 1}\right) - [ \mathbf {K} _ {\mathbf {x} _ {n}} q _ {\mathbf {x} _ {n + 1}} ] \left(X _ {n + 1}\right)\right)} _ {r _ {n} ^ {(\mathbf {x}, 1)}} \tag {35b} \\ + \gamma_ {n + 1} \underbrace {\left((\mathbf {K} _ {\mathbf {x} _ {n}} q _ {\mathbf {x} _ {n}}) (X _ {n}) - (\mathbf {K} _ {\mathbf {x} _ {n + 1}} q _ {\mathbf {x} _ {n + 1}}) (X _ {n + 1})\right)} _ {r _ {n} ^ {(\mathbf {x}, 2)}}. \\ \end{array}
+$$
+
+such that
+
+$$
+\boldsymbol {\theta} _ {n + 1} = \boldsymbol {\theta} _ {n} + \beta_ {n + 1} \mathbf {h} \left(\boldsymbol {\theta} _ {n}, \mathbf {x} _ {n}\right) + \beta_ {n + 1} M _ {n + 1} ^ {(\boldsymbol {\theta})} + \beta_ {n + 1} r _ {n} ^ {(\boldsymbol {\theta}, 1)} + \beta_ {n + 1} r _ {n} ^ {(\boldsymbol {\theta}, 2)}, \tag {36a}
+$$
+
+$$
+\mathbf {x} _ {n + 1} = \mathbf {x} _ {n} + \gamma_ {n + 1} \left(\boldsymbol {\pi} \left(\mathbf {x} _ {n}\right) - \mathbf {x} _ {n}\right) + \gamma_ {n + 1} M _ {n + 1} ^ {(\mathbf {x})} + \gamma_ {n + 1} r _ {n} ^ {(\mathbf {x}, 1)} + \gamma_ {n + 1} r _ {n} ^ {(\mathbf {x}, 2)}. \tag {36b}
+$$
+
+We can observe that (36) differs from the expression in Konda & Tsitsiklis (2004); Mokkadem & Pelletier (2006), which studied the two-timescale SA with Martingale difference noise. Here, due to the presence of the iterate-dependent Markovian noise and the application of the Poisson equation technique, we have additional non-vanishing terms  $r_n^{(\theta, 2)}$ ,  $r_n^{(\mathbf{x}, 2)}$ , which will be further examined in Lemma E.2. Additionally, when we apply the Poisson equation to the Martingale difference terms  $M_{n+1}^{(\theta)}$ ,  $M_{n+1}^{(\mathbf{x})}$ , we find that there are some covariances that are also non-vanishing as in Lemma E.1. We will mention this again when we obtain those covariances. These extra non-zero noise terms make our analysis distinct from the previous ones since the key assumption (A4) in Mokkadem & Pelletier (2006) is not satisfied. We demonstrate that the long-term average performance of these terms can be managed so that they do not affect the final CLT result.
+
+Analysis of Terms  $M_{n + 1}^{(\theta)},M_{n + 1}^{(\mathbf{x})}$
+
+Consider the filtration  $\mathcal{F}_n \triangleq \sigma(\pmb{\theta}_0, \mathbf{x}_0, X_0, \dots, \pmb{\theta}_n, \mathbf{x}_n, X_n)$ , it is evident that  $M_{n+1}^{(\pmb{\theta})}, M_{n+1}^{(\mathbf{x})}$  are Martingale difference sequences adapted to  $\mathcal{F}_n$ . Then, we have
+
+$$
+\begin{array}{l} \mathbb {E} \left[ M _ {n + 1} ^ {(\mathbf {x})} (M _ {n + 1} ^ {(\mathbf {x})}) ^ {T} \Big | \mathcal {F} _ {n} \right] \\ = \mathbb {E} \left[ q _ {\mathbf {x} _ {n}} \left(X _ {n + 1}\right) q _ {\mathbf {x} _ {n}} \left(X _ {n + 1}\right) ^ {T} \mid \mathcal {F} _ {n} \right] + \left(\mathbf {K} _ {\mathbf {x} _ {n}} q _ {\mathbf {x} _ {n}}\right) \left(X _ {n}\right) \left(\left(\mathbf {K} _ {\mathbf {x} _ {n}} q _ {\mathbf {x} _ {n}}\right) \left(X _ {n}\right)\right) ^ {T} \tag {37} \\ - \mathbb {E} \left[ q _ {\mathbf {x} _ {n}} \left(X _ {n + 1}\right) \mid \mathcal {F} _ {n} \right] \left(\mathbf {K} _ {\mathbf {x} _ {n}} q _ {\mathbf {x} _ {n}}\right) \left(X _ {n}\right)) ^ {T} - \left(\mathbf {K} _ {\mathbf {x} _ {n}} q _ {\mathbf {x} _ {n}}\right) \left(X _ {n}\right) \mathbb {E} \left[ q _ {\mathbf {x} _ {n}} \left(X _ {n + 1}\right) ^ {T} \mid \mathcal {F} _ {n} \right] \\ = \mathbb {E} \left[ q _ {\mathbf {x} _ {n}} \left(X _ {n + 1}\right) q _ {\mathbf {x} _ {n}} \left(X _ {n + 1}\right) ^ {T} \mid \mathcal {F} _ {n} \right] - \left(\mathbf {K} _ {\mathbf {x} _ {n}} q _ {\mathbf {x} _ {n}}\right) \left(X _ {n}\right) \left(\left(\mathbf {K} _ {\mathbf {x} _ {n}} q _ {\mathbf {x} _ {n}}\right) \left(X _ {n}\right)\right) ^ {T}. \\ \end{array}
+$$
+
+Similarly, we have
+
+$$
+\begin{array}{l} \mathbb {E} \left[ M _ {n + 1} ^ {(\boldsymbol {\theta})} \left(M _ {n + 1} ^ {(\boldsymbol {\theta})}\right) ^ {T} \mid \mathcal {F} _ {n} \right] \tag {38} \\ = \mathbb {E} \left[ \tilde {H} _ {\boldsymbol {\theta} _ {n}, \mathbf {x} _ {n}} \left(X _ {n + 1}\right) \tilde {H} _ {\boldsymbol {\theta} _ {n}, \mathbf {x} _ {n}} \left(X _ {n + 1}\right) ^ {T} \mid \mathcal {F} _ {n} \right] - \left(\mathbf {K} _ {\mathbf {x} _ {n}} \tilde {H} _ {\boldsymbol {\theta} _ {n}, \mathbf {x} _ {n}}\right) \left(X _ {n}\right) \left(\left(\mathbf {K} _ {\mathbf {x} _ {n}} \tilde {H} _ {\boldsymbol {\theta} _ {n}, \mathbf {x} _ {n}}\right) \left(X _ {n}\right)\right) ^ {T}, \\ \end{array}
+$$
+
+and
+
+$$
+\begin{array}{l} \mathbb {E} \left[ M _ {n + 1} ^ {(\mathbf {x})} (M _ {n + 1} ^ {(\boldsymbol {\theta})}) ^ {T} \Big | \mathcal {F} _ {n} \right] \\ = \mathbb {E} [ q _ {\mathbf {x} _ {n}} (X _ {n + 1}) \tilde {H} _ {\pmb {\theta} _ {n}, \mathbf {x} _ {n}} (X _ {n + 1}) ^ {T} | \mathcal {F} _ {n} ] - (\mathbf {K} _ {\mathbf {x} _ {n}} q _ {\mathbf {x} _ {n}}) (X _ {n}) ((\mathbf {K} _ {\mathbf {x} _ {n}} \tilde {H} _ {\pmb {\theta} _ {n}, \mathbf {x} _ {n}}) (X _ {n})) ^ {T}. \\ \end{array}
+$$
+
+We now focus on  $\mathbb{E}\left[M_{n + 1}^{(\mathbf{x})}(M_{n + 1}^{(\mathbf{x})})^T\Big|\mathcal{F}_n\right]$ . Denote by
+
+$$
+V _ {1} (\mathbf {x}, i) \triangleq \sum_ {j \in \mathcal {N}} \mathbf {K} _ {i, j} [ \mathbf {x} ] q _ {\mathbf {x}} (j) q _ {\mathbf {x}} (j) ^ {T} - \left(\mathbf {K} _ {\mathbf {x}} q _ {\mathbf {x}}\right) (i) \left(\left(\mathbf {K} _ {\mathbf {x}} q _ {\mathbf {x}}\right) (i)\right) ^ {T}, \tag {39}
+$$
+
+and let its expectation w.r.t the stationary distribution  $\pi (\mathbf{x})$  be  $v_{1}(\mathbf{x})\triangleq \mathbb{E}_{i\sim \pi (\mathbf{x})}[V_{1}(\mathbf{x},i)]$ , we can construct another Poisson equation, i.e.,
+
+$$
+\begin{array}{l} \mathbb {E} \left[ M _ {n + 1} ^ {(\mathbf {x})} (M _ {n + 1} ^ {(\mathbf {x})}) ^ {T} \Big | \mathcal {F} _ {n} \right] - \sum_ {X _ {n} \in \mathcal {N}} \pi_ {X _ {n}} (\mathbf {x} _ {n}) \mathbb {E} \left[ M _ {n + 1} ^ {(\mathbf {x})} (M _ {n + 1} ^ {(\mathbf {x})}) ^ {T} \Big | \mathcal {F} _ {n} \right] \\ = V _ {1} \left(\mathbf {x} _ {n}, X _ {n + 1}\right) - v _ {1} \left(\mathbf {x} _ {n}\right) \\ = \varphi_ {\mathbf {x}} ^ {(1)} (X _ {n + 1}) - (\mathbf {K} _ {\mathbf {x} _ {n}} \varphi_ {\mathbf {x} _ {n}} ^ {(1)}) (X _ {n + 1}), \\ \end{array}
+$$
+
+for some matrix-valued function  $\varphi_{\mathbf{x}}^{(1)}:\mathcal{N}\to \mathbb{R}^{N\times N}$ . Since  $q_{\mathbf{x}}$  and  $\mathbf{K}[\mathbf{x}]$  are continuous in  $\mathbf{x}$ , functions  $V_{1},v_{1}$  are also continuous in  $\mathbf{x}$ . Then, we can decompose (39) into
+
+$$
+\begin{array}{l} V _ {1} (\mathbf {x} _ {n}, X _ {n + 1}) = \underbrace {v _ {1} (\boldsymbol {\mu})} _ {\mathbf {U} _ {2 2}} + \underbrace {v _ {1} (\mathbf {x} _ {n}) - v _ {1} (\boldsymbol {\mu})} _ {\mathbf {D} _ {n} ^ {(1)}} + \underbrace {\varphi_ {\mathbf {x} _ {n}} ^ {(1)} (X _ {n + 1}) - (\mathbf {K} _ {\mathbf {x} _ {n}} \varphi_ {\mathbf {x} _ {n}} ^ {(1)}) (X _ {n})} _ {\mathbf {J} _ {n} ^ {(1, a)}} \\ + \underbrace {\left(\mathbf {K} _ {\mathbf {x} _ {n}} \varphi_ {\mathbf {x} _ {n}} ^ {(1)}\right) \left(X _ {n}\right) - \left(\mathbf {K} _ {\mathbf {x} _ {n}} \varphi_ {\mathbf {x} _ {n}} ^ {(1)}\right) \left(X _ {n + 1}\right)} _ {\mathbf {J} _ {n} ^ {(1, b)}}. \tag {40} \\ \end{array}
+$$
+
+Thus, we have
+
+$$
+\mathbb {E} \left[ M _ {n + 1} ^ {(\mathbf {x})} \left(M _ {n + 1} ^ {(\mathbf {x})}\right) ^ {T} \mid \mathcal {F} _ {n} \right] = \mathbf {U} _ {2 2} + \mathbf {D} _ {n} ^ {(1)} + \mathbf {J} _ {n} ^ {(1)}, \tag {41}
+$$
+
+where  $\mathbf{J}_n^{(1)} = \mathbf{J}_n^{(1,a)} + \mathbf{J}_n^{(1,b)}$ .
+
+Following the similar steps above, we can decompose  $\mathbb{E}\left[M_{n + 1}^{(\mathbf{x})}(M_{n + 1}^{(\boldsymbol{\theta})})^T\Big|\mathcal{F}_n\right]$  and  $\mathbb{E}\left[M_{n + 1}^{(\boldsymbol{\theta})}(M_{n + 1}^{(\boldsymbol{\theta})})^T\Big|\mathcal{F}_n\right]$  as
+
+$$
+\mathbb {E} \left[ M _ {n + 1} ^ {(\mathbf {x})} \left(M _ {n + 1} ^ {(\boldsymbol {\theta})}\right) ^ {T} \mid \mathcal {F} _ {n} \right] = \mathbf {U} _ {2 1} + \mathbf {D} _ {n} ^ {(2)} + \mathbf {J} _ {n} ^ {(2)}, \tag {42a}
+$$
+
+$$
+\mathbb {E} \left[ M _ {n + 1} ^ {(\boldsymbol {\theta})} \left(M _ {n + 1} ^ {(\boldsymbol {\theta})}\right) ^ {T} \mid \mathcal {F} _ {n} \right] = \mathbf {U} _ {1 1} + \mathbf {D} _ {n} ^ {(3)} + \mathbf {J} _ {n} ^ {(3)}. \tag {42b}
+$$
+
+where  $\mathbf{J}_n^{(2)} = \mathbf{J}_n^{(2,a)} + \mathbf{J}_n^{(2,b)}$  and  $\mathbf{J}_n^{(3)} = \mathbf{J}_n^{(3,a)} + \mathbf{J}_n^{(3,b)}$ . Here, we note that matrices  $\mathbf{J}_n^i$  for  $i = 1,2,3$  are in presence of the current CLT analysis of the two-timescale SA with Martingale difference noise. In addition,  $\mathbf{U}_{11}$ ,  $\mathbf{U}_{12}$  and  $\mathbf{U}_{22}$  inherently include the information of the underlying Markov chain (with its eigenpair  $(\lambda_i,\mathbf{u}_i)$ ), which is an extension of the previous works (Konda & Tsitsiklis, 2004; Mokkadem & Pelletier, 2006).
+
+Lemma E.1. For  $M_{n + 1}^{(\theta)},M_{n + 1}^{(\mathbf{x})}$  defined in (35) and their decomposition in (41) and (42), we have
+
+$$
+\mathbf {U} _ {1 1} = \sum_ {i = 1} ^ {N - 1} \frac {1 + \lambda_ {i}}{1 - \lambda_ {i}} \mathbf {u} _ {i} \mathbf {u} _ {i} ^ {T}, \quad \mathbf {U} _ {2 1} = \sum_ {i = 1} ^ {N - 1} \frac {1 + \lambda_ {i}}{1 - \lambda_ {i}} \mathbf {u} _ {i} \mathbf {u} _ {i} ^ {T} \mathbf {H}, \quad \mathbf {U} _ {1 1} = \sum_ {i = 1} ^ {N - 1} \frac {1 + \lambda_ {i}}{1 - \lambda_ {i}} \mathbf {H} ^ {T} \mathbf {u} _ {i} \mathbf {u} _ {i} ^ {T} \mathbf {H}, \tag {43a}
+$$
+
+$$
+\lim  _ {n \rightarrow \infty} \mathbf {D} _ {n} ^ {(i)} = 0 a. s. f o r i = 1, 2, 3, \tag {43b}
+$$
+
+$$
+\lim  _ {n \rightarrow \infty} \gamma_ {n} \mathbb {E} \left[\left\| \sum_ {k = 1} ^ {n} \mathbf {J} _ {k} ^ {(i)} \right\|\right] = 0, \quad f o r \quad i = 1, 2, 3. \tag {43c}
+$$
+
+Proof. We now provide the properties of the four terms inside (41) as an example. Note that
+
+$$
+\begin{array}{l} \mathbf {U} _ {1 1} = \mathbb {E} _ {i \sim \boldsymbol {\mu}} [ V _ {1} (\boldsymbol {\mu}, i) ] = \sum_ {i \in \mathcal {N}} \mu_ {i} \left[ \sum_ {j \in \mathcal {N}} \mathbf {P} (i, j) q _ {\boldsymbol {\mu}} (j) q _ {\boldsymbol {\mu}} (j) ^ {T} - (\mathbf {P} q _ {\boldsymbol {\mu}}) (i) ((\mathbf {P} q _ {\boldsymbol {\mu}}) (i)) ^ {T} \right] \\ = \sum_ {j \in \mathcal {N}} \mu_ {j} q _ {\pmb {\mu}} (j) q _ {\pmb {\mu}} (j) ^ {T} - (\mathbf {P} q _ {\pmb {\mu}}) (j) ((\mathbf {P} q _ {\pmb {\mu}}) (j)) ^ {T}. \\ \end{array}
+$$
+
+We can see that it has exactly the same structure as matrix  $\mathbf{U}$  in (27). Following the similar steps in deducing the explicit form of  $\mathbf{U}$  from (28) to (30), we get
+
+$$
+\mathbf {U} _ {1 1} = \sum_ {i = 1} ^ {N - 1} \frac {1 + \lambda_ {i}}{1 - \lambda_ {i}} \mathbf {u} _ {i} \mathbf {u} _ {i} ^ {T}. \tag {44}
+$$
+
+By the almost sure convergence result  $\mathbf{x}_n\to \pmb{\mu}$  in Lemma 3.1,  $v_{1}(\mathbf{x}_{n})\rightarrow v_{1}(\pmb {\mu})$  a.s. such that  $\lim_{n\to \infty}\mathbf{D}_n^{(1)} = 0$  a.s.
+
+We next prove that  $\lim_{n\to \infty}\gamma_n\mathbb{E}\left[\left\| \sum_{k = 1}^n{\bf J}_k^{(1,a)}\right\|\right] = 0$  and  $\lim_{n\to \infty}\gamma_n\mathbb{E}\left[\left\| \sum_{k = 1}^n{\bf J}_k^{(1,b)}\right\|\right] = 0.$
+
+Since  $\{\mathbf{J}_n^{(1,a)}\}$  is a Martingale difference sequence adapted to  $\mathcal{F}_n$ , with the Burkholder inequality in Lemma G.2 and  $p = 1$ , we show that
+
+$$
+\mathbb {E} \left[ \left\| \sum_ {k = 1} ^ {n} \mathbf {J} _ {k} ^ {(1, a)} \right\| \right] \leq C _ {1} \mathbb {E} \left[ \sqrt {\left(\sum_ {k = 1} ^ {n} \left\| \mathbf {J} _ {k} ^ {(1 , a)} \right\| ^ {2}\right)} \right]. \tag {45}
+$$
+
+By assumption A4,  $\mathbf{x}_n$  is always within some compact set  $\Omega$  such that  $\sup_n\| \mathbf{J}_n^{(1,a)}\| \leq C_\Omega < \infty$  and for a given trajectory  $\omega$  of  $\mathbf{x}_n(\omega)$
+
+$$
+\gamma_ {n} C _ {p} \sqrt {\left(\sum_ {k = 1} ^ {n} \left\| \mathbf {J} _ {k} ^ {(1 , a)} \right\| ^ {2}\right)} \leq C _ {p} C _ {\Omega} \gamma_ {n} \sqrt {n}, \tag {46}
+$$
+
+and the last term decreases to zero in  $n$  since  $a > 1 / 2$ .
+
+For  $\mathbf{J}_n^{(1,b)}$ , we use Abel transformation and obtain
+
+$$
+\begin{array}{l} \sum_ {k = 1} ^ {n} \mathbf {J} _ {k} ^ {(1, b)} = \sum_ {k = 1} ^ {n} ((\mathbf {K} _ {\mathbf {x} _ {k}} \varphi_ {\mathbf {x} _ {k}} ^ {(1)}) (X _ {k - 1}) - (\mathbf {K} _ {\mathbf {x} _ {k - 1}} \varphi_ {\mathbf {x} _ {k - 1}} ^ {(1)}) (X _ {k - 1})) \\ + \left(\mathbf {K} _ {\mathbf {x} _ {0}} \varphi_ {\mathbf {x} _ {0}} ^ {(1)}\right) (X _ {0}) - \left(\mathbf {K} _ {\mathbf {x} _ {n}} \varphi_ {\mathbf {x} _ {n}} ^ {(1)}\right) (X _ {n}). \\ \end{array}
+$$
+
+Since  $(\mathbf{K}_{\mathbf{x}}\varphi_{\mathbf{x}}^{(1)})(X)$  is continuous in  $\mathbf{x}$ , for  $\mathbf{x}_n$  within a compact set  $\Omega$  (assumption A4), it is local Lipschitz with a constant  $L_{\Omega}$  such that
+
+$$
+\left\| \right.\left(\mathbf {K} _ {\mathbf {x} _ {k}} \varphi_ {\mathbf {x} _ {k}} ^ {(1)}\right)\left(X _ {k - 1}\right) - \mathbf {K} _ {\mathbf {x} _ {k - 1}} \varphi_ {\mathbf {x} _ {k - 1}} ^ {(1)}\left. \right)\left(X _ {k - 1}\right)\left. \right\| \leq L _ {\Omega} \| \mathbf {x} _ {k} - \mathbf {x} _ {k - 1} \| \leq 2 L _ {\Omega} \gamma_ {k}.
+$$
+
+where the last inequality arises from (4b), i.e.,  $\mathbf{x}_k - \mathbf{x}_{k-1} = \gamma_k (\delta_{X_k} - \mathbf{x}_{k-1})$  and  $\|\delta_{X_k} - \mathbf{x}_{k-1}\| \leq 2$  because  $\mathbf{x}_n \in \mathrm{Int}(\Sigma)$ . Also,  $\|(K_{\mathbf{x}_0} \varphi_{\mathbf{x}_0}^{(1)})(X_0)\| + \|(K_{\mathbf{x}_n} \varphi_{\mathbf{x}_n}^{(1)})(X_n)\|$  are upper-bounded by some positive constant  $C_\Omega'$ . This implies that
+
+$$
+\left\| \sum_ {k = 1} ^ {n} \mathbf {J} _ {k} ^ {(1, b)} \right\| \leq C _ {\Omega} ^ {\prime} + 2 L _ {\Omega} \sum_ {k = 1} ^ {n} \gamma_ {k}.
+$$
+
+Note that
+
+$$
+\gamma_ {n} \left\| \sum_ {k = 1} ^ {n} \mathbf {J} _ {k} ^ {(1, b)} \right\| \leq \gamma_ {n} C _ {\Omega} ^ {\prime} + 2 L _ {\Omega} \gamma_ {n} \sum_ {k = 1} ^ {n} \gamma_ {k} \leq \gamma_ {n} C _ {\Omega} ^ {\prime} + \frac {2 L _ {\Omega}}{a} n ^ {1 - 2 a}, \tag {47}
+$$
+
+where the last inequality is from  $\sum_{k=1}^{n} \gamma_k < \frac{1}{a} n^{1-a}$ . We observe that the last term in (47) is decreasing to zero in  $n$  because  $a > 1/2$ .
+
+Note that  $\mathbf{J}_k^{(1)} = \mathbf{J}_k^{(1,a)} + \mathbf{J}_k^{(1,b)}$ , by triangular inequality we have
+
+$$
+\begin{array}{l} \gamma_ {n} \mathbb {E} \left[ \left\| \sum_ {k = 1} ^ {n} \mathbf {J} _ {k} ^ {(1 1)} \right\| \right] \leq \gamma_ {n} \mathbb {E} \left[ \left\| \sum_ {k = 1} ^ {n} \mathbf {J} _ {k} ^ {(1 1, A)} \right\| \right] + \gamma_ {n} \mathbb {E} \left[ \left\| \sum_ {k = 1} ^ {n} \mathbf {J} _ {k} ^ {(1 1, B)} \right\| \right] \\ \leq \gamma_ {n} C _ {1} \mathbb {E} \left[ \sqrt {\left(\sum_ {k = 1} ^ {n} \left\| \mathbf {J} _ {k} ^ {(1 1 , A)} \right\| ^ {2}\right)} \right] + \gamma_ {n} \mathbb {E} \left[ \left\| \sum_ {k = 1} ^ {n} \mathbf {J} _ {k} ^ {(1 1, B)} \right\| \right] \tag {48} \\ = \mathbb {E} \left[ \gamma_ {n} C _ {1} \sqrt {\left(\sum_ {k = 1} ^ {n} \left\| \mathbf {J} _ {k} ^ {(1 1 , A)} \right\| ^ {2}\right)} + \gamma_ {n} \left\| \sum_ {k = 1} ^ {n} \mathbf {J} _ {k} ^ {(1 1 , B)} \right\| \right], \\ \end{array}
+$$
+
+where the second inequality comes from (45). By (46) and (47) we know that both terms in the last line of (48) are uniformly bounded by constants over time  $n$  that depend on the set  $\Omega$ . Therefore, by dominated convergence theorem, taking the limit over the last line of (48) gives
+
+$$
+\begin{array}{l} \lim  _ {n \rightarrow \infty} \mathbb {E} \left[ \gamma_ {n} C _ {1} \sqrt {\left(\sum_ {k = 1} ^ {n} \left\| \mathbf {J} _ {k} ^ {(1 1 , A)} \right\| ^ {2}\right)} + \gamma_ {n} \left\| \sum_ {k = 1} ^ {n} \mathbf {J} _ {k} ^ {(1 1, B)} \right\|\right] \\ = \mathbb {E} \left[ \lim  _ {n \rightarrow \infty} \gamma_ {n} C _ {1} \sqrt {\left(\sum_ {k = 1} ^ {n} \left\| \mathbf {J} _ {k} ^ {(1 1 , A)} \right\| ^ {2}\right)} + \gamma_ {n} \left\| \sum_ {k = 1} ^ {n} \mathbf {J} _ {k} ^ {(1 1 , B)} \right\|\right] = 0. \\ \end{array}
+$$
+
+Therefore, we have
+
+$$
+\lim  _ {n \rightarrow \infty} \gamma_ {n} \mathbb {E} \left[\left\| \sum_ {k = 1} ^ {n} \mathbf {J} _ {k} ^ {(1)} \right\|\right] = 0,
+$$
+
+In sum, in terms of  $\mathbb{E}[M_{n + 1}^{(\mathbf{x})}(M_{n + 1}^{(\mathbf{x})})^T|\mathcal{F}_n]$  in (41), we have  $\mathbf{U}_{11}$  in (44),  $\lim_{n\to \infty}\mathbf{D}_n^{(1)} = 0$  a.s. and  $\lim_{n\to \infty}\gamma_n\mathbb{E}\left[\left\| \sum_{k = 1}^n\mathbf{J}_k^{(1)}\right\|\right] = 0.$
+
+We can apply the same steps as above for the other two terms  $i = 2,3$  in (42) and obtain the results.
+
+Analysis of Terms  $r_n^{(\pmb {\theta},1)},r_n^{(\pmb {\theta},2)},r_n^{(\mathbf{x},1)},r_n^{(\mathbf{x},2)}$
+
+Lemma E.2. For  $r_n^{(\theta, 1)}, r_n^{(\theta, 2)}, r_n^{(\mathbf{x}, 1)}, r_n^{(\mathbf{x}, 2)}$  defined in (35), we have the following results:
+
+$$
+\left\| r _ {n} ^ {(\boldsymbol {\theta}, 1)} \right\| = O (\gamma_ {n}) = o \left(\sqrt {\beta_ {n}}\right), \quad \sqrt {\gamma_ {n}} \left\| \sum_ {k = 1} ^ {n} r _ {k} ^ {(\boldsymbol {\theta}, 2)} \right\| = O \left(\sqrt {\gamma_ {n}}\right) = o (1). \tag {49a}
+$$
+
+$$
+\left\| r _ {n} ^ {(\mathbf {x}, 1)} \right\| = O \left(\gamma_ {n}\right) = o \left(\sqrt {\beta_ {n}}\right), \quad \sqrt {\gamma_ {n}} \left\| \sum_ {k = 1} ^ {n} r _ {k} ^ {(\mathbf {x}, 2)} \right\| = O \left(\sqrt {\gamma_ {n}}\right) = o (1). \tag {49b}
+$$
+
+Proof. For  $r_n^{(\theta, 1)}$ , note that
+
+$$
+\begin{array}{l} r _ {n} ^ {(\boldsymbol {\theta}, 1)} = \left(\mathbf {K} _ {\mathbf {x} _ {n + 1}} \tilde {H} _ {\boldsymbol {\theta} _ {n + 1}, \mathbf {x} _ {n + 1}}\right) \left(X _ {n + 1}\right) - \left(\mathbf {K} _ {\mathbf {x} _ {n}} \tilde {H} _ {\boldsymbol {\theta} _ {n}, \mathbf {x} _ {n}}\right) \left(X _ {n + 1}\right) \\ = \sum_ {j \in \mathcal {N}} \left(\mathbf {K} _ {X _ {n}, j} [ \mathbf {x} _ {n + 1} ] \tilde {H} _ {\boldsymbol {\theta} _ {n + 1}, \mathbf {x} _ {n + 1}} (j) - \mathbf {K} _ {X _ {n}, j} [ \mathbf {x} _ {n} ] \tilde {H} _ {\boldsymbol {\theta} _ {n}, \mathbf {x} _ {n}} (j)\right) \tag {50} \\ \leq \sum_ {j \in \mathcal {N}} L _ {\mathcal {C}} \left(\left\| \boldsymbol {\theta} _ {n + 1} - \boldsymbol {\theta} _ {n} \right\| + \left\| \mathbf {x} _ {n + 1} - \mathbf {x} _ {n} \right\|\right) \\ \leq N L _ {\mathcal {C}} \left(C _ {\mathcal {C}} \beta_ {n + 1} + 2 \gamma_ {n + 1}\right) \\ \end{array}
+$$
+
+where the second last inequality is because  $\mathbf{K}_{i,j}[\mathbf{x}]\tilde{H}_{\boldsymbol{\theta},\mathbf{x}}(j)$  is continuous in  $\pmb {\theta},\mathbf{x}$ $\mathbf{K}[\mathbf{x}]$ , which stems from continuous functions  $\pmb {K}[\mathbf{x}]$  and  $\tilde{H}_{\pmb{\theta},\mathbf{x}}$ . The last inequality is from update rules (4) and  $(\pmb {\theta}_n,\mathbf{x}_n)\in \Omega$  for some compact subset  $\Omega$  by assumption A4. Then, we have  $\| r_n^{(\pmb {\theta},1)}\| = O(\gamma_n) = o(\sqrt{\beta_n})$  because of  $a > 1 / 2\geq b / 2$  by assumption A2.
+
+We let  $\nu_{n} \triangleq (\mathbf{K}_{\mathbf{x}_{n}}\tilde{H}_{\boldsymbol{\theta}_{n},\mathbf{x}_{n}})(X_{n})$  such that  $r_n^{(\pmb{\theta},2)} = \nu_n - \nu_{n + 1}$ . Note that  $\sum_{k = 1}^{n}r_k^{(\pmb{\theta},2)} = \nu_1 - \nu_{n + 1}$  and by assumption A4,  $\| \nu_{n}\|$  is upper bounded by a constant dependent on the compact set, which leads to
+
+$$
+\sqrt {\gamma_ {n}} \left\| \sum_ {k = 1} ^ {n} r _ {k} ^ {(\boldsymbol {\theta}, 2)} \right\| = \sqrt {\gamma_ {n}} \| \nu_ {1} - \nu_ {n + 1} \| = O (\sqrt {\gamma_ {n}}) = o (1).
+$$
+
+Similarly, we can also obtain  $\| r_n^{(\mathbf{x},1)}\| = o(\sqrt{\beta_n})$  and  $\sqrt{\gamma_n}\left\| \sum_{k = 1}^n r_k^{(\mathbf{x},2)}\right\| = O(\sqrt{\gamma_n}) = o(1)$ .
+
+# E.2.2 EFFECT OF SRRW ITERATION ON SA ITERATION
+
+In view of the almost sure convergence results in Lemma 3.1 and Lemma 3.2, for large enough  $n$  so that both iterations  $\theta_{n},\mathbf{x}_{n}$  are close to the equilibrium  $(\theta^{*},\mu)$ , we can apply the Taylor expansion to functions  $\mathbf{h}(\theta ,\mathbf{x})$  and  $\pi [\mathbf{x}] - \mathbf{x}$  in (36) at the point  $(\theta^{*},\mu)$ , which results in
+
+$$
+\mathbf {h} (\boldsymbol {\theta}, \mathbf {x}) = \mathbf {h} \left(\boldsymbol {\theta} ^ {*}, \boldsymbol {\mu}\right) + \nabla_ {\boldsymbol {\theta}} \mathbf {h} \left(\boldsymbol {\theta} ^ {*}, \boldsymbol {\mu}\right) \left(\boldsymbol {\theta} - \boldsymbol {\theta} ^ {*}\right) + \nabla_ {\mathbf {x}} \mathbf {h} \left(\boldsymbol {\theta} ^ {*}, \boldsymbol {\mu}\right) (\mathbf {x} - \boldsymbol {\mu}) + O \left(\left\| \boldsymbol {\theta} - \boldsymbol {\theta} ^ {*} \right\| ^ {2} + \| \mathbf {x} - \boldsymbol {\mu} \| ^ {2}\right), \tag {51a}
+$$
+
+$$
+\pi [ \mathbf {x} ] - \mathbf {x} = \pi [ \boldsymbol {\mu} ] - \boldsymbol {\mu} + \nabla_ {\mathbf {x}} (\pi (\mathbf {x}) - \mathbf {x}) | _ {\mathbf {x} = \boldsymbol {\mu}} (\mathbf {x} - \boldsymbol {\mu}) + O (\| \mathbf {x} - \boldsymbol {\mu} \| ^ {2}). \tag {51b}
+$$
+
+With matrix  $\mathbf{J}(\alpha)$ , we have the following:
+
+$$
+\mathbf {J} _ {1 1} = \nabla_ {\boldsymbol {\theta}} \mathbf {h} \left(\boldsymbol {\theta} ^ {*}, \boldsymbol {\mu}\right) = \nabla \mathbf {h} \left(\boldsymbol {\theta} ^ {*}\right),
+$$
+
+$$
+\mathbf {J} _ {1 2} (\alpha) = \nabla_ {\mathbf {x}} \mathbf {h} \left(\boldsymbol {\theta} ^ {*}, \boldsymbol {\mu}\right) = - \alpha \mathbf {H} ^ {T} \left(\mathbf {P} ^ {T} + \mathbf {I}\right), \tag {52}
+$$
+
+$$
+\mathbf {J} _ {2 2} (\alpha) = \nabla_ {\mathbf {x}} (\boldsymbol {\pi} (\mathbf {x}) - \mathbf {x}) | _ {\mathbf {x} = \boldsymbol {\mu}} = 2 \alpha \boldsymbol {\mu} \mathbf {1} ^ {T} - \alpha \mathbf {P} ^ {T} - (\alpha + 1) \mathbf {I}.
+$$
+
+Then, (36) becomes
+
+$$
+\boldsymbol {\theta} _ {n + 1} = \boldsymbol {\theta} _ {n} + \beta_ {n + 1} \left(\mathbf {J} _ {1 1} \left(\boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*}\right) + \mathbf {J} _ {1 2} (\alpha) \left(\mathbf {x} _ {n} - \boldsymbol {\mu}\right) + r _ {n} ^ {(\boldsymbol {\theta}, 1)} + r _ {n} ^ {(\boldsymbol {\theta}, 2)} + M _ {n + 1} ^ {(\boldsymbol {\theta})} + \eta_ {n} ^ {(\boldsymbol {\theta})}\right), \tag {53a}
+$$
+
+$$
+\mathbf {x} _ {n + 1} = \mathbf {x} _ {n} + \gamma_ {n + 1} \left(\mathbf {J} _ {2 2} (\alpha) \left(\mathbf {x} _ {n} - \boldsymbol {\mu}\right) + r _ {n} ^ {(\mathbf {x}, 1)} + r _ {n} ^ {(\mathbf {x}, 2)} + M _ {n + 1} ^ {(\mathbf {x})} + \eta_ {n} ^ {(\mathbf {x})}\right), \tag {53b}
+$$
+
+where  $\eta_n^{(\pmb{\theta})} = O(\| \mathbf{x}_n\|^2 + \| \pmb{\theta}_n\|^2)$  and  $\eta_n^{(\mathbf{x})} = O(\| \mathbf{x}_n\|^2)$ .
+
+Then, inspired by Mokkadem & Pelletier (2006), we decompose iterates  $\{\mathbf{x}_n\}$  and  $\{\pmb {\theta}_n\}$  into  $\mathbf{x}_n =$ $L_{n}^{(\pmb {x})} + \Delta_{n}^{(\pmb {x})}$  and  $\pmb {\theta}_n = L_n^{(\pmb {\theta})} + R_n^{(\pmb {\theta})} + \Delta_n^{(\pmb {\theta})}$ . Rewriting (53b) gives
+
+$$
+\mathbf {x} _ {n} - \boldsymbol {\mu} = \gamma_ {n + 1} ^ {- 1} \mathbf {J} _ {2 2} (\alpha) ^ {- 1} (\mathbf {x} _ {n + 1} - \mathbf {x} _ {n}) - \mathbf {J} _ {2 2} (\alpha) ^ {- 1} (r _ {n} ^ {(\mathbf {x}, 1)} + r _ {n} ^ {(\mathbf {x}, 2)} + M _ {n + 1} ^ {(\mathbf {x})} + \eta_ {n} ^ {(\mathbf {x})}),
+$$
+
+and substituting the above equation back in (53a) gives
+
+$$
+\begin{array}{l} \boldsymbol {\theta} _ {n + 1} - \boldsymbol {\theta} ^ {*} = \boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*} + \beta_ {n + 1} \left(\mathbf {J} _ {1 1} \left(\boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*}\right) + \gamma_ {n + 1} ^ {- 1} \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} \left(\mathbf {x} _ {n + 1} - \mathbf {x} _ {n}\right) \right. \\ \left. - \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} \left(r _ {n} ^ {(\mathbf {x}, 1)} + r _ {n} ^ {(\mathbf {x}, 2)} + M _ {n + 1} ^ {(\mathbf {x})} + \eta_ {n} ^ {(\mathbf {x})}\right) + r _ {n} ^ {(\boldsymbol {\theta}, 1)} + r _ {n} ^ {(\boldsymbol {\theta}, 2)} + M _ {n + 1} ^ {(\boldsymbol {\theta})} + \eta_ {n} ^ {(\boldsymbol {\theta})}\right) \\ = (\mathbf {I} + \beta_ {n + 1} \mathbf {J} _ {1 1}) \left(\boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*}\right) + \left[ \beta_ {n + 1} \gamma_ {n + 1} ^ {- 1} \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} \left(\mathbf {x} _ {n + 1} - \mathbf {x} _ {n}\right) \right] \\ + \beta_ {n + 1} (M _ {n + 1} ^ {(\boldsymbol {\theta})} - \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} M _ {n + 1} ^ {(\mathbf {x})}) \\ + \beta_ {n + 1} \left(r _ {n} ^ {(\boldsymbol {\theta}, 1)} + r _ {n} ^ {(\boldsymbol {\theta}, 2)} + \eta_ {n} ^ {(\boldsymbol {\theta})} - \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} \left(r _ {n} ^ {(\mathbf {x}, 1)} + r _ {n} ^ {(\mathbf {x}, 2)} + \eta_ {n} ^ {(\mathbf {x})}\right)\right), \tag {54} \\ \end{array}
+$$
+
+From (54) we can see the iteration  $\{\theta_n\}$  implicitly embeds the recursions of three sequences
+
+-  $\beta_{n+1} \gamma_{n+1}^{-1} \mathbf{J}_{12}(\alpha) \mathbf{J}_{22}(\alpha)^{-1} (\mathbf{x}_{n+1} - \mathbf{x}_n)$ ;  
+-  $\beta_{n+1}(M_{n+1}^{(\boldsymbol{\theta})} - \mathbf{J}_{12}(\alpha)\mathbf{J}_{22}(\alpha)^{-1}M_{n+1}^{(\mathbf{x})})$ ;  
+-  $\beta_{n+1}(r_n^{(\pmb{\theta}, 1)} + r_n^{(\pmb{\theta}, 2)} + \eta_n^{(\pmb{\theta})} - \mathbf{J}_{12}(\alpha) \mathbf{J}_{22}(\alpha)^{-1}(r_n^{(\mathbf{x}, 1)} + r_n^{(\mathbf{x}, 2)} + \eta_n^{(\mathbf{x}))})$ .
+
+Let  $u_{n} \triangleq \sum_{k=1}^{n} \beta_{k}$  and  $s_{n} \triangleq \sum_{k=1}^{n} \gamma_{k}$ . Below we define two iterations:
+
+$$
+\begin{array}{l} L _ {n} ^ {(\boldsymbol {\theta})} = e ^ {\beta_ {n} \mathbf {J} _ {1 1}} L _ {n - 1} ^ {(\boldsymbol {\theta})} + \beta_ {n} \left(M _ {n} ^ {(\boldsymbol {\theta})} - \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} M _ {n} ^ {(\mathbf {x})}\right) \\ = \sum_ {k = 1} ^ {n} e ^ {\left(u _ {n} - u _ {k}\right) \mathbf {J} _ {1 1}} \beta_ {k} \left(M _ {k} ^ {(\boldsymbol {\theta})} - \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} M _ {k} ^ {(\mathbf {x})}\right) \tag {55a} \\ \end{array}
+$$
+
+$$
+\begin{array}{l} R _ {n} ^ {(\boldsymbol {\theta})} = e ^ {\beta_ {n} \mathbf {J} _ {1 1}} R _ {n - 1} ^ {(\boldsymbol {\theta})} + \beta_ {n} \gamma_ {n} ^ {- 1} \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} (\mathbf {x} _ {n} - \mathbf {x} _ {n - 1}) \\ = \sum_ {k = 1} ^ {n} e ^ {\left(u _ {n} - u _ {k}\right) \mathbf {J} _ {1 1}} \beta_ {k} \gamma_ {k} ^ {- 1} \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} \left(\mathbf {x} _ {k} - \mathbf {x} _ {k - 1}\right) \tag {55b} \\ \end{array}
+$$
+
+and a remaining term  $\Delta_n^{(\pmb{\theta})} \triangleq \pmb{\theta}_n - \pmb{\theta}^* - L_n^{(\pmb{\theta})} - R_n^{(\pmb{\theta})}$ .
+
+Similarly, for iteration  $\mathbf{x}_n$ , define the sequence  $L_n^{(\mathbf{x})}$  such that
+
+$$
+L _ {n} ^ {(\mathbf {x})} = e ^ {\gamma_ {n} \mathbf {J} _ {2 2} (\alpha)} L _ {n - 1} ^ {(\mathbf {x})} + \gamma_ {n} M _ {n} ^ {(\mathbf {x})} = \sum_ {k = 1} ^ {n} e ^ {(s _ {n} - s _ {k}) \mathbf {J} _ {2 2} (\alpha)} \gamma_ {k} M _ {k} ^ {(\mathbf {x})}, \tag {56}
+$$
+
+and a remaining term
+
+$$
+\Delta_ {n} ^ {(\mathbf {x})} \triangleq \mathbf {x} _ {n} - \boldsymbol {\mu} - L _ {n} ^ {(\boldsymbol {\theta})} \tag {57}
+$$
+
+The decomposition of  $\theta_{n} - \theta^{*}$  and  $\mathbf{x}_n - \pmb{\mu}$  in the above form is also standard in the single-timescale SA literature (Delyon, 2000; Fort, 2015).
+
+Characterization of Sequences  $\{L_n^{(\theta)}\}$  and  $\{L_n^{(\mathbf{x})}\}$
+
+we set a Martingale  $Z^{(n)} = \{Z_k^{(n)}\}_{k\geq 1}$  such that
+
+$$
+Z _ {k} ^ {(n)} = \left( \begin{array}{c c} \beta_ {n} ^ {- 1 / 2} e ^ {u _ {n} \mathbf {J} _ {1 1}} & 0 \\ 0 & \gamma_ {n} ^ {- 1 / 2} e ^ {s _ {n} \mathbf {J} _ {2 2} (\alpha)} \end{array} \right) \times \sum_ {j = 1} ^ {k} \left( \begin{array}{c} e ^ {- u _ {k} \mathbf {J} _ {1 1}} \beta_ {k} (M _ {k} ^ {(\boldsymbol {\theta})} - \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} M _ {k} ^ {(\mathbf {x})}) \\ e ^ {- s _ {k} \mathbf {J} _ {2 2} (\alpha)} \gamma_ {k} M _ {k} ^ {(\mathbf {x})} \end{array} \right).
+$$
+
+Then, the Martingale difference array  $Z_{k}^{(n)} - Z_{k - 1}^{(n)}$  becomes
+
+$$
+Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)} = \left( \begin{array}{c} \beta_ {n} ^ {- 1 / 2} e ^ {(u _ {n} - u _ {k}) \mathbf {J} _ {1 1}} \beta_ {k} (M _ {k} ^ {(\boldsymbol {\theta})} - \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} M _ {k} ^ {(\mathbf {x})}) \\ \gamma_ {n} ^ {- 1 / 2} e ^ {(s _ {n} - s _ {k}) \mathbf {J} _ {2 2} (\alpha)} \gamma_ {k} M _ {k} ^ {(\mathbf {x})} \end{array} \right)
+$$
+
+and
+
+$$
+\sum_ {k = 1} ^ {n} \mathbb {E} \left[ (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) ^ {T} | \mathcal {F} _ {k - 1} \right] = \left( \begin{array}{c c} A _ {1, n} & A _ {2, n} \\ A _ {2, n} ^ {T} & A _ {4, n} \end{array} \right),
+$$
+
+where, in view of decomposition of  $M_{n}^{(\pmb{\theta})}$  and  $M_{n}^{(\mathbf{x})}$  in (41) and (42), respectively,
+
+$$
+\begin{array}{l} A _ {1, n} = \beta_ {n} ^ {- 1} \sum_ {k = 1} ^ {n} \beta_ {k} ^ {2} e ^ {(u _ {n} - u _ {k}) \mathbf {J} _ {1 1}} \left(\mathbf {U} _ {2 2} + \mathbf {D} _ {k} ^ {(1)} + \mathbf {J} _ {k} ^ {(1)} - \left(\mathbf {U} _ {2 1} + \mathbf {D} _ {k} ^ {(2)} + \mathbf {J} _ {k} ^ {(2)}\right) \left(\mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1}\right) ^ {T} \right. \\ + \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} \left(\mathbf {U} _ {1 1} + \mathbf {D} _ {k} ^ {(3)} + \mathbf {J} _ {k} ^ {(3)}\right) \left(\mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1}\right) ^ {T} \\ \left. - \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} \left(\mathbf {U} _ {2 1} + \mathbf {D} _ {k} ^ {(2)} + \mathbf {J} _ {k} ^ {(2)}\right) ^ {T}\right) e ^ {\left(u _ {n} - u _ {k}\right) \left(\mathbf {J} _ {1 1}\right) ^ {T}}, \tag {58a} \\ \end{array}
+$$
+
+$$
+A _ {2, n} = \beta_ {n} ^ {- 1 / 2} \gamma_ {n} ^ {- 1 / 2} \sum_ {k = 1} ^ {n} \beta_ {k} \gamma_ {k} e ^ {\left(u _ {n} - u _ {k}\right) \mathbf {J} _ {1 1}} \left(\mathbf {U} _ {2 1} - \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} \mathbf {U} _ {1 1}\right) e ^ {\left(s _ {n} - s _ {k}\right) \mathbf {J} _ {2 2} (\alpha) ^ {T}}, \tag {58b}
+$$
+
+$$
+A _ {4, n} = \gamma_ {n} ^ {- 1} \sum_ {k = 1} ^ {n} \gamma_ {k} ^ {2} e ^ {\left(s _ {n} - s _ {k}\right) \mathbf {J} _ {2 2} (\alpha)} \left(\mathbf {U} _ {1 1} + \mathbf {D} _ {k} ^ {(3)} + \mathbf {J} _ {k} ^ {(3)}\right) e ^ {\left(s _ {n} - s _ {k}\right) \mathbf {J} _ {2 2} (\alpha) ^ {T}}. \tag {58c}
+$$
+
+We further decompose  $A_{1,n}$  into three parts:
+
+$$
+\begin{array}{l} A _ {1, n} = \beta_ {n} ^ {- 1} \sum_ {k = 1} ^ {n} \left(\beta_ {k} ^ {2} e ^ {\left(u _ {n} - u _ {k}\right) \mathbf {J} _ {1 1}} \left(\mathbf {U} _ {2 2} - \mathbf {U} _ {2 1} \left(\mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1}\right) ^ {T} \right. \right. \\ \left. \left. - \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} \mathbf {U} _ {1 2} + \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} \mathbf {U} _ {1 1} \left(\mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1}\right) ^ {T}\right) e ^ {\left(u _ {n} - u _ {k}\right) \left(\mathbf {J} _ {1 1}\right) ^ {T}}\right) \\ + \beta_ {n} ^ {- 1} \sum_ {k = 1} ^ {n} \left(\beta_ {k} ^ {2} e ^ {\left(u _ {n} - u _ {k}\right) \mathbf {J} _ {1 1}} \left(\mathbf {D} _ {k} ^ {(1)} + \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} \mathbf {D} _ {k} ^ {(3)} \left(\mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1}\right) ^ {T} \right. \right. \\ \left. - \mathbf {D} _ {k} ^ {(2)} \left(\mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1}\right) ^ {T} - \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} \left(\mathbf {D} _ {k} ^ {(2)}\right) ^ {T}\right) e ^ {\left(u _ {n} - u _ {k}\right) \left(\mathbf {J} _ {1 1}\right) ^ {T}} \\ + \beta_ {n} ^ {- 1} \sum_ {k = 1} ^ {n} \left(\beta_ {k} ^ {2} e ^ {(u _ {n} - u _ {k}) \mathbf {J} _ {1 1}} (\mathbf {J} _ {k} ^ {(1)} + \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} \mathbf {J} _ {k} ^ {(3)} (\mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1}) ^ {T} \right. \\ \left. - \mathbf {J} _ {k} ^ {(2)} \left(\mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1}\right) ^ {T} - \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} \left(\mathbf {J} _ {k} ^ {(2)}\right) ^ {T}\right) e ^ {\left(u _ {n} - u _ {k}\right) \left(\mathbf {J} _ {1 1}\right) ^ {T}} \\ \triangleq A _ {1, n} ^ {(a)} + A _ {1, n} ^ {(b)} + A _ {1, n} ^ {(c)}. \tag {59} \\ \end{array}
+$$
+
+Here, we define  $\mathbf{U}_{\pmb{\theta}}(\alpha) \triangleq \mathbf{U}_{22} - \mathbf{U}_{21}(\mathbf{J}_{12}(\alpha)\mathbf{J}_{22}(\alpha)^{-1})^T - \mathbf{J}_{12}(\alpha)\mathbf{J}_{22}(\alpha)^{-1}\mathbf{U}_{12} + \mathbf{J}_{12}(\alpha)\mathbf{J}_{22}(\alpha)^{-1}\mathbf{U}_{11} + \mathbf{J}_{12}(\alpha)\mathbf{J}_{22}(\alpha)^{-1}\mathbf{J}_{12}$ . By (52) and (43a) in Lemma E.1, we have
+
+$$
+\mathbf {U} _ {\boldsymbol {\theta}} (\alpha) = \sum_ {i = 1} ^ {N - 1} \frac {1}{(\alpha (1 + \lambda_ {i}) + 1) ^ {2}} \cdot \frac {1 + \lambda_ {i}}{1 - \lambda_ {i}} \mathbf {H} ^ {T} \mathbf {u} _ {i} \mathbf {u} _ {i} ^ {T} \mathbf {H}. \tag {60}
+$$
+
+Then, we have the following lemma.
+
+Lemma E.3. For  $A_{1,n}^{(a)}, A_{1,n}^{(b)}, A_{1,n}^{(c)}$  defined in (59), we have
+
+$$
+\lim  _ {n \rightarrow \infty} A _ {1, n} ^ {(a)} = \mathbf {V} _ {\boldsymbol {\theta}} (\alpha), \quad \lim  _ {n \rightarrow \infty} \| A _ {1, n} ^ {(b)} \| = 0, \quad \lim  _ {n \rightarrow \infty} \| A _ {1, n} ^ {(c)} \| = 0, \tag {61}
+$$
+
+where  $\mathbf{V}_{\theta}(\alpha)$  is the solution to the Lyapunov equation
+
+$$
+\left(\mathbf {J} _ {1 1} + \frac {\mathbb {1} _ {\{b = 1 \}}}{2} \boldsymbol {I}\right) \mathbf {V} _ {\boldsymbol {\theta}} (\alpha) + \mathbf {V} _ {\boldsymbol {\theta}} (\alpha) \left(\mathbf {J} _ {1 1} + \frac {\mathbb {1} _ {\{b = 1 \}}}{2} \boldsymbol {I}\right) ^ {T} + \mathbf {U} _ {\boldsymbol {\theta}} (\alpha) = 0.
+$$
+
+Proof. First, from Lemma G.4, we have for some  $c, T > 0$  such that
+
+$$
+\begin{array}{l} \| A _ {1, n} ^ {(b)} \| \leq \beta_ {n} ^ {- 1} \sum_ {k = 1} ^ {n} \left\| \mathbf {D} _ {k} ^ {(1)} + \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} \mathbf {D} _ {k} ^ {(3)} (\mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1}) ^ {T} - \mathbf {D} _ {k} ^ {(2)} (\mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1}) ^ {T} \right. \\ \left. - \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} \left(\mathbf {D} _ {k} ^ {(2)}\right) ^ {T} \right\| \cdot \beta_ {k} ^ {2} c ^ {2} e ^ {- 2 T \left(u _ {n} - u _ {k}\right)}. \\ \end{array}
+$$
+
+Applying Lemma G.6, together with  $\mathbf{D}_n^{(i)}\to 0$  a.s. in Lemma E.1, gives
+
+$$
+\begin{array}{l} \lim \sup \| A _ {1, n} ^ {(b)} \| \\ \leq \frac {1}{C (b , p)} \lim  _ {n} \sup  _ {n} \| (\mathbf {D} _ {n} ^ {(1)} + \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} \mathbf {D} _ {n} ^ {(4)} (\mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1}) ^ {T} \\ \left. - \mathbf {D} _ {n} ^ {(2)} \left(\mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1}\right) ^ {T} - \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} \mathbf {D} _ {n} ^ {(3)}\right) \| \\ = 0. \\ \end{array}
+$$
+
+We now consider  $\| A_{1,n}^{(c)}\|$ . Set
+
+$$
+\begin{array}{l} \Xi_ {n} \triangleq \sum_ {k = 1} ^ {n} \left(\mathbf {J} _ {n} ^ {(1)} + \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} \mathbf {J} _ {k} ^ {(3)} (\mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1}) ^ {T} \right. \\ - \mathbf {J} _ {k} ^ {(2)} (\mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1}) ^ {T} - \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} (\mathbf {J} _ {k} ^ {(2)}) ^ {T}), \\ \end{array}
+$$
+
+we can rewrite  $A_{1,n}^{(c)}$  as
+
+$$
+A _ {1, n} ^ {(c)} = \beta_ {n} ^ {- 1} \sum_ {k = 1} ^ {n} \beta_ {k} ^ {2} e ^ {(u _ {n} - u _ {k}) \mathbf {J} _ {1 1}} (\Xi_ {k} - \Xi_ {k - 1}) e ^ {(u _ {n} - u _ {k}) (\mathbf {J} _ {1 1}) ^ {T}}.
+$$
+
+By the Abel transformation, we have
+
+$$
+\begin{array}{l} A _ {1, n} ^ {(c)} = \beta_ {n} \Xi_ {n} + \beta_ {n} ^ {- 1} \sum_ {k = 1} ^ {n - 1} \left[ \beta_ {k} ^ {2} e ^ {\left(u _ {n} - u _ {k}\right) \mathbf {J} _ {1 1}} \Xi_ {k} e ^ {\left(u _ {n} - u _ {k}\right) \left(\mathbf {J} _ {1 1}\right) ^ {T}} \right. \tag {62} \\ \left. - \beta_ {k + 1} ^ {2} e ^ {(u _ {n} - u _ {k + 1}) \mathbf {J} _ {1 1}} \Xi_ {k} e ^ {(u _ {n} - u _ {k + 1}) (\mathbf {J} _ {1 1}) ^ {T}} \right]. \\ \end{array}
+$$
+
+We know from Lemma E.1 that  $\beta_{n}\Xi_{n}\to 0$  a.s. because  $\Xi_n = o(\gamma_n^{-1})$ . Besides,
+
+$$
+\begin{array}{l} \left\| \beta_ {k} e ^ {\left(u _ {n} - u _ {k}\right) \mathbf {J} _ {1 1}} - \beta_ {k + 1} e ^ {\left(u _ {n} - u _ {k + 1}\right) \mathbf {J} _ {1 1}} \right\| \\ = \| \left(\beta_ {k} - \beta_ {k + 1}\right) e ^ {\left(u _ {n} - u _ {k}\right) \mathbf {J} _ {1 1}} + \beta_ {k + 1} e ^ {\left(u _ {n} - u _ {k}\right) \mathbf {J} _ {1 1}} \left(\mathbf {I} - e ^ {- \beta_ {k + 1} \mathbf {J} _ {1 1}}\right) \| \\ \leq C _ {1} \beta_ {k} ^ {2} e ^ {- (u _ {n} - u _ {k}) T} \\ \end{array}
+$$
+
+for some constant  $C_1 > 0$  because  $\beta_{n} - \beta_{n + 1}\leq C_{2}\beta_{n}^{2}$  and  $\| \mathbf{I} - e^{-\beta_{k + 1}\mathbf{J}_{11}}\| \leq C_3\beta_{k + 1}$ . Moreover,
+
+$$
+\begin{array}{l} \left\| \beta_ {k} e ^ {\left(u _ {n} - u _ {k}\right) \mathbf {J} _ {1 1}} \right\| + \left\| \beta_ {k + 1} e ^ {\left(u _ {n} - u _ {k + 1}\right) \mathbf {J} _ {1 1}} \right\| \\ \leq \beta_ {k} \| e ^ {(u _ {n} - u _ {k}) \mathbf {J} _ {1 1}} \| + \beta_ {k} \| e ^ {(u _ {n} - u _ {k}) \mathbf {J} _ {1 1}} \| \cdot \| e ^ {- \beta_ {k + 1} \mathbf {J} _ {1 1}} \| \\ \leq C _ {4} \beta_ {k} e ^ {- (u _ {n} - u _ {k}) T}. \\ \end{array}
+$$
+
+Using Lemma G.7 on (62) gives
+
+$$
+\| A _ {1, n} ^ {(c)} \| \leq C _ {1} C _ {4} \beta_ {n} ^ {- 1} \sum_ {k = 1} ^ {n - 1} \beta_ {k} ^ {2} e ^ {- 2 (u _ {n} - u _ {k}) T} \| \beta_ {k} \Xi_ {k} \| + \| \beta_ {n} \Xi_ {n} \|.
+$$
+
+Applying Lemma G.6 again gives
+
+$$
+\lim  _ {n} \sup  _ {n} \| A _ {1, n} ^ {(c)} \| \leq C _ {5} \lim  _ {n} \sup  _ {n} \| \beta_ {n} \Xi_ {n} \| = 0
+$$
+
+for some constant  $C_5 > 0$
+
+Finally, we provide an existing lemma below.
+
+Lemma E.4 (Mokkadem & Pelletier (2005) Lemma 4). For a sequence with decreasing step size  $\beta_{n} = (n + 1)^{-b}$  for  $b\in (1 / 2,1]$ ,  $u_{n} = \sum_{k = 1}^{n}\beta_{k}$ , a positive semi-definite matrix  $\Gamma$  and a Hurwitz matrix  $\mathbf{Q}$ , which is given by
+
+$$
+\beta_ {n} ^ {- 1} \sum_ {k = 1} ^ {n} \beta_ {n} ^ {2} e ^ {(u _ {n} - u _ {k}) \mathbf {Q}} \Gamma e ^ {(u _ {n} - u _ {k}) \mathbf {Q} ^ {T}},
+$$
+
+we have
+
+$$
+\lim  _ {n \rightarrow \infty} \beta_ {n} ^ {- 1} \sum_ {k = 1} ^ {n} \beta_ {n} ^ {2} e ^ {(u _ {n} - u _ {k}) \mathbf {Q}} \Gamma e ^ {(u _ {n} - u _ {k}) \mathbf {Q} ^ {T}} = \mathbf {V}
+$$
+
+where  $\mathbf{V}$  is the solution of the Lyapunov equation
+
+$$
+\left(\mathbf {Q} + \frac {\mathbb {1} _ {\{b = 1 \}}}{2} \mathbf {I}\right) \mathbf {V} + \mathbf {V} \left(\mathbf {Q} ^ {T} + \frac {\mathbb {1} _ {\{b = 1 \}}}{2} \mathbf {I}\right) + \Gamma = 0.
+$$
+
+Then,  $\lim_{n\to \infty}A_{1,n}^{(a)} = \mathbf{V}_{\pmb{\theta}}(\alpha)$  is a direct application of Lemma E.4.
+
+![](images/7b45744ab21e80417e29886cf76609edd54e24c713dd773e4b043bc07136d5b3.jpg)
+
+We can follow the similar steps in Lemma E.3 to obtain
+
+$$
+\lim  _ {n \to \infty} A _ {4, n} = \mathbf {V} _ {\mathbf {x}} (\alpha),
+$$
+
+where  $\mathbf{V}_{\mathbf{x}}(\alpha)$  is in the form of (32).
+
+The last step is to show  $\lim_{n\to \infty}A_{2,n} = 0$  . Note that
+
+$$
+\begin{array}{l} \| A _ {2, n} \| = O \left(\beta_ {n} ^ {- 1 / 2} \gamma_ {n} ^ {- 1 / 2} \sum_ {k = 1} ^ {n} \beta_ {k} \gamma_ {k} \| e ^ {(u _ {n} - u _ {k}) \mathbf {J} _ {1 1}} \| \| e ^ {(s _ {n} - s _ {k}) \mathbf {J} _ {2 2} (\alpha) ^ {T}} \|\right) \\ = O \left(\beta_ {n} ^ {- 1 / 2} \gamma_ {n} ^ {- 1 / 2} \sum_ {k = 1} ^ {n} \beta_ {k} \gamma_ {k} e ^ {- (u _ {n} - u _ {k}) T} e ^ {- (s _ {n} - s _ {k}) T ^ {\prime}}\right) \\ = O \left(\beta_ {n} ^ {- 1 / 2} \gamma_ {n} ^ {- 1 / 2} \sum_ {k = 1} ^ {n} \beta_ {k} \gamma_ {k} e ^ {- (s _ {n} - s _ {k}) T ^ {\prime}}\right), \\ \end{array}
+$$
+
+where the second equality is from Lemma G.4. Then, we use Lemma G.6 with  $p = 0$  to obtain
+
+$$
+\sum_ {k = 1} ^ {n} \beta_ {k} \gamma_ {k} e ^ {- \left(s _ {n} - s _ {k}\right) T ^ {\prime}} = O \left(\beta_ {n}\right) \tag {63}
+$$
+
+Additionally, since  $\beta_{n} = o(\gamma_{n})$ , we have
+
+$$
+\beta_ {n} ^ {- 1 / 2} \gamma_ {n} ^ {- 1 / 2} \sum_ {k = 1} ^ {n} \beta_ {k} \gamma_ {k} ^ {- 1 / 2} \gamma_ {k} ^ {3 / 2} e ^ {- (s _ {n} - s _ {k}) T ^ {\prime}} = O \left(\beta_ {n} ^ {1 / 2} \gamma_ {n} ^ {- 1 / 2}\right) = o (1).
+$$
+
+Then, it follows that  $\lim_{n\to \infty}A_{2,n} = 0$ . Therefore, we obtain
+
+$$
+\lim  _ {n \to \infty} \sum_ {k = 1} ^ {n} \mathbb {E} \left[ (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) ^ {T} | \mathcal {F} _ {k - 1} \right] = \left( \begin{array}{c c} \mathbf {V} _ {\boldsymbol {\theta}} (\alpha) & 0 \\ 0 & \mathbf {V} _ {\mathbf {x}} (\alpha) \end{array} \right).
+$$
+
+Now, we turn to verifying the conditions in Theorem G.3. For some  $\tau > 0$ , we have
+
+$$
+\begin{array}{l} \sum_ {k = 1} ^ {n} \mathbb {E} \left[ \| Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)} \| ^ {2 + \tau} | \mathcal {F} _ {k - 1} \right] \\ = O \left(\beta_ {n} ^ {- (1 + \frac {\tau}{2})} \sum_ {k = 1} ^ {n} \beta_ {k} ^ {2 + \frac {\tau}{2}} \beta_ {k} ^ {\frac {\tau}{2}} e ^ {- (2 + \tau) \left(u _ {n} - u _ {k}\right) T} + \gamma_ {n} ^ {- (1 + \frac {\tau}{2})} \sum_ {k = 1} ^ {n} \gamma_ {k} ^ {2 + \frac {\tau}{2}} \gamma_ {k} ^ {\frac {\tau}{2}} e ^ {- (2 + \tau) \left(s _ {n} - s _ {k}\right) T ^ {\prime}}\right) \tag {64} \\ = O \left(\beta_ {n} ^ {\frac {\tau}{2}} + \gamma_ {n} ^ {\frac {\tau}{2}}\right) \\ \end{array}
+$$
+
+where the last equality comes from Lemma G.6. Since (64) also holds for  $\tau = 0$ , we have
+
+$$
+\sum_ {k = 1} ^ {n} \mathbb {E} \left[ \| Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)} \| ^ {2} | \mathcal {F} _ {k - 1} \right] = O (1) <   \infty .
+$$
+
+Therefore, all the conditions in Theorem G.3 are satisfied and its application then gives
+
+$$
+Z ^ {(n)} = \left(\begin{array}{c}\sqrt {\beta_ {n} ^ {- 1}} L _ {n} ^ {(\boldsymbol {\theta})}\\\sqrt {\gamma_ {n} ^ {- 1}} L _ {n} ^ {(\mathbf {x})}\end{array}\right) \xrightarrow [ d i s t . ]{n \rightarrow \infty} N \left(0, \left(\begin{array}{c c}\mathbf {V} _ {\boldsymbol {\theta}} (\alpha)&0\\0&\mathbf {V} _ {\mathbf {x}} (\alpha)\end{array}\right)\right). \tag {65}
+$$
+
+Furthermore, we have the following lemma about the strong convergence rate of  $\{L_n^{(\pmb{\theta})}\}$  and  $\{L_n^{(\mathbf{x})}\}$ .
+
+Lemma E.5.
+
+$$
+\left\| L _ {n} ^ {(\boldsymbol {\theta})} \right\| = O \left(\sqrt {\beta_ {n} \log \left(u _ {n}\right)}\right) \quad a. s. \tag {66a}
+$$
+
+$$
+\left\| L _ {n} ^ {(\mathbf {x})} \right\| = O \left(\sqrt {\gamma_ {n} \log \left(s _ {n}\right)}\right) \quad a. s. \tag {66b}
+$$
+
+Proof. This proof follows Pelletier (1998, Lemma 1). We only need the special case of Pelletier (1998, Lemma 1) that fits our scenario; e.g., we let the two types of step sizes therein be the same. Specifically, we attach the following lemma.
+
+Lemma E.6 (Pelletier (1998) Lemma 1). Consider a sequence
+
+$$
+L _ {n + 1} = e ^ {u _ {n} \mathbf {H}} \sum_ {k = 1} ^ {n} e ^ {- u _ {k} \mathbf {H}} \beta_ {k} M _ {k + 1},
+$$
+
+where  $\beta_{n} = n^{-b}$ ,  $1/2 < b \leq 1$ , and  $\{M_n\}$  is a Martingale difference sequence adapted to the filtration  $\mathcal{F}$  such that, almost surely,  $\lim \sup_{n} \mathbb{E}[\| M_{n+1} \|^2 | \mathcal{F}_n] \leq M^2$  and there exists  $\tau \in (0,2)$ ,  $b(2 + \tau) > 2$ , such that  $\sup_{n} \mathbb{E}[\| M_{n+1} \|^2 + \tau | \mathcal{F}_n] < \infty$ . Then, almost surely,
+
+$$
+\lim  _ {n} \sup  \frac {\left\| L _ {n} \right\|}{\sqrt {\beta_ {n} \log \left(u _ {n}\right)}} \leq C _ {M}, \tag {67}
+$$
+
+where  $C_M$  is a constant dependent on  $M$ .
+
+By assumption A4, the iterates  $(\pmb{\theta}_n,\mathbf{x}_n)$  are bounded within a compact subset  $\Omega$ . Recall the form of  $M_{n + 1}^{(\pmb{\theta})},M_{n + 1}^{(\mathbf{x})}$  defined in (35), it comprises the functions  $\tilde{H}_{\pmb{\theta}_n,\mathbf{x}_n}(i)$  and  $(\mathbf{K}_{\mathbf{x}_n}\tilde{H}_{\pmb{\theta}_n,\mathbf{x}_n})(i)$ , which in turn include the function  $H(\pmb {\theta},i)$ . We know that  $H(\pmb {\theta},i)$  is bounded for  $\pmb{\theta}$  in some compact set  $\mathcal{C}$ . Thus, for any  $(\pmb {\theta}_n,\mathbf{x}_n)\in \Omega$  for some compact set  $\Omega$ ,  $M_{n + 1}^{(\pmb{\theta})},M_{n + 1}^{(\mathbf{x})}$  are bounded and we denote by  $c_{\pmb{\theta}}$  and  $c_{\mathbf{x}}$  as their upper bounds, i.e.,  $\mathbb{E}[\| M_{n + 1}^{(\pmb{\theta})}\| ^2 |\mathcal{F}_n]\leq c_\Omega^{(\pmb{\theta})}$  and  $\mathbb{E}[\| M_{n + 1}^{(\mathbf{x})}\| ^2 |\mathcal{F}_n]\leq c_\Omega^{(\mathbf{x})}$ . We only need to replace the upper bound  $c$  in Lemma E.6 by  $c_{\Omega}^{(\pmb{\theta})}$  for the sequence  $\{L_n^{(\pmb{\theta})}\}$  (resp.  $c_{\Omega}^{(\mathbf{x})}$  for the sequence  $\{L_n^{(\mathbf{x})}\}$ ), i.e.,
+
+$$
+\lim  _ {n} \sup  \frac {\left\| L _ {n} ^ {(\boldsymbol {\theta})} \right\|}{\sqrt {\beta_ {n} \log (u _ {n})}} \leq C _ {\Omega} ^ {(\boldsymbol {\theta})}, \tag {68a}
+$$
+
+$$
+\lim  _ {n} \sup  \frac {\| L _ {n} ^ {(\mathbf {x})} \|}{\sqrt {\gamma_ {n} \log (s _ {n})}} \leq C _ {\Omega} ^ {(\mathbf {x})}, \tag {68b}
+$$
+
+such that  $\| L_n^{(\pmb{\theta})}\| = O\big(\sqrt{\beta_n\log(u_n)}\big)$  a.s. and  $\| L_{n}^{(\mathbf{x})}\| = O\big(\sqrt{\gamma_{n}\log(s_{n})}\big)$  a.s. which completes the proof.
+
+Note that we have  $\mathbf{x}_n - \boldsymbol{\mu}$  and  $L_{n}^{(\mathbf{x})}$  weakly converge to the same Gaussian distribution from Remark E.1 and (65). Then,  $\gamma_n^{-1 / 2}\Delta_n^{(\mathbf{x})}$  weakly converges to zero, implying that  $\gamma_n^{-1 / 2}\Delta_n^{(\mathbf{x})}$  converges to zero with probability 1. Therefore, together with  $\{\gamma_n\}$  being strictly positive, we have
+
+$$
+\Delta_ {n} ^ {(\mathbf {x})} = o \left(\sqrt {\gamma_ {n}}\right) \quad a. s. \tag {69}
+$$
+
+Characterization of Sequences  $\{R_n^{(\theta)}\}$  and  $\{\Delta_n^{(\theta)}\}$
+
+We first consider the sequence  $\{R_n^{(\theta)}\}$ . We assume a positive real-valued bounded sequence  $\{w_{n}\}$  under the same conditions as in Mokkadem & Pelletier (2006, Definition 1), i.e.,
+
+Definition E.1. In the case  $b < 1$ ,  $\frac{w_n}{w_{n+1}} = 1 + o(\beta_n)$ , which also implies  $\frac{w_n}{w_{n+1}} = 1 + o(\gamma_n)$ .
+
+In the case  $b = 1$ , there exist  $\epsilon \geq 0$  and a nondecreasing slowly varying function  $l(n)$  such that  $w_{n} = n^{-\epsilon}l(n)$ . When  $\epsilon = 0$ , we require function  $l(n)$  to be bounded.
+
+Since  $\| \mathbf{x}_n - \boldsymbol{\mu} \| = o(1)$  by a.s. convergence result, we can assume that there exists  $\{w_n\}$  such that  $\| \mathbf{x}_n - \boldsymbol{\mu} \| = O(w_n)$ . Then, from (55b), we can use the Abel transformation and obtain
+
+$$
+\begin{array}{l} R _ {n} ^ {(\boldsymbol {\theta})} = \beta_ {n} \gamma_ {n} ^ {- 1} \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} (\mathbf {x} _ {n} - \boldsymbol {\mu}) - e ^ {(u _ {n} - u _ {1}) \mathbf {J} _ {1 1}} \beta_ {1} \gamma_ {1} ^ {- 1} \mathbf {U} _ {1 1} \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} (\mathbf {x} _ {1} - \boldsymbol {\mu}) \\ + e ^ {u _ {n} \mathbf {J} _ {1 1}} \sum_ {k = 1} ^ {n - 1} \left(e ^ {- u _ {k} \mathbf {J} _ {1 1}} \beta_ {k} \gamma_ {k} ^ {- 1} - e ^ {- u _ {k + 1} \mathbf {J} _ {1 1}} \beta_ {k + 1} \gamma_ {k + 1} ^ {- 1}\right) \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} (\mathbf {x} _ {k + 1} - \boldsymbol {\mu}), \\ \end{array}
+$$
+
+where the last term on the RHS can be rewritten as
+
+$$
+W _ {n} = \sum_ {k = 1} ^ {n - 1} e ^ {(u _ {n} - u _ {k + 1}) \mathbf {J} _ {1 1}} \beta_ {k + 1} \gamma_ {k + 1} ^ {- 1} \left(e ^ {\beta_ {k + 1} \mathbf {J} _ {1 1}} \beta_ {k} \beta_ {k + 1} ^ {- 1} \gamma_ {k} ^ {- 1} \gamma_ {k + 1} - \mathbf {I}\right) \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} (\mathbf {x} _ {k + 1} - \boldsymbol {\mu}).
+$$
+
+Using Lemma G.6 on  $W_{n}$  gives  $\| W_n\| = O(\gamma_n^{-1}\| e^{\beta_n\mathbf{J}_{11}} - \mathbf{I}\| \| \mathbf{x}_n - \pmb {\mu}\|) = O(\gamma_n^{-1}\beta_n\omega_n)$ . Then, it follows that for some  $T > 0$
+
+$$
+\left\| R _ {n} ^ {(\boldsymbol {\theta})} \right\| = O \left(\beta_ {n} \gamma_ {n} ^ {- 1} \omega_ {n} + \left\| e ^ {u _ {n} \mathbf {J} _ {1 1}} \right\|\right) = O \left(\beta_ {n} \gamma_ {n} ^ {- 1} \omega_ {n} + e ^ {- u _ {n} T}\right) \tag {70}
+$$
+
+with the application of Lemma G.4 to the second equality.
+
+Then, we shift our focus on  $\{\Delta_n^{(\theta)}\}$ . Specifically, we take (54), (55a), and (56) back to  $\Delta_{n}^{(\theta)} = \pmb{\theta}_{n} - \pmb{\theta}^{*} - L_{n}^{(\theta)} - R_{n}^{(\theta)}$ , and obtain
+
+$$
+\begin{array}{l} \Delta_ {n + 1} ^ {(\boldsymbol {\theta})} = (\mathbf {I} + \beta_ {n + 1} \mathbf {J} _ {1 1}) (\boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*}) \\ + \beta_ {n + 1} \left(r _ {n} ^ {(\boldsymbol {\theta}, 1)} + r _ {n} ^ {(\boldsymbol {\theta}, 2)} + \eta_ {n} ^ {(\boldsymbol {\theta})} - \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} \left(r _ {n} ^ {(\mathbf {x}, 1)} + r _ {n} ^ {(\mathbf {x}, 2)} + \eta_ {n} ^ {(\mathbf {x})}\right)\right) \\ - e ^ {\beta_ {n + 1} \mathbf {J} _ {1 1}} L _ {n} ^ {(\boldsymbol {\theta})} - e ^ {\beta_ {n + 1} \mathbf {J} _ {1 1}} R _ {n} ^ {(\boldsymbol {\theta})} \\ = \left(\mathbf {I} + \beta_ {n + 1} \mathbf {J} _ {1 1}\right) \left(\boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*}\right) + \beta_ {n + 1} \left(r _ {x} ^ {(\boldsymbol {\theta}, 1)} + r _ {x} ^ {(\boldsymbol {\theta}, 2)} + \eta_ {x} ^ {(\boldsymbol {\theta})} - \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} \left(r _ {x} ^ {(\mathbf {x}, 1)} + r _ {x} ^ {(\mathbf {x}, 2)} + \eta_ {x} ^ {(\mathbf {x})}\right)\right) \tag {71} \\ - \left(\mathbf {I} + \beta_ {n + 1} \mathbf {J} _ {1 1} + O \left(\beta_ {n + 1} ^ {2}\right)\right) L _ {n} ^ {(\boldsymbol {\theta})} - \left(\mathbf {I} + \beta_ {n + 1} \mathbf {J} _ {1 1} + O \left(\beta_ {n + 1} ^ {2}\right)\right) R _ {n} ^ {(\boldsymbol {\theta})} \\ = (\mathbf {I} + \beta_ {n + 1} \mathbf {J} _ {1 1}) \Delta_ {n} ^ {(\pmb {\theta})} + O (\beta_ {n + 1} ^ {2}) (L _ {n} ^ {(\pmb {\theta})} + R _ {n} ^ {(\pmb {\theta}))}) \\ + \beta_ {n + 1} (r _ {n} ^ {(\pmb {\theta}, 1)} + r _ {n} ^ {(\pmb {\theta}, 2)} + \eta_ {n} ^ {(\pmb {\theta})} - \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} (r _ {n} ^ {(\mathbf {x}, 1)} + r _ {n} ^ {(\mathbf {x}, 2)} + \eta_ {n} ^ {(\mathbf {x}))}), \\ \end{array}
+$$
+
+where the second equality is by taking the Taylor expansion  $e^{\beta_{n + 1}\mathbf{J}_{11}} = \mathbf{I} + \beta_{n + 1}\mathbf{J}_{11} + O(\beta_{n + 1}^2)$ .
+
+Define  $\Phi_{k,n} \triangleq \prod_{j=k+1}^{n} (\mathbf{I} + \beta_j \mathbf{J}_{11})$  and by convention  $\Phi_{n,n} = \mathbf{I}$ . Then, we rewrite (71) as
+
+$$
+\begin{array}{l} \Delta_ {n + 1} ^ {(\boldsymbol {\theta})} = \sum_ {k = 1} ^ {n} \Phi_ {k, n} \beta_ {k + 1} \left(O (\beta_ {k + 1}) L _ {k} ^ {(\boldsymbol {\theta})} + O (\beta_ {k + 1}) R _ {k} ^ {(\boldsymbol {\theta})}\right) \\ + \sum_ {k = 1} ^ {n} \Phi_ {k, n} \beta_ {k + 1} (r _ {k} ^ {(\boldsymbol {\theta}, 1)} + r _ {k} ^ {(\boldsymbol {\theta}, 2)} + \eta_ {k} ^ {(\boldsymbol {\theta})} - \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} (r _ {k} ^ {(\mathbf {x}, 1)} + r _ {k} ^ {(\mathbf {x}, 2)} + \eta_ {k} ^ {(\mathbf {x}))}) \\ = \sum_ {k = 1} ^ {n} \Phi_ {k, n} \beta_ {k + 1} \left(O \left(\beta_ {k + 1}\right) L _ {k} ^ {(\boldsymbol {\theta})} + O \left(\beta_ {k + 1}\right) R _ {k} ^ {(\boldsymbol {\theta})}\right) \\ + \sum_ {k = 1} ^ {n} \Phi_ {k, n} \beta_ {k + 1} (r _ {k} ^ {(\pmb {\theta}, 1)} + \eta_ {k} ^ {(\pmb {\theta})} - \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} (r _ {k} ^ {(\mathbf {x}, 1)} + \eta_ {k} ^ {(\mathbf {x})})) \\ + \sum_ {k = 1} ^ {n} \Phi_ {k, n} \beta_ {k + 1} \left(r _ {k} ^ {(\boldsymbol {\theta}, 2)} - \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} r _ {k} ^ {(\mathbf {x}, 2)}\right). \tag {72} \\ \end{array}
+$$
+
+From (72), we can indeed decompose  $\Delta_{n+1}^{(\boldsymbol{x})}$  into two parts  $\Delta_{n+1}^{(\boldsymbol{\theta})} = \Delta_{n+1}^{(\boldsymbol{\theta}, 1)} + \Delta_{n+1}^{(\boldsymbol{\theta}, 2)}$ , where
+
+$$
+\begin{array}{l} \Delta_ {n + 1} ^ {(\boldsymbol {\theta}, 1)} \triangleq \sum_ {k = 1} ^ {n} \Phi_ {k, n} \beta_ {k + 1} \left(O \left(\beta_ {k + 1}\right) L _ {k} ^ {(\boldsymbol {\theta})} + O \left(\beta_ {k + 1}\right) R _ {k} ^ {(\boldsymbol {\theta})}\right) (73a) \\ + \sum_ {k = 1} ^ {n} \Phi_ {k, n} \beta_ {k + 1} (r _ {k} ^ {(\pmb {\theta}, 1)} + \eta_ {k} ^ {(\pmb {\theta})} - \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} (r _ {k} ^ {(\mathbf {x}, 1)} + \eta_ {k} ^ {(\mathbf {x})})), \\ \Delta_ {n + 1} ^ {(\boldsymbol {\theta}, 2)} \triangleq \sum_ {k = 1} ^ {n} \Phi_ {k, n} \beta_ {k + 1} \left(r _ {k} ^ {(\boldsymbol {\theta}, 2)} - \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} r _ {k} ^ {(\mathbf {x}, 2)}\right). (73b) \\ \end{array}
+$$
+
+This term  $\Delta_{n+1}^{(\theta,1)}$  shares the same recursive form as in the sequence defined in Mokkadem & Pelletier (2006, Lemma 6), which is given below.
+
+Lemma E.7 (Mokkadem & Pelletier (2006) Lemma 6). For  $\Delta_{n+1}^{(\theta,1)}$  in the form of (73a), assume  $\| \mathbf{x}_n - \boldsymbol{\mu} \| = O(\omega_n)$  and  $\| \Delta_n^{(\mathbf{x})} \| = O(\delta_n)$  for the sequences  $\omega_n, \delta_n$  defined in (E.1). Then, we have
+
+$$
+\| \Delta_ {n + 1} ^ {(\boldsymbol {\theta}, 1)} \| = O \left(\beta_ {n} ^ {2} \gamma_ {n} ^ {- 2} \omega_ {n} ^ {2} + \beta_ {n} \gamma_ {n} ^ {- 1} \delta_ {n}\right) + o (\sqrt {\beta_ {n}}) \quad a. s.
+$$
+
+Since we already have  $\Delta_n^{(\mathbf{x})} = o(\sqrt{\gamma_n})$  in (69), together with Lemma E.7, we have
+
+$$
+\| \Delta_ {n + 1} ^ {(\boldsymbol {\theta}, 1)} \| = O (\beta_ {n} ^ {2} \gamma_ {n} ^ {- 2} \omega_ {n} ^ {2}) + o (\beta_ {n} \gamma_ {n} ^ {- 1 / 2}) + o (\sqrt {\beta_ {n}}) = O (\beta_ {n} ^ {2} \gamma_ {n} ^ {- 2} \omega_ {n} ^ {2}) + o (\sqrt {\beta_ {n}})
+$$
+
+where the second equality comes from  $o(\beta_n\gamma_n^{-1 / 2}) = o(\beta_n^{1 / 2}(\beta_n\gamma_n^{-1})^{1 / 2}) = o(\beta_n^{1 / 2}).$
+
+We now focus on  $\Delta_{n + 1}^{(\theta ,2)}$ . Define a sequence
+
+$$
+\Psi_ {n} \triangleq \sum_ {k = 1} ^ {n} r _ {k} ^ {(\boldsymbol {\theta}, 2)} - \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} r _ {k} ^ {(\mathbf {x}, 2)}, \tag {74}
+$$
+
+and we have
+
+$$
+\begin{array}{l} \beta_ {n + 1} ^ {- 1 / 2} \sum_ {k = 1} ^ {n} \Phi_ {k, n} \beta_ {k + 1} \left(r _ {k} ^ {\left(\boldsymbol {\theta}, 2\right)} - \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} r _ {k} ^ {\left(\mathbf {x}, 2\right)}\right) \\ = \beta_ {n + 1} ^ {- 1 / 2} \sum_ {k = 1} ^ {n} \Phi_ {k, n} \beta_ {k + 1} (\Psi_ {k} - \Psi_ {k - 1}) \\ = \beta_ {n + 1} ^ {1 / 2} \Psi_ {n} + \beta_ {n + 1} ^ {- 1 / 2} \sum_ {k = 1} ^ {n - 1} \left(\beta_ {k} \Phi_ {k, n} - \beta_ {k + 1} \Phi_ {k + 1, n}\right) \Psi_ {k} \\ \end{array}
+$$
+
+where the last equality comes from the Abel transformation. Note that
+
+$$
+\begin{array}{l} \left\| \beta_ {k} \Phi_ {k, n} - \beta_ {k + 1} \Phi_ {k + 1, n} \right\| \leq \beta_ {k + 1} \left\| \Phi_ {k, n} - \Phi_ {k + 1, n} \right\| + \left(\beta_ {k} - \beta_ {k + 1}\right) \left\| \Phi_ {k, n} \right\| \\ \leq \beta_ {k + 1} \| \Phi_ {k + 1, n} \| \beta_ {k} \| \mathbf {J} _ {1 1} \| + C _ {7} \beta_ {k} ^ {2} \| \Phi_ {k, n} \| \\ \leq C _ {8} \beta_ {k} ^ {2} e ^ {- (u _ {n} - u _ {k}) T} \\ \end{array}
+$$
+
+for some constant  $C_7, C_8 > 0$ , where the last inequality is from Lemma G.4 and  $\| \Phi_{k+1,n} \| \leq C_9 \| \Phi_{k,n} \|$  for some constant  $C_9 > 0$  that depends on  $e^{\beta_0 T}$ . Then,
+
+$$
+\begin{array}{l} \beta_ {n + 1} ^ {- 1 / 2} \left\| \sum_ {k = 1} ^ {n} \Phi_ {k, n} \beta_ {k + 1} \left(r _ {k} ^ {\left(\boldsymbol {\theta}, 2\right)} - \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} r _ {k} ^ {\left(\mathbf {x}, 2\right)}\right) \right\| \\ \leq \| \beta_ {n + 1} ^ {1 / 2} \Psi_ {n} \| + \left(\frac {\beta_ {n + 1}}{\beta_ {n}}\right) ^ {1 / 2} \beta_ {n} ^ {- 1 / 2} \sum_ {k = 1} ^ {n} \| \beta_ {k} \Phi_ {k, n} - \beta_ {k + 1} \Phi_ {k + 1, n} \| \| \Psi_ {k} \| \\ \leq \| \beta_ {n + 1} ^ {1 / 2} \Psi_ {n} \| + C _ {8} \left(\frac {\beta_ {n + 1}}{\beta_ {n}}\right) ^ {1 / 2} \beta_ {n} ^ {- 1 / 2} \sum_ {k = 1} ^ {n} \beta_ {k} ^ {3 / 2} e ^ {- (u _ {n} - u _ {k}) T} \| \beta_ {k} ^ {1 / 2} \Psi_ {k} \|. \\ \end{array}
+$$
+
+By Lemma E.1, we have  $\beta_n^{1/2} \Psi_n \to 0$  a.s. such that by Lemma G.6, it follows that
+
+$$
+\lim _ {n} \sup  _ {\boldsymbol {n}} \beta_ {n + 1} ^ {- 1 / 2} \left\| \sum_ {k = 1} ^ {n} \Phi_ {k, n} \beta_ {k + 1} (r _ {k} ^ {(\boldsymbol {\theta}, 2)} - \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} r _ {k} ^ {(\mathbf {x}, 2)}) \right\| \leq \frac {\lim  \sup  _ {n} \| \beta_ {n} ^ {1 / 2} \Psi_ {n} \|}{C (T , 1 / 2)} = 0.
+$$
+
+Therefore, we have
+
+$$
+\Delta_ {n + 1} ^ {(\boldsymbol {\theta}, 2)} = \sum_ {k = 1} ^ {n} \Phi_ {k, n} \beta_ {k + 1} \left(r _ {k} ^ {(\boldsymbol {\theta}, 2)} - \mathbf {J} _ {1 2} (\alpha) \mathbf {J} _ {2 2} (\alpha) ^ {- 1} r _ {k} ^ {(\mathbf {x}, 2)}\right) = o \left(\sqrt {\beta_ {n}}\right). \tag {75}
+$$
+
+Consequently,  $\Delta_{n + 1}^{(\theta)} = O(\beta_n^2\gamma_n^{-2}\omega_n^2) + o(\sqrt{\beta_n})$  almost surely.
+
+Now we are dealing with  $\mathbf{x}_n - \pmb{\mu}$  and its related sequence  $\omega_{n}$ . Note that by Lemma E.5 and (69), we have almost surely,
+
+$$
+\begin{array}{l} \left\| \mathbf {x} _ {n} - \boldsymbol {\mu} \right\| = O \left(\left\| L _ {n} ^ {(\mathbf {x})} \right\| + \left\| \Delta_ {n} ^ {\mathbf {x}} \right\|\right) \\ = O \left(\sqrt {\gamma_ {n} \log \left(s _ {n}\right)} + o \left(\sqrt {\gamma_ {n}}\right)\right) \tag {76} \\ = O \left(\sqrt {\gamma_ {n} \log \left(s _ {n}\right)}\right). \\ \end{array}
+$$
+
+Thus, we can set  $\omega_{n}\equiv O(\sqrt{\gamma_{n}\log(s_{n})})$  such that  $\| R_n^{(\pmb {\theta})}\|$  in (70) can be written as
+
+$$
+\left\| R _ {n} ^ {(\boldsymbol {\theta})} \right\| = O \left(n ^ {a / 2 - b} \sqrt {\log \left(s _ {n}\right)} + e ^ {- u _ {n} T}\right),
+$$
+
+and
+
+$$
+\left\| \Delta_ {n + 1} ^ {(\boldsymbol {\theta})} \right\| = O \left(n ^ {a - 2 b} \log \left(s _ {n}\right)\right) + o \left(\sqrt {\beta_ {n}}\right).
+$$
+
+In view of assumption A2 and  $\beta_{n} = o(\gamma_{n})$ ,  $a / 2 - b < -b / 2$  and  $a - 2b < -b$ , there exists a  $c > b / 2$  such that almost surely,
+
+$$
+\left\| R _ {n} ^ {\left(\boldsymbol {\theta}\right)} \right\| = O \left(n ^ {- s}\right), \quad \left\| \Delta_ {n + 1} ^ {\left(\boldsymbol {\theta}\right)} \right\| = o (\sqrt {\beta_ {n}}).
+$$
+
+Therefore,  $\beta_{n}^{-1 / 2}(R_{n}^{(\pmb{\theta})} + \Delta_{n + 1}^{(\pmb{\theta})})\to 0$  almost surely. This completes the proof of Scenario 2.
+
+# E.3 CASE (III):  $\gamma_{n} = o(\beta_{n})$
+
+For  $\gamma_{n} = o(\beta_{n})$ , we can see that the roles of  $\theta_{n}$  and  $\mathbf{x}_n$  are flipped, i.e.,  $\theta_{n}$  is now on fast timescale while  $\mathbf{x}_n$  is on slow timescale.
+
+We still decompose  $\mathbf{x}_n$  as  $\mathbf{x}_n - \boldsymbol{\mu} = L_n^{(\mathbf{x})} + \Delta_n^{(\mathbf{x})}$ , where  $L_{n}^{(\mathbf{x})},\Delta_{n}^{(\mathbf{x})}$  are defined in (56) and (57), respectively. Since  $\mathbf{x}_n$  is independent of  $\theta_{n}$ , the results of  $L_{n}^{(\mathbf{x})}$  and  $\Delta_{n}^{(\mathbf{x})}$  remain the same, i.e., almost surely,  $L_{n}^{(\mathbf{x})} = O\bigl (\sqrt{\gamma_{n}\log(s_{n})}\bigr)$  from Lemma E.5 and  $\Delta_n^{(\mathbf{x})} = o(\sqrt{\gamma_n})$  from (69). Then, we define sequences  $\hat{L}_n^{(\pmb {\theta})}$  and  $\hat{R}_n^{(\pmb {\theta})}$  as follows.
+
+$$
+\hat {L} _ {n} ^ {(\boldsymbol {\theta})} \triangleq e ^ {\beta_ {n} \mathbf {J} _ {1 1}} \hat {L} _ {n - 1} ^ {(\boldsymbol {\theta})} + \beta_ {n} M _ {n} ^ {(\boldsymbol {\theta})} = \sum_ {k = 1} ^ {n} e ^ {(u _ {n} - u _ {k}) \mathbf {J} _ {1 1}} \beta_ {k} M _ {k} ^ {(\boldsymbol {\theta})}, \tag {77a}
+$$
+
+$$
+\hat {R} _ {n} ^ {(\boldsymbol {\theta})} \triangleq e ^ {\beta_ {n} \mathbf {J} _ {1 1}} \hat {R} _ {n - 1} ^ {(\boldsymbol {\theta})} + \beta_ {n} \mathbf {J} _ {1 2} (\alpha) \left(L _ {n - 1} ^ {(\mathbf {x})} + R _ {n - 1} ^ {(\mathbf {x})}\right) = \sum_ {k = 1} ^ {n} e ^ {\left(u _ {n} - u _ {k}\right) \mathbf {J} _ {1 1}} \beta_ {k} \mathbf {J} _ {1 2} (\alpha) \left(L _ {k - 1} ^ {(\mathbf {x})} + R _ {k - 1} ^ {(\mathbf {x})}\right). \tag {77b}
+$$
+
+Moreover, the remaining term  $\hat{\Delta}_n^{(\pmb {\theta})}\triangleq \pmb {\theta}_n - \pmb {\theta}^* -\hat{L}_n^{(\pmb {\theta})} - \hat{R}_n^{(\pmb {\theta})}.$
+
+The proof outline is the same as in the previous scenario:
+
+- We first show  $\beta_n^{-1/2} \hat{\Delta}_n^{(\theta)}$  weakly converges to the distribution  $N(0, \mathbf{V}_{\pmb{\theta}}^{(3)})(\alpha)$ ;  
+- We analyse  $\hat{L}_n^{(\theta)}$  and  $\hat{R}_n^{(\theta)}$  to ensure that these two terms decrease faster than the CLT scale  $\beta_n^{-1/2}$ , i.e.,  $\lim_{n\to\infty}\beta_n^{-1/2}(\hat{L}_n^{(\theta)} - \hat{R}_n^{(\theta)}) = 0$ ;
+
+- With above two steps, we can show that  $\beta_{n}^{-1 / 2}(\pmb{\theta}_{n} - \pmb{\theta}^{*})$  weakly converges to the distribution  $N(0,\mathbf{V}_{\pmb{\theta}}^{(3)})(\alpha)$ .
+
+Analysis of  $\hat{L}_n^{(\theta)}$
+
+We first focus on  $\hat{L}_n^{(\theta)}$  and follow similar steps as we did when we analysed  $L_{n}^{(\theta)}$  in the previous scenario. We set a Martingale  $Z^{(n)} = \{Z_k^{(n)}\}_{k\geq 1}$  such that
+
+$$
+Z _ {k} ^ {(n)} = \beta_ {n} ^ {- 1 / 2} \sum_ {k = 1} ^ {n} e ^ {\left(u _ {n} - u _ {k}\right) \mathbf {J} _ {1 1} \beta_ {k} M _ {k} ^ {(\boldsymbol {\theta})}}.
+$$
+
+Then,
+
+$$
+A _ {n} \triangleq \sum_ {k = 1} ^ {n} \mathbb {E} \left[ (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) ^ {T} \Big | \mathcal {F} _ {k - 1} \right].
+$$
+
+Following the similar steps in (59) to decompose  $M_{k}^{(\pmb{\theta})}$  with (42b), we have
+
+$$
+\begin{array}{l} A _ {n} = \beta_ {n} ^ {- 1} \sum_ {k = 1} ^ {n} \beta_ {k} ^ {2} e ^ {(u _ {n} - u _ {k}) \mathbf {J} _ {1 1}} \left(\mathbf {U} _ {1 1} + \mathbf {D} _ {k} ^ {(3)} + \mathbf {J} _ {k} ^ {(3)}\right) e ^ {(u _ {n} - u _ {k}) \mathbf {J} _ {1 1} ^ {T}} \\ = \underbrace {\beta_ {n} ^ {- 1} \sum_ {k = 1} ^ {n} \beta_ {k} ^ {2} e ^ {(u _ {n} - u _ {k}) \mathbf {J} _ {1 1}} \mathbf {U} _ {1 1} e ^ {(u _ {n} - u _ {k}) \mathbf {J} _ {1 1} ^ {T}}} _ {A _ {n} ^ {(a)}} + \underbrace {\beta_ {n} ^ {- 1} \sum_ {k = 1} ^ {n} \beta_ {k} ^ {2} e ^ {(u _ {n} - u _ {k}) \mathbf {J} _ {1 1}} \mathbf {D} _ {k} ^ {(3)} e ^ {(u _ {n} - u _ {k}) \mathbf {J} _ {1 1} ^ {T}}} _ {A _ {n} ^ {(b)}} \tag {78} \\ + \underbrace {\beta_ {n} ^ {- 1} \sum_ {k = 1} ^ {n} \beta_ {k} ^ {2} e ^ {(u _ {n} - u _ {k}) \mathbf {J} _ {1 1}} \mathbf {J} _ {k} ^ {(3)} e ^ {(u _ {n} - u _ {k}) \mathbf {J} _ {1 1} ^ {T}}} _ {A _ {n} ^ {(c)}} \\ \end{array}
+$$
+
+Since  $A_{n}^{(a)}, A_{n}^{(b)}, A_{n}^{(c)}$  share similar forms as in Lemma E.3, we follow the same steps as the proof therein, with the application of Lemma E.1. To avoid repetition, we omit the proof and directly give the following lemma.
+
+Lemma E.8. For  $A_{n}^{(a)}, A_{n}^{(b)}, A_{n}^{(c)}$  defined in (78), we have
+
+$$
+\lim  _ {n \rightarrow \infty} A _ {n} ^ {(a)} = \mathbf {V} _ {\boldsymbol {\theta}} ^ {(3)} (\alpha), \quad \lim  _ {n \rightarrow \infty} \| A _ {n} ^ {(b)} \| = 0, \quad \lim  _ {n \rightarrow \infty} \| A _ {n} ^ {(c)} \| = 0, \tag {79}
+$$
+
+where  $\mathbf{V}_{\pmb{\theta}}^{(3)}(\alpha)$  is the solution to the Lyapunov equation
+
+$$
+\mathbf {J} _ {1 1} \mathbf {V} + \mathbf {V} \mathbf {J} _ {1 1} ^ {T} + \mathbf {U} _ {1 1} = 0.
+$$
+
+Note that here we don't have the term  $\frac{\mathbb{1}_{\{b = 1\}}}{2}\pmb{I}$  in above lemma, compared to Lemma E.3, because in the case of  $\gamma_{n} = o(\beta_{n})$ ,  $b < 1$  such that  $\mathbb{1}_{\{b = 1\}} = 0$ . Then, applying Lemma G.1 to derive the closed form of  $\mathbf{V}_{\pmb{\theta}}^{(3)}(\alpha)$  gives
+
+$$
+\mathbf {V} _ {\boldsymbol {\theta}} ^ {(3)} (\alpha) = \int_ {0} ^ {\infty} e ^ {t \nabla_ {\boldsymbol {\theta}} \mathbf {h} (\boldsymbol {\theta} ^ {*})} \mathbf {U} _ {1 1} e ^ {t \nabla_ {\boldsymbol {\theta}} \mathbf {h} (\boldsymbol {\theta} ^ {*})} d t.
+$$
+
+Thus, it follows that
+
+$$
+\lim  _ {n \rightarrow \infty} \sum_ {k = 1} ^ {n} \mathbb {E} \left[ (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) ^ {T} | \mathcal {F} _ {k - 1} \right] = \mathbf {V} _ {\boldsymbol {\theta}} ^ {(3)} (\alpha).
+$$
+
+Again, we use the Martingale CLT result in Theorem G.3 and have the following result.
+
+$$
+Z ^ {n} = \beta_ {n} ^ {- 1 / 2} \hat {L} _ {n} ^ {(\boldsymbol {\theta})} \xrightarrow [ d i s t . ]{n \to \infty} N \left(0, \mathbf {V} _ {\boldsymbol {\theta}} ^ {(3)} (\alpha)\right).
+$$
+
+Moreover, similar to the tighter upper bound of  $L_{n}^{(\mathbf{x})}$  proved in Lemma E.5, we utilize the tighter upper bound Lemma E.6 in the proof thereof, and obtain  $\hat{L}_n^{(\pmb{\theta})} = O(\sqrt{\beta_n\log(u_n)})$ .
+
+Analysis of  $\hat{R}_n^{(\theta)}$
+
+Next, we turn to the term  $\hat{R}_n^{(\theta)}$  in (77b). Taking the norm gives the following inequality for some constant  $C, T > 0$  by applying Lemma G.4,
+
+$$
+\| \hat {R} _ {n} ^ {(\boldsymbol {\theta})} \| \leq C \sum_ {k = 1} ^ {n} e ^ {- (u _ {n} - u _ {k}) T} \beta_ {k} (\| L _ {k - 1} ^ {(\mathbf {x})} \| + \| R _ {k - 1} ^ {(\mathbf {x})} \)).
+$$
+
+Using Lemma G.6 gives
+
+$$
+\sum_ {k = 1} ^ {n} e ^ {- (u _ {n} - u _ {k}) T} \beta_ {k} (\| L _ {k - 1} ^ {(\mathbf {x})} \| + \| R _ {k - 1} ^ {(\mathbf {x})} \|) = O (\| L _ {k - 1} ^ {(\mathbf {x})} \| + \| R _ {n - 1} ^ {(\mathbf {x})} \)).
+$$
+
+Thus,  $\beta_n^{-1/2} \|\hat{R}_n^{(\pmb{\theta})}\| = o(\sqrt{\gamma_n \beta_n^{-1}}) + O\left(\sqrt{\gamma_n \beta_n^{-1} \log(s_n)}\right)$ . Since  $\gamma_n = o(\beta_n), \gamma_n \beta_n^{-1} = (n + 1)^{b - a}$ , where  $b - a < 0$ . Then, there exists some  $s > 0$  such that  $b - a < -s < 0$ . Together with  $\log(s_n) = O(\log(n))$ , we have  $O\left(\sqrt{\gamma_n \beta_n^{-1} \log(s_n)}\right) = O(\sqrt{n^{-s} \log(n)}) = o(1)$ . Therefore, we have
+
+$$
+\lim _ {n \to \infty} \beta_ {n} ^ {- 1 / 2} \hat {R} _ {n} ^ {(\boldsymbol {\theta})} = 0.
+$$
+
+Analysis of  $\hat{\Delta}_n^{(\theta)}$
+
+Lastly, let's focus on the term  $\hat{\Delta}_n^{(\theta)}$ . We have
+
+$$
+\begin{array}{l} \hat {\Delta} _ {n + 1} ^ {(\boldsymbol {\theta})} = \boldsymbol {\theta} _ {n + 1} - \boldsymbol {\theta} ^ {*} - \hat {L} _ {n + 1} ^ {(\boldsymbol {\theta})} - \hat {R} _ {n + 1} ^ {(\boldsymbol {\theta})} \\ = \boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*} + \beta_ {n + 1} \left(\mathbf {J} _ {1 1} (\boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*}) + \mathbf {J} _ {1 2} (\alpha) (\mathbf {x} _ {n} - \boldsymbol {\mu}) + M _ {n + 1} ^ {(\boldsymbol {\theta})} + r _ {n} ^ {(\boldsymbol {\theta}, 1)} + r _ {n} ^ {(\boldsymbol {\theta}, 2)} + \eta_ {n} ^ {(\boldsymbol {\theta})}\right) \\ - e ^ {\beta_ {n + 1} \mathbf {J} _ {1 1}} \hat {L} _ {n} ^ {(\boldsymbol {\theta})} - \beta_ {n + 1} M _ {n + 1} ^ {(\boldsymbol {\theta})} - e ^ {\beta_ {n + 1} \mathbf {J} _ {1 1}} \hat {R} _ {n} ^ {(\boldsymbol {\theta})} - \beta_ {n + 1} \mathbf {J} _ {1 2} (\alpha) (L _ {n} ^ {(\mathbf {x})} + R _ {n} ^ {(\mathbf {x})}) \\ = (\mathbf {I} + \beta_ {n + 1} \mathbf {J} _ {1 1}) \left(\boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*}\right) + \beta_ {n + 1} \mathbf {J} _ {1 2} (\alpha) \Delta_ {n} ^ {(\mathbf {x})} + \beta_ {n + 1} \left(r _ {n} ^ {(\boldsymbol {\theta}, 1)} + r _ {n} ^ {(\boldsymbol {\theta}, 2)} + \eta_ {n} ^ {(\boldsymbol {\theta})}\right) \\ - \left(\mathbf {I} + \beta_ {n + 1} \mathbf {J} _ {1 1} + O \left(\beta_ {n + 1} ^ {2}\right)\right) \left(\hat {L} _ {n} ^ {(\boldsymbol {\theta})} + \hat {R} _ {n} ^ {(\boldsymbol {\theta})}\right) \\ = \left(\mathbf {I} + \beta_ {n + 1} \mathbf {J} _ {1 1}\right) \hat {\Delta} _ {n} ^ {(\boldsymbol {\theta})} + \beta_ {n + 1} \mathbf {J} _ {1 2} (\alpha) \Delta_ {n} ^ {(\mathbf {x})} + \beta_ {n + 1} \left(r _ {n} ^ {(\boldsymbol {\theta}, 1)} + r _ {n} ^ {(\boldsymbol {\theta}, 2)} + \eta_ {n} ^ {(\boldsymbol {\theta})}\right) \\ + O \left(\beta_ {n + 1} ^ {2}\right) \left(\hat {L} _ {n} ^ {(\boldsymbol {\theta})} + \hat {R} _ {n} ^ {(\boldsymbol {\theta})}\right). \\ \end{array}
+$$
+
+where the second equality is from (53a), the third equality stems from the approximation of  $e^{\beta_{n+1}\mathbf{J}_{11}}$ . Then, we again use the definition  $\Phi_{k,n} \triangleq \prod_{j=k+1}^{n}(\mathbf{I} + \beta_{j}\mathbf{J}_{11})$  and reiterate the above equation as
+
+$$
+\begin{array}{l} \hat {\Delta} _ {n + 1} ^ {(\boldsymbol {\theta})} = \sum_ {k = 1} ^ {n} \Phi_ {k, n} \beta_ {k + 1} \left(O (\beta_ {k + 1}) L _ {k} ^ {(\boldsymbol {\theta})} + O (\beta_ {k + 1}) R _ {k} ^ {(\boldsymbol {\theta})}\right) \\ + \sum_ {k = 1} ^ {n} \Phi_ {k, n} \beta_ {k + 1} \mathbf {J} _ {1 2} (\alpha) \Delta_ {n} ^ {(\mathbf {x})} + \sum_ {k = 1} ^ {n} \Phi_ {k, n} \beta_ {k + 1} \left(r _ {k} ^ {(\boldsymbol {\theta}, 1)} + \eta_ {k} ^ {(\boldsymbol {\theta})}\right) \\ + \sum_ {k = 1} ^ {n} \Phi_ {k, n} \beta_ {k + 1} r _ {k} ^ {(\boldsymbol {\theta}, 2)} \\ \triangleq \hat {\Delta} _ {n + 1} ^ {(\boldsymbol {\theta}, 1)} + \hat {\Delta} _ {n + 1} ^ {(\boldsymbol {\theta}, 2)}, \\ \end{array}
+$$
+
+where  $\hat{\Delta}_{n + 1}^{(\pmb {\theta},2)} = \sum_{k = 1}^{n}\Phi_{k,n}\beta_{k + 1}r_{k}^{(\pmb {\theta},2)}$  and
+
+$$
+\begin{array}{l} \hat {\Delta} _ {n + 1} ^ {(\boldsymbol {\theta}, 1)} = \sum_ {k = 1} ^ {n} \Phi_ {k, n} \beta_ {k + 1} \left(O \left(\beta_ {k + 1}\right) L _ {k} ^ {(\boldsymbol {\theta})} + O \left(\beta_ {k + 1}\right) R _ {k} ^ {(\boldsymbol {\theta})}\right) \tag {80} \\ + \sum_ {k = 1} ^ {n} \Phi_ {k, n} \beta_ {k + 1} \left(r _ {k} ^ {\left(\boldsymbol {\theta}, 1\right)} + \eta_ {k} ^ {\left(\boldsymbol {\theta}\right)} + \mathbf {J} _ {1 2} (\alpha) \Delta_ {n} ^ {\left(\mathbf {x}\right)}\right). \\ \end{array}
+$$
+
+For  $\hat{\Delta}_{n+1}^{(\theta,2)}$ , we follow the same steps from (74) to (75), and obtain  $\hat{\Delta}_{n+1}^{(\theta,2)} = o\left(\sqrt{\beta_n}\right)$ .
+
+Next, we consider  $\hat{\Delta}_{n+1}^{(\pmb{\theta},1)}$  and want to show that  $\hat{\Delta}_{n+1}^{(\pmb{\theta},1)} = o(\sqrt{\beta_n})$ . Again, we utilize Mokkadem & Pelletier (2006, Lemma 6) for  $\hat{\Delta}_{n+1}^{(\pmb{\theta},1)}$  and adapt the notation here for the case  $\gamma_n = o(\beta_n)$ .
+
+Lemma E.9. For  $\hat{\Delta}_{n+1}^{(\pmb{\theta}, 1)}$  in the form of (80), assume  $\|\pmb{\theta}_n - \pmb{\theta}^*\| = O(\omega_n)$  and  $\|\hat{\Delta}_n^{(\pmb{\theta}, 1)}\| = O(\delta_n)$  for the sequences  $\omega_n, \delta_n$  defined in (E.1). Then, we have
+
+$$
+\left\| \hat {\Delta} _ {n + 1} ^ {(\boldsymbol {\theta}, 1)} \right\| = O \left(\gamma_ {n} ^ {2} \beta_ {n} ^ {- 2} \omega_ {n} ^ {2} + \gamma_ {n} \beta_ {n} ^ {- 1} \delta_ {n}\right) + o (\sqrt {\gamma_ {n}}) \quad a. s. \tag {81}
+$$
+
+Now we need to further analyse  $\delta_{n}$  and tighten its big O form, starting from  $\delta_{n} \equiv 1$ , so that we can finally obtain the big O form of  $\|\hat{\Delta}_{n+1}^{(\theta,1)}\|$ . The following steps are borrowed from the ideas in Mokkadem & Pelletier (2006, Section 2.3.2).
+
+By almost sure convergence result  $\lim_{n\to \infty}\pmb{\theta}_n = \pmb{\theta}^*$ , we have  $\lim_{n\to \infty}\Delta_n^{(\pmb{\theta})} = 0$  a.s. such that we can first set  $\delta_{n}\equiv 1$ , and  $\| \hat{\Delta}_{n + 1}^{(\pmb {\theta},1)}\| = O(\gamma_n^2\beta_n^{-2}\omega_n^2 +\gamma_n\beta_n^{-1}) + o(\sqrt{\gamma_n})$ . Then, we redefine
+
+$$
+\delta_ {n} \equiv O (\gamma_ {n} ^ {2} \beta_ {n} ^ {- 2} \omega_ {n} ^ {2} + \gamma_ {n} \beta_ {n} ^ {- 1}) + o (\sqrt {\gamma_ {n}}),
+$$
+
+and notice that it still satisfies definition E.1. Then, reapplying this  $\delta_{n}$  form to (81) gives
+
+$$
+\| \hat {\Delta} _ {n + 1} ^ {(\pmb {\theta}, 1)} \| = O (\gamma_ {n} ^ {2} \beta_ {n} ^ {- 2} \omega_ {n} ^ {2} + [ \gamma_ {n} \beta_ {n} ^ {- 1} ] ^ {2}) + o (\sqrt {\gamma_ {n}})
+$$
+
+and by induction we have for all integers  $k \geq 1$ ,
+
+$$
+\| \hat {\Delta} _ {n + 1} ^ {(\pmb {\theta}, 1)} \| = O (\gamma_ {n} ^ {2} \beta_ {n} ^ {- 2} \omega_ {n} ^ {2} + [ \gamma_ {n} \beta_ {n} ^ {- 1} ] ^ {k}) + o (\sqrt {\gamma_ {n}}).
+$$
+
+Since  $[\gamma_n\beta_n^{-1}]^k = n^{(b - a)k}$ , there exists  $k_0 > a / 2(a - b)$  such that  $[\gamma_n\beta_n^{-1}]^{k_0} = o(\sqrt{\gamma_n})$ , and
+
+$$
+\left\| \hat {\Delta} _ {n + 1} ^ {(\boldsymbol {\theta}, 1)} \right\| = O \left(\gamma_ {n} ^ {2} \beta_ {n} ^ {- 2} \omega_ {n} ^ {2}\right) + o (\sqrt {\gamma_ {n}}). \tag {82}
+$$
+
+Then, as suggested in Mokkadem & Pelletier (2006, Section 2.3.2), we can choose  $\omega_{n} = O(\sqrt{\beta_{n}\log(u_{n})} +[\gamma_{n}\beta_{n}^{-1}]^{k})$ , which also satisfies definition E.1. Then,
+
+$$
+\begin{array}{l} \left\| \boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*} \right\| = \left\| \hat {L} _ {n} ^ {(\boldsymbol {\theta})} + \hat {R} _ {n} ^ {(\boldsymbol {\theta})} + \hat {\Delta} _ {n} ^ {(\boldsymbol {\theta})} \right\| \\ = O \left(\sqrt {\beta_ {n} \log (u _ {n})} + \sqrt {\gamma_ {n} \beta_ {n} ^ {- 1} \log (s _ {n})} + \left(\left[ \gamma_ {n} \beta_ {n} ^ {- 1} \right] ^ {k + 1} + \gamma_ {n} \beta_ {n} ^ {- 1} \sqrt {\beta_ {n} \log (u _ {n})}\right) ^ {2}\right) \\ + o (\sqrt {\beta_ {n}} + \sqrt {\gamma_ {n}}) \\ = O (\sqrt {\beta_ {n} \log (u _ {n})} + [ \gamma_ {n} \beta_ {n} ^ {- 1} ] ^ {k + 1}). \\ \end{array}
+$$
+
+By induction, this holds for all  $k \geq 1$  such that there exists  $k_0$ ,  $[\gamma_n\beta_n^{-1}]^{k_0} = o(\sqrt{\beta_n})$  and  $\| \pmb{\theta}_n - \pmb{\theta}^*\| = O(\sqrt{\beta_n\log(u_n)})$ . Equivalently,  $\omega_{n} = \sqrt{\beta_{n}\log(u_{n})}$ . Therefore, from (82) we have
+
+$$
+\| \hat {\Delta} _ {n + 1} ^ {(\boldsymbol {\theta}, 1)} \| = O (\gamma_ {n} ^ {2} \beta_ {n} ^ {- 1} \log (u _ {n})) + o (\sqrt {\gamma_ {n}}) = o (\sqrt {\gamma_ {n}}).
+$$
+
+Together with  $\| \hat{\Delta}_{n + 1}^{(\pmb {\theta},2)}\| = o(\sqrt{\beta_n})$  , we have  $\beta_{n}^{-1 / 2}\| \hat{\Delta}_{n + 1}^{(\pmb {\theta})}\| = o(\sqrt{\gamma_{n}\beta_{n}^{-1}}) + 1)$  such that
+
+$$
+\lim  _ {n \to \infty} \beta_ {n} ^ {- 1 / 2} \hat {\Delta} _ {n + 1} ^ {(\boldsymbol {\theta})} = 0.
+$$
+
+Thus, we have finished the proof according to the proof outline mentioned at the beginning of this part.
+
+# F DISCUSSION OF COVARIANCE ORDERING OF SA-SRRW
+
+# F.1 PROOF OF PROPOSITION 3.4
+
+For any  $\alpha > 0$  and any vector  $\mathbf{x} \in \mathbb{R}^d$ , we have
+
+$$
+\mathbf {x} ^ {T} \mathbf {V} _ {\boldsymbol {\theta}} ^ {(1)} (\alpha) \mathbf {x} = \int_ {0} ^ {\infty} \mathbf {x} ^ {T} e ^ {t (\nabla_ {\boldsymbol {\theta}} \mathbf {h} (\boldsymbol {\theta} ^ {*}) + \frac {1}{2} \mathbf {\Sigma} _ {\{b = 1 \}} ^ {*} \boldsymbol {I})} \mathbf {U} _ {\boldsymbol {\theta}} (\alpha) e ^ {t (\nabla_ {\boldsymbol {\theta}} \mathbf {h} (\boldsymbol {\theta} ^ {*}) + \frac {1}{2} \mathbf {\Sigma} _ {\{b = 1 \}} ^ {*} \boldsymbol {I}) ^ {T}} \mathbf {x} d t
+$$
+
+where the first equality is from the form of  $\mathbf{V}_{\theta}^{(1)}(\alpha)$  in Theorem 3.3. Let  $\mathbf{y} \triangleq e^{t(\nabla_{\theta}\mathbf{h}(\boldsymbol{\theta}^{*}) + \frac{1}{2}\left\{b = 1\right\} I)}\mathbf{x}$ , with the dependence on variable  $t$  left implicit. The matrix  $\mathbf{U}_{\theta}(\alpha)$ , given explicitly in (11) positive semi definite, since  $\lambda_i \in (-1,1)$  for all  $i \in \{1,\dots,N-1\}$ . Thus, the terms  $\mathbf{y}^T\mathbf{U}_{\theta}(\alpha)\mathbf{y}$  inside the integral are non-negative, and it is enough to provide an ordering on  $\mathbf{y}^T\mathbf{U}_{\theta}(\alpha)\mathbf{y}$  with respect to  $\alpha$ .
+
+For any  $\alpha_{2} > \alpha_{1} > 0$
+
+$$
+\begin{array}{l} \mathbf {y} ^ {T} \mathbf {U} _ {\boldsymbol {\theta}} (\alpha_ {2}) \mathbf {y} = \sum_ {i = 1} ^ {N - 1} \frac {1}{(\alpha_ {2} (1 + \lambda_ {i}) + 1) ^ {2}} \cdot \frac {1 + \lambda_ {i}}{1 - \lambda_ {i}} \mathbf {y} ^ {T} \mathbf {H} ^ {T} \mathbf {u} _ {i} \mathbf {u} _ {i} ^ {T} \mathbf {H} \mathbf {y} \\ <   \sum_ {i = 1} ^ {N - 1} \frac {1}{\left(\alpha_ {1} \left(1 + \lambda_ {i}\right) + 1\right) ^ {2}} \cdot \frac {1 + \lambda_ {i}}{1 - \lambda_ {i}} \mathbf {y} ^ {T} \mathbf {H} ^ {T} \mathbf {u} _ {i} \mathbf {u} _ {i} ^ {T} \mathbf {H} \mathbf {y} = \mathbf {y} ^ {T} \mathbf {U} _ {\boldsymbol {\theta}} (\alpha_ {1}) \mathbf {y} \\ <   \sum_ {i = 1} ^ {N - 1} \cdot \frac {1 + \lambda_ {i}}{1 - \lambda_ {i}} \mathbf {y} ^ {T} \mathbf {H} ^ {T} \mathbf {u} _ {i} \mathbf {u} _ {i} ^ {T} \mathbf {H} \mathbf {y} = \mathbf {y} ^ {T} \mathbf {U} _ {\boldsymbol {\theta}} (0) \mathbf {y}, \\ \end{array}
+$$
+
+where the inequality $^{13}$  is because  $\alpha(1 + \lambda_i) > 0$  for all  $i \in \{1, \dots, N\}$  and any  $\alpha > 0$ . In fact, the ordering is monotone in  $\alpha$ , and  $\mathbf{y}^T \mathbf{U}_{\theta}(\alpha_2) \mathbf{y}$  decreases at rate  $1 / \alpha^2$  as seen from its form in the equation above. This completes the proof.
+
+# F.2 DISCUSSION REGARDING PROPOSITION 3.4 AND MSE ORDERING
+
+We can use Proposition 3.4 to show that the MSE of SA iterates of (4c) driven by SRRW eventually becomes smaller than that SA iterates when the stochastic noise is driven by an i.i.d. sequence of random variables. The diagonal entries of  $\mathbf{V}_{\theta}^{(1)}(\alpha)$  are obtained by evaluating  $\mathbf{e}_i^T\mathbf{V}_{\theta}^{(1)}(\alpha)\mathbf{e}_i$ , where  $\mathbf{e}_i$  is the  $i$ 'th standard basis vector. $^{14}$  These diagonal entries are the asymptotic variance corresponding to the element-wise iterate errors, and for large enough  $n$ , we have  $\mathbf{e}_i^T\mathbf{V}_{\theta}^{(1)}(\alpha)\mathbf{e}_i \approx \mathbb{E}[(\boldsymbol{\theta}_n - \boldsymbol{\theta}^*)_i^2] / \beta_n$  for all  $i \in \{1, \dots, D\}$ . Thus, the trace of matrix  $\mathbf{V}_{\theta}^{(1)}(\alpha)$  approximates the scaled MSE, that is  $\mathrm{Tr}(\mathbf{V}_{\theta}^{(1)}(\alpha)) = \sum_i \mathbf{e}_i^T\mathbf{V}_{\theta}^{(1)}(\alpha)\mathbf{e}_i \approx \sum_i \mathbb{E}[(\boldsymbol{\theta}_n - \boldsymbol{\theta}^*)_i^2] / \beta_n = \mathbb{E}[||\boldsymbol{\theta}_n - \boldsymbol{\theta}^*||^2] / \beta_n$  for large  $n$ . Since all entries of  $\mathbf{V}_{\theta}^{(1)}(\alpha)$  go to zero as  $\alpha$  increases, they get smaller than the corresponding term for the SA algorithm with i.i.d. input for large enough  $\alpha$ , which achieves a constant MSE in the similarly scaled limit, since the asymptotic covariance is not a function of  $\alpha$ . Moreover, the value of  $\alpha$  only needs to be moderately large, since the asymptotic covariance terms decrease at rate  $O(1 / \alpha^2)$  as shown in Proposition 3.4.
+
+# F.3 PROOF OF COROLLARY 3.5
+
+We see that  $\mathbf{V}_{\pmb{\theta}}^{(3)}(\alpha) = \mathbf{V}_{\pmb{\theta}}^{(3)}(0)$  for all  $\alpha > 0$ , because the form of  $\mathbf{V}_{\pmb{\theta}}^{(3)}(\alpha)$  in Theorem 3.3 is independent of  $\alpha$ . To prove that  $\mathbf{V}_{\pmb{\theta}}^{(1)}(\alpha) <_{L} \mathbf{V}_{\pmb{\theta}}^{(3)}(0)$ , it is enough to show that  $\mathbf{V}_{\pmb{\theta}}^{(1)}(0) = \mathbf{V}_{\pmb{\theta}}^{(3)}(0)$ , since  $\mathbf{V}_{\pmb{\theta}}^{(1)}(\alpha) <_{L} \mathbf{V}_{\pmb{\theta}}^{(1)}(0)$  from Proposition 3.4. This is easily checked by substituting  $\alpha = 0$  in 11, for which  $\mathbf{U}_{\pmb{\theta}}(0) = \mathbf{U}_{11}$ . Substituting in the respective forms of  $\mathbf{V}_{\pmb{\theta}}^{(1)}(0)$  and  $\mathbf{V}_{\pmb{\theta}}^{(3)}(0)$  in Theorem 3.3, we get equivalence. This completes the proof.
+
+# G BACKGROUND THEORY
+
+# G.1 TECHNICAL LEMMAS
+
+Lemma G.1 (Solution to the Lyapunov Equation). If all the eigenvalues of matrix  $\mathbf{M}$  have negative real part, then for every positive semi-definite matrix  $\mathbf{U}$  there exists a unique positive semi-definite
+
+matrix  $\mathbf{V}$  satisfying the Lyapunov equation  $\mathbf{U} + \mathbf{M}\mathbf{V} + \mathbf{V}\mathbf{M}^T = \mathbf{0}$ . The explicit solution  $\mathbf{V}$  is given as
+
+$$
+\mathbf {V} = \int_ {0} ^ {\infty} e ^ {\mathbf {M} t} \mathbf {U} e ^ {(\mathbf {M} ^ {T}) t} d t. \tag {83}
+$$
+
+Chellaboina & Haddad (2008, Theorem 3.16) states that for a positive definite matrix  $\mathbf{U}$ , there exists a positive definite matrix  $\mathbf{V}$ . The reason they focus on the positive definite matrix  $\mathbf{U}$  is that they require the related autonomous ODE system to be asymptotically stable. However, in this paper we don't need this requirement. The same steps therein can be used to prove Lemma G.1 and show that if  $\mathbf{U}$  is positive semi-definite, then  $\mathbf{V}$  in the form of (83) is unique and also positive semi-definite.
+
+Lemma G.2 (Burkholder Inequality, Davis (1970), Hall et al. (2014) Theorem 2.10). Given a Martingale difference sequence  $\{M_{i,n}\}_{i = 1}^{n}$ , for  $p\geq 1$  and some positive constant  $C_p$ , we have
+
+$$
+\mathbb {E} \left[ \left\| \sum_ {i = 1} ^ {n} M _ {i, n} \right\| ^ {p} \right] \leq C _ {p} \mathbb {E} \left[ \left(\sum_ {i = 1} ^ {n} \| M _ {i, n} \| ^ {2}\right) ^ {p / 2} \right] \tag {84}
+$$
+
+Theorem G.3 (Martingale CLT, Delyon (2000) Theorem 30). If a Martingale difference array  $\{X_{n,i}\}$  satisfies the following condition: for some  $\tau > 0$ ,
+
+$$
+\sum_ {k = 1} ^ {n} \mathbb {E} \left[ \| X _ {n, k} \| ^ {2 + \tau} \mid \mathcal {F} _ {k - 1} \right] \xrightarrow {\mathbb {P}} 0, \tag {85}
+$$
+
+$$
+\sup  _ {n} \sum_ {k = 1} ^ {n} \mathbb {E} \left[ \| X _ {n, k} \| ^ {2} \mid \mathcal {F} _ {k - 1} \right] <   \infty , \tag {86}
+$$
+
+and
+
+$$
+\sum_ {k = 1} ^ {n} \mathbb {E} \left[ X _ {n, k} X _ {n, k} ^ {T} \mid \mathcal {F} _ {k - 1} \right] \xrightarrow {\mathbb {P}} \boldsymbol {V}, \tag {87}
+$$
+
+then
+
+$$
+\sum_ {i = 1} ^ {n} X _ {n, i} \xrightarrow {\text {d i s t .}} N (0, V). \tag {88}
+$$
+
+![](images/d95b76f2eb15428d0dc289e8c29dfd2ff318ad2f40ca5095ea67f83225e15a5e.jpg)
+
+Lemma G.4 (Duflo (1996) Proposition 3.I.2). For a Hurwitz matrix  $\mathbf{H}$ , there exist some positive constants  $C, b$  such that for any  $n$ ,
+
+$$
+\left\| e ^ {\mathbf {H} n} \right\| \leq C e ^ {- b n}. \tag {89}
+$$
+
+Lemma G.5 (Fort (2015) Lemma 5.8). For a Hurwitz matrix  $\mathbf{A}$ , denote by  $-r$ ,  $r > 0$ , the largest real part of its eigenvalues. Let a positive sequence  $\{\gamma_n\}$  such that  $\lim_{n} \gamma_{n} = 0$ . Then for any  $0 < r' < r$ , there exists a positive constant  $C$  such that for any  $k < n$ ,
+
+$$
+\left\| \prod_ {j = k} ^ {n} (\boldsymbol {I} + \gamma_ {j} \boldsymbol {A}) \right\| \leq C e ^ {- r ^ {\prime} \sum_ {j = k} ^ {n} \gamma_ {j}}. \tag {90}
+$$
+
+Lemma G.6 (Fort (2015) Lemma 5.9, Mokkadem & Pelletier (2006) Lemma 10). Let  $\{\gamma_n\}$  be a positive sequence such that  $\lim_{n} \gamma_{n} = 0$  and  $\sum_{n} \gamma_{n} = \infty$ . Let  $\{\epsilon_{n}, n \geq 0\}$  be a nonnegative sequence. Then, for  $b > 0$ ,  $p \geq 0$ ,
+
+$$
+\lim  _ {n} \sup  _ {n} \gamma_ {n} ^ {- p} \sum_ {k = 1} ^ {n} \gamma_ {k} ^ {p + 1} e ^ {- b \sum_ {j = k + 1} ^ {n} \gamma_ {j}} \epsilon_ {k} \leq \frac {1}{C (b , p)} \lim  _ {n} \sup  _ {n} \epsilon_ {n} \tag {91}
+$$
+
+for some constant  $C(b,p) > 0$
+
+When  $p = 0$  and define a positive sequence  $\{w_{n}\}$  satisfying  $w_{n - 1} / w_{n} = 1 + o(\gamma_{n})$ , we have
+
+$$
+\sum_ {k = 1} ^ {n} \gamma_ {k} e ^ {- b \sum_ {j = k + 1} ^ {n} \gamma_ {j}} \epsilon_ {k} = \left\{ \begin{array}{l l} O (w _ {n}), & \text {i f} \epsilon_ {n} = O (w _ {n}), \\ o (w _ {n}), & \text {i f} \epsilon_ {n} = o (w _ {n}). \end{array} \right. \tag {92}
+$$
+
+Lemma G.7 (Fort (2015) Lemma 5.10). For any matrices  $A, B, C$ ,
+
+$$
+\left\| A B A ^ {T} - C B C ^ {T} \right\| \leq \| A - C \| \| B \| \left(\| A \| + \| C \|\right). \tag {93}
+$$
+
+# G.2 ASYMPTOTIC RESULTS OF SINGLE-TIMESCALE SA
+
+Consider the stochastic approximation in the form of
+
+$$
+\boldsymbol {z} _ {n + 1} = \boldsymbol {z} _ {n} + \gamma_ {n + 1} G \left(\boldsymbol {z} _ {n}, X _ {n + 1}\right). \tag {94}
+$$
+
+Let  $\mathbf{K}_z$  be the transition kernel of the underlying Markov chain  $\{X_n\}_{n\geq 0}$  with stationary distribution  $\pi (z)$  such that  $g(z)\triangleq \mathbb{E}_{X\sim \pi (z)}[G(z,X)]$  with domain  $\mathcal{O}\subseteq \mathbb{R}^d$ . Define an operator  $\mathbf{K}_zf$  for any function  $f:\mathcal{N}\to \mathbb{R}^{D}$  such that
+
+$$
+\left(\mathbf {K} _ {\boldsymbol {z}} f\right) (i) = \sum_ {j \in \mathcal {N}} f (j) \mathbf {K} _ {\boldsymbol {z}} (i, j). \tag {95}
+$$
+
+Assume that
+
+C1. W.p.1, the closure of  $\{z_{n}\}_{n\geq 0}$  is a compact subset of  $\mathcal{O}$ .  
+C2.  $\gamma_{n} = \gamma_{0} / n^{a}, a\in (1 / 2,1]$  
+C3. Function  $g$  is continuous on  $\mathcal{O}$  and there exists a non-negative  $C^1$  function  $w$  and a compact set  $\mathcal{K} \subset \mathcal{O}$  such that
+
+-  $\nabla w(\boldsymbol{z})^T g(\boldsymbol{z}) \leq 0$  for all  $\boldsymbol{z} \in \mathcal{O}$  and  $\nabla w(\boldsymbol{z})^T g(\boldsymbol{z}) < 0$  if  $\boldsymbol{z} \notin \mathcal{K}$ ;  
+- the set  $S \triangleq \{z \mid \nabla w(z)^T g(z) = 0\}$  is such that  $w(S)$  has an empty interior;
+
+C4. For every  $z$ , there exists a solution  $m_z: \mathcal{N} \to \mathbb{R}^d$  for the following Poisson equation
+
+$$
+m _ {z} (i) - \left(\mathbf {K} _ {z} m _ {z}\right) (i) = G (z, i) - g (z) \tag {96}
+$$
+
+for any  $i\in \mathcal{N}$  ; for any compact set  $\mathcal{C}\subset \mathcal{O}$
+
+$$
+\sup  _ {\boldsymbol {z} \in \mathcal {C}, i \in \mathcal {N}} \| \left(\mathbf {K} _ {\boldsymbol {z}} m _ {\boldsymbol {z}}\right) (i) \| + \| m _ {\boldsymbol {z}} (i) \| <   \infty \tag {97}
+$$
+
+and there exist a continuous function  $\phi_{\mathcal{C}},\phi_{\mathcal{C}}(0) = 0$  , such that for any  $\pmb {z},\pmb{z}^{\prime}\in \mathcal{C}$
+
+$$
+\sup  _ {i \in \mathcal {N}} \| \left(\mathbf {K} _ {\boldsymbol {z}} m _ {\boldsymbol {z}}\right) (i) - \left(\mathbf {K} _ {\boldsymbol {z} ^ {\prime}} m _ {\boldsymbol {z} ^ {\prime}}\right) (i) \| \leq \phi_ {\mathcal {C}} (\| \boldsymbol {z} - \boldsymbol {z} ^ {\prime} \|). \tag {98}
+$$
+
+C5. Denote by  $-r$  the largest real part of the eigenvalues of the Jacobian matrix  $\nabla g(z^{*})$  and assume  $r > \frac{\mathbb{1}_{\{a = 1\}}}{2}$ .  
+C6. For every  $z$ , there exists a solution  $Q_{z}:\mathcal{N}\to \mathbb{R}^{d\times d}$  for the following Poisson equation
+
+$$
+Q _ {z} (i) - \left(\mathbf {K} _ {z} Q _ {z}\right) (i) = F (z, i) - \mathbb {E} _ {j \sim \pi (z)} [ F (z, j) ] \tag {99}
+$$
+
+for any  $i\in \mathcal{N}$  , where
+
+$$
+F (\boldsymbol {z}, i) \triangleq \sum_ {j \in \mathcal {N}} m _ {\boldsymbol {z}} (j) m _ {\boldsymbol {z}} (j) ^ {T} \mathbf {K} _ {\boldsymbol {z}} (i, j) - \left(\mathbf {K} _ {\boldsymbol {z}} m _ {\boldsymbol {z}}\right) (i) \left(\mathbf {K} _ {\boldsymbol {z}} m _ {\boldsymbol {z}}\right) (i) ^ {T}. \tag {100}
+$$
+
+For any compact set  $\mathcal{C} \subset \mathcal{O}$ ,
+
+$$
+\sup  _ {\boldsymbol {z} \in \mathcal {C}, i \in \mathcal {N}} \| Q _ {\boldsymbol {z}} (i) \| + \| \left(\mathbf {K} _ {\boldsymbol {z}} Q _ {\boldsymbol {z}}\right) (i) \| <   \infty \tag {101}
+$$
+
+and there exist  $p, C_{\mathcal{C}} > 0$ , such that for any  $\mathbf{z}, \mathbf{z}' \in \mathcal{C}$
+
+$$
+\sup  _ {i \in \mathcal {N}} \| \left(\mathbf {K} _ {\boldsymbol {z}} Q _ {\boldsymbol {z}}\right) (i) - \left(\mathbf {K} _ {\boldsymbol {z} ^ {\prime}} Q _ {\boldsymbol {z} ^ {\prime}}\right) (i) \| \leq C _ {\mathcal {C}} \| \boldsymbol {z} - \boldsymbol {z} ^ {\prime} \| ^ {p}. \tag {102}
+$$
+
+Theorem G.8 (Delyon et al. (1999) Theorem 2). Consider (94) and assume C1 - C4. Then, w.p.1,  $\lim \sup_{n}d(\pmb{z}_n,S) = 0$ .
+
+Theorem G.9 (Fort (2015) Theorem 2.1 & Proposition 4.1). Consider (94) and assume  $C1 - C6$ . Then, given the condition that  $z_{n}$  converges to one point  $z^{*} \in S$ , we have
+
+$$
+\gamma_ {n} ^ {- 1 / 2} \left(\boldsymbol {z} _ {n} - \boldsymbol {z} ^ {*}\right) \xrightarrow [ n \rightarrow \infty ]{\text {d i s t .}} N (0, \mathbf {V}), \tag {103}
+$$
+
+where
+
+$$
+\mathbf {V} \left(\frac {\mathbb {1} _ {\{b = 1 \}}}{2} \mathbf {I} + \nabla g \left(\boldsymbol {z} ^ {*}\right) ^ {T}\right) + \left(\frac {\mathbb {1} _ {\{b = 1 \}}}{2} \mathbf {I} + \nabla g \left(\boldsymbol {z} ^ {*}\right)\right) \mathbf {V} + \mathbf {U} = 0, \tag {104}
+$$
+
+and
+
+$$
+\mathbf {U} \triangleq \sum_ {i \in \mathcal {N}} \mu_ {i} \left(m _ {\mathbf {z} ^ {*}} (i) m _ {\mathbf {z} ^ {*}} (i) ^ {T} - \left(\mathbf {K} _ {\mathbf {z} ^ {*}} m _ {\mathbf {z} ^ {*}}\right) (i) \left(\mathbf {K} _ {\mathbf {z} ^ {*}} m _ {\mathbf {z} ^ {*}}\right) (i) ^ {T}\right). \tag {105}
+$$
+
+# G.3 ASYMPTOTIC RESULTS OF TWO-TIMESCALE SA
+
+For the two-timescale SA with iterate-dependent Markov chain, we have the following iterations:
+
+$$
+\mathbf {z} _ {n + 1} = \mathbf {z} _ {n} + \beta_ {n + 1} G _ {1} \left(\mathbf {z} _ {n}, \mathbf {y} _ {n} X _ {n + 1}\right), \tag {106a}
+$$
+
+$$
+\mathbf {y} _ {n + 1} = \mathbf {y} _ {n} + \gamma_ {n + 1} G _ {2} \left(\mathbf {z} _ {n}, \mathbf {y} _ {n}, X _ {n + 1}\right), \tag {106b}
+$$
+
+with the goal of finding the root  $(\mathbf{z}^{*},\mathbf{y}^{*})$  such that
+
+$$
+g _ {1} \left(\mathbf {z} ^ {*}, \mathbf {y} ^ {*}\right) = \mathbb {E} _ {X \sim \boldsymbol {\mu}} \left[ G _ {1} \left(\mathbf {z} ^ {*}, \mathbf {y} ^ {*}, X\right) \right] = 0, \quad g _ {2} \left(\mathbf {z} ^ {*}, \mathbf {y} ^ {*}\right) = \mathbb {E} _ {X \sim \boldsymbol {\mu}} \left[ G _ {2} \left(\mathbf {z} ^ {*}, \mathbf {y} ^ {*}, X\right) \right] = 0. \tag {107}
+$$
+
+We present here a simplified version of the assumptions for single-valued functions  $G_{1}, G_{2}$  that are necessary for the almost sure convergence result in Yaji & Bhatnagar (2020, Theorem 4). The original assumptions are intended for more general set-valued functions  $G_{1}, G_{2}$ .
+
+(B1) The step sizes  $\beta_{n} \triangleq n^{-b}$  and  $\gamma_{n} \triangleq n^{-a}$ , where  $0.5 < a < b \leq 1$ .  
+(B2) Assume the function  $G_{1}(\mathbf{z},\mathbf{y},X)$  is continuous and differentiable with respect to  $\mathbf{z},\mathbf{y}$ . There exists a positive constant  $L_{1}$  such that  $\| G_{1}(\mathbf{z},\mathbf{y},X)\| \leq L_{1}(1 + \| \mathbf{z}\| +\| \mathbf{y}\|)$  for every  $\mathbf{z}\in \mathbb{R}^{d_1},\mathbf{y}\in \mathbb{R}^{d_2},X\in \mathcal{N}$ . The same condition holds for the function  $G_{2}$  as well.  
+(B3) Assume there exists a function  $\rho : \mathbb{R}^{d_1} \to \mathbb{R}^{d_2}$  such that the following three properties hold: (i)  $\| \rho(\mathbf{z}) \| \leq L_2(1 + \| \mathbf{z} \|)$  for some positive constant  $L_2$ ; (ii) the ODE  $\dot{\mathbf{y}} = g_2(\mathbf{z}, \mathbf{y})$  has a globally asymptotically stable equilibrium  $\lambda(\mathbf{z})$  such that  $g_2(\mathbf{z}, \rho(\mathbf{z})) = 0$ . Additionally, let  $\hat{g}_1(\mathbf{z}) \triangleq g_1(\mathbf{z}, \rho(\mathbf{z}))$ , there exists a set of disjoint roots  $\Lambda \triangleq \{\mathbf{z}^* : \hat{g}_1(\mathbf{z}^*) = 0\}$ , which is the set of globally asymptotically stable equilibria of the ODE  $\dot{\mathbf{z}} = \hat{g}_1(\mathbf{z})$ .  
+(B4)  $\{X_{n}\}_{n\geq 0}$  is an iterate-dependent Markov process in finite state space  $\mathcal{N}$ . For every  $n\geq 0$ ,  $P(X_{n + 1} = j|\mathbf{z}_m,\mathbf{y}_m,X_m,0\leq m\leq n) = P(X_{n + 1} = j|\mathbf{z}_n,\mathbf{y}_n,X_n = i) = \mathbf{P}_{i,j}[\mathbf{z}_n,\mathbf{y}_n]$ , where the transition kernel  $\mathbf{P}[\mathbf{z},\mathbf{y}]$  is continuous in  $\mathbf{z},\mathbf{y}$ , and the Markov chain generated by  $\mathbf{P}[\mathbf{z},\mathbf{y}]$  is ergodic so that it admits a stationary distribution  $\pi (\mathbf{z},\mathbf{y})$ , and  $\pi (\mathbf{z}^{*},\rho (\mathbf{z}^{*})) = \pmb{\mu}$ .  
+(B5)  $\sup_{n\geq 0}\bigl (\| \mathbf{z}_n\| +\| \mathbf{y}_n\| \bigr) <   \infty$  a.s.
+
+Yaji & Bhatnagar (2020) included assumptions A1 - A9 and A11 for the following Theorem G.10. We briefly show the correspondence of our assumptions (B1) - (B5) and theirs: (B1) with A5, (B2) with A1 and A2, (B3) with A9 and A11, (B4) with A3 and A4, and (B5) with A8. Given that our two-timescale SA framework (106) excludes additional noises (setting them to zero), A6 and A7 therein are inherently met.
+
+Theorem G.10 (Yaji & Bhatnagar (2020) Theorem 4). Under Assumptions (B1) - (B5), iterations  $(\mathbf{z}_n, \mathbf{y}_n)$  in (106) almost surely converge to a set of roots, i.e.,  $(\mathbf{z}_n, \mathbf{y}_n) \to \bigcup_{\mathbf{z}^* \in \Lambda} (\mathbf{z}^*, \rho(\mathbf{z}^*))$  a.s.
+
+# H ADDITIONAL SIMULATION RESULTS
+
+# H.1 BINARY CLASSIFICATION ON ADDITIONAL DATSETS
+
+In this part, we perform the binary classification task as in Section 4 on additional datasets, i.e.,  $a9a$  (with 123 features) and splice (with 60 features) from LIBSVM (Chang & Lin, 2011). Figure 4 provides the performance ordering of different  $\alpha$  values, and we empirically demonstrate that the curves with  $\alpha \geq 5$  still outperform the i.i.d. counterpart. Additionally, Figure 5 compare cases (i) - (iii) under both  $a9a$  and splice datasets, and case (i) consistently perform the best.
+
+# H.2 NON-CONVEX LINEAR REGRESSION
+
+We further test SGD-SRRW and SHB-SRRW algorithms with a non-convex function to demonstrate the efficiency of our SA-SRRW algorithm beyond the convex setting. In this task, we simulate the following linear regression problem in Khaled & Richtárik (2023) with non-convex regularization
+
+$$
+\min  _ {\boldsymbol {\theta} \in \mathbb {R} ^ {d}} \left\{f (\boldsymbol {\theta}) \triangleq \frac {1}{N} \sum_ {i = 1} ^ {N} l _ {i} (\boldsymbol {\theta}) + \kappa \sum_ {j = 1} ^ {d} \frac {\boldsymbol {\theta} _ {j} ^ {2}}{\boldsymbol {\theta} _ {j} ^ {2} + 1} \right\} \tag {108}
+$$
+
+![](images/2d1e8f8f246d13e6aa251844ff8a067dbf0170e28fea236f3258cbba8058ba8c.jpg)  
+(a) SGD-SRRW,  $a9a$
+
+![](images/8f2c6102f9a6088003a51d97deaa629169541c9c71233b2cc0570ed68c7d2053.jpg)  
+(b) SGD-SRRW, splice
+
+![](images/b8866562c6e07806e4f4c55c24f8e8e309d3bf7ec91a0ddef4f6ad41baa82448.jpg)  
+(a)  $\alpha = 5$  ,SGD-SRRW,  $a9a$
+
+![](images/264b693785fa52b8afe5e2db5f54679f226463c06f8cfe1225ab7f1d6d6e6e1c.jpg)  
+Figure 4: Simulation results with various  $\alpha$  values in  $a9a$  and splice datasets.
+
+![](images/1a0d5f354d77a5b47896bcd75d825c3586cd1b099c55f28b0ee7c7f21c729e01.jpg)
+
+![](images/1577c5522b60a09974502a32cefeb168ec93c72b2edbe4bcdf77210e4c4f7713.jpg)  
+(d)  $\alpha = 5$  ,SGD-SRRW, splice
+
+![](images/8c84d1d6c94b4e3086c8b7bf6884cb09a72b02849cfc7f603b16018cd8ee3b18.jpg)  
+(b)  $\alpha = 10$  ,SGD-SRRW,  $a9a$  
+(e)  $\alpha = 10$  ,SGD-SRRW,splice
+
+![](images/9c36ee1f937b69d21db470aff9b2d69cb83fc14bcd7e80899e2e16ddb03e694c.jpg)  
+(c)  $\alpha = 20$  ,SGD-SRRW,  $a9a$  
+(f)  $\alpha = 20$  ,SGD-SRRW,splice  
+Figure 5: Performance comparison among cases (i) - (iii) for  $\alpha \in \{5,10,20\}$  in  $a9a$  and splice datasets.
+
+where the loss function  $l_{i}(\pmb{\theta}) = \|\mathbf{s}_{i}^{T}\pmb{\theta} - y_{i}\|^{2}$  and  $\kappa = 1$ , with the data points  $\{(\mathbf{s}_i,y_i)\}_{i\in \mathcal{N}}$  from the ijCNN1 dataset of LIBVIM (Chang & Lin, 2011). We still perform the optimization over the wikiVote graph, as done in Section 4.
+
+The numerical results for the non-convex linear regression taks are presented in Figures 6 and 7, where each experiment is repeated 100 times. Figures 6a and 6b show that the performance ordering across different  $\alpha$  values is still preserved for both algorithms over almost all time, and curves for  $\alpha \geq 5$  outperform that of the i.i.d. sampling (in black) under the graph topological constraints. Additionally, among the three cases examined at identical  $\alpha$  values, Figures 7a - 7c confirm that case (i) performs consistently better than the other two cases, implying that case (i) can even become the best choice for non-convex distributed optimization tasks.
+
+![](images/94f57fdbc14f2a8f7f300bdc7855b74e0bc099ed55eed335cf7c26a74ed1b3d7.jpg)  
+(a) SGD-SRRW
+
+![](images/d29eb500eb3ae4ad0a5543bcd74368c38296c152e0e34ac58bbf76d0c4291400.jpg)  
+(b) SHB-SRRW  
+Figure 6: Simulation results for non-convex linear regression under case (i) with various  $\alpha$  values.
+
+![](images/ad925edb5647cb031d4b4cc76c2d11ec1c343d050407da3fa9f661246eec1754.jpg)  
+(a)  $\alpha = 5$  ,SGD-SRRW  
+Figure 7: Performance comparison among cases (i) - (iii) for non-convex regression.
+
+![](images/6a8587f7d29fcf98434402e7aae5e6b9d8ef46bfadc5b1dd5e7d86ebbc2a8788.jpg)  
+(b)  $\alpha = 10$  ,SGD-SRRW
+
+![](images/a78add41258ce9f7def976cf23892a9d4df59d33a6f273c56037fad8cff23d92.jpg)  
+(c)  $\alpha = 20$  ,SGD-SRRW
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+                                    "type": "text",
+                                    "content": "ACCELERATING DISTRIBUTED STOCHASTIC OPTIMIZATION VIA SELF-REPELLENT RANDOM WALKS"
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+                                    "type": "text",
+                                    "content": "Jie Hu\\*1, Vishwaraj Doshi\\*2, Do Young Eun"
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+                                    "type": "text",
+                                    "content": "North Carolina State University, "
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+                                    "content": "{jhu29, dyeun}@ncsu.edu, vishwaraj.doshi@iqvia.com"
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+                                    "content": "ABSTRACT"
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+                                    "type": "text",
+                                    "content": "We study a family of distributed stochastic optimization algorithms where gradients are sampled by a token traversing a network of agents in random-walk fashion. Typically, these random-walks are chosen to be Markov chains that asymptotically sample from a desired target distribution, and play a critical role in the convergence of the optimization iterates. In this paper, we take a novel approach by replacing the standard linear Markovian token by one which follows a nonlinear Markov chain - namely the Self-Repellent Radom Walk (SRRW). Defined for any given 'base' Markov chain, the SRRW, parameterized by a positive scalar "
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+                                    "content": "\\alpha"
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+                                    "type": "text",
+                                    "content": ", is less likely to transition to states that were highly visited in the past, thus the name. In the context of MCMC sampling on a graph, a recent breakthrough in Doshi et al. (2023) shows that the SRRW achieves "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "O(1 / \\alpha)"
+                                },
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+                                    "bbox": [
+                                        140,
+                                        222,
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+                                    "type": "text",
+                                    "content": " decrease in the asymptotic variance for sampling. We propose the use of a 'generalized' version of the SRRW to drive token algorithms for distributed stochastic optimization in the form of stochastic approximation, termed SA-SRRW. We prove that the optimization iterate errors of the resulting SA-SRRW converge to zero almost surely and prove a central limit theorem, deriving the explicit form of the resulting asymptotic covariance matrix corresponding to iterate errors. This asymptotic covariance is always smaller than that of an algorithm driven by the base Markov chain and decreases at rate "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "O(1 / \\alpha^2)"
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+                                    "type": "text",
+                                    "content": " - the performance benefit of using SRRW thereby amplified in the stochastic optimization context. Empirical results support our theoretical findings."
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+                                    "content": "1 INTRODUCTION"
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+                                    "content": "Stochastic optimization algorithms solve optimization problems of the form"
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+                                    "type": "interline_equation",
+                                    "content": "\\boldsymbol {\\theta} ^ {*} \\in \\underset {\\boldsymbol {\\theta} \\in \\mathbb {R} ^ {d}} {\\arg \\min } f (\\boldsymbol {\\theta}), \\quad \\text {w h e r e} f (\\boldsymbol {\\theta}) \\triangleq \\mathbb {E} _ {X \\sim \\boldsymbol {\\mu}} [ F (\\boldsymbol {\\theta}, X) ] = \\sum_ {i \\in \\mathcal {N}} \\mu_ {i} F (\\boldsymbol {\\theta}, i), \\tag {1}",
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+                                    "type": "text",
+                                    "content": "with the objective function "
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+                                    "type": "text",
+                                    "content": " taking values in a finite state space "
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+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{N}"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " with distribution "
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+                                    "type": "inline_equation",
+                                    "content": "\\mu \\triangleq [\\mu_i]_{i \\in \\mathcal{N}}"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ". Leveraging partial gradient information per iteration, these algorithms have been recognized for their scalability and efficiency with large datasets (Bottou et al., 2018; Even, 2023). For any given noise sequence "
+                                },
+                                {
+                                    "bbox": [
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+                                    "content": "\\{X_n\\}_{n \\geq 0} \\subset \\mathcal{N}"
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+                                    "type": "text",
+                                    "content": ", and step size sequence "
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+                                    "type": "inline_equation",
+                                    "content": "\\{\\beta_n\\}_{n \\geq 0} \\subset \\mathbb{R}_+"
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+                                    "type": "text",
+                                    "content": ", most stochastic optimization algorithms can be classified as stochastic approximations (SA) of the form"
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+                                    "type": "interline_equation",
+                                    "content": "\\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} + \\beta_ {n + 1} H \\left(\\boldsymbol {\\theta} _ {n}, X _ {n + 1}\\right), \\quad \\forall n \\geq 0, \\tag {2}",
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+                                    "content": "where, roughly speaking, "
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+                                    "type": "inline_equation",
+                                    "content": "H(\\pmb{\\theta}, i)"
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+                                    "content": " contains gradient information "
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+                                    "content": ", such that "
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+                                    "type": "text",
+                                    "content": " solves "
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+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{h}(\\pmb{\\theta}) \\triangleq \\mathbb{E}_{X \\sim \\pmb{\\mu}}[H(\\pmb{\\theta}, X)] = \\sum_{i \\in \\mathcal{N}} \\mu_i H(\\pmb{\\theta}, i) = \\mathbf{0}"
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+                                    "type": "text",
+                                    "content": ". Such SA iterations include the well-known stochastic gradient descent (SGD), stochastic heavy ball (SHB) (Gadat et al., 2018; Li et al., 2022), and some SGD-type algorithms employing additional auxiliary variables (Barakat et al., 2021). These algorithms typically have the stochastic noise term "
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": " generated by i.i.d. random variables with probability distribution "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\mu}"
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+                                    "type": "text",
+                                    "content": " in each iteration. In this paper, we study a stochastic optimization algorithm where the noise sequence governing access to the gradient information is generated from general stochastic processes in place of i.i.d. random variables."
+                                }
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "type": "text",
+                                    "content": "* Equal contributors."
+                                }
+                            ]
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+                                    "type": "text",
+                                    "content": " is a (typically Markovian) random walk, and should asymptotically be able to sample the gradients from the desired probability distribution "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ". This is equivalent to saying that the random walker's empirical distribution converges to "
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+                                    "type": "text",
+                                    "content": " almost surely (a.s.); that is, "
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+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n \\triangleq \\frac{1}{n+1} \\sum_{k=0}^{n} \\delta_{X_k} \\xrightarrow[n\\to\\infty]{a.s.} \\mu"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " for any initial "
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+                                    "type": "inline_equation",
+                                    "content": "X_0 \\in \\mathcal{N}"
+                                },
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ", where "
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+                                    "type": "inline_equation",
+                                    "content": "\\delta_{X_k}"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " is the delta measure whose "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "X_k"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "'th entry is one, the rest being zero. Such convergence is most commonly achieved by employing the Metropolis Hastings random walk (MHRW) which can be designed to sample from any target measure "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
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+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " and implemented in a scalable manner (Sun et al., 2018). Unsurprisingly, convergence characteristics of the employed Markov chain affect that of the SA sequence (2), and appear in both finite-time and asymptotic analyses. Finite-time bounds typically involve the second largest eigenvalue in modulus (SLEM) of the Markov chain's transition kernel "
+                                },
+                                {
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": ", which is critically connected to the mixing time of a Markov chain (Levin & Peres, 2017); whereas asymptotic results such as central limit theorems (CLT) involve asymptotic covariance matrices that embed information regarding the entire spectrum of "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
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+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ", i.e., all eigenvalues as well as eigenvectors (Bremaud, 2013), which are key to understanding the sampling efficiency of a Markov chain. Thus, the choice of random walker can significantly impact the performance of (2), and simply ensuring that it samples from "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
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+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " asymptotically is not enough to achieve optimal algorithmic performance. In this paper, we take a closer look at the distributed stochastic optimization problem through the lens of a non-linear Markov chain, known as the Self Repellent Random Walk (SRRW), which was shown in Doshi et al. (2023) to achieve asymptotically minimal sampling variance for large values of "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\alpha"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ", a positive scalar controlling the strength of the random walker's self-repellence behaviour. Our proposed modification of (2) can be implemented within the settings of decentralized learning applications in a scalable manner, while also enjoying significant performance benefit over distributed stochastic optimization algorithms driven by vanilla Markov chains."
+                                }
+                            ]
+                        }
+                    ],
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+                                    "type": "text",
+                                    "content": "Token Algorithms for Decentralized Learning. In decentralized learning, agents like smartphones or IoT devices, each containing a subset of data, collaboratively train models on a graph "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{G}(\\mathcal{N},\\mathcal{E})"
+                                },
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+                                    "type": "text",
+                                    "content": " by sharing information locally without a central server (McMahan et al., 2017). In this setup, "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "N = |\\mathcal{N}|"
+                                },
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " agents correspond to nodes "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "i\\in \\mathcal{N}"
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+                                    "type": "text",
+                                    "content": ", and an edge "
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+                                {
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+                                    "type": "inline_equation",
+                                    "content": "(i,j)\\in \\mathcal{E}"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " indicates direct communication between agents "
+                                },
+                                {
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+                                    "type": "inline_equation",
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+                                {
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+                                    "type": "text",
+                                    "content": " and "
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+                                    "type": "inline_equation",
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ". This decentralized approach offers several advantages compared to the traditional centralized learning setting, promoting data privacy and security by eliminating the need for raw data to be aggregated centrally and thus reducing the risk of data breach or misuse (Bottou et al., 2018; Nedic, 2020). Additionally, decentralized approaches are more scalable and can handle vast amounts of heterogeneous data from distributed agents without overwhelming a central server, alleviating concerns about single point of failure (Vogels et al., 2021)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
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+                {
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+                                    "type": "text",
+                                    "content": "Among decentralized learning approaches, the class of 'Token' algorithms can be expressed as stochastic approximation iterations of the type (2), wherein the sequence "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "\\{X_{n}\\}"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " is realized as the sample path of a token that stochastically traverses the graph "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{G}"
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+                                {
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+                                    "type": "text",
+                                    "content": ", carrying with it the iterate "
+                                },
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": " for any time "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "n\\geq 0"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " and allowing each visited node (agent) to incrementally update "
+                                },
+                                {
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+                                },
+                                {
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+                                    "type": "text",
+                                    "content": " using locally available gradient information. Token algorithms have gained popularity in recent years (Hu et al., 2022; Triastcyn et al., 2022; Hendrikx, 2023), and are provably more communication efficient (Even, 2023) when compared to consensus-based algorithms - another popular approach for solving distributed optimization problems (Boyd et al., 2006; Morral et al., 2017; Olshevsky, 2022). The construction of token algorithms means that they do not suffer from expensive costs of synchronization and communication that are typical of consensus-based approaches, where all agents (or a subset of agents selected by a coordinator (Boyd et al., 2006; Wang et al., 2019)) on the graph are required to take simultaneous actions, such as communicating on the graph at each iteration. While decentralized Federated learning has indeed helped mitigate the communication overhead by processing multiple SGD iterations prior to each aggregation (Lalitha et al., 2018; Ye et al., 2022; Chellapandi et al., 2023), they still cannot overcome challenges such as synchronization and straggler issues."
+                                }
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+                                    "type": "text",
+                                    "content": "Self Repellent Random Walk. As mentioned earlier, sample paths "
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": " of token algorithms are usually generated using Markov chains with "
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+                                    "type": "inline_equation",
+                                    "content": "\\mu \\in \\mathrm{Int}(\\Sigma)"
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+                                    "type": "text",
+                                    "content": " as their limiting distribution. Here, "
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+                                    "type": "text",
+                                    "content": " denotes the "
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+                                    "type": "inline_equation",
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+                                    "type": "text",
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": " representing its interior. A recent work by Doshi et al. (2023) pioneers the use of non-linear Markov chains to, in some sense, improve upon any given time-reversible Markov chain with transition kernel "
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": " whose stationary distribution is "
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+                                    "content": "\\mathbf{K}[\\cdot ]:\\mathrm{Int}(\\Sigma)\\to [0,1]^{N\\times N}"
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+                                    "content": "K _ {i j} [ \\mathbf {x} ] \\triangleq \\frac {P _ {i j} \\left(x _ {j} / \\mu_ {j}\\right) ^ {- \\alpha}}{\\sum_ {k \\in \\mathcal {N}} P _ {i k} \\left(x _ {k} / \\mu_ {k}\\right) ^ {- \\alpha}}, \\quad \\forall i, j \\in \\mathcal {N}, \\tag {3}",
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+                                    "content": ", when simulated as a self-interacting random walk (Del Moral & Miclo, 2006; Del Moral & Doucet, 2010), can achieve smaller asymptotic variance than the base Markov chain when sampling over a graph "
+                                },
+                                {
+                                    "bbox": [
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+                                        122,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{G}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        122,
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+                                        277
+                                    ],
+                                    "type": "text",
+                                    "content": ", for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        122,
+                                        506,
+                                        277
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        122,
+                                        506,
+                                        277
+                                    ],
+                                    "type": "text",
+                                    "content": ". The argument "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        122,
+                                        506,
+                                        277
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        122,
+                                        506,
+                                        277
+                                    ],
+                                    "type": "text",
+                                    "content": " for the kernel "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        122,
+                                        506,
+                                        277
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{K}[x]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        122,
+                                        506,
+                                        277
+                                    ],
+                                    "type": "text",
+                                    "content": " is taken to be the empirical distribution "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        122,
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+                                        277
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " at each time step "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        122,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n \\geq 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        122,
+                                        506,
+                                        277
+                                    ],
+                                    "type": "text",
+                                    "content": ". For instance, if node "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "j"
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+                                {
+                                    "bbox": [
+                                        104,
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+                                    "type": "text",
+                                    "content": " has been visited more often than other nodes so far, the entry "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "x_j"
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " becomes larger (than target value "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mu_j"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        122,
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+                                        277
+                                    ],
+                                    "type": "text",
+                                    "content": "), resulting in a smaller transition probability from "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "i"
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+                                {
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+                                    "type": "text",
+                                    "content": " to "
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+                                    "type": "inline_equation",
+                                    "content": "j"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " under "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{K}[\\mathbf{x}]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        122,
+                                        506,
+                                        277
+                                    ],
+                                    "type": "text",
+                                    "content": " in (3) compared to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        122,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{ij}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                        277
+                                    ],
+                                    "type": "text",
+                                    "content": ". This ensures that a random walker prioritizes more seldom visited nodes in the process, and is thus 'self-repellent'. This effect is made more drastic by increasing "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        122,
+                                        506,
+                                        277
+                                    ],
+                                    "type": "text",
+                                    "content": ", and leads to asymptotically near-zero variance at a rate of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        122,
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+                                        277
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "O(1 / \\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        122,
+                                        506,
+                                        277
+                                    ],
+                                    "type": "text",
+                                    "content": ". Moreover, the polynomial function "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        122,
+                                        506,
+                                        277
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(x_i / \\mu_i)^{-\\alpha}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        122,
+                                        506,
+                                        277
+                                    ],
+                                    "type": "text",
+                                    "content": " chosen to encode self-repellent behaviour is shown in Doshi et al. (2023) to be the only one that allows the SRRW to inherit the so-called 'scale-invariance' property of the underlying Markov chain - a necessary component for the scalable implementation of a random walker over a large network without requiring knowledge of any graph-related global constants. Conclusively, such attributes render SRRW especially suitable for distributed optimization."
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        122,
+                                        506,
+                                        277
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "^3"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        282,
+                        504,
+                        403
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                        282,
+                                        504,
+                                        403
+                                    ],
+                                    "type": "text",
+                                    "content": "Effect of Stochastic Noise - Finite time and Asymptotic Approaches. Most contemporary token algorithms driven by Markov chains are analyzed using the finite-time bounds approach for obtaining insights into their convergence rates (Sun et al., 2018; Doan et al., 2019; Triastcyn et al., 2022; Hendrikx, 2023). However, as also explained in Even (2023), in most cases these bounds are overly dependent on mixing time properties of the specific Markov chain employed therein. This makes them largely ineffective in capturing the exact contribution of the underlying random walk in a manner which is qualitative enough to be used for algorithm design; and performance enhancements are typically achieved via application of techniques such as variance reduction (Defazio et al., 2014; Schmidt et al., 2017), momentum/Nesterov's acceleration (Gadat et al., 2018; Li et al., 2022), adaptive step size (Kingma & Ba, 2015; Reddi et al., 2018), which work by modifying the algorithm iterations themselves, and never consider potential improvements to the stochastic input itself."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        408,
+                        504,
+                        486
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
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+                                        408,
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+                                        486
+                                    ],
+                                    "type": "text",
+                                    "content": "Complement to finite-time approaches, asymptotic analysis using CLT has proven to be an excellent tool to approach the design of stochastic algorithms (Hu et al., 2022; Devraj & Meyn, 2017; Morral et al., 2017; Chen et al., 2020a; Mou et al., 2020; Devraj & Meyn, 2021). Hu et al. (2022) shows how asymptotic analysis can be used to compare the performance of SGD algorithms for various stochastic inputs using their notion of efficiency ordering, and, as mentioned in Devraj & Meyn (2017), the asymptotic benefits from minimizing the limiting covariance matrix are known to be a good predictor of finite-time algorithmic performance, also observed empirically in Section 4."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        491,
+                        504,
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                        504,
+                                        591
+                                    ],
+                                    "type": "text",
+                                    "content": "From the perspective of both finite-time analysis as well as asymptotic analysis of token algorithms, it is now well established that employing 'better' Markov chains can enhance the performance of stochastic optimization algorithm. For instance, Markov chains with smaller SLEMs yield tighter finite-time upper bounds (Sun et al., 2018; Ayache & El Rouayheb, 2021; Even, 2023). Similarly, Markov chains with smaller asymptotic variance for MCMC sampling tasks also provide better performance, resulting in smaller covariance matrix of SGD algorithms (Hu et al., 2022). Therefore, with these breakthrough results via SRRW achieving near-zero sampling variance, it is within reason to ask: Can we achieve near-zero variance in distributed stochastic optimization driven by SRRW-like token algorithms on any general graph?4 In this paper, we answer in the affirmative."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
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+                                    "type": "text",
+                                    "content": "SRRW Driven Algorithm and Analysis Approach. For any ergodic time-reversible Markov chain with transition probability matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        596,
+                                        504,
+                                        632
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{P} \\triangleq [P_{ij}]_{i,j \\in \\mathcal{N}}"
+                                },
+                                {
+                                    "bbox": [
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+                                        596,
+                                        504,
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+                                    ],
+                                    "type": "text",
+                                    "content": " and stationary distribution "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\mu} \\in \\mathrm{Int}(\\Sigma)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        596,
+                                        504,
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+                                    ],
+                                    "type": "text",
+                                    "content": ", we consider a general step size version of the SRRW stochastic process analysed in Doshi et al. (2023) and"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                }
+            ],
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
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+                                    "type": "text",
+                                    "content": "2Here, non-linearity in the transition kernel implies that "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{K}[\\mathbf{x}]"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " takes probability distribution "
+                                },
+                                {
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " as the argument (Andrieu et al., 2007), as opposed to the kernel being a linear operator "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{K}[\\mathbf{x}] = \\mathbf{P}"
+                                },
+                                {
+                                    "bbox": [
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+                                        639,
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+                                    ],
+                                    "type": "text",
+                                    "content": " for a constant stochastic matrix "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{P}"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " in a standard (linear) Markovian setting."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
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+                    "type": "page_footnote",
+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "<sup>3</sup>Recently, Guo et al. (2020) introduce an optimization scheme, which designs self-repellence into the perturbation of the gradient descent iterates (Jin et al., 2017; 2018; 2021) with the goal of escaping saddle points. This notion of self-repellence is distinct from the SRRW, which is a probability kernel designed specifically for a token to sample from a target distribution "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\mu"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " over a set of nodes on an arbitrary graph."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
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+                                    "type": "text",
+                                    "content": "This near-zero sampling variance implies a significantly smaller variance than even an i.i.d. sampling counterpart, while adhering to graph topological constraints of token algorithms."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
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+                                    "content": "3"
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+                            "type": "image_body"
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+                                            "type": "text",
+                                            "content": "Figure 1: Visualization of token algorithms using SRRW versus traditional MC in distributed learning. Our CLT analysis, extended from SRRW itself to distributed stochastic approximation, leads to near-zero variance for the SA iteration "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
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+                                            "type": "inline_equation",
+                                            "content": "\\theta_{n}"
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+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": ". Node numbers on the left denote visit counts."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 2,
+                            "angle": 0,
+                            "type": "image_caption"
+                        }
+                    ],
+                    "index": 1
+                },
+                {
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+                                    "type": "text",
+                                    "content": "use it to drive the noise sequence in (2). Our SA-SRRW algorithm is as follows:"
+                                }
+                            ]
+                        }
+                    ],
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        210,
+                                        205,
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+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": "Draw: "
+                                },
+                                {
+                                    "bbox": [
+                                        210,
+                                        205,
+                                        504,
+                                        217
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "X_{n + 1}\\sim \\mathbf{K}_{X_n,\\cdot}[\\mathbf{x}_n]"
+                                },
+                                {
+                                    "bbox": [
+                                        210,
+                                        205,
+                                        504,
+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": " (4a)"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        210,
+                        218,
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+                        232
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            ],
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+                                    "bbox": [
+                                        210,
+                                        218,
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+                                        232
+                                    ],
+                                    "type": "text",
+                                    "content": "Update: "
+                                },
+                                {
+                                    "bbox": [
+                                        210,
+                                        218,
+                                        504,
+                                        232
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_{n + 1} = \\mathbf{x}_n + \\gamma_{n + 1}(\\pmb{\\delta}_{X_{n + 1}} - \\mathbf{x}_n),"
+                                },
+                                {
+                                    "bbox": [
+                                        210,
+                                        218,
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+                                        232
+                                    ],
+                                    "type": "text",
+                                    "content": " (4b)"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                    "bbox": [
+                                        258,
+                                        233,
+                                        504,
+                                        246
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} + \\beta_ {n + 1} H \\left(\\boldsymbol {\\theta} _ {n}, X _ {n + 1}\\right), \\tag {4c}",
+                                    "image_path": "434df46668dcb056f5bd35c5cbc397157a877686e07acaaf871c74ac041433af.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                247,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        247,
+                                        506,
+                                        369
+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        247,
+                                        506,
+                                        369
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\beta_n\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        247,
+                                        506,
+                                        369
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        247,
+                                        506,
+                                        369
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\gamma_n\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        247,
+                                        506,
+                                        369
+                                    ],
+                                    "type": "text",
+                                    "content": " are step size sequences decreasing to zero, and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        247,
+                                        506,
+                                        369
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{K}[\\mathbf{x}]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        247,
+                                        506,
+                                        369
+                                    ],
+                                    "type": "text",
+                                    "content": " is the SRRW kernel in (3). Current non-asymptotic analyses require globally Lipschitz mean field function (Chen et al., 2020b; Doan, 2021; Zeng et al., 2021; Even, 2023) and is thus inapplicable to SA-SRRW since the mean field function of the SRRW iterates (4b) is only locally Lipschitz (details deferred to Appendix B). Instead, we successfully obtain non-trivial results by taking an asymptotic CLT-based approach to analyze (4). This goes beyond just analyzing the asymptotic sampling covariance<sup>5</sup> as in Doshi et al. (2023), the result therein forming a special case of ours by setting "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        247,
+                                        506,
+                                        369
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\gamma_{n} = 1 / (n + 1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        247,
+                                        506,
+                                        369
+                                    ],
+                                    "type": "text",
+                                    "content": " and considering only (4a) and (4b), that is, in the absence of optimization iteration (4c). Specifically, we capture the effect of SRRW's hyper-parameter "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        247,
+                                        506,
+                                        369
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        247,
+                                        506,
+                                        369
+                                    ],
+                                    "type": "text",
+                                    "content": " on the asymptotic speed of convergence of the optimization error term "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        247,
+                                        506,
+                                        369
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta_{n} - \\theta^{*}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        247,
+                                        506,
+                                        369
+                                    ],
+                                    "type": "text",
+                                    "content": " to zero via explicit deduction of its asymptotic covariance matrix. See Figure 1 for illustration."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
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+                        374,
+                        191,
+                        385
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                191,
+                                385
+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
+                                        374,
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+                                        385
+                                    ],
+                                    "type": "text",
+                                    "content": "Our Contributions."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        391,
+                        504,
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+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 11,
+                    "blocks": [
+                        {
+                            "bbox": [
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+                                460
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                104,
+                                                391,
+                                                504,
+                                                460
+                                            ],
+                                            "type": "text",
+                                            "content": "1. Given any time-reversible 'base' Markov chain with transition kernel "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                391,
+                                                504,
+                                                460
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathbf{P}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                391,
+                                                504,
+                                                460
+                                            ],
+                                            "type": "text",
+                                            "content": " and stationary distribution "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                391,
+                                                504,
+                                                460
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mu"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                391,
+                                                504,
+                                                460
+                                            ],
+                                            "type": "text",
+                                            "content": ", we generalize first and second order convergence results of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                391,
+                                                504,
+                                                460
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathbf{x}_n"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                391,
+                                                504,
+                                                460
+                                            ],
+                                            "type": "text",
+                                            "content": " to target measure "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                391,
+                                                504,
+                                                460
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pmb{\\mu}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                391,
+                                                504,
+                                                460
+                                            ],
+                                            "type": "text",
+                                            "content": " (Theorems 4.1 and 4.2 in Doshi et al., 2023) to a class of weighted empirical measures, through the use of more general step sizes "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                391,
+                                                504,
+                                                460
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\gamma_{n}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                391,
+                                                504,
+                                                460
+                                            ],
+                                            "type": "text",
+                                            "content": ". This includes showing that the asymptotic sampling covariance terms decrease to zero at rate "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                391,
+                                                504,
+                                                460
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "O(1 / \\alpha)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                391,
+                                                504,
+                                                460
+                                            ],
+                                            "type": "text",
+                                            "content": ", thus quantifying the effect of self-repellent on "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                391,
+                                                504,
+                                                460
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathbf{x}_n"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                391,
+                                                504,
+                                                460
+                                            ],
+                                            "type": "text",
+                                            "content": ". Our generalization is not for the sake thereof and is shown in Section 3 to be crucial for the design of step sizes "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                391,
+                                                504,
+                                                460
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_{n},\\gamma_{n}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                391,
+                                                504,
+                                                460
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 9
+                        },
+                        {
+                            "bbox": [
+                                104,
+                                463,
+                                504,
+                                486
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        463,
+                                        504,
+                                        486
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                104,
+                                                463,
+                                                504,
+                                                486
+                                            ],
+                                            "type": "text",
+                                            "content": "2. Building upon the convergence results for iterates "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                463,
+                                                504,
+                                                486
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathbf{x}_n"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                463,
+                                                504,
+                                                486
+                                            ],
+                                            "type": "text",
+                                            "content": ", we analyze the algorithm (4) driven by the SRRW kernel in (3) with step sizes "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                463,
+                                                504,
+                                                486
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_{n}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                463,
+                                                504,
+                                                486
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                463,
+                                                504,
+                                                486
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\gamma_{n}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                463,
+                                                504,
+                                                486
+                                            ],
+                                            "type": "text",
+                                            "content": " separated into three disjoint cases:"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 10
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        121,
+                        489,
+                        446,
+                        525
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 15,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                125,
+                                489,
+                                446,
+                                501
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        125,
+                                        489,
+                                        446,
+                                        501
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                125,
+                                                489,
+                                                446,
+                                                501
+                                            ],
+                                            "type": "text",
+                                            "content": "(i) "
+                                        },
+                                        {
+                                            "bbox": [
+                                                125,
+                                                489,
+                                                446,
+                                                501
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_{n} = o(\\gamma_{n})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                125,
+                                                489,
+                                                446,
+                                                501
+                                            ],
+                                            "type": "text",
+                                            "content": ", and we say that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                125,
+                                                489,
+                                                446,
+                                                501
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\theta_{n}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                125,
+                                                489,
+                                                446,
+                                                501
+                                            ],
+                                            "type": "text",
+                                            "content": " is on the slower timescale compared to "
+                                        },
+                                        {
+                                            "bbox": [
+                                                125,
+                                                489,
+                                                446,
+                                                501
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathbf{x}_n"
+                                        },
+                                        {
+                                            "bbox": [
+                                                125,
+                                                489,
+                                                446,
+                                                501
+                                            ],
+                                            "type": "text",
+                                            "content": ";"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 12
+                        },
+                        {
+                            "bbox": [
+                                123,
+                                501,
+                                397,
+                                513
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        123,
+                                        501,
+                                        397,
+                                        513
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                123,
+                                                501,
+                                                397,
+                                                513
+                                            ],
+                                            "type": "text",
+                                            "content": "(ii) "
+                                        },
+                                        {
+                                            "bbox": [
+                                                123,
+                                                501,
+                                                397,
+                                                513
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_{n} = \\gamma_{n}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                123,
+                                                501,
+                                                397,
+                                                513
+                                            ],
+                                            "type": "text",
+                                            "content": ", and we say that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                123,
+                                                501,
+                                                397,
+                                                513
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\theta_{n}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                123,
+                                                501,
+                                                397,
+                                                513
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                123,
+                                                501,
+                                                397,
+                                                513
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathbf{x}_n"
+                                        },
+                                        {
+                                            "bbox": [
+                                                123,
+                                                501,
+                                                397,
+                                                513
+                                            ],
+                                            "type": "text",
+                                            "content": " are on the same timescale;"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 13
+                        },
+                        {
+                            "bbox": [
+                                121,
+                                513,
+                                441,
+                                525
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        513,
+                                        441,
+                                        525
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                121,
+                                                513,
+                                                441,
+                                                525
+                                            ],
+                                            "type": "text",
+                                            "content": "(iii) "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                513,
+                                                441,
+                                                525
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\gamma_{n} = o(\\beta_{n})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                513,
+                                                441,
+                                                525
+                                            ],
+                                            "type": "text",
+                                            "content": ", and we say that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                513,
+                                                441,
+                                                525
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\theta_{n}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                513,
+                                                441,
+                                                525
+                                            ],
+                                            "type": "text",
+                                            "content": " is on the faster timescale compared to "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                513,
+                                                441,
+                                                525
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathbf{x}_n"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                513,
+                                                441,
+                                                525
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 14
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        104,
+                        528,
+                        501,
+                        540
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                528,
+                                501,
+                                540
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        501,
+                                        540
+                                    ],
+                                    "type": "text",
+                                    "content": "For any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        501,
+                                        540
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha \\geq 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        501,
+                                        540
+                                    ],
+                                    "type": "text",
+                                    "content": " and let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        501,
+                                        540
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k = 1,2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        501,
+                                        540
+                                    ],
+                                    "type": "text",
+                                    "content": " and 3 refer to the corresponding cases (i), (ii) and (iii), we show that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        172,
+                        542,
+                        436,
+                        563
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        172,
+                                        542,
+                                        436,
+                                        563
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\boldsymbol {\\theta} _ {n} \\xrightarrow [ n \\to \\infty ]{a . s .} \\boldsymbol {\\theta} ^ {*} \\quad \\text {a n d} \\quad (\\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*}) / \\sqrt {\\beta_ {n}} \\xrightarrow [ n \\to \\infty ]{d i s t.} N \\left(\\mathbf {0}, \\mathbf {V} _ {\\boldsymbol {\\theta}} ^ {(k)} (\\alpha)\\right),",
+                                    "image_path": "3d32d063848472525b6c51134063f844aa1f2572dfd767fcde06204931318c1a.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        104,
+                        567,
+                        504,
+                        615
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                567,
+                                504,
+                                615
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        504,
+                                        615
+                                    ],
+                                    "type": "text",
+                                    "content": "featuring distinct asymptotic covariance matrices "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        504,
+                                        615
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\theta}^{(1)}(\\alpha), \\mathbf{V}_{\\theta}^{(2)}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        504,
+                                        615
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        504,
+                                        615
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\theta}^{(3)}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        504,
+                                        615
+                                    ],
+                                    "type": "text",
+                                    "content": ", respectively. The three matrices coincide when "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        504,
+                                        615
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha = 0,6"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        504,
+                                        615
+                                    ],
+                                    "type": "text",
+                                    "content": ". Moreover, the derivation of the CLT for cases (i) and (iii), for which (4) corresponds to two-timescale SA with controlled Markov noise, is the first of its kind and thus a key technical contribution in this paper, as expanded upon in Section 3."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            ],
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+                                {
+                                    "bbox": [
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+                                        693
+                                    ],
+                                    "type": "text",
+                                    "content": "3. For case (i), we show that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        621,
+                                        505,
+                                        693
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\theta}^{(1)}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        621,
+                                        505,
+                                        693
+                                    ],
+                                    "type": "text",
+                                    "content": " decreases to zero (in the sense of Loewner ordering introduced in Section 2.1) as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        621,
+                                        505,
+                                        693
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        621,
+                                        505,
+                                        693
+                                    ],
+                                    "type": "text",
+                                    "content": " increases, with rate "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        621,
+                                        505,
+                                        693
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "O(1 / \\alpha^2)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        621,
+                                        505,
+                                        693
+                                    ],
+                                    "type": "text",
+                                    "content": ". This is especially surprising, since the asymptotic performance benefit from using the SRRW kernel with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        621,
+                                        505,
+                                        693
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        621,
+                                        505,
+                                        693
+                                    ],
+                                    "type": "text",
+                                    "content": " in (3), to drive the noise terms "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        621,
+                                        505,
+                                        693
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "X_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        621,
+                                        505,
+                                        693
+                                    ],
+                                    "type": "text",
+                                    "content": ", is amplified in the context of distributed learning and estimating "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        621,
+                                        505,
+                                        693
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta^{*}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        621,
+                                        505,
+                                        693
+                                    ],
+                                    "type": "text",
+                                    "content": "; compared to the sampling case, for which the rate is "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        621,
+                                        505,
+                                        693
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "O(1 / \\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        621,
+                                        505,
+                                        693
+                                    ],
+                                    "type": "text",
+                                    "content": " as mentioned earlier. For case (iii), we show that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        621,
+                                        505,
+                                        693
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\theta}^{(3)}(\\alpha) = \\mathbf{V}_{\\theta}^{(3)}(0)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        621,
+                                        505,
+                                        693
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        621,
+                                        505,
+                                        693
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha \\geq 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        621,
+                                        505,
+                                        693
+                                    ],
+                                    "type": "text",
+                                    "content": ", implying that using the SRRW in this case provides no asymptotic benefit than the"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                38
+                            ],
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+                                {
+                                    "bbox": [
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+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        116,
+                        700,
+                        407,
+                        711
+                    ],
+                    "type": "page_footnote",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        700,
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+                                        711
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "{}^{5}"
+                                },
+                                {
+                                    "bbox": [
+                                        116,
+                                        700,
+                                        407,
+                                        711
+                                    ],
+                                    "type": "text",
+                                    "content": " Sampling covariance corresponds to only the empirical distribution "
+                                },
+                                {
+                                    "bbox": [
+                                        116,
+                                        700,
+                                        407,
+                                        711
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "{\\mathbf{x}}_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        116,
+                                        700,
+                                        407,
+                                        711
+                                    ],
+                                    "type": "text",
+                                    "content": " in (4b)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                },
+                {
+                    "bbox": [
+                        104,
+                        710,
+                        504,
+                        732
+                    ],
+                    "type": "page_footnote",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                504,
+                                732
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        710,
+                                        504,
+                                        732
+                                    ],
+                                    "type": "text",
+                                    "content": "The "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        710,
+                                        504,
+                                        732
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        710,
+                                        504,
+                                        732
+                                    ],
+                                    "type": "text",
+                                    "content": " case is equivalent to simply running the base Markov chain, since from (3) we have "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        710,
+                                        504,
+                                        732
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{K}[\\cdot ] = \\mathbf{P}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        710,
+                                        504,
+                                        732
+                                    ],
+                                    "type": "text",
+                                    "content": ", thus bypassing the SRRW's effect and rendering all three cases nearly the same."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 21
+                },
+                {
+                    "bbox": [
+                        302,
+                        751,
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+                        760
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        302,
+                                        751,
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+                                        760
+                                    ],
+                                    "type": "text",
+                                    "content": "4"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 23
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 3
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        82,
+                        504,
+                        109
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                                109
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        109
+                                    ],
+                                    "type": "text",
+                                    "content": "original base Markov chain, and thus performs worse than case (i). In summary, we deduce that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        109
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(\\alpha_2) < _L\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(\\alpha_1) < _L\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(0) = \\mathbf{V}_{\\pmb{\\theta}}^{(3)}(0) = \\mathbf{V}_{\\pmb{\\theta}}^{(3)}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        109
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        109
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha_{2} > \\alpha_{1} > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        109
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        109
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha >0"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        113,
+                        504,
+                        180
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                                504,
+                                180
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        113,
+                                        504,
+                                        180
+                                    ],
+                                    "type": "text",
+                                    "content": "4. We numerically simulate our SA-SRRW algorithm on various real-world datasets, focusing on a binary classification task, to evaluate its performance across all three cases. By carefully choosing the function "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        113,
+                                        504,
+                                        180
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        113,
+                                        504,
+                                        180
+                                    ],
+                                    "type": "text",
+                                    "content": " in SA-SRRW, we test the SGD and algorithms driven by SRRW. Our findings consistently highlight the superiority of case (i) over cases (ii) and (iii) for diverse "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        113,
+                                        504,
+                                        180
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        113,
+                                        504,
+                                        180
+                                    ],
+                                    "type": "text",
+                                    "content": " values, even in their finite time performance. Notably, our tests validate the variance reduction at a rate of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        113,
+                                        504,
+                                        180
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "O(1 / \\alpha^2)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        113,
+                                        504,
+                                        180
+                                    ],
+                                    "type": "text",
+                                    "content": " for case (i), suggesting it as the best algorithmic choice among the three cases."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        105,
+                        195,
+                        313,
+                        209
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
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+                                313,
+                                209
+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
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+                                        209
+                                    ],
+                                    "type": "text",
+                                    "content": "2 PRELIMINARIES AND MODEL SETUP"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        220,
+                        504,
+                        266
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                220,
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+                                266
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        220,
+                                        504,
+                                        266
+                                    ],
+                                    "type": "text",
+                                    "content": "In Section 2.1, we first standardize the notations used throughout the paper, and define key mathematical terms and quantities used in our theoretical analyses. Then, in Section 2.2, we consolidate the model assumptions of our SA-SRRW algorithm (4). We then go on to discuss our assumptions, and provide additional interpretations of our use of generalized step-sizes."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        105,
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+                        289
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
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+                            ],
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+                                {
+                                    "bbox": [
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+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": "2.1 BASIC NOTATIONS AND DEFINITIONS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        297,
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+                        399
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "text",
+                                    "content": "Vectors are denoted by lower-case bold letters, e.g., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{v} \\triangleq [v_i] \\in \\mathbb{R}^D"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "text",
+                                    "content": ", and matrices by upper-case bold, e.g., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{M} \\triangleq [M_{ij}] \\in \\mathbb{R}^{D \\times D}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "text",
+                                    "content": ". "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{M}^{-T}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "text",
+                                    "content": " is the transpose of the matrix inverse "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{M}^{-1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "text",
+                                    "content": ". The diagonal matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{D}_{\\mathbf{v}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "text",
+                                    "content": " is formed by vector "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{v}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "text",
+                                    "content": " with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v_i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "text",
+                                    "content": " as the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "text",
+                                    "content": "'th diagonal entry. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{0}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "text",
+                                    "content": " denote vectors of all ones and zeros, respectively. The identity matrix is represented by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{I}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "text",
+                                    "content": ", with subscripts indicating dimensions as needed. A matrix is Hurwitz if all its eigenvalues possess strictly negative real parts. "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{1}_{\\{\\cdot\\}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "text",
+                                    "content": " denotes an indicator function with condition in parentheses. We use "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\|\\cdot\\|"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "text",
+                                    "content": " to denote both the Euclidean norm of vectors and the spectral norm of matrices. Two symmetric matrices "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{M}_1, \\mathbf{M}_2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "text",
+                                    "content": " follow Loewner ordering "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{M}_1 <_L \\mathbf{M}_2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "text",
+                                    "content": " if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{M}_2 - \\mathbf{M}_1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "text",
+                                    "content": " is positive semi-definite and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{M}_1 \\neq \\mathbf{M}_2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "text",
+                                    "content": ". This slightly differs from the conventional definition with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\leq_L"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "text",
+                                    "content": ", which allows "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{M}_1 = \\mathbf{M}_2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        297,
+                                        506,
+                                        399
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        405,
+                        506,
+                        507
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                405,
+                                506,
+                                507
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": "Throughout the paper, the matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{P} \\triangleq [P_{i,j}]_{i,j \\in \\mathcal{N}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": " and vector "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\mu} \\triangleq [\\mu_i]_{i \\in \\mathcal{N}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": " are used exclusively to denote an "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "N \\times N"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": "-dimensional transition kernel of an ergodic Markov chain, and its stationary distribution, respectively. Without loss of generality, we assume "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{ij} > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": " if and only if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "a_{ij} > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": ". Markov chains satisfying the detailed balance equation, where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mu_i P_{ij} = \\mu_j P_{ji}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i, j \\in \\mathcal{N}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": ", are termed time-reversible. For such chains, we use "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\lambda_i, \\mathbf{u}_i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": " (resp. "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\lambda_i, \\mathbf{v}_i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": ") to denote the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": "'th left (resp. right) eigenpair where the eigenvalues are ordered: "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "-1 < \\lambda_1 \\leq \\dots \\leq \\lambda_{N-1} < \\lambda_N = 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": ", with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{u}_N = \\pmb{\\mu}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{v}_N = \\mathbf{1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": " in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{R}^N"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": ". We assume eigenvectors to be normalized such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{u}_i^T \\mathbf{v}_i = 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": ", and we have "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{u}_i = \\mathbf{D}_{\\pmb{\\mu}} \\mathbf{v}_i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{u}_i^T \\mathbf{v}_j = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i, j \\in \\mathcal{N}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        506,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": ". We direct the reader to Aldous & Fill (2002, Chapter 3.4) for a detailed exposition on spectral properties of time-reversible Markov chains."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        105,
+                        519,
+                        351,
+                        530
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                519,
+                                351,
+                                530
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        519,
+                                        351,
+                                        530
+                                    ],
+                                    "type": "text",
+                                    "content": "2.2 SA-SRRW: KEY ASSUMPTIONS AND DISCUSSIONS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        105,
+                        540,
+                        441,
+                        552
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                540,
+                                441,
+                                552
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        540,
+                                        441,
+                                        552
+                                    ],
+                                    "type": "text",
+                                    "content": "Assumptions: All results in our paper are proved under the following assumptions."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        116,
+                        554,
+                        504,
+                        666
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 14,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                116,
+                                554,
+                                504,
+                                578
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        554,
+                                        504,
+                                        578
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                554,
+                                                504,
+                                                578
+                                            ],
+                                            "type": "text",
+                                            "content": "(A1) The function "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                554,
+                                                504,
+                                                578
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "H: \\mathbb{R}^D \\times \\mathcal{N} \\to \\mathbb{R}^D"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                554,
+                                                504,
+                                                578
+                                            ],
+                                            "type": "text",
+                                            "content": ", is a continuous at every "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                554,
+                                                504,
+                                                578
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pmb{\\theta} \\in \\mathbb{R}^D"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                554,
+                                                504,
+                                                578
+                                            ],
+                                            "type": "text",
+                                            "content": ", and there exists a positive constant "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                554,
+                                                504,
+                                                578
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "L"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                554,
+                                                504,
+                                                578
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                554,
+                                                504,
+                                                578
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\| H(\\pmb{\\theta}, i) \\| \\leq L (1 + \\| \\pmb{\\theta} \\|)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                554,
+                                                504,
+                                                578
+                                            ],
+                                            "type": "text",
+                                            "content": " for every "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                554,
+                                                504,
+                                                578
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pmb{\\theta} \\in \\mathbb{R}^D, i \\in \\mathcal{N}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                554,
+                                                504,
+                                                578
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 10
+                        },
+                        {
+                            "bbox": [
+                                116,
+                                580,
+                                487,
+                                593
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        580,
+                                        487,
+                                        593
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                580,
+                                                487,
+                                                593
+                                            ],
+                                            "type": "text",
+                                            "content": "(A2) Step sizes "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                580,
+                                                487,
+                                                593
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_{n}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                580,
+                                                487,
+                                                593
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                580,
+                                                487,
+                                                593
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\gamma_{n}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                580,
+                                                487,
+                                                593
+                                            ],
+                                            "type": "text",
+                                            "content": " follow "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                580,
+                                                487,
+                                                593
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_{n} = (n + 1)^{-b}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                580,
+                                                487,
+                                                593
+                                            ],
+                                            "type": "text",
+                                            "content": ", and "
+                                        },
+                                        {
+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": ", where "
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+                                            "type": "text",
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+                                            "type": "text",
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+                                            "content": "\\Theta \\triangleq \\left\\{\\pmb{\\theta}^{*}|\\mathbf{h}(\\pmb{\\theta}^{*}) = \\mathbf{0}, \\nabla \\mathbf{h}(\\pmb{\\theta}^{*}) + \\frac{\\mathbb{1}_{\\{b=1\\}}}{2}\\mathbf{I}\\text{ is Hurwitz}\\right\\} \\neq \\emptyset"
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+                                            "type": "text",
+                                            "content": " of the associated ordinary differential equation (ODE) for iteration (4c), given by "
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+                                            "content": "d\\pmb{\\theta}(t)/dt = \\mathbf{h}(\\pmb{\\theta}(t))"
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+                                            "type": "text",
+                                            "content": "."
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+                            "index": 12
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+                                            "content": "(A4) For any "
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+                                            "content": "(\\pmb{\\theta}_0, \\mathbf{x}_0, X_0) \\in \\mathbb{R}^D \\times \\mathrm{Int}(\\Sigma) \\times \\mathcal{N}"
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+                                            "type": "text",
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+                                            "content": "\\mathbb{P}_{\\pmb{\\theta}_0, \\mathbf{x}_0, X_0}"
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+                                            "content": "-almost surely contained within a compact subset of "
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+                                            "content": "\\mathbb{R}^D"
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+                                            "type": "text",
+                                            "content": " (resp. "
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+                                            "type": "text",
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+                                    "type": "text",
+                                    "content": "Discussions on Assumptions: Assumption A1 requires "
+                                },
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": " to only be locally Lipschitz albeit with linear growth, and is less stringent than the globally Lipschitz assumption prevalent in optimization literature (Li & Wai, 2022; Hendrikx, 2023; Even, 2023)."
+                                }
+                            ]
+                        }
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+                                    "type": "text",
+                                    "content": "In particular, this is the reason why we advocate for a more general step size "
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+                                    "content": "\\gamma_{n} = (n + 1)^{-a}"
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+                                    "type": "text",
+                                    "content": " in the SRRW iterates with "
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+                                    "type": "text",
+                                    "content": ", allowing us to choose "
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+                                    "type": "inline_equation",
+                                    "content": "\\beta_{n} = (n + 1)^{-b}"
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+                                {
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+                                    "type": "inline_equation",
+                                    "content": "b \\in (a, 1]"
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+                                    "type": "text",
+                                    "content": " to satisfy "
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+                                    "type": "inline_equation",
+                                    "content": "\\beta_{n} = o(\\gamma_{n})"
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": " for case (i)."
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+        {
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+                                    "type": "text",
+                                    "content": "Assumption A2 is the general umbrella assumption under which cases (i), (ii) and (iii) mentioned in Section 1 are extracted by setting: (i) "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "a < b"
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+                                    "type": "text",
+                                    "content": ", (ii) "
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+                                    "type": "inline_equation",
+                                    "content": "a = b"
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+                                {
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+                                    "type": "text",
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+                                    "type": "text",
+                                    "content": ". Cases (i) and (iii) render "
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta_{n}, \\mathbf{x}_{n}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " on different timescales; the polynomial form of "
+                                },
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_{n}, \\gamma_{n}"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " widely assumed in the two-timescale SA literature (Mokkadem & Pelletier, 2006; Zeng et al., 2021; Hong et al., 2023). Case (ii) characterizes the SA-SRRW algorithm (4) as a single-timescale SA with polynomially decreasing step size, and is among the most common assumptions in the SA literature (Borkar, 2022; Fort, 2015; Li et al., 2023). In all three cases, the form of "
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+                                    "type": "text",
+                                    "content": " ensures "
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+                                    "type": "inline_equation",
+                                    "content": "\\gamma_{n} \\leq 1"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " such that the SRRW iterates "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_{n}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " in (4b) is within "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{Int}(\\Sigma)"
+                                },
+                                {
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+                                    ],
+                                    "type": "text",
+                                    "content": ", ensuring that "
+                                },
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+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{K}[\\mathbf{x}_n]"
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": " is well-defined for all "
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+                                    "type": "inline_equation",
+                                    "content": "n \\geq 0"
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+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
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+                },
+                {
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "In Assumption A3, limiting dynamics of SA iterations "
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+                                    "type": "inline_equation",
+                                    "content": "\\{\\pmb{\\theta}_n\\}_{n\\geq 0}"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " closely follow trajectories "
+                                },
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\pmb {\\theta}(t)\\}_{t\\geq 0}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " of their associated ODE, and assuming the existence of globally stable equilibria is standard (Borkar, 2022; Fort, 2015; Li et al., 2023). In optimization problems, this is equivalent to assuming the existence of at most countably many local minima."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
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+                {
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+                    "angle": 0,
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Assumption A4 assumes almost sure boundedness of iterates "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\theta_{n}"
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n"
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ", which is a common assumption in SA algorithms (Kushner & Yin, 2003; Chen, 2006; Borkar, 2022; Karmakar & Bhatnagar, 2018; Li et al., 2023) for the stability of the SA iterations by ensuring the well-definiteness of all quantities involved. Stability of the weighted empirical measure "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n"
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+                                    ],
+                                    "type": "text",
+                                    "content": " of the SRRW process is practically ensured by studying (4b) with a truncation-based procedure (see Doshi et al., 2023, Remark 4.5 and Appendix E for a comprehensive explanation), while that for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        226,
+                                        504,
+                                        326
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        226,
+                                        504,
+                                        326
+                                    ],
+                                    "type": "text",
+                                    "content": " is usually ensured either as a by-product of the algorithm design, or via mechanisms such as projections onto a compact subset of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        226,
+                                        504,
+                                        326
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{R}^D"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        226,
+                                        504,
+                                        326
+                                    ],
+                                    "type": "text",
+                                    "content": ", depending on the application context. We now provide additional discussions regarding the step-size assumptions and their implications on the SRRW iteration (4b)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
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+                        330,
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+                        365
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        330,
+                                        504,
+                                        365
+                                    ],
+                                    "type": "text",
+                                    "content": "SRRW with General Step Size: As shown in Benaim & Cloez (2015, Remark 1.1), albeit for a completely different non-linear Markov kernel driving the algorithm therein, iterates "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        330,
+                                        504,
+                                        365
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        330,
+                                        504,
+                                        365
+                                    ],
+                                    "type": "text",
+                                    "content": " of (4b) can also be expressed as weighted empirical measures of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        330,
+                                        504,
+                                        365
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{X_n\\}_{n\\geq 0}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        330,
+                                        504,
+                                        365
+                                    ],
+                                    "type": "text",
+                                    "content": ", in the following form:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        153,
+                        367,
+                        504,
+                        394
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        153,
+                                        367,
+                                        504,
+                                        394
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {x} _ {n} = \\frac {\\sum_ {i = 1} ^ {n} \\omega_ {i} \\delta_ {X _ {i}} + \\omega_ {0} \\mathbf {x} _ {0}}{\\sum_ {i = 0} ^ {n} \\omega_ {i}}, \\quad \\text {w h e r e} \\omega_ {0} = 1, \\text {a n d} \\omega_ {n} = \\frac {\\gamma_ {n}}{\\prod_ {i = 1} ^ {n} (1 - \\gamma_ {i})}, \\tag {5}",
+                                    "image_path": "e92f978e101f57a0d374332192bbebb81b82e928e1c01c2a6eac540fbd4e8c04.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "text",
+                                    "content": "for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "text",
+                                    "content": ". For the special case when "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\gamma_{n} = 1 / (n + 1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "text",
+                                    "content": " as in Doshi et al. (2023), we have "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\omega_{n} = 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n \\geq 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "text",
+                                    "content": " is the typical, unweighted empirical measure. For the additional case considered in our paper, when "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "a < 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "text",
+                                    "content": " for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\gamma_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "text",
+                                    "content": " as in assumption A2, we can approximate "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "1 - \\gamma_{n} \\approx e^{-\\gamma_{n}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\omega_{n} \\approx n^{-a} e^{n^{(1 - a)} / (1 - a)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "text",
+                                    "content": ". This implies that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\omega_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "text",
+                                    "content": " will increase at sub-exponential rate, giving more weight to recent visit counts and allowing it to quickly 'forget' the poor initial measure "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "text",
+                                    "content": " and shed the correlation with the initial choice of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "X_0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "text",
+                                    "content": ". This 'speed up' effect by setting "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "a < 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "text",
+                                    "content": " is guaranteed in case (i) irrespective of the choice of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "b"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "text",
+                                    "content": " in Assumption A2, and in Section 3 we show how this can lead to further reduction in covariance of optimization error "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta_{n} = \\theta^{*}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        487
+                                    ],
+                                    "type": "text",
+                                    "content": " in the asymptotic regime."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        492,
+                        504,
+                        516
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                492,
+                                504,
+                                516
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        492,
+                                        504,
+                                        516
+                                    ],
+                                    "type": "text",
+                                    "content": "Additional assumption for case (iii): Before moving on to Section 3, we take another look at the case when "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        492,
+                                        504,
+                                        516
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\gamma_{n} = o(\\beta_{n})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        492,
+                                        504,
+                                        516
+                                    ],
+                                    "type": "text",
+                                    "content": ", and replace A3 with the following, stronger assumption only for case (iii)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        113,
+                        521,
+                        504,
+                        548
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                113,
+                                521,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        113,
+                                        521,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "text",
+                                    "content": "(A3') For any "
+                                },
+                                {
+                                    "bbox": [
+                                        113,
+                                        521,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x} \\in \\mathrm{Int}(\\Sigma)"
+                                },
+                                {
+                                    "bbox": [
+                                        113,
+                                        521,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "text",
+                                    "content": ", there exists a function "
+                                },
+                                {
+                                    "bbox": [
+                                        113,
+                                        521,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho: \\mathrm{Int}(\\Sigma) \\to \\mathbb{R}^D"
+                                },
+                                {
+                                    "bbox": [
+                                        113,
+                                        521,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        113,
+                                        521,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\|\\rho(\\mathbf{x})\\| \\leq L_2(1 + \\| \\mathbf{x}\\|)"
+                                },
+                                {
+                                    "bbox": [
+                                        113,
+                                        521,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "text",
+                                    "content": " for some "
+                                },
+                                {
+                                    "bbox": [
+                                        113,
+                                        521,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "L_2 > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        113,
+                                        521,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        113,
+                                        521,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{E}_{i \\sim \\pi[\\mathbf{x}]}[H(\\rho(\\mathbf{x}), i)] = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        113,
+                                        521,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        113,
+                                        521,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{E}_{i \\sim \\pi[\\mathbf{x}]}[\\nabla H(\\rho(\\mathbf{x}), i)] + \\frac{\\mathbf{1}_{\\{b=1\\}}}{2} \\mathbf{I}"
+                                },
+                                {
+                                    "bbox": [
+                                        113,
+                                        521,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "text",
+                                    "content": " is Hurwitz."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        552,
+                        504,
+                        586
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                                504,
+                                586
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        552,
+                                        504,
+                                        586
+                                    ],
+                                    "type": "text",
+                                    "content": "While Assumption A3' for case (iii) is much stronger than A3, it is not detrimental to the overall results of our paper, since case (i) is of far greater interest as impressed upon in Section 1. This is discussed further in Appendix C."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        105,
+                        601,
+                        417,
+                        615
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
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+                                        615
+                                    ],
+                                    "type": "text",
+                                    "content": "3 ASYMPTOTIC ANALYSIS OF THE SA-SRRW ALGORITHM"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        626,
+                        504,
+                        694
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                            ],
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+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        694
+                                    ],
+                                    "type": "text",
+                                    "content": "In this section, we provide the main results for the SA-SRRW algorithm (4). We first present the a.s. convergence and the CLT result for SRRW with generalized step size, extending the results in Doshi et al. (2023). Building upon this, we present the a.s. convergence and the CLT result for the SA iterate "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        694
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        694
+                                    ],
+                                    "type": "text",
+                                    "content": " under different settings of step sizes. We then shift our focus to the analysis of the different asymptotic covariance matrices emerging out of the CLT result, and capture the effect of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        694
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        694
+                                    ],
+                                    "type": "text",
+                                    "content": " and the step sizes, particularly in cases (i) and (iii), on "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        694
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta_{n} - \\theta^{*}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        694
+                                    ],
+                                    "type": "text",
+                                    "content": " via performance ordering."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        104,
+                        698,
+                        504,
+                        733
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                                504,
+                                733
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        698,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "Almost Sure convergence and CLT: The following result establishes first and second order convergence of the sequence "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        698,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\mathbf{x}_n\\}_{n\\geq 0}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        698,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ", which represents the weighted empirical measures of the SRRW process "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        698,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{X_{n}\\}_{n\\geq 0}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        698,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ", based on the update rule in (4b)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
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+                        26,
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+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                38
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        302,
+                        751,
+                        309,
+                        760
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                302,
+                                751,
+                                309,
+                                760
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        302,
+                                        751,
+                                        309,
+                                        760
+                                    ],
+                                    "type": "text",
+                                    "content": "6"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 5
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        82,
+                        451,
+                        94
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                82,
+                                451,
+                                94
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        451,
+                                        94
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 3.1. Under Assumptions A1, A2 and A4, for the SRRW iterates (4b), we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        180,
+                        95,
+                        428,
+                        113
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                180,
+                                95,
+                                428,
+                                113
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        180,
+                                        95,
+                                        428,
+                                        113
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {x} _ {n} \\xrightarrow [ n \\to \\infty ]{a . s .} \\boldsymbol {\\mu}, \\quad a n d \\quad \\gamma_ {n} ^ {- 1 / 2} (\\mathbf {x} _ {n} - \\boldsymbol {\\mu}) \\xrightarrow [ n \\to \\infty ]{d i s t .} N (\\mathbf {0}, \\mathbf {V} _ {\\mathbf {x}} (\\alpha)),",
+                                    "image_path": "9bf33707f7f490972b26dfd20995dd733a1040e72866bb0e7b1ae16b7a5f08b3.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        172,
+                        113,
+                        504,
+                        144
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                172,
+                                113,
+                                504,
+                                144
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        172,
+                                        113,
+                                        504,
+                                        144
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "w h e r e \\quad \\mathbf {V} _ {\\mathbf {x}} (\\alpha) = \\sum_ {i = 1} ^ {N - 1} \\frac {1}{2 \\alpha \\left(1 + \\lambda_ {i}\\right) + 2 - \\mathbb {1} _ {\\{a = 1 \\}}} \\cdot \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T}. \\tag {6}",
+                                    "image_path": "4efd4c5715bcf15fb500b9091d96eb3af8c78735e22d6f3e54d4c5ce2939f8d6.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        107,
+                        147,
+                        406,
+                        159
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                107,
+                                147,
+                                406,
+                                159
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        107,
+                                        147,
+                                        406,
+                                        159
+                                    ],
+                                    "type": "text",
+                                    "content": "Moreover, for all "
+                                },
+                                {
+                                    "bbox": [
+                                        107,
+                                        147,
+                                        406,
+                                        159
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha_{2} > \\alpha_{1} > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        107,
+                                        147,
+                                        406,
+                                        159
+                                    ],
+                                    "type": "text",
+                                    "content": ", we have "
+                                },
+                                {
+                                    "bbox": [
+                                        107,
+                                        147,
+                                        406,
+                                        159
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\mathbf{x}}(\\alpha_{2}) < _{L}\\mathbf{V}_{\\mathbf{x}}(\\alpha_{1}) < _{L}\\mathbf{V}_{\\mathbf{x}}(0)"
+                                },
+                                {
+                                    "bbox": [
+                                        107,
+                                        147,
+                                        406,
+                                        159
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        162,
+                        506,
+                        273
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                162,
+                                506,
+                                273
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        162,
+                                        506,
+                                        273
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 3.1 shows that the SRRW iterates "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        162,
+                                        506,
+                                        273
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        162,
+                                        506,
+                                        273
+                                    ],
+                                    "type": "text",
+                                    "content": " converges to the target distribution "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        162,
+                                        506,
+                                        273
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mu"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        162,
+                                        506,
+                                        273
+                                    ],
+                                    "type": "text",
+                                    "content": " a.s. even under the general step size "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        162,
+                                        506,
+                                        273
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\gamma_n = (n + 1)^{-a}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        162,
+                                        506,
+                                        273
+                                    ],
+                                    "type": "text",
+                                    "content": " for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        162,
+                                        506,
+                                        273
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "a\\in (0.5,1]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        162,
+                                        506,
+                                        273
+                                    ],
+                                    "type": "text",
+                                    "content": ". We also observe that the asymptotic covariance matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        162,
+                                        506,
+                                        273
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\mathbf{x}}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        162,
+                                        506,
+                                        273
+                                    ],
+                                    "type": "text",
+                                    "content": " decreases at rate "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        162,
+                                        506,
+                                        273
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "O(1 / \\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        162,
+                                        506,
+                                        273
+                                    ],
+                                    "type": "text",
+                                    "content": ". Lemma 3.1 aligns with Doshi et al. (2023, Theorem 4.2 and Corollary 4.3) for the special case of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        162,
+                                        506,
+                                        273
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "a = 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        162,
+                                        506,
+                                        273
+                                    ],
+                                    "type": "text",
+                                    "content": ", and is therefore more general. Critically, it helps us establish our next result regarding the first-order convergence for the optimization iterate sequence "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        162,
+                                        506,
+                                        273
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\theta_{n}\\}_{n\\geq 0}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        162,
+                                        506,
+                                        273
+                                    ],
+                                    "type": "text",
+                                    "content": " following update rule (4c), as well as its second-order convergence result, which follows shortly after. The proofs of Lemma 3.1 and our next result, Theorem 3.2, are deferred to Appendix D. In what follows, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        162,
+                                        506,
+                                        273
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k = 1,2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        162,
+                                        506,
+                                        273
+                                    ],
+                                    "type": "text",
+                                    "content": ", and 3 refer to cases (i), (ii), and (iii) in Section 2.2, respectively. All subsequent results are proven under Assumptions A1 to A4, with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        162,
+                                        506,
+                                        273
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{A}3^{\\prime}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        162,
+                                        506,
+                                        273
+                                    ],
+                                    "type": "text",
+                                    "content": " replacing A3 only when the step sizes "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        162,
+                                        506,
+                                        273
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_{n},\\gamma_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        162,
+                                        506,
+                                        273
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfy case (iii)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        274,
+                        504,
+                        300
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                274,
+                                504,
+                                300
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        504,
+                                        300
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem 3.2. For "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        504,
+                                        300
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k \\in \\{1,2,3\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        504,
+                                        300
+                                    ],
+                                    "type": "text",
+                                    "content": ", and any initial "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        504,
+                                        300
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\pmb{\\theta}_0,\\mathbf{x}_0,X_0) \\in \\mathbb{R}^D \\times \\mathrm{Int}(\\Sigma) \\times \\mathcal{N}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        504,
+                                        300
+                                    ],
+                                    "type": "text",
+                                    "content": ", we have "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        504,
+                                        300
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\theta}_n \\to \\pmb{\\theta}^*"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        504,
+                                        300
+                                    ],
+                                    "type": "text",
+                                    "content": " as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        504,
+                                        300
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n \\to \\infty"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        504,
+                                        300
+                                    ],
+                                    "type": "text",
+                                    "content": " for some "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        504,
+                                        300
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\theta}^* \\in \\Theta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        504,
+                                        300
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        504,
+                                        300
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{P}_{\\pmb{\\theta}_0,\\mathbf{x}_0,X_0}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        504,
+                                        300
+                                    ],
+                                    "type": "text",
+                                    "content": "-almost surely."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        304,
+                        504,
+                        348
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                304,
+                                504,
+                                348
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        348
+                                    ],
+                                    "type": "text",
+                                    "content": "In the stochastic optimization context, the above result ensures convergence of iterates "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        348
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\theta}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        348
+                                    ],
+                                    "type": "text",
+                                    "content": " to a local minimizer "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        348
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\theta}^*"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        348
+                                    ],
+                                    "type": "text",
+                                    "content": ". Loosely speaking, the first-order convergence of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        348
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        348
+                                    ],
+                                    "type": "text",
+                                    "content": " in Lemma 3.1 as well as that of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        348
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\theta}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        348
+                                    ],
+                                    "type": "text",
+                                    "content": " are closely related to the convergence of trajectories "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        348
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\mathbf{z}(t) \\triangleq (\\pmb{\\theta}(t), \\mathbf{x}(t))\\}_{t \\geq 0}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        348
+                                    ],
+                                    "type": "text",
+                                    "content": " of the (coupled) mean-field ODE, written in a matrix-vector form as"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        200,
+                        348,
+                        504,
+                        374
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                200,
+                                348,
+                                504,
+                                374
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        200,
+                                        348,
+                                        504,
+                                        374
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\frac {d}{d t} \\mathbf {z} (t) = \\mathbf {g} (\\mathbf {z} (t)) \\triangleq \\left[ \\begin{array}{c} \\mathbf {H} (\\boldsymbol {\\theta} (t)) ^ {T} \\boldsymbol {\\pi} [ \\mathbf {x} (t) ] \\\\ \\boldsymbol {\\pi} [ \\mathbf {x} (t) ] - \\mathbf {x} (t) \\end{array} \\right] \\in \\mathbb {R} ^ {D + N}. \\tag {7}",
+                                    "image_path": "da0f3bc518b7c42b3fe9793e4ce7575cdc02cd576140dd5c6beaa675f1a04375.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        374,
+                        504,
+                        443
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                374,
+                                504,
+                                443
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        504,
+                                        443
+                                    ],
+                                    "type": "text",
+                                    "content": "where matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        504,
+                                        443
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{H}(\\pmb{\\theta}) \\triangleq [H(\\pmb{\\theta}, 1), \\dots, H(\\pmb{\\theta}, N)]^T \\in \\mathbb{R}^{N \\times D}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        504,
+                                        443
+                                    ],
+                                    "type": "text",
+                                    "content": " for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        504,
+                                        443
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\theta} \\in \\mathbb{R}^D"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        504,
+                                        443
+                                    ],
+                                    "type": "text",
+                                    "content": ". Here, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        504,
+                                        443
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\pi}[\\mathbf{x}] \\in \\mathrm{Int}(\\Sigma)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        504,
+                                        443
+                                    ],
+                                    "type": "text",
+                                    "content": " is the stationary distribution of the SRRW kernel "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        504,
+                                        443
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{K}[\\mathbf{x}]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        504,
+                                        443
+                                    ],
+                                    "type": "text",
+                                    "content": " and is shown in Doshi et al. (2023) to be given by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        504,
+                                        443
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi_i[\\mathbf{x}] \\propto \\sum_{j \\in \\mathcal{N}} \\mu_i P_{ij}(x_i / \\mu_i)^{-\\alpha}(x_j / \\mu_j)^{-\\alpha}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        504,
+                                        443
+                                    ],
+                                    "type": "text",
+                                    "content": ". The Jacobian matrix of (7) when evaluated at equilibria "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        504,
+                                        443
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{z}^* = (\\pmb{\\theta}^*, \\pmb{\\mu})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        504,
+                                        443
+                                    ],
+                                    "type": "text",
+                                    "content": " for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        504,
+                                        443
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\theta}^* \\in \\Theta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        504,
+                                        443
+                                    ],
+                                    "type": "text",
+                                    "content": " captures the behaviour of solutions of the mean-field in their vicinity, and plays an important role in the asymptotic covariance matrices arising out of our CLT results. We evaluate this Jacobian matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        504,
+                                        443
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{J}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        504,
+                                        443
+                                    ],
+                                    "type": "text",
+                                    "content": " as a function of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        504,
+                                        443
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha \\geq 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        504,
+                                        443
+                                    ],
+                                    "type": "text",
+                                    "content": " to be given by"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        154,
+                        443,
+                        504,
+                        470
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                154,
+                                443,
+                                504,
+                                470
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        154,
+                                        443,
+                                        504,
+                                        470
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {J} (\\alpha) \\triangleq \\nabla g \\left(\\mathbf {z} ^ {*}\\right) = \\left[ \\begin{array}{c c} \\nabla \\mathbf {h} \\left(\\boldsymbol {\\theta} ^ {*}\\right) & - \\alpha \\mathbf {H} \\left(\\boldsymbol {\\theta} ^ {*}\\right) ^ {T} \\left(\\mathbf {P} ^ {T} + \\mathbf {I}\\right) \\\\ \\mathbf {0} _ {N \\times D} & 2 \\alpha \\boldsymbol {\\mu} \\mathbf {1} ^ {T} - \\alpha \\mathbf {P} ^ {T} - (\\alpha + 1) \\mathbf {I} \\end{array} \\right] \\triangleq \\left[ \\begin{array}{l l} \\mathbf {J} _ {1 1} & \\mathbf {J} _ {1 2} (\\alpha) \\\\ \\mathbf {J} _ {2 1} & \\mathbf {J} _ {2 2} (\\alpha) \\end{array} \\right]. \\tag {8}",
+                                    "image_path": "fdb335b85e04d6e3b7e39377f02696669b83fb2745393315bd25c223532ecb74.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        470,
+                        504,
+                        525
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                470,
+                                504,
+                                525
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        504,
+                                        525
+                                    ],
+                                    "type": "text",
+                                    "content": "The derivation of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        504,
+                                        525
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{J}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        504,
+                                        525
+                                    ],
+                                    "type": "text",
+                                    "content": " is referred to Appendix E.1.8 Here, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        504,
+                                        525
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{J}_{21}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        504,
+                                        525
+                                    ],
+                                    "type": "text",
+                                    "content": " is a zero matrix since "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        504,
+                                        525
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi[\\mathbf{x}] - \\mathbf{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        504,
+                                        525
+                                    ],
+                                    "type": "text",
+                                    "content": " is devoid of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        504,
+                                        525
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        504,
+                                        525
+                                    ],
+                                    "type": "text",
+                                    "content": ". While matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        504,
+                                        525
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{J}_{22}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        504,
+                                        525
+                                    ],
+                                    "type": "text",
+                                    "content": " is exactly of the form in Doshi et al. (2023, Lemma 3.4) to characterize the SRRW performance, our analysis includes an additional matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        504,
+                                        525
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{J}_{12}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        504,
+                                        525
+                                    ],
+                                    "type": "text",
+                                    "content": ", which captures the effect of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        504,
+                                        525
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}(t)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        504,
+                                        525
+                                    ],
+                                    "type": "text",
+                                    "content": " on "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        504,
+                                        525
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta(t)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        504,
+                                        525
+                                    ],
+                                    "type": "text",
+                                    "content": " in the ODE (7), which translates to the influence of our generalized SRRW empirical measure "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        504,
+                                        525
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        504,
+                                        525
+                                    ],
+                                    "type": "text",
+                                    "content": " on the SA iterates "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        504,
+                                        525
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        504,
+                                        525
+                                    ],
+                                    "type": "text",
+                                    "content": " in (4)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        104,
+                        530,
+                        504,
+                        565
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                530,
+                                504,
+                                565
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        530,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "text",
+                                    "content": "For notational simplicity, and without loss of generality, all our remaining results are stated while conditioning on the event that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        530,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\pmb {\\theta}_n\\to \\pmb {\\theta}^*\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        530,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "text",
+                                    "content": ", for some "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        530,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb {\\theta}^{*}\\in \\Theta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        530,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "text",
+                                    "content": ". We also adopt the shorthand notation "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        530,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{H}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        530,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "text",
+                                    "content": " to represent "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        530,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{H}(\\pmb {\\theta}^{*})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        530,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "text",
+                                    "content": ". Our main CLT result is as follows, with its proof deferred to Appendix E."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        104,
+                        567,
+                        489,
+                        580
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                567,
+                                489,
+                                580
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        489,
+                                        580
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem 3.3. For any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        489,
+                                        580
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha \\geq 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        489,
+                                        580
+                                    ],
+                                    "type": "text",
+                                    "content": ", we have: (a) There exists "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        489,
+                                        580
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}^{(k)}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        489,
+                                        580
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        489,
+                                        580
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k \\in \\{1,2,3\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        489,
+                                        580
+                                    ],
+                                    "type": "text",
+                                    "content": " such that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        214,
+                        580,
+                        397,
+                        611
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                214,
+                                580,
+                                397,
+                                611
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        214,
+                                        580,
+                                        397,
+                                        611
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left[ \\begin{array}{c} \\beta_ {n} ^ {- 1 / 2} (\\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*}) \\\\ \\gamma_ {n} ^ {- 1 / 2} (\\mathbf {x} _ {n} - \\boldsymbol {\\mu}) \\end{array} \\right] \\xrightarrow [ n \\to \\infty ]{d i s t.} N \\left(\\mathbf {0}, \\mathbf {V} ^ {(k)} (\\alpha)\\right).",
+                                    "image_path": "5b306d37c1ac2ca8bd1bd98f43d2d5e87b7364932392cfa08a6d156c6c3f52ef.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        104,
+                        611,
+                        504,
+                        637
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                611,
+                                504,
+                                637
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
+                                        504,
+                                        637
+                                    ],
+                                    "type": "text",
+                                    "content": "(b) For "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
+                                        504,
+                                        637
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k = 2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
+                                        504,
+                                        637
+                                    ],
+                                    "type": "text",
+                                    "content": ", matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
+                                        504,
+                                        637
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}^{(2)}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
+                                        504,
+                                        637
+                                    ],
+                                    "type": "text",
+                                    "content": " solves the Lyapunov equation "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
+                                        504,
+                                        637
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{J}(\\alpha)\\mathbf{V}^{(2)}(\\alpha) + \\mathbf{V}^{(2)}(\\alpha)\\mathbf{J}(\\alpha)^T + \\mathbb{1}_{\\{b = 1\\}}\\mathbf{V}^{(2)}(\\alpha) = -\\mathbf{U}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
+                                        504,
+                                        637
+                                    ],
+                                    "type": "text",
+                                    "content": ", where the Jacobian matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
+                                        504,
+                                        637
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{J}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
+                                        504,
+                                        637
+                                    ],
+                                    "type": "text",
+                                    "content": " is in (8), and"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        176,
+                        638,
+                        504,
+                        669
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                176,
+                                638,
+                                504,
+                                669
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        176,
+                                        638,
+                                        504,
+                                        669
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {U} \\triangleq \\sum_ {i = 1} ^ {N - 1} \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\cdot \\left[ \\begin{array}{c c} \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H} & \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\\\ \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H} & \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\end{array} \\right] \\triangleq \\left[ \\begin{array}{c c} \\mathbf {U} _ {1 1} & \\mathbf {U} _ {1 2} \\\\ \\mathbf {U} _ {2 1} & \\mathbf {U} _ {2 2} \\end{array} \\right]. \\tag {9}",
+                                    "image_path": "478efe60a913c03229cd5c665277ec6f04cb87a2b0b9903055aeb802f03ef53a.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        104,
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+                        682
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                                392,
+                                682
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        670,
+                                        392,
+                                        682
+                                    ],
+                                    "type": "text",
+                                    "content": "(c) For "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        670,
+                                        392,
+                                        682
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k \\in \\{1, 3\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        670,
+                                        392,
+                                        682
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        670,
+                                        392,
+                                        682
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}^{(k)}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        670,
+                                        392,
+                                        682
+                                    ],
+                                    "type": "text",
+                                    "content": " becomes block diagonal, which is given by"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        235,
+                        682,
+                        504,
+                        708
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                235,
+                                682,
+                                504,
+                                708
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        235,
+                                        682,
+                                        504,
+                                        708
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {V} ^ {(k)} (\\alpha) = \\left[ \\begin{array}{l l} \\mathbf {V} _ {\\boldsymbol {\\theta}} ^ {(k)} (\\alpha) & \\mathbf {0} _ {D \\times N} \\\\ \\mathbf {0} _ {N \\times D} & \\mathbf {V} _ {\\mathbf {x}} (\\alpha) \\end{array} \\right], \\tag {10}",
+                                    "image_path": "5cc4882e8ea7964c06fb899e4a7251ca84a17d537fba9d571ec606a43b8bda15.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
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+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
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+                            ],
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+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        104,
+                        710,
+                        504,
+                        731
+                    ],
+                    "type": "page_footnote",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                710,
+                                504,
+                                731
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        710,
+                                        504,
+                                        731
+                                    ],
+                                    "type": "text",
+                                    "content": "The Jacobian "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        710,
+                                        504,
+                                        731
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{J}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        710,
+                                        504,
+                                        731
+                                    ],
+                                    "type": "text",
+                                    "content": " is "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        710,
+                                        504,
+                                        731
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(D + N)\\times (D + N)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        710,
+                                        504,
+                                        731
+                                    ],
+                                    "type": "text",
+                                    "content": "-dimensional, with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        710,
+                                        504,
+                                        731
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{J}_{11}\\in \\mathbb{R}^{D\\times D}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        710,
+                                        504,
+                                        731
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        710,
+                                        504,
+                                        731
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{J}_{22}(\\alpha)\\in \\mathbb{R}^{N\\times N}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        710,
+                                        504,
+                                        731
+                                    ],
+                                    "type": "text",
+                                    "content": ". Following this, all matrices written in a block form, such as matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        710,
+                                        504,
+                                        731
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{U}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        710,
+                                        504,
+                                        731
+                                    ],
+                                    "type": "text",
+                                    "content": " in (9), will inherit the same dimensional structure."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                },
+                {
+                    "bbox": [
+                        302,
+                        751,
+                        309,
+                        760
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                302,
+                                751,
+                                309,
+                                760
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        302,
+                                        751,
+                                        309,
+                                        760
+                                    ],
+                                    "type": "text",
+                                    "content": "7"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 6
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        80,
+                        499,
+                        97
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                80,
+                                499,
+                                97
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        499,
+                                        97
+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        499,
+                                        97
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\mathbf{x}}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        499,
+                                        97
+                                    ],
+                                    "type": "text",
+                                    "content": " is as in (6), and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        499,
+                                        97
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        499,
+                                        97
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        499,
+                                        97
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\pmb{\\theta}}^{(3)}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        499,
+                                        97
+                                    ],
+                                    "type": "text",
+                                    "content": " can be written in the following explicit form:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        181,
+                        101,
+                        457,
+                        120
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                181,
+                                101,
+                                457,
+                                120
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        181,
+                                        101,
+                                        457,
+                                        120
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {V} _ {\\boldsymbol {\\theta}} ^ {(1)} (\\alpha) = \\int_ {0} ^ {\\infty} e ^ {t (\\nabla_ {\\boldsymbol {\\theta}} \\mathbf {h} (\\boldsymbol {\\theta} ^ {*}) + \\frac {1}{2} \\{b = 1 \\} \\boldsymbol {I})} \\mathbf {U} _ {\\boldsymbol {\\theta}} (\\alpha) e ^ {t (\\nabla_ {\\boldsymbol {\\theta}} \\mathbf {h} (\\boldsymbol {\\theta} ^ {*}) + \\frac {1}{2} \\{b = 1 \\} \\boldsymbol {I}) ^ {T}} d t,",
+                                    "image_path": "10602407dcf1c32e4403354550b6ae51bd2a785e978a28aaaa5d062bde687788.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        181,
+                        121,
+                        357,
+                        137
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                181,
+                                121,
+                                357,
+                                137
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        181,
+                                        121,
+                                        357,
+                                        137
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {V} _ {\\pmb {\\theta}} ^ {(3)} (\\alpha) = \\int_ {0} ^ {\\infty} e ^ {t \\nabla_ {\\pmb {\\theta}} \\mathbf {h} (\\pmb {\\theta} ^ {*})} \\mathbf {U} _ {1 1} e ^ {t \\nabla_ {\\pmb {\\theta}} \\mathbf {h} (\\pmb {\\theta} ^ {*})} d t,",
+                                    "image_path": "5bc7fb2d056188e7499187f5846eab9dd50d57c6ac47a738ca2ef4b410e405a8.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        153,
+                        140,
+                        505,
+                        171
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                153,
+                                140,
+                                505,
+                                171
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        153,
+                                        140,
+                                        505,
+                                        171
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "w h e r e \\mathbf {U} _ {\\boldsymbol {\\theta}} (\\alpha) = \\sum_ {i = 1} ^ {N - 1} \\frac {1}{\\left(\\alpha \\left(1 + \\lambda_ {i}\\right) + 1\\right) ^ {2}} \\cdot \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H}. \\tag {11}",
+                                    "image_path": "156b2e83b7fe3fd59b9ecdb0c9cf7503142f5047259f31bc6bbb195497ae28ab.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        171,
+                        506,
+                        318
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                171,
+                                506,
+                                318
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        506,
+                                        318
+                                    ],
+                                    "type": "text",
+                                    "content": "For "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        506,
+                                        318
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k \\in \\{1,3\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        506,
+                                        318
+                                    ],
+                                    "type": "text",
+                                    "content": ", SA-SRRW in (4) is a two-timescale SA with controlled Markov noise. While a few works study the CLT of two-timescale SA with the stochastic input being a martingale-difference (i.i.d.) noise (Konda & Tsitsiklis, 2004; Mokkadem & Pelletier, 2006), a CLT result covering the case of controlled Markov noise (e.g., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        506,
+                                        318
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k \\in \\{1,3\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        506,
+                                        318
+                                    ],
+                                    "type": "text",
+                                    "content": "), a far more general setting than martingale-difference noise, is still an open problem. Thus, we here prove our CLT for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        506,
+                                        318
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k \\in \\{1,3\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        506,
+                                        318
+                                    ],
+                                    "type": "text",
+                                    "content": " from scratch by a series of careful decompositions of the Markovian noise, ultimately into a martingale-difference term and several non-vanishing noise terms through repeated application of the Poisson equation (Benveniste et al., 2012; Fort, 2015). Although the form of the resulting asymptotic covariance looks similar to that for the martingale-difference case in (Konda & Tsitsiklis, 2004; Mokkadem & Pelletier, 2006) at first glance, they are not equivalent. Specifically, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        506,
+                                        318
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\theta}^{(k)}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        506,
+                                        318
+                                    ],
+                                    "type": "text",
+                                    "content": " captures both the effect of SRRW hyper-parameter "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        506,
+                                        318
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        506,
+                                        318
+                                    ],
+                                    "type": "text",
+                                    "content": ", as well as that of the underlying base Markov chain via eigenpairs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        506,
+                                        318
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\lambda_i, \\mathbf{u}_i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        506,
+                                        318
+                                    ],
+                                    "type": "text",
+                                    "content": " of its transition probability matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        506,
+                                        318
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{P}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        506,
+                                        318
+                                    ],
+                                    "type": "text",
+                                    "content": " in matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        506,
+                                        318
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{U}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        506,
+                                        318
+                                    ],
+                                    "type": "text",
+                                    "content": ", whereas the latter only covers the martingale-difference noise terms as a special case."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        323,
+                        504,
+                        407
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                323,
+                                504,
+                                407
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        323,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "text",
+                                    "content": "When "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        323,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k = 2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        323,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "text",
+                                    "content": ", that is, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        323,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_{n} = \\gamma_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        323,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "text",
+                                    "content": ", algorithm (4) can be regarded as a single-timescale SA algorithm. In this case, we utilize the CLT in Fort (2015, Theorem 2.1) to obtain the implicit form of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        323,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}^{(2)}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        323,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "text",
+                                    "content": " as shown in Theorem 3.3. However, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        323,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{J}_{12}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        323,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "text",
+                                    "content": " being non-zero for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        323,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        323,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "text",
+                                    "content": " restricts us from obtaining an explicit form for the covariance term corresponding to SA iterate errors "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        323,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta_{n} - \\theta^{*}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        323,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "text",
+                                    "content": ". On the other hand, for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        323,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k \\in \\{1,3\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        323,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "text",
+                                    "content": ", the nature of two-timescale structure causes "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        323,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        323,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        323,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        323,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "text",
+                                    "content": " to become asymptotically independent with zero correlation terms inside "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        323,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}^{(k)}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        323,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "text",
+                                    "content": " in (10), and we can explicitly deduce "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        323,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\pmb{\\theta}}^{(k)}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        323,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "text",
+                                    "content": ". We now take a deeper dive into "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        323,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        323,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "text",
+                                    "content": " and study its effect on "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        323,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\pmb{\\theta}}^{(k)}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        323,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        411,
+                        504,
+                        437
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                411,
+                                504,
+                                437
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        504,
+                                        437
+                                    ],
+                                    "type": "text",
+                                    "content": "Covariance Ordering of SA-SRRW: We refer the reader to Appendix F for proofs of all remaining results. We begin by focusing on case (i) and capturing the impact of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        504,
+                                        437
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        504,
+                                        437
+                                    ],
+                                    "type": "text",
+                                    "content": " on "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        504,
+                                        437
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\theta}^{(1)}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        504,
+                                        437
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        104,
+                        441,
+                        504,
+                        471
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                441,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        441,
+                                        504,
+                                        471
+                                    ],
+                                    "type": "text",
+                                    "content": "Proposition 3.4. For all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        441,
+                                        504,
+                                        471
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha_{2} > \\alpha_{1} > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        441,
+                                        504,
+                                        471
+                                    ],
+                                    "type": "text",
+                                    "content": ", we have "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        441,
+                                        504,
+                                        471
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(\\alpha_2) < _L\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(\\alpha_1) < _L\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(0)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        441,
+                                        504,
+                                        471
+                                    ],
+                                    "type": "text",
+                                    "content": ". Furthermore, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        441,
+                                        504,
+                                        471
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        441,
+                                        504,
+                                        471
+                                    ],
+                                    "type": "text",
+                                    "content": " decreases to zero at a rate of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        441,
+                                        504,
+                                        471
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "O(1 / \\alpha^2)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        441,
+                                        504,
+                                        471
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        475,
+                        506,
+                        644
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "text",
+                                    "content": "Proposition 3.4 proves a monotonic reduction (in terms of Loewner ordering) of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\theta}^{(1)}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "text",
+                                    "content": " as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
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+                                        644
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "text",
+                                    "content": " increases. Moreover, the decrease rate "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "O(1 / \\alpha^2)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "text",
+                                    "content": " surpasses the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "O(1 / \\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "text",
+                                    "content": " rate seen in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\mathbf{x}}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "text",
+                                    "content": " and the sampling application in Doshi et al. (2023, Corollary 4.7), and is also empirically observed in our simulation in Section 4. Suppose we consider the same SA now driven by an i.i.d. sequence "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{X_{n}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "text",
+                                    "content": " with the same marginal distribution "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\mu}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, our Proposition 3.4 asserts that a token algorithm employing SRRW (walk on a graph) with large enough "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "text",
+                                    "content": " on a general graph can actually produce better SA iterates with its asymptotic covariance going down to zero, than a 'hypothetical situation' where the walker is able to access any node "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "j"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "text",
+                                    "content": " with probability "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mu_{j}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "text",
+                                    "content": " from anywhere in one step (more like a random jumper). This can be seen by noting that for large time "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "text",
+                                    "content": ", the scaled MSE "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{E}[\\|\\pmb{\\theta}_n - \\pmb{\\theta}^*\\|^2] / \\beta_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "text",
+                                    "content": " is composed of the diagonal entries of the covariance matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\theta}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "text",
+                                    "content": ", which, as we discuss in detail in Appendix F.2, are decreasing in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "text",
+                                    "content": " as a consequence of the Loewner ordering in Proposition 3.4. For large enough "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "text",
+                                    "content": ", the scaled MSE for SA-SRRW becomes smaller than its i.i.d. counterpart, which is always a constant. Although Doshi et al. (2023) alluded this for sampling applications with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\mathbf{x}}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "text",
+                                    "content": ", we broaden its horizons to distributed optimization problems with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\pmb{\\theta}}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        506,
+                                        644
+                                    ],
+                                    "type": "text",
+                                    "content": " using tokens on graphs. Our subsequent result concerns the performance comparison between cases (i) and (iii)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
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+                        663
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "Corollary 3.5. For any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        647,
+                                        405,
+                                        663
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha >0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        647,
+                                        405,
+                                        663
+                                    ],
+                                    "type": "text",
+                                    "content": ", we have "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        647,
+                                        405,
+                                        663
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(\\alpha) < _L\\mathbf{V}_{\\pmb{\\theta}}^{(3)}(\\alpha) = \\mathbf{V}_{\\pmb{\\theta}}^{(3)}(0)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        647,
+                                        405,
+                                        663
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "We show that case (i) is asymptotically better than case (iii) for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": ". In view of Proposition 3.4 and Corollary 3.5, the advantages of case (i) become prominent."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                }
+            ],
+            "discarded_blocks": [
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
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+                                    "type": "text",
+                                    "content": "Further insights of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        700,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "O(1 / \\alpha^2)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        700,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " are tied to the two-timescale structure, particularly "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        700,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_{n} = o(\\gamma_{n})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        700,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " in case (i), which places "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        700,
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+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        700,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " on the slow timescale so that the correlation terms "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        700,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{J}_{12}(\\alpha),\\mathbf{J}_{22}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        700,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " in the Jacobian matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{J}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        700,
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+                                    ],
+                                    "type": "text",
+                                    "content": " in (8) come into play. Technical details are referred to Appendix E.2, where we show the form of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        700,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{U}_{\\theta}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
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+                                    "content": "8"
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+                                            "content": "(a) SGD-SRRW."
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+                                    "content": "4 SIMULATION"
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+                                    "type": "text",
+                                    "content": "In this section, we simulate our SA-SRRW algorithm on the wikiVote graph (Leskovec & Krevl, 2014), comprising 889 nodes and 2914 edges. We configure the SRRW's base Markov chain "
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+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{P}"
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+                                    "type": "text",
+                                    "content": " as the MHRW with a uniform target distribution "
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+                                    "content": "\\mu = \\frac{1}{N} \\mathbf{1}"
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+                                    "type": "text",
+                                    "content": ". For distributed optimization, we consider the following "
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+                                    "type": "inline_equation",
+                                    "content": "L_{2}"
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+                                    "type": "text",
+                                    "content": " regularized binary classification problem:"
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+                                    "type": "interline_equation",
+                                    "content": "\\left. \\min  _ {\\boldsymbol {\\theta} \\in \\mathbb {R} ^ {D}} \\left\\{f (\\boldsymbol {\\theta}) \\triangleq \\frac {1}{N} \\sum_ {i = 1} ^ {N} \\log \\left(1 + e ^ {\\boldsymbol {\\theta} ^ {T} \\mathbf {s} _ {i}}\\right) - y _ {i} \\left(\\boldsymbol {\\theta} ^ {T} \\mathbf {s} _ {i}\\right) + \\frac {\\kappa}{2} \\| \\boldsymbol {\\theta} \\| ^ {2} \\right\\}, \\right. \\tag {12}",
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+                                    "type": "text",
+                                    "content": "where "
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+                                    "type": "inline_equation",
+                                    "content": "\\{(\\mathbf{s}_i, y_i)\\}_{i=1}^N"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " is the ijCNN1 dataset (with 22 features, i.e., "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{s}_i \\in \\mathbb{R}^{22}"
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": ") from LIBSVM (Chang & Lin, 2011), and penalty parameter "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\kappa = 1"
+                                },
+                                {
+                                    "bbox": [
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+                                        407,
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+                                    ],
+                                    "type": "text",
+                                    "content": ". Each node in the wikiVote graph is assigned one data point, thus 889 data points in total. We perform SRRW driven SGD (SGD-SRRW) and SRRW driven stochastic heavy ball (SHB-SRRW) algorithms (see (13) in Appendix A for its algorithm). We fix the step size "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\beta_n = (n + 1)^{-0.9}"
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " for the SA iterates and adjust "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\gamma_n = (n + 1)^{-a}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    "type": "text",
+                                    "content": " in the SRRW iterates to cover all three cases discussed in this paper: (i) "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "a = 0.8"
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": "; (ii) "
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+                                    "type": "inline_equation",
+                                    "content": "a = 0.9"
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": "; (iii) "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "a = 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": ". We use mean square error (MSE), i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{E}[\\|\\pmb{\\theta}_n - \\pmb{\\theta}^*\\|^2]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    "type": "text",
+                                    "content": ", to measure the error on the SA iterates."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
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+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "Our results are presented in Figures 2 and 3, where each experiment is repeated 100 times. Figures 2a and 2b, based on wikiVote graph, highlight the consistent performance ordering across different "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "\\alpha"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " values for both algorithms over almost all time (not just asymptotically). Notably, curves for "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\alpha \\geq 5"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " outperform that of the i.i.d. sampling (in black) even under the graph constraints. Figure 2c on the smaller Dolphin graph (Rossi & Ahmed, 2015) - 62 nodes and 159 edges - illustrates that the points of "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "(\\alpha, \\mathrm{MSE})"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " pair arising from SGD-SRRW at time "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
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+                                        615
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n = 10^7"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        615
+                                    ],
+                                    "type": "text",
+                                    "content": " align with a curve in the form of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
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+                                        615
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g(x) = \\frac{c_1}{(x + c_2)^2} + c_3"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        615
+                                    ],
+                                    "type": "text",
+                                    "content": " to showcase "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        615
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "O(1 / \\alpha^2)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        615
+                                    ],
+                                    "type": "text",
+                                    "content": " rates. This smaller graph allows for longer simulations to observe the asymptotic behaviour. Additionally, among the three cases examined at identical "
+                                },
+                                {
+                                    "bbox": [
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+                                        491,
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+                                    "type": "inline_equation",
+                                    "content": "\\alpha"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " values, Figures 3a - 3c confirm that case (i) performs consistently better than the rest, underscoring its superiority in practice. Further results, including those from non-convex functions and additional datasets, are deferred to Appendix H due to space constraints."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                },
+                {
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+                                    "type": "text",
+                                    "content": "5 CONCLUSION"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                },
+                {
+                    "bbox": [
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+                                    "type": "text",
+                                    "content": "In this paper, we show both theoretically and empirically that the SRRW as a drop-in replacement for Markov chains can provide significant performance improvements when used for token algorithms, where the acceleration comes purely from the careful analysis of the stochastic input of the algorithm, without changing the optimization iteration itself. Our paper is an instance where the asymptotic analysis approach allows the design of better algorithms despite the usage of unconventional noise sequences such as nonlinear Markov chains like the SRRW, for which traditional finite-time analytical approaches fall short, thus advocating their wider adoption."
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+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "content": "9"
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+                                    "type": "text",
+                                    "content": "6 ACKNOWLEDGMENTS AND DISCLOSURE OF FUNDING"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
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+                                    "type": "text",
+                                    "content": "We thank the anonymous reviewers for their constructive comments, especially Reviewer DxLx for raising future directions. This work was supported in part by National Science Foundation under Grant Nos. CNS-2007423 and IIS-1910749."
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+                        }
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+                                    "type": "text",
+                                    "content": "REFERENCES"
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+                                            "content": "Aleksei Triastcyn, Matthias Reisser, and Christos Louizos. Decentralized learning with random walks and communication-efficient adaptive optimization. In Workshop on Federated Learning: Recent Advances and New Challenges (in Conjunction with NeurIPS 2022), 2022."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 11
+                        },
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+                                            "content": "Thijs Vogels, Lie He, Anastasiia Koloskova, Sai Praneeth Karimireddy, Tao Lin, Sebastian U Stich, and Martin Jaggi. Relaysum for decentralized deep learning on heterogeneous data. In Advances in Neural Information Processing Systems, volume 34, pp. 28004-28015, 2021."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 12
+                        },
+                        {
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+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": "Jianyu Wang, Anit Kumar Sahu, Zhouyi Yang, Gauri Joshi, and Soummya Kar. Matcha: Speeding up decentralized sgd via matching decomposition sampling. In 2019 Sixth Indian Control Conference (ICC), pp. 299-300. IEEE, 2019."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 13
+                        },
+                        {
+                            "bbox": [
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+                                            "type": "text",
+                                            "content": "Vinayaka G Yaji and Shalabh Bhatnagar. Stochastic recursive inclusions in two timescales with nonadditive iterate-dependent markov noise. Mathematics of Operations Research, 45(4):1405-1444, 2020."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 14
+                        },
+                        {
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+                                            "type": "text",
+                                            "content": "Hao Ye, Le Liang, and Geoffrey Ye Li. Decentralized federated learning with unreliable communications. IEEE Journal of Selected Topics in Signal Processing, 16(3):487-500, 2022."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 15
+                        },
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+                                            "content": "Sihan Zeng, Thinh T Doan, and Justin Romberg. A two-time-scale stochastic optimization framework with applications in control and reinforcement learning. arXiv preprint arXiv:2109.14756, 2021."
+                                        }
+                                    ]
+                                }
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+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "type": "text",
+                                    "content": "A EXAMPLES OF STOCHASTIC ALGORITHMS OF THE FORM (2)."
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+                    ],
+                    "index": 1
+                },
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+                                    "type": "text",
+                                    "content": "In the literature of stochastic optimizations, many SGD variants have been proposed by introducing an auxiliary variable to improve convergence. In what follows, we present two SGD variants with decreasing step size that can be presented in the form of (2): SHB (Gadat et al., 2018; Li et al., 2022) and momentum-based algorithm (Barakat et al., 2021; Barakat & Bianchi, 2021)."
+                                }
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+                                    "type": "interline_equation",
+                                    "content": "\\left\\{ \\begin{array}{l} \\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} - \\beta_ {n + 1} \\mathbf {m} _ {n} \\\\ \\mathbf {m} _ {n + 1} = \\mathbf {m} _ {n} + \\beta_ {n + 1} (\\nabla F (\\boldsymbol {\\theta} _ {n}, X _ {n + 1}) - \\mathbf {m} _ {n}), \\end{array} \\right. \\left\\{ \\begin{array}{l} \\mathbf {v} _ {n + 1} = \\mathbf {v} _ {n} + \\beta_ {n + 1} (\\nabla F (\\boldsymbol {\\theta} _ {n}, X _ {n + 1}) ^ {2} - \\mathbf {v} _ {n}), \\\\ \\mathbf {m} _ {n + 1} = \\mathbf {m} _ {n} + \\beta_ {n + 1} (\\nabla F (\\boldsymbol {\\theta} _ {n}, X _ {n + 1}) - \\mathbf {m} _ {n}), \\\\ \\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} - \\beta_ {n + 1} \\mathbf {m} _ {n} / \\sqrt {\\mathbf {v} _ {n} + \\epsilon}, \\end{array} \\right.",
+                                    "image_path": "285e91b951b057bb7f4396d669d97c343efb126ca81b7d819a511eff58384f90.jpg"
+                                }
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "(a). SHB"
+                                }
+                            ]
+                        }
+                    ],
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+                },
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+                                    "type": "text",
+                                    "content": "(b). Momentum-based Algorithm"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
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+                    ],
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+                                    "type": "text",
+                                    "content": "(13)"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        234,
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+                                        258
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\epsilon > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        234,
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+                                        258
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        234,
+                                        504,
+                                        258
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\theta}_n, \\mathbf{m}_n, \\mathbf{v}_n, \\nabla F(\\pmb{\\theta}, X) \\in \\mathbb{R}^d"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        234,
+                                        504,
+                                        258
+                                    ],
+                                    "type": "text",
+                                    "content": ", and the square and square root in (13) (b) are element-wise operators."
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "^{10}"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "For SHB, we introduce an augmented variable "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        262,
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+                                        274
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{z}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        262,
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+                                    ],
+                                    "type": "text",
+                                    "content": " and function "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        262,
+                                        494,
+                                        274
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H(\\mathbf{z}_n,X_{n + 1})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        262,
+                                        494,
+                                        274
+                                    ],
+                                    "type": "text",
+                                    "content": " defined as follows:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {z} _ {n} \\triangleq \\left[ \\begin{array}{c} \\boldsymbol {\\theta} _ {n} \\\\ \\mathbf {m} _ {n} \\end{array} \\right] \\in \\mathbb {R} ^ {2 d}, \\quad H (\\mathbf {z} _ {n}, X _ {n + 1}) \\triangleq \\left[ \\begin{array}{c} - \\mathbf {m} _ {n} \\\\ \\nabla F (\\boldsymbol {\\theta} _ {n}, X _ {n + 1}) - \\mathbf {m} _ {n} \\end{array} \\right] \\in \\mathbb {R} ^ {2 d}.",
+                                    "image_path": "6ae72185ea9d68adf9e54c0511ccaa1ccbdefd1a696c08b6ab47072e7fc92f7a.jpg"
+                                }
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+                {
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "For the general momentum-based algorithm, we define"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
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+                                    "bbox": [
+                                        161,
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+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {z} _ {n} \\triangleq \\left[ \\begin{array}{c} \\mathbf {v} _ {n} \\\\ \\mathbf {m} _ {n} \\\\ \\boldsymbol {\\theta} _ {n} \\end{array} \\right] \\in \\mathbb {R} ^ {3 d}, \\quad H (\\mathbf {z} _ {n}, X) \\triangleq \\left[ \\begin{array}{c} \\nabla F (\\boldsymbol {\\theta} _ {n}, X _ {n + 1}) ^ {2} - \\mathbf {v} _ {n} \\\\ \\nabla F (\\boldsymbol {\\theta} _ {n}, X _ {n + 1}) - \\mathbf {m} _ {n} \\\\ - \\mathbf {m} _ {n} / \\sqrt {\\mathbf {v} _ {n} + \\epsilon} \\end{array} \\right] \\in \\mathbb {R} ^ {3 d}.",
+                                    "image_path": "31d4615478441bf8bf2cf85a5488f216195f74571dd6c157183c2154a36c308c.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        104,
+                        376,
+                        504,
+                        456
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "Thus, we can reformulate both algorithms in (13) as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        376,
+                                        504,
+                                        456
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{z}_{n + 1} = \\mathbf{z}_n + \\beta_{n + 1}H(\\mathbf{z}_n,X_{n + 1})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        376,
+                                        504,
+                                        456
+                                    ],
+                                    "type": "text",
+                                    "content": ". This augmentation approach was previously adopted in (Gadat et al., 2018; Barakat et al., 2021; Barakat & Bianchi, 2021; Li et al., 2022) to analyze the asymptotic performance of algorithms in (13) using an i.i.d. sequence "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        376,
+                                        504,
+                                        456
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{X_{n}\\}_{n\\geq 0}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        376,
+                                        504,
+                                        456
+                                    ],
+                                    "type": "text",
+                                    "content": ". Consequently, the general SA iteration (2) includes these SGD variants. However, we mainly focus on the CLT for the general SA driven by SRRW in this paper. Pursuing the explicit CLT results of these SGD variants with specific form of function "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        376,
+                                        504,
+                                        456
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H(\\theta ,X)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        376,
+                                        504,
+                                        456
+                                    ],
+                                    "type": "text",
+                                    "content": " driven by the SRRW sequence "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        376,
+                                        504,
+                                        456
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{X_{n}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        376,
+                                        504,
+                                        456
+                                    ],
+                                    "type": "text",
+                                    "content": " is out of the scope of this paper."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        104,
+                        460,
+                        504,
+                        495
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                        495
+                                    ],
+                                    "type": "text",
+                                    "content": "When we numerically test the SHB algorithm in Section 4, we use the exact form of (13) (a) and the stochastic sequence "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        460,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{X_{n}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        460,
+                                        504,
+                                        495
+                                    ],
+                                    "type": "text",
+                                    "content": " is now driven by the SRRW. Specifically, we consider MHRW with transition kernel "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        460,
+                                        504,
+                                        495
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{P}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        460,
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+                                        495
+                                    ],
+                                    "type": "text",
+                                    "content": " as the base Markov chain of the SRRW process, e.g.,"
+                                }
+                            ]
+                        }
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+                    "index": 13
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+                                    "content": "P _ {i j} = \\left\\{ \\begin{array}{l l} \\min  \\left\\{\\frac {1}{d _ {i}}, \\frac {1}{d _ {j}} \\right\\} & \\text {i f n o d e j i s t h e n e i g h b o r o f n o d e i ,} \\\\ 0 & \\text {o t h e r w i s e ,} \\end{array} \\right.",
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+                                    "content": "P _ {i i} = 1 - \\sum_ {j \\in \\mathcal {N}} P _ {i j}.",
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+                                    "type": "text",
+                                    "content": "Then, at each time step "
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+                                    "type": "text",
+                                    "content": "Draw: "
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+                                    "type": "inline_equation",
+                                    "content": "X_{n + 1}\\sim \\mathbf{K}_{X_n,\\cdot}[\\mathbf{x}_n],"
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+                                    "type": "text",
+                                    "content": "where "
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+                                    "type": "inline_equation",
+                                    "content": "K_{ij}[\\mathbf{x}]\\triangleq \\frac{P_{ij}(x_j / \\mu_j)^{-\\alpha}}{\\sum_{k\\in\\mathcal{N}}P_{ik}(x_k / \\mu_k)^{-\\alpha}},\\quad \\forall i,j\\in \\mathcal{N},"
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+                                    "type": "text",
+                                    "content": "Update: "
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+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_{n + 1} = \\mathbf{x}_n + \\gamma_{n + 1}(\\pmb{\\delta}_{X_{n + 1}} - \\mathbf{x}_n)"
+                                }
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+                                    "type": "interline_equation",
+                                    "content": "\\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} - \\beta_ {n + 1} \\mathbf {m} _ {n},",
+                                    "image_path": "7de60a39abc572f2b77286724362c08652d22f935bab821d3dd177d36ec8dcf1.jpg"
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+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {m} _ {n + 1} = \\mathbf {m} _ {n} + \\beta_ {n + 1} (\\nabla F (\\boldsymbol {\\theta} _ {n}, X _ {n + 1}) - \\mathbf {m} _ {n}).",
+                                    "image_path": "737e3efe76b68846091e8060f62646d079c4a5ef09b80043ef637fd1abccb5cf.jpg"
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
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+                                    "type": "text",
+                                    "content": "For ease of expression, we simplify the original SHB and momentum-based algorithms from Gadat et al. (2018); Li et al. (2022); Barakat et al. (2021); Barakat & Bianchi (2021), setting all tunable parameters to 1 and resulting in (13)."
+                                }
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+                                    "content": "14"
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+                                    "type": "text",
+                                    "content": "B DISCUSSION ON MEAN FIELD FUNCTION OF SRRW ITERATES (4b)"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
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+                {
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+                                    "type": "text",
+                                    "content": "Non-asymptotic analyses have seen extensive attention recently in both single-timescale SA literature (Sun et al., 2018; Karimi et al., 2019; Chen et al., 2020b; 2022) and two-timescale SA literature (Doan, 2021; Zeng et al., 2021). Specifically, single-timescale SA has the following form:"
+                                }
+                            ]
+                        }
+                    ],
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+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {x} _ {n + 1} = \\mathbf {x} _ {n} + \\beta_ {n + 1} H (\\mathbf {x} _ {n}, X _ {n + 1}),",
+                                    "image_path": "3d438966751635a864e5254a9b4f5b0944d069c971395df28064b12a7a4a4818.jpg"
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+                                    "type": "text",
+                                    "content": "and function "
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+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "h(\\mathbf{x}) \\triangleq \\mathbb{E}_{X \\sim \\mu}[H(\\mathbf{x}, X)]"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " is the mean field of function "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H(\\mathbf{x}, X)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": ". Similarly, for two-timescale SA, we have the following recursions:"
+                                }
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+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {x} _ {n + 1} = \\mathbf {x} _ {n} + \\beta_ {n + 1} H _ {1} (\\mathbf {x} _ {n}, \\mathbf {y} _ {n}, X _ {n + 1}),",
+                                    "image_path": "e549040e7896772ac23a58a58397d66ee3f752b44a261099540ca3286b12c779.jpg"
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+                                    "content": "\\mathbf {y} _ {n + 1} = \\mathbf {y} _ {n} + \\gamma_ {n + 1} H _ {2} (\\mathbf {x} _ {n}, \\mathbf {y} _ {n}, X _ {n + 1}),",
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+                                    "content": "where "
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+                                    "type": "inline_equation",
+                                    "content": "\\{\\beta_n\\}"
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+                                    "type": "text",
+                                    "content": " and "
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+                                    "type": "inline_equation",
+                                    "content": "\\{\\gamma_n\\}"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " are on different timescales, and function "
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+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "h_i(\\mathbf{x},\\mathbf{y})\\triangleq \\mathbb{E}_{X\\sim \\mu}[H_i(\\mathbf{x},\\mathbf{y},X)]"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " is the mean field of function "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "H_{i}(\\mathbf{x},\\mathbf{y},X)"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " for "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "i = \\{1,2\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": ". All the aforementioned works require the mean field function "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                        504,
+                                        288
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h(\\mathbf{x})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
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+                                        288
+                                    ],
+                                    "type": "text",
+                                    "content": " in the single-timescale SA (or "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "h_1(\\mathbf{x},\\mathbf{y}),h_2(\\mathbf{x},\\mathbf{y})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " in the two-timescale SA) to be globally Lipschitz with a Lipschitz constant "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "L"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " to proceed with the derivation of finite-time bounds including the constant "
+                                },
+                                {
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+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
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+                    "angle": 0,
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Here, we show that the mean field function "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi[\\mathbf{x}] - \\mathbf{x}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " in the SRRW iterates (4b) is not globally Lipschitz, where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        293,
+                                        505,
+                                        350
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi[\\mathbf{x}]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        293,
+                                        505,
+                                        350
+                                    ],
+                                    "type": "text",
+                                    "content": " is the stationary distribution of the SRRW kernel "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        293,
+                                        505,
+                                        350
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{K}[\\mathbf{x}]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        293,
+                                        505,
+                                        350
+                                    ],
+                                    "type": "text",
+                                    "content": " defined in (3). To this end, we show that each entry of Jacobian matrix of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        293,
+                                        505,
+                                        350
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi[\\mathbf{x}] - \\mathbf{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        293,
+                                        505,
+                                        350
+                                    ],
+                                    "type": "text",
+                                    "content": " goes unbounded because a multivariate function is Lipschitz if and only if it has bounded partial derivatives. Note that from Doshi et al. (2023, Proposition 2.1), for the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        293,
+                                        505,
+                                        350
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        293,
+                                        505,
+                                        350
+                                    ],
+                                    "type": "text",
+                                    "content": "-th entry of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        293,
+                                        505,
+                                        350
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi[\\mathbf{x}]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        293,
+                                        505,
+                                        350
+                                    ],
+                                    "type": "text",
+                                    "content": ", we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        196,
+                        356,
+                        504,
+                        388
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                196,
+                                356,
+                                504,
+                                388
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        196,
+                                        356,
+                                        504,
+                                        388
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\boldsymbol {\\pi} _ {i} [ \\mathbf {x} ] = \\frac {\\sum_ {j \\in \\mathcal {N}} \\mu_ {i} P _ {i j} \\left(x _ {i} / \\mu_ {i}\\right) ^ {- \\alpha} \\left(x _ {j} / \\mu_ {j}\\right) ^ {- \\alpha}}{\\sum_ {i \\in \\mathcal {N}} \\sum_ {j \\in \\mathcal {N}} \\mu_ {i} P _ {i j} \\left(x _ {i} / \\mu_ {i}\\right) ^ {- \\alpha} \\left(x _ {j} / \\mu_ {j}\\right) ^ {- \\alpha}}. \\tag {16}",
+                                    "image_path": "5f57652b7d57bb72a4c883a2dfc385cc58696a83a402c3f3f09d1b47f852286c.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
+                        394,
+                        504,
+                        417
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                394,
+                                504,
+                                417
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        394,
+                                        504,
+                                        417
+                                    ],
+                                    "type": "text",
+                                    "content": "Then, the Jacobian matrix of the mean field function "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        394,
+                                        504,
+                                        417
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi[\\mathbf{x}] - \\mathbf{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        394,
+                                        504,
+                                        417
+                                    ],
+                                    "type": "text",
+                                    "content": ", which has been derived in Doshi et al. (2023, Proof of Lemma 3.4 in Appendix B), is given as follows:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        124,
+                        423,
+                        504,
+                        514
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        124,
+                                        423,
+                                        504,
+                                        514
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\frac {\\partial (\\boldsymbol {\\pi} _ {i} [ \\mathbf {x} ] - x _ {i})}{\\partial x _ {j}} \\\\ = \\frac {2 \\alpha}{x _ {j}} \\cdot \\frac {\\left(\\sum_ {k \\in \\mathcal {N}} \\mu_ {i} P _ {i k} \\left(x _ {i} / \\mu_ {i}\\right) ^ {- \\alpha} \\left(x _ {k} / \\mu_ {k}\\right) ^ {- \\alpha}\\right) \\left(\\sum_ {k \\in \\mathcal {N}} \\mu_ {j} P _ {j k} \\left(x _ {j} / \\mu_ {j}\\right) ^ {- \\alpha} \\left(x _ {k} / \\mu_ {k}\\right) ^ {- \\alpha}\\right)}{\\left(\\sum_ {l \\in \\mathcal {N}} \\sum_ {k \\in \\mathcal {N}} \\mu_ {l} P _ {l k} \\left(x _ {l} / \\mu_ {l}\\right) ^ {- \\alpha} \\left(x _ {k} / \\mu_ {k}\\right) ^ {- \\alpha}\\right) ^ {2}} \\tag {17} \\\\ - \\frac {\\alpha}{x _ {j}} \\cdot \\frac {\\mu_ {i} P _ {i j} (x _ {i} / \\mu_ {i}) ^ {- \\alpha} (x _ {j} / \\mu_ {j}) ^ {- \\alpha}}{\\sum_ {l \\in \\mathcal {N}} \\sum_ {k \\in \\mathcal {N}} \\mu_ {l} P _ {l k} (x _ {l} / \\mu_ {l}) ^ {- \\alpha} (x _ {k} / \\mu_ {k}) ^ {- \\alpha}} \\\\ \\end{array}",
+                                    "image_path": "a021c506ca2eeebb75214b0cb8a93574bc24dc70ad54dd1244504f8baee6f7c9.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        105,
+                        519,
+                        203,
+                        532
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                519,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        519,
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+                                        532
+                                    ],
+                                    "type": "text",
+                                    "content": "for "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        519,
+                                        203,
+                                        532
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i,j\\in \\mathcal{N},i\\neq j"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
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+                                        203,
+                                        532
+                                    ],
+                                    "type": "text",
+                                    "content": " , and"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        162,
+                        537,
+                        505,
+                        628
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                162,
+                                537,
+                                505,
+                                628
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        162,
+                                        537,
+                                        505,
+                                        628
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\frac {\\partial \\left(\\boldsymbol {\\pi} _ {i} [ \\mathbf {x} ] - x _ {i}\\right)}{\\partial x _ {i}} \\\\ = \\frac {2 \\alpha}{x _ {i}} \\cdot \\frac {\\left(\\sum_ {k \\in \\mathcal {N}} \\mu_ {i} P _ {i k} \\left(x _ {i} / \\mu_ {i}\\right) ^ {- \\alpha} \\left(x _ {k} / \\mu_ {k}\\right) ^ {- \\alpha}\\right) ^ {2}}{\\left(\\sum_ {l \\in \\mathcal {N}} \\sum_ {k \\in \\mathcal {N}} \\mu_ {l} P _ {l k} \\left(x _ {l} / \\mu_ {l}\\right) ^ {- \\alpha} \\left(x _ {k} / \\mu_ {k}\\right) ^ {- \\alpha}\\right) ^ {2}} \\tag {18} \\\\ - \\frac {\\alpha}{x _ {i}} \\cdot \\frac {\\sum_ {k \\in \\mathcal {N}} \\mu_ {i} P _ {i k} (x _ {i} / \\mu_ {i}) ^ {- \\alpha} (x _ {k} / \\mu_ {k}) ^ {- \\alpha} + \\mu_ {i} P _ {i i} (x _ {i} / \\mu_ {i}) ^ {- 2 \\alpha}}{\\sum_ {l \\in \\mathcal {N}} \\sum_ {k \\in \\mathcal {N}} \\mu_ {l} P _ {l k} (x _ {l} / \\mu_ {l}) ^ {- \\alpha} (x _ {k} / \\mu_ {k}) ^ {- \\alpha}} - 1 \\\\ \\end{array}",
+                                    "image_path": "88784c666fc077a3888333251c3629aa25919ba4711d8570344a6fcc7908ae63.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        104,
+                        633,
+                        504,
+                        678
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                633,
+                                504,
+                                678
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        633,
+                                        504,
+                                        678
+                                    ],
+                                    "type": "text",
+                                    "content": "for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        633,
+                                        504,
+                                        678
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i \\in \\mathcal{N}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        633,
+                                        504,
+                                        678
+                                    ],
+                                    "type": "text",
+                                    "content": ". Since the empirical distribution "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        633,
+                                        504,
+                                        678
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x} \\in \\mathrm{Int}(\\Sigma)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        633,
+                                        504,
+                                        678
+                                    ],
+                                    "type": "text",
+                                    "content": ", we have "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        633,
+                                        504,
+                                        678
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_i \\in (0,1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        633,
+                                        504,
+                                        678
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        633,
+                                        504,
+                                        678
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i \\in \\mathcal{N}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        633,
+                                        504,
+                                        678
+                                    ],
+                                    "type": "text",
+                                    "content": ". For fixed "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        633,
+                                        504,
+                                        678
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        633,
+                                        504,
+                                        678
+                                    ],
+                                    "type": "text",
+                                    "content": ", assume "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        633,
+                                        504,
+                                        678
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_i = x_j"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        633,
+                                        504,
+                                        678
+                                    ],
+                                    "type": "text",
+                                    "content": " and as they approach zero, the terms "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        633,
+                                        504,
+                                        678
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(x_i / \\mu_i)^{-\\alpha}, (x_j / \\mu_j)^{-\\alpha}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        633,
+                                        504,
+                                        678
+                                    ],
+                                    "type": "text",
+                                    "content": " dominate the fraction in (17) and both the numerator and the denominator of the fraction have the same order in terms of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        633,
+                                        504,
+                                        678
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_i, x_j"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        633,
+                                        504,
+                                        678
+                                    ],
+                                    "type": "text",
+                                    "content": ". Thus, we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        249,
+                        679,
+                        360,
+                        707
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                249,
+                                679,
+                                360,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        249,
+                                        679,
+                                        360,
+                                        707
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\frac {\\partial (\\pmb {\\pi} _ {i} [ \\mathbf {x} ] - x _ {i})}{\\partial x _ {j}} = O \\left(\\frac {1}{x _ {j}}\\right)",
+                                    "image_path": "c2a6fb6f6ae5b5a2c263696fe9e58a1d3fd56e0cdd1968cc2233e504560b7591.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
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+                        733
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                                504,
+                                733
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        709,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "such that the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        709,
+                                        504,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(i,j)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        709,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "-th entry of the Jacobian matrix can go unbounded as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        709,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_{j} \\to 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        709,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ". Consequently, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        709,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi[\\mathbf{x}] - \\mathbf{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        709,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " is not globally Lipschitz for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        709,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x} \\in \\operatorname{Int}(\\Sigma)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        709,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
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+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
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+                            ],
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+                                    "bbox": [
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+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
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+                    ],
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+                                    ],
+                                    "type": "text",
+                                    "content": "15"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                }
+            ],
+            "page_size": [
+                612,
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+            ],
+            "page_idx": 14
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
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+                        94
+                    ],
+                    "type": "title",
+                    "angle": 0,
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+                                    ],
+                                    "type": "text",
+                                    "content": "C DISCUSSION ON ASSUMPTION A3'"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
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+                        163
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                106,
+                                504,
+                                163
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
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+                                        163
+                                    ],
+                                    "type": "text",
+                                    "content": "When "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\gamma_{n} = o(\\beta_{n})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "text",
+                                    "content": ", iterates "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "text",
+                                    "content": " has smaller step size compared to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\theta}_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "text",
+                                    "content": ", thus converges 'slower' than "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\theta}_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "text",
+                                    "content": ". From Assumption A3', "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\theta}_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "text",
+                                    "content": " will intuitively converge to some point "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho (\\mathbf{x})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "text",
+                                    "content": " with the current value "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "text",
+                                    "content": " from the iteration "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "text",
+                                    "content": ", i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{E}_{X\\sim \\pi [\\mathbf{x}]}[H(\\rho (\\mathbf{x}),X)] = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "text",
+                                    "content": ", while the Hurwitz condition is to ensure the stability around "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho (\\mathbf{x})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "text",
+                                    "content": ". We can see that Assumption A3 is less stringent than A3' in that it only assumes such condition when "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x} = \\pmb{\\mu}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho (\\pmb {\\mu}) = \\pmb{\\theta}^{*}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "text",
+                                    "content": " rather than for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}\\in \\mathrm{Int}(\\Sigma)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        167,
+                        505,
+                        205
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                167,
+                                505,
+                                205
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        167,
+                                        505,
+                                        205
+                                    ],
+                                    "type": "text",
+                                    "content": "One special instance of Assumption A3' is by assuming the linear SA, e.g., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        167,
+                                        505,
+                                        205
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H(\\pmb{\\theta}, i) = A_i\\pmb{\\theta} + b_i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        167,
+                                        505,
+                                        205
+                                    ],
+                                    "type": "text",
+                                    "content": ". In this case, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        167,
+                                        505,
+                                        205
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{E}_{X \\sim \\pi[\\mathbf{x}]}[H(\\rho(\\mathbf{x}), X)] = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        167,
+                                        505,
+                                        205
+                                    ],
+                                    "type": "text",
+                                    "content": " is equivalent to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        167,
+                                        505,
+                                        205
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{E}_{i \\sim \\pi[\\mathbf{x}]}[A_i]\\rho(\\mathbf{x}) + \\mathbb{E}_{i \\sim \\pi[\\mathbf{x}]}[b_i] = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        167,
+                                        505,
+                                        205
+                                    ],
+                                    "type": "text",
+                                    "content": ". Under the condition that for every "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        167,
+                                        505,
+                                        205
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x} \\in \\mathrm{Int}(\\Sigma)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        167,
+                                        505,
+                                        205
+                                    ],
+                                    "type": "text",
+                                    "content": ", matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        167,
+                                        505,
+                                        205
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{E}_{i \\sim \\pi[\\mathbf{x}]}[A_i]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        167,
+                                        505,
+                                        205
+                                    ],
+                                    "type": "text",
+                                    "content": " is invertible, we then have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        222,
+                        210,
+                        386,
+                        224
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                222,
+                                210,
+                                386,
+                                224
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        222,
+                                        210,
+                                        386,
+                                        224
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\rho (\\mathbf {x}) = - \\left(\\mathbb {E} _ {i \\sim \\boldsymbol {\\pi} [ \\mathbf {x} ]} [ A _ {i} ]\\right) ^ {- 1} \\cdot \\mathbb {E} _ {i \\sim \\boldsymbol {\\pi} [ \\mathbf {x} ]} [ b _ {i} ].",
+                                    "image_path": "256e34f43b3112b9f3eb4fc9ab7c51ff48b08efba1403f4b8faac34a17d7fba2.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        230,
+                        504,
+                        284
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                230,
+                                504,
+                                284
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        230,
+                                        504,
+                                        284
+                                    ],
+                                    "type": "text",
+                                    "content": "However, this condition is quite strict. Loosely speaking, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        230,
+                                        504,
+                                        284
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{E}_{i\\sim \\pi [\\mathbf{x}]}[A_i]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        230,
+                                        504,
+                                        284
+                                    ],
+                                    "type": "text",
+                                    "content": " being invertible for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        230,
+                                        504,
+                                        284
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        230,
+                                        504,
+                                        284
+                                    ],
+                                    "type": "text",
+                                    "content": " is similar to saying that any convex combination of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        230,
+                                        504,
+                                        284
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{A_i\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        230,
+                                        504,
+                                        284
+                                    ],
+                                    "type": "text",
+                                    "content": " is invertible. For example, if we assume "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        230,
+                                        504,
+                                        284
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{A_i\\}_{i\\in \\mathcal{N}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        230,
+                                        504,
+                                        284
+                                    ],
+                                    "type": "text",
+                                    "content": " are negative definite and they all share the same eigenbasis "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        230,
+                                        504,
+                                        284
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\mathbf{u}_i\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        230,
+                                        504,
+                                        284
+                                    ],
+                                    "type": "text",
+                                    "content": ", e.g., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        230,
+                                        504,
+                                        284
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A_{i} = \\sum_{j = 1}^{D}\\lambda_{j}^{i}\\mathbf{u}_{i}\\mathbf{u}_{i}^{T}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        230,
+                                        504,
+                                        284
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        230,
+                                        504,
+                                        284
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\lambda_j^i < 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        230,
+                                        504,
+                                        284
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        230,
+                                        504,
+                                        284
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i\\in \\mathcal{N}, j\\in [D]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        230,
+                                        504,
+                                        284
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        230,
+                                        504,
+                                        284
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{E}_{i\\sim \\pi [\\mathbf{x}]}[A_i]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        230,
+                                        504,
+                                        284
+                                    ],
+                                    "type": "text",
+                                    "content": " is invertible."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        287,
+                        504,
+                        322
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                287,
+                                504,
+                                322
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        287,
+                                        504,
+                                        322
+                                    ],
+                                    "type": "text",
+                                    "content": "Another example for Assumption A3' is when "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        287,
+                                        504,
+                                        322
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H(\\pmb{\\theta}, i) = H(\\pmb{\\theta}, j)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        287,
+                                        504,
+                                        322
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        287,
+                                        504,
+                                        322
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i, j \\in \\mathcal{N}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        287,
+                                        504,
+                                        322
+                                    ],
+                                    "type": "text",
+                                    "content": ", which implies that each agent in the distributed learning has the same local dataset to collaboratively train the model. In this example, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        287,
+                                        504,
+                                        322
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho(\\mathbf{x}) = \\pmb{\\theta}^*"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        287,
+                                        504,
+                                        322
+                                    ],
+                                    "type": "text",
+                                    "content": " such that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        155,
+                        327,
+                        455,
+                        371
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                155,
+                                327,
+                                455,
+                                371
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        155,
+                                        327,
+                                        455,
+                                        371
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\mathbb {E} _ {i \\sim \\pi [ \\mathbf {x} ]} [ H (\\rho (\\mathbf {x}), i) ] = h (\\boldsymbol {\\theta} ^ {*}) = 0, \\\\ \\mathbb {E} _ {i \\sim \\pi [ \\mathbf {x} ]} [ \\nabla H (\\rho (\\mathbf {x}), i) ] + \\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I} = \\nabla h (\\boldsymbol {\\theta} ^ {*}) + \\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I} \\quad \\text {b e i n g H u r w i t z}. \\\\ \\end{array}",
+                                    "image_path": "c611517c59b166c7a0bd32a9dc94526f3088dd8bd174e6b1f456a6084cdcf593.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        105,
+                        384,
+                        332,
+                        396
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                384,
+                                332,
+                                396
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        384,
+                                        332,
+                                        396
+                                    ],
+                                    "type": "text",
+                                    "content": "D PROOF OF LEMMA 3.1 AND LEMMA 3.2"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        409,
+                        504,
+                        464
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                409,
+                                504,
+                                464
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        409,
+                                        504,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": "In this section, we demonstrate the almost sure convergence of both "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        409,
+                                        504,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        409,
+                                        504,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        409,
+                                        504,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        409,
+                                        504,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": " together. This proof naturally incorporates the almost certain convergence of the SRRW iteration in Lemma 3.1, since "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        409,
+                                        504,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        409,
+                                        504,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": " is independent of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        409,
+                                        504,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        409,
+                                        504,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": " (as indicated in (4)), allowing us to separate out its asymptotic results. The same reason applies to the CLT analysis of SRRW iterates and we refer the reader to Section E.1 for the CLT result of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        409,
+                                        504,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        409,
+                                        504,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": " in Lemma 3.1."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
+                        469,
+                        504,
+                        493
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                469,
+                                504,
+                                493
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        504,
+                                        493
+                                    ],
+                                    "type": "text",
+                                    "content": "We will use different techniques for different settings of step sizes in Assumption A2. Specifically, for step sizes "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        504,
+                                        493
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\gamma_{n} = (n + 1)^{-a},\\beta_{n} = (n + 1)^{-b}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        504,
+                                        493
+                                    ],
+                                    "type": "text",
+                                    "content": ", we consider the following scenarios:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        501,
+                        504,
+                        535
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                501,
+                                504,
+                                535
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        501,
+                                        504,
+                                        535
+                                    ],
+                                    "type": "text",
+                                    "content": "Scenario 1: We consider case(ii): "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        501,
+                                        504,
+                                        535
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "1/2 < a = b \\leq 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        501,
+                                        504,
+                                        535
+                                    ],
+                                    "type": "text",
+                                    "content": ", and will apply the almost sure convergence result of the single-timescale stochastic approximation in Theorem G.8 and verify all the conditions therein."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        104,
+                        538,
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        538,
+                                        504,
+                                        605
+                                    ],
+                                    "type": "text",
+                                    "content": "Scenario 2: We consider both case(i): "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        538,
+                                        504,
+                                        605
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "1/2 < a < b \\leq 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        538,
+                                        504,
+                                        605
+                                    ],
+                                    "type": "text",
+                                    "content": " and case (iii): "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        538,
+                                        504,
+                                        605
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "1/2 < b < a \\leq 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        538,
+                                        504,
+                                        605
+                                    ],
+                                    "type": "text",
+                                    "content": ". In these two cases, step sizes "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        538,
+                                        504,
+                                        605
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\gamma_{n}, \\beta_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        538,
+                                        504,
+                                        605
+                                    ],
+                                    "type": "text",
+                                    "content": " decrease at different rates, thereby putting iterates "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        538,
+                                        504,
+                                        605
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_{n}, \\pmb{\\theta}_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        538,
+                                        504,
+                                        605
+                                    ],
+                                    "type": "text",
+                                    "content": " on different timescales and resulting in a two-timescale structure. We will apply the existing almost sure convergence result of the two-timescale stochastic approximation with iteratedependent Markov chain in Yaji & Bhatnagar (2020, Theorem 4) where our SA-SRRW algorithm can be regarded as a special instance.[11]"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        105,
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+                        629
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "D.1 SCENARIO 1"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "In Scenario 1, we have "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_{n} = \\gamma_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        639,
+                                        333,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": ". First, we rewrite (4) as"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left[ \\begin{array}{l} \\boldsymbol {\\theta} _ {n + 1} \\\\ \\mathbf {x} _ {n + 1} \\end{array} \\right] = \\left[ \\begin{array}{l} \\boldsymbol {\\theta} _ {n} \\\\ \\mathbf {x} _ {n} \\end{array} \\right] + \\gamma_ {n + 1} \\left[ \\begin{array}{c} H \\left(\\boldsymbol {\\theta} _ {n}. X _ {n + 1}\\right) \\\\ \\boldsymbol {\\delta} _ {X _ {n + 1}} - \\mathbf {x} _ {n} \\end{array} \\right]. \\tag {19}",
+                                    "image_path": "3a60be1996fd9201222cde083f98864fe81d16d5d3c6e0daee7c00695a999f84.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                26,
+                                293,
+                                38
+                            ],
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "page_footnote",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "<sup>11</sup>However, Yaji & Bhatnagar (2020) paper only analysed the almost sure convergence. The central limit theorem analysis remains unknown in the literature for the two-timescale stochastic approximation with iteratedependent Markov chains. Thus, our CLT analysis in Section E for this two-timescale structure with iteratedependent Markov chain is still novel and recognized as our contribution."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "16"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                }
+            ],
+            "page_size": [
+                612,
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+            ],
+            "page_idx": 15
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "By augmentations, we define the variable "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        506,
+                                        172
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{z}_n \\triangleq \\left[ \\begin{array}{c} \\boldsymbol{\\theta}_n \\\\ \\mathbf{x}_n \\end{array} \\right] \\in \\mathbb{R}^{(N + D) \\times 1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        506,
+                                        172
+                                    ],
+                                    "type": "text",
+                                    "content": " and the function "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        506,
+                                        172
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G(\\mathbf{z}_n, i) \\triangleq \\left[ \\begin{array}{c} H(\\boldsymbol{\\theta}_n, i) \\\\ \\delta_i - \\mathbf{x}_n \\end{array} \\right] \\in \\mathbb{R}^{(N + d) \\times 1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        506,
+                                        172
+                                    ],
+                                    "type": "text",
+                                    "content": ". In addition, we define a new Markov chain "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        506,
+                                        172
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{Y_n\\}_{n \\geq 0}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        506,
+                                        172
+                                    ],
+                                    "type": "text",
+                                    "content": " in the same state space "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        506,
+                                        172
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{N}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        506,
+                                        172
+                                    ],
+                                    "type": "text",
+                                    "content": " as SRRW sequence "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        506,
+                                        172
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{X_n\\}_{n \\geq 0}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        506,
+                                        172
+                                    ],
+                                    "type": "text",
+                                    "content": ". With slight abuse of notation, the transition kernel of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        506,
+                                        172
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{Y_n\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        506,
+                                        172
+                                    ],
+                                    "type": "text",
+                                    "content": " is denoted by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        506,
+                                        172
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{K}'[\\mathbf{z}_n] \\equiv \\mathbf{K}[\\mathbf{x}_n]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        506,
+                                        172
+                                    ],
+                                    "type": "text",
+                                    "content": " and its stationary distribution "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        506,
+                                        172
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi'[\\mathbf{z}_n] \\equiv \\pi[\\mathbf{x}_n]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        506,
+                                        172
+                                    ],
+                                    "type": "text",
+                                    "content": ", where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        506,
+                                        172
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{K}[\\mathbf{x}_n]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        506,
+                                        172
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        506,
+                                        172
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi(\\mathbf{x}_n)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        506,
+                                        172
+                                    ],
+                                    "type": "text",
+                                    "content": " are the transition kernel and its corresponding stationary distribution of SRRW, with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        506,
+                                        172
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi[\\mathbf{x}]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        506,
+                                        172
+                                    ],
+                                    "type": "text",
+                                    "content": " of the form"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
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+                                    "bbox": [
+                                        220,
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+                                        196
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\pi_ {i} [ \\mathbf {x} ] \\propto \\sum_ {j \\in \\mathcal {N}} \\mu_ {i} P _ {i j} \\left(x _ {i} / \\mu_ {i}\\right) ^ {- \\alpha} \\left(x _ {j} / \\mu_ {j}\\right) ^ {- \\alpha}. \\tag {20}",
+                                    "image_path": "9ee85facf3bd183218e8a2043929aa1993d2f9415ea7e92a694f360d00c86249.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
+                                        104,
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+                                        220
+                                    ],
+                                    "type": "text",
+                                    "content": "Recall that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        220
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\mu}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        220
+                                    ],
+                                    "type": "text",
+                                    "content": " is the fixed point, i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
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+                                        220
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi[\\pmb{\\mu}] = \\pmb{\\mu}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        220
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        220
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{P}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        220
+                                    ],
+                                    "type": "text",
+                                    "content": " is the base Markov chain inside SRRW (see (3)). Then, the mean field"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "g (\\mathbf {z}) = \\mathbb {E} _ {Y \\sim \\boldsymbol {\\pi^ {\\prime}} (\\mathbf {z})} [ G (\\mathbf {z}, Y) ] = \\left[ \\begin{array}{c} \\sum_ {i \\in \\mathcal {N}} \\pi_ {i} [ \\mathbf {x} ] H (\\boldsymbol {\\theta}, i) \\\\ \\boldsymbol {\\pi} [ \\mathbf {x} ] - \\mathbf {x} \\end{array} \\right],",
+                                    "image_path": "939440a4a9336620ba8e65d7dee6d4b60b03aaed51949a9116a9cff3fe7b35cc.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        249,
+                                        504,
+                                        270
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{z}^{*} = (\\pmb{\\theta}^{*},\\pmb{\\mu})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        249,
+                                        504,
+                                        270
+                                    ],
+                                    "type": "text",
+                                    "content": " for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        249,
+                                        504,
+                                        270
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\theta}^{*}\\in \\Theta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        249,
+                                        504,
+                                        270
+                                    ],
+                                    "type": "text",
+                                    "content": " in Assumption A3 is the root of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        249,
+                                        504,
+                                        270
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g(\\mathbf{z})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        249,
+                                        504,
+                                        270
+                                    ],
+                                    "type": "text",
+                                    "content": ", i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        249,
+                                        504,
+                                        270
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g(\\mathbf{z}^{*}) = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        249,
+                                        504,
+                                        270
+                                    ],
+                                    "type": "text",
+                                    "content": ". The augmented iteration (19) becomes"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {z} _ {n + 1} = \\mathbf {z} _ {n} + \\gamma_ {n + 1} G \\left(\\mathbf {z} _ {n}, Y _ {n + 1}\\right) \\tag {21}",
+                                    "image_path": "150aff2cab0bbc821bae69b68d23fa818729dba398623cc1a0c5373f2b5fe593.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        283,
+                        504,
+                        350
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                283,
+                                504,
+                                350
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        350
+                                    ],
+                                    "type": "text",
+                                    "content": "with the goal of solving "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        350
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g(\\mathbf{z}) = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        350
+                                    ],
+                                    "type": "text",
+                                    "content": ". Therefore, we can treat (21) as an SA algorithm driven by a Markov chain "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        350
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{Y_{n}\\}_{n\\geq 0}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        350
+                                    ],
+                                    "type": "text",
+                                    "content": " with its kernel "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        350
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{K}'[\\mathbf{z}]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        350
+                                    ],
+                                    "type": "text",
+                                    "content": " and stationary distribution "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        350
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi^{\\prime}[\\mathbf{z}]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        350
+                                    ],
+                                    "type": "text",
+                                    "content": ", which has been widely studied in the literature (e.g., Delyon (2000); Benveniste et al. (2012); Fort (2015); Li et al. (2023)). In what follows, we demonstrate that for any initial point "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        350
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{z}_0 = (\\pmb {\\theta}_0,\\mathbf{x}_0)\\in \\mathbb{R}^D\\times \\mathrm{Int}(\\Sigma)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        350
+                                    ],
+                                    "type": "text",
+                                    "content": ", the SRRW iteration "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        350
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\mathbf{x}_n\\}_{n\\geq 0}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        350
+                                    ],
+                                    "type": "text",
+                                    "content": " will almost surely converge to the target distribution "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        350
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\mu}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        350
+                                    ],
+                                    "type": "text",
+                                    "content": ", and the SA iteration "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        350
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\pmb {\\theta}_n\\}_{n\\geq 0}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        350
+                                    ],
+                                    "type": "text",
+                                    "content": " will almost surely converge to the set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        350
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Theta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        350
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        104,
+                        354,
+                        505,
+                        453
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                354,
+                                505,
+                                453
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        505,
+                                        453
+                                    ],
+                                    "type": "text",
+                                    "content": "Now we verify conditions C1 - C4 in Theorem G.8. Our assumption A4 is equivalent to condition C1 and assumption A2 corresponds to condition C2. For condition C3, we set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        505,
+                                        453
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\nabla w(\\mathbf{z}) \\equiv -g(\\mathbf{z})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        505,
+                                        453
+                                    ],
+                                    "type": "text",
+                                    "content": ", and the set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        505,
+                                        453
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S \\equiv \\{\\mathbf{z}^* | \\boldsymbol{\\theta}^* \\in \\Theta, \\mathbf{x}^* = \\boldsymbol{\\mu}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        505,
+                                        453
+                                    ],
+                                    "type": "text",
+                                    "content": ", including disjoint points. For condition C4, since "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        505,
+                                        453
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{K}'[\\mathbf{z}]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        505,
+                                        453
+                                    ],
+                                    "type": "text",
+                                    "content": ", or equivalently "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        505,
+                                        453
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{K}[\\mathbf{x}]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        505,
+                                        453
+                                    ],
+                                    "type": "text",
+                                    "content": ", is ergodic and time-reversible for a given "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        505,
+                                        453
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{z}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        505,
+                                        453
+                                    ],
+                                    "type": "text",
+                                    "content": ", as shown in the SRRW work Doshi et al. (2023), it automatically ensures a solution to the Poisson equation, which has been well discussed in Chen et al. (2020a, Section 2) and Benveniste et al. (2012); Meyn (2022). To show (97) and (98) in condition C4, for each given "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        505,
+                                        453
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{z}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        505,
+                                        453
+                                    ],
+                                    "type": "text",
+                                    "content": " and any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        505,
+                                        453
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i \\in \\mathcal{N}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        505,
+                                        453
+                                    ],
+                                    "type": "text",
+                                    "content": ", we need to give the explicit solution "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        505,
+                                        453
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m_{\\mathbf{z}}(i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        505,
+                                        453
+                                    ],
+                                    "type": "text",
+                                    "content": " to the Poisson equation "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        505,
+                                        453
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m_{\\mathbf{z}}(i) - (\\mathbf{K}_{\\mathbf{z}}' m_{\\mathbf{z}})(i) = G(\\mathbf{z}, i) - g(\\mathbf{z})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        505,
+                                        453
+                                    ],
+                                    "type": "text",
+                                    "content": " in (96). The notation "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        505,
+                                        453
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\mathbf{K}_{\\mathbf{z}}' m_{\\mathbf{z}})(i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        505,
+                                        453
+                                    ],
+                                    "type": "text",
+                                    "content": " is defined as follows."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        230,
+                        451,
+                        379,
+                        476
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                230,
+                                451,
+                                379,
+                                476
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        230,
+                                        451,
+                                        379,
+                                        476
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left(\\mathbf {K} _ {\\mathbf {z}} ^ {\\prime} m _ {\\mathbf {z}}\\right) (i) = \\sum_ {j \\in \\mathcal {N}} \\mathbf {K} _ {\\mathbf {z}} ^ {\\prime} (i, j) m (\\mathbf {z}, j).",
+                                    "image_path": "30d312714068d1591fdaa22ab1d46bb74cc3c7f9ee328e1c717f3628af4d31d2.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
+                        484,
+                        504,
+                        508
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                484,
+                                504,
+                                508
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        484,
+                                        504,
+                                        508
+                                    ],
+                                    "type": "text",
+                                    "content": "Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        484,
+                                        504,
+                                        508
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{G}(\\mathbf{z})\\triangleq [G(\\mathbf{z},1),\\dots ,G(\\mathbf{z},N)]^T\\in \\mathbb{R}^{N\\times D}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        484,
+                                        504,
+                                        508
+                                    ],
+                                    "type": "text",
+                                    "content": ". We use "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        484,
+                                        504,
+                                        508
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "[\\mathbf{A}]_{:,i}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        484,
+                                        504,
+                                        508
+                                    ],
+                                    "type": "text",
+                                    "content": " to denote the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        484,
+                                        504,
+                                        508
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        484,
+                                        504,
+                                        508
+                                    ],
+                                    "type": "text",
+                                    "content": "-th column of matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        484,
+                                        504,
+                                        508
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{A}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        484,
+                                        504,
+                                        508
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, we let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        484,
+                                        504,
+                                        508
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m_{\\mathbf{z}}(i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        484,
+                                        504,
+                                        508
+                                    ],
+                                    "type": "text",
+                                    "content": " such that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        125,
+                        510,
+                        504,
+                        541
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                125,
+                                510,
+                                504,
+                                541
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        125,
+                                        510,
+                                        504,
+                                        541
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "m _ {\\mathbf {z}} (i) = \\sum_ {k = 0} ^ {\\infty} \\left(\\left[ \\mathbf {G} (\\mathbf {z}) \\left(\\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] ^ {k}\\right) ^ {T} \\right] _ {[:, i ]} - g (\\mathbf {z})\\right) = \\sum_ {k = 0} ^ {\\infty} \\left[ \\mathbf {G} (\\mathbf {z}) \\left(\\left(\\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] ^ {k}\\right) ^ {T} - \\boldsymbol {\\pi} ^ {\\prime} [ \\mathbf {z} ] \\mathbf {1} ^ {T}\\right) \\right] _ {[:, i ]}. \\tag {22}",
+                                    "image_path": "94cb18a37e44c481b6e6a1dc498347b4ac570b4c0b54a4953c0f5b12b0da6181.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        105,
+                        543,
+                        155,
+                        553
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                543,
+                                155,
+                                553
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        543,
+                                        155,
+                                        553
+                                    ],
+                                    "type": "text",
+                                    "content": "In addition,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        205,
+                        552,
+                        504,
+                        583
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                205,
+                                552,
+                                504,
+                                583
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        205,
+                                        552,
+                                        504,
+                                        583
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left(\\mathbf {K} _ {\\mathbf {z}} ^ {\\prime} m _ {\\mathbf {z}}\\right) (i) = \\sum_ {k = 1} ^ {\\infty} \\left[ \\mathbf {G} (\\mathbf {z}) \\left(\\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] ^ {T} - \\boldsymbol {\\pi} ^ {\\prime} [ \\mathbf {z} ] \\mathbf {1} ^ {T}\\right) \\right] _ {[ i, i ]}. \\tag {23}",
+                                    "image_path": "25f25f50c4b829d66a563bc5f5dd58a0945430be2fe1583d7b6aff301ff8cd44.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        104,
+                        584,
+                        504,
+                        618
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                584,
+                                504,
+                                618
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        584,
+                                        504,
+                                        618
+                                    ],
+                                    "type": "text",
+                                    "content": "We can check that the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        584,
+                                        504,
+                                        618
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m_{\\mathbf{z}}(i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        584,
+                                        504,
+                                        618
+                                    ],
+                                    "type": "text",
+                                    "content": " form in (22) is indeed the solution of the above Poisson equation. Now, by induction, we get "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        584,
+                                        504,
+                                        618
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{K}'[\\mathbf{z}]^k - \\mathbf{1}\\pi'[\\mathbf{z}]^T = (\\mathbf{K}'[\\mathbf{z}] - \\mathbf{1}\\pi'[\\mathbf{z}]^T)^k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        584,
+                                        504,
+                                        618
+                                    ],
+                                    "type": "text",
+                                    "content": " for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        584,
+                                        504,
+                                        618
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k \\geq 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        584,
+                                        504,
+                                        618
+                                    ],
+                                    "type": "text",
+                                    "content": " and for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        584,
+                                        504,
+                                        618
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        584,
+                                        504,
+                                        618
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        584,
+                                        504,
+                                        618
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{K}'[\\mathbf{z}]^0 - \\mathbf{1}\\pi'[\\mathbf{z}]^T = (\\mathbf{K}'[\\mathbf{z}] - \\mathbf{1}\\pi'[\\mathbf{z}]^T)^0 - \\mathbf{1}\\pi'[\\mathbf{z}]^T"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        584,
+                                        504,
+                                        618
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        183,
+                        620,
+                        504,
+                        734
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                183,
+                                620,
+                                504,
+                                734
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        183,
+                                        620,
+                                        504,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} m _ {\\mathbf {z}} (i) = \\sum_ {k = 0} ^ {\\infty} [ \\mathbf {G} (\\mathbf {z}) (\\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] ^ {T} - \\boldsymbol {\\pi} ^ {\\prime} [ \\mathbf {z} ] \\mathbf {1} ^ {T}) ^ {k} ] _ {[; i ]} - g (\\mathbf {z}) \\\\ = \\left[ \\mathbf {G} (\\mathbf {z}) \\sum_ {k = 0} ^ {\\infty} \\left(\\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] ^ {T} - \\boldsymbol {\\pi} ^ {\\prime} [ \\mathbf {z} ] \\mathbf {1} ^ {T}\\right) ^ {k} \\right] _ {[;, i ]} - g (\\mathbf {z}) \\tag {24} \\\\ = \\left[ \\mathbf {G} (\\mathbf {z}) \\left(\\mathbf {I} - \\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] ^ {T} + \\boldsymbol {\\pi} ^ {\\prime} [ \\mathbf {z} ] \\mathbf {1} ^ {T}\\right) ^ {- 1} \\right] _ {[,: i ]} - g (\\mathbf {z}) \\\\ = \\sum_ {j \\in \\mathcal {N}} \\left(\\mathbf {I} - \\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] + \\mathbf {1} \\boldsymbol {\\pi} ^ {\\prime} [ \\mathbf {z} ] ^ {T}\\right) ^ {- 1} (i, j) G (\\mathbf {z}, j) - g (\\mathbf {z}). \\\\ \\end{array}",
+                                    "image_path": "116fb196673c4043653b777c4388eaacaf75be108f6ee4eeb1b14d46bb0ea8c9.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
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+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        300,
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+                    ],
+                    "type": "page_number",
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+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "17"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 16
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        81,
+                        504,
+                        138
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                81,
+                                504,
+                                138
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        138
+                                    ],
+                                    "type": "text",
+                                    "content": "Here, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        138
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\mathbf{I} - \\mathbf{K}'[\\mathbf{z}] + 1\\pi'[\\mathbf{z}]^T)^{-1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        138
+                                    ],
+                                    "type": "text",
+                                    "content": " is well defined because "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        138
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{K}'[\\mathbf{z}]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        138
+                                    ],
+                                    "type": "text",
+                                    "content": " is ergodic and time-reversible for any given "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        138
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{z}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        138
+                                    ],
+                                    "type": "text",
+                                    "content": " (proved in Doshi et al. (2023, Appendix A)). Now that both functions "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        138
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H(\\theta, i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        138
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        138
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\delta_i - \\mathbf{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        138
+                                    ],
+                                    "type": "text",
+                                    "content": " are bounded for each compact subset of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        138
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{R}^D \\times \\Sigma"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        138
+                                    ],
+                                    "type": "text",
+                                    "content": " by our assumption A1, function "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        138
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G(\\mathbf{z}, i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        138
+                                    ],
+                                    "type": "text",
+                                    "content": " is also bounded within the compact subset of its domain. Thus, function "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        138
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m_{\\mathbf{z}}(i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        138
+                                    ],
+                                    "type": "text",
+                                    "content": " is bounded, and (97) is verified. Moreover, for a fixed "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        138
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i \\in \\mathcal{N}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        138
+                                    ],
+                                    "type": "text",
+                                    "content": ","
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        113,
+                        144,
+                        496,
+                        170
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                113,
+                                144,
+                                496,
+                                170
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        113,
+                                        144,
+                                        496,
+                                        170
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\sum_ {j \\in \\mathcal {N}} (\\mathbf {I} - \\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] + \\mathbf {1} \\boldsymbol {\\pi} ^ {\\prime} [ \\mathbf {z} ] ^ {T}) ^ {- 1} (i, j) \\boldsymbol {\\delta} _ {j} = (\\mathbf {I} - \\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] + \\mathbf {1} \\boldsymbol {\\pi} ^ {\\prime} [ \\mathbf {z} ] ^ {T}) _ {[,: i ]} ^ {- 1} = (\\mathbf {I} - \\mathbf {K} [ \\mathbf {x} ] + \\mathbf {1} \\boldsymbol {\\pi} [ \\mathbf {x} ] ^ {T}) _ {[,: i ]} ^ {- 1}",
+                                    "image_path": "422a8494aa630f52ed5bcb8afe287036dbce2bb4b3ff5a6d66abdf2fddb842e3.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        176,
+                        504,
+                        199
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                176,
+                                504,
+                                199
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        176,
+                                        504,
+                                        199
+                                    ],
+                                    "type": "text",
+                                    "content": "and this vector-valued function is continuous in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        176,
+                                        504,
+                                        199
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        176,
+                                        504,
+                                        199
+                                    ],
+                                    "type": "text",
+                                    "content": " because "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        176,
+                                        504,
+                                        199
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{K}[\\mathbf{x}], \\pi[\\mathbf{x}]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        176,
+                                        504,
+                                        199
+                                    ],
+                                    "type": "text",
+                                    "content": " are continuous. We then rewrite (24) as"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        135,
+                        205,
+                        474,
+                        239
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                135,
+                                205,
+                                474,
+                                239
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        135,
+                                        205,
+                                        474,
+                                        239
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "m _ {\\mathbf {z}} (i) = \\left[ \\begin{array}{c} \\sum_ {j \\in \\mathcal {N}} (\\mathbf {I} - \\mathbf {K} [ \\mathbf {x} ] + \\mathbf {1} \\pi [ \\mathbf {x} ] ^ {T}) ^ {- 1} (i, j) H (\\mathbf {x}, j) \\\\ (\\mathbf {I} - \\mathbf {K} [ \\mathbf {x} ] ^ {T} + \\pi [ \\mathbf {x} ] \\mathbf {1} ^ {T}) _ {[,: i ]} ^ {- 1} \\end{array} \\right] - \\left[ \\begin{array}{c} \\sum_ {i \\in \\mathcal {N}} \\pi_ {i} [ \\mathbf {x} ] H (\\pmb {\\theta}, i) \\\\ \\pmb {\\pi} [ \\mathbf {x} ] - \\mathbf {x} \\end{array} \\right].",
+                                    "image_path": "40ce12f126086ae1102dec2baefde8290441bbd1a29ce9c30b5ae6f7f532e7db.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        243,
+                        505,
+                        300
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                243,
+                                505,
+                                300
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        505,
+                                        300
+                                    ],
+                                    "type": "text",
+                                    "content": "With continuous functions "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        505,
+                                        300
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H(\\theta, i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        505,
+                                        300
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        505,
+                                        300
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{K}[\\mathbf{x}]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        505,
+                                        300
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        505,
+                                        300
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi[\\mathbf{x}]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        505,
+                                        300
+                                    ],
+                                    "type": "text",
+                                    "content": ", we have "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        505,
+                                        300
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m_{\\mathbf{z}}(i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        505,
+                                        300
+                                    ],
+                                    "type": "text",
+                                    "content": " continuous with respect to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        505,
+                                        300
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{z}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        505,
+                                        300
+                                    ],
+                                    "type": "text",
+                                    "content": ", so does "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        505,
+                                        300
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\mathbf{K}_{\\mathbf{z}}^{\\prime} m_{\\mathbf{z}})(i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        505,
+                                        300
+                                    ],
+                                    "type": "text",
+                                    "content": ". This implies that functions "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        505,
+                                        300
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m_{\\mathbf{z}}(i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        505,
+                                        300
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        505,
+                                        300
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\mathbf{K}_{\\mathbf{z}}^{\\prime} m_{\\mathbf{z}})(i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        505,
+                                        300
+                                    ],
+                                    "type": "text",
+                                    "content": " are locally Lipschitz, which satisfies (98) with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        505,
+                                        300
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\phi_{\\mathcal{C}}(x) = C_{\\mathcal{C}} x"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        505,
+                                        300
+                                    ],
+                                    "type": "text",
+                                    "content": " for some constant "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        505,
+                                        300
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_{\\mathcal{C}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        505,
+                                        300
+                                    ],
+                                    "type": "text",
+                                    "content": " that depends on the compact set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        505,
+                                        300
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{C}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        505,
+                                        300
+                                    ],
+                                    "type": "text",
+                                    "content": ". Therefore, condition C4 is checked, and we can apply Theorem G.8 to show the almost convergence result of (19), i.e., almost surely,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        195,
+                        305,
+                        414,
+                        324
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                195,
+                                305,
+                                414,
+                                324
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        195,
+                                        305,
+                                        414,
+                                        324
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\lim  _ {n \\to \\infty} \\mathbf {x} _ {n} = \\boldsymbol {\\mu}, \\quad \\text {a n d} \\quad \\lim  _ {n \\to \\infty} \\sup  _ {\\boldsymbol {\\theta} ^ {*} \\in \\Theta} \\inf  _ {\\boldsymbol {\\theta}} \\| \\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*} \\| = 0.",
+                                    "image_path": "54d9c3147d3b616fb428ad0a1f8c8c7d3f4801b71aa6808a35a6ce8d5711ad75.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        330,
+                        506,
+                        353
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                330,
+                                506,
+                                353
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        330,
+                                        506,
+                                        353
+                                    ],
+                                    "type": "text",
+                                    "content": "Therefore, the almost sure convergence of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        330,
+                                        506,
+                                        353
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        330,
+                                        506,
+                                        353
+                                    ],
+                                    "type": "text",
+                                    "content": " in Lemma 3.1 is also proved. This finishes the proof in Scenario 1."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        105,
+                        366,
+                        187,
+                        377
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                366,
+                                187,
+                                377
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        366,
+                                        187,
+                                        377
+                                    ],
+                                    "type": "text",
+                                    "content": "D.2 SCENARIO 2"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        387,
+                        504,
+                        422
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                387,
+                                504,
+                                422
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        387,
+                                        504,
+                                        422
+                                    ],
+                                    "type": "text",
+                                    "content": "Now in this subsection, we consider the steps sizes "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        387,
+                                        504,
+                                        422
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\gamma_{n},\\beta_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        387,
+                                        504,
+                                        422
+                                    ],
+                                    "type": "text",
+                                    "content": " with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        387,
+                                        504,
+                                        422
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "1 / 2 < a < b\\leq 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        387,
+                                        504,
+                                        422
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        387,
+                                        504,
+                                        422
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "1 / 2 < b < a\\leq 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        387,
+                                        504,
+                                        422
+                                    ],
+                                    "type": "text",
+                                    "content": ". We will frequently use assumptions (B1) - (B5) in Section G.3 and Theorem G.10 to prove the almost sure convergence."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        105,
+                        434,
+                        255,
+                        445
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                434,
+                                255,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        434,
+                                        255,
+                                        445
+                                    ],
+                                    "type": "text",
+                                    "content": "D.2.1 CASE (I): "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        434,
+                                        255,
+                                        445
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "1/2 < a < b \\leq 1"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        453,
+                        504,
+                        485
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                453,
+                                504,
+                                485
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        453,
+                                        504,
+                                        485
+                                    ],
+                                    "type": "text",
+                                    "content": "In case (i), "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        453,
+                                        504,
+                                        485
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        453,
+                                        504,
+                                        485
+                                    ],
+                                    "type": "text",
+                                    "content": " is on the slow timescale and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        453,
+                                        504,
+                                        485
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        453,
+                                        504,
+                                        485
+                                    ],
+                                    "type": "text",
+                                    "content": " is on the fast timescale because iteration "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        453,
+                                        504,
+                                        485
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        453,
+                                        504,
+                                        485
+                                    ],
+                                    "type": "text",
+                                    "content": " has smaller step size than "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        453,
+                                        504,
+                                        485
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        453,
+                                        504,
+                                        485
+                                    ],
+                                    "type": "text",
+                                    "content": ", making "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        453,
+                                        504,
+                                        485
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        453,
+                                        504,
+                                        485
+                                    ],
+                                    "type": "text",
+                                    "content": " converge slower than "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        453,
+                                        504,
+                                        485
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        453,
+                                        504,
+                                        485
+                                    ],
+                                    "type": "text",
+                                    "content": ". Here, we consider the two-timescale SA of the form:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        236,
+                        486,
+                        373,
+                        499
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                236,
+                                486,
+                                373,
+                                499
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        236,
+                                        486,
+                                        373,
+                                        499
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} + \\beta_ {n + 1} H (\\boldsymbol {\\theta}, X _ {n + 1}),",
+                                    "image_path": "fc514dedcb10ce61f9617ef7e921fa9f5cc4289d594652408ccb1ef663d58481.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        236,
+                        502,
+                        373,
+                        517
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                236,
+                                502,
+                                373,
+                                517
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        236,
+                                        502,
+                                        373,
+                                        517
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {x} _ {n + 1} = \\mathbf {x} _ {n} + \\gamma_ {n + 1} (\\boldsymbol {\\delta} _ {X _ {n + 1}} - \\mathbf {x}).",
+                                    "image_path": "51a8b5853da9f78d8b8a588079665c6dc48ba25c9a132cade258877722afe433.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        104,
+                        525,
+                        354,
+                        537
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                525,
+                                354,
+                                537
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        354,
+                                        537
+                                    ],
+                                    "type": "text",
+                                    "content": "Now, we verify assumptions (B1) - (B5) listed in Section G.3."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        132,
+                        546,
+                        504,
+                        622
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 18,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                132,
+                                546,
+                                432,
+                                558
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        132,
+                                        546,
+                                        432,
+                                        558
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                132,
+                                                546,
+                                                432,
+                                                558
+                                            ],
+                                            "type": "text",
+                                            "content": "- Assumptions (B1) and (B5) are satisfied by our assumptions A2 and A4."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 15
+                        },
+                        {
+                            "bbox": [
+                                132,
+                                561,
+                                504,
+                                594
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        132,
+                                        561,
+                                        504,
+                                        594
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                132,
+                                                561,
+                                                504,
+                                                594
+                                            ],
+                                            "type": "text",
+                                            "content": "- Our assumption A3 shows that the function "
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                561,
+                                                504,
+                                                594
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "H(\\pmb{\\theta}, X)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                561,
+                                                504,
+                                                594
+                                            ],
+                                            "type": "text",
+                                            "content": " is continuous and differentiable w.r.t "
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                561,
+                                                504,
+                                                594
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pmb{\\theta}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                561,
+                                                504,
+                                                594
+                                            ],
+                                            "type": "text",
+                                            "content": " and grows linearly with "
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                561,
+                                                504,
+                                                594
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\| \\pmb{\\theta} \\|"
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                561,
+                                                504,
+                                                594
+                                            ],
+                                            "type": "text",
+                                            "content": ". In addition, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                561,
+                                                504,
+                                                594
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\delta_X - \\mathbf{x}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                561,
+                                                504,
+                                                594
+                                            ],
+                                            "type": "text",
+                                            "content": " also satisfies this property. Therefore, (B2) is satisfied."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 16
+                        },
+                        {
+                            "bbox": [
+                                132,
+                                598,
+                                504,
+                                622
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        132,
+                                        598,
+                                        504,
+                                        622
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                132,
+                                                598,
+                                                504,
+                                                622
+                                            ],
+                                            "type": "text",
+                                            "content": "- Now that the function "
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                598,
+                                                504,
+                                                622
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pi[\\mathbf{x}] - \\mathbf{x}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                598,
+                                                504,
+                                                622
+                                            ],
+                                            "type": "text",
+                                            "content": " is independent of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                598,
+                                                504,
+                                                622
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pmb{\\theta}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                598,
+                                                504,
+                                                622
+                                            ],
+                                            "type": "text",
+                                            "content": ", we can set "
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                598,
+                                                504,
+                                                622
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho(\\pmb{\\theta}) = \\pmb{\\mu}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                598,
+                                                504,
+                                                622
+                                            ],
+                                            "type": "text",
+                                            "content": " for any "
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                598,
+                                                504,
+                                                622
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pmb{\\theta} \\in \\mathbb{R}^D"
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                598,
+                                                504,
+                                                622
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                598,
+                                                504,
+                                                622
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pi[\\pmb{\\mu}] - \\pmb{\\mu} = 0"
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                598,
+                                                504,
+                                                622
+                                            ],
+                                            "type": "text",
+                                            "content": " from Doshi et al. (2023, Proposition 3.1), and"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 17
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        219,
+                        628,
+                        427,
+                        643
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                219,
+                                628,
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+                                643
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        219,
+                                        628,
+                                        427,
+                                        643
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\nabla_ {\\mathbf {x}} (\\boldsymbol {\\pi} (\\mathbf {x}) - \\mathbf {x}) | _ {\\mathbf {x} = \\boldsymbol {\\mu}} = 2 \\alpha \\mathbf {u} \\mathbf {1} ^ {T} - \\alpha \\mathbf {P} ^ {T} - (\\alpha + 1) \\mathbf {I}",
+                                    "image_path": "e4b1d7da2e4689283447ae8c96ef3a8f325c6a2241067ffc771f32ab2377ec24.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                },
+                {
+                    "bbox": [
+                        139,
+                        647,
+                        504,
+                        706
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                139,
+                                647,
+                                504,
+                                706
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        139,
+                                        647,
+                                        504,
+                                        706
+                                    ],
+                                    "type": "text",
+                                    "content": "from Doshi et al. (2023, Lemma 3.4), which is Hurwitz. Furthermore, "
+                                },
+                                {
+                                    "bbox": [
+                                        139,
+                                        647,
+                                        504,
+                                        706
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho(\\pmb{\\theta}) = \\pmb{\\mu}"
+                                },
+                                {
+                                    "bbox": [
+                                        139,
+                                        647,
+                                        504,
+                                        706
+                                    ],
+                                    "type": "text",
+                                    "content": " inherently satisfies the condition "
+                                },
+                                {
+                                    "bbox": [
+                                        139,
+                                        647,
+                                        504,
+                                        706
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\|\\rho(\\pmb{\\theta})\\| \\leq L_2(1 + \\|\\pmb{\\theta}\\|)"
+                                },
+                                {
+                                    "bbox": [
+                                        139,
+                                        647,
+                                        504,
+                                        706
+                                    ],
+                                    "type": "text",
+                                    "content": " for any "
+                                },
+                                {
+                                    "bbox": [
+                                        139,
+                                        647,
+                                        504,
+                                        706
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "L_2 \\geq \\|\\pmb{\\mu}\\|"
+                                },
+                                {
+                                    "bbox": [
+                                        139,
+                                        647,
+                                        504,
+                                        706
+                                    ],
+                                    "type": "text",
+                                    "content": ". Thus, conditions (i) - (iii) in (B3) are satisfied. Additionally, "
+                                },
+                                {
+                                    "bbox": [
+                                        139,
+                                        647,
+                                        504,
+                                        706
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sum_{i \\in \\mathcal{N}} \\pi_i[\\rho(\\pmb{\\theta})] H(\\pmb{\\theta}, i) = \\sum_{i \\in \\mathcal{N}} \\pi_i[\\mathbf{x}] = \\mathbf{h}(\\pmb{\\theta})"
+                                },
+                                {
+                                    "bbox": [
+                                        139,
+                                        647,
+                                        504,
+                                        706
+                                    ],
+                                    "type": "text",
+                                    "content": " such that for "
+                                },
+                                {
+                                    "bbox": [
+                                        139,
+                                        647,
+                                        504,
+                                        706
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\theta}^* \\in \\Theta"
+                                },
+                                {
+                                    "bbox": [
+                                        139,
+                                        647,
+                                        504,
+                                        706
+                                    ],
+                                    "type": "text",
+                                    "content": " defined in assumption A3, "
+                                },
+                                {
+                                    "bbox": [
+                                        139,
+                                        647,
+                                        504,
+                                        706
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sum_{i \\in \\mathcal{N}} \\pi_i[\\rho(\\pmb{\\theta}^*)] H(\\pmb{\\theta}^*, i) = \\mathbf{h}(\\pmb{\\theta}^*) = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        139,
+                                        647,
+                                        504,
+                                        706
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        139,
+                                        647,
+                                        504,
+                                        706
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\nabla_\\theta \\mathbf{h}(\\pmb{\\theta}^*)"
+                                },
+                                {
+                                    "bbox": [
+                                        139,
+                                        647,
+                                        504,
+                                        706
+                                    ],
+                                    "type": "text",
+                                    "content": " is Hurwitz. Therefore, (B3) is checked."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                },
+                {
+                    "bbox": [
+                        132,
+                        709,
+                        504,
+                        733
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                132,
+                                709,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        132,
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+                                    ],
+                                    "type": "text",
+                                    "content": "- Assumption (B4) is verified by the nature of SRRW, i.e., its transition kernel "
+                                },
+                                {
+                                    "bbox": [
+                                        132,
+                                        709,
+                                        504,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{K}[\\mathbf{x}]"
+                                },
+                                {
+                                    "bbox": [
+                                        132,
+                                        709,
+                                        504,
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+                                    ],
+                                    "type": "text",
+                                    "content": " and the corresponding stationary distribution "
+                                },
+                                {
+                                    "bbox": [
+                                        132,
+                                        709,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi[\\mathbf{x}]"
+                                },
+                                {
+                                    "bbox": [
+                                        132,
+                                        709,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " with "
+                                },
+                                {
+                                    "bbox": [
+                                        132,
+                                        709,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi[\\boldsymbol{\\mu}] = \\boldsymbol{\\mu}"
+                                },
+                                {
+                                    "bbox": [
+                                        132,
+                                        709,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 21
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
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+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
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+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
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+                                750,
+                                311,
+                                761
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        300,
+                                        750,
+                                        311,
+                                        761
+                                    ],
+                                    "type": "text",
+                                    "content": "18"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 22
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 17
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        82,
+                        504,
+                        106
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                82,
+                                504,
+                                106
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": "Consequently, assumptions (B1) - (B5) are satisfied by our assumeproins A1 - A4 and by Theorem G.10, we have "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\lim_{n\\to \\infty}\\mathbf{x}_n = \\pmb{\\mu}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta_{n}\\rightarrow \\Theta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": " almost surely."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        110,
+                        506,
+                        212
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                110,
+                                506,
+                                212
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        506,
+                                        212
+                                    ],
+                                    "type": "text",
+                                    "content": "Next, we consider "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        506,
+                                        212
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "1/2 < b < a \\leq 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        506,
+                                        212
+                                    ],
+                                    "type": "text",
+                                    "content": ". As discussed before, (B1), (B2), (B4) and (B5) are satisfied by our assumptions A1 - A4 and the properties of SRRW. The only difference for this step size setting, compared to the previous one "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        506,
+                                        212
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "1/2 < a < b \\leq 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        506,
+                                        212
+                                    ],
+                                    "type": "text",
+                                    "content": ", is that the roles of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        506,
+                                        212
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta_{n}, \\mathbf{x}_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        506,
+                                        212
+                                    ],
+                                    "type": "text",
+                                    "content": " are now flipped, that is, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        506,
+                                        212
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        506,
+                                        212
+                                    ],
+                                    "type": "text",
+                                    "content": " is now on the fast timescale while "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        506,
+                                        212
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        506,
+                                        212
+                                    ],
+                                    "type": "text",
+                                    "content": " is on the slow timescale. By a much stronger Assumption A3', for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        506,
+                                        212
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x} \\in \\mathrm{Int}(\\Sigma)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        506,
+                                        212
+                                    ],
+                                    "type": "text",
+                                    "content": ", (i) "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        506,
+                                        212
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{E}_{X \\sim \\pi[\\mathbf{x}]}[H(\\rho(\\mathbf{x}), X)] = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        506,
+                                        212
+                                    ],
+                                    "type": "text",
+                                    "content": "; (ii) "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        506,
+                                        212
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{E}_{X \\sim \\pi[\\mathbf{x}]}[\\nabla H(\\rho(\\mathbf{x}), X)]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        506,
+                                        212
+                                    ],
+                                    "type": "text",
+                                    "content": " is Hurwitz; (iii) "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        506,
+                                        212
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\|\\rho(\\mathbf{x})\\| \\leq L_{2}(1 + \\| \\mathbf{x}\\|)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        506,
+                                        212
+                                    ],
+                                    "type": "text",
+                                    "content": ". Hence, conditions (i) - (iii) in (B3) are satisfied. Moreover, we have "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        506,
+                                        212
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi[\\pmb{\\mu}] - \\pmb{\\mu} = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        506,
+                                        212
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        506,
+                                        212
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\nabla(\\pi[\\mathbf{x}] - \\mathbf{x})|_{\\mathbf{x} = \\pmb{\\mu}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        506,
+                                        212
+                                    ],
+                                    "type": "text",
+                                    "content": " being Hurwitz, as mentioned in the previous part. Therefore, (B3) is verified. Accordingly, (B1) - (B5) are checked by our assumptions A1, A2, A3', A4. By Theorem G.10, we have "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        506,
+                                        212
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\lim_{n \\to \\infty} \\mathbf{x}_n = \\pmb{\\mu}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        506,
+                                        212
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        506,
+                                        212
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta_{n} \\to \\Theta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        506,
+                                        212
+                                    ],
+                                    "type": "text",
+                                    "content": " almost surely."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        105,
+                        232,
+                        254,
+                        244
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                232,
+                                254,
+                                244
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        232,
+                                        254,
+                                        244
+                                    ],
+                                    "type": "text",
+                                    "content": "E PROOF OF THEOREM 3.3"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        259,
+                        504,
+                        327
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                259,
+                                504,
+                                327
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        259,
+                                        504,
+                                        327
+                                    ],
+                                    "type": "text",
+                                    "content": "This section is devoted to the proof of Theorem 3.3, which also includes the proof of the CLT results for the SRRW iteration "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        259,
+                                        504,
+                                        327
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        259,
+                                        504,
+                                        327
+                                    ],
+                                    "type": "text",
+                                    "content": " in Lemma 3.1. We will use different techniques depending on the step sizes in Assumption A2. Specifically, for step sizes "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        259,
+                                        504,
+                                        327
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\gamma_n = (n + 1)^{-a}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        259,
+                                        504,
+                                        327
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        259,
+                                        504,
+                                        327
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_n = (n + 1)^{-b}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        259,
+                                        504,
+                                        327
+                                    ],
+                                    "type": "text",
+                                    "content": ", we will consider three cases: case (i): "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        259,
+                                        504,
+                                        327
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_n = o(\\gamma_n)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        259,
+                                        504,
+                                        327
+                                    ],
+                                    "type": "text",
+                                    "content": "; case (ii): "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        259,
+                                        504,
+                                        327
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_n = \\gamma_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        259,
+                                        504,
+                                        327
+                                    ],
+                                    "type": "text",
+                                    "content": "; and case (iii): "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        259,
+                                        504,
+                                        327
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\gamma_n = o(\\beta_n)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        259,
+                                        504,
+                                        327
+                                    ],
+                                    "type": "text",
+                                    "content": ". For case (ii), we will use the existing CLT result for single-timescale SA in Theorem G.9. For cases (i) and (iii), we will construct our own CLT analysis for the two-timescale structure. We start with case (ii)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        105,
+                        344,
+                        215,
+                        357
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                344,
+                                215,
+                                357
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        344,
+                                        215,
+                                        357
+                                    ],
+                                    "type": "text",
+                                    "content": "E.1 CASE (II): "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        344,
+                                        215,
+                                        357
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_{n} = \\gamma_{n}"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        366,
+                        504,
+                        400
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                366,
+                                504,
+                                400
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        366,
+                                        504,
+                                        400
+                                    ],
+                                    "type": "text",
+                                    "content": "In this part, we stick to the notations for single-timescale SA studied in Section D.1. To utilize Theorem G.9, apart from Conditions C1 - C4 that have been checked in Section D.1, we still need to check conditions C5 and C6 listed in Section G.2."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        405,
+                        504,
+                        429
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                405,
+                                504,
+                                429
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        504,
+                                        429
+                                    ],
+                                    "type": "text",
+                                    "content": "Assumption A3 corresponds to condition C5. For condition C6, we need to obtain the explicit form of function "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        504,
+                                        429
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Q_{\\mathbf{z}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        405,
+                                        504,
+                                        429
+                                    ],
+                                    "type": "text",
+                                    "content": " to the Poisson equation defined in (96), that is,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        202,
+                        437,
+                        407,
+                        452
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                202,
+                                437,
+                                407,
+                                452
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        202,
+                                        437,
+                                        407,
+                                        452
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "Q _ {\\mathbf {z}} (i) - \\left(\\mathbf {K} _ {\\mathbf {z}} ^ {\\prime} Q _ {\\mathbf {z}}\\right) (i) = \\psi (\\mathbf {z}, i) - \\mathbb {E} _ {j \\sim \\pi [ \\mathbf {z} ]} [ \\psi (\\mathbf {z}, j) ]",
+                                    "image_path": "4d46269a4f227e56b0d571f101d0a4db76206c713d2b8a5aaf79cd56db94590a.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        105,
+                        461,
+                        133,
+                        471
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                461,
+                                133,
+                                471
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        461,
+                                        133,
+                                        471
+                                    ],
+                                    "type": "text",
+                                    "content": "where"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        171,
+                        475,
+                        438,
+                        502
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                171,
+                                475,
+                                438,
+                                502
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        171,
+                                        475,
+                                        438,
+                                        502
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\psi (\\mathbf {z}, i) \\triangleq \\sum_ {j \\in \\mathcal {N}} \\mathbf {K} _ {\\mathbf {z}} ^ {\\prime} (i, j) m _ {\\mathbf {z}} (j) m _ {\\mathbf {z}} (j) ^ {T} - (\\mathbf {K} _ {\\mathbf {z}} ^ {\\prime} m _ {\\mathbf {z}}) (i) (\\mathbf {K} _ {\\mathbf {z}} ^ {\\prime} m _ {\\mathbf {z}}) (i) ^ {T}.",
+                                    "image_path": "f121dcaa700ba9ff6c581701b078b2bd14804ab02e16801e4bdfad8c60b4c0b8.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        511,
+                        430,
+                        524
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                511,
+                                430,
+                                524
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        511,
+                                        430,
+                                        524
+                                    ],
+                                    "type": "text",
+                                    "content": "Following the similar steps in the derivation of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        511,
+                                        430,
+                                        524
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m_{\\mathbf{z}}(i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        511,
+                                        430,
+                                        524
+                                    ],
+                                    "type": "text",
+                                    "content": " from (22) to (24), we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        171,
+                        534,
+                        438,
+                        561
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                171,
+                                534,
+                                438,
+                                561
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        171,
+                                        534,
+                                        438,
+                                        561
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "Q _ {\\mathbf {z}} (i) = \\sum_ {j \\in \\mathcal {N}} (\\mathbf {I} - \\mathbf {K} ^ {\\prime} [ \\mathbf {z} ] + \\mathbf {1} \\pmb {\\pi} ^ {\\prime} [ \\mathbf {z} ] ^ {T}) ^ {- 1} (i, j) m _ {\\mathbf {z}} (j) - \\pi_ {j} ^ {\\prime} [ \\mathbf {z} ] m _ {\\mathbf {z}} (j).",
+                                    "image_path": "7ef3491fd579abc367841b84da3f34b082d4decf2337fee7d402d31bd3b2fa1e.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        104,
+                        571,
+                        504,
+                        628
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                571,
+                                504,
+                                628
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        571,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": "We also know that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        571,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Q_{\\mathbf{z}}(i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        571,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        571,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\mathbf{K}_{\\mathbf{z}}^{\\prime}Q_{\\mathbf{z}})(i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        571,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": " are continuous in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        571,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{z}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        571,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": " for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        571,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i \\in \\mathcal{N}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        571,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": ". For any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        571,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{z}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        571,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": " in a compact set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        571,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Omega"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        571,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        571,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Q_{\\mathbf{z}}(i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        571,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        571,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\mathbf{K}_{\\mathbf{z}}^{\\prime}Q_{\\mathbf{z}})(i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        571,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": " are bounded because function "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        571,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m_{\\mathbf{z}}(i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        571,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": " is bounded. Therefore, C6 is checked. By Theorem G.9, assume "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        571,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{z}_n = \\begin{bmatrix} \\theta_n \\\\ \\mathbf{x}_n \\end{bmatrix}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        571,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": " converges to a point "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        571,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{z}^* = \\begin{bmatrix} \\theta^* \\\\ \\mu \\end{bmatrix}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        571,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": " for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        571,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta^* \\in \\Theta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        571,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": ", we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        235,
+                        632,
+                        505,
+                        654
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                235,
+                                632,
+                                505,
+                                654
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        235,
+                                        632,
+                                        505,
+                                        654
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\gamma_ {n} ^ {- 1 / 2} \\left(\\mathbf {z} _ {n} - \\mathbf {z} ^ {*}\\right) \\xrightarrow [ n \\rightarrow \\infty ]{d i s t .} N (0, \\mathbf {V}), \\tag {26}",
+                                    "image_path": "36f65b4e76faea1105cbe35c06da51bae3013a92bb70ca656bec906e5beb7fb1.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
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+                        349,
+                        674
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                        674
+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        661,
+                                        349,
+                                        674
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        661,
+                                        349,
+                                        674
+                                    ],
+                                    "type": "text",
+                                    "content": " is the solution of the following Lyapunov equation"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        168,
+                        682,
+                        505,
+                        711
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                168,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        168,
+                                        682,
+                                        505,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {V} \\left(\\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I} + \\nabla g \\left(\\boldsymbol {z} ^ {*}\\right) ^ {T}\\right) + \\left(\\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I} + \\nabla g \\left(\\boldsymbol {z} ^ {*}\\right)\\right) \\mathbf {V} + \\mathbf {U} = 0, \\tag {27}",
+                                    "image_path": "d9f693ad7d388e60420dc056bdf385050c552e54893cad9ea394535eb15739a7.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        104,
+                        719,
+                        389,
+                        735
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        719,
+                                        389,
+                                        735
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{U} = \\sum_{i\\in \\mathcal{N}}\\mu_i\\left(m_{\\mathbf{z}^*}(i)m_{\\mathbf{z}^*}(i)\\right)^T - (\\mathbf{K}_{\\mathbf{z}^*}m_{\\mathbf{z}^*})(i)(\\mathbf{K}_{\\mathbf{z}^*}m_{\\mathbf{z}^*})(i)^T."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
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+                                38
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        300,
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+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        300,
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+                                        761
+                                    ],
+                                    "type": "text",
+                                    "content": "19"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 18
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        82,
+                        504,
+                        105
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                82,
+                                504,
+                                105
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "text",
+                                    "content": "By algebraic calculations of derivative of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi[\\mathbf{x}]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "text",
+                                    "content": " with respect to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "text",
+                                    "content": " in (20), we can rewrite "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\nabla g(\\mathbf{z}^*)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "text",
+                                    "content": " in terms of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}, \\pmb{\\theta}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "text",
+                                    "content": ", i.e.,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        140,
+                        110,
+                        468,
+                        174
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                140,
+                                110,
+                                468,
+                                174
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        140,
+                                        110,
+                                        468,
+                                        174
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\mathbf {J} (\\alpha) \\triangleq \\nabla g (\\mathbf {z} ^ {*}) = \\left[ \\begin{array}{c c} \\frac {\\partial \\sum_ {i \\in \\mathcal {N}} \\pi_ {i} [ \\mathbf {x} ] H (\\boldsymbol {\\theta} , i)}{\\partial \\boldsymbol {\\theta}} & \\frac {\\partial \\sum_ {i \\in \\mathcal {N}} \\pi_ {i} [ \\mathbf {x} ] H (\\boldsymbol {\\theta} , i)}{\\partial \\mathbf {x}} \\\\ \\frac {\\partial (\\boldsymbol {\\pi} [ \\mathbf {x} ] - \\mathbf {x})}{\\partial \\boldsymbol {\\theta}} & \\frac {\\partial \\boldsymbol {\\pi} [ \\mathbf {x} ] - \\mathbf {x}}{\\partial \\mathbf {x}} \\end{array} \\right] _ {\\mathbf {z} = \\mathbf {z} ^ {*}} \\\\ = \\left[ \\begin{array}{c c} \\nabla \\mathbf {h} (\\boldsymbol {\\theta} ^ {*}) & - \\alpha \\mathbf {H} ^ {T} (\\mathbf {P} ^ {T} + \\mathbf {I}) \\\\ \\mathbf {0} & 2 \\alpha \\boldsymbol {\\mu} \\mathbf {1} ^ {T} - \\alpha \\mathbf {P} ^ {T} - (\\alpha + 1) \\mathbf {I} \\end{array} \\right] \\triangleq \\left[ \\begin{array}{c c} \\mathbf {J} _ {1 1} & \\mathbf {J} _ {1 2} (\\alpha) \\\\ \\mathbf {J} _ {2 1} & \\mathbf {J} _ {2 2} (\\alpha) \\end{array} \\right], \\\\ \\end{array}",
+                                    "image_path": "3ae4f2fe00b13966cd40836e4c9de3d6c2faf94bcbb2ab36de81c15a1e840ad9.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        105,
+                        181,
+                        493,
+                        194
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                181,
+                                493,
+                                194
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        181,
+                                        493,
+                                        194
+                                    ],
+                                    "type": "text",
+                                    "content": "where matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        181,
+                                        493,
+                                        194
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{H} = [H(\\pmb{\\theta}^*, 1), \\dots, H(\\pmb{\\theta}, N)]^T"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        181,
+                                        493,
+                                        194
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, we further clarify the matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        181,
+                                        493,
+                                        194
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{U}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        181,
+                                        493,
+                                        194
+                                    ],
+                                    "type": "text",
+                                    "content": ". Note that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        201,
+                        504,
+                        243
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                201,
+                                504,
+                                243
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        201,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "m _ {\\mathbf {z} ^ {*}} (i) = \\sum_ {k = 0} ^ {\\infty} [ \\mathbf {G} (\\mathbf {z} ^ {*}) ((\\mathbf {P} ^ {k}) ^ {T} - \\boldsymbol {\\mu} \\mathbf {1} ^ {T}) ] _ {[:, i ]} = \\sum_ {k = 0} ^ {\\infty} [ \\mathbf {G} (\\mathbf {z} ^ {*}) (\\mathbf {P} ^ {k}) ^ {T} ] _ {[:, i ]} = \\mathbb {E} \\left[ \\sum_ {k = 0} ^ {\\infty} [ G (\\mathbf {z} ^ {*}, X _ {k}) ] \\right| X _ {0} = i ], \\tag {28}",
+                                    "image_path": "b8ac79b9bbaac8997119a4054e42d8da09a205487543a018103d19eac7c56423.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        243,
+                        504,
+                        289
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                243,
+                                504,
+                                289
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": "where the first equality holds because "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{K}^{\\prime}[\\pmb {\\mu}] = \\mathbf{P}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": " from the definition of SRRW kernel (3), the second equality stems from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{G}(\\mathbf{z}^{*})\\pmb {\\mu} = g(\\mathbf{z}^{*}) = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": " , and the last term is a conditional expectation over the base Markov chain "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{X_{k}\\}_{k\\geq 0}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": " (with transition kernel "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{P}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": ") conditioned on "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "X_0 = i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": " . Similarly, with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\mathbf{K}_{\\mathbf{z}}^{\\prime}m_{\\mathbf{z}})(i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        243,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": " in the form of (23), we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        212,
+                        295,
+                        398,
+                        329
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                212,
+                                295,
+                                398,
+                                329
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        212,
+                                        295,
+                                        398,
+                                        329
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "(\\mathbf {K} _ {\\mathbf {z}} ^ {\\prime} m _ {\\mathbf {z}}) (i) = \\mathbb {E} \\left[ \\sum_ {k = 1} ^ {\\infty} [ G (\\mathbf {z} ^ {*}, X _ {k}) ] \\Bigg | X _ {0} = i \\right].",
+                                    "image_path": "e5cfcaa30d57310cae7f1cf4b4bab8b626d3c7cb5101dd23de569122c2fdcada.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        342,
+                        504,
+                        365
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                342,
+                                504,
+                                365
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        342,
+                                        504,
+                                        365
+                                    ],
+                                    "type": "text",
+                                    "content": "From the form "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        342,
+                                        504,
+                                        365
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sum_{i\\in \\mathcal{N}}\\mu_i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        342,
+                                        504,
+                                        365
+                                    ],
+                                    "type": "text",
+                                    "content": " inside the matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        342,
+                                        504,
+                                        365
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{U}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        342,
+                                        504,
+                                        365
+                                    ],
+                                    "type": "text",
+                                    "content": ", the Markov chain "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        342,
+                                        504,
+                                        365
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{X_k\\}_{k\\geq 0}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        342,
+                                        504,
+                                        365
+                                    ],
+                                    "type": "text",
+                                    "content": " is in its stationary regime from the beginning, i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        342,
+                                        504,
+                                        365
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "X_{k}\\sim \\pmb{\\mu}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        342,
+                                        504,
+                                        365
+                                    ],
+                                    "type": "text",
+                                    "content": " for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        342,
+                                        504,
+                                        365
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k\\geq 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        342,
+                                        504,
+                                        365
+                                    ],
+                                    "type": "text",
+                                    "content": ". Hence,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        110,
+                        372,
+                        504,
+                        572
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                110,
+                                372,
+                                504,
+                                572
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        110,
+                                        372,
+                                        504,
+                                        572
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\mathbf {U} = \\mathbb {E} \\left[ \\left(\\sum_ {k = 0} ^ {\\infty} [ G (\\mathbf {z} ^ {*}, X _ {k}) ]\\right) \\left(\\sum_ {k = 0} ^ {\\infty} [ G (\\mathbf {z} ^ {*}, X _ {k}) ]\\right) ^ {T} \\right] \\\\ - \\mathbb {E} \\left[ \\left(\\sum_ {k = 1} ^ {\\infty} [ G (\\mathbf {z} ^ {*}, X _ {k}) ]\\right) \\left(\\sum_ {k = 1} ^ {\\infty} [ G (\\mathbf {z} ^ {*}, X _ {k}) ]\\right) ^ {T} \\right] \\\\ = \\mathbb {E} \\left[ G \\left(\\mathbf {z} ^ {*}, X _ {0}\\right) G \\left(\\mathbf {z} ^ {*}, X _ {0}\\right) ^ {T} \\right] + \\mathbb {E} \\left[ G \\left(\\mathbf {z} ^ {*}, X _ {0}\\right) \\left(\\sum_ {k = 1} ^ {\\infty} G \\left(\\mathbf {z} ^ {*}, X _ {k}\\right)\\right) ^ {T} \\right] \\tag {29} \\\\ + \\mathbb {E} \\left[ \\left(\\sum_ {k = 1} ^ {\\infty} G (\\mathbf {z} ^ {*}, X _ {k})\\right) G (\\mathbf {z} ^ {*}, X _ {0}) ^ {T} \\right] \\\\ = \\operatorname {C o v} \\left(G \\left(\\mathbf {z} ^ {*}, X _ {0}\\right), G \\left(\\mathbf {z} ^ {*}, X _ {0}\\right)\\right) \\\\ + \\sum_ {k = 1} ^ {\\infty} \\left[ \\operatorname {C o v} \\left(G \\left(\\mathbf {z} ^ {*}, X _ {0}\\right), G \\left(\\mathbf {z} ^ {*}, X _ {k}\\right)\\right) + \\operatorname {C o v} \\left(G \\left(\\mathbf {z} ^ {*}, X _ {k}\\right), G \\left(\\mathbf {z} ^ {*}, X _ {0}\\right)\\right) \\right], \\\\ \\end{array}",
+                                    "image_path": "50c28883d50e2525009dee5dc98389ba4b5b115baa4fcd9ee63681eef3e2a312.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        581,
+                        504,
+                        637
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                581,
+                                504,
+                                637
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        581,
+                                        504,
+                                        637
+                                    ],
+                                    "type": "text",
+                                    "content": "where the covariance between "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        581,
+                                        504,
+                                        637
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G(\\mathbf{z}^*, X_0)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        581,
+                                        504,
+                                        637
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        581,
+                                        504,
+                                        637
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G(\\mathbf{z}^*, X_k)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        581,
+                                        504,
+                                        637
+                                    ],
+                                    "type": "text",
+                                    "content": " for the Markov chain "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        581,
+                                        504,
+                                        637
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{X_n\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        581,
+                                        504,
+                                        637
+                                    ],
+                                    "type": "text",
+                                    "content": " in the stationary regime is "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        581,
+                                        504,
+                                        637
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{Cov}(G(\\mathbf{z}^*, X_0), G(\\mathbf{z}^*, X_k))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        581,
+                                        504,
+                                        637
+                                    ],
+                                    "type": "text",
+                                    "content": ". By Brémaud (2013, Theorem 6.3.7), it is demonstrated that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        581,
+                                        504,
+                                        637
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{U}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        581,
+                                        504,
+                                        637
+                                    ],
+                                    "type": "text",
+                                    "content": " is the sampling covariance of the base Markov chain "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        581,
+                                        504,
+                                        637
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{P}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        581,
+                                        504,
+                                        637
+                                    ],
+                                    "type": "text",
+                                    "content": " for the test function "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        581,
+                                        504,
+                                        637
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G(\\mathbf{z}^*, \\cdot)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        581,
+                                        504,
+                                        637
+                                    ],
+                                    "type": "text",
+                                    "content": ". Moreover, Brémaud (2013, equation (6.34)) states that this sampling covariance "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        581,
+                                        504,
+                                        637
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{U}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        581,
+                                        504,
+                                        637
+                                    ],
+                                    "type": "text",
+                                    "content": " can be rewritten in the following form:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        115,
+                        644,
+                        505,
+                        678
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                115,
+                                644,
+                                505,
+                                678
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        115,
+                                        644,
+                                        505,
+                                        678
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {U} = \\sum_ {i = 1} ^ {N - 1} \\mathbf {G} \\left(\\mathbf {z} ^ {*}\\right) ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} \\mathbf {G} \\left(\\mathbf {z} ^ {*}\\right) = \\sum_ {i = 1} ^ {N - 1} \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\left[ \\begin{array}{c c} \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H} & \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\\\ \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H} & \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\end{array} \\right] \\triangleq \\left[ \\begin{array}{l l} \\mathbf {U} _ {1 1} & \\mathbf {U} _ {1 2} \\\\ \\mathbf {U} _ {2 1} & \\mathbf {U} _ {2 2} \\end{array} \\right], \\tag {30}",
+                                    "image_path": "3038a30f32c3665b62bee81f3062376051d58f5b9d5f1f0493c07067f783eb8e.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        685,
+                        504,
+                        709
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                685,
+                                504,
+                                709
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        685,
+                                        504,
+                                        709
+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        685,
+                                        504,
+                                        709
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{(\\lambda_i, \\mathbf{u}_i)\\}_{i \\in \\mathcal{N}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        685,
+                                        504,
+                                        709
+                                    ],
+                                    "type": "text",
+                                    "content": " is the eigenpair of the transition kernel "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        685,
+                                        504,
+                                        709
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{P}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        685,
+                                        504,
+                                        709
+                                    ],
+                                    "type": "text",
+                                    "content": " of the ergodic and time-reversible base Markov chain. This completes the proof of case 1."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        114,
+                        719,
+                        498,
+                        734
+                    ],
+                    "type": "page_footnote",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                114,
+                                719,
+                                498,
+                                734
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        114,
+                                        719,
+                                        498,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": "12One may refer to Doshi et al. (2023, Appendix B, Proof of Lemma 3.4) for the computation of "
+                                },
+                                {
+                                    "bbox": [
+                                        114,
+                                        719,
+                                        498,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\frac{\\partial\\pi|\\mathbf{x}| - \\mathbf{x}}{\\partial\\mathbf{x}}"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        299,
+                        750,
+                        312,
+                        760
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                299,
+                                750,
+                                312,
+                                760
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        299,
+                                        750,
+                                        312,
+                                        760
+                                    ],
+                                    "type": "text",
+                                    "content": "20"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 19
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        82,
+                        504,
+                        120
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                82,
+                                504,
+                                120
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        120
+                                    ],
+                                    "type": "text",
+                                    "content": "Remark E.1. For the CLT result (26), we can look further into the asymptotic covariance matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        120
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        120
+                                    ],
+                                    "type": "text",
+                                    "content": " as in (27). For convenience, we denote "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        120
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V} = \\begin{bmatrix} \\mathbf{V}_{11} & \\mathbf{V}_{12}\\\\ \\mathbf{V}_{21} & \\mathbf{V}_{22} \\end{bmatrix}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        120
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        120
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{U}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        120
+                                    ],
+                                    "type": "text",
+                                    "content": " in the form of (30) such that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        127,
+                        123,
+                        505,
+                        152
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                127,
+                                123,
+                                505,
+                                152
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        127,
+                                        123,
+                                        505,
+                                        152
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left[ \\begin{array}{l l} \\mathbf {V} _ {1 1} & \\mathbf {V} _ {1 2} \\\\ \\mathbf {V} _ {2 1} & \\mathbf {V} _ {2 2} \\end{array} \\right] \\left(\\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I} + \\mathbf {J} (\\alpha) ^ {T}\\right) + \\left(\\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I} + \\mathbf {J} (\\alpha)\\right) \\left[ \\begin{array}{l l} \\mathbf {V} _ {1 1} & \\mathbf {V} _ {1 2} \\\\ \\mathbf {V} _ {2 1} & \\mathbf {V} _ {2 2} \\end{array} \\right] + \\mathbf {U} = 0. \\tag {31}",
+                                    "image_path": "8f22fa78ea635f68735f99f51f3b5aedfbdb06d19335d533986841f1e086b940.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        157,
+                        506,
+                        196
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                157,
+                                506,
+                                196
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        157,
+                                        506,
+                                        196
+                                    ],
+                                    "type": "text",
+                                    "content": "For the SRRW iteration "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        157,
+                                        506,
+                                        196
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        157,
+                                        506,
+                                        196
+                                    ],
+                                    "type": "text",
+                                    "content": ", from (26) we know that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        157,
+                                        506,
+                                        196
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\gamma_n^{-1/2}(\\mathbf{x}_n - \\boldsymbol{\\mu}) \\xrightarrow[n\\to\\infty]{dist.} N(\\mathbf{0}, \\mathbf{V}_4)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        157,
+                                        506,
+                                        196
+                                    ],
+                                    "type": "text",
+                                    "content": ". Thus, in this remark, we want to obtain the closed form of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        157,
+                                        506,
+                                        196
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{22}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        157,
+                                        506,
+                                        196
+                                    ],
+                                    "type": "text",
+                                    "content": ". By algebraic computations of the bottom-right sub-block matrix, we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        202,
+                        201,
+                        409,
+                        271
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                201,
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+                                271
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        202,
+                                        201,
+                                        409,
+                                        271
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\left(2 \\alpha \\boldsymbol {\\mu} \\mathbf {1} ^ {T} - \\alpha \\mathbf {P} ^ {T} - \\left(\\alpha + 1 - \\frac {\\mathbb {1} _ {\\{a = 1 \\}}}{2}\\right) \\mathbf {I}\\right) \\mathbf {V} _ {2 2} \\\\ + \\mathbf {V} _ {2 2} \\left(2 \\alpha \\boldsymbol {\\mu} \\mathbf {1} ^ {T} - \\alpha \\mathbf {P} ^ {T} - \\left(\\alpha + 1 - \\frac {\\mathbb {1} _ {\\{a = 1 \\}}}{2}\\right) \\mathbf {I}\\right) ^ {T} \\\\ + \\mathbf {U} _ {2 2} = 0. \\\\ \\end{array}",
+                                    "image_path": "1df75f511069627e9ceab4433417311f40a791bd64d5288447136f6c89211c3a.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
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+                        300
+                    ],
+                    "type": "text",
+                    "angle": 0,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                        277,
+                                        504,
+                                        300
+                                    ],
+                                    "type": "text",
+                                    "content": "By using result of the closed form solution to the Lyapunov equation (e.g., Lemma G.1) and the eigendecomposition of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        277,
+                                        504,
+                                        300
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{P}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        277,
+                                        504,
+                                        300
+                                    ],
+                                    "type": "text",
+                                    "content": ", we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        194,
+                        305,
+                        505,
+                        337
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                            ],
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+                                    "bbox": [
+                                        194,
+                                        305,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {V} _ {2 2} = \\sum_ {i = 1} ^ {N - 1} \\frac {1}{2 \\alpha \\left(1 + \\lambda_ {i}\\right) + 2 - \\mathbb {1} _ {\\{a = 1 \\}}} \\cdot \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T}. \\tag {32}",
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+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
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+                        363
+                    ],
+                    "type": "title",
+                    "angle": 0,
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+                            ],
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+                                    ],
+                                    "type": "text",
+                                    "content": "E.2 CASE (I): "
+                                },
+                                {
+                                    "bbox": [
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+                                        350,
+                                        224,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_{n} = o(\\gamma_{n})"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "In this part, we mainly focus on the CLT of the SA iteration "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        370,
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+                                    ],
+                                    "type": "text",
+                                    "content": " because the SRRW iteration "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        370,
+                                        504,
+                                        395
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        370,
+                                        504,
+                                        395
+                                    ],
+                                    "type": "text",
+                                    "content": " is independent of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        370,
+                                        504,
+                                        395
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        370,
+                                        504,
+                                        395
+                                    ],
+                                    "type": "text",
+                                    "content": " and its CLT result has been shown in Remark E.1."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        105,
+                        405,
+                        342,
+                        417
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
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+                            ],
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+                                    ],
+                                    "type": "text",
+                                    "content": "E.2.1 DECOMPOSITION OF SA-SRRW ITERATION (4)"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
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+                            ],
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "We slightly abuse the math notation and define the function"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
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+                    ],
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+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {h} (\\boldsymbol {\\theta}, \\mathbf {x}) \\triangleq \\mathbb {E} _ {i \\sim \\pi [ \\mathbf {x} ]} H (\\boldsymbol {\\theta}, i) = \\sum_ {i \\in \\mathcal {N}} \\pi_ {i} [ \\mathbf {x} ] H (\\boldsymbol {\\theta}, i)",
+                                    "image_path": "b7884add204a1b08417a9844f512aa1894eeb65ef83c62f9d0e8619b4163357e.jpg"
+                                }
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+                    "index": 11
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "such that "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{h}(\\pmb {\\theta},\\pmb {\\mu})\\equiv \\mathbf{h}(\\pmb {\\theta})"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        472,
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+                                    ],
+                                    "type": "text",
+                                    "content": " . Then, we reformulate (25) as"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
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+                    ],
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+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} + \\beta_ {n + 1} \\mathbf {h} \\left(\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}\\right) + \\beta_ {n + 1} \\left(H \\left(\\boldsymbol {\\theta} _ {n}, X _ {n + 1}\\right) - \\mathbf {h} \\left(\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}\\right)\\right). \\tag {33a}",
+                                    "image_path": "56c776a5c1e74ea7611e9f820136b815dcb78eb2290f10a0ddc6fc67a8a6bb1c.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
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+                        520
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        180,
+                                        506,
+                                        504,
+                                        520
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {x} _ {n + 1} = \\mathbf {x} _ {n} + \\gamma_ {n + 1} \\left(\\boldsymbol {\\pi} \\left[ \\mathbf {x} _ {n} \\right] - \\mathbf {x} _ {n}\\right) + \\gamma_ {n + 1} \\left(\\boldsymbol {\\delta} _ {X _ {n + 1}}\\right) - \\boldsymbol {\\pi} \\left[ \\mathbf {x} _ {n} \\right]). \\tag {33b}",
+                                    "image_path": "e3741a309871ff0774ee91ffba4100ed56ed37314e487d065d5f51fcc8f60a87.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        105,
+                        523,
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+                        536
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "There exist functions "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        523,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "q_{\\mathbf{x}}:\\mathcal{N}\\to \\mathbb{R}^{N},\\tilde{H}_{\\boldsymbol{\\theta},\\mathbf{x}}:\\mathcal{N}\\to \\mathbb{R}^{D}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "type": "text",
+                                    "content": " satisfying the following Poisson equations"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        237,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\boldsymbol {\\delta} _ {i} - \\boldsymbol {\\pi} (\\mathbf {x}) = q _ {\\mathbf {x}} (i) - \\left(\\mathbf {K} _ {\\mathbf {x}} q _ {\\mathbf {x}}\\right) (i) \\tag {34a}",
+                                    "image_path": "4c8e72b8afc25b261367277abac94d661899ece69af210de8ba1ed268a638648.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "H (\\boldsymbol {\\theta}, i) - \\mathbf {h} (\\boldsymbol {\\theta}, \\mathbf {x}) = \\tilde {H} _ {\\boldsymbol {\\theta}, \\mathbf {x}} (i) - \\left(\\mathbf {K} _ {\\mathbf {x}} \\tilde {H} _ {\\boldsymbol {\\theta}, \\mathbf {x}}\\right) (i), \\tag {34b}",
+                                    "image_path": "4903bc7b897f6a0f725c7c177aaa1599bd7206b8e50a97d0fe693732be43ca3b.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        506,
+                                        626
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\theta} \\in \\mathbb{R}^{D}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        506,
+                                        626
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        506,
+                                        626
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x} \\in \\mathrm{Int}(\\Sigma)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        506,
+                                        626
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        506,
+                                        626
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i \\in \\mathcal{N}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
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+                                        626
+                                    ],
+                                    "type": "text",
+                                    "content": ", where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        506,
+                                        626
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\mathbf{K}_{\\mathbf{x}}q_{\\mathbf{x}})(i) \\triangleq \\sum_{j \\in \\mathcal{N}} K_{ij}[\\mathbf{x}] q_{\\mathbf{x}}(j)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        506,
+                                        626
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        506,
+                                        626
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\mathbf{K}_{\\mathbf{x}}\\tilde{H}_{\\pmb{\\theta},\\mathbf{x}})(j) \\triangleq \\sum_{j \\in \\mathcal{N}} K_{ij}[\\mathbf{x}]\\tilde{H}_{\\pmb{\\theta},\\mathbf{x}}(j)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        506,
+                                        626
+                                    ],
+                                    "type": "text",
+                                    "content": ". The existence and explicit form of the solutions "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        506,
+                                        626
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "q_{\\mathbf{x}}, \\tilde{H}_{\\pmb{\\theta},\\mathbf{x}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        506,
+                                        626
+                                    ],
+                                    "type": "text",
+                                    "content": ", which are continuous w.r.t "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        506,
+                                        626
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        506,
+                                        626
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
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+                                        626
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\theta}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        506,
+                                        626
+                                    ],
+                                    "type": "text",
+                                    "content": ", follow the similar steps that can be found in Section D.1 from (22) to (24). Thus, we can further decompose (33) into"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
+                        146,
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+                        735
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        146,
+                                        631,
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+                                        735
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} + \\beta_ {n + 1} \\mathbf {h} (\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}) + \\beta_ {n + 1} \\underbrace {(\\tilde {H} _ {\\boldsymbol {\\theta} _ {n} , \\mathbf {x} _ {n}} (X _ {n + 1}) - (\\mathbf {K} _ {\\mathbf {x} _ {n}} \\tilde {H} _ {\\boldsymbol {\\theta} _ {n} , \\mathbf {x} _ {n}}) (X _ {n}))} _ {M _ {n + 1} ^ {(\\boldsymbol {\\theta})}} \\\\ + \\beta_ {n + 1} \\underbrace {\\left(\\left(\\mathbf {K} _ {\\mathbf {x} _ {n + 1}} \\tilde {H} _ {\\boldsymbol {\\theta} _ {n + 1} , \\mathbf {x} _ {n + 1}}\\right) \\left(X _ {n + 1}\\right) - \\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} \\tilde {H} _ {\\boldsymbol {\\theta} _ {n} , \\mathbf {x} _ {n}}\\right) \\left(X _ {n + 1}\\right)\\right)} _ {r _ {n} ^ {(\\boldsymbol {\\theta}, 1)}} \\tag {35a} \\\\ + \\beta_ {n + 1} \\underbrace {((\\mathbf {K _ {x}} _ {n} \\tilde {H} ) (X _ {n}) - (\\mathbf {K _ {x}} _ {n + 1} \\tilde {H} ) (X _ {n + 1}))} _ {r _ {n} ^ {(\\pmb {\\theta}, 2)}}, \\\\ \\end{array}",
+                                    "image_path": "843b1add4990762b473a118611afc536e0d6748486d6968d44778201fd6d42e6.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                }
+            ],
+            "discarded_blocks": [
+                {
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+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
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+                                    "type": "text",
+                                    "content": "21"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                }
+            ],
+            "page_size": [
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+            ],
+            "page_idx": 20
+        },
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\mathbf {x} _ {n + 1} = \\mathbf {x} _ {n} + \\gamma_ {n + 1} (\\boldsymbol {\\pi} (\\mathbf {x} _ {n}) - \\mathbf {x} _ {n}) + \\gamma_ {n + 1} \\underbrace {(q _ {\\mathbf {x} _ {n}} (X _ {n + 1}) - (\\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}}) (X _ {n}))} _ {M _ {n + 1} ^ {(\\mathbf {x})}} \\\\ + \\gamma_ {n + 1} \\underbrace {\\left([ \\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}} ] \\left(X _ {n + 1}\\right) - [ \\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n + 1}} ] \\left(X _ {n + 1}\\right)\\right)} _ {r _ {n} ^ {(\\mathbf {x}, 1)}} \\tag {35b} \\\\ + \\gamma_ {n + 1} \\underbrace {\\left((\\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}}) (X _ {n}) - (\\mathbf {K} _ {\\mathbf {x} _ {n + 1}} q _ {\\mathbf {x} _ {n + 1}}) (X _ {n + 1})\\right)} _ {r _ {n} ^ {(\\mathbf {x}, 2)}}. \\\\ \\end{array}",
+                                    "image_path": "5691ddceaa66461551f483ea6975fcf6184238789d5d6d424422d3a57abf3440.jpg"
+                                }
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+                {
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+                    "type": "text",
+                    "angle": 0,
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "such that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} + \\beta_ {n + 1} \\mathbf {h} \\left(\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}\\right) + \\beta_ {n + 1} M _ {n + 1} ^ {(\\boldsymbol {\\theta})} + \\beta_ {n + 1} r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} + \\beta_ {n + 1} r _ {n} ^ {(\\boldsymbol {\\theta}, 2)}, \\tag {36a}",
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {x} _ {n + 1} = \\mathbf {x} _ {n} + \\gamma_ {n + 1} \\left(\\boldsymbol {\\pi} \\left(\\mathbf {x} _ {n}\\right) - \\mathbf {x} _ {n}\\right) + \\gamma_ {n + 1} M _ {n + 1} ^ {(\\mathbf {x})} + \\gamma_ {n + 1} r _ {n} ^ {(\\mathbf {x}, 1)} + \\gamma_ {n + 1} r _ {n} ^ {(\\mathbf {x}, 2)}. \\tag {36b}",
+                                    "image_path": "7e6c3d36e9825a85cc61b194ceddc06232365822778db1325618d440cdd1ce28.jpg"
+                                }
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+                },
+                {
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+                    ],
+                    "type": "text",
+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "We can observe that (36) differs from the expression in Konda & Tsitsiklis (2004); Mokkadem & Pelletier (2006), which studied the two-timescale SA with Martingale difference noise. Here, due to the presence of the iterate-dependent Markovian noise and the application of the Poisson equation technique, we have additional non-vanishing terms "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
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+                                        340
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r_n^{(\\theta, 2)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        504,
+                                        340
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        504,
+                                        340
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r_n^{(\\mathbf{x}, 2)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        504,
+                                        340
+                                    ],
+                                    "type": "text",
+                                    "content": ", which will be further examined in Lemma E.2. Additionally, when we apply the Poisson equation to the Martingale difference terms "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        504,
+                                        340
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M_{n+1}^{(\\theta)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        504,
+                                        340
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        504,
+                                        340
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M_{n+1}^{(\\mathbf{x})}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
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+                                        340
+                                    ],
+                                    "type": "text",
+                                    "content": ", we find that there are some covariances that are also non-vanishing as in Lemma E.1. We will mention this again when we obtain those covariances. These extra non-zero noise terms make our analysis distinct from the previous ones since the key assumption (A4) in Mokkadem & Pelletier (2006) is not satisfied. We demonstrate that the long-term average performance of these terms can be managed so that they do not affect the final CLT result."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
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+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Analysis of Terms "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M_{n + 1}^{(\\theta)},M_{n + 1}^{(\\mathbf{x})}"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
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+                        392
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": "Consider the filtration "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        365,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}_n \\triangleq \\sigma(\\pmb{\\theta}_0, \\mathbf{x}_0, X_0, \\dots, \\pmb{\\theta}_n, \\mathbf{x}_n, X_n)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": ", it is evident that "
+                                },
+                                {
+                                    "bbox": [
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+                                        392
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M_{n+1}^{(\\pmb{\\theta})}, M_{n+1}^{(\\mathbf{x})}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        365,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": " are Martingale difference sequences adapted to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        365,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        365,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                            ],
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+                                    "bbox": [
+                                        143,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\mathbb {E} \\left[ M _ {n + 1} ^ {(\\mathbf {x})} (M _ {n + 1} ^ {(\\mathbf {x})}) ^ {T} \\Big | \\mathcal {F} _ {n} \\right] \\\\ = \\mathbb {E} \\left[ q _ {\\mathbf {x} _ {n}} \\left(X _ {n + 1}\\right) q _ {\\mathbf {x} _ {n}} \\left(X _ {n + 1}\\right) ^ {T} \\mid \\mathcal {F} _ {n} \\right] + \\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}}\\right) \\left(X _ {n}\\right) \\left(\\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}}\\right) \\left(X _ {n}\\right)\\right) ^ {T} \\tag {37} \\\\ - \\mathbb {E} \\left[ q _ {\\mathbf {x} _ {n}} \\left(X _ {n + 1}\\right) \\mid \\mathcal {F} _ {n} \\right] \\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}}\\right) \\left(X _ {n}\\right)) ^ {T} - \\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}}\\right) \\left(X _ {n}\\right) \\mathbb {E} \\left[ q _ {\\mathbf {x} _ {n}} \\left(X _ {n + 1}\\right) ^ {T} \\mid \\mathcal {F} _ {n} \\right] \\\\ = \\mathbb {E} \\left[ q _ {\\mathbf {x} _ {n}} \\left(X _ {n + 1}\\right) q _ {\\mathbf {x} _ {n}} \\left(X _ {n + 1}\\right) ^ {T} \\mid \\mathcal {F} _ {n} \\right] - \\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}}\\right) \\left(X _ {n}\\right) \\left(\\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}}\\right) \\left(X _ {n}\\right)\\right) ^ {T}. \\\\ \\end{array}",
+                                    "image_path": "3aef51bc38631ea2e82d08d1dc39191ffa13fcaf1979e69b90416d80c604d5c2.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Similarly, we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
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+                    ],
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+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
+                                        117,
+                                        483,
+                                        504,
+                                        526
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\mathbb {E} \\left[ M _ {n + 1} ^ {(\\boldsymbol {\\theta})} \\left(M _ {n + 1} ^ {(\\boldsymbol {\\theta})}\\right) ^ {T} \\mid \\mathcal {F} _ {n} \\right] \\tag {38} \\\\ = \\mathbb {E} \\left[ \\tilde {H} _ {\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}} \\left(X _ {n + 1}\\right) \\tilde {H} _ {\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}} \\left(X _ {n + 1}\\right) ^ {T} \\mid \\mathcal {F} _ {n} \\right] - \\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} \\tilde {H} _ {\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}}\\right) \\left(X _ {n}\\right) \\left(\\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} \\tilde {H} _ {\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}}\\right) \\left(X _ {n}\\right)\\right) ^ {T}, \\\\ \\end{array}",
+                                    "image_path": "05b7a153220deef72711570d39ec83dfc6ab6914ebb09ba937920588c299b3d1.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "and"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                140,
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+                                470,
+                                586
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        140,
+                                        542,
+                                        470,
+                                        586
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\mathbb {E} \\left[ M _ {n + 1} ^ {(\\mathbf {x})} (M _ {n + 1} ^ {(\\boldsymbol {\\theta})}) ^ {T} \\Big | \\mathcal {F} _ {n} \\right] \\\\ = \\mathbb {E} [ q _ {\\mathbf {x} _ {n}} (X _ {n + 1}) \\tilde {H} _ {\\pmb {\\theta} _ {n}, \\mathbf {x} _ {n}} (X _ {n + 1}) ^ {T} | \\mathcal {F} _ {n} ] - (\\mathbf {K} _ {\\mathbf {x} _ {n}} q _ {\\mathbf {x} _ {n}}) (X _ {n}) ((\\mathbf {K} _ {\\mathbf {x} _ {n}} \\tilde {H} _ {\\pmb {\\theta} _ {n}, \\mathbf {x} _ {n}}) (X _ {n})) ^ {T}. \\\\ \\end{array}",
+                                    "image_path": "f1d23be5a578c0ab647f4fa8d4c1c98291647b0f3b7a5aa8dc664bffecb86060.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "We now focus on "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{E}\\left[M_{n + 1}^{(\\mathbf{x})}(M_{n + 1}^{(\\mathbf{x})})^T\\Big|\\mathcal{F}_n\\right]"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        594,
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+                                    ],
+                                    "type": "text",
+                                    "content": ". Denote by"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        172,
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        172,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "V _ {1} (\\mathbf {x}, i) \\triangleq \\sum_ {j \\in \\mathcal {N}} \\mathbf {K} _ {i, j} [ \\mathbf {x} ] q _ {\\mathbf {x}} (j) q _ {\\mathbf {x}} (j) ^ {T} - \\left(\\mathbf {K} _ {\\mathbf {x}} q _ {\\mathbf {x}}\\right) (i) \\left(\\left(\\mathbf {K} _ {\\mathbf {x}} q _ {\\mathbf {x}}\\right) (i)\\right) ^ {T}, \\tag {39}",
+                                    "image_path": "c049ac80c407456b5f0ea7b3e51af2fac8d99fd3091773de4f811f71018aec3b.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
+                                        104,
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+                                        673
+                                    ],
+                                    "type": "text",
+                                    "content": "and let its expectation w.r.t the stationary distribution "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi (\\mathbf{x})"
+                                },
+                                {
+                                    "bbox": [
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+                                        673
+                                    ],
+                                    "type": "text",
+                                    "content": " be "
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+                                {
+                                    "bbox": [
+                                        104,
+                                        649,
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+                                        673
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v_{1}(\\mathbf{x})\\triangleq \\mathbb{E}_{i\\sim \\pi (\\mathbf{x})}[V_{1}(\\mathbf{x},i)]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        649,
+                                        504,
+                                        673
+                                    ],
+                                    "type": "text",
+                                    "content": ", we can construct another Poisson equation, i.e.,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
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+                    "bbox": [
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+                                    "bbox": [
+                                        162,
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+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\mathbb {E} \\left[ M _ {n + 1} ^ {(\\mathbf {x})} (M _ {n + 1} ^ {(\\mathbf {x})}) ^ {T} \\Big | \\mathcal {F} _ {n} \\right] - \\sum_ {X _ {n} \\in \\mathcal {N}} \\pi_ {X _ {n}} (\\mathbf {x} _ {n}) \\mathbb {E} \\left[ M _ {n + 1} ^ {(\\mathbf {x})} (M _ {n + 1} ^ {(\\mathbf {x})}) ^ {T} \\Big | \\mathcal {F} _ {n} \\right] \\\\ = V _ {1} \\left(\\mathbf {x} _ {n}, X _ {n + 1}\\right) - v _ {1} \\left(\\mathbf {x} _ {n}\\right) \\\\ = \\varphi_ {\\mathbf {x}} ^ {(1)} (X _ {n + 1}) - (\\mathbf {K} _ {\\mathbf {x} _ {n}} \\varphi_ {\\mathbf {x} _ {n}} ^ {(1)}) (X _ {n + 1}), \\\\ \\end{array}",
+                                    "image_path": "5117bd4996943f3570440d391b3d6d2c357255bd26a19fe19eb28f196e85c844.jpg"
+                                }
+                            ]
+                        }
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+            ],
+            "discarded_blocks": [
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
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+                                    "type": "text",
+                                    "content": "22"
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "for some matrix-valued function "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\varphi_{\\mathbf{x}}^{(1)}:\\mathcal{N}\\to \\mathbb{R}^{N\\times N}"
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ". Since "
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+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "q_{\\mathbf{x}}"
+                                },
+                                {
+                                    "bbox": [
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+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{K}[\\mathbf{x}]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": " are continuous in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
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+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": ", functions "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "V_{1},v_{1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": " are also continuous in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, we can decompose (39) into"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        148,
+                        110,
+                        504,
+                        180
+                    ],
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+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                180
+                            ],
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+                                    "bbox": [
+                                        148,
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+                                        180
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} V _ {1} (\\mathbf {x} _ {n}, X _ {n + 1}) = \\underbrace {v _ {1} (\\boldsymbol {\\mu})} _ {\\mathbf {U} _ {2 2}} + \\underbrace {v _ {1} (\\mathbf {x} _ {n}) - v _ {1} (\\boldsymbol {\\mu})} _ {\\mathbf {D} _ {n} ^ {(1)}} + \\underbrace {\\varphi_ {\\mathbf {x} _ {n}} ^ {(1)} (X _ {n + 1}) - (\\mathbf {K} _ {\\mathbf {x} _ {n}} \\varphi_ {\\mathbf {x} _ {n}} ^ {(1)}) (X _ {n})} _ {\\mathbf {J} _ {n} ^ {(1, a)}} \\\\ + \\underbrace {\\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} \\varphi_ {\\mathbf {x} _ {n}} ^ {(1)}\\right) \\left(X _ {n}\\right) - \\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} \\varphi_ {\\mathbf {x} _ {n}} ^ {(1)}\\right) \\left(X _ {n + 1}\\right)} _ {\\mathbf {J} _ {n} ^ {(1, b)}}. \\tag {40} \\\\ \\end{array}",
+                                    "image_path": "4deb995f50433451a5b55eac494340b4df2cd7e276618f6bb3641641450205b6.jpg"
+                                }
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Thus, we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
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+                                    "type": "interline_equation",
+                                    "content": "\\mathbb {E} \\left[ M _ {n + 1} ^ {(\\mathbf {x})} \\left(M _ {n + 1} ^ {(\\mathbf {x})}\\right) ^ {T} \\mid \\mathcal {F} _ {n} \\right] = \\mathbf {U} _ {2 2} + \\mathbf {D} _ {n} ^ {(1)} + \\mathbf {J} _ {n} ^ {(1)}, \\tag {41}",
+                                    "image_path": "411ee766d34e70c64f4d332ef493699825b6126b70eae394cf067c906ca248d8.jpg"
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+                                    "type": "text",
+                                    "content": "where "
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+                                    "bbox": [
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+                                        232
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{J}_n^{(1)} = \\mathbf{J}_n^{(1,a)} + \\mathbf{J}_n^{(1,b)}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
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+                            "bbox": [
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+                                    "bbox": [
+                                        104,
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+                                        274
+                                    ],
+                                    "type": "text",
+                                    "content": "Following the similar steps above, we can decompose "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                        274
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{E}\\left[M_{n + 1}^{(\\mathbf{x})}(M_{n + 1}^{(\\boldsymbol{\\theta})})^T\\Big|\\mathcal{F}_n\\right]"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                        274
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{E}\\left[M_{n + 1}^{(\\boldsymbol{\\theta})}(M_{n + 1}^{(\\boldsymbol{\\theta})})^T\\Big|\\mathcal{F}_n\\right]"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " as"
+                                }
+                            ]
+                        }
+                    ],
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+                },
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+                    ],
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+                    "angle": 0,
+                    "lines": [
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+                            ],
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+                                    "bbox": [
+                                        205,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbb {E} \\left[ M _ {n + 1} ^ {(\\mathbf {x})} \\left(M _ {n + 1} ^ {(\\boldsymbol {\\theta})}\\right) ^ {T} \\mid \\mathcal {F} _ {n} \\right] = \\mathbf {U} _ {2 1} + \\mathbf {D} _ {n} ^ {(2)} + \\mathbf {J} _ {n} ^ {(2)}, \\tag {42a}",
+                                    "image_path": "c690ea4fcc45875a0d5b54c68880aa63506cc77f59ac1811a752af7c23e055fc.jpg"
+                                }
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+                    ],
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+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
+                                        205,
+                                        307,
+                                        504,
+                                        328
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbb {E} \\left[ M _ {n + 1} ^ {(\\boldsymbol {\\theta})} \\left(M _ {n + 1} ^ {(\\boldsymbol {\\theta})}\\right) ^ {T} \\mid \\mathcal {F} _ {n} \\right] = \\mathbf {U} _ {1 1} + \\mathbf {D} _ {n} ^ {(3)} + \\mathbf {J} _ {n} ^ {(3)}. \\tag {42b}",
+                                    "image_path": "28adb906448a65caa4641459294819b6c7387f8f0ffcf60c1cca11b09c498c4b.jpg"
+                                }
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+                        }
+                    ],
+                    "index": 8
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+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                331,
+                                504,
+                                390
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        390
+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        390
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{J}_n^{(2)} = \\mathbf{J}_n^{(2,a)} + \\mathbf{J}_n^{(2,b)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        390
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        390
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{J}_n^{(3)} = \\mathbf{J}_n^{(3,a)} + \\mathbf{J}_n^{(3,b)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        390
+                                    ],
+                                    "type": "text",
+                                    "content": ". Here, we note that matrices "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        390
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{J}_n^i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        390
+                                    ],
+                                    "type": "text",
+                                    "content": " for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        390
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i = 1,2,3"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        390
+                                    ],
+                                    "type": "text",
+                                    "content": " are in presence of the current CLT analysis of the two-timescale SA with Martingale difference noise. In addition, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        390
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{U}_{11}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        390
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        390
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{U}_{12}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        390
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        390
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{U}_{22}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        390
+                                    ],
+                                    "type": "text",
+                                    "content": " inherently include the information of the underlying Markov chain (with its eigenpair "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        390
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\lambda_i,\\mathbf{u}_i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        390
+                                    ],
+                                    "type": "text",
+                                    "content": "), which is an extension of the previous works (Konda & Tsitsiklis, 2004; Mokkadem & Pelletier, 2006)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
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+                        501,
+                        408
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        394,
+                                        501,
+                                        408
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma E.1. For "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        394,
+                                        501,
+                                        408
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M_{n + 1}^{(\\theta)},M_{n + 1}^{(\\mathbf{x})}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        394,
+                                        501,
+                                        408
+                                    ],
+                                    "type": "text",
+                                    "content": " defined in (35) and their decomposition in (41) and (42), we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
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+                        457
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
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+                            ],
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {U} _ {1 1} = \\sum_ {i = 1} ^ {N - 1} \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T}, \\quad \\mathbf {U} _ {2 1} = \\sum_ {i = 1} ^ {N - 1} \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H}, \\quad \\mathbf {U} _ {1 1} = \\sum_ {i = 1} ^ {N - 1} \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H}, \\tag {43a}",
+                                    "image_path": "fd75968468fbf9242a5016bcb3c1085f71bc5ab146c03d710be4186c487264d5.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
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+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\lim  _ {n \\rightarrow \\infty} \\mathbf {D} _ {n} ^ {(i)} = 0 a. s. f o r i = 1, 2, 3, \\tag {43b}",
+                                    "image_path": "ee5046c0d9ed97ef1cb66653318efde3d4400d45125af51d0a8d300f1f204623.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        206,
+                                        479,
+                                        504,
+                                        514
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\lim  _ {n \\rightarrow \\infty} \\gamma_ {n} \\mathbb {E} \\left[\\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(i)} \\right\\|\\right] = 0, \\quad f o r \\quad i = 1, 2, 3. \\tag {43c}",
+                                    "image_path": "6b30df50a1c532bb1adcf9da9a22d06557ff07487bbe1a0a667590c67e05c0ac.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                        538
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. We now provide the properties of the four terms inside (41) as an example. Note that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        544,
+                                        477,
+                                        611
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\mathbf {U} _ {1 1} = \\mathbb {E} _ {i \\sim \\boldsymbol {\\mu}} [ V _ {1} (\\boldsymbol {\\mu}, i) ] = \\sum_ {i \\in \\mathcal {N}} \\mu_ {i} \\left[ \\sum_ {j \\in \\mathcal {N}} \\mathbf {P} (i, j) q _ {\\boldsymbol {\\mu}} (j) q _ {\\boldsymbol {\\mu}} (j) ^ {T} - (\\mathbf {P} q _ {\\boldsymbol {\\mu}}) (i) ((\\mathbf {P} q _ {\\boldsymbol {\\mu}}) (i)) ^ {T} \\right] \\\\ = \\sum_ {j \\in \\mathcal {N}} \\mu_ {j} q _ {\\pmb {\\mu}} (j) q _ {\\pmb {\\mu}} (j) ^ {T} - (\\mathbf {P} q _ {\\pmb {\\mu}}) (j) ((\\mathbf {P} q _ {\\pmb {\\mu}}) (j)) ^ {T}. \\\\ \\end{array}",
+                                    "image_path": "52a16c65e38bdb1ad96e83d5c2b18036a6eba2edfdcbb52b5050142665b5ab2a.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                640
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        504,
+                                        640
+                                    ],
+                                    "type": "text",
+                                    "content": "We can see that it has exactly the same structure as matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        504,
+                                        640
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{U}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        504,
+                                        640
+                                    ],
+                                    "type": "text",
+                                    "content": " in (27). Following the similar steps in deducing the explicit form of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        504,
+                                        640
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{U}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        504,
+                                        640
+                                    ],
+                                    "type": "text",
+                                    "content": " from (28) to (30), we get"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
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+                        504,
+                        678
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        250,
+                                        646,
+                                        504,
+                                        678
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {U} _ {1 1} = \\sum_ {i = 1} ^ {N - 1} \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T}. \\tag {44}",
+                                    "image_path": "c735f395149ebd53b869515878c2a4e6381623ad204ad84637904c5104a3ad1c.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        104,
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+                        711
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        685,
+                                        504,
+                                        711
+                                    ],
+                                    "type": "text",
+                                    "content": "By the almost sure convergence result "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        685,
+                                        504,
+                                        711
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n\\to \\pmb{\\mu}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        685,
+                                        504,
+                                        711
+                                    ],
+                                    "type": "text",
+                                    "content": " in Lemma 3.1, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        685,
+                                        504,
+                                        711
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v_{1}(\\mathbf{x}_{n})\\rightarrow v_{1}(\\pmb {\\mu})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        685,
+                                        504,
+                                        711
+                                    ],
+                                    "type": "text",
+                                    "content": " a.s. such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        685,
+                                        504,
+                                        711
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\lim_{n\\to \\infty}\\mathbf{D}_n^{(1)} = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        685,
+                                        504,
+                                        711
+                                    ],
+                                    "type": "text",
+                                    "content": " a.s."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "We next prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\lim_{n\\to \\infty}\\gamma_n\\mathbb{E}\\left[\\left\\| \\sum_{k = 1}^n{\\bf J}_k^{(1,a)}\\right\\|\\right] = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        715,
+                                        497,
+                                        736
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        715,
+                                        497,
+                                        736
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\lim_{n\\to \\infty}\\gamma_n\\mathbb{E}\\left[\\left\\| \\sum_{k = 1}^n{\\bf J}_k^{(1,b)}\\right\\|\\right] = 0."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                26,
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+                                38
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
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+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "23"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                }
+            ],
+            "page_size": [
+                612,
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+            ],
+            "page_idx": 22
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        81,
+                        504,
+                        105
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                81,
+                                504,
+                                105
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "text",
+                                    "content": "Since "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\mathbf{J}_n^{(1,a)}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "text",
+                                    "content": " is a Martingale difference sequence adapted to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "text",
+                                    "content": ", with the Burkholder inequality in Lemma G.2 and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p = 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "text",
+                                    "content": ", we show that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        203,
+                        109,
+                        503,
+                        146
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                203,
+                                109,
+                                503,
+                                146
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        203,
+                                        109,
+                                        503,
+                                        146
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbb {E} \\left[ \\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1, a)} \\right\\| \\right] \\leq C _ {1} \\mathbb {E} \\left[ \\sqrt {\\left(\\sum_ {k = 1} ^ {n} \\left\\| \\mathbf {J} _ {k} ^ {(1 , a)} \\right\\| ^ {2}\\right)} \\right]. \\tag {45}",
+                                    "image_path": "4c731d4769996063bd1a3856989173153f78529bbaf6083595372bf43c49dd9b.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
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+                        504,
+                        174
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                149,
+                                504,
+                                174
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        149,
+                                        504,
+                                        174
+                                    ],
+                                    "type": "text",
+                                    "content": "By assumption A4, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        149,
+                                        504,
+                                        174
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        149,
+                                        504,
+                                        174
+                                    ],
+                                    "type": "text",
+                                    "content": " is always within some compact set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        149,
+                                        504,
+                                        174
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Omega"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        149,
+                                        504,
+                                        174
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        149,
+                                        504,
+                                        174
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sup_n\\| \\mathbf{J}_n^{(1,a)}\\| \\leq C_\\Omega < \\infty"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        149,
+                                        504,
+                                        174
+                                    ],
+                                    "type": "text",
+                                    "content": " and for a given trajectory "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        149,
+                                        504,
+                                        174
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\omega"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        149,
+                                        504,
+                                        174
+                                    ],
+                                    "type": "text",
+                                    "content": " of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        149,
+                                        504,
+                                        174
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n(\\omega)"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        219,
+                        177,
+                        503,
+                        214
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                219,
+                                177,
+                                503,
+                                214
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        219,
+                                        177,
+                                        503,
+                                        214
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\gamma_ {n} C _ {p} \\sqrt {\\left(\\sum_ {k = 1} ^ {n} \\left\\| \\mathbf {J} _ {k} ^ {(1 , a)} \\right\\| ^ {2}\\right)} \\leq C _ {p} C _ {\\Omega} \\gamma_ {n} \\sqrt {n}, \\tag {46}",
+                                    "image_path": "fe7d55bd4d8887622bc843e68f3bfed414a6024cb8f9ebe683e1384389264a13.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
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+                        216,
+                        324,
+                        228
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                216,
+                                324,
+                                228
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        216,
+                                        324,
+                                        228
+                                    ],
+                                    "type": "text",
+                                    "content": "and the last term decreases to zero in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        216,
+                                        324,
+                                        228
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        216,
+                                        324,
+                                        228
+                                    ],
+                                    "type": "text",
+                                    "content": " since "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        216,
+                                        324,
+                                        228
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "a > 1 / 2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        216,
+                                        324,
+                                        228
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        233,
+                        307,
+                        247
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                233,
+                                307,
+                                247
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        307,
+                                        247
+                                    ],
+                                    "type": "text",
+                                    "content": "For "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        307,
+                                        247
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{J}_n^{(1,b)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        307,
+                                        247
+                                    ],
+                                    "type": "text",
+                                    "content": ", we use Abel transformation and obtain"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        179,
+                        250,
+                        431,
+                        297
+                    ],
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+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                250,
+                                431,
+                                297
+                            ],
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+                                {
+                                    "bbox": [
+                                        179,
+                                        250,
+                                        431,
+                                        297
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1, b)} = \\sum_ {k = 1} ^ {n} ((\\mathbf {K} _ {\\mathbf {x} _ {k}} \\varphi_ {\\mathbf {x} _ {k}} ^ {(1)}) (X _ {k - 1}) - (\\mathbf {K} _ {\\mathbf {x} _ {k - 1}} \\varphi_ {\\mathbf {x} _ {k - 1}} ^ {(1)}) (X _ {k - 1})) \\\\ + \\left(\\mathbf {K} _ {\\mathbf {x} _ {0}} \\varphi_ {\\mathbf {x} _ {0}} ^ {(1)}\\right) (X _ {0}) - \\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} \\varphi_ {\\mathbf {x} _ {n}} ^ {(1)}\\right) (X _ {n}). \\\\ \\end{array}",
+                                    "image_path": "60c070c41888811723e2ece9ffa1a454117c1346ef7d204b340fdbc36cdb8366.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        104,
+                        300,
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+                        324
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                300,
+                                504,
+                                324
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        504,
+                                        324
+                                    ],
+                                    "type": "text",
+                                    "content": "Since "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        504,
+                                        324
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\mathbf{K}_{\\mathbf{x}}\\varphi_{\\mathbf{x}}^{(1)})(X)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        504,
+                                        324
+                                    ],
+                                    "type": "text",
+                                    "content": " is continuous in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        504,
+                                        324
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        504,
+                                        324
+                                    ],
+                                    "type": "text",
+                                    "content": ", for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        504,
+                                        324
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        504,
+                                        324
+                                    ],
+                                    "type": "text",
+                                    "content": " within a compact set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        504,
+                                        324
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Omega"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        504,
+                                        324
+                                    ],
+                                    "type": "text",
+                                    "content": " (assumption A4), it is local Lipschitz with a constant "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        504,
+                                        324
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "L_{\\Omega}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        504,
+                                        324
+                                    ],
+                                    "type": "text",
+                                    "content": " such that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        151,
+                        327,
+                        460,
+                        341
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                151,
+                                327,
+                                460,
+                                341
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        151,
+                                        327,
+                                        460,
+                                        341
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left\\| \\right.\\left(\\mathbf {K} _ {\\mathbf {x} _ {k}} \\varphi_ {\\mathbf {x} _ {k}} ^ {(1)}\\right)\\left(X _ {k - 1}\\right) - \\mathbf {K} _ {\\mathbf {x} _ {k - 1}} \\varphi_ {\\mathbf {x} _ {k - 1}} ^ {(1)}\\left. \\right)\\left(X _ {k - 1}\\right)\\left. \\right\\| \\leq L _ {\\Omega} \\| \\mathbf {x} _ {k} - \\mathbf {x} _ {k - 1} \\| \\leq 2 L _ {\\Omega} \\gamma_ {k}.",
+                                    "image_path": "cd0b58ae7f1b4ea5d28898c7f489e7f55b35d33e16350de534cba0710d55c20c.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
+                        343,
+                        504,
+                        380
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                343,
+                                504,
+                                380
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        343,
+                                        504,
+                                        380
+                                    ],
+                                    "type": "text",
+                                    "content": "where the last inequality arises from (4b), i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        343,
+                                        504,
+                                        380
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_k - \\mathbf{x}_{k-1} = \\gamma_k (\\delta_{X_k} - \\mathbf{x}_{k-1})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        343,
+                                        504,
+                                        380
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        343,
+                                        504,
+                                        380
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\|\\delta_{X_k} - \\mathbf{x}_{k-1}\\| \\leq 2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        343,
+                                        504,
+                                        380
+                                    ],
+                                    "type": "text",
+                                    "content": " because "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        343,
+                                        504,
+                                        380
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n \\in \\mathrm{Int}(\\Sigma)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        343,
+                                        504,
+                                        380
+                                    ],
+                                    "type": "text",
+                                    "content": ". Also, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        343,
+                                        504,
+                                        380
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\|(K_{\\mathbf{x}_0} \\varphi_{\\mathbf{x}_0}^{(1)})(X_0)\\| + \\|(K_{\\mathbf{x}_n} \\varphi_{\\mathbf{x}_n}^{(1)})(X_n)\\|"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        343,
+                                        504,
+                                        380
+                                    ],
+                                    "type": "text",
+                                    "content": " are upper-bounded by some positive constant "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        343,
+                                        504,
+                                        380
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_\\Omega'"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        343,
+                                        504,
+                                        380
+                                    ],
+                                    "type": "text",
+                                    "content": ". This implies that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        237,
+                        383,
+                        374,
+                        415
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                237,
+                                383,
+                                374,
+                                415
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        237,
+                                        383,
+                                        374,
+                                        415
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1, b)} \\right\\| \\leq C _ {\\Omega} ^ {\\prime} + 2 L _ {\\Omega} \\sum_ {k = 1} ^ {n} \\gamma_ {k}.",
+                                    "image_path": "d3a6397d9a10c1d7f672cd754c0f9d352c235e1e0e5cfd33f2d3a1a4367ccfee.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
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+                        416,
+                        145,
+                        426
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                416,
+                                145,
+                                426
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        416,
+                                        145,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": "Note that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        172,
+                        426,
+                        503,
+                        456
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                172,
+                                426,
+                                503,
+                                456
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        172,
+                                        426,
+                                        503,
+                                        456
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\gamma_ {n} \\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1, b)} \\right\\| \\leq \\gamma_ {n} C _ {\\Omega} ^ {\\prime} + 2 L _ {\\Omega} \\gamma_ {n} \\sum_ {k = 1} ^ {n} \\gamma_ {k} \\leq \\gamma_ {n} C _ {\\Omega} ^ {\\prime} + \\frac {2 L _ {\\Omega}}{a} n ^ {1 - 2 a}, \\tag {47}",
+                                    "image_path": "53049da21f3861e748d7a6227e2e2ee9ab6cb200ec554ff88e9ca2fa42902f48.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        104,
+                        458,
+                        504,
+                        482
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                458,
+                                504,
+                                482
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        458,
+                                        504,
+                                        482
+                                    ],
+                                    "type": "text",
+                                    "content": "where the last inequality is from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        458,
+                                        504,
+                                        482
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sum_{k=1}^{n} \\gamma_k < \\frac{1}{a} n^{1-a}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        458,
+                                        504,
+                                        482
+                                    ],
+                                    "type": "text",
+                                    "content": ". We observe that the last term in (47) is decreasing to zero in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        458,
+                                        504,
+                                        482
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        458,
+                                        504,
+                                        482
+                                    ],
+                                    "type": "text",
+                                    "content": " because "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        458,
+                                        504,
+                                        482
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "a > 1/2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        458,
+                                        504,
+                                        482
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        104,
+                        487,
+                        369,
+                        502
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                487,
+                                369,
+                                502
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        369,
+                                        502
+                                    ],
+                                    "type": "text",
+                                    "content": "Note that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        369,
+                                        502
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{J}_k^{(1)} = \\mathbf{J}_k^{(1,a)} + \\mathbf{J}_k^{(1,b)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        369,
+                                        502
+                                    ],
+                                    "type": "text",
+                                    "content": ", by triangular inequality we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        144,
+                        505,
+                        504,
+                        618
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                144,
+                                505,
+                                504,
+                                618
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        144,
+                                        505,
+                                        504,
+                                        618
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\gamma_ {n} \\mathbb {E} \\left[ \\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1 1)} \\right\\| \\right] \\leq \\gamma_ {n} \\mathbb {E} \\left[ \\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1 1, A)} \\right\\| \\right] + \\gamma_ {n} \\mathbb {E} \\left[ \\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1 1, B)} \\right\\| \\right] \\\\ \\leq \\gamma_ {n} C _ {1} \\mathbb {E} \\left[ \\sqrt {\\left(\\sum_ {k = 1} ^ {n} \\left\\| \\mathbf {J} _ {k} ^ {(1 1 , A)} \\right\\| ^ {2}\\right)} \\right] + \\gamma_ {n} \\mathbb {E} \\left[ \\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1 1, B)} \\right\\| \\right] \\tag {48} \\\\ = \\mathbb {E} \\left[ \\gamma_ {n} C _ {1} \\sqrt {\\left(\\sum_ {k = 1} ^ {n} \\left\\| \\mathbf {J} _ {k} ^ {(1 1 , A)} \\right\\| ^ {2}\\right)} + \\gamma_ {n} \\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1 1 , B)} \\right\\| \\right], \\\\ \\end{array}",
+                                    "image_path": "41c65ebc10814aa3e117d7cee04cc81669be3f29e1512304e9ecc43500b6f492.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        104,
+                        620,
+                        504,
+                        653
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                620,
+                                504,
+                                653
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        620,
+                                        504,
+                                        653
+                                    ],
+                                    "type": "text",
+                                    "content": "where the second inequality comes from (45). By (46) and (47) we know that both terms in the last line of (48) are uniformly bounded by constants over time "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        620,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        620,
+                                        504,
+                                        653
+                                    ],
+                                    "type": "text",
+                                    "content": " that depend on the set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        620,
+                                        504,
+                                        653
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Omega"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        620,
+                                        504,
+                                        653
+                                    ],
+                                    "type": "text",
+                                    "content": ". Therefore, by dominated convergence theorem, taking the limit over the last line of (48) gives"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        176,
+                        656,
+                        435,
+                        735
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        176,
+                                        656,
+                                        435,
+                                        735
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\lim  _ {n \\rightarrow \\infty} \\mathbb {E} \\left[ \\gamma_ {n} C _ {1} \\sqrt {\\left(\\sum_ {k = 1} ^ {n} \\left\\| \\mathbf {J} _ {k} ^ {(1 1 , A)} \\right\\| ^ {2}\\right)} + \\gamma_ {n} \\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1 1, B)} \\right\\|\\right] \\\\ = \\mathbb {E} \\left[ \\lim  _ {n \\rightarrow \\infty} \\gamma_ {n} C _ {1} \\sqrt {\\left(\\sum_ {k = 1} ^ {n} \\left\\| \\mathbf {J} _ {k} ^ {(1 1 , A)} \\right\\| ^ {2}\\right)} + \\gamma_ {n} \\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1 1 , B)} \\right\\|\\right] = 0. \\\\ \\end{array}",
+                                    "image_path": "eb4c8d102e3f0c92e5ada74cdd2e9d6cdc52b7b82f534ded8522dbbb7fa8c413.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
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+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                38
+                            ],
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+                                    "bbox": [
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+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "24"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                }
+            ],
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+            ],
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+        {
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+                                    "type": "text",
+                                    "content": "Therefore, we have"
+                                }
+                            ]
+                        }
+                    ],
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        244,
+                                        91,
+                                        365,
+                                        125
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\lim  _ {n \\rightarrow \\infty} \\gamma_ {n} \\mathbb {E} \\left[\\left\\| \\sum_ {k = 1} ^ {n} \\mathbf {J} _ {k} ^ {(1)} \\right\\|\\right] = 0,",
+                                    "image_path": "e9b83e2285b0dd1da8e0eb14704a201f46763041420cfbbf82bbd7f97594f407.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        133,
+                        504,
+                        167
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                133,
+                                504,
+                                167
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        133,
+                                        504,
+                                        167
+                                    ],
+                                    "type": "text",
+                                    "content": "In sum, in terms of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        133,
+                                        504,
+                                        167
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{E}[M_{n + 1}^{(\\mathbf{x})}(M_{n + 1}^{(\\mathbf{x})})^T|\\mathcal{F}_n]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        133,
+                                        504,
+                                        167
+                                    ],
+                                    "type": "text",
+                                    "content": " in (41), we have "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        133,
+                                        504,
+                                        167
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{U}_{11}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        133,
+                                        504,
+                                        167
+                                    ],
+                                    "type": "text",
+                                    "content": " in (44), "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        133,
+                                        504,
+                                        167
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\lim_{n\\to \\infty}\\mathbf{D}_n^{(1)} = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        133,
+                                        504,
+                                        167
+                                    ],
+                                    "type": "text",
+                                    "content": " a.s. and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        133,
+                                        504,
+                                        167
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\lim_{n\\to \\infty}\\gamma_n\\mathbb{E}\\left[\\left\\| \\sum_{k = 1}^n\\mathbf{J}_k^{(1)}\\right\\|\\right] = 0."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        171,
+                        505,
+                        194
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                171,
+                                505,
+                                194
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        505,
+                                        194
+                                    ],
+                                    "type": "text",
+                                    "content": "We can apply the same steps as above for the other two terms "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        505,
+                                        194
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i = 2,3"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        505,
+                                        194
+                                    ],
+                                    "type": "text",
+                                    "content": " in (42) and obtain the results."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        105,
+                        202,
+                        290,
+                        217
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                202,
+                                290,
+                                217
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        202,
+                                        290,
+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": "Analysis of Terms "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        202,
+                                        290,
+                                        217
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r_n^{(\\pmb {\\theta},1)},r_n^{(\\pmb {\\theta},2)},r_n^{(\\mathbf{x},1)},r_n^{(\\mathbf{x},2)}"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        219,
+                        469,
+                        235
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                219,
+                                469,
+                                235
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        219,
+                                        469,
+                                        235
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma E.2. For "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        219,
+                                        469,
+                                        235
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r_n^{(\\theta, 1)}, r_n^{(\\theta, 2)}, r_n^{(\\mathbf{x}, 1)}, r_n^{(\\mathbf{x}, 2)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        219,
+                                        469,
+                                        235
+                                    ],
+                                    "type": "text",
+                                    "content": " defined in (35), we have the following results:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        160,
+                        237,
+                        504,
+                        269
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                160,
+                                237,
+                                504,
+                                269
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        160,
+                                        237,
+                                        504,
+                                        269
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left\\| r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} \\right\\| = O (\\gamma_ {n}) = o \\left(\\sqrt {\\beta_ {n}}\\right), \\quad \\sqrt {\\gamma_ {n}} \\left\\| \\sum_ {k = 1} ^ {n} r _ {k} ^ {(\\boldsymbol {\\theta}, 2)} \\right\\| = O \\left(\\sqrt {\\gamma_ {n}}\\right) = o (1). \\tag {49a}",
+                                    "image_path": "b6011623ca3e5369fa5d278c0104272ae91ff4a79752ee83f6e45e6a188c207d.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        160,
+                        273,
+                        504,
+                        305
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                160,
+                                273,
+                                504,
+                                305
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        160,
+                                        273,
+                                        504,
+                                        305
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left\\| r _ {n} ^ {(\\mathbf {x}, 1)} \\right\\| = O \\left(\\gamma_ {n}\\right) = o \\left(\\sqrt {\\beta_ {n}}\\right), \\quad \\sqrt {\\gamma_ {n}} \\left\\| \\sum_ {k = 1} ^ {n} r _ {k} ^ {(\\mathbf {x}, 2)} \\right\\| = O \\left(\\sqrt {\\gamma_ {n}}\\right) = o (1). \\tag {49b}",
+                                    "image_path": "772707df987383bac9422ed549468359512649c1e0b4aaf6cdd48621bb32c920.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        317,
+                        217,
+                        331
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                317,
+                                217,
+                                331
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        317,
+                                        217,
+                                        331
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. For "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        317,
+                                        217,
+                                        331
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r_n^{(\\theta, 1)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        317,
+                                        217,
+                                        331
+                                    ],
+                                    "type": "text",
+                                    "content": ", note that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        163,
+                        334,
+                        504,
+                        422
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                163,
+                                334,
+                                504,
+                                422
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        163,
+                                        334,
+                                        504,
+                                        422
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} = \\left(\\mathbf {K} _ {\\mathbf {x} _ {n + 1}} \\tilde {H} _ {\\boldsymbol {\\theta} _ {n + 1}, \\mathbf {x} _ {n + 1}}\\right) \\left(X _ {n + 1}\\right) - \\left(\\mathbf {K} _ {\\mathbf {x} _ {n}} \\tilde {H} _ {\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}}\\right) \\left(X _ {n + 1}\\right) \\\\ = \\sum_ {j \\in \\mathcal {N}} \\left(\\mathbf {K} _ {X _ {n}, j} [ \\mathbf {x} _ {n + 1} ] \\tilde {H} _ {\\boldsymbol {\\theta} _ {n + 1}, \\mathbf {x} _ {n + 1}} (j) - \\mathbf {K} _ {X _ {n}, j} [ \\mathbf {x} _ {n} ] \\tilde {H} _ {\\boldsymbol {\\theta} _ {n}, \\mathbf {x} _ {n}} (j)\\right) \\tag {50} \\\\ \\leq \\sum_ {j \\in \\mathcal {N}} L _ {\\mathcal {C}} \\left(\\left\\| \\boldsymbol {\\theta} _ {n + 1} - \\boldsymbol {\\theta} _ {n} \\right\\| + \\left\\| \\mathbf {x} _ {n + 1} - \\mathbf {x} _ {n} \\right\\|\\right) \\\\ \\leq N L _ {\\mathcal {C}} \\left(C _ {\\mathcal {C}} \\beta_ {n + 1} + 2 \\gamma_ {n + 1}\\right) \\\\ \\end{array}",
+                                    "image_path": "d3c2c1183c36c3caceede7743ed380e7de64f4ca1e5b741beaf322aed630b8ba.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        426,
+                        504,
+                        479
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                426,
+                                504,
+                                479
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "text",
+                                    "content": "where the second last inequality is because "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{K}_{i,j}[\\mathbf{x}]\\tilde{H}_{\\boldsymbol{\\theta},\\mathbf{x}}(j)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "text",
+                                    "content": " is continuous in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb {\\theta},\\mathbf{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{K}[\\mathbf{x}]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "text",
+                                    "content": ", which stems from continuous functions "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb {K}[\\mathbf{x}]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{H}_{\\pmb{\\theta},\\mathbf{x}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "text",
+                                    "content": ". The last inequality is from update rules (4) and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\pmb {\\theta}_n,\\mathbf{x}_n)\\in \\Omega"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "text",
+                                    "content": " for some compact subset "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Omega"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "text",
+                                    "content": " by assumption A4. Then, we have "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\| r_n^{(\\pmb {\\theta},1)}\\| = O(\\gamma_n) = o(\\sqrt{\\beta_n})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "text",
+                                    "content": " because of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "a > 1 / 2\\geq b / 2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "text",
+                                    "content": " by assumption A2."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        104,
+                        483,
+                        504,
+                        517
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                483,
+                                504,
+                                517
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        483,
+                                        504,
+                                        517
+                                    ],
+                                    "type": "text",
+                                    "content": "We let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        483,
+                                        504,
+                                        517
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\nu_{n} \\triangleq (\\mathbf{K}_{\\mathbf{x}_{n}}\\tilde{H}_{\\boldsymbol{\\theta}_{n},\\mathbf{x}_{n}})(X_{n})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        483,
+                                        504,
+                                        517
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        483,
+                                        504,
+                                        517
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r_n^{(\\pmb{\\theta},2)} = \\nu_n - \\nu_{n + 1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        483,
+                                        504,
+                                        517
+                                    ],
+                                    "type": "text",
+                                    "content": ". Note that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        483,
+                                        504,
+                                        517
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sum_{k = 1}^{n}r_k^{(\\pmb{\\theta},2)} = \\nu_1 - \\nu_{n + 1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        483,
+                                        504,
+                                        517
+                                    ],
+                                    "type": "text",
+                                    "content": " and by assumption A4, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        483,
+                                        504,
+                                        517
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\| \\nu_{n}\\|"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        483,
+                                        504,
+                                        517
+                                    ],
+                                    "type": "text",
+                                    "content": " is upper bounded by a constant dependent on the compact set, which leads to"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        185,
+                        517,
+                        424,
+                        548
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        185,
+                                        517,
+                                        424,
+                                        548
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\sqrt {\\gamma_ {n}} \\left\\| \\sum_ {k = 1} ^ {n} r _ {k} ^ {(\\boldsymbol {\\theta}, 2)} \\right\\| = \\sqrt {\\gamma_ {n}} \\| \\nu_ {1} - \\nu_ {n + 1} \\| = O (\\sqrt {\\gamma_ {n}}) = o (1).",
+                                    "image_path": "2fa90a0beacb246143674ca688cbbf3a5304076216a7e3c46b656e4a96f3b2db.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        104,
+                        556,
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+                        577
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                                577
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        556,
+                                        504,
+                                        577
+                                    ],
+                                    "type": "text",
+                                    "content": "Similarly, we can also obtain "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        556,
+                                        504,
+                                        577
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\| r_n^{(\\mathbf{x},1)}\\| = o(\\sqrt{\\beta_n})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        556,
+                                        504,
+                                        577
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        556,
+                                        504,
+                                        577
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sqrt{\\gamma_n}\\left\\| \\sum_{k = 1}^n r_k^{(\\mathbf{x},2)}\\right\\| = O(\\sqrt{\\gamma_n}) = o(1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        556,
+                                        504,
+                                        577
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        105,
+                        586,
+                        350,
+                        597
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "E.2.2 EFFECT OF SRRW ITERATION ON SA ITERATION"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
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+                    ],
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+                    "angle": 0,
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+                                    ],
+                                    "type": "text",
+                                    "content": "In view of the almost sure convergence results in Lemma 3.1 and Lemma 3.2, for large enough "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        606,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        606,
+                                        504,
+                                        640
+                                    ],
+                                    "type": "text",
+                                    "content": " so that both iterations "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        606,
+                                        504,
+                                        640
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta_{n},\\mathbf{x}_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        606,
+                                        504,
+                                        640
+                                    ],
+                                    "type": "text",
+                                    "content": " are close to the equilibrium "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        606,
+                                        504,
+                                        640
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\theta^{*},\\mu)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        606,
+                                        504,
+                                        640
+                                    ],
+                                    "type": "text",
+                                    "content": ", we can apply the Taylor expansion to functions "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        606,
+                                        504,
+                                        640
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{h}(\\theta ,\\mathbf{x})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        606,
+                                        504,
+                                        640
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        606,
+                                        504,
+                                        640
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi [\\mathbf{x}] - \\mathbf{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        606,
+                                        504,
+                                        640
+                                    ],
+                                    "type": "text",
+                                    "content": " in (36) at the point "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        606,
+                                        504,
+                                        640
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\theta^{*},\\mu)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        606,
+                                        504,
+                                        640
+                                    ],
+                                    "type": "text",
+                                    "content": ", which results in"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
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+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {h} (\\boldsymbol {\\theta}, \\mathbf {x}) = \\mathbf {h} \\left(\\boldsymbol {\\theta} ^ {*}, \\boldsymbol {\\mu}\\right) + \\nabla_ {\\boldsymbol {\\theta}} \\mathbf {h} \\left(\\boldsymbol {\\theta} ^ {*}, \\boldsymbol {\\mu}\\right) \\left(\\boldsymbol {\\theta} - \\boldsymbol {\\theta} ^ {*}\\right) + \\nabla_ {\\mathbf {x}} \\mathbf {h} \\left(\\boldsymbol {\\theta} ^ {*}, \\boldsymbol {\\mu}\\right) (\\mathbf {x} - \\boldsymbol {\\mu}) + O \\left(\\left\\| \\boldsymbol {\\theta} - \\boldsymbol {\\theta} ^ {*} \\right\\| ^ {2} + \\| \\mathbf {x} - \\boldsymbol {\\mu} \\| ^ {2}\\right), \\tag {51a}",
+                                    "image_path": "5d1ca90d8e1ea9f002d178b9d8ab1bcdc9a6d8800d1e1078e4f834f0d0f46dec.jpg"
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+                            "bbox": [
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\pi [ \\mathbf {x} ] - \\mathbf {x} = \\pi [ \\boldsymbol {\\mu} ] - \\boldsymbol {\\mu} + \\nabla_ {\\mathbf {x}} (\\pi (\\mathbf {x}) - \\mathbf {x}) | _ {\\mathbf {x} = \\boldsymbol {\\mu}} (\\mathbf {x} - \\boldsymbol {\\mu}) + O (\\| \\mathbf {x} - \\boldsymbol {\\mu} \\| ^ {2}). \\tag {51b}",
+                                    "image_path": "eebc5c1440ef3a6de1997866f108cc829bc86bd662269cf26f349ed11f61da20.jpg"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "With matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{J}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ", we have the following:"
+                                }
+                            ]
+                        }
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {J} _ {1 1} = \\nabla_ {\\boldsymbol {\\theta}} \\mathbf {h} \\left(\\boldsymbol {\\theta} ^ {*}, \\boldsymbol {\\mu}\\right) = \\nabla \\mathbf {h} \\left(\\boldsymbol {\\theta} ^ {*}\\right),",
+                                    "image_path": "3207efd6d2d9328cd7ad54b4b1eb7f2ba20820735fb3d2411c7c8825e0bfd45d.jpg"
+                                }
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+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {J} _ {1 2} (\\alpha) = \\nabla_ {\\mathbf {x}} \\mathbf {h} \\left(\\boldsymbol {\\theta} ^ {*}, \\boldsymbol {\\mu}\\right) = - \\alpha \\mathbf {H} ^ {T} \\left(\\mathbf {P} ^ {T} + \\mathbf {I}\\right), \\tag {52}",
+                                    "image_path": "fbf23134c926996c690cd17ae09829dd357eb55855a1780077cd9046df45cda0.jpg"
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {J} _ {2 2} (\\alpha) = \\nabla_ {\\mathbf {x}} (\\boldsymbol {\\pi} (\\mathbf {x}) - \\mathbf {x}) | _ {\\mathbf {x} = \\boldsymbol {\\mu}} = 2 \\alpha \\boldsymbol {\\mu} \\mathbf {1} ^ {T} - \\alpha \\mathbf {P} ^ {T} - (\\alpha + 1) \\mathbf {I}.",
+                                    "image_path": "758e2c1ca251b45ca365ff2161b3e2b8382eec30fd4d431954de95891d2e5817.jpg"
+                                }
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+            "discarded_blocks": [
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
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+                                    "type": "text",
+                                    "content": "25"
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Then, (36) becomes"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
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+                                    "type": "interline_equation",
+                                    "content": "\\boldsymbol {\\theta} _ {n + 1} = \\boldsymbol {\\theta} _ {n} + \\beta_ {n + 1} \\left(\\mathbf {J} _ {1 1} \\left(\\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*}\\right) + \\mathbf {J} _ {1 2} (\\alpha) \\left(\\mathbf {x} _ {n} - \\boldsymbol {\\mu}\\right) + r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} + r _ {n} ^ {(\\boldsymbol {\\theta}, 2)} + M _ {n + 1} ^ {(\\boldsymbol {\\theta})} + \\eta_ {n} ^ {(\\boldsymbol {\\theta})}\\right), \\tag {53a}",
+                                    "image_path": "ffb958925b73d59babad3e4232be0a543ebd33faa21eaf99ab8df23905b3a866.jpg"
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {x} _ {n + 1} = \\mathbf {x} _ {n} + \\gamma_ {n + 1} \\left(\\mathbf {J} _ {2 2} (\\alpha) \\left(\\mathbf {x} _ {n} - \\boldsymbol {\\mu}\\right) + r _ {n} ^ {(\\mathbf {x}, 1)} + r _ {n} ^ {(\\mathbf {x}, 2)} + M _ {n + 1} ^ {(\\mathbf {x})} + \\eta_ {n} ^ {(\\mathbf {x})}\\right), \\tag {53b}",
+                                    "image_path": "de29703282f8c5cea9232c3418f2f05be92f143adab719351d6b890e70813114.jpg"
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+                            "bbox": [
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+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\eta_n^{(\\pmb{\\theta})} = O(\\| \\mathbf{x}_n\\|^2 + \\| \\pmb{\\theta}_n\\|^2)"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "type": "text",
+                                    "content": " and "
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+                                    "bbox": [
+                                        105,
+                                        134,
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+                                        150
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\eta_n^{(\\mathbf{x})} = O(\\| \\mathbf{x}_n\\|^2)"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        105,
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Then, inspired by Mokkadem & Pelletier (2006), we decompose iterates "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\{\\mathbf{x}_n\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
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+                                        179
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
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+                                        179
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\pmb {\\theta}_n\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "type": "text",
+                                    "content": " into "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        153,
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+                                        179
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n ="
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        153,
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+                                        179
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "L_{n}^{(\\pmb {x})} + \\Delta_{n}^{(\\pmb {x})}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        153,
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+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        153,
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+                                        179
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb {\\theta}_n = L_n^{(\\pmb {\\theta})} + R_n^{(\\pmb {\\theta})} + \\Delta_n^{(\\pmb {\\theta})}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        153,
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+                                        179
+                                    ],
+                                    "type": "text",
+                                    "content": ". Rewriting (53b) gives"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
+                                        132,
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+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {x} _ {n} - \\boldsymbol {\\mu} = \\gamma_ {n + 1} ^ {- 1} \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (\\mathbf {x} _ {n + 1} - \\mathbf {x} _ {n}) - \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (r _ {n} ^ {(\\mathbf {x}, 1)} + r _ {n} ^ {(\\mathbf {x}, 2)} + M _ {n + 1} ^ {(\\mathbf {x})} + \\eta_ {n} ^ {(\\mathbf {x})}),",
+                                    "image_path": "d7cc5a62b973d1f5d9f1ee1a37393ecaf06fa19e2bd330ec8031120bb6f1388c.jpg"
+                                }
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+                    "index": 6
+                },
+                {
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "type": "text",
+                                    "content": "and substituting the above equation back in (53a) gives"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\boldsymbol {\\theta} _ {n + 1} - \\boldsymbol {\\theta} ^ {*} = \\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*} + \\beta_ {n + 1} \\left(\\mathbf {J} _ {1 1} \\left(\\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*}\\right) + \\gamma_ {n + 1} ^ {- 1} \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(\\mathbf {x} _ {n + 1} - \\mathbf {x} _ {n}\\right) \\right. \\\\ \\left. - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(r _ {n} ^ {(\\mathbf {x}, 1)} + r _ {n} ^ {(\\mathbf {x}, 2)} + M _ {n + 1} ^ {(\\mathbf {x})} + \\eta_ {n} ^ {(\\mathbf {x})}\\right) + r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} + r _ {n} ^ {(\\boldsymbol {\\theta}, 2)} + M _ {n + 1} ^ {(\\boldsymbol {\\theta})} + \\eta_ {n} ^ {(\\boldsymbol {\\theta})}\\right) \\\\ = (\\mathbf {I} + \\beta_ {n + 1} \\mathbf {J} _ {1 1}) \\left(\\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*}\\right) + \\left[ \\beta_ {n + 1} \\gamma_ {n + 1} ^ {- 1} \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(\\mathbf {x} _ {n + 1} - \\mathbf {x} _ {n}\\right) \\right] \\\\ + \\beta_ {n + 1} (M _ {n + 1} ^ {(\\boldsymbol {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} M _ {n + 1} ^ {(\\mathbf {x})}) \\\\ + \\beta_ {n + 1} \\left(r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} + r _ {n} ^ {(\\boldsymbol {\\theta}, 2)} + \\eta_ {n} ^ {(\\boldsymbol {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(r _ {n} ^ {(\\mathbf {x}, 1)} + r _ {n} ^ {(\\mathbf {x}, 2)} + \\eta_ {n} ^ {(\\mathbf {x})}\\right)\\right), \\tag {54} \\\\ \\end{array}",
+                                    "image_path": "c7dd12588bad81855c51e1cddf72b28045719e817be1bdeb18958b820f890988.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
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+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "From (54) we can see the iteration "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\{\\theta_n\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " implicitly embeds the recursions of three sequences"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
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+                    "type": "list",
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+                            "angle": 0,
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+                                            "type": "text",
+                                            "content": "- "
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+                                            "type": "inline_equation",
+                                            "content": "\\beta_{n+1} \\gamma_{n+1}^{-1} \\mathbf{J}_{12}(\\alpha) \\mathbf{J}_{22}(\\alpha)^{-1} (\\mathbf{x}_{n+1} - \\mathbf{x}_n)"
+                                        },
+                                        {
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+                                            "type": "text",
+                                            "content": ";"
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+                            "type": "text",
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+                                            "type": "text",
+                                            "content": "- "
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+                                            "bbox": [
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+                                            "type": "inline_equation",
+                                            "content": "\\beta_{n+1}(M_{n+1}^{(\\boldsymbol{\\theta})} - \\mathbf{J}_{12}(\\alpha)\\mathbf{J}_{22}(\\alpha)^{-1}M_{n+1}^{(\\mathbf{x})})"
+                                        },
+                                        {
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+                                            "type": "text",
+                                            "content": ";"
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+                                    ]
+                                }
+                            ],
+                            "index": 11
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": "- "
+                                        },
+                                        {
+                                            "bbox": [
+                                                133,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_{n+1}(r_n^{(\\pmb{\\theta}, 1)} + r_n^{(\\pmb{\\theta}, 2)} + \\eta_n^{(\\pmb{\\theta})} - \\mathbf{J}_{12}(\\alpha) \\mathbf{J}_{22}(\\alpha)^{-1}(r_n^{(\\mathbf{x}, 1)} + r_n^{(\\mathbf{x}, 2)} + \\eta_n^{(\\mathbf{x}))})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                133,
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+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 12
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        104,
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+                        429
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        415,
+                                        399,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u_{n} \\triangleq \\sum_{k=1}^{n} \\beta_{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        415,
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+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        415,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s_{n} \\triangleq \\sum_{k=1}^{n} \\gamma_{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        415,
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+                                    ],
+                                    "type": "text",
+                                    "content": ". Below we define two iterations:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} L _ {n} ^ {(\\boldsymbol {\\theta})} = e ^ {\\beta_ {n} \\mathbf {J} _ {1 1}} L _ {n - 1} ^ {(\\boldsymbol {\\theta})} + \\beta_ {n} \\left(M _ {n} ^ {(\\boldsymbol {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} M _ {n} ^ {(\\mathbf {x})}\\right) \\\\ = \\sum_ {k = 1} ^ {n} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} \\beta_ {k} \\left(M _ {k} ^ {(\\boldsymbol {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} M _ {k} ^ {(\\mathbf {x})}\\right) \\tag {55a} \\\\ \\end{array}",
+                                    "image_path": "58650d6456b41a08249a23e8944e25c6ddb689410a01d01cf3f96b0536a7e8f4.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        164,
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+                                        503,
+                                        533
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} R _ {n} ^ {(\\boldsymbol {\\theta})} = e ^ {\\beta_ {n} \\mathbf {J} _ {1 1}} R _ {n - 1} ^ {(\\boldsymbol {\\theta})} + \\beta_ {n} \\gamma_ {n} ^ {- 1} \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (\\mathbf {x} _ {n} - \\mathbf {x} _ {n - 1}) \\\\ = \\sum_ {k = 1} ^ {n} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} \\beta_ {k} \\gamma_ {k} ^ {- 1} \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(\\mathbf {x} _ {k} - \\mathbf {x} _ {k - 1}\\right) \\tag {55b} \\\\ \\end{array}",
+                                    "image_path": "bdbaa99919d9aa09aa6ca6226f062b660136c0cc44a1d7c08e25c71b7ca8f4ab.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                        551
+                                    ],
+                                    "type": "text",
+                                    "content": "and a remaining term "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta_n^{(\\pmb{\\theta})} \\triangleq \\pmb{\\theta}_n - \\pmb{\\theta}^* - L_n^{(\\pmb{\\theta})} - R_n^{(\\pmb{\\theta})}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        105,
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Similarly, for iteration "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        557,
+                                        353,
+                                        570
+                                    ],
+                                    "type": "text",
+                                    "content": ", define the sequence "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        557,
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+                                        570
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "L_n^{(\\mathbf{x})}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
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+                                        570
+                                    ],
+                                    "type": "text",
+                                    "content": " such that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "L _ {n} ^ {(\\mathbf {x})} = e ^ {\\gamma_ {n} \\mathbf {J} _ {2 2} (\\alpha)} L _ {n - 1} ^ {(\\mathbf {x})} + \\gamma_ {n} M _ {n} ^ {(\\mathbf {x})} = \\sum_ {k = 1} ^ {n} e ^ {(s _ {n} - s _ {k}) \\mathbf {J} _ {2 2} (\\alpha)} \\gamma_ {k} M _ {k} ^ {(\\mathbf {x})}, \\tag {56}",
+                                    "image_path": "79a9e5b6ca07a2e7fe305cadbd43bd3bf5770620c7208216dc59173f87630023.jpg"
+                                }
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+                        }
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+                },
+                {
+                    "bbox": [
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+                        620
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "and a remaining term"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\Delta_ {n} ^ {(\\mathbf {x})} \\triangleq \\mathbf {x} _ {n} - \\boldsymbol {\\mu} - L _ {n} ^ {(\\boldsymbol {\\theta})} \\tag {57}",
+                                    "image_path": "f0adcd769318afa6d81a1ca7e206be4b89d77a67130c511ec5b53b9b04456aee.jpg"
+                                }
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+                        }
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+                    "index": 21
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "The decomposition of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta_{n} - \\theta^{*}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        634,
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+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n - \\pmb{\\mu}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        634,
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+                                    ],
+                                    "type": "text",
+                                    "content": " in the above form is also standard in the single-timescale SA literature (Delyon, 2000; Fort, 2015)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 22
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Characterization of Sequences "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{L_n^{(\\theta)}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        662,
+                                        317,
+                                        677
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{L_n^{(\\mathbf{x})}\\}"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 23
+                },
+                {
+                    "bbox": [
+                        105,
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+                        696
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "type": "text",
+                                    "content": "we set a Martingale "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Z^{(n)} = \\{Z_k^{(n)}\\}_{k\\geq 1}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "type": "text",
+                                    "content": " such that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 24
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "Z _ {k} ^ {(n)} = \\left( \\begin{array}{c c} \\beta_ {n} ^ {- 1 / 2} e ^ {u _ {n} \\mathbf {J} _ {1 1}} & 0 \\\\ 0 & \\gamma_ {n} ^ {- 1 / 2} e ^ {s _ {n} \\mathbf {J} _ {2 2} (\\alpha)} \\end{array} \\right) \\times \\sum_ {j = 1} ^ {k} \\left( \\begin{array}{c} e ^ {- u _ {k} \\mathbf {J} _ {1 1}} \\beta_ {k} (M _ {k} ^ {(\\boldsymbol {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} M _ {k} ^ {(\\mathbf {x})}) \\\\ e ^ {- s _ {k} \\mathbf {J} _ {2 2} (\\alpha)} \\gamma_ {k} M _ {k} ^ {(\\mathbf {x})} \\end{array} \\right).",
+                                    "image_path": "294515e03603c0aa6a46e9fdf29860dd03339406e6685bda8f9a480d14cffddb.jpg"
+                                }
+                            ]
+                        }
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
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+                                    "type": "text",
+                                    "content": "26"
+                                }
+                            ]
+                        }
+                    ],
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+                }
+            ],
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Then, the Martingale difference array "
+                                },
+                                {
+                                    "bbox": [
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+                                        80,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Z_{k}^{(n)} - Z_{k - 1}^{(n)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        352,
+                                        96
+                                    ],
+                                    "type": "text",
+                                    "content": " becomes"
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)} = \\left( \\begin{array}{c} \\beta_ {n} ^ {- 1 / 2} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\beta_ {k} (M _ {k} ^ {(\\boldsymbol {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} M _ {k} ^ {(\\mathbf {x})}) \\\\ \\gamma_ {n} ^ {- 1 / 2} e ^ {(s _ {n} - s _ {k}) \\mathbf {J} _ {2 2} (\\alpha)} \\gamma_ {k} M _ {k} ^ {(\\mathbf {x})} \\end{array} \\right)",
+                                    "image_path": "6e44aaa70c58f0ec56b4da59d7cae10abef5cf54aba66aae5cc768dab65cbff9.jpg"
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+                    "index": 2
+                },
+                {
+                    "bbox": [
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+                        143
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": "and"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                172
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        168,
+                                        141,
+                                        440,
+                                        172
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\sum_ {k = 1} ^ {n} \\mathbb {E} \\left[ (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) ^ {T} | \\mathcal {F} _ {k - 1} \\right] = \\left( \\begin{array}{c c} A _ {1, n} & A _ {2, n} \\\\ A _ {2, n} ^ {T} & A _ {4, n} \\end{array} \\right),",
+                                    "image_path": "58e10116b893d2aceb1b670a840ef3eb8f299c7bd799781bc2e1dd8bbe4a36ba.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
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+                                        189
+                                    ],
+                                    "type": "text",
+                                    "content": "where, in view of decomposition of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        436,
+                                        189
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M_{n}^{(\\pmb{\\theta})}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        436,
+                                        189
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        436,
+                                        189
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M_{n}^{(\\mathbf{x})}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        436,
+                                        189
+                                    ],
+                                    "type": "text",
+                                    "content": " in (41) and (42), respectively,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        116,
+                        191,
+                        504,
+                        278
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                116,
+                                191,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        191,
+                                        504,
+                                        278
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} A _ {1, n} = \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\beta_ {k} ^ {2} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\left(\\mathbf {U} _ {2 2} + \\mathbf {D} _ {k} ^ {(1)} + \\mathbf {J} _ {k} ^ {(1)} - \\left(\\mathbf {U} _ {2 1} + \\mathbf {D} _ {k} ^ {(2)} + \\mathbf {J} _ {k} ^ {(2)}\\right) \\left(\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}\\right) ^ {T} \\right. \\\\ + \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(\\mathbf {U} _ {1 1} + \\mathbf {D} _ {k} ^ {(3)} + \\mathbf {J} _ {k} ^ {(3)}\\right) \\left(\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}\\right) ^ {T} \\\\ \\left. - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(\\mathbf {U} _ {2 1} + \\mathbf {D} _ {k} ^ {(2)} + \\mathbf {J} _ {k} ^ {(2)}\\right) ^ {T}\\right) e ^ {\\left(u _ {n} - u _ {k}\\right) \\left(\\mathbf {J} _ {1 1}\\right) ^ {T}}, \\tag {58a} \\\\ \\end{array}",
+                                    "image_path": "417ff6e1540c6bf7c46dc7dcb5013bb543c3e926fc405257e55add10b06bb2cf.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
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+                        313
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                114,
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+                                504,
+                                313
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        114,
+                                        282,
+                                        504,
+                                        313
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "A _ {2, n} = \\beta_ {n} ^ {- 1 / 2} \\gamma_ {n} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} \\beta_ {k} \\gamma_ {k} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} \\left(\\mathbf {U} _ {2 1} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\mathbf {U} _ {1 1}\\right) e ^ {\\left(s _ {n} - s _ {k}\\right) \\mathbf {J} _ {2 2} (\\alpha) ^ {T}}, \\tag {58b}",
+                                    "image_path": "e1a9559afe5316cfff141088878b5a6c549e47fed62a57ed880235711f6ba293.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        159,
+                        316,
+                        504,
+                        346
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                159,
+                                316,
+                                504,
+                                346
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        159,
+                                        316,
+                                        504,
+                                        346
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "A _ {4, n} = \\gamma_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\gamma_ {k} ^ {2} e ^ {\\left(s _ {n} - s _ {k}\\right) \\mathbf {J} _ {2 2} (\\alpha)} \\left(\\mathbf {U} _ {1 1} + \\mathbf {D} _ {k} ^ {(3)} + \\mathbf {J} _ {k} ^ {(3)}\\right) e ^ {\\left(s _ {n} - s _ {k}\\right) \\mathbf {J} _ {2 2} (\\alpha) ^ {T}}. \\tag {58c}",
+                                    "image_path": "41bbbcec6a4cc5779409468e6a05a9358b954899c0fd1c4e9ba4e25fbedec5af.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
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+                        358
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        346,
+                                        285,
+                                        358
+                                    ],
+                                    "type": "text",
+                                    "content": "We further decompose "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        346,
+                                        285,
+                                        358
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A_{1,n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        346,
+                                        285,
+                                        358
+                                    ],
+                                    "type": "text",
+                                    "content": " into three parts:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        108,
+                        361,
+                        504,
+                        572
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        108,
+                                        361,
+                                        504,
+                                        572
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} A _ {1, n} = \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\left(\\beta_ {k} ^ {2} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} \\left(\\mathbf {U} _ {2 2} - \\mathbf {U} _ {2 1} \\left(\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}\\right) ^ {T} \\right. \\right. \\\\ \\left. \\left. - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\mathbf {U} _ {1 2} + \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\mathbf {U} _ {1 1} \\left(\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}\\right) ^ {T}\\right) e ^ {\\left(u _ {n} - u _ {k}\\right) \\left(\\mathbf {J} _ {1 1}\\right) ^ {T}}\\right) \\\\ + \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\left(\\beta_ {k} ^ {2} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} \\left(\\mathbf {D} _ {k} ^ {(1)} + \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\mathbf {D} _ {k} ^ {(3)} \\left(\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}\\right) ^ {T} \\right. \\right. \\\\ \\left. - \\mathbf {D} _ {k} ^ {(2)} \\left(\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}\\right) ^ {T} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(\\mathbf {D} _ {k} ^ {(2)}\\right) ^ {T}\\right) e ^ {\\left(u _ {n} - u _ {k}\\right) \\left(\\mathbf {J} _ {1 1}\\right) ^ {T}} \\\\ + \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\left(\\beta_ {k} ^ {2} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} (\\mathbf {J} _ {k} ^ {(1)} + \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\mathbf {J} _ {k} ^ {(3)} (\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}) ^ {T} \\right. \\\\ \\left. - \\mathbf {J} _ {k} ^ {(2)} \\left(\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}\\right) ^ {T} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(\\mathbf {J} _ {k} ^ {(2)}\\right) ^ {T}\\right) e ^ {\\left(u _ {n} - u _ {k}\\right) \\left(\\mathbf {J} _ {1 1}\\right) ^ {T}} \\\\ \\triangleq A _ {1, n} ^ {(a)} + A _ {1, n} ^ {(b)} + A _ {1, n} ^ {(c)}. \\tag {59} \\\\ \\end{array}",
+                                    "image_path": "c3e0d71dbfb35b8499c80a75dce4d9bf45dd13f8f1ebb142ff6dc7071282d7c9.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        576,
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+                        601
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                576,
+                                504,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        576,
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+                                        601
+                                    ],
+                                    "type": "text",
+                                    "content": "Here, we define "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        576,
+                                        504,
+                                        601
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{U}_{\\pmb{\\theta}}(\\alpha) \\triangleq \\mathbf{U}_{22} - \\mathbf{U}_{21}(\\mathbf{J}_{12}(\\alpha)\\mathbf{J}_{22}(\\alpha)^{-1})^T - \\mathbf{J}_{12}(\\alpha)\\mathbf{J}_{22}(\\alpha)^{-1}\\mathbf{U}_{12} + \\mathbf{J}_{12}(\\alpha)\\mathbf{J}_{22}(\\alpha)^{-1}\\mathbf{U}_{11} + \\mathbf{J}_{12}(\\alpha)\\mathbf{J}_{22}(\\alpha)^{-1}\\mathbf{J}_{12}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        576,
+                                        504,
+                                        601
+                                    ],
+                                    "type": "text",
+                                    "content": ". By (52) and (43a) in Lemma E.1, we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        194,
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+                        504,
+                        635
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        194,
+                                        604,
+                                        504,
+                                        635
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {U} _ {\\boldsymbol {\\theta}} (\\alpha) = \\sum_ {i = 1} ^ {N - 1} \\frac {1}{(\\alpha (1 + \\lambda_ {i}) + 1) ^ {2}} \\cdot \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H}. \\tag {60}",
+                                    "image_path": "aed35f67be2884c5b51faf7c883f0886907c81e99b925e2015539f835ed2470b.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        104,
+                        638,
+                        254,
+                        649
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                638,
+                                254,
+                                649
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        638,
+                                        254,
+                                        649
+                                    ],
+                                    "type": "text",
+                                    "content": "Then, we have the following lemma."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        104,
+                        651,
+                        346,
+                        667
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        346,
+                                        667
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma E.3. For "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        346,
+                                        667
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A_{1,n}^{(a)}, A_{1,n}^{(b)}, A_{1,n}^{(c)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        346,
+                                        667
+                                    ],
+                                    "type": "text",
+                                    "content": " defined in (59), we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
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+                        689
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\lim  _ {n \\rightarrow \\infty} A _ {1, n} ^ {(a)} = \\mathbf {V} _ {\\boldsymbol {\\theta}} (\\alpha), \\quad \\lim  _ {n \\rightarrow \\infty} \\| A _ {1, n} ^ {(b)} \\| = 0, \\quad \\lim  _ {n \\rightarrow \\infty} \\| A _ {1, n} ^ {(c)} \\| = 0, \\tag {61}",
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+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\theta}(\\alpha)"
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+                                    "type": "text",
+                                    "content": " is the solution to the Lyapunov equation"
+                                }
+                            ]
+                        }
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+                                    "content": "\\left(\\mathbf {J} _ {1 1} + \\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\boldsymbol {I}\\right) \\mathbf {V} _ {\\boldsymbol {\\theta}} (\\alpha) + \\mathbf {V} _ {\\boldsymbol {\\theta}} (\\alpha) \\left(\\mathbf {J} _ {1 1} + \\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\boldsymbol {I}\\right) ^ {T} + \\mathbf {U} _ {\\boldsymbol {\\theta}} (\\alpha) = 0.",
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
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+                                    "content": "27"
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+                                    "type": "text",
+                                    "content": "Proof. First, from Lemma G.4, we have for some "
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+                                    "content": "c, T > 0"
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+                                    "type": "text",
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+                                    "content": "\\begin{array}{l} \\| A _ {1, n} ^ {(b)} \\| \\leq \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\left\\| \\mathbf {D} _ {k} ^ {(1)} + \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\mathbf {D} _ {k} ^ {(3)} (\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}) ^ {T} - \\mathbf {D} _ {k} ^ {(2)} (\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}) ^ {T} \\right. \\\\ \\left. - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(\\mathbf {D} _ {k} ^ {(2)}\\right) ^ {T} \\right\\| \\cdot \\beta_ {k} ^ {2} c ^ {2} e ^ {- 2 T \\left(u _ {n} - u _ {k}\\right)}. \\\\ \\end{array}",
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+                                    "type": "text",
+                                    "content": "Applying Lemma G.6, together with "
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+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{D}_n^{(i)}\\to 0"
+                                },
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+                                    "type": "text",
+                                    "content": " a.s. in Lemma E.1, gives"
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+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\lim \\sup \\| A _ {1, n} ^ {(b)} \\| \\\\ \\leq \\frac {1}{C (b , p)} \\lim  _ {n} \\sup  _ {n} \\| (\\mathbf {D} _ {n} ^ {(1)} + \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\mathbf {D} _ {n} ^ {(4)} (\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}) ^ {T} \\\\ \\left. - \\mathbf {D} _ {n} ^ {(2)} \\left(\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}\\right) ^ {T} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\mathbf {D} _ {n} ^ {(3)}\\right) \\| \\\\ = 0. \\\\ \\end{array}",
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+                                    "type": "text",
+                                    "content": "We now consider "
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+                                    "type": "inline_equation",
+                                    "content": "\\| A_{1,n}^{(c)}\\|"
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+                                    "type": "text",
+                                    "content": ". Set"
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+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\Xi_ {n} \\triangleq \\sum_ {k = 1} ^ {n} \\left(\\mathbf {J} _ {n} ^ {(1)} + \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\mathbf {J} _ {k} ^ {(3)} (\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}) ^ {T} \\right. \\\\ - \\mathbf {J} _ {k} ^ {(2)} (\\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1}) ^ {T} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (\\mathbf {J} _ {k} ^ {(2)}) ^ {T}), \\\\ \\end{array}",
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+                                    "content": "we can rewrite "
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+                                    "type": "text",
+                                    "content": " as"
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+                                    "type": "interline_equation",
+                                    "content": "A _ {1, n} ^ {(c)} = \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\beta_ {k} ^ {2} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} (\\Xi_ {k} - \\Xi_ {k - 1}) e ^ {(u _ {n} - u _ {k}) (\\mathbf {J} _ {1 1}) ^ {T}}.",
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+                                    "type": "text",
+                                    "content": "By the Abel transformation, we have"
+                                }
+                            ]
+                        }
+                    ],
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+                },
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+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} A _ {1, n} ^ {(c)} = \\beta_ {n} \\Xi_ {n} + \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n - 1} \\left[ \\beta_ {k} ^ {2} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} \\Xi_ {k} e ^ {\\left(u _ {n} - u _ {k}\\right) \\left(\\mathbf {J} _ {1 1}\\right) ^ {T}} \\right. \\tag {62} \\\\ \\left. - \\beta_ {k + 1} ^ {2} e ^ {(u _ {n} - u _ {k + 1}) \\mathbf {J} _ {1 1}} \\Xi_ {k} e ^ {(u _ {n} - u _ {k + 1}) (\\mathbf {J} _ {1 1}) ^ {T}} \\right]. \\\\ \\end{array}",
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+                                    "type": "text",
+                                    "content": "We know from Lemma E.1 that "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "\\beta_{n}\\Xi_{n}\\to 0"
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": " a.s. because "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "\\Xi_n = o(\\gamma_n^{-1})"
+                                },
+                                {
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+                                    ],
+                                    "type": "text",
+                                    "content": ". Besides,"
+                                }
+                            ]
+                        }
+                    ],
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+                },
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+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\left\\| \\beta_ {k} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} - \\beta_ {k + 1} e ^ {\\left(u _ {n} - u _ {k + 1}\\right) \\mathbf {J} _ {1 1}} \\right\\| \\\\ = \\| \\left(\\beta_ {k} - \\beta_ {k + 1}\\right) e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} + \\beta_ {k + 1} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} \\left(\\mathbf {I} - e ^ {- \\beta_ {k + 1} \\mathbf {J} _ {1 1}}\\right) \\| \\\\ \\leq C _ {1} \\beta_ {k} ^ {2} e ^ {- (u _ {n} - u _ {k}) T} \\\\ \\end{array}",
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+                                }
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+                                    "type": "text",
+                                    "content": "for some constant "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "C_1 > 0"
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": " because "
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+                                    "type": "inline_equation",
+                                    "content": "\\beta_{n} - \\beta_{n + 1}\\leq C_{2}\\beta_{n}^{2}"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " and "
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+                                    "type": "inline_equation",
+                                    "content": "\\| \\mathbf{I} - e^{-\\beta_{k + 1}\\mathbf{J}_{11}}\\| \\leq C_3\\beta_{k + 1}"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ". Moreover,"
+                                }
+                            ]
+                        }
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+                },
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                        558,
+                                        443,
+                                        604
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\left\\| \\beta_ {k} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} \\right\\| + \\left\\| \\beta_ {k + 1} e ^ {\\left(u _ {n} - u _ {k + 1}\\right) \\mathbf {J} _ {1 1}} \\right\\| \\\\ \\leq \\beta_ {k} \\| e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\| + \\beta_ {k} \\| e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\| \\cdot \\| e ^ {- \\beta_ {k + 1} \\mathbf {J} _ {1 1}} \\| \\\\ \\leq C _ {4} \\beta_ {k} e ^ {- (u _ {n} - u _ {k}) T}. \\\\ \\end{array}",
+                                    "image_path": "156ebd29c867a1376b8426e55e7218491abf8bcab522d3b8b3085cad529b80fd.jpg"
+                                }
+                            ]
+                        }
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+                {
+                    "bbox": [
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+                        621
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+                    "angle": 0,
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Using Lemma G.7 on (62) gives"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\| A _ {1, n} ^ {(c)} \\| \\leq C _ {1} C _ {4} \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n - 1} \\beta_ {k} ^ {2} e ^ {- 2 (u _ {n} - u _ {k}) T} \\| \\beta_ {k} \\Xi_ {k} \\| + \\| \\beta_ {n} \\Xi_ {n} \\|.",
+                                    "image_path": "cd319195a465906270f48defc602e404ef9bb1d69d8cce08a5528cd255bd8fc4.jpg"
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Applying Lemma G.6 again gives"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
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+                            "bbox": [
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\lim  _ {n} \\sup  _ {n} \\| A _ {1, n} ^ {(c)} \\| \\leq C _ {5} \\lim  _ {n} \\sup  _ {n} \\| \\beta_ {n} \\Xi_ {n} \\| = 0",
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "for some constant "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "C_5 > 0"
+                                }
+                            ]
+                        }
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+                                    "type": "text",
+                                    "content": "Finally, we provide an existing lemma below."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                }
+            ],
+            "discarded_blocks": [
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
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+                                    "type": "text",
+                                    "content": "28"
+                                }
+                            ]
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma E.4 (Mokkadem & Pelletier (2005) Lemma 4). For a sequence with decreasing step size "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_{n} = (n + 1)^{-b}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " for "
+                                },
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+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "b\\in (1 / 2,1]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
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+                                        117
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
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+                                        117
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u_{n} = \\sum_{k = 1}^{n}\\beta_{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        117
+                                    ],
+                                    "type": "text",
+                                    "content": ", a positive semi-definite matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        117
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Gamma"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                        117
+                                    ],
+                                    "type": "text",
+                                    "content": " and a Hurwitz matrix "
+                                },
+                                {
+                                    "bbox": [
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+                                        117
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{Q}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
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+                                        117
+                                    ],
+                                    "type": "text",
+                                    "content": ", which is given by"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
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+                    "angle": 0,
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+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
+                                        230,
+                                        121,
+                                        380,
+                                        152
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\beta_ {n} ^ {2} e ^ {(u _ {n} - u _ {k}) \\mathbf {Q}} \\Gamma e ^ {(u _ {n} - u _ {k}) \\mathbf {Q} ^ {T}},",
+                                    "image_path": "6e4020bc97601f7177c49bd353b037b836a44a2a79c62c0f5a2385a44bf8b937.jpg"
+                                }
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+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
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+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        209,
+                                        167,
+                                        401,
+                                        198
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\lim  _ {n \\rightarrow \\infty} \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\beta_ {n} ^ {2} e ^ {(u _ {n} - u _ {k}) \\mathbf {Q}} \\Gamma e ^ {(u _ {n} - u _ {k}) \\mathbf {Q} ^ {T}} = \\mathbf {V}",
+                                    "image_path": "a9a19b58b7ca7b4e85e5a50e371bbc72f9cdc075d4346be9a5162e9f03734b96.jpg"
+                                }
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+                {
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+                    ],
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+                    "angle": 0,
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+                            ],
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
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+                                        212
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        200,
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+                                        212
+                                    ],
+                                    "type": "text",
+                                    "content": " is the solution of the Lyapunov equation"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        194,
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+                    ],
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+                    "angle": 0,
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+                            "bbox": [
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+                            ],
+                            "spans": [
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+                                    "bbox": [
+                                        194,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left(\\mathbf {Q} + \\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I}\\right) \\mathbf {V} + \\mathbf {V} \\left(\\mathbf {Q} ^ {T} + \\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I}\\right) + \\Gamma = 0.",
+                                    "image_path": "b07be4931b591f37f87687163dfdddf3caf0f13f1cd3c94f6a0fe1ae4864f98f.jpg"
+                                }
+                            ]
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+                },
+                {
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+                    ],
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+                    "angle": 0,
+                    "lines": [
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+                            ],
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Then, "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        255,
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+                                        271
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\lim_{n\\to \\infty}A_{1,n}^{(a)} = \\mathbf{V}_{\\pmb{\\theta}}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        255,
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+                                        271
+                                    ],
+                                    "type": "text",
+                                    "content": " is a direct application of Lemma E.4."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
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+                    "bbox": [
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+                    ],
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+                            "bbox": [
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+                            ],
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+                                    "bbox": [
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+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 8,
+                            "angle": 0,
+                            "type": "image_body"
+                        }
+                    ],
+                    "index": 8
+                },
+                {
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
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+                                        293
+                                    ],
+                                    "type": "text",
+                                    "content": "We can follow the similar steps in Lemma E.3 to obtain"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
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+                    ],
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+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
+                                        260,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\lim  _ {n \\to \\infty} A _ {4, n} = \\mathbf {V} _ {\\mathbf {x}} (\\alpha),",
+                                    "image_path": "cc74467d891300b9310ace9f24cccb093e2c78a2d3de8354b4bfce77f74e821c.jpg"
+                                }
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+                    "index": 10
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+                {
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+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
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+                                        333
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\mathbf{x}}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
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+                                        333
+                                    ],
+                                    "type": "text",
+                                    "content": " is in the form of (32)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "The last step is to show "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\lim_{n\\to \\infty}A_{2,n} = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        338,
+                                        323,
+                                        350
+                                    ],
+                                    "type": "text",
+                                    "content": " . Note that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\| A _ {2, n} \\| = O \\left(\\beta_ {n} ^ {- 1 / 2} \\gamma_ {n} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} \\beta_ {k} \\gamma_ {k} \\| e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\| \\| e ^ {(s _ {n} - s _ {k}) \\mathbf {J} _ {2 2} (\\alpha) ^ {T}} \\|\\right) \\\\ = O \\left(\\beta_ {n} ^ {- 1 / 2} \\gamma_ {n} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} \\beta_ {k} \\gamma_ {k} e ^ {- (u _ {n} - u _ {k}) T} e ^ {- (s _ {n} - s _ {k}) T ^ {\\prime}}\\right) \\\\ = O \\left(\\beta_ {n} ^ {- 1 / 2} \\gamma_ {n} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} \\beta_ {k} \\gamma_ {k} e ^ {- (s _ {n} - s _ {k}) T ^ {\\prime}}\\right), \\\\ \\end{array}",
+                                    "image_path": "3588a9fb13c5655975843e43858e893284b1a02374af8c2431aa769a7f83b8bd.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        105,
+                        462,
+                        486,
+                        474
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                462,
+                                486,
+                                474
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        462,
+                                        486,
+                                        474
+                                    ],
+                                    "type": "text",
+                                    "content": "where the second equality is from Lemma G.4. Then, we use Lemma G.6 with "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        462,
+                                        486,
+                                        474
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        462,
+                                        486,
+                                        474
+                                    ],
+                                    "type": "text",
+                                    "content": " to obtain"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        239,
+                        479,
+                        504,
+                        510
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                239,
+                                479,
+                                504,
+                                510
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        239,
+                                        479,
+                                        504,
+                                        510
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\sum_ {k = 1} ^ {n} \\beta_ {k} \\gamma_ {k} e ^ {- \\left(s _ {n} - s _ {k}\\right) T ^ {\\prime}} = O \\left(\\beta_ {n}\\right) \\tag {63}",
+                                    "image_path": "63be6c94cca67ee7095c46359ee0203e09c7886913ebe48cd43275a7e874c3f5.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        105,
+                        515,
+                        272,
+                        528
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                515,
+                                272,
+                                528
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        515,
+                                        272,
+                                        528
+                                    ],
+                                    "type": "text",
+                                    "content": "Additionally, since "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        515,
+                                        272,
+                                        528
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_{n} = o(\\gamma_{n})"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        515,
+                                        272,
+                                        528
+                                    ],
+                                    "type": "text",
+                                    "content": ", we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        165,
+                        533,
+                        444,
+                        564
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                165,
+                                533,
+                                444,
+                                564
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        165,
+                                        533,
+                                        444,
+                                        564
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\beta_ {n} ^ {- 1 / 2} \\gamma_ {n} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} \\beta_ {k} \\gamma_ {k} ^ {- 1 / 2} \\gamma_ {k} ^ {3 / 2} e ^ {- (s _ {n} - s _ {k}) T ^ {\\prime}} = O \\left(\\beta_ {n} ^ {1 / 2} \\gamma_ {n} ^ {- 1 / 2}\\right) = o (1).",
+                                    "image_path": "3e8302d57e0cea8339732d79a1c5ed74432bf4a69aab8008112b8db81ab76b27.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                        582
+                                    ],
+                                    "type": "text",
+                                    "content": "Then, it follows that "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        570,
+                                        357,
+                                        582
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\lim_{n\\to \\infty}A_{2,n} = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        570,
+                                        357,
+                                        582
+                                    ],
+                                    "type": "text",
+                                    "content": ". Therefore, we obtain"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
+                        148,
+                        588,
+                        460,
+                        619
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                148,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        148,
+                                        588,
+                                        460,
+                                        619
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\lim  _ {n \\to \\infty} \\sum_ {k = 1} ^ {n} \\mathbb {E} \\left[ (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) ^ {T} | \\mathcal {F} _ {k - 1} \\right] = \\left( \\begin{array}{c c} \\mathbf {V} _ {\\boldsymbol {\\theta}} (\\alpha) & 0 \\\\ 0 & \\mathbf {V} _ {\\mathbf {x}} (\\alpha) \\end{array} \\right).",
+                                    "image_path": "b7b1490f968fe7b8e4b2b2c843b0e72d6e8e21639f06f69ba07e38ebae101c46.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                443,
+                                642
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        630,
+                                        443,
+                                        642
+                                    ],
+                                    "type": "text",
+                                    "content": "Now, we turn to verifying the conditions in Theorem G.3. For some "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        630,
+                                        443,
+                                        642
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        630,
+                                        443,
+                                        642
+                                    ],
+                                    "type": "text",
+                                    "content": ", we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                },
+                {
+                    "bbox": [
+                        113,
+                        647,
+                        504,
+                        736
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        113,
+                                        647,
+                                        504,
+                                        736
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\sum_ {k = 1} ^ {n} \\mathbb {E} \\left[ \\| Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)} \\| ^ {2 + \\tau} | \\mathcal {F} _ {k - 1} \\right] \\\\ = O \\left(\\beta_ {n} ^ {- (1 + \\frac {\\tau}{2})} \\sum_ {k = 1} ^ {n} \\beta_ {k} ^ {2 + \\frac {\\tau}{2}} \\beta_ {k} ^ {\\frac {\\tau}{2}} e ^ {- (2 + \\tau) \\left(u _ {n} - u _ {k}\\right) T} + \\gamma_ {n} ^ {- (1 + \\frac {\\tau}{2})} \\sum_ {k = 1} ^ {n} \\gamma_ {k} ^ {2 + \\frac {\\tau}{2}} \\gamma_ {k} ^ {\\frac {\\tau}{2}} e ^ {- (2 + \\tau) \\left(s _ {n} - s _ {k}\\right) T ^ {\\prime}}\\right) \\tag {64} \\\\ = O \\left(\\beta_ {n} ^ {\\frac {\\tau}{2}} + \\gamma_ {n} ^ {\\frac {\\tau}{2}}\\right) \\\\ \\end{array}",
+                                    "image_path": "a5eccb48b57675ba41bd18a946494761e009312e1442bf236650c469bc43b1f9.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 21
+                }
+            ],
+            "discarded_blocks": [
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+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
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+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "29"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 22
+                }
+            ],
+            "page_size": [
+                612,
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+            ],
+            "page_idx": 28
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        82,
+                        466,
+                        95
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                82,
+                                466,
+                                95
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        466,
+                                        95
+                                    ],
+                                    "type": "text",
+                                    "content": "where the last equality comes from Lemma G.6. Since (64) also holds for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        466,
+                                        95
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        466,
+                                        95
+                                    ],
+                                    "type": "text",
+                                    "content": ", we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        209,
+                        100,
+                        400,
+                        130
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                209,
+                                100,
+                                400,
+                                130
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        209,
+                                        100,
+                                        400,
+                                        130
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\sum_ {k = 1} ^ {n} \\mathbb {E} \\left[ \\| Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)} \\| ^ {2} | \\mathcal {F} _ {k - 1} \\right] = O (1) <   \\infty .",
+                                    "image_path": "4dd4e1c786ab35030693135e01aa4655fe488d7cc69156f2ace4492f9ac6dd81.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        136,
+                        456,
+                        148
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                                456,
+                                148
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        136,
+                                        456,
+                                        148
+                                    ],
+                                    "type": "text",
+                                    "content": "Therefore, all the conditions in Theorem G.3 are satisfied and its application then gives"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        179,
+                        153,
+                        504,
+                        186
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                179,
+                                153,
+                                504,
+                                186
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        179,
+                                        153,
+                                        504,
+                                        186
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "Z ^ {(n)} = \\left(\\begin{array}{c}\\sqrt {\\beta_ {n} ^ {- 1}} L _ {n} ^ {(\\boldsymbol {\\theta})}\\\\\\sqrt {\\gamma_ {n} ^ {- 1}} L _ {n} ^ {(\\mathbf {x})}\\end{array}\\right) \\xrightarrow [ d i s t . ]{n \\rightarrow \\infty} N \\left(0, \\left(\\begin{array}{c c}\\mathbf {V} _ {\\boldsymbol {\\theta}} (\\alpha)&0\\\\0&\\mathbf {V} _ {\\mathbf {x}} (\\alpha)\\end{array}\\right)\\right). \\tag {65}",
+                                    "image_path": "d8e0aa3a691396553a4dc52b3b552337f251ebe8c9de6974c50f7d73186684c9.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        198,
+                        504,
+                        213
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                198,
+                                504,
+                                213
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": "Furthermore, we have the following lemma about the strong convergence rate of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{L_n^{(\\pmb{\\theta})}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{L_n^{(\\mathbf{x})}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        105,
+                        214,
+                        160,
+                        225
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                214,
+                                160,
+                                225
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        214,
+                                        160,
+                                        225
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma E.5."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        231,
+                        224,
+                        504,
+                        243
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                231,
+                                224,
+                                504,
+                                243
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        231,
+                                        224,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left\\| L _ {n} ^ {(\\boldsymbol {\\theta})} \\right\\| = O \\left(\\sqrt {\\beta_ {n} \\log \\left(u _ {n}\\right)}\\right) \\quad a. s. \\tag {66a}",
+                                    "image_path": "e1e065e8483351cb050523ba88e786087e2534501aebf2858bc06b03e6999543.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        232,
+                        246,
+                        504,
+                        266
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                232,
+                                246,
+                                504,
+                                266
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        232,
+                                        246,
+                                        504,
+                                        266
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left\\| L _ {n} ^ {(\\mathbf {x})} \\right\\| = O \\left(\\sqrt {\\gamma_ {n} \\log \\left(s _ {n}\\right)}\\right) \\quad a. s. \\tag {66b}",
+                                    "image_path": "aacf46fbb51402743bc17a4a862dab62a9e5f684dc2cf97ad4a2142684370310.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        277,
+                        504,
+                        311
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                277,
+                                504,
+                                311
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        277,
+                                        504,
+                                        311
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. This proof follows Pelletier (1998, Lemma 1). We only need the special case of Pelletier (1998, Lemma 1) that fits our scenario; e.g., we let the two types of step sizes therein be the same. Specifically, we attach the following lemma."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
+                        316,
+                        361,
+                        328
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                316,
+                                361,
+                                328
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        316,
+                                        361,
+                                        328
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma E.6 (Pelletier (1998) Lemma 1). Consider a sequence"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "L _ {n + 1} = e ^ {u _ {n} \\mathbf {H}} \\sum_ {k = 1} ^ {n} e ^ {- u _ {k} \\mathbf {H}} \\beta_ {k} M _ {k + 1},",
+                                    "image_path": "9d96c64e964d11be5658da038c14f798e2870bb1a48a7c1340de8feb121d8cef.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        103,
+                        371,
+                        504,
+                        407
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                103,
+                                371,
+                                504,
+                                407
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        103,
+                                        371,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        371,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_{n} = n^{-b}"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        371,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        371,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "1/2 < b \\leq 1"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        371,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        371,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{M_n\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        371,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "text",
+                                    "content": " is a Martingale difference sequence adapted to the filtration "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        371,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        371,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "text",
+                                    "content": " such that, almost surely, "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        371,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\lim \\sup_{n} \\mathbb{E}[\\| M_{n+1} \\|^2 | \\mathcal{F}_n] \\leq M^2"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        371,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "text",
+                                    "content": " and there exists "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        371,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau \\in (0,2)"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        371,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        371,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "b(2 + \\tau) > 2"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        371,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "text",
+                                    "content": ", such that "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        371,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sup_{n} \\mathbb{E}[\\| M_{n+1} \\|^2 + \\tau | \\mathcal{F}_n] < \\infty"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        371,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, almost surely,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        242,
+                        411,
+                        505,
+                        439
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                242,
+                                411,
+                                505,
+                                439
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        242,
+                                        411,
+                                        505,
+                                        439
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\lim  _ {n} \\sup  \\frac {\\left\\| L _ {n} \\right\\|}{\\sqrt {\\beta_ {n} \\log \\left(u _ {n}\\right)}} \\leq C _ {M}, \\tag {67}",
+                                    "image_path": "16dd03166ee39b7583d82dc031000b6377d244d0e57d312e49d11e474d27c290.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        104,
+                        443,
+                        275,
+                        455
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                443,
+                                275,
+                                455
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        443,
+                                        275,
+                                        455
+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        443,
+                                        275,
+                                        455
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        443,
+                                        275,
+                                        455
+                                    ],
+                                    "type": "text",
+                                    "content": " is a constant dependent on "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        443,
+                                        275,
+                                        455
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        443,
+                                        275,
+                                        455
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        104,
+                        466,
+                        506,
+                        561
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                466,
+                                506,
+                                561
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "text",
+                                    "content": "By assumption A4, the iterates "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\pmb{\\theta}_n,\\mathbf{x}_n)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "text",
+                                    "content": " are bounded within a compact subset "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Omega"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "text",
+                                    "content": ". Recall the form of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M_{n + 1}^{(\\pmb{\\theta})},M_{n + 1}^{(\\mathbf{x})}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "text",
+                                    "content": " defined in (35), it comprises the functions "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{H}_{\\pmb{\\theta}_n,\\mathbf{x}_n}(i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\mathbf{K}_{\\mathbf{x}_n}\\tilde{H}_{\\pmb{\\theta}_n,\\mathbf{x}_n})(i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "text",
+                                    "content": ", which in turn include the function "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H(\\pmb {\\theta},i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "text",
+                                    "content": ". We know that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H(\\pmb {\\theta},i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "text",
+                                    "content": " is bounded for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\theta}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "text",
+                                    "content": " in some compact set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{C}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "text",
+                                    "content": ". Thus, for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\pmb {\\theta}_n,\\mathbf{x}_n)\\in \\Omega"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "text",
+                                    "content": " for some compact set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Omega"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M_{n + 1}^{(\\pmb{\\theta})},M_{n + 1}^{(\\mathbf{x})}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "text",
+                                    "content": " are bounded and we denote by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "c_{\\pmb{\\theta}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "c_{\\mathbf{x}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "text",
+                                    "content": " as their upper bounds, i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{E}[\\| M_{n + 1}^{(\\pmb{\\theta})}\\| ^2 |\\mathcal{F}_n]\\leq c_\\Omega^{(\\pmb{\\theta})}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{E}[\\| M_{n + 1}^{(\\mathbf{x})}\\| ^2 |\\mathcal{F}_n]\\leq c_\\Omega^{(\\mathbf{x})}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "text",
+                                    "content": ". We only need to replace the upper bound "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "c"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "text",
+                                    "content": " in Lemma E.6 by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "c_{\\Omega}^{(\\pmb{\\theta})}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "text",
+                                    "content": " for the sequence "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{L_n^{(\\pmb{\\theta})}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "text",
+                                    "content": " (resp. "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "c_{\\Omega}^{(\\mathbf{x})}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "text",
+                                    "content": " for the sequence "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{L_n^{(\\mathbf{x})}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        466,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "text",
+                                    "content": "), i.e.,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        240,
+                        567,
+                        505,
+                        598
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                240,
+                                567,
+                                505,
+                                598
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        240,
+                                        567,
+                                        505,
+                                        598
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\lim  _ {n} \\sup  \\frac {\\left\\| L _ {n} ^ {(\\boldsymbol {\\theta})} \\right\\|}{\\sqrt {\\beta_ {n} \\log (u _ {n})}} \\leq C _ {\\Omega} ^ {(\\boldsymbol {\\theta})}, \\tag {68a}",
+                                    "image_path": "2b9c9db6dae216f8baf38214c4d0d4472293cc1377c75364bba0ae63b7685e92.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        241,
+                        603,
+                        504,
+                        634
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                241,
+                                603,
+                                504,
+                                634
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        241,
+                                        603,
+                                        504,
+                                        634
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\lim  _ {n} \\sup  \\frac {\\| L _ {n} ^ {(\\mathbf {x})} \\|}{\\sqrt {\\gamma_ {n} \\log (s _ {n})}} \\leq C _ {\\Omega} ^ {(\\mathbf {x})}, \\tag {68b}",
+                                    "image_path": "96df7574f559ef5aa15b73d8739097f8955018371ac47ca19c281c3aa8ba4c9e.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        104,
+                        636,
+                        506,
+                        663
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                636,
+                                506,
+                                663
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        636,
+                                        506,
+                                        663
+                                    ],
+                                    "type": "text",
+                                    "content": "such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        636,
+                                        506,
+                                        663
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\| L_n^{(\\pmb{\\theta})}\\| = O\\big(\\sqrt{\\beta_n\\log(u_n)}\\big)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        636,
+                                        506,
+                                        663
+                                    ],
+                                    "type": "text",
+                                    "content": " a.s. and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        636,
+                                        506,
+                                        663
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\| L_{n}^{(\\mathbf{x})}\\| = O\\big(\\sqrt{\\gamma_{n}\\log(s_{n})}\\big)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        636,
+                                        506,
+                                        663
+                                    ],
+                                    "type": "text",
+                                    "content": " a.s. which completes the proof."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
+                        104,
+                        675,
+                        505,
+                        714
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                675,
+                                505,
+                                714
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        675,
+                                        505,
+                                        714
+                                    ],
+                                    "type": "text",
+                                    "content": "Note that we have "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        675,
+                                        505,
+                                        714
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n - \\boldsymbol{\\mu}"
+                                },
+                                {
+                                    "bbox": [
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+                                        675,
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+                                        714
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        675,
+                                        505,
+                                        714
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "L_{n}^{(\\mathbf{x})}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        675,
+                                        505,
+                                        714
+                                    ],
+                                    "type": "text",
+                                    "content": " weakly converge to the same Gaussian distribution from Remark E.1 and (65). Then, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        675,
+                                        505,
+                                        714
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\gamma_n^{-1 / 2}\\Delta_n^{(\\mathbf{x})}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        675,
+                                        505,
+                                        714
+                                    ],
+                                    "type": "text",
+                                    "content": " weakly converges to zero, implying that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        675,
+                                        505,
+                                        714
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\gamma_n^{-1 / 2}\\Delta_n^{(\\mathbf{x})}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        675,
+                                        505,
+                                        714
+                                    ],
+                                    "type": "text",
+                                    "content": " converges to zero with probability 1. Therefore, together with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        675,
+                                        505,
+                                        714
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\gamma_n\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        675,
+                                        505,
+                                        714
+                                    ],
+                                    "type": "text",
+                                    "content": " being strictly positive, we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                },
+                {
+                    "bbox": [
+                        258,
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+                                    "type": "interline_equation",
+                                    "content": "\\Delta_ {n} ^ {(\\mathbf {x})} = o \\left(\\sqrt {\\gamma_ {n}}\\right) \\quad a. s. \\tag {69}",
+                                    "image_path": "771002381f991494d68973bfc10121f8b560d6269752f4952a61e46ca7ea34c4.jpg"
+                                }
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+            "discarded_blocks": [
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
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+                                    "type": "text",
+                                    "content": "30"
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+                }
+            ],
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+        },
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+            "para_blocks": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Characterization of Sequences "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        80,
+                                        320,
+                                        95
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{R_n^{(\\theta)}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        80,
+                                        320,
+                                        95
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        80,
+                                        320,
+                                        95
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\Delta_n^{(\\theta)}\\}"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
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+                        125
+                    ],
+                    "type": "text",
+                    "angle": 0,
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+                            "bbox": [
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+                                    "bbox": [
+                                        104,
+                                        102,
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+                                        125
+                                    ],
+                                    "type": "text",
+                                    "content": "We first consider the sequence "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        125
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{R_n^{(\\theta)}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        125
+                                    ],
+                                    "type": "text",
+                                    "content": ". We assume a positive real-valued bounded sequence "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        125
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{w_{n}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        125
+                                    ],
+                                    "type": "text",
+                                    "content": " under the same conditions as in Mokkadem & Pelletier (2006, Definition 1), i.e.,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        126,
+                        483,
+                        142
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                126,
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+                                142
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        126,
+                                        483,
+                                        142
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition E.1. In the case "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        126,
+                                        483,
+                                        142
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "b < 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        126,
+                                        483,
+                                        142
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        126,
+                                        483,
+                                        142
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\frac{w_n}{w_{n+1}} = 1 + o(\\beta_n)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        126,
+                                        483,
+                                        142
+                                    ],
+                                    "type": "text",
+                                    "content": ", which also implies "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        126,
+                                        483,
+                                        142
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\frac{w_n}{w_{n+1}} = 1 + o(\\gamma_n)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        126,
+                                        483,
+                                        142
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        146,
+                        505,
+                        170
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                146,
+                                505,
+                                170
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        146,
+                                        505,
+                                        170
+                                    ],
+                                    "type": "text",
+                                    "content": "In the case "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        146,
+                                        505,
+                                        170
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "b = 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        146,
+                                        505,
+                                        170
+                                    ],
+                                    "type": "text",
+                                    "content": ", there exist "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        146,
+                                        505,
+                                        170
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\epsilon \\geq 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        146,
+                                        505,
+                                        170
+                                    ],
+                                    "type": "text",
+                                    "content": " and a nondecreasing slowly varying function "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        146,
+                                        505,
+                                        170
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "l(n)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        146,
+                                        505,
+                                        170
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        146,
+                                        505,
+                                        170
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_{n} = n^{-\\epsilon}l(n)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        146,
+                                        505,
+                                        170
+                                    ],
+                                    "type": "text",
+                                    "content": ". When "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        146,
+                                        505,
+                                        170
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\epsilon = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        146,
+                                        505,
+                                        170
+                                    ],
+                                    "type": "text",
+                                    "content": ", we require function "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        146,
+                                        505,
+                                        170
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "l(n)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        146,
+                                        505,
+                                        170
+                                    ],
+                                    "type": "text",
+                                    "content": " to be bounded."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        177,
+                        504,
+                        201
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                177,
+                                504,
+                                201
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        177,
+                                        504,
+                                        201
+                                    ],
+                                    "type": "text",
+                                    "content": "Since "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        177,
+                                        504,
+                                        201
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\| \\mathbf{x}_n - \\boldsymbol{\\mu} \\| = o(1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        177,
+                                        504,
+                                        201
+                                    ],
+                                    "type": "text",
+                                    "content": " by a.s. convergence result, we can assume that there exists "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        177,
+                                        504,
+                                        201
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{w_n\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        177,
+                                        504,
+                                        201
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        177,
+                                        504,
+                                        201
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\| \\mathbf{x}_n - \\boldsymbol{\\mu} \\| = O(w_n)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        177,
+                                        504,
+                                        201
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, from (55b), we can use the Abel transformation and obtain"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
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+                        203,
+                        493,
+                        252
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                    "bbox": [
+                                        118,
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+                                        252
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} R _ {n} ^ {(\\boldsymbol {\\theta})} = \\beta_ {n} \\gamma_ {n} ^ {- 1} \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (\\mathbf {x} _ {n} - \\boldsymbol {\\mu}) - e ^ {(u _ {n} - u _ {1}) \\mathbf {J} _ {1 1}} \\beta_ {1} \\gamma_ {1} ^ {- 1} \\mathbf {U} _ {1 1} \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (\\mathbf {x} _ {1} - \\boldsymbol {\\mu}) \\\\ + e ^ {u _ {n} \\mathbf {J} _ {1 1}} \\sum_ {k = 1} ^ {n - 1} \\left(e ^ {- u _ {k} \\mathbf {J} _ {1 1}} \\beta_ {k} \\gamma_ {k} ^ {- 1} - e ^ {- u _ {k + 1} \\mathbf {J} _ {1 1}} \\beta_ {k + 1} \\gamma_ {k + 1} ^ {- 1}\\right) \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (\\mathbf {x} _ {k + 1} - \\boldsymbol {\\mu}), \\\\ \\end{array}",
+                                    "image_path": "6d67dde09cf425ae8be81720ab7ce8e37da50a115c753b29553b265434fd2fa0.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        105,
+                        254,
+                        312,
+                        265
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                254,
+                                312,
+                                265
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        254,
+                                        312,
+                                        265
+                                    ],
+                                    "type": "text",
+                                    "content": "where the last term on the RHS can be rewritten as"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        108,
+                        266,
+                        501,
+                        299
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                266,
+                                501,
+                                299
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        108,
+                                        266,
+                                        501,
+                                        299
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "W _ {n} = \\sum_ {k = 1} ^ {n - 1} e ^ {(u _ {n} - u _ {k + 1}) \\mathbf {J} _ {1 1}} \\beta_ {k + 1} \\gamma_ {k + 1} ^ {- 1} \\left(e ^ {\\beta_ {k + 1} \\mathbf {J} _ {1 1}} \\beta_ {k} \\beta_ {k + 1} ^ {- 1} \\gamma_ {k} ^ {- 1} \\gamma_ {k + 1} - \\mathbf {I}\\right) \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (\\mathbf {x} _ {k + 1} - \\boldsymbol {\\mu}).",
+                                    "image_path": "8bfb03bfce9dbabe0ab56696a0892fcdcc9994d823299be51fe739c5318ba717.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        301,
+                        504,
+                        324
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                301,
+                                504,
+                                324
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        301,
+                                        504,
+                                        324
+                                    ],
+                                    "type": "text",
+                                    "content": "Using Lemma G.6 on "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        301,
+                                        504,
+                                        324
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "W_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        301,
+                                        504,
+                                        324
+                                    ],
+                                    "type": "text",
+                                    "content": " gives "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        301,
+                                        504,
+                                        324
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\| W_n\\| = O(\\gamma_n^{-1}\\| e^{\\beta_n\\mathbf{J}_{11}} - \\mathbf{I}\\| \\| \\mathbf{x}_n - \\pmb {\\mu}\\|) = O(\\gamma_n^{-1}\\beta_n\\omega_n)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        301,
+                                        504,
+                                        324
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, it follows that for some "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        301,
+                                        504,
+                                        324
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T > 0"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        176,
+                        326,
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+                        341
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                            ],
+                            "spans": [
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+                                    "bbox": [
+                                        176,
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+                                        341
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left\\| R _ {n} ^ {(\\boldsymbol {\\theta})} \\right\\| = O \\left(\\beta_ {n} \\gamma_ {n} ^ {- 1} \\omega_ {n} + \\left\\| e ^ {u _ {n} \\mathbf {J} _ {1 1}} \\right\\|\\right) = O \\left(\\beta_ {n} \\gamma_ {n} ^ {- 1} \\omega_ {n} + e ^ {- u _ {n} T}\\right) \\tag {70}",
+                                    "image_path": "2cba0c7b175fc028fc46f5559b624b424b4918f42ced5c7cbd8342a65edf9e86.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
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+                    ],
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+                                    ],
+                                    "type": "text",
+                                    "content": "with the application of Lemma G.4 to the second equality."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        104,
+                        359,
+                        505,
+                        387
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                359,
+                                505,
+                                387
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        359,
+                                        505,
+                                        387
+                                    ],
+                                    "type": "text",
+                                    "content": "Then, we shift our focus on "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        359,
+                                        505,
+                                        387
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\Delta_n^{(\\theta)}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        359,
+                                        505,
+                                        387
+                                    ],
+                                    "type": "text",
+                                    "content": ". Specifically, we take (54), (55a), and (56) back to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        359,
+                                        505,
+                                        387
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta_{n}^{(\\theta)} = \\pmb{\\theta}_{n} - \\pmb{\\theta}^{*} - L_{n}^{(\\theta)} - R_{n}^{(\\theta)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        359,
+                                        505,
+                                        387
+                                    ],
+                                    "type": "text",
+                                    "content": ", and obtain"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        127,
+                        390,
+                        504,
+                        519
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                127,
+                                390,
+                                504,
+                                519
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        127,
+                                        390,
+                                        504,
+                                        519
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\Delta_ {n + 1} ^ {(\\boldsymbol {\\theta})} = (\\mathbf {I} + \\beta_ {n + 1} \\mathbf {J} _ {1 1}) (\\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*}) \\\\ + \\beta_ {n + 1} \\left(r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} + r _ {n} ^ {(\\boldsymbol {\\theta}, 2)} + \\eta_ {n} ^ {(\\boldsymbol {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(r _ {n} ^ {(\\mathbf {x}, 1)} + r _ {n} ^ {(\\mathbf {x}, 2)} + \\eta_ {n} ^ {(\\mathbf {x})}\\right)\\right) \\\\ - e ^ {\\beta_ {n + 1} \\mathbf {J} _ {1 1}} L _ {n} ^ {(\\boldsymbol {\\theta})} - e ^ {\\beta_ {n + 1} \\mathbf {J} _ {1 1}} R _ {n} ^ {(\\boldsymbol {\\theta})} \\\\ = \\left(\\mathbf {I} + \\beta_ {n + 1} \\mathbf {J} _ {1 1}\\right) \\left(\\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*}\\right) + \\beta_ {n + 1} \\left(r _ {x} ^ {(\\boldsymbol {\\theta}, 1)} + r _ {x} ^ {(\\boldsymbol {\\theta}, 2)} + \\eta_ {x} ^ {(\\boldsymbol {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} \\left(r _ {x} ^ {(\\mathbf {x}, 1)} + r _ {x} ^ {(\\mathbf {x}, 2)} + \\eta_ {x} ^ {(\\mathbf {x})}\\right)\\right) \\tag {71} \\\\ - \\left(\\mathbf {I} + \\beta_ {n + 1} \\mathbf {J} _ {1 1} + O \\left(\\beta_ {n + 1} ^ {2}\\right)\\right) L _ {n} ^ {(\\boldsymbol {\\theta})} - \\left(\\mathbf {I} + \\beta_ {n + 1} \\mathbf {J} _ {1 1} + O \\left(\\beta_ {n + 1} ^ {2}\\right)\\right) R _ {n} ^ {(\\boldsymbol {\\theta})} \\\\ = (\\mathbf {I} + \\beta_ {n + 1} \\mathbf {J} _ {1 1}) \\Delta_ {n} ^ {(\\pmb {\\theta})} + O (\\beta_ {n + 1} ^ {2}) (L _ {n} ^ {(\\pmb {\\theta})} + R _ {n} ^ {(\\pmb {\\theta}))}) \\\\ + \\beta_ {n + 1} (r _ {n} ^ {(\\pmb {\\theta}, 1)} + r _ {n} ^ {(\\pmb {\\theta}, 2)} + \\eta_ {n} ^ {(\\pmb {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (r _ {n} ^ {(\\mathbf {x}, 1)} + r _ {n} ^ {(\\mathbf {x}, 2)} + \\eta_ {n} ^ {(\\mathbf {x}))}), \\\\ \\end{array}",
+                                    "image_path": "a95e37496bfa219ab483203b298315ceea7d35c8b9628610f679f9984a6dca0f.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        104,
+                        521,
+                        501,
+                        536
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                521,
+                                501,
+                                536
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        521,
+                                        501,
+                                        536
+                                    ],
+                                    "type": "text",
+                                    "content": "where the second equality is by taking the Taylor expansion "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        521,
+                                        501,
+                                        536
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "e^{\\beta_{n + 1}\\mathbf{J}_{11}} = \\mathbf{I} + \\beta_{n + 1}\\mathbf{J}_{11} + O(\\beta_{n + 1}^2)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        521,
+                                        501,
+                                        536
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        104,
+                        540,
+                        472,
+                        555
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                540,
+                                472,
+                                555
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        472,
+                                        555
+                                    ],
+                                    "type": "text",
+                                    "content": "Define "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        472,
+                                        555
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Phi_{k,n} \\triangleq \\prod_{j=k+1}^{n} (\\mathbf{I} + \\beta_j \\mathbf{J}_{11})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        472,
+                                        555
+                                    ],
+                                    "type": "text",
+                                    "content": " and by convention "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        472,
+                                        555
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Phi_{n,n} = \\mathbf{I}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        472,
+                                        555
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, we rewrite (71) as"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        119,
+                        559,
+                        504,
+                        734
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                119,
+                                559,
+                                504,
+                                734
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        119,
+                                        559,
+                                        504,
+                                        734
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\Delta_ {n + 1} ^ {(\\boldsymbol {\\theta})} = \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(O (\\beta_ {k + 1}) L _ {k} ^ {(\\boldsymbol {\\theta})} + O (\\beta_ {k + 1}) R _ {k} ^ {(\\boldsymbol {\\theta})}\\right) \\\\ + \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} (r _ {k} ^ {(\\boldsymbol {\\theta}, 1)} + r _ {k} ^ {(\\boldsymbol {\\theta}, 2)} + \\eta_ {k} ^ {(\\boldsymbol {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (r _ {k} ^ {(\\mathbf {x}, 1)} + r _ {k} ^ {(\\mathbf {x}, 2)} + \\eta_ {k} ^ {(\\mathbf {x}))}) \\\\ = \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(O \\left(\\beta_ {k + 1}\\right) L _ {k} ^ {(\\boldsymbol {\\theta})} + O \\left(\\beta_ {k + 1}\\right) R _ {k} ^ {(\\boldsymbol {\\theta})}\\right) \\\\ + \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} (r _ {k} ^ {(\\pmb {\\theta}, 1)} + \\eta_ {k} ^ {(\\pmb {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (r _ {k} ^ {(\\mathbf {x}, 1)} + \\eta_ {k} ^ {(\\mathbf {x})})) \\\\ + \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(r _ {k} ^ {(\\boldsymbol {\\theta}, 2)} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} r _ {k} ^ {(\\mathbf {x}, 2)}\\right). \\tag {72} \\\\ \\end{array}",
+                                    "image_path": "d7801000fa5ae18f23234eb779af6179b7a4fbebb686153b7a71f3503b0eeb12.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
+                        750,
+                        310,
+                        760
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                299,
+                                750,
+                                310,
+                                760
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        299,
+                                        750,
+                                        310,
+                                        760
+                                    ],
+                                    "type": "text",
+                                    "content": "31"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 30
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        79,
+                        476,
+                        96
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                79,
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+                                96
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        476,
+                                        96
+                                    ],
+                                    "type": "text",
+                                    "content": "From (72), we can indeed decompose "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        476,
+                                        96
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta_{n+1}^{(\\boldsymbol{x})}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        476,
+                                        96
+                                    ],
+                                    "type": "text",
+                                    "content": " into two parts "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        476,
+                                        96
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta_{n+1}^{(\\boldsymbol{\\theta})} = \\Delta_{n+1}^{(\\boldsymbol{\\theta}, 1)} + \\Delta_{n+1}^{(\\boldsymbol{\\theta}, 2)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        476,
+                                        96
+                                    ],
+                                    "type": "text",
+                                    "content": ", where"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        152,
+                        102,
+                        504,
+                        202
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                152,
+                                102,
+                                504,
+                                202
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        152,
+                                        102,
+                                        504,
+                                        202
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\Delta_ {n + 1} ^ {(\\boldsymbol {\\theta}, 1)} \\triangleq \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(O \\left(\\beta_ {k + 1}\\right) L _ {k} ^ {(\\boldsymbol {\\theta})} + O \\left(\\beta_ {k + 1}\\right) R _ {k} ^ {(\\boldsymbol {\\theta})}\\right) (73a) \\\\ + \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} (r _ {k} ^ {(\\pmb {\\theta}, 1)} + \\eta_ {k} ^ {(\\pmb {\\theta})} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} (r _ {k} ^ {(\\mathbf {x}, 1)} + \\eta_ {k} ^ {(\\mathbf {x})})), \\\\ \\Delta_ {n + 1} ^ {(\\boldsymbol {\\theta}, 2)} \\triangleq \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(r _ {k} ^ {(\\boldsymbol {\\theta}, 2)} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} r _ {k} ^ {(\\mathbf {x}, 2)}\\right). (73b) \\\\ \\end{array}",
+                                    "image_path": "b1d717f08745f756c7df9aa283f730189501cbfd714154686496e6686b5e0179.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        206,
+                        504,
+                        232
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                206,
+                                504,
+                                232
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        504,
+                                        232
+                                    ],
+                                    "type": "text",
+                                    "content": "This term "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        504,
+                                        232
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta_{n+1}^{(\\theta,1)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        504,
+                                        232
+                                    ],
+                                    "type": "text",
+                                    "content": " shares the same recursive form as in the sequence defined in Mokkadem & Pelletier (2006, Lemma 6), which is given below."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        236,
+                        505,
+                        265
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                236,
+                                505,
+                                265
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        236,
+                                        505,
+                                        265
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma E.7 (Mokkadem & Pelletier (2006) Lemma 6). For "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        236,
+                                        505,
+                                        265
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta_{n+1}^{(\\theta,1)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        236,
+                                        505,
+                                        265
+                                    ],
+                                    "type": "text",
+                                    "content": " in the form of (73a), assume "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        236,
+                                        505,
+                                        265
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\| \\mathbf{x}_n - \\boldsymbol{\\mu} \\| = O(\\omega_n)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        236,
+                                        505,
+                                        265
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        236,
+                                        505,
+                                        265
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\| \\Delta_n^{(\\mathbf{x})} \\| = O(\\delta_n)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        236,
+                                        505,
+                                        265
+                                    ],
+                                    "type": "text",
+                                    "content": " for the sequences "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        236,
+                                        505,
+                                        265
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\omega_n, \\delta_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        236,
+                                        505,
+                                        265
+                                    ],
+                                    "type": "text",
+                                    "content": " defined in (E.1). Then, we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        192,
+                        269,
+                        417,
+                        285
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                192,
+                                269,
+                                417,
+                                285
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        192,
+                                        269,
+                                        417,
+                                        285
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\| \\Delta_ {n + 1} ^ {(\\boldsymbol {\\theta}, 1)} \\| = O \\left(\\beta_ {n} ^ {2} \\gamma_ {n} ^ {- 2} \\omega_ {n} ^ {2} + \\beta_ {n} \\gamma_ {n} ^ {- 1} \\delta_ {n}\\right) + o (\\sqrt {\\beta_ {n}}) \\quad a. s.",
+                                    "image_path": "64c83f60fcf94ddacf5a997648be8677810dd59de6a39cc2bd0a5ca9390b8de4.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        298,
+                        439,
+                        312
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                298,
+                                439,
+                                312
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        298,
+                                        439,
+                                        312
+                                    ],
+                                    "type": "text",
+                                    "content": "Since we already have "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        298,
+                                        439,
+                                        312
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta_n^{(\\mathbf{x})} = o(\\sqrt{\\gamma_n})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        298,
+                                        439,
+                                        312
+                                    ],
+                                    "type": "text",
+                                    "content": " in (69), together with Lemma E.7, we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        141,
+                        317,
+                        468,
+                        334
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                141,
+                                317,
+                                468,
+                                334
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        317,
+                                        468,
+                                        334
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\| \\Delta_ {n + 1} ^ {(\\boldsymbol {\\theta}, 1)} \\| = O (\\beta_ {n} ^ {2} \\gamma_ {n} ^ {- 2} \\omega_ {n} ^ {2}) + o (\\beta_ {n} \\gamma_ {n} ^ {- 1 / 2}) + o (\\sqrt {\\beta_ {n}}) = O (\\beta_ {n} ^ {2} \\gamma_ {n} ^ {- 2} \\omega_ {n} ^ {2}) + o (\\sqrt {\\beta_ {n}})",
+                                    "image_path": "ad2a300b7ec3fbf224f22924760ff4efc2b55466ea3e35464815e84aeaefa8de.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
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+                        {
+                            "bbox": [
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+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "where the second equality comes from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        339,
+                                        452,
+                                        355
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "o(\\beta_n\\gamma_n^{-1 / 2}) = o(\\beta_n^{1 / 2}(\\beta_n\\gamma_n^{-1})^{1 / 2}) = o(\\beta_n^{1 / 2})."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "We now focus on "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta_{n + 1}^{(\\theta ,2)}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        359,
+                                        285,
+                                        374
+                                    ],
+                                    "type": "text",
+                                    "content": ". Define a sequence"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        218,
+                        380,
+                        504,
+                        411
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                218,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        218,
+                                        380,
+                                        504,
+                                        411
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\Psi_ {n} \\triangleq \\sum_ {k = 1} ^ {n} r _ {k} ^ {(\\boldsymbol {\\theta}, 2)} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} r _ {k} ^ {(\\mathbf {x}, 2)}, \\tag {74}",
+                                    "image_path": "f98dd2a811e2d2cd412ce65faef1417bbba018c337b15030d5d3a371e7b6e2e9.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        105,
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+                        426
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                417,
+                                158,
+                                426
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        417,
+                                        158,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": "and we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                {
+                                    "bbox": [
+                                        194,
+                                        433,
+                                        417,
+                                        531
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\beta_ {n + 1} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(r _ {k} ^ {\\left(\\boldsymbol {\\theta}, 2\\right)} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} r _ {k} ^ {\\left(\\mathbf {x}, 2\\right)}\\right) \\\\ = \\beta_ {n + 1} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} (\\Psi_ {k} - \\Psi_ {k - 1}) \\\\ = \\beta_ {n + 1} ^ {1 / 2} \\Psi_ {n} + \\beta_ {n + 1} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n - 1} \\left(\\beta_ {k} \\Phi_ {k, n} - \\beta_ {k + 1} \\Phi_ {k + 1, n}\\right) \\Psi_ {k} \\\\ \\end{array}",
+                                    "image_path": "d1997f6cef5b3a1c42ecd14ab863c03b470f2d33d0dfa75626d802fa0b113bf9.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        105,
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        536,
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+                                        548
+                                    ],
+                                    "type": "text",
+                                    "content": "where the last equality comes from the Abel transformation. Note that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\left\\| \\beta_ {k} \\Phi_ {k, n} - \\beta_ {k + 1} \\Phi_ {k + 1, n} \\right\\| \\leq \\beta_ {k + 1} \\left\\| \\Phi_ {k, n} - \\Phi_ {k + 1, n} \\right\\| + \\left(\\beta_ {k} - \\beta_ {k + 1}\\right) \\left\\| \\Phi_ {k, n} \\right\\| \\\\ \\leq \\beta_ {k + 1} \\| \\Phi_ {k + 1, n} \\| \\beta_ {k} \\| \\mathbf {J} _ {1 1} \\| + C _ {7} \\beta_ {k} ^ {2} \\| \\Phi_ {k, n} \\| \\\\ \\leq C _ {8} \\beta_ {k} ^ {2} e ^ {- (u _ {n} - u _ {k}) T} \\\\ \\end{array}",
+                                    "image_path": "84e23743f14c2dccc5fdd733d17313bf9e537a78178733132b92ba20cdd6e499.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": "for some constant "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        604,
+                                        504,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_7, C_8 > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        604,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": ", where the last inequality is from Lemma G.4 and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        604,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\| \\Phi_{k+1,n} \\| \\leq C_9 \\| \\Phi_{k,n} \\|"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        604,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": " for some constant "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        604,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_9 > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        604,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": " that depends on "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        604,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "e^{\\beta_0 T}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        604,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        160,
+                        635,
+                        449,
+                        734
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                160,
+                                635,
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+                                734
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        160,
+                                        635,
+                                        449,
+                                        734
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\beta_ {n + 1} ^ {- 1 / 2} \\left\\| \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(r _ {k} ^ {\\left(\\boldsymbol {\\theta}, 2\\right)} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} r _ {k} ^ {\\left(\\mathbf {x}, 2\\right)}\\right) \\right\\| \\\\ \\leq \\| \\beta_ {n + 1} ^ {1 / 2} \\Psi_ {n} \\| + \\left(\\frac {\\beta_ {n + 1}}{\\beta_ {n}}\\right) ^ {1 / 2} \\beta_ {n} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} \\| \\beta_ {k} \\Phi_ {k, n} - \\beta_ {k + 1} \\Phi_ {k + 1, n} \\| \\| \\Psi_ {k} \\| \\\\ \\leq \\| \\beta_ {n + 1} ^ {1 / 2} \\Psi_ {n} \\| + C _ {8} \\left(\\frac {\\beta_ {n + 1}}{\\beta_ {n}}\\right) ^ {1 / 2} \\beta_ {n} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} \\beta_ {k} ^ {3 / 2} e ^ {- (u _ {n} - u _ {k}) T} \\| \\beta_ {k} ^ {1 / 2} \\Psi_ {k} \\|. \\\\ \\end{array}",
+                                    "image_path": "3d40c34f93ed04efdcb31fc91d68e37f2f0536541bf8c93c84f96c8dedf892bc.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
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+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
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+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "32"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 31
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
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+                        81,
+                        440,
+                        95
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                95
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        440,
+                                        95
+                                    ],
+                                    "type": "text",
+                                    "content": "By Lemma E.1, we have "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        440,
+                                        95
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_n^{1/2} \\Psi_n \\to 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        440,
+                                        95
+                                    ],
+                                    "type": "text",
+                                    "content": " a.s. such that by Lemma G.6, it follows that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        115,
+                        100,
+                        495,
+                        133
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                115,
+                                100,
+                                495,
+                                133
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        115,
+                                        100,
+                                        495,
+                                        133
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\lim _ {n} \\sup  _ {\\boldsymbol {n}} \\beta_ {n + 1} ^ {- 1 / 2} \\left\\| \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} (r _ {k} ^ {(\\boldsymbol {\\theta}, 2)} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} r _ {k} ^ {(\\mathbf {x}, 2)}) \\right\\| \\leq \\frac {\\lim  \\sup  _ {n} \\| \\beta_ {n} ^ {1 / 2} \\Psi_ {n} \\|}{C (T , 1 / 2)} = 0.",
+                                    "image_path": "3681203f9cbd22963fc17df488c38dadc76b1b31b7477fb0c73bc50163c075a2.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        105,
+                        139,
+                        186,
+                        150
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                139,
+                                186,
+                                150
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        139,
+                                        186,
+                                        150
+                                    ],
+                                    "type": "text",
+                                    "content": "Therefore, we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        164,
+                        156,
+                        504,
+                        188
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                164,
+                                156,
+                                504,
+                                188
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        164,
+                                        156,
+                                        504,
+                                        188
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\Delta_ {n + 1} ^ {(\\boldsymbol {\\theta}, 2)} = \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(r _ {k} ^ {(\\boldsymbol {\\theta}, 2)} - \\mathbf {J} _ {1 2} (\\alpha) \\mathbf {J} _ {2 2} (\\alpha) ^ {- 1} r _ {k} ^ {(\\mathbf {x}, 2)}\\right) = o \\left(\\sqrt {\\beta_ {n}}\\right). \\tag {75}",
+                                    "image_path": "f6913f491d330deba5ce8be0eb10837ad98854efb10077dea4272f1409df5c14.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        201,
+                        362,
+                        217
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                201,
+                                362,
+                                217
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        201,
+                                        362,
+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": "Consequently, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        201,
+                                        362,
+                                        217
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta_{n + 1}^{(\\theta)} = O(\\beta_n^2\\gamma_n^{-2}\\omega_n^2) + o(\\sqrt{\\beta_n})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        201,
+                                        362,
+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": " almost surely."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        220,
+                        504,
+                        243
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                220,
+                                504,
+                                243
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        220,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "text",
+                                    "content": "Now we are dealing with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        220,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n - \\pmb{\\mu}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        220,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "text",
+                                    "content": " and its related sequence "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        220,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\omega_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        220,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "text",
+                                    "content": ". Note that by Lemma E.5 and (69), we have almost surely,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
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+                        249,
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+                        298
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        220,
+                                        249,
+                                        504,
+                                        298
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\left\\| \\mathbf {x} _ {n} - \\boldsymbol {\\mu} \\right\\| = O \\left(\\left\\| L _ {n} ^ {(\\mathbf {x})} \\right\\| + \\left\\| \\Delta_ {n} ^ {\\mathbf {x}} \\right\\|\\right) \\\\ = O \\left(\\sqrt {\\gamma_ {n} \\log \\left(s _ {n}\\right)} + o \\left(\\sqrt {\\gamma_ {n}}\\right)\\right) \\tag {76} \\\\ = O \\left(\\sqrt {\\gamma_ {n} \\log \\left(s _ {n}\\right)}\\right). \\\\ \\end{array}",
+                                    "image_path": "7f8ddd47d864fa09ac487c55e2744fc09d2c1ea097f79fee148c908b0b7755f7.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            ],
+                            "spans": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Thus, we can set "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\omega_{n}\\equiv O(\\sqrt{\\gamma_{n}\\log(s_{n})})"
+                                },
+                                {
+                                    "bbox": [
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+                                        305,
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+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\| R_n^{(\\pmb {\\theta})}\\|"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        305,
+                                        440,
+                                        320
+                                    ],
+                                    "type": "text",
+                                    "content": " in (70) can be written as"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
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+                                    "type": "interline_equation",
+                                    "content": "\\left\\| R _ {n} ^ {(\\boldsymbol {\\theta})} \\right\\| = O \\left(n ^ {a / 2 - b} \\sqrt {\\log \\left(s _ {n}\\right)} + e ^ {- u _ {n} T}\\right),",
+                                    "image_path": "22cb03b189728907332f985414ec511191e7733fa8e20a28c3fb3bb2699c2bcc.jpg"
+                                }
+                            ]
+                        }
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+                },
+                {
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+                    ],
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+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "and"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                            ],
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left\\| \\Delta_ {n + 1} ^ {(\\boldsymbol {\\theta})} \\right\\| = O \\left(n ^ {a - 2 b} \\log \\left(s _ {n}\\right)\\right) + o \\left(\\sqrt {\\beta_ {n}}\\right).",
+                                    "image_path": "5481f046fe4d76dd0f8c173b9900c159066e0ad65c4cc8dabccdffc55af9658c.jpg"
+                                }
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+                {
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+                    ],
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+                    "angle": 0,
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+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
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+                                        398
+                                    ],
+                                    "type": "text",
+                                    "content": "In view of assumption A2 and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        504,
+                                        398
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_{n} = o(\\gamma_{n})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        504,
+                                        398
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        504,
+                                        398
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "a / 2 - b < -b / 2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        504,
+                                        398
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        504,
+                                        398
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "a - 2b < -b"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        504,
+                                        398
+                                    ],
+                                    "type": "text",
+                                    "content": ", there exists a "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        504,
+                                        398
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "c > b / 2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        504,
+                                        398
+                                    ],
+                                    "type": "text",
+                                    "content": " such that almost surely,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        217,
+                        403,
+                        392,
+                        420
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                    "bbox": [
+                                        217,
+                                        403,
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+                                        420
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left\\| R _ {n} ^ {\\left(\\boldsymbol {\\theta}\\right)} \\right\\| = O \\left(n ^ {- s}\\right), \\quad \\left\\| \\Delta_ {n + 1} ^ {\\left(\\boldsymbol {\\theta}\\right)} \\right\\| = o (\\sqrt {\\beta_ {n}}).",
+                                    "image_path": "05fb74fb5350df073216bae2c2da5fbbf9757bbe1719a0ada4c8a2ea7659a119.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
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+                        442
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                426,
+                                479,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
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+                                        442
+                                    ],
+                                    "type": "text",
+                                    "content": "Therefore, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
+                                        479,
+                                        442
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_{n}^{-1 / 2}(R_{n}^{(\\pmb{\\theta})} + \\Delta_{n + 1}^{(\\pmb{\\theta})})\\to 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
+                                        479,
+                                        442
+                                    ],
+                                    "type": "text",
+                                    "content": " almost surely. This completes the proof of Scenario 2."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
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+                        454,
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+                        467
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                454,
+                                230,
+                                467
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                        467
+                                    ],
+                                    "type": "text",
+                                    "content": "E.3 CASE (III): "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        454,
+                                        230,
+                                        467
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\gamma_{n} = o(\\beta_{n})"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        104,
+                        475,
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+                        498
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                                504,
+                                498
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
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+                                        498
+                                    ],
+                                    "type": "text",
+                                    "content": "For "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        504,
+                                        498
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\gamma_{n} = o(\\beta_{n})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        504,
+                                        498
+                                    ],
+                                    "type": "text",
+                                    "content": ", we can see that the roles of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        504,
+                                        498
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        504,
+                                        498
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        504,
+                                        498
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        504,
+                                        498
+                                    ],
+                                    "type": "text",
+                                    "content": " are flipped, i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        504,
+                                        498
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        504,
+                                        498
+                                    ],
+                                    "type": "text",
+                                    "content": " is now on fast timescale while "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        504,
+                                        498
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        475,
+                                        504,
+                                        498
+                                    ],
+                                    "type": "text",
+                                    "content": " is on slow timescale."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        104,
+                        503,
+                        505,
+                        559
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                503,
+                                505,
+                                559
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        503,
+                                        505,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": "We still decompose "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        503,
+                                        505,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        503,
+                                        505,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": " as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        503,
+                                        505,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n - \\boldsymbol{\\mu} = L_n^{(\\mathbf{x})} + \\Delta_n^{(\\mathbf{x})}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        503,
+                                        505,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": ", where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        503,
+                                        505,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "L_{n}^{(\\mathbf{x})},\\Delta_{n}^{(\\mathbf{x})}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        503,
+                                        505,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": " are defined in (56) and (57), respectively. Since "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        503,
+                                        505,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        503,
+                                        505,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": " is independent of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        503,
+                                        505,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        503,
+                                        505,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": ", the results of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        503,
+                                        505,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "L_{n}^{(\\mathbf{x})}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        503,
+                                        505,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        503,
+                                        505,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta_{n}^{(\\mathbf{x})}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        503,
+                                        505,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": " remain the same, i.e., almost surely, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        503,
+                                        505,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "L_{n}^{(\\mathbf{x})} = O\\bigl (\\sqrt{\\gamma_{n}\\log(s_{n})}\\bigr)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        503,
+                                        505,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": " from Lemma E.5 and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        503,
+                                        505,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta_n^{(\\mathbf{x})} = o(\\sqrt{\\gamma_n})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        503,
+                                        505,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": " from (69). Then, we define sequences "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        503,
+                                        505,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{L}_n^{(\\pmb {\\theta})}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        503,
+                                        505,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        503,
+                                        505,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{R}_n^{(\\pmb {\\theta})}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        503,
+                                        505,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": " as follows."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        183,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\hat {L} _ {n} ^ {(\\boldsymbol {\\theta})} \\triangleq e ^ {\\beta_ {n} \\mathbf {J} _ {1 1}} \\hat {L} _ {n - 1} ^ {(\\boldsymbol {\\theta})} + \\beta_ {n} M _ {n} ^ {(\\boldsymbol {\\theta})} = \\sum_ {k = 1} ^ {n} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\beta_ {k} M _ {k} ^ {(\\boldsymbol {\\theta})}, \\tag {77a}",
+                                    "image_path": "ae474434db0612cab0af485cc8fbd942816756f46b35ac15a3d580d4bef768f3.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                111,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        111,
+                                        605,
+                                        504,
+                                        645
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\hat {R} _ {n} ^ {(\\boldsymbol {\\theta})} \\triangleq e ^ {\\beta_ {n} \\mathbf {J} _ {1 1}} \\hat {R} _ {n - 1} ^ {(\\boldsymbol {\\theta})} + \\beta_ {n} \\mathbf {J} _ {1 2} (\\alpha) \\left(L _ {n - 1} ^ {(\\mathbf {x})} + R _ {n - 1} ^ {(\\mathbf {x})}\\right) = \\sum_ {k = 1} ^ {n} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1}} \\beta_ {k} \\mathbf {J} _ {1 2} (\\alpha) \\left(L _ {k - 1} ^ {(\\mathbf {x})} + R _ {k - 1} ^ {(\\mathbf {x})}\\right). \\tag {77b}",
+                                    "image_path": "d75c028e541c9e8028a2fab8da75f43673f192bb744d9b6fbcee8b63a64b5bac.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "Moreover, the remaining term "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\Delta}_n^{(\\pmb {\\theta})}\\triangleq \\pmb {\\theta}_n - \\pmb {\\theta}^* -\\hat{L}_n^{(\\pmb {\\theta})} - \\hat{R}_n^{(\\pmb {\\theta})}."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
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+                    "lines": [
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+                            "bbox": [
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+                                335,
+                                676
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        664,
+                                        335,
+                                        676
+                                    ],
+                                    "type": "text",
+                                    "content": "The proof outline is the same as in the previous scenario:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 21
+                },
+                {
+                    "bbox": [
+                        132,
+                        685,
+                        504,
+                        733
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 24,
+                    "blocks": [
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+                            "bbox": [
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+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": "- We first show "
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                685,
+                                                457,
+                                                700
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_n^{-1/2} \\hat{\\Delta}_n^{(\\theta)}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                685,
+                                                457,
+                                                700
+                                            ],
+                                            "type": "text",
+                                            "content": " weakly converges to the distribution "
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                685,
+                                                457,
+                                                700
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "N(0, \\mathbf{V}_{\\pmb{\\theta}}^{(3)})(\\alpha)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                685,
+                                                457,
+                                                700
+                                            ],
+                                            "type": "text",
+                                            "content": ";"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 22
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "spans": [
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+                                            "bbox": [
+                                                132,
+                                                704,
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+                                            ],
+                                            "type": "text",
+                                            "content": "- We analyse "
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                704,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\hat{L}_n^{(\\theta)}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                704,
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+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                704,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\hat{R}_n^{(\\theta)}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                704,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "text",
+                                            "content": " to ensure that these two terms decrease faster than the CLT scale "
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                704,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_n^{-1/2}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
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+                                            ],
+                                            "type": "text",
+                                            "content": ", i.e., "
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                704,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\lim_{n\\to\\infty}\\beta_n^{-1/2}(\\hat{L}_n^{(\\theta)} - \\hat{R}_n^{(\\theta)}) = 0"
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
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+                                            ],
+                                            "type": "text",
+                                            "content": ";"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 23
+                        }
+                    ],
+                    "sub_type": "text"
+                }
+            ],
+            "discarded_blocks": [
+                {
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+                    ],
+                    "type": "header",
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+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
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+                },
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+                                    "content": "33"
+                                }
+                            ]
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+                }
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+                                    "bbox": [
+                                        132,
+                                        81,
+                                        504,
+                                        110
+                                    ],
+                                    "type": "text",
+                                    "content": "- With above two steps, we can show that "
+                                },
+                                {
+                                    "bbox": [
+                                        132,
+                                        81,
+                                        504,
+                                        110
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_{n}^{-1 / 2}(\\pmb{\\theta}_{n} - \\pmb{\\theta}^{*})"
+                                },
+                                {
+                                    "bbox": [
+                                        132,
+                                        81,
+                                        504,
+                                        110
+                                    ],
+                                    "type": "text",
+                                    "content": " weakly converges to the distribution "
+                                },
+                                {
+                                    "bbox": [
+                                        132,
+                                        81,
+                                        504,
+                                        110
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "N(0,\\mathbf{V}_{\\pmb{\\theta}}^{(3)})(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        132,
+                                        81,
+                                        504,
+                                        110
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
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+                        118,
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+                        133
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
+                                        118,
+                                        176,
+                                        133
+                                    ],
+                                    "type": "text",
+                                    "content": "Analysis of "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        118,
+                                        176,
+                                        133
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{L}_n^{(\\theta)}"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        137,
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+                        167
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                504,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        137,
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+                                        167
+                                    ],
+                                    "type": "text",
+                                    "content": "We first focus on "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        137,
+                                        504,
+                                        167
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{L}_n^{(\\theta)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        137,
+                                        504,
+                                        167
+                                    ],
+                                    "type": "text",
+                                    "content": " and follow similar steps as we did when we analysed "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        137,
+                                        504,
+                                        167
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "L_{n}^{(\\theta)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        137,
+                                        504,
+                                        167
+                                    ],
+                                    "type": "text",
+                                    "content": " in the previous scenario. We set a Martingale "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        137,
+                                        504,
+                                        167
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Z^{(n)} = \\{Z_k^{(n)}\\}_{k\\geq 1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        137,
+                                        504,
+                                        167
+                                    ],
+                                    "type": "text",
+                                    "content": " such that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        226,
+                        172,
+                        383,
+                        202
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                226,
+                                172,
+                                383,
+                                202
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        226,
+                                        172,
+                                        383,
+                                        202
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "Z _ {k} ^ {(n)} = \\beta_ {n} ^ {- 1 / 2} \\sum_ {k = 1} ^ {n} e ^ {\\left(u _ {n} - u _ {k}\\right) \\mathbf {J} _ {1 1} \\beta_ {k} M _ {k} ^ {(\\boldsymbol {\\theta})}}.",
+                                    "image_path": "51e394abc13f970e2700d7dea7e90402720e8cafdf3c8cfb7d17aaa3e829b71a.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        105,
+                        208,
+                        132,
+                        218
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                208,
+                                132,
+                                218
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        208,
+                                        132,
+                                        218
+                                    ],
+                                    "type": "text",
+                                    "content": "Then,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        192,
+                        218,
+                        416,
+                        249
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                192,
+                                218,
+                                416,
+                                249
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        192,
+                                        218,
+                                        416,
+                                        249
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "A _ {n} \\triangleq \\sum_ {k = 1} ^ {n} \\mathbb {E} \\left[ (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) ^ {T} \\Big | \\mathcal {F} _ {k - 1} \\right].",
+                                    "image_path": "f1976b147cdf08c2a34ed087bb8ad8d74f64cf6306682db14dd76398e548e070.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        252,
+                        411,
+                        268
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                252,
+                                411,
+                                268
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        252,
+                                        411,
+                                        268
+                                    ],
+                                    "type": "text",
+                                    "content": "Following the similar steps in (59) to decompose "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        252,
+                                        411,
+                                        268
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M_{k}^{(\\pmb{\\theta})}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        252,
+                                        411,
+                                        268
+                                    ],
+                                    "type": "text",
+                                    "content": " with (42b), we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        115,
+                        273,
+                        504,
+                        407
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                115,
+                                273,
+                                504,
+                                407
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        115,
+                                        273,
+                                        504,
+                                        407
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} A _ {n} = \\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\beta_ {k} ^ {2} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\left(\\mathbf {U} _ {1 1} + \\mathbf {D} _ {k} ^ {(3)} + \\mathbf {J} _ {k} ^ {(3)}\\right) e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1} ^ {T}} \\\\ = \\underbrace {\\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\beta_ {k} ^ {2} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\mathbf {U} _ {1 1} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1} ^ {T}}} _ {A _ {n} ^ {(a)}} + \\underbrace {\\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\beta_ {k} ^ {2} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\mathbf {D} _ {k} ^ {(3)} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1} ^ {T}}} _ {A _ {n} ^ {(b)}} \\tag {78} \\\\ + \\underbrace {\\beta_ {n} ^ {- 1} \\sum_ {k = 1} ^ {n} \\beta_ {k} ^ {2} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1}} \\mathbf {J} _ {k} ^ {(3)} e ^ {(u _ {n} - u _ {k}) \\mathbf {J} _ {1 1} ^ {T}}} _ {A _ {n} ^ {(c)}} \\\\ \\end{array}",
+                                    "image_path": "eea6547639edc478c3b7ab3e1d5badfc92872c2622ed4b6cb0d0b2ab86250334.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        413,
+                        506,
+                        449
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                413,
+                                506,
+                                449
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        413,
+                                        506,
+                                        449
+                                    ],
+                                    "type": "text",
+                                    "content": "Since "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        413,
+                                        506,
+                                        449
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A_{n}^{(a)}, A_{n}^{(b)}, A_{n}^{(c)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        413,
+                                        506,
+                                        449
+                                    ],
+                                    "type": "text",
+                                    "content": " share similar forms as in Lemma E.3, we follow the same steps as the proof therein, with the application of Lemma E.1. To avoid repetition, we omit the proof and directly give the following lemma."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
+                        452,
+                        345,
+                        467
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                452,
+                                345,
+                                467
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        452,
+                                        345,
+                                        467
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma E.8. For "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        452,
+                                        345,
+                                        467
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A_{n}^{(a)}, A_{n}^{(b)}, A_{n}^{(c)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        452,
+                                        345,
+                                        467
+                                    ],
+                                    "type": "text",
+                                    "content": " defined in (78), we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        175,
+                        472,
+                        504,
+                        491
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                175,
+                                472,
+                                504,
+                                491
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        175,
+                                        472,
+                                        504,
+                                        491
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\lim  _ {n \\rightarrow \\infty} A _ {n} ^ {(a)} = \\mathbf {V} _ {\\boldsymbol {\\theta}} ^ {(3)} (\\alpha), \\quad \\lim  _ {n \\rightarrow \\infty} \\| A _ {n} ^ {(b)} \\| = 0, \\quad \\lim  _ {n \\rightarrow \\infty} \\| A _ {n} ^ {(c)} \\| = 0, \\tag {79}",
+                                    "image_path": "96ea89a18072820f451a2e98e58bc751e859aded77d124f85c913c9d02e9b22d.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        104,
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+                        331,
+                        512
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                497,
+                                331,
+                                512
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        497,
+                                        331,
+                                        512
+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        497,
+                                        331,
+                                        512
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\pmb{\\theta}}^{(3)}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        497,
+                                        331,
+                                        512
+                                    ],
+                                    "type": "text",
+                                    "content": " is the solution to the Lyapunov equation"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        248,
+                        517,
+                        361,
+                        531
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                248,
+                                517,
+                                361,
+                                531
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        248,
+                                        517,
+                                        361,
+                                        531
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {J} _ {1 1} \\mathbf {V} + \\mathbf {V} \\mathbf {J} _ {1 1} ^ {T} + \\mathbf {U} _ {1 1} = 0.",
+                                    "image_path": "c7199be7f52b2fb87939aa9380ddc86ee0462b8a1d3f5859e2a5ac9c28a90a09.jpg"
+                                }
+                            ]
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "Note that here we don't have the term "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        543,
+                                        504,
+                                        583
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\frac{\\mathbb{1}_{\\{b = 1\\}}}{2}\\pmb{I}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        543,
+                                        504,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": " in above lemma, compared to Lemma E.3, because in the case of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        543,
+                                        504,
+                                        583
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\gamma_{n} = o(\\beta_{n})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        543,
+                                        504,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        543,
+                                        504,
+                                        583
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "b < 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        543,
+                                        504,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        543,
+                                        504,
+                                        583
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{1}_{\\{b = 1\\}} = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        543,
+                                        504,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, applying Lemma G.1 to derive the closed form of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        543,
+                                        504,
+                                        583
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\pmb{\\theta}}^{(3)}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        543,
+                                        504,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": " gives"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        217,
+                        588,
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+                        605
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
+                                        217,
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+                                        605
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {V} _ {\\boldsymbol {\\theta}} ^ {(3)} (\\alpha) = \\int_ {0} ^ {\\infty} e ^ {t \\nabla_ {\\boldsymbol {\\theta}} \\mathbf {h} (\\boldsymbol {\\theta} ^ {*})} \\mathbf {U} _ {1 1} e ^ {t \\nabla_ {\\boldsymbol {\\theta}} \\mathbf {h} (\\boldsymbol {\\theta} ^ {*})} d t.",
+                                    "image_path": "e7fc7606ae4455c94a0265d7b16c83019347a9661ae26999486ab5aa8fb34f6a.jpg"
+                                }
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+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "type": "text",
+                                    "content": "Thus, it follows that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        173,
+                                        625,
+                                        436,
+                                        655
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\lim  _ {n \\rightarrow \\infty} \\sum_ {k = 1} ^ {n} \\mathbb {E} \\left[ (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) (Z _ {k} ^ {(n)} - Z _ {k - 1} ^ {(n)}) ^ {T} | \\mathcal {F} _ {k - 1} \\right] = \\mathbf {V} _ {\\boldsymbol {\\theta}} ^ {(3)} (\\alpha).",
+                                    "image_path": "da388415963e1a5cdaab2df5988b1189199b7c6c9c3a319ac7a5478b0001d1df.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "Again, we use the Martingale CLT result in Theorem G.3 and have the following result."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
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+                        677,
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "Z ^ {n} = \\beta_ {n} ^ {- 1 / 2} \\hat {L} _ {n} ^ {(\\boldsymbol {\\theta})} \\xrightarrow [ d i s t . ]{n \\to \\infty} N \\left(0, \\mathbf {V} _ {\\boldsymbol {\\theta}} ^ {(3)} (\\alpha)\\right).",
+                                    "image_path": "04545ec0d58b52c159da7d1013eaedf995fcc4b5b8348a7b5bb56af2429bd33f.jpg"
+                                }
+                            ]
+                        }
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+                    "index": 19
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        706,
+                                        504,
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+                                    ],
+                                    "type": "text",
+                                    "content": "Moreover, similar to the tighter upper bound of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "L_{n}^{(\\mathbf{x})}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        706,
+                                        504,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": " proved in Lemma E.5, we utilize the tighter upper bound Lemma E.6 in the proof thereof, and obtain "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        706,
+                                        504,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{L}_n^{(\\pmb{\\theta})} = O(\\sqrt{\\beta_n\\log(u_n)})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        706,
+                                        504,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                }
+            ],
+            "discarded_blocks": [
+                {
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+                    ],
+                    "type": "header",
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+                                    "bbox": [
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+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
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+                                    "type": "text",
+                                    "content": "34"
+                                }
+                            ]
+                        }
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+                    "index": 21
+                }
+            ],
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+        },
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+            "para_blocks": [
+                {
+                    "bbox": [
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+                    ],
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+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                {
+                                    "bbox": [
+                                        105,
+                                        79,
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+                                        95
+                                    ],
+                                    "type": "text",
+                                    "content": "Analysis of "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        79,
+                                        176,
+                                        95
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{R}_n^{(\\theta)}"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        100,
+                        504,
+                        126
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                100,
+                                504,
+                                126
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        504,
+                                        126
+                                    ],
+                                    "type": "text",
+                                    "content": "Next, we turn to the term "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        504,
+                                        126
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{R}_n^{(\\theta)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        504,
+                                        126
+                                    ],
+                                    "type": "text",
+                                    "content": " in (77b). Taking the norm gives the following inequality for some constant "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        504,
+                                        126
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C, T > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        504,
+                                        126
+                                    ],
+                                    "type": "text",
+                                    "content": " by applying Lemma G.4,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        195,
+                        132,
+                        414,
+                        163
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                195,
+                                132,
+                                414,
+                                163
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        195,
+                                        132,
+                                        414,
+                                        163
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\| \\hat {R} _ {n} ^ {(\\boldsymbol {\\theta})} \\| \\leq C \\sum_ {k = 1} ^ {n} e ^ {- (u _ {n} - u _ {k}) T} \\beta_ {k} (\\| L _ {k - 1} ^ {(\\mathbf {x})} \\| + \\| R _ {k - 1} ^ {(\\mathbf {x})} \\)).",
+                                    "image_path": "d67a819557715efaf5a919fd4633bbddcf4dc10b3ab0a726371677799323c717.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        105,
+                        168,
+                        206,
+                        180
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                168,
+                                206,
+                                180
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        168,
+                                        206,
+                                        180
+                                    ],
+                                    "type": "text",
+                                    "content": "Using Lemma G.6 gives"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        168,
+                        186,
+                        441,
+                        217
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                168,
+                                186,
+                                441,
+                                217
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        168,
+                                        186,
+                                        441,
+                                        217
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\sum_ {k = 1} ^ {n} e ^ {- (u _ {n} - u _ {k}) T} \\beta_ {k} (\\| L _ {k - 1} ^ {(\\mathbf {x})} \\| + \\| R _ {k - 1} ^ {(\\mathbf {x})} \\|) = O (\\| L _ {k - 1} ^ {(\\mathbf {x})} \\| + \\| R _ {n - 1} ^ {(\\mathbf {x})} \\)).",
+                                    "image_path": "febc1563401540c6c4d765f0565c8102fe63065c7badd11f3f38262d3d719a80.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        224,
+                        504,
+                        295
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                224,
+                                504,
+                                295
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        504,
+                                        295
+                                    ],
+                                    "type": "text",
+                                    "content": "Thus, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        504,
+                                        295
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_n^{-1/2} \\|\\hat{R}_n^{(\\pmb{\\theta})}\\| = o(\\sqrt{\\gamma_n \\beta_n^{-1}}) + O\\left(\\sqrt{\\gamma_n \\beta_n^{-1} \\log(s_n)}\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        504,
+                                        295
+                                    ],
+                                    "type": "text",
+                                    "content": ". Since "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        504,
+                                        295
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\gamma_n = o(\\beta_n), \\gamma_n \\beta_n^{-1} = (n + 1)^{b - a}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        504,
+                                        295
+                                    ],
+                                    "type": "text",
+                                    "content": ", where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        504,
+                                        295
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "b - a < 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        504,
+                                        295
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, there exists some "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        504,
+                                        295
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        504,
+                                        295
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        504,
+                                        295
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "b - a < -s < 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        504,
+                                        295
+                                    ],
+                                    "type": "text",
+                                    "content": ". Together with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        504,
+                                        295
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\log(s_n) = O(\\log(n))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        504,
+                                        295
+                                    ],
+                                    "type": "text",
+                                    "content": ", we have "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        504,
+                                        295
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "O\\left(\\sqrt{\\gamma_n \\beta_n^{-1} \\log(s_n)}\\right) = O(\\sqrt{n^{-s} \\log(n)}) = o(1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        504,
+                                        295
+                                    ],
+                                    "type": "text",
+                                    "content": ". Therefore, we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        259,
+                        295,
+                        350,
+                        313
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                259,
+                                295,
+                                350,
+                                313
+                            ],
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+                                {
+                                    "bbox": [
+                                        259,
+                                        295,
+                                        350,
+                                        313
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\lim _ {n \\to \\infty} \\beta_ {n} ^ {- 1 / 2} \\hat {R} _ {n} ^ {(\\boldsymbol {\\theta})} = 0.",
+                                    "image_path": "8cef54c3b69157e04aaf463d27c2f31c5c88e624343d446d789d4ecbbf952ecc.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
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+                        337
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
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+                                176,
+                                337
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        323,
+                                        176,
+                                        337
+                                    ],
+                                    "type": "text",
+                                    "content": "Analysis of "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        323,
+                                        176,
+                                        337
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\Delta}_n^{(\\theta)}"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
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+                        342,
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+                        357
+                    ],
+                    "type": "text",
+                    "angle": 0,
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+                        {
+                            "bbox": [
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+                                    "bbox": [
+                                        105,
+                                        342,
+                                        288,
+                                        357
+                                    ],
+                                    "type": "text",
+                                    "content": "Lastly, let's focus on the term "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        342,
+                                        288,
+                                        357
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\Delta}_n^{(\\theta)}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        342,
+                                        288,
+                                        357
+                                    ],
+                                    "type": "text",
+                                    "content": ". We have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        110,
+                        363,
+                        498,
+                        485
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                110,
+                                363,
+                                498,
+                                485
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        110,
+                                        363,
+                                        498,
+                                        485
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\hat {\\Delta} _ {n + 1} ^ {(\\boldsymbol {\\theta})} = \\boldsymbol {\\theta} _ {n + 1} - \\boldsymbol {\\theta} ^ {*} - \\hat {L} _ {n + 1} ^ {(\\boldsymbol {\\theta})} - \\hat {R} _ {n + 1} ^ {(\\boldsymbol {\\theta})} \\\\ = \\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*} + \\beta_ {n + 1} \\left(\\mathbf {J} _ {1 1} (\\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*}) + \\mathbf {J} _ {1 2} (\\alpha) (\\mathbf {x} _ {n} - \\boldsymbol {\\mu}) + M _ {n + 1} ^ {(\\boldsymbol {\\theta})} + r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} + r _ {n} ^ {(\\boldsymbol {\\theta}, 2)} + \\eta_ {n} ^ {(\\boldsymbol {\\theta})}\\right) \\\\ - e ^ {\\beta_ {n + 1} \\mathbf {J} _ {1 1}} \\hat {L} _ {n} ^ {(\\boldsymbol {\\theta})} - \\beta_ {n + 1} M _ {n + 1} ^ {(\\boldsymbol {\\theta})} - e ^ {\\beta_ {n + 1} \\mathbf {J} _ {1 1}} \\hat {R} _ {n} ^ {(\\boldsymbol {\\theta})} - \\beta_ {n + 1} \\mathbf {J} _ {1 2} (\\alpha) (L _ {n} ^ {(\\mathbf {x})} + R _ {n} ^ {(\\mathbf {x})}) \\\\ = (\\mathbf {I} + \\beta_ {n + 1} \\mathbf {J} _ {1 1}) \\left(\\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*}\\right) + \\beta_ {n + 1} \\mathbf {J} _ {1 2} (\\alpha) \\Delta_ {n} ^ {(\\mathbf {x})} + \\beta_ {n + 1} \\left(r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} + r _ {n} ^ {(\\boldsymbol {\\theta}, 2)} + \\eta_ {n} ^ {(\\boldsymbol {\\theta})}\\right) \\\\ - \\left(\\mathbf {I} + \\beta_ {n + 1} \\mathbf {J} _ {1 1} + O \\left(\\beta_ {n + 1} ^ {2}\\right)\\right) \\left(\\hat {L} _ {n} ^ {(\\boldsymbol {\\theta})} + \\hat {R} _ {n} ^ {(\\boldsymbol {\\theta})}\\right) \\\\ = \\left(\\mathbf {I} + \\beta_ {n + 1} \\mathbf {J} _ {1 1}\\right) \\hat {\\Delta} _ {n} ^ {(\\boldsymbol {\\theta})} + \\beta_ {n + 1} \\mathbf {J} _ {1 2} (\\alpha) \\Delta_ {n} ^ {(\\mathbf {x})} + \\beta_ {n + 1} \\left(r _ {n} ^ {(\\boldsymbol {\\theta}, 1)} + r _ {n} ^ {(\\boldsymbol {\\theta}, 2)} + \\eta_ {n} ^ {(\\boldsymbol {\\theta})}\\right) \\\\ + O \\left(\\beta_ {n + 1} ^ {2}\\right) \\left(\\hat {L} _ {n} ^ {(\\boldsymbol {\\theta})} + \\hat {R} _ {n} ^ {(\\boldsymbol {\\theta})}\\right). \\\\ \\end{array}",
+                                    "image_path": "f697674966549c15f93d5f231f73f41e7565397dc5fd7907aa3120e00a016aa9.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        492,
+                        502,
+                        519
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                492,
+                                502,
+                                519
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        492,
+                                        502,
+                                        519
+                                    ],
+                                    "type": "text",
+                                    "content": "where the second equality is from (53a), the third equality stems from the approximation of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        492,
+                                        502,
+                                        519
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "e^{\\beta_{n+1}\\mathbf{J}_{11}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        492,
+                                        502,
+                                        519
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, we again use the definition "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        492,
+                                        502,
+                                        519
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Phi_{k,n} \\triangleq \\prod_{j=k+1}^{n}(\\mathbf{I} + \\beta_{j}\\mathbf{J}_{11})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        492,
+                                        502,
+                                        519
+                                    ],
+                                    "type": "text",
+                                    "content": " and reiterate the above equation as"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        165,
+                        526,
+                        444,
+                        641
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                165,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        165,
+                                        526,
+                                        444,
+                                        641
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\hat {\\Delta} _ {n + 1} ^ {(\\boldsymbol {\\theta})} = \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(O (\\beta_ {k + 1}) L _ {k} ^ {(\\boldsymbol {\\theta})} + O (\\beta_ {k + 1}) R _ {k} ^ {(\\boldsymbol {\\theta})}\\right) \\\\ + \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\mathbf {J} _ {1 2} (\\alpha) \\Delta_ {n} ^ {(\\mathbf {x})} + \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(r _ {k} ^ {(\\boldsymbol {\\theta}, 1)} + \\eta_ {k} ^ {(\\boldsymbol {\\theta})}\\right) \\\\ + \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} r _ {k} ^ {(\\boldsymbol {\\theta}, 2)} \\\\ \\triangleq \\hat {\\Delta} _ {n + 1} ^ {(\\boldsymbol {\\theta}, 1)} + \\hat {\\Delta} _ {n + 1} ^ {(\\boldsymbol {\\theta}, 2)}, \\\\ \\end{array}",
+                                    "image_path": "7f8b1331ed8ea6976b535a0e93a1929ef46f271e546e72f2402768d3cd7d4478.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        105,
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+                        663
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
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+                                        663
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\Delta}_{n + 1}^{(\\pmb {\\theta},2)} = \\sum_{k = 1}^{n}\\Phi_{k,n}\\beta_{k + 1}r_{k}^{(\\pmb {\\theta},2)}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        647,
+                                        282,
+                                        663
+                                    ],
+                                    "type": "text",
+                                    "content": " and"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        190,
+                        671,
+                        504,
+                        734
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                190,
+                                671,
+                                504,
+                                734
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        190,
+                                        671,
+                                        504,
+                                        734
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\hat {\\Delta} _ {n + 1} ^ {(\\boldsymbol {\\theta}, 1)} = \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(O \\left(\\beta_ {k + 1}\\right) L _ {k} ^ {(\\boldsymbol {\\theta})} + O \\left(\\beta_ {k + 1}\\right) R _ {k} ^ {(\\boldsymbol {\\theta})}\\right) \\tag {80} \\\\ + \\sum_ {k = 1} ^ {n} \\Phi_ {k, n} \\beta_ {k + 1} \\left(r _ {k} ^ {\\left(\\boldsymbol {\\theta}, 1\\right)} + \\eta_ {k} ^ {\\left(\\boldsymbol {\\theta}\\right)} + \\mathbf {J} _ {1 2} (\\alpha) \\Delta_ {n} ^ {\\left(\\mathbf {x}\\right)}\\right). \\\\ \\end{array}",
+                                    "image_path": "74f3137ed9528498b49538a3e08758f412a30df68256978aca960f35738e7beb.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
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+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
+                        750,
+                        310,
+                        760
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "35"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 34
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        79,
+                        449,
+                        97
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                79,
+                                449,
+                                97
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        449,
+                                        97
+                                    ],
+                                    "type": "text",
+                                    "content": "For "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        449,
+                                        97
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\Delta}_{n+1}^{(\\theta,2)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        449,
+                                        97
+                                    ],
+                                    "type": "text",
+                                    "content": ", we follow the same steps from (74) to (75), and obtain "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        449,
+                                        97
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\Delta}_{n+1}^{(\\theta,2)} = o\\left(\\sqrt{\\beta_n}\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        79,
+                                        449,
+                                        97
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        101,
+                        504,
+                        133
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                101,
+                                504,
+                                133
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        101,
+                                        504,
+                                        133
+                                    ],
+                                    "type": "text",
+                                    "content": "Next, we consider "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        101,
+                                        504,
+                                        133
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\Delta}_{n+1}^{(\\pmb{\\theta},1)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        101,
+                                        504,
+                                        133
+                                    ],
+                                    "type": "text",
+                                    "content": " and want to show that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        101,
+                                        504,
+                                        133
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\Delta}_{n+1}^{(\\pmb{\\theta},1)} = o(\\sqrt{\\beta_n})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        101,
+                                        504,
+                                        133
+                                    ],
+                                    "type": "text",
+                                    "content": ". Again, we utilize Mokkadem & Pelletier (2006, Lemma 6) for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        101,
+                                        504,
+                                        133
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\Delta}_{n+1}^{(\\pmb{\\theta},1)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        101,
+                                        504,
+                                        133
+                                    ],
+                                    "type": "text",
+                                    "content": " and adapt the notation here for the case "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        101,
+                                        504,
+                                        133
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\gamma_n = o(\\beta_n)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        101,
+                                        504,
+                                        133
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        103,
+                        134,
+                        504,
+                        160
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                103,
+                                134,
+                                504,
+                                160
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        103,
+                                        134,
+                                        504,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma E.9. For "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        134,
+                                        504,
+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\Delta}_{n+1}^{(\\pmb{\\theta}, 1)}"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        134,
+                                        504,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": " in the form of (80), assume "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        134,
+                                        504,
+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\|\\pmb{\\theta}_n - \\pmb{\\theta}^*\\| = O(\\omega_n)"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        134,
+                                        504,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        134,
+                                        504,
+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\|\\hat{\\Delta}_n^{(\\pmb{\\theta}, 1)}\\| = O(\\delta_n)"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        134,
+                                        504,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": " for the sequences "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        134,
+                                        504,
+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\omega_n, \\delta_n"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        134,
+                                        504,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": " defined in (E.1). Then, we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        193,
+                        164,
+                        505,
+                        182
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                193,
+                                164,
+                                505,
+                                182
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        193,
+                                        164,
+                                        505,
+                                        182
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left\\| \\hat {\\Delta} _ {n + 1} ^ {(\\boldsymbol {\\theta}, 1)} \\right\\| = O \\left(\\gamma_ {n} ^ {2} \\beta_ {n} ^ {- 2} \\omega_ {n} ^ {2} + \\gamma_ {n} \\beta_ {n} ^ {- 1} \\delta_ {n}\\right) + o (\\sqrt {\\gamma_ {n}}) \\quad a. s. \\tag {81}",
+                                    "image_path": "a36b1712701b772a01271fcd12c4cca8a98dbb48940a2c9126fb0c608f5acaf6.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        190,
+                        506,
+                        228
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                190,
+                                506,
+                                228
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        228
+                                    ],
+                                    "type": "text",
+                                    "content": "Now we need to further analyse "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        228
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\delta_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        228
+                                    ],
+                                    "type": "text",
+                                    "content": " and tighten its big O form, starting from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        228
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\delta_{n} \\equiv 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        228
+                                    ],
+                                    "type": "text",
+                                    "content": ", so that we can finally obtain the big O form of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        228
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\|\\hat{\\Delta}_{n+1}^{(\\theta,1)}\\|"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        228
+                                    ],
+                                    "type": "text",
+                                    "content": ". The following steps are borrowed from the ideas in Mokkadem & Pelletier (2006, Section 2.3.2)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        233,
+                        505,
+                        262
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                233,
+                                505,
+                                262
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        262
+                                    ],
+                                    "type": "text",
+                                    "content": "By almost sure convergence result "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        262
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\lim_{n\\to \\infty}\\pmb{\\theta}_n = \\pmb{\\theta}^*"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        262
+                                    ],
+                                    "type": "text",
+                                    "content": ", we have "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        262
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\lim_{n\\to \\infty}\\Delta_n^{(\\pmb{\\theta})} = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        262
+                                    ],
+                                    "type": "text",
+                                    "content": " a.s. such that we can first set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        262
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\delta_{n}\\equiv 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        262
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        262
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\| \\hat{\\Delta}_{n + 1}^{(\\pmb {\\theta},1)}\\| = O(\\gamma_n^2\\beta_n^{-2}\\omega_n^2 +\\gamma_n\\beta_n^{-1}) + o(\\sqrt{\\gamma_n})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        262
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, we redefine"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        220,
+                        266,
+                        389,
+                        281
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                220,
+                                266,
+                                389,
+                                281
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        220,
+                                        266,
+                                        389,
+                                        281
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\delta_ {n} \\equiv O (\\gamma_ {n} ^ {2} \\beta_ {n} ^ {- 2} \\omega_ {n} ^ {2} + \\gamma_ {n} \\beta_ {n} ^ {- 1}) + o (\\sqrt {\\gamma_ {n}}),",
+                                    "image_path": "7235d3ad86e213510b40595a6502d9aab9a3c8c04a2b0826171d936cebdf3a29.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        104,
+                        285,
+                        461,
+                        297
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                285,
+                                461,
+                                297
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        285,
+                                        461,
+                                        297
+                                    ],
+                                    "type": "text",
+                                    "content": "and notice that it still satisfies definition E.1. Then, reapplying this "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        285,
+                                        461,
+                                        297
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\delta_{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        285,
+                                        461,
+                                        297
+                                    ],
+                                    "type": "text",
+                                    "content": " form to (81) gives"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        204,
+                        301,
+                        406,
+                        318
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                204,
+                                301,
+                                406,
+                                318
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        204,
+                                        301,
+                                        406,
+                                        318
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\| \\hat {\\Delta} _ {n + 1} ^ {(\\pmb {\\theta}, 1)} \\| = O (\\gamma_ {n} ^ {2} \\beta_ {n} ^ {- 2} \\omega_ {n} ^ {2} + [ \\gamma_ {n} \\beta_ {n} ^ {- 1} ] ^ {2}) + o (\\sqrt {\\gamma_ {n}})",
+                                    "image_path": "fbffa6f0bfaaf5c6fa15112843740e71fc0e851d3e6d3f23a3c1893be92e0ff5.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
+                        322,
+                        301,
+                        334
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                322,
+                                301,
+                                334
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        322,
+                                        301,
+                                        334
+                                    ],
+                                    "type": "text",
+                                    "content": "and by induction we have for all integers "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        322,
+                                        301,
+                                        334
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k \\geq 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        322,
+                                        301,
+                                        334
+                                    ],
+                                    "type": "text",
+                                    "content": ","
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        203,
+                        338,
+                        407,
+                        355
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                203,
+                                338,
+                                407,
+                                355
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        203,
+                                        338,
+                                        407,
+                                        355
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\| \\hat {\\Delta} _ {n + 1} ^ {(\\pmb {\\theta}, 1)} \\| = O (\\gamma_ {n} ^ {2} \\beta_ {n} ^ {- 2} \\omega_ {n} ^ {2} + [ \\gamma_ {n} \\beta_ {n} ^ {- 1} ] ^ {k}) + o (\\sqrt {\\gamma_ {n}}).",
+                                    "image_path": "454e58bf5f495a198a21c2fd39b295338abf38440bdb0e1126e4f2982dcf0414.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        104,
+                        359,
+                        483,
+                        374
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                359,
+                                483,
+                                374
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        359,
+                                        483,
+                                        374
+                                    ],
+                                    "type": "text",
+                                    "content": "Since "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        359,
+                                        483,
+                                        374
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "[\\gamma_n\\beta_n^{-1}]^k = n^{(b - a)k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        359,
+                                        483,
+                                        374
+                                    ],
+                                    "type": "text",
+                                    "content": ", there exists "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        359,
+                                        483,
+                                        374
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k_0 > a / 2(a - b)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        359,
+                                        483,
+                                        374
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        359,
+                                        483,
+                                        374
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "[\\gamma_n\\beta_n^{-1}]^{k_0} = o(\\sqrt{\\gamma_n})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        359,
+                                        483,
+                                        374
+                                    ],
+                                    "type": "text",
+                                    "content": ", and"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        228,
+                        378,
+                        504,
+                        396
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                228,
+                                378,
+                                504,
+                                396
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        228,
+                                        378,
+                                        504,
+                                        396
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left\\| \\hat {\\Delta} _ {n + 1} ^ {(\\boldsymbol {\\theta}, 1)} \\right\\| = O \\left(\\gamma_ {n} ^ {2} \\beta_ {n} ^ {- 2} \\omega_ {n} ^ {2}\\right) + o (\\sqrt {\\gamma_ {n}}). \\tag {82}",
+                                    "image_path": "f05185836a974c84c7486bf30c2dbeff0f72ff522e5db202554ba0a1b5fb6ba1.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        104,
+                        399,
+                        504,
+                        425
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                399,
+                                504,
+                                425
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        399,
+                                        504,
+                                        425
+                                    ],
+                                    "type": "text",
+                                    "content": "Then, as suggested in Mokkadem & Pelletier (2006, Section 2.3.2), we can choose "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        399,
+                                        504,
+                                        425
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\omega_{n} = O(\\sqrt{\\beta_{n}\\log(u_{n})} +[\\gamma_{n}\\beta_{n}^{-1}]^{k})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        399,
+                                        504,
+                                        425
+                                    ],
+                                    "type": "text",
+                                    "content": ", which also satisfies definition E.1. Then,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        140,
+                        429,
+                        469,
+                        507
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                140,
+                                429,
+                                469,
+                                507
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        140,
+                                        429,
+                                        469,
+                                        507
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\left\\| \\boldsymbol {\\theta} _ {n} - \\boldsymbol {\\theta} ^ {*} \\right\\| = \\left\\| \\hat {L} _ {n} ^ {(\\boldsymbol {\\theta})} + \\hat {R} _ {n} ^ {(\\boldsymbol {\\theta})} + \\hat {\\Delta} _ {n} ^ {(\\boldsymbol {\\theta})} \\right\\| \\\\ = O \\left(\\sqrt {\\beta_ {n} \\log (u _ {n})} + \\sqrt {\\gamma_ {n} \\beta_ {n} ^ {- 1} \\log (s _ {n})} + \\left(\\left[ \\gamma_ {n} \\beta_ {n} ^ {- 1} \\right] ^ {k + 1} + \\gamma_ {n} \\beta_ {n} ^ {- 1} \\sqrt {\\beta_ {n} \\log (u _ {n})}\\right) ^ {2}\\right) \\\\ + o (\\sqrt {\\beta_ {n}} + \\sqrt {\\gamma_ {n}}) \\\\ = O (\\sqrt {\\beta_ {n} \\log (u _ {n})} + [ \\gamma_ {n} \\beta_ {n} ^ {- 1} ] ^ {k + 1}). \\\\ \\end{array}",
+                                    "image_path": "e682575b653e2429d40ee9af6a727886ce50c3540dc460f86df60b7043cc1040.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        104,
+                        511,
+                        504,
+                        538
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                511,
+                                504,
+                                538
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        511,
+                                        504,
+                                        538
+                                    ],
+                                    "type": "text",
+                                    "content": "By induction, this holds for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        511,
+                                        504,
+                                        538
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k \\geq 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        511,
+                                        504,
+                                        538
+                                    ],
+                                    "type": "text",
+                                    "content": " such that there exists "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        511,
+                                        504,
+                                        538
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k_0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        511,
+                                        504,
+                                        538
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        511,
+                                        504,
+                                        538
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "[\\gamma_n\\beta_n^{-1}]^{k_0} = o(\\sqrt{\\beta_n})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        511,
+                                        504,
+                                        538
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        511,
+                                        504,
+                                        538
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\| \\pmb{\\theta}_n - \\pmb{\\theta}^*\\| = O(\\sqrt{\\beta_n\\log(u_n)})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        511,
+                                        504,
+                                        538
+                                    ],
+                                    "type": "text",
+                                    "content": ". Equivalently, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        511,
+                                        504,
+                                        538
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\omega_{n} = \\sqrt{\\beta_{n}\\log(u_{n})}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        511,
+                                        504,
+                                        538
+                                    ],
+                                    "type": "text",
+                                    "content": ". Therefore, from (82) we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        195,
+                        543,
+                        415,
+                        559
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                195,
+                                543,
+                                415,
+                                559
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        195,
+                                        543,
+                                        415,
+                                        559
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\| \\hat {\\Delta} _ {n + 1} ^ {(\\boldsymbol {\\theta}, 1)} \\| = O (\\gamma_ {n} ^ {2} \\beta_ {n} ^ {- 1} \\log (u _ {n})) + o (\\sqrt {\\gamma_ {n}}) = o (\\sqrt {\\gamma_ {n}}).",
+                                    "image_path": "5bee32347c79df31e13b51d110518597fbec6e74f599b33e710e71c6b50c7532.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        104,
+                        563,
+                        470,
+                        581
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                563,
+                                470,
+                                581
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        470,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": "Together with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        470,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\| \\hat{\\Delta}_{n + 1}^{(\\pmb {\\theta},2)}\\| = o(\\sqrt{\\beta_n})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        470,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": " , we have "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        470,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_{n}^{-1 / 2}\\| \\hat{\\Delta}_{n + 1}^{(\\pmb {\\theta})}\\| = o(\\sqrt{\\gamma_{n}\\beta_{n}^{-1}}) + 1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        470,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": " such that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
+                        257,
+                        584,
+                        352,
+                        605
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                257,
+                                584,
+                                352,
+                                605
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        257,
+                                        584,
+                                        352,
+                                        605
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\lim  _ {n \\to \\infty} \\beta_ {n} ^ {- 1 / 2} \\hat {\\Delta} _ {n + 1} ^ {(\\boldsymbol {\\theta})} = 0.",
+                                    "image_path": "823298955f6080f81efe031674716c4d87245300f81d02e041591eeafd0d7c33.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                },
+                {
+                    "bbox": [
+                        104,
+                        609,
+                        504,
+                        633
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                609,
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+                                633
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        609,
+                                        504,
+                                        633
+                                    ],
+                                    "type": "text",
+                                    "content": "Thus, we have finished the proof according to the proof outline mentioned at the beginning of this part."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                },
+                {
+                    "bbox": [
+                        104,
+                        647,
+                        414,
+                        659
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                        659
+                                    ],
+                                    "type": "text",
+                                    "content": "F DISCUSSION OF COVARIANCE ORDERING OF SA-SRRW"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 21
+                },
+                {
+                    "bbox": [
+                        105,
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+                        683
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "type": "text",
+                                    "content": "F.1 PROOF OF PROPOSITION 3.4"
+                                }
+                            ]
+                        }
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+                                    "type": "text",
+                                    "content": "For any "
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+                                    "type": "inline_equation",
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " and any vector "
+                                },
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+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x} \\in \\mathbb{R}^d"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        693,
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+                                    ],
+                                    "type": "text",
+                                    "content": ", we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 23
+                },
+                {
+                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {x} ^ {T} \\mathbf {V} _ {\\boldsymbol {\\theta}} ^ {(1)} (\\alpha) \\mathbf {x} = \\int_ {0} ^ {\\infty} \\mathbf {x} ^ {T} e ^ {t (\\nabla_ {\\boldsymbol {\\theta}} \\mathbf {h} (\\boldsymbol {\\theta} ^ {*}) + \\frac {1}{2} \\mathbf {\\Sigma} _ {\\{b = 1 \\}} ^ {*} \\boldsymbol {I})} \\mathbf {U} _ {\\boldsymbol {\\theta}} (\\alpha) e ^ {t (\\nabla_ {\\boldsymbol {\\theta}} \\mathbf {h} (\\boldsymbol {\\theta} ^ {*}) + \\frac {1}{2} \\mathbf {\\Sigma} _ {\\{b = 1 \\}} ^ {*} \\boldsymbol {I}) ^ {T}} \\mathbf {x} d t",
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+                                }
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+                    ],
+                    "type": "header",
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
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+                    ],
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+                                    "type": "text",
+                                    "content": "36"
+                                }
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+                    ],
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+                }
+            ],
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+                                    "bbox": [
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+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": "where the first equality is from the form of "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\theta}^{(1)}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
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+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": " in Theorem 3.3. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
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+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{y} \\triangleq e^{t(\\nabla_{\\theta}\\mathbf{h}(\\boldsymbol{\\theta}^{*}) + \\frac{1}{2}\\left\\{b = 1\\right\\} I)}\\mathbf{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
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+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": ", with the dependence on variable "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": " left implicit. The matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{U}_{\\theta}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": ", given explicitly in (11) positive semi definite, since "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\lambda_i \\in (-1,1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i \\in \\{1,\\dots,N-1\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": ". Thus, the terms "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{y}^T\\mathbf{U}_{\\theta}(\\alpha)\\mathbf{y}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": " inside the integral are non-negative, and it is enough to provide an ordering on "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{y}^T\\mathbf{U}_{\\theta}(\\alpha)\\mathbf{y}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": " with respect to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
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+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        105,
+                        134,
+                        197,
+                        146
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                134,
+                                197,
+                                146
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        134,
+                                        197,
+                                        146
+                                    ],
+                                    "type": "text",
+                                    "content": "For any "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        134,
+                                        197,
+                                        146
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha_{2} > \\alpha_{1} > 0"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        138,
+                        153,
+                        470,
+                        256
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                138,
+                                153,
+                                470,
+                                256
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        138,
+                                        153,
+                                        470,
+                                        256
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\mathbf {y} ^ {T} \\mathbf {U} _ {\\boldsymbol {\\theta}} (\\alpha_ {2}) \\mathbf {y} = \\sum_ {i = 1} ^ {N - 1} \\frac {1}{(\\alpha_ {2} (1 + \\lambda_ {i}) + 1) ^ {2}} \\cdot \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {y} ^ {T} \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H} \\mathbf {y} \\\\ <   \\sum_ {i = 1} ^ {N - 1} \\frac {1}{\\left(\\alpha_ {1} \\left(1 + \\lambda_ {i}\\right) + 1\\right) ^ {2}} \\cdot \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {y} ^ {T} \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H} \\mathbf {y} = \\mathbf {y} ^ {T} \\mathbf {U} _ {\\boldsymbol {\\theta}} (\\alpha_ {1}) \\mathbf {y} \\\\ <   \\sum_ {i = 1} ^ {N - 1} \\cdot \\frac {1 + \\lambda_ {i}}{1 - \\lambda_ {i}} \\mathbf {y} ^ {T} \\mathbf {H} ^ {T} \\mathbf {u} _ {i} \\mathbf {u} _ {i} ^ {T} \\mathbf {H} \\mathbf {y} = \\mathbf {y} ^ {T} \\mathbf {U} _ {\\boldsymbol {\\theta}} (0) \\mathbf {y}, \\\\ \\end{array}",
+                                    "image_path": "e84efe72a9ff1d00c045917b0d94d5eb40ce8f46ceebbabdd5d68202e74289c1.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
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+                        305
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                270,
+                                504,
+                                305
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
+                                        504,
+                                        305
+                                    ],
+                                    "type": "text",
+                                    "content": "where the inequality"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
+                                        504,
+                                        305
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "^{13}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
+                                        504,
+                                        305
+                                    ],
+                                    "type": "text",
+                                    "content": " is because "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
+                                        504,
+                                        305
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha(1 + \\lambda_i) > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
+                                        504,
+                                        305
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
+                                        504,
+                                        305
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i \\in \\{1, \\dots, N\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
+                                        504,
+                                        305
+                                    ],
+                                    "type": "text",
+                                    "content": " and any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
+                                        504,
+                                        305
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
+                                        504,
+                                        305
+                                    ],
+                                    "type": "text",
+                                    "content": ". In fact, the ordering is monotone in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
+                                        504,
+                                        305
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
+                                        504,
+                                        305
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
+                                        504,
+                                        305
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{y}^T \\mathbf{U}_{\\theta}(\\alpha_2) \\mathbf{y}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
+                                        504,
+                                        305
+                                    ],
+                                    "type": "text",
+                                    "content": " decreases at rate "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
+                                        504,
+                                        305
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "1 / \\alpha^2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
+                                        504,
+                                        305
+                                    ],
+                                    "type": "text",
+                                    "content": " as seen from its form in the equation above. This completes the proof."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        105,
+                        319,
+                        409,
+                        331
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                319,
+                                409,
+                                331
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                        331
+                                    ],
+                                    "type": "text",
+                                    "content": "F.2 DISCUSSION REGARDING PROPOSITION 3.4 AND MSE ORDERING"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        340,
+                        506,
+                        491
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                340,
+                                506,
+                                491
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "text",
+                                    "content": "We can use Proposition 3.4 to show that the MSE of SA iterates of (4c) driven by SRRW eventually becomes smaller than that SA iterates when the stochastic noise is driven by an i.i.d. sequence of random variables. The diagonal entries of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\theta}^{(1)}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "text",
+                                    "content": " are obtained by evaluating "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{e}_i^T\\mathbf{V}_{\\theta}^{(1)}(\\alpha)\\mathbf{e}_i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "text",
+                                    "content": ", where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{e}_i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "text",
+                                    "content": " is the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "text",
+                                    "content": "'th standard basis vector."
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "^{14}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "text",
+                                    "content": " These diagonal entries are the asymptotic variance corresponding to the element-wise iterate errors, and for large enough "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "text",
+                                    "content": ", we have "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{e}_i^T\\mathbf{V}_{\\theta}^{(1)}(\\alpha)\\mathbf{e}_i \\approx \\mathbb{E}[(\\boldsymbol{\\theta}_n - \\boldsymbol{\\theta}^*)_i^2] / \\beta_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i \\in \\{1, \\dots, D\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "text",
+                                    "content": ". Thus, the trace of matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\theta}^{(1)}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "text",
+                                    "content": " approximates the scaled MSE, that is "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{Tr}(\\mathbf{V}_{\\theta}^{(1)}(\\alpha)) = \\sum_i \\mathbf{e}_i^T\\mathbf{V}_{\\theta}^{(1)}(\\alpha)\\mathbf{e}_i \\approx \\sum_i \\mathbb{E}[(\\boldsymbol{\\theta}_n - \\boldsymbol{\\theta}^*)_i^2] / \\beta_n = \\mathbb{E}[||\\boldsymbol{\\theta}_n - \\boldsymbol{\\theta}^*||^2] / \\beta_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "text",
+                                    "content": " for large "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "text",
+                                    "content": ". Since all entries of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\theta}^{(1)}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "text",
+                                    "content": " go to zero as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "text",
+                                    "content": " increases, they get smaller than the corresponding term for the SA algorithm with i.i.d. input for large enough "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "text",
+                                    "content": ", which achieves a constant MSE in the similarly scaled limit, since the asymptotic covariance is not a function of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "text",
+                                    "content": ". Moreover, the value of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "text",
+                                    "content": " only needs to be moderately large, since the asymptotic covariance terms decrease at rate "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "O(1 / \\alpha^2)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        506,
+                                        491
+                                    ],
+                                    "type": "text",
+                                    "content": " as shown in Proposition 3.4."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        105,
+                        505,
+                        247,
+                        516
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                505,
+                                247,
+                                516
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        505,
+                                        247,
+                                        516
+                                    ],
+                                    "type": "text",
+                                    "content": "F.3 PROOF OF COROLLARY 3.5"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        104,
+                        523,
+                        504,
+                        593
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                523,
+                                504,
+                                593
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        523,
+                                        504,
+                                        593
+                                    ],
+                                    "type": "text",
+                                    "content": "We see that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        523,
+                                        504,
+                                        593
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\pmb{\\theta}}^{(3)}(\\alpha) = \\mathbf{V}_{\\pmb{\\theta}}^{(3)}(0)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        523,
+                                        504,
+                                        593
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        523,
+                                        504,
+                                        593
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        523,
+                                        504,
+                                        593
+                                    ],
+                                    "type": "text",
+                                    "content": ", because the form of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        523,
+                                        504,
+                                        593
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\pmb{\\theta}}^{(3)}(\\alpha)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        523,
+                                        504,
+                                        593
+                                    ],
+                                    "type": "text",
+                                    "content": " in Theorem 3.3 is independent of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        523,
+                                        504,
+                                        593
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        523,
+                                        504,
+                                        593
+                                    ],
+                                    "type": "text",
+                                    "content": ". To prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        523,
+                                        504,
+                                        593
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(\\alpha) <_{L} \\mathbf{V}_{\\pmb{\\theta}}^{(3)}(0)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        523,
+                                        504,
+                                        593
+                                    ],
+                                    "type": "text",
+                                    "content": ", it is enough to show that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        523,
+                                        504,
+                                        593
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(0) = \\mathbf{V}_{\\pmb{\\theta}}^{(3)}(0)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        523,
+                                        504,
+                                        593
+                                    ],
+                                    "type": "text",
+                                    "content": ", since "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        523,
+                                        504,
+                                        593
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(\\alpha) <_{L} \\mathbf{V}_{\\pmb{\\theta}}^{(1)}(0)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        523,
+                                        504,
+                                        593
+                                    ],
+                                    "type": "text",
+                                    "content": " from Proposition 3.4. This is easily checked by substituting "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        523,
+                                        504,
+                                        593
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        523,
+                                        504,
+                                        593
+                                    ],
+                                    "type": "text",
+                                    "content": " in 11, for which "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        523,
+                                        504,
+                                        593
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{U}_{\\pmb{\\theta}}(0) = \\mathbf{U}_{11}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        523,
+                                        504,
+                                        593
+                                    ],
+                                    "type": "text",
+                                    "content": ". Substituting in the respective forms of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        523,
+                                        504,
+                                        593
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\pmb{\\theta}}^{(1)}(0)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        523,
+                                        504,
+                                        593
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        523,
+                                        504,
+                                        593
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}_{\\pmb{\\theta}}^{(3)}(0)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        523,
+                                        504,
+                                        593
+                                    ],
+                                    "type": "text",
+                                    "content": " in Theorem 3.3, we get equivalence. This completes the proof."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        105,
+                        609,
+                        250,
+                        621
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                609,
+                                250,
+                                621
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        609,
+                                        250,
+                                        621
+                                    ],
+                                    "type": "text",
+                                    "content": "G BACKGROUND THEORY"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        105,
+                        635,
+                        226,
+                        647
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
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+                                226,
+                                647
+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
+                                        635,
+                                        226,
+                                        647
+                                    ],
+                                    "type": "text",
+                                    "content": "G.1 TECHNICAL LEMMAS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
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+                        681
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                504,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        657,
+                                        504,
+                                        681
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma G.1 (Solution to the Lyapunov Equation). If all the eigenvalues of matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        657,
+                                        504,
+                                        681
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        657,
+                                        504,
+                                        681
+                                    ],
+                                    "type": "text",
+                                    "content": " have negative real part, then for every positive semi-definite matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        657,
+                                        504,
+                                        681
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{U}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        657,
+                                        504,
+                                        681
+                                    ],
+                                    "type": "text",
+                                    "content": " there exists a unique positive semi-definite"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        104,
+                        689,
+                        504,
+                        719
+                    ],
+                    "type": "page_footnote",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                {
+                                    "bbox": [
+                                        104,
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+                                        504,
+                                        719
+                                    ],
+                                    "type": "text",
+                                    "content": "<sup>13</sup>The inequality may not be strict when "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        689,
+                                        504,
+                                        719
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        689,
+                                        504,
+                                        719
+                                    ],
+                                    "type": "text",
+                                    "content": " is low rank, however it will always be true for some choice of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        689,
+                                        504,
+                                        719
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        689,
+                                        504,
+                                        719
+                                    ],
+                                    "type": "text",
+                                    "content": ", since "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        689,
+                                        504,
+                                        719
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{H}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        689,
+                                        504,
+                                        719
+                                    ],
+                                    "type": "text",
+                                    "content": " is not a zero matrix. Thus, the ordering derived still follows our definition of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        689,
+                                        504,
+                                        719
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "<_{L}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        689,
+                                        504,
+                                        719
+                                    ],
+                                    "type": "text",
+                                    "content": " in Section 1, footnote 6."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        115,
+                        720,
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+                        732
+                    ],
+                    "type": "page_footnote",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "^{14}D"
+                                },
+                                {
+                                    "bbox": [
+                                        115,
+                                        720,
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+                                    ],
+                                    "type": "text",
+                                    "content": "-dimensional vector of all zeros except at the "
+                                },
+                                {
+                                    "bbox": [
+                                        115,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i"
+                                },
+                                {
+                                    "bbox": [
+                                        115,
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+                                    ],
+                                    "type": "text",
+                                    "content": "'th position which is 1."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        299,
+                        750,
+                        311,
+                        760
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "37"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 36
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        82,
+                        504,
+                        103
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                82,
+                                504,
+                                103
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                        103
+                                    ],
+                                    "type": "text",
+                                    "content": "matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        103
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        103
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfying the Lyapunov equation "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        103
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{U} + \\mathbf{M}\\mathbf{V} + \\mathbf{V}\\mathbf{M}^T = \\mathbf{0}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        103
+                                    ],
+                                    "type": "text",
+                                    "content": ". The explicit solution "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        103
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        103
+                                    ],
+                                    "type": "text",
+                                    "content": " is given as"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        249,
+                        102,
+                        505,
+                        128
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                249,
+                                102,
+                                505,
+                                128
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        249,
+                                        102,
+                                        505,
+                                        128
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {V} = \\int_ {0} ^ {\\infty} e ^ {\\mathbf {M} t} \\mathbf {U} e ^ {(\\mathbf {M} ^ {T}) t} d t. \\tag {83}",
+                                    "image_path": "8a57c6405c15c65700ce68ef5ced7a7ac8d7445ff548510f861a42b3da62ed2e.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
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+                        190
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        134,
+                                        504,
+                                        190
+                                    ],
+                                    "type": "text",
+                                    "content": "Chellaboina & Haddad (2008, Theorem 3.16) states that for a positive definite matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        134,
+                                        504,
+                                        190
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{U}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        134,
+                                        504,
+                                        190
+                                    ],
+                                    "type": "text",
+                                    "content": ", there exists a positive definite matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        134,
+                                        504,
+                                        190
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        134,
+                                        504,
+                                        190
+                                    ],
+                                    "type": "text",
+                                    "content": ". The reason they focus on the positive definite matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        134,
+                                        504,
+                                        190
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{U}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        134,
+                                        504,
+                                        190
+                                    ],
+                                    "type": "text",
+                                    "content": " is that they require the related autonomous ODE system to be asymptotically stable. However, in this paper we don't need this requirement. The same steps therein can be used to prove Lemma G.1 and show that if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        134,
+                                        504,
+                                        190
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{U}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        134,
+                                        504,
+                                        190
+                                    ],
+                                    "type": "text",
+                                    "content": " is positive semi-definite, then "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        134,
+                                        504,
+                                        190
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{V}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        134,
+                                        504,
+                                        190
+                                    ],
+                                    "type": "text",
+                                    "content": " in the form of (83) is unique and also positive semi-definite."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
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+                        504,
+                        216
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        191,
+                                        504,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma G.2 (Burkholder Inequality, Davis (1970), Hall et al. (2014) Theorem 2.10). Given a Martingale difference sequence "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        191,
+                                        504,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{M_{i,n}\\}_{i = 1}^{n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        191,
+                                        504,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": ", for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        191,
+                                        504,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p\\geq 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        191,
+                                        504,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": " and some positive constant "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        191,
+                                        504,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_p"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        191,
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+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": ", we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
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+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                            ],
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+                                    "bbox": [
+                                        203,
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+                                        258
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbb {E} \\left[ \\left\\| \\sum_ {i = 1} ^ {n} M _ {i, n} \\right\\| ^ {p} \\right] \\leq C _ {p} \\mathbb {E} \\left[ \\left(\\sum_ {i = 1} ^ {n} \\| M _ {i, n} \\| ^ {2}\\right) ^ {p / 2} \\right] \\tag {84}",
+                                    "image_path": "d68fd44e8d36888c26f74a6897db386b0becb9843ea3078315e194a2c008b908.jpg"
+                                }
+                            ]
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+                {
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+                    ],
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+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
+                                        105,
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+                                        283
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem G.3 (Martingale CLT, Delyon (2000) Theorem 30). If a Martingale difference array "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        259,
+                                        504,
+                                        283
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{X_{n,i}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        259,
+                                        504,
+                                        283
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfies the following condition: for some "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        259,
+                                        504,
+                                        283
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        259,
+                                        504,
+                                        283
+                                    ],
+                                    "type": "text",
+                                    "content": ","
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
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+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
+                                        239,
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+                                        318
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\sum_ {k = 1} ^ {n} \\mathbb {E} \\left[ \\| X _ {n, k} \\| ^ {2 + \\tau} \\mid \\mathcal {F} _ {k - 1} \\right] \\xrightarrow {\\mathbb {P}} 0, \\tag {85}",
+                                    "image_path": "a5cd7443eb865866924e8ebb09fab8f8b18b302595ba50554e12ffab98d1d490.jpg"
+                                }
+                            ]
+                        }
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+                    "index": 7
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+                {
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+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                                {
+                                    "bbox": [
+                                        235,
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+                                        354
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\sup  _ {n} \\sum_ {k = 1} ^ {n} \\mathbb {E} \\left[ \\| X _ {n, k} \\| ^ {2} \\mid \\mathcal {F} _ {k - 1} \\right] <   \\infty , \\tag {86}",
+                                    "image_path": "a076794a3da85a2787545f36ad884af71d5cb80d60e58995cb4f7a04e751ccdf.jpg"
+                                }
+                            ]
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+                    "index": 8
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+                {
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+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "and"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
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+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                            ],
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+                                    "bbox": [
+                                        239,
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+                                        394
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\sum_ {k = 1} ^ {n} \\mathbb {E} \\left[ X _ {n, k} X _ {n, k} ^ {T} \\mid \\mathcal {F} _ {k - 1} \\right] \\xrightarrow {\\mathbb {P}} \\boldsymbol {V}, \\tag {87}",
+                                    "image_path": "27b72a6079a32b1c36991be90c147b6cc59b1a4097d8bbd4ecb9b76470f82818.jpg"
+                                }
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "then"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                        435
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\sum_ {i = 1} ^ {n} X _ {n, i} \\xrightarrow {\\text {d i s t .}} N (0, V). \\tag {88}",
+                                    "image_path": "92b58af508a2c1fa46212978a432e861a2a812735c1c9291b5b2f6e62407b02d.jpg"
+                                }
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+                        }
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+                {
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+                    ],
+                    "blocks": [
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+                            "bbox": [
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+                                445
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+                                    "bbox": [
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+                                        }
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+                            ],
+                            "index": 13,
+                            "angle": 0,
+                            "type": "image_body"
+                        }
+                    ],
+                    "index": 13
+                },
+                {
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        504,
+                                        472
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma G.4 (Duflo (1996) Proposition 3.I.2). For a Hurwitz matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        504,
+                                        472
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{H}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        504,
+                                        472
+                                    ],
+                                    "type": "text",
+                                    "content": ", there exist some positive constants "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        504,
+                                        472
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C, b"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        504,
+                                        472
+                                    ],
+                                    "type": "text",
+                                    "content": " such that for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        504,
+                                        472
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        504,
+                                        472
+                                    ],
+                                    "type": "text",
+                                    "content": ","
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        268,
+                        471,
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+                        486
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                268,
+                                471,
+                                504,
+                                486
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        268,
+                                        471,
+                                        504,
+                                        486
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left\\| e ^ {\\mathbf {H} n} \\right\\| \\leq C e ^ {- b n}. \\tag {89}",
+                                    "image_path": "05ccf950fa9407ceff51b59407542988ab827f056aa61da164570bb1205eb275.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
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+                        521
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                487,
+                                504,
+                                521
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        521
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma G.5 (Fort (2015) Lemma 5.8). For a Hurwitz matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        521
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{A}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        521
+                                    ],
+                                    "type": "text",
+                                    "content": ", denote by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        521
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "-r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        521
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        521
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        521
+                                    ],
+                                    "type": "text",
+                                    "content": ", the largest real part of its eigenvalues. Let a positive sequence "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        521
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\gamma_n\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        521
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        521
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\lim_{n} \\gamma_{n} = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        521
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        521
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "0 < r' < r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        521
+                                    ],
+                                    "type": "text",
+                                    "content": ", there exists a positive constant "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        521
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        521
+                                    ],
+                                    "type": "text",
+                                    "content": " such that for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        521
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k < n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        521
+                                    ],
+                                    "type": "text",
+                                    "content": ","
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        231,
+                        523,
+                        504,
+                        562
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                231,
+                                523,
+                                504,
+                                562
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        231,
+                                        523,
+                                        504,
+                                        562
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left\\| \\prod_ {j = k} ^ {n} (\\boldsymbol {I} + \\gamma_ {j} \\boldsymbol {A}) \\right\\| \\leq C e ^ {- r ^ {\\prime} \\sum_ {j = k} ^ {n} \\gamma_ {j}}. \\tag {90}",
+                                    "image_path": "d96e91b971c8a0aa4ba861470fc4c47637605cc70ec83f5d167f8cac7aa7c2f9.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        104,
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+                        504,
+                        600
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                600
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        565,
+                                        504,
+                                        600
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma G.6 (Fort (2015) Lemma 5.9, Mokkadem & Pelletier (2006) Lemma 10). Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        565,
+                                        504,
+                                        600
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\gamma_n\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        565,
+                                        504,
+                                        600
+                                    ],
+                                    "type": "text",
+                                    "content": " be a positive sequence such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        565,
+                                        504,
+                                        600
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\lim_{n} \\gamma_{n} = 0"
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+                                    "content": "\\sum_{n} \\gamma_{n} = \\infty"
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+                                    "type": "text",
+                                    "content": ". Let "
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+                                    "type": "inline_equation",
+                                    "content": "\\{\\epsilon_{n}, n \\geq 0\\}"
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+                                    "type": "text",
+                                    "content": " be a nonnegative sequence. Then, for "
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+                                    "type": "inline_equation",
+                                    "content": "b > 0"
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+                                    "type": "text",
+                                    "content": ", "
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+                                    "type": "inline_equation",
+                                    "content": "p \\geq 0"
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+                                    "type": "text",
+                                    "content": ","
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+                                    "type": "interline_equation",
+                                    "content": "\\lim  _ {n} \\sup  _ {n} \\gamma_ {n} ^ {- p} \\sum_ {k = 1} ^ {n} \\gamma_ {k} ^ {p + 1} e ^ {- b \\sum_ {j = k + 1} ^ {n} \\gamma_ {j}} \\epsilon_ {k} \\leq \\frac {1}{C (b , p)} \\lim  _ {n} \\sup  _ {n} \\epsilon_ {n} \\tag {91}",
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+                                    "type": "text",
+                                    "content": " and define a positive sequence "
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+                                    "type": "text",
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+                                    "content": "w_{n - 1} / w_{n} = 1 + o(\\gamma_{n})"
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+                                    "type": "text",
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+                                    "content": "\\sum_ {k = 1} ^ {n} \\gamma_ {k} e ^ {- b \\sum_ {j = k + 1} ^ {n} \\gamma_ {j}} \\epsilon_ {k} = \\left\\{ \\begin{array}{l l} O (w _ {n}), & \\text {i f} \\epsilon_ {n} = O (w _ {n}), \\\\ o (w _ {n}), & \\text {i f} \\epsilon_ {n} = o (w _ {n}). \\end{array} \\right. \\tag {92}",
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+                                    "content": "Lemma G.7 (Fort (2015) Lemma 5.10). For any matrices "
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+                                    "content": "A, B, C"
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+                                    "content": "\\left\\| A B A ^ {T} - C B C ^ {T} \\right\\| \\leq \\| A - C \\| \\| B \\| \\left(\\| A \\| + \\| C \\|\\right). \\tag {93}",
+                                    "image_path": "28decb0cf95cfc1b2a08d2eea9249729e76d680f8876315df3a76c24108feacb.jpg"
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+                                    "content": "Consider the stochastic approximation in the form of"
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+                                    "content": "\\boldsymbol {z} _ {n + 1} = \\boldsymbol {z} _ {n} + \\gamma_ {n + 1} G \\left(\\boldsymbol {z} _ {n}, X _ {n + 1}\\right). \\tag {94}",
+                                    "image_path": "ed3a65a00bc162e3af95317af90a071167dfcc99450bc188bae0a6f976380340.jpg"
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+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{K}_z"
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+                                    "type": "text",
+                                    "content": " be the transition kernel of the underlying Markov chain "
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+                                    "type": "inline_equation",
+                                    "content": "\\{X_n\\}_{n\\geq 0}"
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+                                    "type": "text",
+                                    "content": " with stationary distribution "
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+                                    "type": "inline_equation",
+                                    "content": "\\pi (z)"
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+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
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+                                    "type": "inline_equation",
+                                    "content": "g(z)\\triangleq \\mathbb{E}_{X\\sim \\pi (z)}[G(z,X)]"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " with domain "
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+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{O}\\subseteq \\mathbb{R}^d"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ". Define an operator "
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+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{K}_zf"
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+                                    "type": "text",
+                                    "content": " for any function "
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+                                    "type": "inline_equation",
+                                    "content": "f:\\mathcal{N}\\to \\mathbb{R}^{D}"
+                                },
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+                                    "type": "text",
+                                    "content": " such that"
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+                                    "content": "\\left(\\mathbf {K} _ {\\boldsymbol {z}} f\\right) (i) = \\sum_ {j \\in \\mathcal {N}} f (j) \\mathbf {K} _ {\\boldsymbol {z}} (i, j). \\tag {95}",
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+                                    "type": "text",
+                                    "content": "Assume that"
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+                                            "content": "C1. W.p.1, the closure of "
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+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{O}"
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+                                            "type": "text",
+                                            "content": "."
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+                                            "content": "C2. "
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+                                            "type": "inline_equation",
+                                            "content": "\\gamma_{n} = \\gamma_{0} / n^{a}, a\\in (1 / 2,1]"
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+                        },
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+                                            "type": "text",
+                                            "content": "C3. Function "
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+                                            "type": "inline_equation",
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+                                            "content": "\\nabla w(\\boldsymbol{z})^T g(\\boldsymbol{z}) \\leq 0"
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+                                            "type": "inline_equation",
+                                            "content": "\\boldsymbol{z} \\notin \\mathcal{K}"
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+                                            "type": "text",
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+                                            "type": "inline_equation",
+                                            "content": "S \\triangleq \\{z \\mid \\nabla w(z)^T g(z) = 0\\}"
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+                                            "type": "text",
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+                                            "type": "text",
+                                            "content": " has an empty interior;"
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+                {
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+                                    "type": "text",
+                                    "content": "C4. For every "
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+                                    "type": "inline_equation",
+                                    "content": "z"
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+                                    "type": "text",
+                                    "content": ", there exists a solution "
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+                                    "type": "inline_equation",
+                                    "content": "m_z: \\mathcal{N} \\to \\mathbb{R}^d"
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": " for the following Poisson equation"
+                                }
+                            ]
+                        }
+                    ],
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+                },
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+                                    "type": "interline_equation",
+                                    "content": "m _ {z} (i) - \\left(\\mathbf {K} _ {z} m _ {z}\\right) (i) = G (z, i) - g (z) \\tag {96}",
+                                    "image_path": "b57c8c3060e6e2340973e43b7e142f8479a1dabc7e9f575a4d3d3bc44fe8434f.jpg"
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+                                    "type": "text",
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+                                    "type": "inline_equation",
+                                    "content": "i\\in \\mathcal{N}"
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": " ; for any compact set "
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+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{C}\\subset \\mathcal{O}"
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+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
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+                                    "type": "interline_equation",
+                                    "content": "\\sup  _ {\\boldsymbol {z} \\in \\mathcal {C}, i \\in \\mathcal {N}} \\| \\left(\\mathbf {K} _ {\\boldsymbol {z}} m _ {\\boldsymbol {z}}\\right) (i) \\| + \\| m _ {\\boldsymbol {z}} (i) \\| <   \\infty \\tag {97}",
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "and there exist a continuous function "
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+                                    "type": "inline_equation",
+                                    "content": "\\phi_{\\mathcal{C}},\\phi_{\\mathcal{C}}(0) = 0"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " , such that for any "
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+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\pmb {z},\\pmb{z}^{\\prime}\\in \\mathcal{C}"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\sup  _ {i \\in \\mathcal {N}} \\| \\left(\\mathbf {K} _ {\\boldsymbol {z}} m _ {\\boldsymbol {z}}\\right) (i) - \\left(\\mathbf {K} _ {\\boldsymbol {z} ^ {\\prime}} m _ {\\boldsymbol {z} ^ {\\prime}}\\right) (i) \\| \\leq \\phi_ {\\mathcal {C}} (\\| \\boldsymbol {z} - \\boldsymbol {z} ^ {\\prime} \\|). \\tag {98}",
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+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": "C5. Denote by "
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "-r"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": " the largest real part of the eigenvalues of the Jacobian matrix "
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+                                            "type": "inline_equation",
+                                            "content": "\\nabla g(z^{*})"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": " and assume "
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+                                            "bbox": [
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+                                            "type": "inline_equation",
+                                            "content": "r > \\frac{\\mathbb{1}_{\\{a = 1\\}}}{2}"
+                                        },
+                                        {
+                                            "bbox": [
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+                                                413,
+                                                504,
+                                                440
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 20
+                        },
+                        {
+                            "bbox": [
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+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
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+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": "C6. For every "
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "z"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": ", there exists a solution "
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "Q_{z}:\\mathcal{N}\\to \\mathbb{R}^{d\\times d}"
+                                        },
+                                        {
+                                            "bbox": [
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+                                                456
+                                            ],
+                                            "type": "text",
+                                            "content": " for the following Poisson equation"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 21
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
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+                    "angle": 0,
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "Q _ {z} (i) - \\left(\\mathbf {K} _ {z} Q _ {z}\\right) (i) = F (z, i) - \\mathbb {E} _ {j \\sim \\pi (z)} [ F (z, j) ] \\tag {99}",
+                                    "image_path": "93d36411b24a5ae2446ba0f445c14ecea664686b84670d02968a76ca9fb58cee.jpg"
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+                                    "type": "text",
+                                    "content": "for any "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "i\\in \\mathcal{N}"
+                                },
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+                                    "type": "text",
+                                    "content": " , where"
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+                                    "type": "interline_equation",
+                                    "content": "F (\\boldsymbol {z}, i) \\triangleq \\sum_ {j \\in \\mathcal {N}} m _ {\\boldsymbol {z}} (j) m _ {\\boldsymbol {z}} (j) ^ {T} \\mathbf {K} _ {\\boldsymbol {z}} (i, j) - \\left(\\mathbf {K} _ {\\boldsymbol {z}} m _ {\\boldsymbol {z}}\\right) (i) \\left(\\mathbf {K} _ {\\boldsymbol {z}} m _ {\\boldsymbol {z}}\\right) (i) ^ {T}. \\tag {100}",
+                                    "image_path": "4ac8eac19b586e15df67060a8924229b39e5a7ada276111de55136676daa0a5c.jpg"
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+                                    "type": "inline_equation",
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+                                    "type": "text",
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+                                    "type": "interline_equation",
+                                    "content": "\\sup  _ {\\boldsymbol {z} \\in \\mathcal {C}, i \\in \\mathcal {N}} \\| Q _ {\\boldsymbol {z}} (i) \\| + \\| \\left(\\mathbf {K} _ {\\boldsymbol {z}} Q _ {\\boldsymbol {z}}\\right) (i) \\| <   \\infty \\tag {101}",
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+                                    "type": "text",
+                                    "content": "and there exist "
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+                                    "type": "inline_equation",
+                                    "content": "p, C_{\\mathcal{C}} > 0"
+                                },
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+                                    "type": "text",
+                                    "content": ", such that for any "
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+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{z}, \\mathbf{z}' \\in \\mathcal{C}"
+                                }
+                            ]
+                        }
+                    ],
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+                },
+                {
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+                            "bbox": [
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\sup  _ {i \\in \\mathcal {N}} \\| \\left(\\mathbf {K} _ {\\boldsymbol {z}} Q _ {\\boldsymbol {z}}\\right) (i) - \\left(\\mathbf {K} _ {\\boldsymbol {z} ^ {\\prime}} Q _ {\\boldsymbol {z} ^ {\\prime}}\\right) (i) \\| \\leq C _ {\\mathcal {C}} \\| \\boldsymbol {z} - \\boldsymbol {z} ^ {\\prime} \\| ^ {p}. \\tag {102}",
+                                    "image_path": "7c2d4635fed279776d87ba903616fc850d491950a1df23c480065d8775590355.jpg"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Theorem G.8 (Delyon et al. (1999) Theorem 2). Consider (94) and assume C1 - C4. Then, w.p.1, "
+                                },
+                                {
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+                                    "content": "\\lim \\sup_{n}d(\\pmb{z}_n,S) = 0"
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+                                    "type": "text",
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+                                    "type": "text",
+                                    "content": "Theorem G.9 (Fort (2015) Theorem 2.1 & Proposition 4.1). Consider (94) and assume "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "C1 - C6"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                        640
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, given the condition that "
+                                },
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+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "z_{n}"
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": " converges to one point "
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+                                    "type": "inline_equation",
+                                    "content": "z^{*} \\in S"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ", we have"
+                                }
+                            ]
+                        }
+                    ],
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+                },
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+                                    "type": "interline_equation",
+                                    "content": "\\gamma_ {n} ^ {- 1 / 2} \\left(\\boldsymbol {z} _ {n} - \\boldsymbol {z} ^ {*}\\right) \\xrightarrow [ n \\rightarrow \\infty ]{\\text {d i s t .}} N (0, \\mathbf {V}), \\tag {103}",
+                                    "image_path": "a608b03bf15c355cc739d02bc5a5495321e31304ed26d71114b72a24ca11fa71.jpg"
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+                                    "type": "text",
+                                    "content": "where"
+                                }
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+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {V} \\left(\\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I} + \\nabla g \\left(\\boldsymbol {z} ^ {*}\\right) ^ {T}\\right) + \\left(\\frac {\\mathbb {1} _ {\\{b = 1 \\}}}{2} \\mathbf {I} + \\nabla g \\left(\\boldsymbol {z} ^ {*}\\right)\\right) \\mathbf {V} + \\mathbf {U} = 0, \\tag {104}",
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+                                    "type": "text",
+                                    "content": "and"
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+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {U} \\triangleq \\sum_ {i \\in \\mathcal {N}} \\mu_ {i} \\left(m _ {\\mathbf {z} ^ {*}} (i) m _ {\\mathbf {z} ^ {*}} (i) ^ {T} - \\left(\\mathbf {K} _ {\\mathbf {z} ^ {*}} m _ {\\mathbf {z} ^ {*}}\\right) (i) \\left(\\mathbf {K} _ {\\mathbf {z} ^ {*}} m _ {\\mathbf {z} ^ {*}}\\right) (i) ^ {T}\\right). \\tag {105}",
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
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+                                    "content": "39"
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+                                    "type": "text",
+                                    "content": "G.3 ASYMPTOTIC RESULTS OF TWO-TIMESCALE SA"
+                                }
+                            ]
+                        }
+                    ],
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+                                    "type": "text",
+                                    "content": "For the two-timescale SA with iterate-dependent Markov chain, we have the following iterations:"
+                                }
+                            ]
+                        }
+                    ],
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+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {z} _ {n + 1} = \\mathbf {z} _ {n} + \\beta_ {n + 1} G _ {1} \\left(\\mathbf {z} _ {n}, \\mathbf {y} _ {n} X _ {n + 1}\\right), \\tag {106a}",
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+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {y} _ {n + 1} = \\mathbf {y} _ {n} + \\gamma_ {n + 1} G _ {2} \\left(\\mathbf {z} _ {n}, \\mathbf {y} _ {n}, X _ {n + 1}\\right), \\tag {106b}",
+                                    "image_path": "1b7e27a3a8653ed771432cd02f14c1dc645973f4b6ff629df06d6df9dce802ca.jpg"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "with the goal of finding the root "
+                                },
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\mathbf{z}^{*},\\mathbf{y}^{*})"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " such that"
+                                }
+                            ]
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+                    ],
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+                                    "type": "interline_equation",
+                                    "content": "g _ {1} \\left(\\mathbf {z} ^ {*}, \\mathbf {y} ^ {*}\\right) = \\mathbb {E} _ {X \\sim \\boldsymbol {\\mu}} \\left[ G _ {1} \\left(\\mathbf {z} ^ {*}, \\mathbf {y} ^ {*}, X\\right) \\right] = 0, \\quad g _ {2} \\left(\\mathbf {z} ^ {*}, \\mathbf {y} ^ {*}\\right) = \\mathbb {E} _ {X \\sim \\boldsymbol {\\mu}} \\left[ G _ {2} \\left(\\mathbf {z} ^ {*}, \\mathbf {y} ^ {*}, X\\right) \\right] = 0. \\tag {107}",
+                                    "image_path": "604efaf3f591b4d18ff10f07e987214c5f6bb36fdcd71d1fe4996400e74e67f4.jpg"
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+                                    "type": "text",
+                                    "content": "We present here a simplified version of the assumptions for single-valued functions "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "G_{1}, G_{2}"
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": " that are necessary for the almost sure convergence result in Yaji & Bhatnagar (2020, Theorem 4). The original assumptions are intended for more general set-valued functions "
+                                },
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+                                    "type": "text",
+                                    "content": "."
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+                                            "type": "text",
+                                            "content": "(B1) The step sizes "
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+                                            "type": "inline_equation",
+                                            "content": "\\beta_{n} \\triangleq n^{-b}"
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+                                            "type": "text",
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+                                            "type": "inline_equation",
+                                            "content": "\\gamma_{n} \\triangleq n^{-a}"
+                                        },
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+                                            "type": "text",
+                                            "content": ", where "
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+                                            "type": "inline_equation",
+                                            "content": "0.5 < a < b \\leq 1"
+                                        },
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+                                            "type": "text",
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+                                        250,
+                                        504,
+                                        283
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                117,
+                                                250,
+                                                504,
+                                                283
+                                            ],
+                                            "type": "text",
+                                            "content": "(B2) Assume the function "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                250,
+                                                504,
+                                                283
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "G_{1}(\\mathbf{z},\\mathbf{y},X)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                250,
+                                                504,
+                                                283
+                                            ],
+                                            "type": "text",
+                                            "content": " is continuous and differentiable with respect to "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                250,
+                                                504,
+                                                283
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathbf{z},\\mathbf{y}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                250,
+                                                504,
+                                                283
+                                            ],
+                                            "type": "text",
+                                            "content": ". There exists a positive constant "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                250,
+                                                504,
+                                                283
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "L_{1}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                250,
+                                                504,
+                                                283
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                250,
+                                                504,
+                                                283
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\| G_{1}(\\mathbf{z},\\mathbf{y},X)\\| \\leq L_{1}(1 + \\| \\mathbf{z}\\| +\\| \\mathbf{y}\\|)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                250,
+                                                504,
+                                                283
+                                            ],
+                                            "type": "text",
+                                            "content": " for every "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                250,
+                                                504,
+                                                283
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathbf{z}\\in \\mathbb{R}^{d_1},\\mathbf{y}\\in \\mathbb{R}^{d_2},X\\in \\mathcal{N}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                250,
+                                                504,
+                                                283
+                                            ],
+                                            "type": "text",
+                                            "content": ". The same condition holds for the function "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                250,
+                                                504,
+                                                283
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "G_{2}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                250,
+                                                504,
+                                                283
+                                            ],
+                                            "type": "text",
+                                            "content": " as well."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 9
+                        },
+                        {
+                            "bbox": [
+                                117,
+                                286,
+                                504,
+                                345
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        286,
+                                        504,
+                                        345
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                117,
+                                                286,
+                                                504,
+                                                345
+                                            ],
+                                            "type": "text",
+                                            "content": "(B3) Assume there exists a function "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                286,
+                                                504,
+                                                345
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho : \\mathbb{R}^{d_1} \\to \\mathbb{R}^{d_2}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                286,
+                                                504,
+                                                345
+                                            ],
+                                            "type": "text",
+                                            "content": " such that the following three properties hold: (i) "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                286,
+                                                504,
+                                                345
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\| \\rho(\\mathbf{z}) \\| \\leq L_2(1 + \\| \\mathbf{z} \\|)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                286,
+                                                504,
+                                                345
+                                            ],
+                                            "type": "text",
+                                            "content": " for some positive constant "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                286,
+                                                504,
+                                                345
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "L_2"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                286,
+                                                504,
+                                                345
+                                            ],
+                                            "type": "text",
+                                            "content": "; (ii) the ODE "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                286,
+                                                504,
+                                                345
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\dot{\\mathbf{y}} = g_2(\\mathbf{z}, \\mathbf{y})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                286,
+                                                504,
+                                                345
+                                            ],
+                                            "type": "text",
+                                            "content": " has a globally asymptotically stable equilibrium "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                286,
+                                                504,
+                                                345
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\lambda(\\mathbf{z})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                286,
+                                                504,
+                                                345
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                286,
+                                                504,
+                                                345
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "g_2(\\mathbf{z}, \\rho(\\mathbf{z})) = 0"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                286,
+                                                504,
+                                                345
+                                            ],
+                                            "type": "text",
+                                            "content": ". Additionally, let "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                286,
+                                                504,
+                                                345
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\hat{g}_1(\\mathbf{z}) \\triangleq g_1(\\mathbf{z}, \\rho(\\mathbf{z}))"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                286,
+                                                504,
+                                                345
+                                            ],
+                                            "type": "text",
+                                            "content": ", there exists a set of disjoint roots "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                286,
+                                                504,
+                                                345
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\Lambda \\triangleq \\{\\mathbf{z}^* : \\hat{g}_1(\\mathbf{z}^*) = 0\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                286,
+                                                504,
+                                                345
+                                            ],
+                                            "type": "text",
+                                            "content": ", which is the set of globally asymptotically stable equilibria of the ODE "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                286,
+                                                504,
+                                                345
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\dot{\\mathbf{z}} = \\hat{g}_1(\\mathbf{z})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                286,
+                                                504,
+                                                345
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 10
+                        },
+                        {
+                            "bbox": [
+                                117,
+                                347,
+                                504,
+                                403
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        347,
+                                        504,
+                                        403
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                117,
+                                                347,
+                                                504,
+                                                403
+                                            ],
+                                            "type": "text",
+                                            "content": "(B4) "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                347,
+                                                504,
+                                                403
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{X_{n}\\}_{n\\geq 0}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                347,
+                                                504,
+                                                403
+                                            ],
+                                            "type": "text",
+                                            "content": " is an iterate-dependent Markov process in finite state space "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                347,
+                                                504,
+                                                403
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{N}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                347,
+                                                504,
+                                                403
+                                            ],
+                                            "type": "text",
+                                            "content": ". For every "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                347,
+                                                504,
+                                                403
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "n\\geq 0"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                347,
+                                                504,
+                                                403
+                                            ],
+                                            "type": "text",
+                                            "content": ", "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                347,
+                                                504,
+                                                403
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P(X_{n + 1} = j|\\mathbf{z}_m,\\mathbf{y}_m,X_m,0\\leq m\\leq n) = P(X_{n + 1} = j|\\mathbf{z}_n,\\mathbf{y}_n,X_n = i) = \\mathbf{P}_{i,j}[\\mathbf{z}_n,\\mathbf{y}_n]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                347,
+                                                504,
+                                                403
+                                            ],
+                                            "type": "text",
+                                            "content": ", where the transition kernel "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                347,
+                                                504,
+                                                403
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathbf{P}[\\mathbf{z},\\mathbf{y}]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                347,
+                                                504,
+                                                403
+                                            ],
+                                            "type": "text",
+                                            "content": " is continuous in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                347,
+                                                504,
+                                                403
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathbf{z},\\mathbf{y}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                347,
+                                                504,
+                                                403
+                                            ],
+                                            "type": "text",
+                                            "content": ", and the Markov chain generated by "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                347,
+                                                504,
+                                                403
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathbf{P}[\\mathbf{z},\\mathbf{y}]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                347,
+                                                504,
+                                                403
+                                            ],
+                                            "type": "text",
+                                            "content": " is ergodic so that it admits a stationary distribution "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                347,
+                                                504,
+                                                403
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pi (\\mathbf{z},\\mathbf{y})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                347,
+                                                504,
+                                                403
+                                            ],
+                                            "type": "text",
+                                            "content": ", and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                347,
+                                                504,
+                                                403
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pi (\\mathbf{z}^{*},\\rho (\\mathbf{z}^{*})) = \\pmb{\\mu}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                347,
+                                                504,
+                                                403
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 11
+                        },
+                        {
+                            "bbox": [
+                                118,
+                                406,
+                                276,
+                                419
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        118,
+                                        406,
+                                        276,
+                                        419
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                118,
+                                                406,
+                                                276,
+                                                419
+                                            ],
+                                            "type": "text",
+                                            "content": "(B5) "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                406,
+                                                276,
+                                                419
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\sup_{n\\geq 0}\\bigl (\\| \\mathbf{z}_n\\| +\\| \\mathbf{y}_n\\| \\bigr) <   \\infty"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                406,
+                                                276,
+                                                419
+                                            ],
+                                            "type": "text",
+                                            "content": " a.s."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 12
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        104,
+                        426,
+                        506,
+                        483
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                426,
+                                506,
+                                483
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
+                                        506,
+                                        483
+                                    ],
+                                    "type": "text",
+                                    "content": "Yaji & Bhatnagar (2020) included assumptions A1 - A9 and A11 for the following Theorem G.10. We briefly show the correspondence of our assumptions (B1) - (B5) and theirs: (B1) with A5, (B2) with A1 and A2, (B3) with A9 and A11, (B4) with A3 and A4, and (B5) with A8. Given that our two-timescale SA framework (106) excludes additional noises (setting them to zero), A6 and A7 therein are inherently met."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        104,
+                        485,
+                        504,
+                        510
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                485,
+                                504,
+                                510
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        485,
+                                        504,
+                                        510
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem G.10 (Yaji & Bhatnagar (2020) Theorem 4). Under Assumptions (B1) - (B5), iterations "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        485,
+                                        504,
+                                        510
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\mathbf{z}_n, \\mathbf{y}_n)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        485,
+                                        504,
+                                        510
+                                    ],
+                                    "type": "text",
+                                    "content": " in (106) almost surely converge to a set of roots, i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        485,
+                                        504,
+                                        510
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\mathbf{z}_n, \\mathbf{y}_n) \\to \\bigcup_{\\mathbf{z}^* \\in \\Lambda} (\\mathbf{z}^*, \\rho(\\mathbf{z}^*))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        485,
+                                        504,
+                                        510
+                                    ],
+                                    "type": "text",
+                                    "content": " a.s."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        105,
+                        524,
+                        314,
+                        536
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                524,
+                                314,
+                                536
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        524,
+                                        314,
+                                        536
+                                    ],
+                                    "type": "text",
+                                    "content": "H ADDITIONAL SIMULATION RESULTS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        105,
+                        548,
+                        362,
+                        559
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                548,
+                                362,
+                                559
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        548,
+                                        362,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": "H.1 BINARY CLASSIFICATION ON ADDITIONAL DATSETS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        104,
+                        569,
+                        504,
+                        625
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                569,
+                                504,
+                                625
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
+                                        504,
+                                        625
+                                    ],
+                                    "type": "text",
+                                    "content": "In this part, we perform the binary classification task as in Section 4 on additional datasets, i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
+                                        504,
+                                        625
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "a9a"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
+                                        504,
+                                        625
+                                    ],
+                                    "type": "text",
+                                    "content": " (with 123 features) and splice (with 60 features) from LIBSVM (Chang & Lin, 2011). Figure 4 provides the performance ordering of different "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
+                                        504,
+                                        625
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
+                                        504,
+                                        625
+                                    ],
+                                    "type": "text",
+                                    "content": " values, and we empirically demonstrate that the curves with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
+                                        504,
+                                        625
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha \\geq 5"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
+                                        504,
+                                        625
+                                    ],
+                                    "type": "text",
+                                    "content": " still outperform the i.i.d. counterpart. Additionally, Figure 5 compare cases (i) - (iii) under both "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
+                                        504,
+                                        625
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "a9a"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
+                                        504,
+                                        625
+                                    ],
+                                    "type": "text",
+                                    "content": " and splice datasets, and case (i) consistently perform the best."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
+                        105,
+                        638,
+                        288,
+                        649
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                638,
+                                288,
+                                649
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        638,
+                                        288,
+                                        649
+                                    ],
+                                    "type": "text",
+                                    "content": "H.2 NON-CONVEX LINEAR REGRESSION"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                },
+                {
+                    "bbox": [
+                        104,
+                        658,
+                        504,
+                        692
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                    "content": " values, Figures 7a - 7c confirm that case (i) performs consistently better than the other two cases, implying that case (i) can even become the best choice for non-convex distributed optimization tasks."
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diff --git a/2024/Amortizing intractable inference in large language models/7ed5418b-3773-431a-ac5f-3cceb14f6956_content_list.json b/2024/Amortizing intractable inference in large language models/7ed5418b-3773-431a-ac5f-3cceb14f6956_content_list.json
new file mode 100644
index 0000000000000000000000000000000000000000..fb6b8fabe3bcfcd434b84164e5d6a873bb5b510b
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+++ b/2024/Amortizing intractable inference in large language models/7ed5418b-3773-431a-ac5f-3cceb14f6956_content_list.json	
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+[
+    {
+        "type": "text",
+        "text": "AMORTIZING INTRACTABLE INFERENCE IN LARGE LANGUAGE MODELS",
+        "text_level": 1,
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+    {
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+    {
+        "type": "text",
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+    },
+    {
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+    {
+        "type": "text",
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+        "type": "text",
+        "text": "University of Oxford",
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+    {
+        "type": "text",
+        "text": "younesse.kaddar@chch.ox.ac.uk",
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+    {
+        "type": "text",
+        "text": "Guillaume Lajoie†, Yoshua Bengio°, Nikolay Malkin",
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+    {
+        "type": "text",
+        "text": "Mila - Quebec AI Institute, Université de Montréal",
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+        "type": "text",
+        "text": "...,guillaume.lajoie,yoshua.bengio,nikolay.malkin}@mila.quebec",
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+    {
+        "type": "text",
+        "text": "ABSTRACT",
+        "text_level": 1,
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+        "text": "Autoregressive large language models (LLMs) compress knowledge from their training data through next-token conditional distributions. This limits tractable querying of this knowledge to start-to-end autoregressive sampling. However, many tasks of interest—including sequence continuation, infilling, and other forms of constrained generation—involve sampling from intractable posterior distributions. We address this limitation by using amortized Bayesian inference to sample from these intractable posteriors. Such amortization is algorithmically achieved by fin-tuning LLMs via diversity-seeking reinforcement learning algorithms: generative flow networks (GFlowNets). We empirically demonstrate that this distribution-matching paradigm of LLM fine-tuning can serve as an effective alternative to maximum-likelihood training and reward-maximizing policy optimization. As an important application, we interpret chain-of-thought reasoning as a latent variable modeling problem and demonstrate that our approach enables data-efficient adaptation of LLMs to tasks that require multi-step rationalization and tool use.",
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+    {
+        "type": "text",
+        "text": "Code: https://github.com/GFNOrg/gfn-lm-tuning.",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Autoregressive large language models (LLMs) trained on general-domain data are vast stores of world knowledge (Petroni et al., 2019). They are typically optimized by predicting a token given its preceding context; therefore, tractable inference over this knowledge is limited to sampling conditioned on a prefix. Many useful tasks, such as infilling (Zhu et al., 2019; Liu et al., 2019), generating text conditioned on length or lexical constraints (Hokamp & Liu, 2017; Hu et al., 2019), and finding the most likely sequence continuation, involve intractable inference in LLMs.",
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+        "type": "text",
+        "text": "Such tasks are related to the problem of reasoning, which has been framed as one of probabilistic inference (Gershman & Goodman, 2014). Correspondingly, the linguistic expression of reasoning can be seen as inference over language. For example, we can interpret chain-of-thought reasoning (Wei et al., 2022; Kojima et al., 2022), a paradigm of reasoning in language models, as a problem of intractable posterior inference. Given a question-answer pair  $(X,Y)$ , we are interested in finding latent chains of thought - token sequences  $Z$  that contribute the most to the conditional likelihood",
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+    {
+        "type": "equation",
+        "text": "\n$$\np (Y \\mid X) = \\sum_ {Z} p _ {\\mathrm {L M}} (Z Y \\mid X) = \\sum_ {Z} p _ {\\mathrm {L M}} (Y \\mid X Z) p _ {\\mathrm {L M}} (Z \\mid X), \\tag {1}\n$$\n",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where  $p_{\\mathrm{LM}}$  denotes the likelihood assigned to a sequence by a language model and apposition of variables (e.g., XZY) denotes the concatenation of the token sequences.",
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+    {
+        "type": "text",
+        "text": "While past work has relied on prompting and in-context learning to produce  $Z$ 's that lead to the correct  $Y$ , treating  $Z$  as a hidden variable in a latent variable model (LVM) renders chain-of-thought reasoning a Bayesian inference problem (Fig. 1). For this LVM, the distribution we must sample from is the posterior  $p_{\\mathrm{LM}}(Z \\mid X, Y) = \\frac{p_{\\mathrm{LM}}(XZY)}{\\sum_{Z'} p_{\\mathrm{LM}}(XZ'Y)}$ . Such sampling is intractable: while it is easy to evaluate  $p_{\\mathrm{LM}}(XYZ)$ , the conditional distributions needed to sample  $Z$  from  $p_{\\mathrm{LM}}(Z \\mid X, Y)$  one token at a time are not easy to compute.",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+    {
+        "type": "page_footnote",
+        "text": "*Equal contribution. *Work done during internship at Mila. †CIFAR AI Chair. °CIFAR Senior Fellow.",
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+    {
+        "type": "page_number",
+        "text": "1",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/c6cca71215175d04cf3778d8e4c9df6f17270eb4fa3f17f04fe57c211809874f.jpg",
+        "image_caption": [
+            "Figure 1: Left: Three problems of reasoning in language - sentence infilling, chain-of-thought reasoning, and problem-solving with external tool use - can all be seen as instances of the latent variable model at the top left, where an input  $(X)$  generates the output  $(Y)$  via a latent variable  $(Z)$ . Right: We fine-tune an LLM to sample from the Bayesian posterior over  $Z$ , conditioning on  $X$  and optionally on  $Y$ . If conditioned on  $Y$ , the trained policy can be used to sample diverse latent sequences (e.g., for infilling, §4.2). If not conditioned on  $Y$ , the policy can sample  $Z$ , and thus predict  $Y$ , for inputs  $X$  not seen during training (e.g., for classification and multi-step reasoning, §4.3, 4.4). As shown in §4.4, modeling the full diversity of the posterior aids generalization."
+        ],
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+        "type": "text",
+        "text": "A standard method to sample approximately from intractable posterior distributions is Markov chain Monte Carlo (MCMC), but it is difficult to craft good proposal distributions for multi-modal distributions over language data (Miao et al., 2019; Zhang et al., 2020a; Lew et al., 2023), and inference on a new input may be prohibitively slow. Alternatively, one can turn to reinforcement learning (RL) approaches such as proximal policy optimization (PPO; Schulman et al., 2017), where the language model is treated as a policy to be fine-tuned. However, these do not aim to model the full diversity of the distribution; instead, learned policies settle around a small number of modes. In both cases, issues with this mode collapse are exacerbated when the target distribution is misspecified, leading to the undesirable behavior of overoptimized samplers (Gao et al., 2023).",
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+    {
+        "type": "text",
+        "text": "Amortized probabilistic inference – that is, training a model to approximate a distribution of interest – provides a principled, efficient, and potentially scalable way to draw samples from the distribution (Beal, 2003). One way to implement amortized inference for high-dimensional discrete data such as text is using generative flow networks (GFlowNets; Bengio et al., 2021), which are diversity-seeking reinforcement learning algorithms that train policies to sample objects (such as a token sequence  $Z$ ) with probability proportional to a given reward function, such as the joint  $p_{\\mathrm{LM}}(XZY)$ .",
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+    {
+        "type": "text",
+        "text": "In this work, we present a method that initializes the GFlowNet policy with a pretrained LLM and continues to train it with a reward objective that can be evaluated with the same LLM. The result is a different type of fine-tuning (FT) procedure for text generation that has a number of advantages, including improved sample diversity, data efficiency, and out-of-distribution generalization. GFlowNet fine-tuning makes the language model sample from the target distribution, enabling amortized inference in a number of applications (Fig. 1).",
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+    {
+        "type": "text",
+        "text": "Leveraging this approach, we empirically demonstrate the possibilities and benefits of learning to sample from intractable distributions over text continuations, latent reasoning chains, and tool use sequences using GFlowNet fine-tuning. Notably, the diversity of samples from the models trained with GFlowNet fine-tuning is beneficial in Bayesian model averaging settings, such as when aggregating answers to questions obtained via multiple reasoning chains. For example, using a pretrained language model with 6B parameters, our method shows an absolute improvement of  $10.9\\%$  over supervised fine-tuning on subjectivity classification with only 10 labeled examples (§4.3) and outperforms supervised fine-tuning and PPO by  $63\\%$  on integer arithmetic with 50 demonstrations, with notable improvements in out-of-distribution generalization (§4.4). Moreover, the benefits of amortized inference allow us to efficiently sample from the fine-tuned model at scale. Our contributions include:",
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+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "(1) A general algorithm for amortized sampling from intractable LLM posteriors.",
+            "(2) A probabilistic approach to fine-tuning LLMs to perform chain-of-thought reasoning.",
+            "(3) Empirical results on sequence continuation, natural language reasoning, integer arithmetic with tool use, and story infilling."
+        ],
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 MOTIVATING EXAMPLE: GENERATING RANDOM NUMBERS WITH LLMS",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We consider a simple task that highlights the limitations of reward-maximizing reinforcement learning (RL) methods in fine-tuning LLMs. For readers unfamiliar with RL, we refer to Sutton & Barto (2018) and include a glossary in §A.1 to define key terms used throughout this paper. The task involves generating random numbers from a given distribution when prompted 'The following is a random integer drawn uniformly between 0 and 100:'. This task is a minimal instantiation of the problem we study in the paper: sample from a target distribution given an unnormalized density.",
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+    },
+    {
+        "type": "page_number",
+        "text": "2",
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/b8bd60645f42132e5ddecdc5a51756e06e79e6081a470c31c756c6e0df7016a4.jpg",
+        "image_caption": [
+            "(a) Base model",
+            "(b) PPO fine-tuning",
+            "(c) GFlowNet fine-tuning",
+            "Figure 2: Empirical distributions of 512,000 integers from 1 to 100 generated by GPT-J fine-tuned with PPO (reward-maximizing; b) and GFlowNet fine-tuning (distribution-matching; c). Note the logarithmic  $y$ -scale."
+        ],
+        "image_footnote": [
+            "$50.5\\%$  of samples are valid numbers.",
+            "$95.8\\%$  of samples are valid numbers.",
+            "$100\\%$  of samples are valid numbers."
+        ],
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Although the target distribution is tractable, making the task seemingly straightforward, it serves as a useful illustration of the behaviors of different fine-tuning methods.",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Renda et al. (2023) found that pretrained LLMs perform quite poorly on this task: the distribution of numbers generated with the above prompt will be far from uniform (Fig. 2a shows an example using an instruction fine-tuned GPT-J 6B (Wang & Komatsuzaki, 2021) $^{1}$ ). There may be many reasons for this, among them the effects of instruction fine-tuning and the LLM's possible bias towards numbers that are more frequent in the training data (e.g., numbers starting with '1' are more frequent due to the properties of many natural data-generating processes (Benford, 1938)).",
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+    {
+        "type": "text",
+        "text": "While reward-maximizing RL can teach the model to generate valid numbers (by penalizing outputs that are not numbers from 1 to 100), it would not resolve the distribution skew introduced during pretraining. Indeed, rewarding all valid integers equally leads to an expected gradient of zero for policy gradient methods. Fig. 2b shows that while most samples are valid numbers after PPO training, the distribution remains highly skewed.",
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+    {
+        "type": "text",
+        "text": "Instead, we can take a principled approach by training the LLM to match the target distribution with a GFlowNet learning objective. Such an objective directly optimizes the likelihood of the model generating a number to be proportional to the reward for that number, which is the number's (potentially unnormalized) probability under the target distribution. When the policy is initialized as the pretrained LLM, the resulting distribution after GFlowNet fine-tuning is shown in Fig. 2c. Quantitatively, the KL divergence from the sampling distribution to the target (uniform) distribution decreases from 3.37 for the original LLM (on the support [0, 100]) to  $9.75 \\cdot 10^{-5}$  for the GFlowNet-fine-tuned model.",
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+    {
+        "type": "text",
+        "text": "This example illustrates a general point: GFlowNet objectives provide a principled and flexible approach to fine-tuning LLMs to match a target distribution where reward-maximizing RL fails to. On this simple task, this distribution matching could also be achieved through supervised fine-tuning; however, this would require access to samples from the target distribution, which are unavailable in general (though not in this simple example). In the following sections, we further illustrate this point in non-trivial problems involving intractable inference, reasoning with latent variables, and tool use.",
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+    {
+        "type": "text",
+        "text": "3 FINE-TUNING LLMS TO SAMPLE FROM INTRACTABLE DISTRIBUTIONS",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We first describe how intractable inference emerges from interesting applications of LLMs, one of which is chain-of-thought reasoning seen through the lens of latent variable models, where the posterior distribution over the latent variable is intractable. We then discuss how GFlowNet objectives can be used to train amortized samplers to perform such intractable inference.",
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+    {
+        "type": "text",
+        "text": "3.1 PROBLEM: INTRACTABLE INFERENCE IN LARGE LANGUAGE MODELS",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Autoregressive language models decompose the distribution over sequences of tokens as a product of ordered conditionals:  $p(w_{1:N}) = p(w_1)p(w_2 \\mid w_1) \\cdots p(w_N \\mid w_{1:N-1})$ . While this decomposition makes left-to-right sampling from the distribution tractable, sampling from other conditional distribu",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+    {
+        "type": "page_footnote",
+        "text": "<sup>1</sup>We use the Instruct-GPT-J model available at <sup>2</sup>.co/nlpcloud/instruct-gpt-j-fp16.",
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+    {
+        "type": "page_number",
+        "text": "3",
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+    {
+        "type": "table",
+        "img_path": "images/d45bb1f63e60c1f77020a7f5c6c05d219f451c88761dfc8afae352e9c06eeac9.jpg",
+        "table_caption": [
+            "Table 1: Objects in language posterior inference. Given a pretrained 'teacher' LM  $p_{\\mathrm{LM}}$ , we train a GFlowNet  $q_{\\mathrm{GFN}}$  to sample the posterior  $p(Z \\mid X, Y)$ . Amortization and generalization are achieved by making  $X$ , and optionally  $Y$ , an input to  $q_{\\mathrm{GFN}}$ ."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Object</td><td>Meaning</td><td>Example 1 (infilling)</td><td>Example 2 (subjectivity classification)</td></tr><tr><td>X</td><td>cause / condition / question</td><td>The cat was hungry.</td><td>A deeply moving storyline.</td></tr><tr><td>Z</td><td>mechanism / reasoning chain</td><td>She ate a mouse.</td><td>This review expresses personal feelings.</td></tr><tr><td>Y</td><td>effect / answer</td><td>Now the cat is sleepy, not hungry.</td><td>Answer: Subjective</td></tr><tr><td>p(Z | X)</td><td>conditional prior</td><td></td><td>pLM(Z | X)</td></tr><tr><td>p(Y | X, Z)</td><td>likelihood of effect given</td><td></td><td>pLM(Y | XZ)</td></tr><tr><td></td><td>cause and mechanism</td><td></td><td></td></tr><tr><td>p(Z, Y | X)</td><td>conditional joint, reward for Z</td><td></td><td>pLM(ZY | X)</td></tr><tr><td>p(Z | X, Y)</td><td>posterior (intractable!)</td><td colspan=\"2\">approximated and amortized by GFlowNet qGFN(Z | X), Y)</td></tr><tr><td>q(Y | X)</td><td>posterior predictive /</td><td colspan=\"2\">approximated as ∑Z qGFN(Z | X)pLM(Y | XZ),</td></tr><tr><td></td><td>Bayesian model average</td><td colspan=\"2\">sampled as Z ~ qGFN(Z | X), Y ~ pLM(Y | XZ)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "tions is intractable. Various problems of language modeling can be viewed as sampling from such intractable conditionals in the distribution over sequences of an LLM; we give two such examples and related terminologies in Table 1. Some tasks we study in §4 are instances of these examples.",
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+    {
+        "type": "text",
+        "text": "Tempered and contrastive sampling. In many applications (e.g., translation, summarization, dialogue systems), one wishes to sample from a low-temperature distribution over sequences  $Z$  conditioned on a prefix  $X$ , i.e.,  $q(Z \\mid X) \\propto p_{\\mathrm{LM}}(XZ)^{1 / T}$  for some temperature  $T < 1$ , as high-likelihood samples are more likely to be fluent or accurate continuations of  $X$  (Tillmann & Ney, 2003). The limit of  $T \\to 0$  gives a distribution that is peaky on the most likely continuation. However, sampling from  $q$ , or finding its mode, is intractable, and it is common to resort to approximations, such as tempering the tokenwise conditional distributions or using beam search to search for a mode. A related problem is sampling a continuation with a correction for its unconditional likelihood, e.g.,  $q(Z \\mid X) \\propto p_{\\mathrm{LM}}(XZ)^{\\alpha}p_{\\mathrm{LM}}(Z)^{\\beta}$  with  $\\beta < 0$  and  $\\alpha > 0$ , where applications again resort to approximating the next-token conditionals of  $q$  by tempering (Malkin et al., 2022b; Li et al., 2023).",
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+    },
+    {
+        "type": "text",
+        "text": "Infilling and reverse generation. Infilling is the task of sampling a sequence of tokens conditioned on both its prior and subsequent context, which can be understood as sampling from the distribution  $q(Z \\mid X, Y) \\propto p_{\\mathrm{LM}}(XZY)$ , where  $X$  and  $Y$  are fixed. Reverse generation is a special case, where  $X$  is an empty sequence. Besides being a meaningful task in its own right (Liu et al., 2019; Zhu et al., 2019; Donahue et al., 2020; Susanto et al., 2020; Lu et al., 2022a), infilling and reverse generation are key components of newly emerging methods of LLM prompting, such as when LLMs are tasked with optimizing their own instruction sequences or reasoning steps (Zhou et al., 2023; Sordoni et al., 2023; Xu et al., 2023). Current applications achieve this by resorting to hand-engineered instructions and inverted prompts.",
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+    {
+        "type": "text",
+        "text": "Constrained generation. Sampling of text with constraints and penalties – for example, those on the presence or the absence of certain words or on the score of an auxiliary classifier evaluated on the text – can be understood as sampling from a distribution  $q(Z) \\propto p_{\\mathrm{LM}}(Z)c(Z)$ , where  $c$  is an externally specified constraint. Current approaches to the problem use tokenwise approximations (Liu et al., 2021), various problem-specific beam search and local search techniques (e.g., Schmaltz et al., 2016; Hokamp & Liu, 2017; Hu et al., 2019; Sha, 2020; Lu et al., 2022b) or classifier-guided conditional generation approaches (e.g., Yang & Klein, 2021; Meng et al., 2022).",
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+    {
+        "type": "text",
+        "text": "3.2 REASONING THROUGH LATENT VARIABLES",
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+    {
+        "type": "text",
+        "text": "Chain-of-thought reasoning (Wei et al., 2022; Kojima et al., 2022) helps LLMs solve complex problems by producing a reasoning chain before giving the final answer. LLMs pretrained on general domain data can learn to produce useful chains of thoughts given demonstrations, which are usually handcrafted or generated by prompting the LM. Interestingly, although the capacity for chain-of-thought reasoning only emerges in large language models, knowledge can also be extracted from smaller language models when they are carefully fine-tuned (Schick & Schütze, 2021).",
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+    {
+        "type": "text",
+        "text": "Motivated by this, we connect chain-of-thought reasoning to the general problem of inference in latent variable models illustrated in Fig. 1. Here, reasoning can be seen as posterior inference: sampling from the posterior distribution over a string of tokens  $Z$  conditioned on a prefix  $X$  and a suffix  $Y$ , given an autoregressive language model  $p_{\\mathrm{LM}}$ . The posterior is defined as",
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+    {
+        "type": "equation",
+        "text": "\n$$\np _ {\\mathrm {L M}} (Z \\mid X, Y) = \\frac {p _ {\\mathrm {L M}} (X Z Y)}{\\sum_ {Z ^ {\\prime}} p _ {\\mathrm {L M}} \\left(X Z ^ {\\prime} Y\\right)} \\propto p _ {\\mathrm {L M}} (X Z Y). \\tag {2}\n$$\n",
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+    {
+        "type": "text",
+        "text": "Our goal is to train models to sample  $Z$  from this posterior distribution. Intuitively, this allows us to sample likely reasoning chains that lead to the desired outcome  $Y$ . Although we take  $Z$  to be a string",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+    },
+    {
+        "type": "page_number",
+        "text": "4",
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+    },
+    {
+        "type": "text",
+        "text": "of tokens, the same formalism and the GFlowNet objectives apply to other structured latent objects, such as trees or sets of natural language statements, as long as one has access to a likelihood model  $p(Y \\mid XZ)$ . While not investigated in this work, these generalizations could be important for formal reasoning and multi-step chains of inference. See, e.g., Yao et al. (2023); Hao et al. (2023); Besta et al. (2024) for approaches to reasoning in language using tree- or list-structured state spaces.",
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+    {
+        "type": "text",
+        "text": "A latent variable model of this form is useful when the marginal distribution  $p_{\\mathrm{LM}}(Y \\mid X)$  is harder to model than  $p_{\\mathrm{LM}}(Z \\mid X)$  and  $p_{\\mathrm{LM}}(Y \\mid XZ)$ , i.e., a difficult inference is broken down into a chain of easier ones. By training a model to match the Bayesian posterior  $p_{\\mathrm{LM}}(Z \\mid X, Y)$ , we can learn to sample latent reasoning chains that increase the likelihood of producing  $Y$  from  $X$  via the sampled  $Z$ .",
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+    },
+    {
+        "type": "text",
+        "text": "However, we can also fine-tune the language model  $p_{\\mathrm{LM}}(Z \\mid XY)$  itself to maximize the likelihood of data pairs  $(X,Y)$  under the LVM. While it is generally intractable to directly maximize the data likelihood  $p_{\\mathrm{LM}}(X,Y) = \\sum_Z p_{\\mathrm{LM}}(XZY)$  because of the summation over  $Z$ , the (variational) expectation-maximization (EM) algorithm (Dempster et al., 1977; Beal, 2003; Koller & Friedman, 2009) can be used for this purpose. In the expectation step (E-step), we draw samples from the posterior over the latent variable  $p_{\\mathrm{LM}}(Z \\mid X,Y)$ , which could come from an amortized sampler of  $Z$ . In the maximization step (M-step), we maximize the log-likelihood of the joint probability of the sampled latent variables  $\\mathbb{E}_{Z \\sim p_{\\mathrm{LM}}(Z|X,Y)} \\log p_{\\mathrm{LM}}(XZY)$  with respect to the parameters of the language model  $p_{\\mathrm{LM}}$ . This combination of amortized inference (learning to sample the chain of thought) and supervised fine-tuning (optimizing the language model with the 'supervision' involving  $Z$  sampled from the amortized posterior) will be illustrated in one of our experiments (§4.3, Table 3).",
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+    {
+        "type": "text",
+        "text": "3.3 AMORTIZED INFERENCE WITH GFLOWNET OBJECTIVES",
+        "text_level": 1,
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+        "type": "text",
+        "text": "For inference in the latent variable model, we leverage the probabilistic framework of generative flow networks (GFlowNets; Bengio et al., 2021; 2023). Using notation from Malkin et al. (2022a), we briefly introduce relevant GFlowNet concepts pertaining to autoregressive sequence generation. Here, GFlowNets learn policies to sample sequences  $Z = z_{1}z_{2}\\dots z_{n}\\top \\in \\mathcal{Z}$  (where  $\\top$  denotes a stop symbol) from a distribution over the space of sequences  $\\mathcal{Z}$ , given an unnormalized density (reward)  $R: \\mathcal{Z} \\to \\mathbb{R}_{>0}$ . The generative process is the same as in autoregressive language models: generation begins with an empty string, and at the  $i$ -th step a token  $z_{i}$  is sampled from a policy  $q_{\\mathrm{GFN}}(z_i \\mid z_{1:i-1})$  which is then appended to the sequence. This process continues until a stop symbol  $\\top$  is generated.",
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+    {
+        "type": "text",
+        "text": "The marginal likelihood  $q_{\\mathrm{GFN}}^{\\top}(Z)$  of sampling a terminal state  $Z = z_{1:n}\\top$  is given by  $\\prod_{i = 1}^{n}q_{\\mathrm{GFN}}(z_i|z_{1:i - 1})q_{\\mathrm{GFN}}(\\top \\mid z)$ , where  $z_{1:0}$  is understood to be the empty string. The goal of GFlowNet training is to fit a parametric policy  $q_{\\mathrm{GFN}}(\\cdot |\\cdot ;\\theta)$  such that  $q_{\\mathrm{GFN}}^{\\top}(Z)\\propto R(Z)$ , i.e., the likelihood of generating a complete sequence is proportional to its reward.",
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+        "text": "Learning objective. We use a modified version of the subtrajectory balance (SubTB; Madan et al., 2023) objective to account for trajectories being terminable at all states (Deleu et al., 2022). The objective for a sequence  $Z = z_{1:n} \\top$  is",
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+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} (Z; \\theta) = \\sum_ {0 \\leq i <   j \\leq n} \\left(\\log \\frac {R \\left(z _ {1 : i} \\top\\right) \\prod_ {k = i + 1} ^ {j} q _ {\\mathrm {G F N}} \\left(z _ {k} \\mid z _ {1 : k - 1}\\right) q _ {\\mathrm {G F N}} (\\top \\mid z _ {1 : j})}{R \\left(z _ {1 : j} \\top\\right) q _ {\\mathrm {G F N}} (\\top \\mid z _ {1 : i})}\\right) ^ {2}, \\tag {3}\n$$\n",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "For sequence generation tasks, the SubTB objective is equivalent to the path consistency objective (Nachum et al., 2017) in max-entropy RL (Haarnoja et al., 2017), which has been previously used in the context of text generation (Guo et al., 2021). See Appendix A.2 for further discussion.",
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+        "type": "text",
+        "text": "Training policy. As the objective in Eq. 3 can be minimized to 0 for all trajectories  $\\tau$  simultaneously given enough model capacity, we can use trajectories sampled from any full-support distribution (training policy) to perform gradient descent on  $\\mathcal{L}(\\tau; \\theta)$  with respect to  $\\theta$ . As the space we are sampling from is combinatorially large, it is important to have a training policy that can efficiently explore  $\\mathcal{Z}$ . To this end, we compose the mini-batch during training using trajectories from three sources: (1) the policy  $q_{\\mathrm{GFN}}$ , (2) a tempered version of the current policy  $q_{\\mathrm{GFN}}$  and (3) a replay buffer storing past trajectories. Replay buffers have been shown to be quite effective in improving GFlowNet training (Jain et al., 2022; Deleu et al., 2022; Shen et al., 2023).",
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+    {
+        "type": "text",
+        "text": "Parametrization, amortization, and generalization. To sample the latent sequence  $Z$  from the posterior defined in Eq. 2, we parametrize the GFlowNet policy as an autoregressive language model that samples the latent  $Z$  one token at a time from left to right. By setting the reward  $R(Z) = p_{\\mathrm{LM}}(XZY) \\propto p_{\\mathrm{LM}}(Z \\mid X,Y)$ , we learn a sampler for the posterior at convergence.",
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "page_number",
+        "text": "5",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As illustrated in Fig. 1, depending on the task, we can condition the GFlowNet policy on either  $X$  or  $X, Y$ . In cases such as reasoning (§3.2), where there is only a single correct  $Y$  for each  $X$  and we interested in predicting  $Y$  for unseen  $X$  at test time, we can simply condition on  $X$ . In this case, the GFlowNet policy is simply a language model that generates  $Z$  as a continuation of  $X$ . To be precise, we initialize  $q_{\\mathrm{GFN}}$  as a copy of  $p_{\\mathrm{LM}}$  that is conditioned on the prefix  $X$ , and then fine-tune<sup>2</sup> it with a GFlowNet objective. With this view, sampling  $Z$  is an inverse problem: we need to infer  $Z$  given a (conditional) prior  $p_{\\mathrm{LM}}(Z \\mid X)$  and an observation  $Y$  under likelihood model  $p_{\\mathrm{LM}}(Y \\mid XZ)$ .",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Allowing the GFlowNet policy to explicitly take  $X$  as input amortizes the sampling procedure and allows generalization to unseen  $X$ . In this sense, the GFlowNet is a Bayesian model (akin to a LM cascade (Dohan et al., 2022) or deep language network (Sordoni et al., 2023)), in which  $Z$  are conditionally sampled 'parameters' that transform  $X$  into  $Y$ . To predict the  $Y$  for an unseen  $X$ , one performs Bayesian model averaging by drawing samples of  $Z$  from  $q_{\\mathrm{GFN}}(Z \\mid X)$  followed by sampling from  $p_{\\mathrm{LM}}(Y \\mid XZ)$ .",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In tasks such as infilling (§4.2), however, the mapping from  $X$  to  $Y$  is one-to-many and  $Y$  is available at test-time. Here, we are interested in  $Z$  itself, rather than using it as an intermediate variable en route to generating  $Y$ . The GFlowNet policy thus has to be conditioned on both  $X$  and  $Y$ . To achieve this, the policy is conditioned on a prompt that contains both  $X$  and  $Y$  (for example, see Appendix C).",
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+    {
+        "type": "text",
+        "text": "4 EMPIRICAL RESULTS",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We first validate GFlowNet fine-tuning on text generation, where we seek to find likely sentence continuation given a prompt (§4.1) or fill in a missing sentence in a story (§4.2). Then, we study reasoning tasks that benefit from chain-of-thought reasoning (§4.3) and external tool use (§4.4).",
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+    {
+        "type": "text",
+        "text": "4.1 SENTENCE CONTINUATION",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Task description. A natural application for autoregressive language models is that of sequence continuation: given a prompt, the model should generate a high-likelihood completion. In applications such as creative writing, we would like the continuations to be semantically diverse while still having a high likelihood under the language model. To demonstrate the benefits of GFlowNet fine-tuning, we consider the task of sampling the next sentence following a prompt.",
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+    {
+        "type": "text",
+        "text": "Sampling autoregressively from the LM until a “.” token is reached is unlikely to produce samples that have a high likelihood because the distribution over sentences has a fat tail. Existing approaches to generate sequence continuations include beam search and its variations (Vijayakumar et al., 2018; Shao et al., 2017), top-  $k$  sampling (Fan et al., 2018), nucleus sampling",
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+    {
+        "type": "text",
+        "text": "(Holtzman et al., 2019), tempered autoregressive sampling, and fine-tuning using importance sampling (Shih et al., 2023), among others. While useful, most of these methods are ultimately hand-crafted heuristics that leave room for improvement. Furthermore, some of these methods (e.g., beam search) involve a computationally expensive search procedure, compared to a single pass of a learned inference model that amortizes over prompts. Our GFlowNet policy autoregressively samples the sequence until a period is sampled, indicating the end of the sentence. Given prompts  $X$ , the LM is fine-tuned to generate the continuations  $Z$  from the tempered posterior by being trained with reward  $R(Z) = p_{\\mathrm{LM}}(Z|X)^{\\frac{1}{T}}$ . When  $T = 1$ , the GFlowNet will trivially sample proportional to  $p_{\\mathrm{LM}}$  without any fine-tuning, so we consider  $0 < T < 1$  to focus on the likely continuations.",
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+    },
+    {
+        "type": "text",
+        "text": "We consider a dataset of prompts from OpenWebText (Gokaslan et al., 2019) with a 1.5B param GPT-2 XL (Radford et al., 2019) as the base model. We draw 8 samples from the fine-tuned model conditioned on a fixed prompt, consider the maximum-likelihood sample under the LM, and report the average over the dataset of prompts. To measure the semantic diversity of the",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/247978d45b5e4c8128a402696a7af7131f16b50fa7345f269a631361bcfee7f0.jpg",
+        "image_caption": [
+            "Figure 3: Maximum log-likelihood and diversity of continuations sampled for fixed prompts. GFlowNet finetuning  $(\\star)$  samples higher log-likelihood sentences while maintaining more sample diversity than the baselines (and ---), even when they are given  $5\\times$  the compute."
+        ],
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+    },
+    {
+        "type": "page_footnote",
+        "text": "2We use LoRA (Hu et al., 2022) instead of full fine-tuning for hardware efficiency in all experiments.",
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+    {
+        "type": "page_number",
+        "text": "6",
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+    },
+    {
+        "type": "text",
+        "text": "samples, we compute the mean pairwise cosine distance between the embeddings (from a pretrained encoder (Reimers & Gurevych, 2019)) of the generated samples and average it over the dataset. We compare to baselines that are commonly used for producing continuations from LMs at inference time (beam search, diverse beam search, nucleus sampling, autoregressive sampling, tempered autoregressive sampling, and greedy generation).",
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+    {
+        "type": "text",
+        "text": "Results. Quantitative results are reported in Fig. 3 and empirical samples are shown in Appendix B. At lower temperatures, our method excels in generating high-likelihood sentences, outperforming the leading baseline, diverse beam search. If we increase the number of beams (and therefore compute) to  $5 \\times$  the number of samples produced by the GFlowNet, our performance remains comparable. Nevertheless, even in this scenario, the GFlowNet's generated samples exhibit notably higher diversity compared to diverse beam search and are on par with the best diversity-scoring benchmarks.",
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+    {
+        "type": "text",
+        "text": "4.2 INFILLING STORIES",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Task description. Next, we consider the story infilling task, a special case of the general infilling problem (§3.1), where given the beginning  $X$  and end  $Y$  of a story, the goal is to generate the middle of the story  $Z$  (Zhu et al., 2019). This is challenging for a language model sampled left to right since continuations  $Z$  conditioned only on  $X$  might be incompatible with the ending  $Y$ . We use the ROCStories corpus (Mostafazadeh et al., 2016), a dataset of short stories containing exactly 5 sentences each. Given the first 3 sentences and the last sentence, the goal is to generate the fourth sentence, which often involves a turning point in the story and is thus challenging to fill in.",
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+    {
+        "type": "text",
+        "text": "As we expect the base model to contain the required knowledge, for this task we use a GPT-2 Large model (Radford et al., 2019) fine-tuned on the entire ROCStories training set as the base model. For evaluating the approach, we consider 900 samples from the dataset as training data to learn  $q_{\\mathrm{GFN}}(Z|X,Y)$  and evaluate the similarity of the generated infills on a dataset of 100 unseen stories. Along with the GFlowNet-fine-tuned model, we also consider two baselines: prompting the model to infill the story and supervised fine-tuning on the same data. Further details are in Appendix C.",
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+    {
+        "type": "text",
+        "text": "Results. To measure the similarity of the generated infills with the reference infills available in the dataset, we compute BERTScore (Zhang et al., 2020b), with DeBERTa (He et al., 2021) - which is correlated with human judgments",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "along with BLEU-4 (Papineni et al., 2002) and GLEU-4 (better suited for sentences; Wu et al., 2016) metrics. Additionally, we also evaluate each method using GPT-4 as a judge. From our results summarized in Table 2, we observe that the infills generated by the model with GFlowNet fine-tuning are closer to the reference infills in the dataset than the baselines. By sampling from  $p_{\\mathrm{LM}}(Z|X,Y)$ , the GFlowNet is able to account for the ending while generating the infill, resulting in infills that link the beginning and the end of the story coherently. For further analysis and details see Appendix C.",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/4bb13ee5bb9b6a093c0dd96e10518e9c5d5530f3d72e39c984f078b0cfa6ca0c.jpg",
+        "table_caption": [
+            "Table 2: Evaluation of the generated infills."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>BERTScore</td><td>BLEU-4</td><td>GLEU-4</td><td>GPT4Eval</td></tr><tr><td>Prompting</td><td>0.081 ± 0.009</td><td>1.3 ± 0.5</td><td>3.2 ± 0.1</td><td>2.4</td></tr><tr><td>Supervised fine-tuning</td><td>0.094 ± 0.007</td><td>1.6 ± 0.8</td><td>3.7 ± 0.4</td><td>2.7</td></tr><tr><td>GFlowNet fine-tuning</td><td>0.184 ± 0.004</td><td>2.1 ± 0.2</td><td>4.2 ± 0.7</td><td>3.4</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.3 SUBJECTIVITY CLASSIFICATION",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Task description. SUBJ (Pang & Lee, 2004) is a binary classification dataset for natural language understanding. It is a collection of movie reviews in which each review is labeled as objective, meaning that it references facts about the movie, or subjective, meaning that it expresses an opinion of the reviewer (see Table D.1 for examples). Given an unlabeled review, the model must predict whether it is objective or subjective. While supervised fine-tuning on the full dataset",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/321b291705dfde9ab92fd5d586069c0a6b383e529e13ce55cd3f5a56888a2b99.jpg",
+        "table_caption": [
+            "Table 3: Test accuracy  $(\\%)$  on SUBJ using an instruct-fine-tuned GPT-J 6B."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td colspan=\"3\">Test accuracy (%) ↑</td></tr><tr><td>Zero-shot prompting</td><td colspan=\"3\">51.7</td></tr><tr><td></td><td colspan=\"3\">Training samples</td></tr><tr><td></td><td>10</td><td>20</td><td>50</td></tr><tr><td>Few-shot prompting</td><td>61.3 ± 6.2</td><td>61.8 ± 5.4</td><td>65.8 ± 10.5</td></tr><tr><td>Supervised fine-tuning</td><td>64.3 ± 2.8</td><td>69.1 ± 0.8</td><td>89.7 ± 0.4</td></tr><tr><td>GFlowNet fine-tuning</td><td>71.4 ± 1.8</td><td>81.1 ± 0.4</td><td>87.7 ± 2.2</td></tr><tr><td>+ Supervised fine-tuning</td><td>75.2 ± 1.8</td><td>78.7 ± 1.6</td><td>89.9 ± 0.2</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "can achieve high test accuracy, we are interested in the low-data regime where we only have tens of labeled examples. We use the same instruction-tuned GPT-J 6B variant as in §2 for this experiment. Without any demonstrations, the model struggles with this task using the prompt in Table D.2 and achieves only  $51.7\\%$  zero-shot accuracy.",
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+        "text": "This task is hard likely because it requires a latent reasoning step. A review could be considered objective because it analyzes the plot or facts about the movie, or it could be subjective because it expresses a personal opinion or makes a judgment. We denote the review  $X$ , the predicted subjectivity  $Y$ , and the latent reason  $Z$ . Then, we GFlowNet-fine-tune the LLM  $q_{\\mathrm{GFN}}(Z \\mid X)$ , initialized with the",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+    {
+        "type": "page_number",
+        "text": "7",
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+    {
+        "type": "text",
+        "text": "base model  $p_{\\mathrm{LM}}$  to match the Bayesian posterior over rationales in Eq. 2. At test time,  $q_{\\mathrm{GFN}}(Z \\mid X)$  generates 10 latent rationales ( $Z$ 's) for an unseen  $X$ . The LLM  $p_{\\mathrm{LM}}$  then autoregressively samples from  $p_{\\mathrm{LM}}(Y \\mid XZ)$  to produce 10 answers, the majority vote of which becomes the final prediction.",
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+        "type": "text",
+        "text": "This posterior inference corresponds to the E-step in the EM algorithm, where the posterior  $p_{\\mathrm{LM}}(Z \\mid X,Y)$  is defined in Eq. 2. Further, as described in §3.2, we can take an M-step by updating  $p_{\\mathrm{LM}}$  to maximize  $\\log p_{\\mathrm{LM}}(XZY)$  over a collection of  $Z$ 's sampled from the amortized posterior  $q_{\\mathrm{GFN}}$ . This is equivalent to applying supervised fine-tuning after GFlowNet fine-tuning.",
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+        "text": "Results. We present few-shot prompting and supervised fine-tuning with LoRA as baselines. In few-shot prompting, we prepend 0, 10, 20, or 50 training examples to each test example using the prompt shown in Table D.2. We randomly shuffle few-shot demonstrations and report the mean and variance in Table 3. In supervised fine-tuning, we directly maximize  $\\log p_{\\mathrm{LM}}(Y \\mid X)$  over the same 10, 20, or 50  $(X, Y)$  pairs. The variance is over model initialization and batch order. All entries except zero-shot prompting are aggregated over 3 random seeds. See Appendix D for experiment details. GFlowNet fine-tuning consistently outperforms supervised fine-tuning in the low-data regime, as shown in Table 3. In some cases, performing supervised fine-tuning on top, which corresponds to running one step of the EM algorithm, further improves the performance.",
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+    {
+        "type": "text",
+        "text": "4.4 SOLVING ARITHMETIC PROBLEMS STEP BY STEP",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Task description. Arithmetic reasoning is a fitting benchmark to evaluate reasoning abilities of large language models as it requires multi-step reasoning and correctness is easy to evaluate (Cobbe et al., 2021). While the distribution of pretraining and fine-tuning data (Magister et al., 2023; Lee et al., 2023; Luo et al., 2023) and prompting choices (Imani et al., 2023) play a critical role in their arithmetic abilities, LLMs are susceptible to poor generalization by learning 'shortcuts' to reasoning (Dziri et al., 2023). We consider a simple integer arithmetic task (Fig. 1), with a general pretrained base model, rather than a one pretrained on mathematical",
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+        "type": "text",
+        "text": "Table 4: Test accuracy  $(\\%)$  on an integer arithmetic task with addition and subtraction using a GPT-J 6B model. Training data only include samples with 3 or 4 operands.",
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+    {
+        "type": "table",
+        "img_path": "images/127cd6035feea1a9fd832fd05a5794ca3b0d56daeb82fc79ae923db422b4c35e.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"3\">Method</td><td></td><td colspan=\"3\">Number of Operands</td></tr><tr><td></td><td colspan=\"2\">In-distribution</td><td>OOD</td></tr><tr><td></td><td>3</td><td>4</td><td>5</td></tr><tr><td rowspan=\"5\">k-shot CoT</td><td>k=0</td><td>10.2</td><td>6.4</td><td>3.2</td></tr><tr><td>k=3</td><td>15.8 ± 3.1</td><td>11 ± 1.7</td><td>5.4 ± 0.2</td></tr><tr><td>k=5</td><td>20.4 ± 10.4</td><td>17.6 ± 0.6</td><td>6.6 ± 1.1</td></tr><tr><td>k=10</td><td>26.5 ± 1.4</td><td>15.2 ± 1.7</td><td>8.9 ± 1.9</td></tr><tr><td>k=20</td><td>35.5 ± 1.9</td><td>21 ± 1.4</td><td>10.5 ± 0.9</td></tr><tr><td colspan=\"2\">Supervised fine-tuning</td><td>72.1 ± 1.3</td><td>19.6 ± 2.2</td><td>12.8 ± 5.7</td></tr><tr><td colspan=\"2\">PPO</td><td>30.6 ± 4.1</td><td>13.7 ± 4.1</td><td>5.6 ± 3.1</td></tr><tr><td colspan=\"2\">GFlowNet fine-tuning</td><td>95.2 ± 1.3</td><td>75.4 ± 2.9</td><td>40.7 ± 9.1</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "tasks (Jelassi et al., 2023). To avoid the pitfalls of symbolic calculations with language models, we adopt the tool use setting (Schick et al., 2023), where the model is equipped with a calculator that can perform parts of the computation, implemented as in Cobbe et al. (2021): when the model outputs  $=$  the expression preceding it is evaluated and appended to the sequence. To prevent the model from \"cheating\", we limit the calculator to evaluate only two-term expressions. Consequently, reasoning here involves learning to plan using a tool with limited capabilities (Hao et al., 2023).",
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+        "type": "text",
+        "text": "For training, we use a synthetic dataset of arithmetic expressions, limited to addition and subtraction. Following Zelikman et al. (2022), we use a small set of 50 demonstrations  $(X,Z,Y)$  to seed the replay buffer in addition to 1000 examples  $(X,Y)$ . We use the same instruction-tuned GPT-J as in §4.3 as the base model. Further details are in Appendix E. We report the accuracy on two types of examples: (1) unseen in-distribution expressions (3 or 4 operands) and (2) longer out-of-distribution expressions (5 operands). As baselines, we consider zero-shot chain-of-thought prompting,  $k$ -shot prompting, supervised fine-tuning on the tool use sequences, and fine-tuning with PPO (Schulman et al., 2017). For all methods, we enable tool use and limit the model to generate only numbers and operators.",
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+        "type": "text",
+        "text": "Results. From the results summarized in Table 4, the base model performs poorly even with chain-of-thought prompts. Including examples in context improves the performance considerably, with monotonic improvements as the number of examples increases. Supervised fine-tuning improves the performance significantly on the in-distribution examples, but the model still struggles to generalize on the out-of-distribution examples. Fine-tuning with PPO results also yields poor performance, caused in part by the poor calibration of the base reward model, i.e. it cannot distinguish good rationales from bad ones. Even though the sequences generated with PPO (illustrated in Appendix E) have high rewards, they are spurious and do not even define valid calls to the tool.",
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+        "type": "text",
+        "text": "Such overoptimization to a misspecified reward is a widely noted issue in LLMs trained with RL (Gao et al., 2023). On the other hand, by matching the entire distribution, GFlowNet fine-tuning avoids collapsing to a single mode of the reward, thereby being robust to the misspecification of",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+        "type": "page_number",
+        "text": "8",
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+    {
+        "type": "text",
+        "text": "the reward (Eysenbach & Levine, 2022) and achieving significantly better performance on in and out-of-distribution examples. See Appendix E for additional results and analysis.",
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+    {
+        "type": "text",
+        "text": "5 FURTHER RELATED WORK",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Sampling from intractable marginals. Beyond the approximations mentioned in §3.1, sampling from intractable posterior distributions given by pretrained models for tasks such as infilling and constrained generation has been an object of study. Miao et al. (2019); Zhang et al. (2020a) use MCMC for these problems, Malkin et al. (2021) used a variable-neighborhood ascent for finding modes, and a sequential Monte Carlo approach was recently proposed by Lew et al. (2023). Others have studied the problem with masked language models, using them to perform variants of Gibbs sampling (Wang & Cho, 2019; Goyal et al., 2022; Yamakoshi et al., 2022) and recovering marginal distributions over small sets of tokens (Torroba Hennigen & Kim, 2023).",
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+    {
+        "type": "text",
+        "text": "GFlowNets. GFlowNets (Bengio et al., 2021) were originally proposed to learn policies for sampling discrete compositional objects from an unnormalized reward distribution, motivated by the need to sample diverse high-reward objects in scientific discovery (Jain et al., 2023), in particular, for biological sequence generation (Jain et al., 2022). The interpretation of GFlowNets as variational inference algorithms (Malkin et al., 2023; Zimmermann et al., 2023) makes them appropriate for sampling Bayesian posterior distributions over structured objects (e.g., Deleu et al., 2022; 2023; van Krieken et al., 2023; Hu et al., 2023).",
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+    {
+        "type": "text",
+        "text": "Chain-of-thought reasoning in LLMs. In recent work on classification and completion with language models, the latent reasoning chain  $Z$ , in the notation of §3.1, is called a 'chain of thought' (Wei et al., 2022). The chain of thought is typically generated by conditioning the language model on  $X$  with the use of specialized demonstrations or prompts (Kojima et al., 2022), with no guarantee of sampling the posterior accurately. Related to our Bayesian formulation, Wang et al. (2023b) noted that appropriately aggregating the conclusions  $Y$  from several latent chains  $Z$  improves predictive performance. In Xu et al. (2023); Zhou et al. (2022), a posterior over latent token sequences is sampled using MCMC, while Zelikman et al. (2022) propose fine-tuning on successful (high-reward, in our language) chains of thought, which achieves reward maximization but gives no guarantee of diversity. In concurrent work, Phan et al. (2023) use MCMC to sample chains-of-thought in problems with binary feedback. We expect these methods to generalize poorly to difficult exploration problems, while GFlowNet fine-tuning takes advantage of generalizable structure in the posterior and has a goal of sampling the full posterior over latent reasoning chains.",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The knowledge compressed in LLMs is crucial for tasks such as infilling and constrained generation, but querying this knowledge involves sampling from intractable posterior distributions. We propose to use GFlowNet objectives to train LLMs to sample from such posterior distributions. Empirical results show that GFlowNet fine-tuning finds a better fidelity-diversity trade-off for text generation and also improves sample efficiency and generalization on downstream tasks compared to maximum-likelihood training or reward-maximizing policy optimization. As an amortized inference algorithm, our method converts computation into better test-time performance without additional data.",
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+    {
+        "type": "text",
+        "text": "Future work should investigate transfer and generalization across tasks, in particular, building a 'universal reasoner' as a model  $q(Z \\mid X)$  shared between  $X$  from different tasks, as was recently considered by Wang et al. (2023a). One should investigate the benefit of using a better knowledge model, e.g., a more capable base LLM, as a starting point for GFlowNet fine-tuning. The ability to draw multiple samples from a GFlowNet can also be used to quantify epistemic uncertainty. Finally, we adopt the GFlowNet formalisms with the perspective of generalizing to latent variables  $Z$  with a richer generative process than left-to-right sampling. We hope that the GFlowNet paradigm will enable more flexible reasoning with LLMs in the future: extending probabilistic programming with language variables (Beurer-Kellner et al., 2023), using structured chains of thought (Yao et al., 2023; Besta et al., 2024), and extending to program synthesis and planning with world models.",
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+    {
+        "type": "text",
+        "text": "Limitations. Due to resource constraints, our experiments use models up to 6B parameters, but we expect the conclusions to hold for larger models. In fact, our method can potentially benefit larger models more: it is harder to optimize a larger model with maximizing objectives on a small amount of data. As with any on-policy method, exploration, especially in problems with more complex latents, remains an open problem. Furthermore, our method improves inference but not the knowledge in the LM. Issues such as hallucination or miscalibration, which are closely related to the knowledge representation, are thus not addressed.",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+    },
+    {
+        "type": "page_number",
+        "text": "9",
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+    {
+        "type": "text",
+        "text": "ETHICS STATEMENT",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "While we foresee no immediate negative societal consequences of our work, we hope that future researchers who build upon it will, as we have, bear in mind the potential of LLMs – and in particular of human-like reasoning in LLMs – to be used both for good and for harm.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Research areas in safe and explainable AI that can benefit from GFlowNet fine-tuning include (1) interpretability of LLMs' reasoning processes and (2) fine-tuning with human feedback or an external reward, where diverse sampling can help prevent 'reward hacking' and overfitting to a misspecified target.",
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+    {
+        "type": "text",
+        "text": "REPRODUCIBILITY",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We discuss the details of the proposed algorithms in §3.3 and provide all the implementation details and hyperparameters for the experiments in the main paper and appendix. Code for our experiments is available at https://github.com/GFNOrg/gfn-lm-tuning.",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The authors are grateful to Bonaventure Dossou and Salem Lahlou for their help in the early stages of this project. We also thank Robert Hawkins, Arian Hosseini, Zhen Wang, and Anirudh Goyal for valuable discussions and suggestions of related work.",
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+    {
+        "type": "text",
+        "text": "GL acknowledges funding from CIFAR, Samsung, and a Canada Research Chair in Neural Computation and Interfacing.",
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+        "type": "text",
+        "text": "YB acknowledges funding from CIFAR, NSERC, IBM, Intel, Genentech, and Samsung.",
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+    {
+        "type": "text",
+        "text": "The research was enabled in part by computational resources provided by the Digital Research Alliance of Canada (https://alliancecan.ca), Mila (https://mila.quebec), and NVIDIA.",
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+    {
+        "type": "text",
+        "text": "REFERENCES",
+        "text_level": 1,
+        "bbox": [
+            171,
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+        ],
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+            "Yongchao Zhou, Andrei Ioan Muresanu, Ziwen Han, Keiran Paster, Silviu Pitis, Harris Chan, and Jimmy Ba. Large language models are human-level prompt engineers. International Conference on Learning Representations (ICLR), 2023.",
+            "Wanrong Zhu, Zhiting Hu, and Eric Xing. Text infilling. arXiv preprint arXiv:1901.00158, 2019.",
+            "Heiko Zimmermann, Fredrik Lindsten, Jan-Willem van de Meent, and Christian A. Naesseth. A variational perspective on generative flow networks. Transactions on Machine Learning Research (TMLR), 2023."
+        ],
+        "bbox": [
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+    },
+    {
+        "type": "page_number",
+        "text": "16",
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+    },
+    {
+        "type": "text",
+        "text": "A ADDITIONAL BACKGROUND",
+        "text_level": 1,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "A.1 GLOSSARY OF RL FOR LLMS",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "We provide definitions for key terms used throughout the paper, with a focus on their relevance to our setting of fine-tuning large language models (LLMs).",
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+    },
+    {
+        "type": "text",
+        "text": "Reinforcement learning. Reinforcement learning (RL) is a branch of machine learning concerned with how agents should take actions in an environment to maximize cumulative rewards. In our context, RL is used to fine-tune the decision-making process of LLMs to improve their performance on specific tasks. For a more comprehensive overview, we refer readers to Sutton & Barto (2018).",
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+    },
+    {
+        "type": "text",
+        "text": "Policy. In RL, a policy is a strategy that defines the behavior of an agent by mapping states of the environment to actions. In our setting, a policy dictates how the language model generates text sequences based on the current context and learned parameters.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Reward. A reward is a signal that evaluates the quality of an action taken by an agent in a particular state. In the fine-tuning of LLMs, rewards are used to guide the model towards generating more desirable text, such as more accurate predictions or more coherent continuations. More specifically, in the context of GFlowNets, the reward corresponds to the unnormalized posterior probability, and the GFlowNet aims to match it by learning a policy that generates samples proportional to their reward.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Matching a distribution. Matching a distribution involves training a model to approximate a target probability distribution. In our work, this concept is applied to fine-tune LLMs so that their generated text matches the desired characteristics, such as adhering to a particular style or content constraint.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Policy gradient methods. Policy gradient methods (such as PPO) are a subset of RL algorithms that optimize a policy by computing gradients of the expected reward with respect to the policy parameters. In the context of LLMs, these methods are used to fine-tune the model's parameters to increase the likelihood of generating high-reward text sequences.",
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+    {
+        "type": "text",
+        "text": "A.2 LEARNING OBJECTIVE",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We use the subtrajectory balance learning objective for training GFlowNets (Madan et al., 2023). In the notation of that paper, with a forward policy  $P_F$ , backward policy  $P_B$  and state flow function  $F$ , the objective over a partial trajectory  $\\tau = s_m \\rightarrow \\dots \\rightarrow s_n$  is defined as follows",
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+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} _ {S u b T B} (Z; \\theta) = \\left(\\log \\frac {F \\left(s _ {m}\\right) \\prod_ {i = m} ^ {n - 1} P _ {F} \\left(s _ {i + 1} \\mid s _ {i}\\right)}{F \\left(s _ {n}\\right) \\prod_ {i = m} ^ {n - 1} P _ {B} \\left(s _ {i} \\mid s _ {i + 1}\\right)}\\right) ^ {2} \\tag {4}\n$$\n",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "In the case of autoregressive generation of a sequence of tokens in a fixed order (left-right), the generative process is a tree, so there is only a single path to each state, and each state has a single parent. Thus,  $P_B(s|s') = 1$  trivially. Additionally, since each state is a valid terminable state, we can incorporate the modification to account for this from Deleu et al. (2022). Specifically, note that at convergence we have  $R(s_n^\\top) = F(s_n)P_F(\\top \\mid s_n)$ . Using this, we can simply substitute  $F(s_{n}) = R(s_{n}^{\\top}) / P_{F}(\\top \\mid s_{n})$  in Eq. (4). This allows us to avoid parameterizing a flow function separately, reducing additional complexity. The only learned object is now the forward sampling policy  $P_F$ , which we refer to as  $q_{GFN}$ . Summing this over all partial trajectories in a trajectory with equal weight  $(\\lambda = 1)$ , we get the final learning objective in Eq. (3).",
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+    {
+        "type": "text",
+        "text": "B SENTENCE CONTINUATION",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Additional details. We choose sentences as the level of granularity for our sequence continuation task because they are a natural unit of generation in language. Moreso than individual words, sentences are analogous to whole thoughts, reasoning steps, etc. because the compositional rules of syntax operate at the level of sentences. Whereas the meaning of a word is often ambiguous and context-dependent, the meaning of a sentence tends to be more self-contained.",
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+    {
+        "type": "text",
+        "text": "A naive solution to the task is to simply sample autoregressively from the LM until a “.” token is reached, marking the end of the sentence. In practice, however, this is unlikely to produce samples that have high likelihood because the distribution over sentences has a long tail: a vast number of sentences have small but non-zero probability, and collectively account for a substantial portion of the total probability mass. Instead, it would be desirable to sample from a low-temperature distribution over sentences, so that the distribution becomes more sparse and highly probable sentences are",
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+        "page_idx": 16
+    },
+    {
+        "type": "page_number",
+        "text": "17",
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+    },
+    {
+        "type": "text",
+        "text": "sampled more often. However, this in itself is a difficult distribution to sample from, and it is different from simply sampling autoregressively from the LM with a lower temperature applied to the distributions over the next words. For instance, as the temperature approaches 0, the desired distribution over sentences should converge to an argmax and produce the single most likely next sentence. However, if we try to apply the temperature to the distribution over the next words and sample autoregressively, then as the temperature approaches 0, the resulting policy will greedily construct a sentence by sequentially picking the most likely next word, which is unlikely to produce the highest probability sentence.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Our GFlowNet sampling policy parametrizes a distribution  $p_{\\mathrm{GFN}}(w_{i+1}|w_{1:i})$  using the same architecture as the LM  $p_{\\mathrm{LM}}(w_{i+1}|w_{1:i})$ , and at the start of training, it is initialized with the same weights. The initial state of the GFlowNet consists of a prompt  $w_{1:k}$  that conditions generation (i.e., the text whose next sentence we are trying to sample). Each subsequent state  $w_{1:k+i}$  is an extension of this text and is obtained by sampling autoregressively from  $p_{\\mathrm{GFN}}$ . This process terminates when a period “.” is sampled, indicating the end of the next sentence. If a predefined maximum sentence length is reached, we force a “.” action to manually terminate the generation.",
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+    {
+        "type": "text",
+        "text": "The GFlowNet policy is trained to sample sentences in proportion to their tempered probability under the original LM given the prompt. Denote the length of the prompt by  $k$ , the length of the sampled sentence by  $m$ , and the temperature by  $T$ . Then, the reward is:",
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+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nR \\left(w _ {1: k + m}\\right) \\stackrel {\\text {d e f}} {=} p _ {\\mathrm {L M}} \\left(w _ {k + 1: k + m} \\mid w _ {1: k}\\right) ^ {\\frac {1}{T}} = \\left(\\prod_ {i = 1} ^ {m} p _ {\\mathrm {L M}} \\left(w _ {k + i} \\mid w _ {1: k + i - 1}\\right)\\right) ^ {\\frac {1}{T}} \\tag {5}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "The intuition behind this reward is that we are assuming that the LM can reliably assign a high likelihood to high-probability sentences, that we wish to preferentially sample over low-probability ones. If we were to set  $T = 1$ , then the solution for the GFlowNet would be to sample proportionally to  $p_{\\mathrm{LM}}$  (which it is initialized to). The preferential sampling of high-probability sentences is therefore obtained by setting  $0 < T < 1$ .",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To run our experiment, we obtained a dataset of 1000 prompts from OpenWebText (Gokaslan et al., 2019) that were each 1-3 sentences long, 50 of which were used for validation. Our LM consisted of a pretrained 1.5B parameter GPT2-XL (Radford et al., 2019), and our GFlowNet was a copy of this model that was fine-tuned in a lower-dimensional space of 80M parameters using LoRA (Hu et al., 2022).",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Additional results. We show examples of empirical next-sentence samples generated by our model in Table B.1 compared to baselines. The model was trained using a reward temperature of 0.875, which achieves a good balance between log-likelihood and diversity.",
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+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "C INFILLING STORIES",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Additional details. See Table C.1 for training examples from the subset of the ROC Stories dataset used for the task. To condition the model on  $X$  and  $Y$ , as well as for the prompting baseline, we use the following prompt:",
+        "bbox": [
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+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "\"Beginning: {X}\\n End: {Y}\\n Middle: \"",
+        "bbox": [
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+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "An assumption we make in this paper is that the base language model already contains the knowledge required for the task, and the goal is to perform inference over this knowledge. However, for this task, none of the pretrained base models were good at assigning high likelihoods to plausible stories. Thus, we fine-tuned GPT-2 Large model (Radford et al., 2019) on the entirety of the stories dataset and used this fine-tuned model as the reward model. This was done with full fine-tuning using theTRL library (von Werra et al., 2020). We trained for 20 epochs with a batch size of 64 and 32 gradient accumulation steps and a learning rate of 0.0005.",
+        "bbox": [
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+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "We detail the hyperparameters used for training GFlowNets in our experiments in Table C.2. During training, we sample  $(X,Y)$  from the dataset and then sample (batch size)  $Zs$  for every  $(X,Y)$ , before using  $p_{\\mathrm{LM}}(XZY)$  as the reward. We use the replay buffer described in §3.3 and seed it with rationales from the dataset. We linearly anneal the reward temperature, the temperature of the behavior policy during training, and the learning rate during warmup. For supervised fine-tuning, we use a batch size of 256 and train for 10 epochs with a learning rate of 0.0001 withTRL and LoRA. At test time, we sample 1024 infills for each example in the test set from all the models at temperature 0.9, and average over 10 such draws.",
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "page_number",
+        "text": "18",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/08782218db0b647adfedf3fde80719bce822bb954b5eb95eba4d8e482ba06e4d.jpg",
+        "table_caption": [
+            "Table B.1: Empirical examples for sequence continuation. Sentences generated from GFlowNet fine-tuning tend to be more reasonable than autoregressively generated samples from the LM with temperature 1.0 and tend to be more diverse than samples generated from diverse beam search."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Input Prompt:</td><td>The matching campaign starts on Friday and will continue through midnight on Monday.</td></tr><tr><td>Sampling w/ T = 1.0:</td><td>(1) Participate with a fairly low $1 donation so we can motivate more volunteers on Sunday afternoon.\n(2) However, the information regarding Cutler&#x27;s suspicious death may not become widely known until early in the week.</td></tr><tr><td>Diverse beam search:</td><td>(1) If you are interested in participating in the matching campaign, you can do so by clicking here.\n(2) There is no limit to the number of times you can enter.</td></tr><tr><td>GFlowNet fine-tuning:</td><td>(1) If you are interested in signing up you can do so here.\n(2) Please share.</td></tr><tr><td>Input Prompt:</td><td>I want hockey to become a huge thing in Houston.</td></tr><tr><td>Sampling w/ T = 1.0:</td><td>(1) We&#x27;ll be here in Texas from late November till end March.\n(2) To that end, I&#x27;ve been following the Houston Aeros (Houston Dynamo&#x27;s minor-league affiliate) and their AHL affiliate (the Texas Stars).</td></tr><tr><td>Diverse beam search:</td><td>(1) That&#x27;s why I&#x27;m here.\n(2) That&#x27;s why I&#x27;m doing this.</td></tr><tr><td>GFlowNet fine-tuning:</td><td>(1) This is something I&#x27;ve always wanted.\n(2) When I was a teenager in middle school, I went to the Ice Arena in University of Houston and loved it.</td></tr><tr><td>Input Prompt:</td><td>That incident got a lot of attention in part because it was captured on video. Israel said he recorded what happened at the synagogue, and made it public, to document it and leave no doubt about what transpired.</td></tr><tr><td>Sampling w/ T = 1.0:</td><td>(1) He blogged about it here as well.\n(2) Israeli TV stations broadcast the video before it aired in Israel, as the country&#x27;s rule requires.</td></tr><tr><td>Diverse beam search:</td><td>(1) However, there is no evidence that he did so.\n(2) However, there is no video of what happened inside the synagogue.</td></tr><tr><td>GFlowNet fine-tuning:</td><td>(1) He is on tape doing all of it.\n(2) It&#x27;s a message countries can use to deter future attacks.</td></tr><tr><td>Input Prompt:</td><td>The Rolling Stones in Concert has since been released solely by ABKCO Records. The band would remain incensed with Klein for decades for that act. Klein died in 2009.</td></tr><tr><td>Sampling w/ T = 1.0:</td><td>(1) Actually art has become as obscene as investment banking.\n(2) Some believe he shot himself in the chest.</td></tr><tr><td>Diverse beam search:</td><td>(1) The Rolling Stones, however, would not be deterred.\n(2) The Rolling Stones would go on to release their own version of The Rolling Stones in Concert.</td></tr><tr><td>GFlowNet fine-tuning:</td><td>(1) He received a lifetime achievement award from the Jazz Times.\n(2) Sometimes it seems like we are destined to repeat our own mistakes.</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "page_number",
+        "text": "19",
+        "bbox": [
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+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "table",
+        "img_path": "images/15b6b83eff4b258503acaca19ec992cdc2dd29ab4b0301059226aab36da37632.jpg",
+        "table_caption": [
+            "Table C.1: Examples of training samples for the stories infilling task."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Beginning (X)</td><td>Middle (Z)</td><td>End (Y)</td></tr><tr><td>I was going to a Halloween party. I looked through my clothes but could not find a costume. I cut up my old clothes and constructed a costume.</td><td>I put my costume on and went to the party.</td><td>My friends loved my costume.</td></tr><tr><td>Allen thought he was a very talented poet. He attended college to study creative writing. In college, he met a boy named Carl.</td><td>Carl told him that he wasn’t very good.</td><td>Because of this, Allen swore off poetry forever.</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "table",
+        "img_path": "images/3c694d8708f02bf0ba0f9c8fe1ac1af50cfe4ffd1107bb871dd2ad3d04a03bf0.jpg",
+        "table_caption": [
+            "Table C.2: Hyperparameters for the story infilling task."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>LoRA rank</td><td>64</td></tr><tr><td>LoRA scaling factor</td><td>16</td></tr><tr><td>LoRA dropout</td><td>0.1</td></tr><tr><td>Batch size</td><td>64</td></tr><tr><td>Gradient accumulation steps</td><td>16</td></tr><tr><td>Learning rate</td><td>0.0001</td></tr><tr><td>Optimizer</td><td>AdamW</td></tr><tr><td>PFtemperature max</td><td>2</td></tr><tr><td>PFtemperature min</td><td>0.5</td></tr><tr><td>Reward temperature start</td><td>1.1</td></tr><tr><td>Reward temperature end</td><td>0.85</td></tr><tr><td>Reward temperature horizon</td><td>100</td></tr><tr><td>Buffer capacity</td><td>25</td></tr><tr><td>Number of training steps</td><td>1000</td></tr><tr><td>Evaluation temperature</td><td>0.8</td></tr><tr><td>Maximum length</td><td>25</td></tr><tr><td>Minimum length</td><td>5</td></tr></table>",
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+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "page_number",
+        "text": "20",
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+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Analysis. Table C.3, C.4, C.5, C.6, C.7, C.8 illustrate some examples of the infills generated with the prompting baseline, supervised fine-tuned and GFlowNet fine-tuned models on the test examples.",
+        "bbox": [
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+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "For evaluating the infills we generate a rating for the stories with infills from each of the methods based on the coherence of the story. We average the rating over 10 infills sampled from each method and average it over all the 100 test examples. The prompt used for evaluation is the following, adapted from Liu et al. (2023). Stories with infills generated by the GFlowNet fine-tuned model on average receive a much higher rating than the baselines. The average rating for the reference infills is 4.3 which should be viewed as an upper bound, as the stories may potentially be present in the GPT-4 training data.",
+        "bbox": [
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+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "You will be given a short story.",
+        "bbox": [
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+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Your task is to rate the story on one metric.",
+        "bbox": [
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+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Please make sure you read and understand these instructions carefully. Please keep this document open while reviewing, and refer to it as needed.",
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+    },
+    {
+        "type": "text",
+        "text": "Evaluation Criteria:",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Coherence (1-5) - the collective quality of all sentences. The story should be well-structured and well-organized. The story should not just be a heap of related information, but should build from sentence to a coherent narrative. The story should also be realistic and all the sentences when put together should make sense.",
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+    },
+    {
+        "type": "text",
+        "text": "Evaluation Steps:",
+        "bbox": [
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+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "1. Read the story carefully. Check if the story is coherent and all the sentences follow a logical order.  \n2. Assign a score for coherence on a scale of 1 to 5, where 1 is the lowest and 5 is the highest based on the Evaluation Criteria. If there are grammatical errors or logical inconsistencies, simply assign a lower score and do not elaborate on the reasoning.",
+        "bbox": [
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+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Example:",
+        "bbox": [
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+            253,
+            679
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Randy had recently separated from his wife.   \nHe felt very lonely and sad all the time.   \nHe considered trying to patch things up with Vera, his ex-wife. He decided to get a puppy instead.   \nHis decision made him happy and he no longer felt sad.",
+        "bbox": [
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+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "- Coherence: 5",
+        "bbox": [
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+            786
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Randy had recently separated from his wife.   \nHe felt very lonely and sad all the time.   \nHe considered trying to patch things up with Vera, his ex-wife.   \nEventually Randy missed Vera very much and developedWhile making memories with   \nHis decision made him happy and he no longer felt sad.",
+        "bbox": [
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+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "- Coherence 1",
+        "bbox": [
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+            896,
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+            909
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
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+            47
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "page_number",
+        "text": "21",
+        "bbox": [
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+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Story:",
+        "bbox": [
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+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "{{Story}}",
+        "bbox": [
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+            131,
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+            146
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "Evaluation Form (scores ONLY):",
+        "bbox": [
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+            157,
+            467,
+            172
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "- Coherence:",
+        "bbox": [
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+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "D SUBJECTIVITY CLASSIFICATION",
+        "text_level": 1,
+        "bbox": [
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+            215,
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+            231
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "Additional details. See Table D.1 for some training examples in the SUBJ dataset. We use the prompts in Table D.2 for all baselines.",
+        "bbox": [
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+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "We run GFlowNet fine-tuning for 1000 steps with a linear warmup over 200 steps, a fixed learning rate of 0.0005, and a batch size of 512 samples; see Table D.3 for all the hyperparameters used. Each batch consists of 8 queries ( $X$ 's), randomly drawn with replacement. We then sample 64 rationales ( $Z$ 's) for every  $X$ , before using  $p_{\\mathrm{LM}}(ZY \\mid X)$  as the reward. We also use the replay buffer described in §3.3 and seed it with potential rationales generated from  $p_{\\mathrm{LM}}$ . For supervised fine-tuning, both on its own and on top of GFlowNet fine-tuning, we use a batch size of 256 with 8 queries, randomly drawn with replacement. We train for 100 steps with a linear warm-up over 20 steps and a constant learning rate. We sweep the learning rate in [0.001, 0.0001] and report the best performance. The reward temperature is annealed from 1.2 down to 1 over the first 150 steps of training.",
+        "bbox": [
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+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "Additional results. To encourage exploration in the space of  $Z$ , we inverse-prompt the model to generate potential chains of thoughts and store them in a replay buffer at the beginning of training. The replay buffer is a priority queue indexed by the reward. Under-performing chains of thoughts are evicted as we collect more trajectories. We ablate the effect of seeding the replay buffer with inverse-prompted chains of thoughts and aggregating multiple chains at test time in Table D.4.",
+        "bbox": [
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+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "E INTEGER ARITHMETIC",
+        "text_level": 1,
+        "bbox": [
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+            483,
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+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "Additional details. See Table E.1 for examples from synthetically generated training data for the integer arithmetic task.",
+        "bbox": [
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+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "As with the infilling task, for the integer arithmetic task, the pretrained base models we tested were imperfect knowledge models, i.e. incorrect rationales are sometimes assigned very high likelihood. To assign the correct rationales higher rewards, we prepend some demonstrations with hand-crafted rationales to the query when computing the reward, i.e.  $p_{\\mathrm{LM}}(XZY|(X_iZ_iY_i)^k)$  where  $k = 3$  in our experiments. This improves the calibration to some extent. Note that these  $(X_i,Z_i,Y_i)$  are taken from the dataset and used only in the reward. The GFlowNet policy here is conditioned only on  $X$ , i.e.,  $q_{\\mathrm{GFN}}(Z|X)$ .",
+        "bbox": [
+            169,
+            541,
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+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "We detail the hyperparameters used for training GFlowNets in our experiments in Table E.2. During training, we sample  $(X,Y)$  from the dataset and then sample (batch size)  $Zs$  for every  $(X,Y)$ , before using  $p_{\\mathrm{LM}}(XZY|(X_iZ_iY_i)^k_{i=1})$  as the reward. During the generation of  $Z$ , as mentioned in §4.4, whenever a “=” is generated we extract the preceding expression and evaluate the last two terms using eval in Python. We use the replay buffer described in §3.3 and seed it with rationales from the dataset  $p_{\\mathrm{LM}}$ . We linearly anneal the reward temperature, and the learning rate during warmup. During evaluation, for all methods, we aggregate the response over multiple samples drawn from the model at some fixed temperature. For the zero-shot baseline, we observe the best performance with the “Let us think step by step.” prompt appended at the end of the question. For supervised fine-tuning and PPO, we use the implementation from trl. For supervised fine-tuning we use a batch size of 128 with 8 gradient accumulation steps and train for 10 epochs with a learning rate of 0.0001 using LoRA. For PPO we use a minibatch size of 64 with 16 gradient accumulation steps, a learning rate of 0.0001, 4 epochs (on the minibatch), a clip range of 0.2, and an adaptive KL coefficient initialized at 0.2.",
+        "bbox": [
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+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "Effect of the number of rationales. As mentioned in §4.4, we seed the buffer with rationales from a dataset. Table E.3 summarizes the results of ablation on the number of examples used to seed the buffer. We observe, as expected, that the performance generally improves as we the number of examples used to seed the buffer grows. When no buffers are used to seed the buffer, the performance is quite poor. We hypothesize that without the good rationales used to seed the buffer, the exploration problem of searching for good rationales becomes very challenging.",
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+            840,
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+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "page_number",
+        "text": "22",
+        "bbox": [
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+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "table",
+        "img_path": "images/c382138eb9598c17aa1cafc1985a2ebcfa5dc18e06aef41e33f9cec3b23b38e7.jpg",
+        "table_caption": [
+            "Table C.3: Examples of infills generated by the prompting baseline along with reference infills for the stories infilling task."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Beginning (X)</td><td>Middle (Z)</td><td>End (Y)</td></tr><tr><td>Randy had recently separated from his wife. He felt very lonely and sad all the time. He considered trying to patch things up with Vera, his ex-wife.</td><td>He decided to get a puppy instead\nHe now happily had a new wife who he really liked.\nHis wife missed him, but for the wrong reasons. Now\nHe finally found someone he could be happy with. It was\nHe began spending more time with his new girlfriend and friends.\nRandy remained single and mostly satis-fied with that. He found his\nWhen Randy was happy he went to the aquarium, a place\nHe started going out more with a display of male affection.\nHe bought Vera a moonstone and gave it to her as\nHe and Vera got back together, even though they had stayed\nHe introduced Paige to his new employ-\nees and they were great.\nRandy is happily married to Vera and they have a healthy relationship\nHe found a new lady with a lot of the same interests</td><td>His decision made him happy and he no longer felt sad.</td></tr><tr><td>Ken put a bottle of beer in the freezer. He heard a popping noise. He looked in the freezer and saw the bottle had burst.</td><td>He didn&#x27;t want to wait for another beer to get cold\nHe went to sleep and came awake with an angry stomach.\nKen had another beer as a treat. After the third beer\nHe stored the beer in the freezer and for-\ngot about it.\nHe walked around the bar a bit. He thinks he&#x27;ll\nWeeks later, Ken preferred his cold beer.\nHowever, he\nHe poured a fresh bottle with a sigh of relief. It\nHe went to the bar for cold beer. No one knew\nKen hung up the phone on his friend. He didn&#x27;t\nHe bought some veggies from the gro-\ncery store. It was warm\nKen watched the movie Goalkeeper. He was glad the cold\nHe laid down for a rest. His belly was full!\nHe laid down for the night and finally sleep again. The</td><td>He drank a warm beer instead.</td></tr></table>",
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+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            173,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "page_number",
+        "text": "23",
+        "bbox": [
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+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "table",
+        "img_path": "images/dbb18837ae8e39bb3b98fe8a2b56237ac88bc643cb7267e5ffaaa116c581c1d1.jpg",
+        "table_caption": [
+            "Table C.4: Additional examples of infills generated by the prompting baseline along with reference infills for the stories infilling task."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Beginning (X)</td><td>Middle (Z)</td><td>End (Y)</td></tr><tr><td>Jasmine had homework to do. She did not have a pencil. She looked around the room for one.</td><td>The room did not have a pencil Jasmine made her pencil and done her work. She looked Her mom brought her a pencil. Jasmine wrote her homework Jasmine took her pencil out of the room. She now Jasmine pretended not to have done her homework. She got Her brother finally brought a pencil. They could finally start! Her brother brought a pencil and solved her problem. Jasmine Her brother brought a pencil and solved her problem. Jasmine Her friend offered her a pencil. Jasmine used the pencil Her teacher saw her looking at her phone. Then, Jas Her mom came in and handed her the extra pencil! Jas Her mom brought her a pencil. Jasmine then got to Jasmine sent her friend money to do it. She was</td><td>Jasmine decided not to do her homework.</td></tr><tr><td>Jane had recently gotten a new job. She was nervous about her first day of work. On the first day of work, Jane overslept.</td><td>Jane arrived at work an hour late Despite arriving late, Jane was greeted warmly by her coworkers Jane learned that you never can predict the future. It was Jane learned not to rush into anything. Jane made a good Many of her peers were not pleased with her. Jane got Jane ended up quitting her new job early. She was happy More time with the wrong people gave her anxiety. She was Jane learned that you do not mess up at a job well Jane realized she needed to find new work contacts. At After her first day of work, Jane was too tired to Secretly, Jane hated her new job. It made her Jane realized she worked at a mid-sized company with long Jane&#x27;s boss realized she did not show up to work on</td><td>Jane did not make a good impression at her new job.</td></tr></table>",
+        "bbox": [
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+            166,
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+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            173,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "page_number",
+        "text": "24",
+        "bbox": [
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+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "table",
+        "img_path": "images/d3c64b45b6d640b5f9d86244e89d1b590367aca5faa06407420dd0e3728bdb13.jpg",
+        "table_caption": [
+            "Table C.5: Examples of infills generated by the supervised fine-tuned model along with reference infills for the stories infilling task."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Beginning (X)</td><td>Middle (Z)</td><td>End (Y)</td></tr><tr><td>Randy had recently separated from his wife. He felt very lonely and sad all the time. He considered trying to patch things up with Vera, his ex-wife.</td><td>He decided to get a puppy instead\nHe started enjoying his time with Vera even more. Now, \nEventually Randy missed Vera very much and developedWhile making memories with \nHe finally decided he had enough pain for no good reason.\nHe began spending more time with his new girlfriend. Now he \nFor the first time since the divorce he started to make new \nSoon Randy and Vera were rekindling their romance. They \nHe started seeing someone new a day.\nNow, he feels \nHe signed up for a class in Same Sex \nLoving Uniqueness \nFor the first time in months, Randy and \nVera celebrated a \nHe introduced Paige to his new employ-\nees. They are now good \nAfter some contemplating, he decided to patch things up with Vera \nHe now looks for another partner to enjoy his life with.</td><td>His decision made him happy and he no longer felt sad.</td></tr><tr><td>Ken put a bottle of beer in the freezer. He heard a popping noise. He looked in the freezer and saw the bottle had burst.</td><td>He didn&#x27;t want to wait for another beer to get cold\nThe popping noise was a freezer needle getting caught. It fell \nKen had a cold for thirty minutes. After the popping noise \nHe stored the beer in the freezer. The pouring sound was \nHe didn&#x27;t know why it didn&#x27;t stop the ringing.\nWeeks later, Ken knew the reason for the freezing. It \nLater Ken learned the gas is out of gas and will end \nHe left the freezer for the beer. It stayed in there \nBefore he knew it, the popping noise was gone. He \nThe popping noise that Ken heard indicated it should have freezer shut \nThe popping noise became louder and more frequent. Now Ken&#x27;s \nHe froze the beer and drank it from a mason jar \nHe went to the store to buy another. Now, he</td><td>He drank a warm beer instead.</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            173,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "page_number",
+        "text": "25",
+        "bbox": [
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+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "table",
+        "img_path": "images/0cd1faba8c3084fa4daed525d16a6efe8c9b7c719941100a5963cdc77fc8c137.jpg",
+        "table_caption": [
+            "Table C.6: Additional examples of infills generated by the supervised fine-tuned model along with reference infills for the stories infilling task."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Beginning (X)</td><td>Middle (Z)</td><td>End (Y)</td></tr><tr><td>Jasmine had homework to do. She did not have a pencil. She looked around the room for one.</td><td>The room did not have a pencil Jasmine decided to talk to her mom about it. They Her dad now has to teach her math! Pick- ing up Jasmine took her math test later in the day. She Jasmine got an A on her assignment. She continued to Her brother finally brought a pencil. They could finally start! Her brother brought a pencil and solved her problem. Jasmine Jasmine used her pink and brown high-lighter instead. Jasmine went to school. She was able to finish her Jasmine made her own pencil. At school, everyone was Her brotherly earrings helped her stay up more. While Jasmine made her own pencil. She ended up getting good Jasmine sent her sister to do it. Now it was</td><td>Jasmine decided not to do her homework.</td></tr><tr><td>Jane had recently gotten a new job. She was nervous about her first day of work. On the first day of work, Jane overslept.</td><td>Jane arrived at work an hour late After arriving home, Jane realized her apartment was not ready. After work, Jane returned home and ate a lousy sandwich. After her first day of work, Jane learned that mistakes happen Due to her bad first day, Jane was fired from her Ending: Jane ended up late for work. Although she On her 2nd day of work, Jane came home late On the second day of work, Jane arrived late and unprepared By the end of the day, Jane was tired and weary After her first day of work, Jane&#x27;s boss recommended her Secretly, Jane hated her new job. Se-cretly, After work, Jane felt worn out and tired. The end By the end of the day Jane felt confident and like she</td><td>Jane did not make a good impression at her new job.</td></tr></table>",
+        "bbox": [
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+            166,
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+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            173,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "page_number",
+        "text": "26",
+        "bbox": [
+            488,
+            946,
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+            960
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "table",
+        "img_path": "images/aa58b45b18a40dceb443fef1b56a18dab4052cd188b22efe49ec7dd8a89302f1.jpg",
+        "table_caption": [
+            "Table C.7: Examples of infills generated by the GFlowNet fine-tuned model along with reference infills for the stories infilling task."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Beginning (X)</td><td>Middle (Z)</td><td>End (Y)</td></tr><tr><td>Randy had recently separated from his wife. He felt very lonely and sad all the time. He considered trying to patch things up with Vera, his ex-wife.</td><td>He decided to get a puppy instead\nHe went to a friend&#x27;s house to talk.\nHe took Vera to the park for a picnic.\nHe finally decided to call Vera on the phone.\nHe asked Vera to meet him and they did.\nHe decided to get a dog for his family.\nHe went to Vera&#x27;s house.\nHe told Vera that he was ready to love again.\nHe bought Vera a nice gift.\nHe had two dogs and felt like he missed Vera.\nHe had a good couple of years.\nHe wanted to get Vera and he eventually did.\nHe wanted to sit in a room alone.</td><td>His decision made him happy and he no longer felt sad.</td></tr><tr><td>Ken put a bottle of beer in the freezer. He heard a popping noise. He looked in the freezer and saw the bottle had burst.</td><td>He didn&#x27;t want to wait for another beer to get cold\nHe decided to stop drinking beer.\nKen suddenly smelled a bottle of beer.\nHe thought it was something\nHe walked around the freezer to see the bottle.\nHe tried to clean up the freezer.\nHe was angry when he had to drink it.\nHe stepped down from the freezer\nHe made a big joke that night.\nHe looked in a jug and see it was, out.\nHe then looked for a beer the freezer.\nHe tried to enjoy the rest.\nHe watered the bottle,</td><td>He drank a warm beer instead.</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            173,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "page_number",
+        "text": "27",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "table",
+        "img_path": "images/298468cbc0411343f34240b04317d7fd3ce0b06b1449588beb058f6087c4f75f.jpg",
+        "table_caption": [
+            "Table C.8: Additional examples of infills generated by the GFlowNet fine-tuned model along with reference infills for the stories infilling task."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Beginning (X)</td><td>Middle (Z)</td><td>End (Y)</td></tr><tr><td>Jasmine had homework to do. She did not have a pencil. She looked around the room for one.</td><td>The room did not have a pencil\nShe was in a panic.\nShe had to place the pencil in her pocket.\nShe had several her classmates.\nShe looked for a neat pencil.\nShe thought she looked everywhere.\nHer brother walked in and borrowed her pencil.\nShe only had a few minutes to do it.\nShe never had her pencil.\nShe saw a pen in her closet.\nShe searched everywhere for a pencil.\nShe went to her room.\nShe didn’t find one.</td><td>Jasmine decided not to do her homework.</td></tr><tr><td>Jane had recently gotten a new job. She was nervous about her first day of work. On the first day of work, Jane overslept.</td><td>Jane arrived at work an hour late\nShe lost her job after the first day of work.\nShe was egged on by her boss.\nShe decided to over hyp her first day.\nShe arrived late and missed her first day of work.\nShe was nervous about going to work.\nShe ended up getting a good job.\nJane waited almost a day to get to work.\nHer first day was very difficult.\nShe made a couple of phone calls.\nShe arrived at the office and was fired.\nShe was all out of coffee.\nJane was surprised she did not make a good impression.</td><td>Jane did not make a good impression at her new job.</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "table",
+        "img_path": "images/509a6d8eb09fa5309662f8f49e0f45a58c6e2194f519d62275eda4c07f17d67d.jpg",
+        "table_caption": [
+            "Table D.1: Two training examples from the SUBJ dataset."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Text (X)</td><td>Label (Y)</td></tr><tr><td>another story follows the relationship between a stepfather ( neeson ) and his young stepson .</td><td>objective</td></tr><tr><td>hoffman ’s performance is authentic to the core of his being .</td><td>subjective</td></tr></table>",
+        "bbox": [
+            176,
+            792,
+            820,
+            842
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            173,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "page_number",
+        "text": "28",
+        "bbox": [
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+            508,
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+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "table",
+        "img_path": "images/c45db547560f2f0a5fd81178ff903eef3275976774b3666da6d4397425ebb775.jpg",
+        "table_caption": [
+            "Table D.2: Prompts used for subjectivity classification."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Few-shot learning / Supervised fine-tuning</td><td>GFlowNet fine-tuning</td></tr><tr><td>Classify this movie review as objective or subjective: “[X]” This review is [Y].</td><td>Classify this movie review as objective or subjective: “[X]” This review is [Z], so it is [Y].</td></tr></table>",
+        "bbox": [
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+            114,
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+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "table",
+        "img_path": "images/c0e48f091ae14a6a7061a780315828f38465e1efa7ba4495f8c4db6879c6a797.jpg",
+        "table_caption": [
+            "Table D.3: Hyperparameters for GFlowNet fine-tuning on subjectivity classification."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>LoRA rank</td><td>256</td></tr><tr><td>LoRA scaling factor</td><td>16</td></tr><tr><td>LoRA dropout</td><td>0.</td></tr><tr><td>Batch size</td><td>16</td></tr><tr><td>Gradient accumulation steps</td><td>32</td></tr><tr><td>Learning rate</td><td>0.0005</td></tr><tr><td>Optimizer</td><td>AdamW</td></tr><tr><td>PFtemperature max</td><td>2</td></tr><tr><td>PFtemperature min</td><td>0.5</td></tr><tr><td>Reward temperature start</td><td>1.2</td></tr><tr><td>Reward temperature end</td><td>1.0</td></tr><tr><td>Reward temperature horizon</td><td>150</td></tr><tr><td>Buffer capacity</td><td>50</td></tr><tr><td>Number of steps</td><td>100</td></tr><tr><td>Number of samples</td><td>10</td></tr><tr><td>Maximum length</td><td>5</td></tr><tr><td>Minimum length</td><td>1</td></tr></table>",
+        "bbox": [
+            359,
+            220,
+            638,
+            460
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "Analysis. In Table E.4 we show some examples generated by PPO and GFlowNet fine-tuned models. We observe that PPO generates sequences with high rewards under the base model. These sequences, however, are not valid expressions, and, in fact, do not even call the tool. Instead, the model learns to simply repeat the expression. Repetitions being assigned high likelihood is an issue that has also been noted by Welleck et al. (2020). On the other hand, GFlowNets are able to generate the correct rationale to evaluate the expression.",
+        "bbox": [
+            169,
+            486,
+            826,
+            570
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "There are still cases where the GFlowNet fails to produce the correct reasoning chain. We illustrate some of these examples in Table E.5. The errors fall mainly into 3 categories: 1) missing operands in longer OOD problems 2) incorrect operand being copied 3) incorrect operator being copied. The potential source for such errors is that the reward model assigns equally high rewards to rationales with these minor mistakes and thus the model generates them with likelihood. These errors can potentially be reduced by using better and potentially larger reward models.",
+        "bbox": [
+            169,
+            574,
+            823,
+            657
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "table",
+        "img_path": "images/81440530b025600a83ae9bfa147f2bce9fc79374b9d8ec9bb4823debbc53eed4.jpg",
+        "table_caption": [
+            "Table D.4: Ablation studies on subjectivity classification using GPT-J 6.8B."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td colspan=\"3\">Test accuracy (%) ↑</td></tr><tr><td></td><td colspan=\"3\">Training samples</td></tr><tr><td></td><td>10</td><td>20</td><td>50</td></tr><tr><td>GFlowNet fine-tuning</td><td>71.4 ± 1.8</td><td>81.1 ± 0.4</td><td>87.7 ± 2.2</td></tr><tr><td>(-) Seed buffer</td><td>64.7 ± 8.9</td><td>68.7 ± 1.7</td><td>77.0 ± 5.5</td></tr><tr><td>(-) Seed buffer (-) Aggregating</td><td>63.9 ± 7.1</td><td>65.9 ± 2.4</td><td>75.4 ± 3.3</td></tr></table>",
+        "bbox": [
+            251,
+            811,
+            743,
+            922
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            173,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "page_number",
+        "text": "29",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "table",
+        "img_path": "images/0126739d32e6a7bc500f80d07da18d4c76d7eee6c8e5959b28a103592696dcca.jpg",
+        "table_caption": [
+            "Table D.5: Top sample rationales for the SUBJ test set using the instruct-fine-tuned GPT-J 6B as both the reward model and the base model for the GFlowNet, which is trained on 50 labeled examples."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>True label</td><td colspan=\"2\">Rationale</td><td>Frequency</td></tr><tr><td rowspan=\"7\">Objective</td><td rowspan=\"7\">This review is</td><td>about a factual event</td><td>11.45%</td></tr><tr><td>describing a factual event</td><td>10.23%</td></tr><tr><td>based on a factual statement</td><td>9.47%</td></tr><tr><td>about a historical event</td><td>8.77%</td></tr><tr><td>about a movie review</td><td>7.41%</td></tr><tr><td>based on facts</td><td>5.64%</td></tr><tr><td>about a factual statement</td><td>4.61%</td></tr><tr><td rowspan=\"7\">Subjective</td><td rowspan=\"7\">This review is</td><td>about a movie review</td><td>26.21%</td></tr><tr><td>about the movie experience</td><td>26.17%</td></tr><tr><td>about the movie</td><td>7.60%</td></tr><tr><td>describing a movie review</td><td>3.44%</td></tr><tr><td>about the movie review</td><td>2.31%</td></tr><tr><td>based on a fictional story</td><td>2.30%</td></tr><tr><td>describing the movie experience</td><td>2.01%</td></tr></table>",
+        "bbox": [
+            183,
+            133,
+            810,
+            359
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "table",
+        "img_path": "images/b50dddaaea545acea310bf7bd54188a5c75b447cc577fcf9e8902d06df387c14.jpg",
+        "table_caption": [
+            "Table E.1: Examples from the integer arithmetic dataset. Note that the results of the two-term expressions are evaluated by the calculator."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Question (X)</td><td>Rationale (Z)</td><td>Answer (Y)</td></tr><tr><td>Question: 6 - 0 - 4 - 8 = ? Answer:</td><td>6 - 0 =, 6 - 4 =, 2 - 8 =</td><td>. The answer is -6.</td></tr><tr><td>Question: 9 + 4 - 8 = ? Answer:</td><td>9 + 4 =, 13 - 8 =</td><td>. The answer is 5.</td></tr></table>",
+        "bbox": [
+            218,
+            404,
+            774,
+            460
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "table",
+        "img_path": "images/94d517ead6b018a36c2d9f4964a50fdcf308cf5702c9f88dd760ae69d6728735.jpg",
+        "table_caption": [
+            "Table E.2: Hyperparameters for the Integer Arithmetic Task"
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>LoRA rank</td><td>64</td></tr><tr><td>LoRA scaling factor</td><td>16</td></tr><tr><td>LoRA dropout</td><td>0.1</td></tr><tr><td>Batch size</td><td>16</td></tr><tr><td>Gradient accumulation steps</td><td>32</td></tr><tr><td>Learning rate</td><td>0.0001</td></tr><tr><td>Optimizer</td><td>AdamW</td></tr><tr><td>PFtemperature max</td><td>2</td></tr><tr><td>PFtemperature min</td><td>0.5</td></tr><tr><td>Reward temperature start</td><td>1.1</td></tr><tr><td>Reward temperature end</td><td>0.5</td></tr><tr><td>Reward Temperature horizon</td><td>150</td></tr><tr><td>Buffer capacity</td><td>50</td></tr><tr><td>Number of training steps</td><td>1000</td></tr><tr><td>Evaluation temperature</td><td>0.1</td></tr><tr><td>Number of samples</td><td>10</td></tr><tr><td>Maximum length</td><td>20</td></tr><tr><td>Minimum length</td><td>5</td></tr></table>",
+        "bbox": [
+            352,
+            494,
+            640,
+            748
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "table",
+        "img_path": "images/753c1c145469337d8df594d90da17f623b02eab580c07acf7eca86d8f1ace56b.jpg",
+        "table_caption": [
+            "Table E.3: Ablation study to understand the effect of the number of rationales used to seed the buffer on the test accuracy  $(\\%)$  for the integer arithmetic task."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">Number of operands</td></tr><tr><td></td><td colspan=\"2\">In-distribution</td><td>OOD</td></tr><tr><td>Number of seed rationales</td><td>3</td><td>4</td><td>5</td></tr><tr><td>0</td><td>22.6</td><td>5.8</td><td>3.4</td></tr><tr><td>10</td><td>58.6</td><td>52.3</td><td>20.2</td></tr><tr><td>25</td><td>48.8</td><td>56.8</td><td>24.6</td></tr><tr><td>50</td><td>95.2</td><td>75.4</td><td>40.7</td></tr></table>",
+        "bbox": [
+            318,
+            792,
+            673,
+            916
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            173,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "page_number",
+        "text": "30",
+        "bbox": [
+            488,
+            946,
+            509,
+            959
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "table",
+        "img_path": "images/e43acff2cd547e6941138c26920c5ecf3658454178abb15fe86a51bde144c96d.jpg",
+        "table_caption": [
+            "Table E.4: Samples generated by PPO fine-tuned and GFlowNet fine-tuned models."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Question (X)</td><td>Generated rationale (Z)</td><td>log R</td></tr><tr><td rowspan=\"2\">Question: 1 - 9 + 8 = ? Answer:</td><td>1 - 9 - 8</td><td>-13.17</td></tr><tr><td>1 - 9 = -8, -8 + 8 = 0</td><td>-27.75</td></tr><tr><td rowspan=\"2\">Question: 8 + 7 + 2 + 7 = ? Answer:</td><td>8 + 7 + 2 + 7</td><td>-2.39</td></tr><tr><td>8 + 7 = 15, 15 + 2 = 17, 17 + 7 = 24</td><td>-11.72</td></tr><tr><td rowspan=\"2\">Question: 7 - 5 + 8 - 0 - 6=? Answer:</td><td>7 - 5 +</td><td>-1.22</td></tr><tr><td>7 - 5 = 2, 2 + 8 = 10, 10 - 0 = 10, 10 - 6 = 4</td><td>-7.99</td></tr></table>",
+        "bbox": [
+            205,
+            239,
+            787,
+            417
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "table",
+        "img_path": "images/792c0c89d68054f41d975c7a6bfc2b7f111692dc1a0ca23b2ffb95a93baf946b.jpg",
+        "table_caption": [
+            "Table E.5: Examples of errors made by the GFlowNet-fine-tuned model."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Question (X)</td><td>Generated rationale (Z)</td></tr><tr><td>Question: 7 + 3 + 4 - 9 + 9 =? Answer:</td><td>7 + 3 = 10, 10 - 9 = 1, 1 + 4 = 5</td></tr><tr><td>Question: 8 + 5 - 1 - 4 + 5 =? Answer:</td><td>8 + 5 = 13, 13 - 1 = 12, 12 + 4 = 16, 16 - 4 = 12</td></tr><tr><td>Question: 0 + 5 - 6 - 1 - 6 =? Answer:</td><td>0 + 5 = 5, 5 - 6 = -1, -1 - 6 = -7, -7 + 1 = -6</td></tr></table>",
+        "bbox": [
+            215,
+            686,
+            777,
+            797
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "page_number",
+        "text": "31",
+        "bbox": [
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+        "page_idx": 30
+    }
+]
\ No newline at end of file
diff --git a/2024/Amortizing intractable inference in large language models/7ed5418b-3773-431a-ac5f-3cceb14f6956_model.json b/2024/Amortizing intractable inference in large language models/7ed5418b-3773-431a-ac5f-3cceb14f6956_model.json
new file mode 100644
index 0000000000000000000000000000000000000000..024491f24086891e8482adc74dadee5e09a2705f
--- /dev/null
+++ b/2024/Amortizing intractable inference in large language models/7ed5418b-3773-431a-ac5f-3cceb14f6956_model.json	
@@ -0,0 +1,4332 @@
+[
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.102,
+                0.655,
+                0.147
+            ],
+            "angle": 0,
+            "content": "AMORTIZING INTRACTABLE INFERENCE IN LARGE LANGUAGE MODELS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.182,
+                0.169,
+                0.506,
+                0.184
+            ],
+            "angle": 0,
+            "content": "Edward J. Hu*, Moksh Jain*, Eric Elmoznino"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.182,
+                0.184,
+                0.521,
+                0.197
+            ],
+            "angle": 0,
+            "content": "Mila - Quebec AI Institute, Université de Montréal"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.182,
+                0.199,
+                0.536,
+                0.212
+            ],
+            "angle": 0,
+            "content": "{edward.hu,moksh.jain,eric.elmoznino,..."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.556,
+                0.169,
+                0.693,
+                0.184
+            ],
+            "angle": 0,
+            "content": "Younesse Kaddar"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.556,
+                0.184,
+                0.697,
+                0.198
+            ],
+            "angle": 0,
+            "content": "University of Oxford"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.556,
+                0.199,
+                0.815,
+                0.212
+            ],
+            "angle": 0,
+            "content": "younesse.kaddar@chch.ox.ac.uk"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.182,
+                0.232,
+                0.551,
+                0.246
+            ],
+            "angle": 0,
+            "content": "Guillaume Lajoie†, Yoshua Bengio°, Nikolay Malkin"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.182,
+                0.247,
+                0.521,
+                0.26
+            ],
+            "angle": 0,
+            "content": "Mila - Quebec AI Institute, Université de Montréal"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.185,
+                0.261,
+                0.731,
+                0.275
+            ],
+            "angle": 0,
+            "content": "...,guillaume.lajoie,yoshua.bengio,nikolay.malkin}@mila.quebec"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.451,
+                0.311,
+                0.547,
+                0.325
+            ],
+            "angle": 0,
+            "content": "ABSTRACT"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.23,
+                0.339,
+                0.77,
+                0.514
+            ],
+            "angle": 0,
+            "content": "Autoregressive large language models (LLMs) compress knowledge from their training data through next-token conditional distributions. This limits tractable querying of this knowledge to start-to-end autoregressive sampling. However, many tasks of interest—including sequence continuation, infilling, and other forms of constrained generation—involve sampling from intractable posterior distributions. We address this limitation by using amortized Bayesian inference to sample from these intractable posteriors. Such amortization is algorithmically achieved by fin-tuning LLMs via diversity-seeking reinforcement learning algorithms: generative flow networks (GFlowNets). We empirically demonstrate that this distribution-matching paradigm of LLM fine-tuning can serve as an effective alternative to maximum-likelihood training and reward-maximizing policy optimization. As an important application, we interpret chain-of-thought reasoning as a latent variable modeling problem and demonstrate that our approach enables data-efficient adaptation of LLMs to tasks that require multi-step rationalization and tool use."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.23,
+                0.513,
+                0.628,
+                0.527
+            ],
+            "angle": 0,
+            "content": "Code: https://github.com/GFNOrg/gfn-lm-tuning."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.174,
+                0.545,
+                0.338,
+                0.559
+            ],
+            "angle": 0,
+            "content": "1 INTRODUCTION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.569,
+                0.828,
+                0.652
+            ],
+            "angle": 0,
+            "content": "Autoregressive large language models (LLMs) trained on general-domain data are vast stores of world knowledge (Petroni et al., 2019). They are typically optimized by predicting a token given its preceding context; therefore, tractable inference over this knowledge is limited to sampling conditioned on a prefix. Many useful tasks, such as infilling (Zhu et al., 2019; Liu et al., 2019), generating text conditioned on length or lexical constraints (Hokamp & Liu, 2017; Hu et al., 2019), and finding the most likely sequence continuation, involve intractable inference in LLMs."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.658,
+                0.825,
+                0.74
+            ],
+            "angle": 0,
+            "content": "Such tasks are related to the problem of reasoning, which has been framed as one of probabilistic inference (Gershman & Goodman, 2014). Correspondingly, the linguistic expression of reasoning can be seen as inference over language. For example, we can interpret chain-of-thought reasoning (Wei et al., 2022; Kojima et al., 2022), a paradigm of reasoning in language models, as a problem of intractable posterior inference. Given a question-answer pair \\((X,Y)\\), we are interested in finding latent chains of thought - token sequences \\(Z\\) that contribute the most to the conditional likelihood"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.292,
+                0.743,
+                0.826,
+                0.775
+            ],
+            "angle": 0,
+            "content": "\\[\np (Y \\mid X) = \\sum_ {Z} p _ {\\mathrm {L M}} (Z Y \\mid X) = \\sum_ {Z} p _ {\\mathrm {L M}} (Y \\mid X Z) p _ {\\mathrm {L M}} (Z \\mid X), \\tag {1}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.779,
+                0.825,
+                0.807
+            ],
+            "angle": 0,
+            "content": "where \\( p_{\\mathrm{LM}} \\) denotes the likelihood assigned to a sequence by a language model and apposition of variables (e.g., XZY) denotes the concatenation of the token sequences."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.813,
+                0.825,
+                0.903
+            ],
+            "angle": 0,
+            "content": "While past work has relied on prompting and in-context learning to produce \\( Z \\)'s that lead to the correct \\( Y \\), treating \\( Z \\) as a hidden variable in a latent variable model (LVM) renders chain-of-thought reasoning a Bayesian inference problem (Fig. 1). For this LVM, the distribution we must sample from is the posterior \\( p_{\\mathrm{LM}}(Z \\mid X, Y) = \\frac{p_{\\mathrm{LM}}(XZY)}{\\sum_{Z'} p_{\\mathrm{LM}}(XZ'Y)} \\). Such sampling is intractable: while it is easy to evaluate \\( p_{\\mathrm{LM}}(XYZ) \\), the conditional distributions needed to sample \\( Z \\) from \\( p_{\\mathrm{LM}}(Z \\mid X, Y) \\) one token at a time are not easy to compute."
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.191,
+                0.91,
+                0.805,
+                0.925
+            ],
+            "angle": 0,
+            "content": "*Equal contribution. *Work done during internship at Mila. †CIFAR AI Chair. °CIFAR Senior Fellow."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
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+            "angle": 0,
+            "content": "1"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
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+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.172,
+                0.089,
+                0.825,
+                0.196
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.17,
+                0.205,
+                0.828,
+                0.294
+            ],
+            "angle": 0,
+            "content": "Figure 1: Left: Three problems of reasoning in language - sentence infilling, chain-of-thought reasoning, and problem-solving with external tool use - can all be seen as instances of the latent variable model at the top left, where an input \\((X)\\) generates the output \\((Y)\\) via a latent variable \\((Z)\\). Right: We fine-tune an LLM to sample from the Bayesian posterior over \\(Z\\), conditioning on \\(X\\) and optionally on \\(Y\\). If conditioned on \\(Y\\), the trained policy can be used to sample diverse latent sequences (e.g., for infilling, §4.2). If not conditioned on \\(Y\\), the policy can sample \\(Z\\), and thus predict \\(Y\\), for inputs \\(X\\) not seen during training (e.g., for classification and multi-step reasoning, §4.3, 4.4). As shown in §4.4, modeling the full diversity of the posterior aids generalization."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.302,
+                0.828,
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+            ],
+            "angle": 0,
+            "content": "A standard method to sample approximately from intractable posterior distributions is Markov chain Monte Carlo (MCMC), but it is difficult to craft good proposal distributions for multi-modal distributions over language data (Miao et al., 2019; Zhang et al., 2020a; Lew et al., 2023), and inference on a new input may be prohibitively slow. Alternatively, one can turn to reinforcement learning (RL) approaches such as proximal policy optimization (PPO; Schulman et al., 2017), where the language model is treated as a policy to be fine-tuned. However, these do not aim to model the full diversity of the distribution; instead, learned policies settle around a small number of modes. In both cases, issues with this mode collapse are exacerbated when the target distribution is misspecified, leading to the undesirable behavior of overoptimized samplers (Gao et al., 2023)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
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+            ],
+            "angle": 0,
+            "content": "Amortized probabilistic inference – that is, training a model to approximate a distribution of interest – provides a principled, efficient, and potentially scalable way to draw samples from the distribution (Beal, 2003). One way to implement amortized inference for high-dimensional discrete data such as text is using generative flow networks (GFlowNets; Bengio et al., 2021), which are diversity-seeking reinforcement learning algorithms that train policies to sample objects (such as a token sequence \\( Z \\)) with probability proportional to a given reward function, such as the joint \\( p_{\\mathrm{LM}}(XZY) \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
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+            ],
+            "angle": 0,
+            "content": "In this work, we present a method that initializes the GFlowNet policy with a pretrained LLM and continues to train it with a reward objective that can be evaluated with the same LLM. The result is a different type of fine-tuning (FT) procedure for text generation that has a number of advantages, including improved sample diversity, data efficiency, and out-of-distribution generalization. GFlowNet fine-tuning makes the language model sample from the target distribution, enabling amortized inference in a number of applications (Fig. 1)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.608,
+                0.828,
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+            ],
+            "angle": 0,
+            "content": "Leveraging this approach, we empirically demonstrate the possibilities and benefits of learning to sample from intractable distributions over text continuations, latent reasoning chains, and tool use sequences using GFlowNet fine-tuning. Notably, the diversity of samples from the models trained with GFlowNet fine-tuning is beneficial in Bayesian model averaging settings, such as when aggregating answers to questions obtained via multiple reasoning chains. For example, using a pretrained language model with 6B parameters, our method shows an absolute improvement of \\(10.9\\%\\) over supervised fine-tuning on subjectivity classification with only 10 labeled examples (§4.3) and outperforms supervised fine-tuning and PPO by \\(63\\%\\) on integer arithmetic with 50 demonstrations, with notable improvements in out-of-distribution generalization (§4.4). Moreover, the benefits of amortized inference allow us to efficiently sample from the fine-tuned model at scale. Our contributions include:"
+        },
+        {
+            "type": "text",
+            "bbox": [
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+            "angle": 0,
+            "content": "(1) A general algorithm for amortized sampling from intractable LLM posteriors."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.766,
+                0.759,
+                0.78
+            ],
+            "angle": 0,
+            "content": "(2) A probabilistic approach to fine-tuning LLMs to perform chain-of-thought reasoning."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.78,
+                0.825,
+                0.807
+            ],
+            "angle": 0,
+            "content": "(3) Empirical results on sequence continuation, natural language reasoning, integer arithmetic with tool use, and story infilling."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.172,
+                0.752,
+                0.825,
+                0.807
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "title",
+            "bbox": [
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+            "angle": 0,
+            "content": "2 MOTIVATING EXAMPLE: GENERATING RANDOM NUMBERS WITH LLMS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.842,
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+            ],
+            "angle": 0,
+            "content": "We consider a simple task that highlights the limitations of reward-maximizing reinforcement learning (RL) methods in fine-tuning LLMs. For readers unfamiliar with RL, we refer to Sutton & Barto (2018) and include a glossary in §A.1 to define key terms used throughout this paper. The task involves generating random numbers from a given distribution when prompted 'The following is a random integer drawn uniformly between 0 and 100:'. This task is a minimal instantiation of the problem we study in the paper: sample from a target distribution given an unnormalized density."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
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+                0.96
+            ],
+            "angle": 0,
+            "content": "2"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.145,
+                0.089,
+                0.825,
+                0.265
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.235,
+                0.273,
+                0.334,
+                0.286
+            ],
+            "angle": 0,
+            "content": "(a) Base model"
+        },
+        {
+            "type": "image_footnote",
+            "bbox": [
+                0.229,
+                0.286,
+                0.34,
+                0.31
+            ],
+            "angle": 0,
+            "content": "\\(50.5\\%\\) of samples are valid numbers."
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.437,
+                0.273,
+                0.562,
+                0.287
+            ],
+            "angle": 0,
+            "content": "(b) PPO fine-tuning"
+        },
+        {
+            "type": "image_footnote",
+            "bbox": [
+                0.444,
+                0.287,
+                0.555,
+                0.31
+            ],
+            "angle": 0,
+            "content": "\\(95.8\\%\\) of samples are valid numbers."
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.633,
+                0.273,
+                0.794,
+                0.287
+            ],
+            "angle": 0,
+            "content": "(c) GFlowNet fine-tuning"
+        },
+        {
+            "type": "image_footnote",
+            "bbox": [
+                0.658,
+                0.287,
+                0.769,
+                0.31
+            ],
+            "angle": 0,
+            "content": "\\(100\\%\\) of samples are valid numbers."
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.177,
+                0.323,
+                0.82,
+                0.35
+            ],
+            "angle": 0,
+            "content": "Figure 2: Empirical distributions of 512,000 integers from 1 to 100 generated by GPT-J fine-tuned with PPO (reward-maximizing; b) and GFlowNet fine-tuning (distribution-matching; c). Note the logarithmic \\(y\\)-scale."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.357,
+                0.825,
+                0.386
+            ],
+            "angle": 0,
+            "content": "Although the target distribution is tractable, making the task seemingly straightforward, it serves as a useful illustration of the behaviors of different fine-tuning methods."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.391,
+                0.825,
+                0.475
+            ],
+            "angle": 0,
+            "content": "Renda et al. (2023) found that pretrained LLMs perform quite poorly on this task: the distribution of numbers generated with the above prompt will be far from uniform (Fig. 2a shows an example using an instruction fine-tuned GPT-J 6B (Wang & Komatsuzaki, 2021)\\(^{1}\\)). There may be many reasons for this, among them the effects of instruction fine-tuning and the LLM's possible bias towards numbers that are more frequent in the training data (e.g., numbers starting with '1' are more frequent due to the properties of many natural data-generating processes (Benford, 1938))."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.48,
+                0.827,
+                0.55
+            ],
+            "angle": 0,
+            "content": "While reward-maximizing RL can teach the model to generate valid numbers (by penalizing outputs that are not numbers from 1 to 100), it would not resolve the distribution skew introduced during pretraining. Indeed, rewarding all valid integers equally leads to an expected gradient of zero for policy gradient methods. Fig. 2b shows that while most samples are valid numbers after PPO training, the distribution remains highly skewed."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.555,
+                0.827,
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+            ],
+            "angle": 0,
+            "content": "Instead, we can take a principled approach by training the LLM to match the target distribution with a GFlowNet learning objective. Such an objective directly optimizes the likelihood of the model generating a number to be proportional to the reward for that number, which is the number's (potentially unnormalized) probability under the target distribution. When the policy is initialized as the pretrained LLM, the resulting distribution after GFlowNet fine-tuning is shown in Fig. 2c. Quantitatively, the KL divergence from the sampling distribution to the target (uniform) distribution decreases from 3.37 for the original LLM (on the support [0, 100]) to \\(9.75 \\cdot 10^{-5}\\) for the GFlowNet-fine-tuned model."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.657,
+                0.828,
+                0.741
+            ],
+            "angle": 0,
+            "content": "This example illustrates a general point: GFlowNet objectives provide a principled and flexible approach to fine-tuning LLMs to match a target distribution where reward-maximizing RL fails to. On this simple task, this distribution matching could also be achieved through supervised fine-tuning; however, this would require access to samples from the target distribution, which are unavailable in general (though not in this simple example). In the following sections, we further illustrate this point in non-trivial problems involving intractable inference, reasoning with latent variables, and tool use."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.75,
+                0.791,
+                0.767
+            ],
+            "angle": 0,
+            "content": "3 FINE-TUNING LLMS TO SAMPLE FROM INTRACTABLE DISTRIBUTIONS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.774,
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+            "angle": 0,
+            "content": "We first describe how intractable inference emerges from interesting applications of LLMs, one of which is chain-of-thought reasoning seen through the lens of latent variable models, where the posterior distribution over the latent variable is intractable. We then discuss how GFlowNet objectives can be used to train amortized samplers to perform such intractable inference."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.837,
+                0.692,
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+            ],
+            "angle": 0,
+            "content": "3.1 PROBLEM: INTRACTABLE INFERENCE IN LARGE LANGUAGE MODELS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.858,
+                0.827,
+                0.902
+            ],
+            "angle": 0,
+            "content": "Autoregressive language models decompose the distribution over sequences of tokens as a product of ordered conditionals: \\( p(w_{1:N}) = p(w_1)p(w_2 \\mid w_1) \\cdots p(w_N \\mid w_{1:N-1}) \\). While this decomposition makes left-to-right sampling from the distribution tractable, sampling from other conditional distribu"
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.191,
+                0.91,
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+            ],
+            "angle": 0,
+            "content": "<sup>1</sup>We use the Instruct-GPT-J model available at <sup>2</sup>.co/nlpcloud/instruct-gpt-j-fp16."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
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+            ],
+            "angle": 0,
+            "content": "3"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.171,
+                0.089,
+                0.825,
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+            ],
+            "angle": 0,
+            "content": "Table 1: Objects in language posterior inference. Given a pretrained 'teacher' LM \\( p_{\\mathrm{LM}} \\), we train a GFlowNet \\( q_{\\mathrm{GFN}} \\) to sample the posterior \\( p(Z \\mid X, Y) \\). Amortization and generalization are achieved by making \\( X \\), and optionally \\( Y \\), an input to \\( q_{\\mathrm{GFN}} \\)."
+        },
+        {
+            "type": "table",
+            "bbox": [
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+            "angle": 0,
+            "content": "<table><tr><td>Object</td><td>Meaning</td><td>Example 1 (infilling)</td><td>Example 2 (subjectivity classification)</td></tr><tr><td>X</td><td>cause / condition / question</td><td>The cat was hungry.</td><td>A deeply moving storyline.</td></tr><tr><td>Z</td><td>mechanism / reasoning chain</td><td>She ate a mouse.</td><td>This review expresses personal feelings.</td></tr><tr><td>Y</td><td>effect / answer</td><td>Now the cat is sleepy, not hungry.</td><td>Answer: Subjective</td></tr><tr><td>p(Z | X)</td><td>conditional prior</td><td></td><td>pLM(Z | X)</td></tr><tr><td>p(Y | X, Z)</td><td>likelihood of effect given</td><td></td><td>pLM(Y | XZ)</td></tr><tr><td></td><td>cause and mechanism</td><td></td><td></td></tr><tr><td>p(Z, Y | X)</td><td>conditional joint, reward for Z</td><td></td><td>pLM(ZY | X)</td></tr><tr><td>p(Z | X, Y)</td><td>posterior (intractable!)</td><td colspan=\"2\">approximated and amortized by GFlowNet qGFN(Z | X), Y)</td></tr><tr><td>q(Y | X)</td><td>posterior predictive /</td><td colspan=\"2\">approximated as ∑Z qGFN(Z | X)pLM(Y | XZ),</td></tr><tr><td></td><td>Bayesian model average</td><td colspan=\"2\">sampled as Z ~ qGFN(Z | X), Y ~ pLM(Y | XZ)</td></tr></table>"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.259,
+                0.825,
+                0.302
+            ],
+            "angle": 0,
+            "content": "tions is intractable. Various problems of language modeling can be viewed as sampling from such intractable conditionals in the distribution over sequences of an LLM; we give two such examples and related terminologies in Table 1. Some tasks we study in §4 are instances of these examples."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.307,
+                0.827,
+                0.448
+            ],
+            "angle": 0,
+            "content": "Tempered and contrastive sampling. In many applications (e.g., translation, summarization, dialogue systems), one wishes to sample from a low-temperature distribution over sequences \\( Z \\) conditioned on a prefix \\( X \\), i.e., \\( q(Z \\mid X) \\propto p_{\\mathrm{LM}}(XZ)^{1 / T} \\) for some temperature \\( T < 1 \\), as high-likelihood samples are more likely to be fluent or accurate continuations of \\( X \\) (Tillmann & Ney, 2003). The limit of \\( T \\to 0 \\) gives a distribution that is peaky on the most likely continuation. However, sampling from \\( q \\), or finding its mode, is intractable, and it is common to resort to approximations, such as tempering the tokenwise conditional distributions or using beam search to search for a mode. A related problem is sampling a continuation with a correction for its unconditional likelihood, e.g., \\( q(Z \\mid X) \\propto p_{\\mathrm{LM}}(XZ)^{\\alpha}p_{\\mathrm{LM}}(Z)^{\\beta} \\) with \\( \\beta < 0 \\) and \\( \\alpha > 0 \\), where applications again resort to approximating the next-token conditionals of \\( q \\) by tempering (Malkin et al., 2022b; Li et al., 2023)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.452,
+                0.828,
+                0.577
+            ],
+            "angle": 0,
+            "content": "Infilling and reverse generation. Infilling is the task of sampling a sequence of tokens conditioned on both its prior and subsequent context, which can be understood as sampling from the distribution \\( q(Z \\mid X, Y) \\propto p_{\\mathrm{LM}}(XZY) \\), where \\( X \\) and \\( Y \\) are fixed. Reverse generation is a special case, where \\( X \\) is an empty sequence. Besides being a meaningful task in its own right (Liu et al., 2019; Zhu et al., 2019; Donahue et al., 2020; Susanto et al., 2020; Lu et al., 2022a), infilling and reverse generation are key components of newly emerging methods of LLM prompting, such as when LLMs are tasked with optimizing their own instruction sequences or reasoning steps (Zhou et al., 2023; Sordoni et al., 2023; Xu et al., 2023). Current applications achieve this by resorting to hand-engineered instructions and inverted prompts."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.581,
+                0.825,
+                0.679
+            ],
+            "angle": 0,
+            "content": "Constrained generation. Sampling of text with constraints and penalties – for example, those on the presence or the absence of certain words or on the score of an auxiliary classifier evaluated on the text – can be understood as sampling from a distribution \\( q(Z) \\propto p_{\\mathrm{LM}}(Z)c(Z) \\), where \\( c \\) is an externally specified constraint. Current approaches to the problem use tokenwise approximations (Liu et al., 2021), various problem-specific beam search and local search techniques (e.g., Schmaltz et al., 2016; Hokamp & Liu, 2017; Hu et al., 2019; Sha, 2020; Lu et al., 2022b) or classifier-guided conditional generation approaches (e.g., Yang & Klein, 2021; Meng et al., 2022)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
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+                0.699
+            ],
+            "angle": 0,
+            "content": "3.2 REASONING THROUGH LATENT VARIABLES"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.706,
+                0.827,
+                0.79
+            ],
+            "angle": 0,
+            "content": "Chain-of-thought reasoning (Wei et al., 2022; Kojima et al., 2022) helps LLMs solve complex problems by producing a reasoning chain before giving the final answer. LLMs pretrained on general domain data can learn to produce useful chains of thoughts given demonstrations, which are usually handcrafted or generated by prompting the LM. Interestingly, although the capacity for chain-of-thought reasoning only emerges in large language models, knowledge can also be extracted from smaller language models when they are carefully fine-tuned (Schick & Schütze, 2021)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.795,
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+            ],
+            "angle": 0,
+            "content": "Motivated by this, we connect chain-of-thought reasoning to the general problem of inference in latent variable models illustrated in Fig. 1. Here, reasoning can be seen as posterior inference: sampling from the posterior distribution over a string of tokens \\( Z \\) conditioned on a prefix \\( X \\) and a suffix \\( Y \\), given an autoregressive language model \\( p_{\\mathrm{LM}} \\). The posterior is defined as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.331,
+                0.858,
+                0.826,
+                0.89
+            ],
+            "angle": 0,
+            "content": "\\[\np _ {\\mathrm {L M}} (Z \\mid X, Y) = \\frac {p _ {\\mathrm {L M}} (X Z Y)}{\\sum_ {Z ^ {\\prime}} p _ {\\mathrm {L M}} \\left(X Z ^ {\\prime} Y\\right)} \\propto p _ {\\mathrm {L M}} (X Z Y). \\tag {2}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
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+            ],
+            "angle": 0,
+            "content": "Our goal is to train models to sample \\( Z \\) from this posterior distribution. Intuitively, this allows us to sample likely reasoning chains that lead to the desired outcome \\( Y \\). Although we take \\( Z \\) to be a string"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
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+            "angle": 0,
+            "content": "4"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.104,
+                0.827,
+                0.175
+            ],
+            "angle": 0,
+            "content": "of tokens, the same formalism and the GFlowNet objectives apply to other structured latent objects, such as trees or sets of natural language statements, as long as one has access to a likelihood model \\( p(Y \\mid XZ) \\). While not investigated in this work, these generalizations could be important for formal reasoning and multi-step chains of inference. See, e.g., Yao et al. (2023); Hao et al. (2023); Besta et al. (2024) for approaches to reasoning in language using tree- or list-structured state spaces."
+        },
+        {
+            "type": "text",
+            "bbox": [
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+                0.179,
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+            ],
+            "angle": 0,
+            "content": "A latent variable model of this form is useful when the marginal distribution \\( p_{\\mathrm{LM}}(Y \\mid X) \\) is harder to model than \\( p_{\\mathrm{LM}}(Z \\mid X) \\) and \\( p_{\\mathrm{LM}}(Y \\mid XZ) \\), i.e., a difficult inference is broken down into a chain of easier ones. By training a model to match the Bayesian posterior \\( p_{\\mathrm{LM}}(Z \\mid X, Y) \\), we can learn to sample latent reasoning chains that increase the likelihood of producing \\( Y \\) from \\( X \\) via the sampled \\( Z \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.24,
+                0.828,
+                0.392
+            ],
+            "angle": 0,
+            "content": "However, we can also fine-tune the language model \\( p_{\\mathrm{LM}}(Z \\mid XY) \\) itself to maximize the likelihood of data pairs \\( (X,Y) \\) under the LVM. While it is generally intractable to directly maximize the data likelihood \\( p_{\\mathrm{LM}}(X,Y) = \\sum_Z p_{\\mathrm{LM}}(XZY) \\) because of the summation over \\( Z \\), the (variational) expectation-maximization (EM) algorithm (Dempster et al., 1977; Beal, 2003; Koller & Friedman, 2009) can be used for this purpose. In the expectation step (E-step), we draw samples from the posterior over the latent variable \\( p_{\\mathrm{LM}}(Z \\mid X,Y) \\), which could come from an amortized sampler of \\( Z \\). In the maximization step (M-step), we maximize the log-likelihood of the joint probability of the sampled latent variables \\( \\mathbb{E}_{Z \\sim p_{\\mathrm{LM}}(Z|X,Y)} \\log p_{\\mathrm{LM}}(XZY) \\) with respect to the parameters of the language model \\( p_{\\mathrm{LM}} \\). This combination of amortized inference (learning to sample the chain of thought) and supervised fine-tuning (optimizing the language model with the 'supervision' involving \\( Z \\) sampled from the amortized posterior) will be illustrated in one of our experiments (§4.3, Table 3)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.399,
+                0.604,
+                0.413
+            ],
+            "angle": 0,
+            "content": "3.3 AMORTIZED INFERENCE WITH GFLOWNET OBJECTIVES"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.42,
+                0.827,
+                0.532
+            ],
+            "angle": 0,
+            "content": "For inference in the latent variable model, we leverage the probabilistic framework of generative flow networks (GFlowNets; Bengio et al., 2021; 2023). Using notation from Malkin et al. (2022a), we briefly introduce relevant GFlowNet concepts pertaining to autoregressive sequence generation. Here, GFlowNets learn policies to sample sequences \\( Z = z_{1}z_{2}\\dots z_{n}\\top \\in \\mathcal{Z} \\) (where \\( \\top \\) denotes a stop symbol) from a distribution over the space of sequences \\( \\mathcal{Z} \\), given an unnormalized density (reward) \\( R: \\mathcal{Z} \\to \\mathbb{R}_{>0} \\). The generative process is the same as in autoregressive language models: generation begins with an empty string, and at the \\( i \\)-th step a token \\( z_{i} \\) is sampled from a policy \\( q_{\\mathrm{GFN}}(z_i \\mid z_{1:i-1}) \\) which is then appended to the sequence. This process continues until a stop symbol \\( \\top \\) is generated."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.536,
+                0.826,
+                0.593
+            ],
+            "angle": 0,
+            "content": "The marginal likelihood \\( q_{\\mathrm{GFN}}^{\\top}(Z) \\) of sampling a terminal state \\( Z = z_{1:n}\\top \\) is given by \\( \\prod_{i = 1}^{n}q_{\\mathrm{GFN}}(z_i|z_{1:i - 1})q_{\\mathrm{GFN}}(\\top \\mid z) \\), where \\( z_{1:0} \\) is understood to be the empty string. The goal of GFlowNet training is to fit a parametric policy \\( q_{\\mathrm{GFN}}(\\cdot |\\cdot ;\\theta) \\) such that \\( q_{\\mathrm{GFN}}^{\\top}(Z)\\propto R(Z) \\), i.e., the likelihood of generating a complete sequence is proportional to its reward."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.598,
+                0.827,
+                0.641
+            ],
+            "angle": 0,
+            "content": "Learning objective. We use a modified version of the subtrajectory balance (SubTB; Madan et al., 2023) objective to account for trajectories being terminable at all states (Deleu et al., 2022). The objective for a sequence \\( Z = z_{1:n} \\top \\) is"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.244,
+                0.645,
+                0.826,
+                0.69
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathcal {L} (Z; \\theta) = \\sum_ {0 \\leq i <   j \\leq n} \\left(\\log \\frac {R \\left(z _ {1 : i} \\top\\right) \\prod_ {k = i + 1} ^ {j} q _ {\\mathrm {G F N}} \\left(z _ {k} \\mid z _ {1 : k - 1}\\right) q _ {\\mathrm {G F N}} (\\top \\mid z _ {1 : j})}{R \\left(z _ {1 : j} \\top\\right) q _ {\\mathrm {G F N}} (\\top \\mid z _ {1 : i})}\\right) ^ {2}, \\tag {3}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.706,
+                0.827,
+                0.749
+            ],
+            "angle": 0,
+            "content": "For sequence generation tasks, the SubTB objective is equivalent to the path consistency objective (Nachum et al., 2017) in max-entropy RL (Haarnoja et al., 2017), which has been previously used in the context of text generation (Guo et al., 2021). See Appendix A.2 for further discussion."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.753,
+                0.826,
+                0.864
+            ],
+            "angle": 0,
+            "content": "Training policy. As the objective in Eq. 3 can be minimized to 0 for all trajectories \\(\\tau\\) simultaneously given enough model capacity, we can use trajectories sampled from any full-support distribution (training policy) to perform gradient descent on \\(\\mathcal{L}(\\tau; \\theta)\\) with respect to \\(\\theta\\). As the space we are sampling from is combinatorially large, it is important to have a training policy that can efficiently explore \\(\\mathcal{Z}\\). To this end, we compose the mini-batch during training using trajectories from three sources: (1) the policy \\(q_{\\mathrm{GFN}}\\), (2) a tempered version of the current policy \\(q_{\\mathrm{GFN}}\\) and (3) a replay buffer storing past trajectories. Replay buffers have been shown to be quite effective in improving GFlowNet training (Jain et al., 2022; Deleu et al., 2022; Shen et al., 2023)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.869,
+                0.826,
+                0.926
+            ],
+            "angle": 0,
+            "content": "Parametrization, amortization, and generalization. To sample the latent sequence \\( Z \\) from the posterior defined in Eq. 2, we parametrize the GFlowNet policy as an autoregressive language model that samples the latent \\( Z \\) one token at a time from left to right. By setting the reward \\( R(Z) = p_{\\mathrm{LM}}(XZY) \\propto p_{\\mathrm{LM}}(Z \\mid X,Y) \\), we learn a sampler for the posterior at convergence."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.505,
+                0.96
+            ],
+            "angle": 0,
+            "content": "5"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.104,
+                0.827,
+                0.203
+            ],
+            "angle": 0,
+            "content": "As illustrated in Fig. 1, depending on the task, we can condition the GFlowNet policy on either \\( X \\) or \\( X, Y \\). In cases such as reasoning (§3.2), where there is only a single correct \\( Y \\) for each \\( X \\) and we interested in predicting \\( Y \\) for unseen \\( X \\) at test time, we can simply condition on \\( X \\). In this case, the GFlowNet policy is simply a language model that generates \\( Z \\) as a continuation of \\( X \\). To be precise, we initialize \\( q_{\\mathrm{GFN}} \\) as a copy of \\( p_{\\mathrm{LM}} \\) that is conditioned on the prefix \\( X \\), and then fine-tune<sup>2</sup> it with a GFlowNet objective. With this view, sampling \\( Z \\) is an inverse problem: we need to infer \\( Z \\) given a (conditional) prior \\( p_{\\mathrm{LM}}(Z \\mid X) \\) and an observation \\( Y \\) under likelihood model \\( p_{\\mathrm{LM}}(Y \\mid XZ) \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.206,
+                0.828,
+                0.29
+            ],
+            "angle": 0,
+            "content": "Allowing the GFlowNet policy to explicitly take \\( X \\) as input amortizes the sampling procedure and allows generalization to unseen \\( X \\). In this sense, the GFlowNet is a Bayesian model (akin to a LM cascade (Dohan et al., 2022) or deep language network (Sordoni et al., 2023)), in which \\( Z \\) are conditionally sampled 'parameters' that transform \\( X \\) into \\( Y \\). To predict the \\( Y \\) for an unseen \\( X \\), one performs Bayesian model averaging by drawing samples of \\( Z \\) from \\( q_{\\mathrm{GFN}}(Z \\mid X) \\) followed by sampling from \\( p_{\\mathrm{LM}}(Y \\mid XZ) \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.295,
+                0.828,
+                0.352
+            ],
+            "angle": 0,
+            "content": "In tasks such as infilling (§4.2), however, the mapping from \\( X \\) to \\( Y \\) is one-to-many and \\( Y \\) is available at test-time. Here, we are interested in \\( Z \\) itself, rather than using it as an intermediate variable en route to generating \\( Y \\). The GFlowNet policy thus has to be conditioned on both \\( X \\) and \\( Y \\). To achieve this, the policy is conditioned on a prompt that contains both \\( X \\) and \\( Y \\) (for example, see Appendix C)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.361,
+                0.382,
+                0.377
+            ],
+            "angle": 0,
+            "content": "4 EMPIRICAL RESULTS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.385,
+                0.825,
+                0.428
+            ],
+            "angle": 0,
+            "content": "We first validate GFlowNet fine-tuning on text generation, where we seek to find likely sentence continuation given a prompt (§4.1) or fill in a missing sentence in a story (§4.2). Then, we study reasoning tasks that benefit from chain-of-thought reasoning (§4.3) and external tool use (§4.4)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.435,
+                0.402,
+                0.448
+            ],
+            "angle": 0,
+            "content": "4.1 SENTENCE CONTINUATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.455,
+                0.487,
+                0.594
+            ],
+            "angle": 0,
+            "content": "Task description. A natural application for autoregressive language models is that of sequence continuation: given a prompt, the model should generate a high-likelihood completion. In applications such as creative writing, we would like the continuations to be semantically diverse while still having a high likelihood under the language model. To demonstrate the benefits of GFlowNet fine-tuning, we consider the task of sampling the next sentence following a prompt."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.599,
+                0.487,
+                0.709
+            ],
+            "angle": 0,
+            "content": "Sampling autoregressively from the LM until a “.” token is reached is unlikely to produce samples that have a high likelihood because the distribution over sentences has a fat tail. Existing approaches to generate sequence continuations include beam search and its variations (Vijayakumar et al., 2018; Shao et al., 2017), top- \\(k\\) sampling (Fan et al., 2018), nucleus sampling"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.709,
+                0.827,
+                0.835
+            ],
+            "angle": 0,
+            "content": "(Holtzman et al., 2019), tempered autoregressive sampling, and fine-tuning using importance sampling (Shih et al., 2023), among others. While useful, most of these methods are ultimately hand-crafted heuristics that leave room for improvement. Furthermore, some of these methods (e.g., beam search) involve a computationally expensive search procedure, compared to a single pass of a learned inference model that amortizes over prompts. Our GFlowNet policy autoregressively samples the sequence until a period is sampled, indicating the end of the sentence. Given prompts \\( X \\), the LM is fine-tuned to generate the continuations \\( Z \\) from the tempered posterior by being trained with reward \\( R(Z) = p_{\\mathrm{LM}}(Z|X)^{\\frac{1}{T}} \\). When \\( T = 1 \\), the GFlowNet will trivially sample proportional to \\( p_{\\mathrm{LM}} \\) without any fine-tuning, so we consider \\( 0 < T < 1 \\) to focus on the likely continuations."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.84,
+                0.828,
+                0.898
+            ],
+            "angle": 0,
+            "content": "We consider a dataset of prompts from OpenWebText (Gokaslan et al., 2019) with a 1.5B param GPT-2 XL (Radford et al., 2019) as the base model. We draw 8 samples from the fine-tuned model conditioned on a fixed prompt, consider the maximum-likelihood sample under the LM, and report the average over the dataset of prompts. To measure the semantic diversity of the"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.517,
+                0.445,
+                0.804,
+                0.634
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.495,
+                0.636,
+                0.828,
+                0.701
+            ],
+            "angle": 0,
+            "content": "Figure 3: Maximum log-likelihood and diversity of continuations sampled for fixed prompts. GFlowNet finetuning \\((\\star)\\) samples higher log-likelihood sentences while maintaining more sample diversity than the baselines (and ---), even when they are given \\(5\\times\\) the compute."
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.191,
+                0.91,
+                0.792,
+                0.925
+            ],
+            "angle": 0,
+            "content": "2We use LoRA (Hu et al., 2022) instead of full fine-tuning for hardware efficiency in all experiments."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.505,
+                0.96
+            ],
+            "angle": 0,
+            "content": "6"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.104,
+                0.825,
+                0.175
+            ],
+            "angle": 0,
+            "content": "samples, we compute the mean pairwise cosine distance between the embeddings (from a pretrained encoder (Reimers & Gurevych, 2019)) of the generated samples and average it over the dataset. We compare to baselines that are commonly used for producing continuations from LMs at inference time (beam search, diverse beam search, nucleus sampling, autoregressive sampling, tempered autoregressive sampling, and greedy generation)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.179,
+                0.827,
+                0.263
+            ],
+            "angle": 0,
+            "content": "Results. Quantitative results are reported in Fig. 3 and empirical samples are shown in Appendix B. At lower temperatures, our method excels in generating high-likelihood sentences, outperforming the leading baseline, diverse beam search. If we increase the number of beams (and therefore compute) to \\(5 \\times\\) the number of samples produced by the GFlowNet, our performance remains comparable. Nevertheless, even in this scenario, the GFlowNet's generated samples exhibit notably higher diversity compared to diverse beam search and are on par with the best diversity-scoring benchmarks."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.269,
+                0.351,
+                0.282
+            ],
+            "angle": 0,
+            "content": "4.2 INFILLING STORIES"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.29,
+                0.825,
+                0.388
+            ],
+            "angle": 0,
+            "content": "Task description. Next, we consider the story infilling task, a special case of the general infilling problem (§3.1), where given the beginning \\( X \\) and end \\( Y \\) of a story, the goal is to generate the middle of the story \\( Z \\) (Zhu et al., 2019). This is challenging for a language model sampled left to right since continuations \\( Z \\) conditioned only on \\( X \\) might be incompatible with the ending \\( Y \\). We use the ROCStories corpus (Mostafazadeh et al., 2016), a dataset of short stories containing exactly 5 sentences each. Given the first 3 sentences and the last sentence, the goal is to generate the fourth sentence, which often involves a turning point in the story and is thus challenging to fill in."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.393,
+                0.827,
+                0.477
+            ],
+            "angle": 0,
+            "content": "As we expect the base model to contain the required knowledge, for this task we use a GPT-2 Large model (Radford et al., 2019) fine-tuned on the entire ROCStories training set as the base model. For evaluating the approach, we consider 900 samples from the dataset as training data to learn \\( q_{\\mathrm{GFN}}(Z|X,Y) \\) and evaluate the similarity of the generated infills on a dataset of 100 unseen stories. Along with the GFlowNet-fine-tuned model, we also consider two baselines: prompting the model to infill the story and supervised fine-tuning on the same data. Further details are in Appendix C."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.482,
+                0.487,
+                0.55
+            ],
+            "angle": 0,
+            "content": "Results. To measure the similarity of the generated infills with the reference infills available in the dataset, we compute BERTScore (Zhang et al., 2020b), with DeBERTa (He et al., 2021) - which is correlated with human judgments"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.55,
+                0.827,
+                0.633
+            ],
+            "angle": 0,
+            "content": "along with BLEU-4 (Papineni et al., 2002) and GLEU-4 (better suited for sentences; Wu et al., 2016) metrics. Additionally, we also evaluate each method using GPT-4 as a judge. From our results summarized in Table 2, we observe that the infills generated by the model with GFlowNet fine-tuning are closer to the reference infills in the dataset than the baselines. By sampling from \\( p_{\\mathrm{LM}}(Z|X,Y) \\), the GFlowNet is able to account for the ending while generating the infill, resulting in infills that link the beginning and the end of the story coherently. For further analysis and details see Appendix C."
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.531,
+                0.486,
+                0.789,
+                0.498
+            ],
+            "angle": 0,
+            "content": "Table 2: Evaluation of the generated infills."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.499,
+                0.498,
+                0.822,
+                0.543
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Method</td><td>BERTScore</td><td>BLEU-4</td><td>GLEU-4</td><td>GPT4Eval</td></tr><tr><td>Prompting</td><td>0.081 ± 0.009</td><td>1.3 ± 0.5</td><td>3.2 ± 0.1</td><td>2.4</td></tr><tr><td>Supervised fine-tuning</td><td>0.094 ± 0.007</td><td>1.6 ± 0.8</td><td>3.7 ± 0.4</td><td>2.7</td></tr><tr><td>GFlowNet fine-tuning</td><td>0.184 ± 0.004</td><td>2.1 ± 0.2</td><td>4.2 ± 0.7</td><td>3.4</td></tr></table>"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.64,
+                0.438,
+                0.653
+            ],
+            "angle": 0,
+            "content": "4.3 SUBJECTIVITY CLASSIFICATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.661,
+                0.486,
+                0.797
+            ],
+            "angle": 0,
+            "content": "Task description. SUBJ (Pang & Lee, 2004) is a binary classification dataset for natural language understanding. It is a collection of movie reviews in which each review is labeled as objective, meaning that it references facts about the movie, or subjective, meaning that it expresses an opinion of the reviewer (see Table D.1 for examples). Given an unlabeled review, the model must predict whether it is objective or subjective. While supervised fine-tuning on the full dataset"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.497,
+                0.665,
+                0.826,
+                0.689
+            ],
+            "angle": 0,
+            "content": "Table 3: Test accuracy \\((\\%)\\) on SUBJ using an instruct-fine-tuned GPT-J 6B."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.498,
+                0.69,
+                0.822,
+                0.795
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Method</td><td colspan=\"3\">Test accuracy (%) ↑</td></tr><tr><td>Zero-shot prompting</td><td colspan=\"3\">51.7</td></tr><tr><td></td><td colspan=\"3\">Training samples</td></tr><tr><td></td><td>10</td><td>20</td><td>50</td></tr><tr><td>Few-shot prompting</td><td>61.3 ± 6.2</td><td>61.8 ± 5.4</td><td>65.8 ± 10.5</td></tr><tr><td>Supervised fine-tuning</td><td>64.3 ± 2.8</td><td>69.1 ± 0.8</td><td>89.7 ± 0.4</td></tr><tr><td>GFlowNet fine-tuning</td><td>71.4 ± 1.8</td><td>81.1 ± 0.4</td><td>87.7 ± 2.2</td></tr><tr><td>+ Supervised fine-tuning</td><td>75.2 ± 1.8</td><td>78.7 ± 1.6</td><td>89.9 ± 0.2</td></tr></table>"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.798,
+                0.827,
+                0.854
+            ],
+            "angle": 0,
+            "content": "can achieve high test accuracy, we are interested in the low-data regime where we only have tens of labeled examples. We use the same instruction-tuned GPT-J 6B variant as in §2 for this experiment. Without any demonstrations, the model struggles with this task using the prompt in Table D.2 and achieves only \\(51.7\\%\\) zero-shot accuracy."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.858,
+                0.826,
+                0.915
+            ],
+            "angle": 0,
+            "content": "This task is hard likely because it requires a latent reasoning step. A review could be considered objective because it analyzes the plot or facts about the movie, or it could be subjective because it expresses a personal opinion or makes a judgment. We denote the review \\( X \\), the predicted subjectivity \\( Y \\), and the latent reason \\( Z \\). Then, we GFlowNet-fine-tune the LLM \\( q_{\\mathrm{GFN}}(Z \\mid X) \\), initialized with the"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.504,
+                0.96
+            ],
+            "angle": 0,
+            "content": "7"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.104,
+                0.825,
+                0.148
+            ],
+            "angle": 0,
+            "content": "base model \\( p_{\\mathrm{LM}} \\) to match the Bayesian posterior over rationales in Eq. 2. At test time, \\( q_{\\mathrm{GFN}}(Z \\mid X) \\) generates 10 latent rationales (\\( Z \\)'s) for an unseen \\( X \\). The LLM \\( p_{\\mathrm{LM}} \\) then autoregressively samples from \\( p_{\\mathrm{LM}}(Y \\mid XZ) \\) to produce 10 answers, the majority vote of which becomes the final prediction."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.152,
+                0.825,
+                0.209
+            ],
+            "angle": 0,
+            "content": "This posterior inference corresponds to the E-step in the EM algorithm, where the posterior \\( p_{\\mathrm{LM}}(Z \\mid X,Y) \\) is defined in Eq. 2. Further, as described in §3.2, we can take an M-step by updating \\( p_{\\mathrm{LM}} \\) to maximize \\( \\log p_{\\mathrm{LM}}(XZY) \\) over a collection of \\( Z \\)'s sampled from the amortized posterior \\( q_{\\mathrm{GFN}} \\). This is equivalent to applying supervised fine-tuning after GFlowNet fine-tuning."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.213,
+                0.827,
+                0.337
+            ],
+            "angle": 0,
+            "content": "Results. We present few-shot prompting and supervised fine-tuning with LoRA as baselines. In few-shot prompting, we prepend 0, 10, 20, or 50 training examples to each test example using the prompt shown in Table D.2. We randomly shuffle few-shot demonstrations and report the mean and variance in Table 3. In supervised fine-tuning, we directly maximize \\(\\log p_{\\mathrm{LM}}(Y \\mid X)\\) over the same 10, 20, or 50 \\((X, Y)\\) pairs. The variance is over model initialization and batch order. All entries except zero-shot prompting are aggregated over 3 random seeds. See Appendix D for experiment details. GFlowNet fine-tuning consistently outperforms supervised fine-tuning in the low-data regime, as shown in Table 3. In some cases, performing supervised fine-tuning on top, which corresponds to running one step of the EM algorithm, further improves the performance."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.344,
+                0.551,
+                0.358
+            ],
+            "angle": 0,
+            "content": "4.4 SOLVING ARITHMETIC PROBLEMS STEP BY STEP"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.366,
+                0.487,
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+            ],
+            "angle": 0,
+            "content": "Task description. Arithmetic reasoning is a fitting benchmark to evaluate reasoning abilities of large language models as it requires multi-step reasoning and correctness is easy to evaluate (Cobbe et al., 2021). While the distribution of pretraining and fine-tuning data (Magister et al., 2023; Lee et al., 2023; Luo et al., 2023) and prompting choices (Imani et al., 2023) play a critical role in their arithmetic abilities, LLMs are susceptible to poor generalization by learning 'shortcuts' to reasoning (Dziri et al., 2023). We consider a simple integer arithmetic task (Fig. 1), with a general pretrained base model, rather than a one pretrained on mathematical"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.495,
+                0.368,
+                0.825,
+                0.407
+            ],
+            "angle": 0,
+            "content": "Table 4: Test accuracy \\((\\%)\\) on an integer arithmetic task with addition and subtraction using a GPT-J 6B model. Training data only include samples with 3 or 4 operands."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.498,
+                0.407,
+                0.822,
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+            ],
+            "angle": 0,
+            "content": "<table><tr><td rowspan=\"3\">Method</td><td></td><td colspan=\"3\">Number of Operands</td></tr><tr><td></td><td colspan=\"2\">In-distribution</td><td>OOD</td></tr><tr><td></td><td>3</td><td>4</td><td>5</td></tr><tr><td rowspan=\"5\">k-shot CoT</td><td>k=0</td><td>10.2</td><td>6.4</td><td>3.2</td></tr><tr><td>k=3</td><td>15.8 ± 3.1</td><td>11 ± 1.7</td><td>5.4 ± 0.2</td></tr><tr><td>k=5</td><td>20.4 ± 10.4</td><td>17.6 ± 0.6</td><td>6.6 ± 1.1</td></tr><tr><td>k=10</td><td>26.5 ± 1.4</td><td>15.2 ± 1.7</td><td>8.9 ± 1.9</td></tr><tr><td>k=20</td><td>35.5 ± 1.9</td><td>21 ± 1.4</td><td>10.5 ± 0.9</td></tr><tr><td colspan=\"2\">Supervised fine-tuning</td><td>72.1 ± 1.3</td><td>19.6 ± 2.2</td><td>12.8 ± 5.7</td></tr><tr><td colspan=\"2\">PPO</td><td>30.6 ± 4.1</td><td>13.7 ± 4.1</td><td>5.6 ± 3.1</td></tr><tr><td colspan=\"2\">GFlowNet fine-tuning</td><td>95.2 ± 1.3</td><td>75.4 ± 2.9</td><td>40.7 ± 9.1</td></tr></table>"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.169,
+                0.558,
+                0.825,
+                0.64
+            ],
+            "angle": 0,
+            "content": "tasks (Jelassi et al., 2023). To avoid the pitfalls of symbolic calculations with language models, we adopt the tool use setting (Schick et al., 2023), where the model is equipped with a calculator that can perform parts of the computation, implemented as in Cobbe et al. (2021): when the model outputs \\(= \\) the expression preceding it is evaluated and appended to the sequence. To prevent the model from \"cheating\", we limit the calculator to evaluate only two-term expressions. Consequently, reasoning here involves learning to plan using a tool with limited capabilities (Hao et al., 2023)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.645,
+                0.827,
+                0.756
+            ],
+            "angle": 0,
+            "content": "For training, we use a synthetic dataset of arithmetic expressions, limited to addition and subtraction. Following Zelikman et al. (2022), we use a small set of 50 demonstrations \\((X,Z,Y)\\) to seed the replay buffer in addition to 1000 examples \\((X,Y)\\). We use the same instruction-tuned GPT-J as in §4.3 as the base model. Further details are in Appendix E. We report the accuracy on two types of examples: (1) unseen in-distribution expressions (3 or 4 operands) and (2) longer out-of-distribution expressions (5 operands). As baselines, we consider zero-shot chain-of-thought prompting, \\(k\\)-shot prompting, supervised fine-tuning on the tool use sequences, and fine-tuning with PPO (Schulman et al., 2017). For all methods, we enable tool use and limit the model to generate only numbers and operators."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.761,
+                0.827,
+                0.871
+            ],
+            "angle": 0,
+            "content": "Results. From the results summarized in Table 4, the base model performs poorly even with chain-of-thought prompts. Including examples in context improves the performance considerably, with monotonic improvements as the number of examples increases. Supervised fine-tuning improves the performance significantly on the in-distribution examples, but the model still struggles to generalize on the out-of-distribution examples. Fine-tuning with PPO results also yields poor performance, caused in part by the poor calibration of the base reward model, i.e. it cannot distinguish good rationales from bad ones. Even though the sequences generated with PPO (illustrated in Appendix E) have high rewards, they are spurious and do not even define valid calls to the tool."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.877,
+                0.825,
+                0.919
+            ],
+            "angle": 0,
+            "content": "Such overoptimization to a misspecified reward is a widely noted issue in LLMs trained with RL (Gao et al., 2023). On the other hand, by matching the entire distribution, GFlowNet fine-tuning avoids collapsing to a single mode of the reward, thereby being robust to the misspecification of"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.504,
+                0.96
+            ],
+            "angle": 0,
+            "content": "8"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.825,
+                0.133
+            ],
+            "angle": 0,
+            "content": "the reward (Eysenbach & Levine, 2022) and achieving significantly better performance on in and out-of-distribution examples. See Appendix E for additional results and analysis."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.143,
+                0.423,
+                0.158
+            ],
+            "angle": 0,
+            "content": "5 FURTHER RELATED WORK"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.167,
+                0.827,
+                0.277
+            ],
+            "angle": 0,
+            "content": "Sampling from intractable marginals. Beyond the approximations mentioned in §3.1, sampling from intractable posterior distributions given by pretrained models for tasks such as infilling and constrained generation has been an object of study. Miao et al. (2019); Zhang et al. (2020a) use MCMC for these problems, Malkin et al. (2021) used a variable-neighborhood ascent for finding modes, and a sequential Monte Carlo approach was recently proposed by Lew et al. (2023). Others have studied the problem with masked language models, using them to perform variants of Gibbs sampling (Wang & Cho, 2019; Goyal et al., 2022; Yamakoshi et al., 2022) and recovering marginal distributions over small sets of tokens (Torroba Hennigen & Kim, 2023)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.283,
+                0.828,
+                0.38
+            ],
+            "angle": 0,
+            "content": "GFlowNets. GFlowNets (Bengio et al., 2021) were originally proposed to learn policies for sampling discrete compositional objects from an unnormalized reward distribution, motivated by the need to sample diverse high-reward objects in scientific discovery (Jain et al., 2023), in particular, for biological sequence generation (Jain et al., 2022). The interpretation of GFlowNets as variational inference algorithms (Malkin et al., 2023; Zimmermann et al., 2023) makes them appropriate for sampling Bayesian posterior distributions over structured objects (e.g., Deleu et al., 2022; 2023; van Krieken et al., 2023; Hu et al., 2023)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.385,
+                0.828,
+                0.565
+            ],
+            "angle": 0,
+            "content": "Chain-of-thought reasoning in LLMs. In recent work on classification and completion with language models, the latent reasoning chain \\( Z \\), in the notation of §3.1, is called a 'chain of thought' (Wei et al., 2022). The chain of thought is typically generated by conditioning the language model on \\( X \\) with the use of specialized demonstrations or prompts (Kojima et al., 2022), with no guarantee of sampling the posterior accurately. Related to our Bayesian formulation, Wang et al. (2023b) noted that appropriately aggregating the conclusions \\( Y \\) from several latent chains \\( Z \\) improves predictive performance. In Xu et al. (2023); Zhou et al. (2022), a posterior over latent token sequences is sampled using MCMC, while Zelikman et al. (2022) propose fine-tuning on successful (high-reward, in our language) chains of thought, which achieves reward maximization but gives no guarantee of diversity. In concurrent work, Phan et al. (2023) use MCMC to sample chains-of-thought in problems with binary feedback. We expect these methods to generalize poorly to difficult exploration problems, while GFlowNet fine-tuning takes advantage of generalizable structure in the posterior and has a goal of sampling the full posterior over latent reasoning chains."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.574,
+                0.321,
+                0.589
+            ],
+            "angle": 0,
+            "content": "6 CONCLUSION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.598,
+                0.827,
+                0.695
+            ],
+            "angle": 0,
+            "content": "The knowledge compressed in LLMs is crucial for tasks such as infilling and constrained generation, but querying this knowledge involves sampling from intractable posterior distributions. We propose to use GFlowNet objectives to train LLMs to sample from such posterior distributions. Empirical results show that GFlowNet fine-tuning finds a better fidelity-diversity trade-off for text generation and also improves sample efficiency and generalization on downstream tasks compared to maximum-likelihood training or reward-maximizing policy optimization. As an amortized inference algorithm, our method converts computation into better test-time performance without additional data."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.7,
+                0.827,
+                0.838
+            ],
+            "angle": 0,
+            "content": "Future work should investigate transfer and generalization across tasks, in particular, building a 'universal reasoner' as a model \\( q(Z \\mid X) \\) shared between \\( X \\) from different tasks, as was recently considered by Wang et al. (2023a). One should investigate the benefit of using a better knowledge model, e.g., a more capable base LLM, as a starting point for GFlowNet fine-tuning. The ability to draw multiple samples from a GFlowNet can also be used to quantify epistemic uncertainty. Finally, we adopt the GFlowNet formalisms with the perspective of generalizing to latent variables \\( Z \\) with a richer generative process than left-to-right sampling. We hope that the GFlowNet paradigm will enable more flexible reasoning with LLMs in the future: extending probabilistic programming with language variables (Beurer-Kellner et al., 2023), using structured chains of thought (Yao et al., 2023; Besta et al., 2024), and extending to program synthesis and planning with world models."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.844,
+                0.828,
+                0.94
+            ],
+            "angle": 0,
+            "content": "Limitations. Due to resource constraints, our experiments use models up to 6B parameters, but we expect the conclusions to hold for larger models. In fact, our method can potentially benefit larger models more: it is harder to optimize a larger model with maximizing objectives on a small amount of data. As with any on-policy method, exploration, especially in problems with more complex latents, remains an open problem. Furthermore, our method improves inference but not the knowledge in the LM. Issues such as hallucination or miscalibration, which are closely related to the knowledge representation, are thus not addressed."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.506,
+                0.96
+            ],
+            "angle": 0,
+            "content": "9"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.103,
+                0.344,
+                0.119
+            ],
+            "angle": 0,
+            "content": "ETHICS STATEMENT"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.127,
+                0.827,
+                0.17
+            ],
+            "angle": 0,
+            "content": "While we foresee no immediate negative societal consequences of our work, we hope that future researchers who build upon it will, as we have, bear in mind the potential of LLMs – and in particular of human-like reasoning in LLMs – to be used both for good and for harm."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.174,
+                0.828,
+                0.232
+            ],
+            "angle": 0,
+            "content": "Research areas in safe and explainable AI that can benefit from GFlowNet fine-tuning include (1) interpretability of LLMs' reasoning processes and (2) fine-tuning with human feedback or an external reward, where diverse sampling can help prevent 'reward hacking' and overfitting to a misspecified target."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.241,
+                0.334,
+                0.256
+            ],
+            "angle": 0,
+            "content": "REPRODUCIBILITY"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.265,
+                0.825,
+                0.308
+            ],
+            "angle": 0,
+            "content": "We discuss the details of the proposed algorithms in §3.3 and provide all the implementation details and hyperparameters for the experiments in the main paper and appendix. Code for our experiments is available at https://github.com/GFNOrg/gfn-lm-tuning."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.318,
+                0.357,
+                0.332
+            ],
+            "angle": 0,
+            "content": "ACKNOWLEDGMENTS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.341,
+                0.826,
+                0.385
+            ],
+            "angle": 0,
+            "content": "The authors are grateful to Bonaventure Dossou and Salem Lahlou for their help in the early stages of this project. We also thank Robert Hawkins, Arian Hosseini, Zhen Wang, and Anirudh Goyal for valuable discussions and suggestions of related work."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.389,
+                0.827,
+                0.418
+            ],
+            "angle": 0,
+            "content": "GL acknowledges funding from CIFAR, Samsung, and a Canada Research Chair in Neural Computation and Interfacing."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.423,
+                0.75,
+                0.438
+            ],
+            "angle": 0,
+            "content": "YB acknowledges funding from CIFAR, NSERC, IBM, Intel, Genentech, and Samsung."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.443,
+                0.826,
+                0.485
+            ],
+            "angle": 0,
+            "content": "The research was enabled in part by computational resources provided by the Digital Research Alliance of Canada (https://alliancecan.ca), Mila (https://mila.quebec), and NVIDIA."
+        },
+        {
+            "type": "title",
+            "bbox": [
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+            "angle": 0,
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+            "angle": 0,
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+        },
+        {
+            "type": "title",
+            "bbox": [
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+            "angle": 0,
+            "content": "A ADDITIONAL BACKGROUND"
+        },
+        {
+            "type": "title",
+            "bbox": [
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+            "angle": 0,
+            "content": "A.1 GLOSSARY OF RL FOR LLMS"
+        },
+        {
+            "type": "text",
+            "bbox": [
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+                0.149,
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+            "angle": 0,
+            "content": "We provide definitions for key terms used throughout the paper, with a focus on their relevance to our setting of fine-tuning large language models (LLMs)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.183,
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+            "angle": 0,
+            "content": "Reinforcement learning. Reinforcement learning (RL) is a branch of machine learning concerned with how agents should take actions in an environment to maximize cumulative rewards. In our context, RL is used to fine-tune the decision-making process of LLMs to improve their performance on specific tasks. For a more comprehensive overview, we refer readers to Sutton & Barto (2018)."
+        },
+        {
+            "type": "text",
+            "bbox": [
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+            "angle": 0,
+            "content": "Policy. In RL, a policy is a strategy that defines the behavior of an agent by mapping states of the environment to actions. In our setting, a policy dictates how the language model generates text sequences based on the current context and learned parameters."
+        },
+        {
+            "type": "text",
+            "bbox": [
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+                0.292,
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+            "angle": 0,
+            "content": "Reward. A reward is a signal that evaluates the quality of an action taken by an agent in a particular state. In the fine-tuning of LLMs, rewards are used to guide the model towards generating more desirable text, such as more accurate predictions or more coherent continuations. More specifically, in the context of GFlowNets, the reward corresponds to the unnormalized posterior probability, and the GFlowNet aims to match it by learning a policy that generates samples proportional to their reward."
+        },
+        {
+            "type": "text",
+            "bbox": [
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+                0.367,
+                0.825,
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+            "angle": 0,
+            "content": "Matching a distribution. Matching a distribution involves training a model to approximate a target probability distribution. In our work, this concept is applied to fine-tune LLMs so that their generated text matches the desired characteristics, such as adhering to a particular style or content constraint."
+        },
+        {
+            "type": "text",
+            "bbox": [
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+                0.414,
+                0.825,
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+            "angle": 0,
+            "content": "Policy gradient methods. Policy gradient methods (such as PPO) are a subset of RL algorithms that optimize a policy by computing gradients of the expected reward with respect to the policy parameters. In the context of LLMs, these methods are used to fine-tune the model's parameters to increase the likelihood of generating high-reward text sequences."
+        },
+        {
+            "type": "title",
+            "bbox": [
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+            "angle": 0,
+            "content": "A.2 LEARNING OBJECTIVE"
+        },
+        {
+            "type": "text",
+            "bbox": [
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+            ],
+            "angle": 0,
+            "content": "We use the subtrajectory balance learning objective for training GFlowNets (Madan et al., 2023). In the notation of that paper, with a forward policy \\(P_F\\), backward policy \\(P_B\\) and state flow function \\(F\\), the objective over a partial trajectory \\(\\tau = s_m \\rightarrow \\dots \\rightarrow s_n\\) is defined as follows"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.333,
+                0.557,
+                0.826,
+                0.602
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathcal {L} _ {S u b T B} (Z; \\theta) = \\left(\\log \\frac {F \\left(s _ {m}\\right) \\prod_ {i = m} ^ {n - 1} P _ {F} \\left(s _ {i + 1} \\mid s _ {i}\\right)}{F \\left(s _ {n}\\right) \\prod_ {i = m} ^ {n - 1} P _ {B} \\left(s _ {i} \\mid s _ {i + 1}\\right)}\\right) ^ {2} \\tag {4}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.608,
+                0.826,
+                0.734
+            ],
+            "angle": 0,
+            "content": "In the case of autoregressive generation of a sequence of tokens in a fixed order (left-right), the generative process is a tree, so there is only a single path to each state, and each state has a single parent. Thus, \\( P_B(s|s') = 1 \\) trivially. Additionally, since each state is a valid terminable state, we can incorporate the modification to account for this from Deleu et al. (2022). Specifically, note that at convergence we have \\( R(s_n^\\top) = F(s_n)P_F(\\top \\mid s_n) \\). Using this, we can simply substitute \\( F(s_{n}) = R(s_{n}^{\\top}) / P_{F}(\\top \\mid s_{n}) \\) in Eq. (4). This allows us to avoid parameterizing a flow function separately, reducing additional complexity. The only learned object is now the forward sampling policy \\( P_F \\), which we refer to as \\( q_{GFN} \\). Summing this over all partial trajectories in a trajectory with equal weight \\( (\\lambda = 1) \\), we get the final learning objective in Eq. (3)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.743,
+                0.434,
+                0.758
+            ],
+            "angle": 0,
+            "content": "B SENTENCE CONTINUATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.767,
+                0.826,
+                0.837
+            ],
+            "angle": 0,
+            "content": "Additional details. We choose sentences as the level of granularity for our sequence continuation task because they are a natural unit of generation in language. Moreso than individual words, sentences are analogous to whole thoughts, reasoning steps, etc. because the compositional rules of syntax operate at the level of sentences. Whereas the meaning of a word is often ambiguous and context-dependent, the meaning of a sentence tends to be more self-contained."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.842,
+                0.827,
+                0.926
+            ],
+            "angle": 0,
+            "content": "A naive solution to the task is to simply sample autoregressively from the LM until a “.” token is reached, marking the end of the sentence. In practice, however, this is unlikely to produce samples that have high likelihood because the distribution over sentences has a long tail: a vast number of sentences have small but non-zero probability, and collectively account for a substantial portion of the total probability mass. Instead, it would be desirable to sample from a low-temperature distribution over sentences, so that the distribution becomes more sparse and highly probable sentences are"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "17"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.104,
+                0.827,
+                0.215
+            ],
+            "angle": 0,
+            "content": "sampled more often. However, this in itself is a difficult distribution to sample from, and it is different from simply sampling autoregressively from the LM with a lower temperature applied to the distributions over the next words. For instance, as the temperature approaches 0, the desired distribution over sentences should converge to an argmax and produce the single most likely next sentence. However, if we try to apply the temperature to the distribution over the next words and sample autoregressively, then as the temperature approaches 0, the resulting policy will greedily construct a sentence by sequentially picking the most likely next word, which is unlikely to produce the highest probability sentence."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.22,
+                0.828,
+                0.318
+            ],
+            "angle": 0,
+            "content": "Our GFlowNet sampling policy parametrizes a distribution \\( p_{\\mathrm{GFN}}(w_{i+1}|w_{1:i}) \\) using the same architecture as the LM \\( p_{\\mathrm{LM}}(w_{i+1}|w_{1:i}) \\), and at the start of training, it is initialized with the same weights. The initial state of the GFlowNet consists of a prompt \\( w_{1:k} \\) that conditions generation (i.e., the text whose next sentence we are trying to sample). Each subsequent state \\( w_{1:k+i} \\) is an extension of this text and is obtained by sampling autoregressively from \\( p_{\\mathrm{GFN}} \\). This process terminates when a period “.” is sampled, indicating the end of the next sentence. If a predefined maximum sentence length is reached, we force a “.” action to manually terminate the generation."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.323,
+                0.827,
+                0.365
+            ],
+            "angle": 0,
+            "content": "The GFlowNet policy is trained to sample sentences in proportion to their tempered probability under the original LM given the prompt. Denote the length of the prompt by \\( k \\), the length of the sampled sentence by \\( m \\), and the temperature by \\( T \\). Then, the reward is:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.273,
+                0.373,
+                0.826,
+                0.411
+            ],
+            "angle": 0,
+            "content": "\\[\nR \\left(w _ {1: k + m}\\right) \\stackrel {\\text {d e f}} {=} p _ {\\mathrm {L M}} \\left(w _ {k + 1: k + m} \\mid w _ {1: k}\\right) ^ {\\frac {1}{T}} = \\left(\\prod_ {i = 1} ^ {m} p _ {\\mathrm {L M}} \\left(w _ {k + i} \\mid w _ {1: k + i - 1}\\right)\\right) ^ {\\frac {1}{T}} \\tag {5}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.418,
+                0.825,
+                0.488
+            ],
+            "angle": 0,
+            "content": "The intuition behind this reward is that we are assuming that the LM can reliably assign a high likelihood to high-probability sentences, that we wish to preferentially sample over low-probability ones. If we were to set \\( T = 1 \\), then the solution for the GFlowNet would be to sample proportionally to \\( p_{\\mathrm{LM}} \\) (which it is initialized to). The preferential sampling of high-probability sentences is therefore obtained by setting \\( 0 < T < 1 \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.492,
+                0.827,
+                0.561
+            ],
+            "angle": 0,
+            "content": "To run our experiment, we obtained a dataset of 1000 prompts from OpenWebText (Gokaslan et al., 2019) that were each 1-3 sentences long, 50 of which were used for validation. Our LM consisted of a pretrained 1.5B parameter GPT2-XL (Radford et al., 2019), and our GFlowNet was a copy of this model that was fine-tuned in a lower-dimensional space of 80M parameters using LoRA (Hu et al., 2022)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.568,
+                0.827,
+                0.61
+            ],
+            "angle": 0,
+            "content": "Additional results. We show examples of empirical next-sentence samples generated by our model in Table B.1 compared to baselines. The model was trained using a reward temperature of 0.875, which achieves a good balance between log-likelihood and diversity."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.62,
+                0.375,
+                0.635
+            ],
+            "angle": 0,
+            "content": "C INFILLING STORIES"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.644,
+                0.825,
+                0.687
+            ],
+            "angle": 0,
+            "content": "Additional details. See Table C.1 for training examples from the subset of the ROC Stories dataset used for the task. To condition the model on \\( X \\) and \\( Y \\), as well as for the prompting baseline, we use the following prompt:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.691,
+                0.564,
+                0.708
+            ],
+            "angle": 0,
+            "content": "\"Beginning: {X}\\n End: {Y}\\n Middle: \""
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.712,
+                0.827,
+                0.809
+            ],
+            "angle": 0,
+            "content": "An assumption we make in this paper is that the base language model already contains the knowledge required for the task, and the goal is to perform inference over this knowledge. However, for this task, none of the pretrained base models were good at assigning high likelihoods to plausible stories. Thus, we fine-tuned GPT-2 Large model (Radford et al., 2019) on the entirety of the stories dataset and used this fine-tuned model as the reward model. This was done with full fine-tuning using theTRL library (von Werra et al., 2020). We trained for 20 epochs with a batch size of 64 and 32 gradient accumulation steps and a learning rate of 0.0005."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.814,
+                0.828,
+                0.926
+            ],
+            "angle": 0,
+            "content": "We detail the hyperparameters used for training GFlowNets in our experiments in Table C.2. During training, we sample \\((X,Y)\\) from the dataset and then sample (batch size) \\(Zs\\) for every \\((X,Y)\\), before using \\(p_{\\mathrm{LM}}(XZY)\\) as the reward. We use the replay buffer described in §3.3 and seed it with rationales from the dataset. We linearly anneal the reward temperature, the temperature of the behavior policy during training, and the learning rate during warmup. For supervised fine-tuning, we use a batch size of 256 and train for 10 epochs with a learning rate of 0.0001 withTRL and LoRA. At test time, we sample 1024 infills for each example in the test set from all the models at temperature 0.9, and average over 10 such draws."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.491,
+                0.949,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "18"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.171,
+                0.107,
+                0.825,
+                0.145
+            ],
+            "angle": 0,
+            "content": "Table B.1: Empirical examples for sequence continuation. Sentences generated from GFlowNet fine-tuning tend to be more reasonable than autoregressively generated samples from the LM with temperature 1.0 and tend to be more diverse than samples generated from diverse beam search."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.182,
+                0.147,
+                0.812,
+                0.916
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Input Prompt:</td><td>The matching campaign starts on Friday and will continue through midnight on Monday.</td></tr><tr><td>Sampling w/ T = 1.0:</td><td>(1) Participate with a fairly low $1 donation so we can motivate more volunteers on Sunday afternoon.\n(2) However, the information regarding Cutler&#x27;s suspicious death may not become widely known until early in the week.</td></tr><tr><td>Diverse beam search:</td><td>(1) If you are interested in participating in the matching campaign, you can do so by clicking here.\n(2) There is no limit to the number of times you can enter.</td></tr><tr><td>GFlowNet fine-tuning:</td><td>(1) If you are interested in signing up you can do so here.\n(2) Please share.</td></tr><tr><td>Input Prompt:</td><td>I want hockey to become a huge thing in Houston.</td></tr><tr><td>Sampling w/ T = 1.0:</td><td>(1) We&#x27;ll be here in Texas from late November till end March.\n(2) To that end, I&#x27;ve been following the Houston Aeros (Houston Dynamo&#x27;s minor-league affiliate) and their AHL affiliate (the Texas Stars).</td></tr><tr><td>Diverse beam search:</td><td>(1) That&#x27;s why I&#x27;m here.\n(2) That&#x27;s why I&#x27;m doing this.</td></tr><tr><td>GFlowNet fine-tuning:</td><td>(1) This is something I&#x27;ve always wanted.\n(2) When I was a teenager in middle school, I went to the Ice Arena in University of Houston and loved it.</td></tr><tr><td>Input Prompt:</td><td>That incident got a lot of attention in part because it was captured on video. Israel said he recorded what happened at the synagogue, and made it public, to document it and leave no doubt about what transpired.</td></tr><tr><td>Sampling w/ T = 1.0:</td><td>(1) He blogged about it here as well.\n(2) Israeli TV stations broadcast the video before it aired in Israel, as the country&#x27;s rule requires.</td></tr><tr><td>Diverse beam search:</td><td>(1) However, there is no evidence that he did so.\n(2) However, there is no video of what happened inside the synagogue.</td></tr><tr><td>GFlowNet fine-tuning:</td><td>(1) He is on tape doing all of it.\n(2) It&#x27;s a message countries can use to deter future attacks.</td></tr><tr><td>Input Prompt:</td><td>The Rolling Stones in Concert has since been released solely by ABKCO Records. The band would remain incensed with Klein for decades for that act. Klein died in 2009.</td></tr><tr><td>Sampling w/ T = 1.0:</td><td>(1) Actually art has become as obscene as investment banking.\n(2) Some believe he shot himself in the chest.</td></tr><tr><td>Diverse beam search:</td><td>(1) The Rolling Stones, however, would not be deterred.\n(2) The Rolling Stones would go on to release their own version of The Rolling Stones in Concert.</td></tr><tr><td>GFlowNet fine-tuning:</td><td>(1) He received a lifetime achievement award from the Jazz Times.\n(2) Sometimes it seems like we are destined to repeat our own mistakes.</td></tr></table>"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.491,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "19"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.294,
+                0.201,
+                0.701,
+                0.215
+            ],
+            "angle": 0,
+            "content": "Table C.1: Examples of training samples for the stories infilling task."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.175,
+                0.215,
+                0.821,
+                0.361
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Beginning (X)</td><td>Middle (Z)</td><td>End (Y)</td></tr><tr><td>I was going to a Halloween party. I looked through my clothes but could not find a costume. I cut up my old clothes and constructed a costume.</td><td>I put my costume on and went to the party.</td><td>My friends loved my costume.</td></tr><tr><td>Allen thought he was a very talented poet. He attended college to study creative writing. In college, he met a boy named Carl.</td><td>Carl told him that he wasn’t very good.</td><td>Because of this, Allen swore off poetry forever.</td></tr></table>"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.336,
+                0.567,
+                0.66,
+                0.581
+            ],
+            "angle": 0,
+            "content": "Table C.2: Hyperparameters for the story infilling task."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.36,
+                0.582,
+                0.639,
+                0.822
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>LoRA rank</td><td>64</td></tr><tr><td>LoRA scaling factor</td><td>16</td></tr><tr><td>LoRA dropout</td><td>0.1</td></tr><tr><td>Batch size</td><td>64</td></tr><tr><td>Gradient accumulation steps</td><td>16</td></tr><tr><td>Learning rate</td><td>0.0001</td></tr><tr><td>Optimizer</td><td>AdamW</td></tr><tr><td>PFtemperature max</td><td>2</td></tr><tr><td>PFtemperature min</td><td>0.5</td></tr><tr><td>Reward temperature start</td><td>1.1</td></tr><tr><td>Reward temperature end</td><td>0.85</td></tr><tr><td>Reward temperature horizon</td><td>100</td></tr><tr><td>Buffer capacity</td><td>25</td></tr><tr><td>Number of training steps</td><td>1000</td></tr><tr><td>Evaluation temperature</td><td>0.8</td></tr><tr><td>Maximum length</td><td>25</td></tr><tr><td>Minimum length</td><td>5</td></tr></table>"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.51,
+                0.96
+            ],
+            "angle": 0,
+            "content": "20"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.828,
+                0.134
+            ],
+            "angle": 0,
+            "content": "Analysis. Table C.3, C.4, C.5, C.6, C.7, C.8 illustrate some examples of the infills generated with the prompting baseline, supervised fine-tuned and GFlowNet fine-tuned models on the test examples."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.152,
+                0.828,
+                0.249
+            ],
+            "angle": 0,
+            "content": "For evaluating the infills we generate a rating for the stories with infills from each of the methods based on the coherence of the story. We average the rating over 10 infills sampled from each method and average it over all the 100 test examples. The prompt used for evaluation is the following, adapted from Liu et al. (2023). Stories with infills generated by the GFlowNet fine-tuned model on average receive a much higher rating than the baselines. The average rating for the reference infills is 4.3 which should be viewed as an upper bound, as the stories may potentially be present in the GPT-4 training data."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.284,
+                0.486,
+                0.298
+            ],
+            "angle": 0,
+            "content": "You will be given a short story."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.311,
+                0.613,
+                0.325
+            ],
+            "angle": 0,
+            "content": "Your task is to rate the story on one metric."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.338,
+                0.753,
+                0.379
+            ],
+            "angle": 0,
+            "content": "Please make sure you read and understand these instructions carefully. Please keep this document open while reviewing, and refer to it as needed."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.393,
+                0.371,
+                0.406
+            ],
+            "angle": 0,
+            "content": "Evaluation Criteria:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.42,
+                0.749,
+                0.502
+            ],
+            "angle": 0,
+            "content": "Coherence (1-5) - the collective quality of all sentences. The story should be well-structured and well-organized. The story should not just be a heap of related information, but should build from sentence to a coherent narrative. The story should also be realistic and all the sentences when put together should make sense."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.516,
+                0.341,
+                0.53
+            ],
+            "angle": 0,
+            "content": "Evaluation Steps:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.543,
+                0.819,
+                0.64
+            ],
+            "angle": 0,
+            "content": "1. Read the story carefully. Check if the story is coherent and all the sentences follow a logical order.  \n2. Assign a score for coherence on a scale of 1 to 5, where 1 is the lowest and 5 is the highest based on the Evaluation Criteria. If there are grammatical errors or logical inconsistencies, simply assign a lower score and do not elaborate on the reasoning."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.665,
+                0.254,
+                0.68
+            ],
+            "angle": 0,
+            "content": "Example:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.693,
+                0.79,
+                0.761
+            ],
+            "angle": 0,
+            "content": "Randy had recently separated from his wife.   \nHe felt very lonely and sad all the time.   \nHe considered trying to patch things up with Vera, his ex-wife. He decided to get a puppy instead.   \nHis decision made him happy and he no longer felt sad."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.169,
+                0.774,
+                0.309,
+                0.787
+            ],
+            "angle": 0,
+            "content": "- Coherence: 5"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.802,
+                0.803,
+                0.884
+            ],
+            "angle": 0,
+            "content": "Randy had recently separated from his wife.   \nHe felt very lonely and sad all the time.   \nHe considered trying to patch things up with Vera, his ex-wife.   \nEventually Randy missed Vera very much and developedWhile making memories with   \nHis decision made him happy and he no longer felt sad."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.169,
+                0.897,
+                0.298,
+                0.91
+            ],
+            "angle": 0,
+            "content": "- Coherence 1"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "21"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.105,
+                0.236,
+                0.12
+            ],
+            "angle": 0,
+            "content": "Story:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.173,
+                0.132,
+                0.263,
+                0.147
+            ],
+            "angle": 0,
+            "content": "{{Story}}"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.159,
+                0.468,
+                0.173
+            ],
+            "angle": 0,
+            "content": "Evaluation Form (scores ONLY):"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.168,
+                0.187,
+                0.289,
+                0.2
+            ],
+            "angle": 0,
+            "content": "- Coherence:"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.216,
+                0.478,
+                0.232
+            ],
+            "angle": 0,
+            "content": "D SUBJECTIVITY CLASSIFICATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.24,
+                0.825,
+                0.269
+            ],
+            "angle": 0,
+            "content": "Additional details. See Table D.1 for some training examples in the SUBJ dataset. We use the prompts in Table D.2 for all baselines."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.274,
+                0.827,
+                0.399
+            ],
+            "angle": 0,
+            "content": "We run GFlowNet fine-tuning for 1000 steps with a linear warmup over 200 steps, a fixed learning rate of 0.0005, and a batch size of 512 samples; see Table D.3 for all the hyperparameters used. Each batch consists of 8 queries (\\(X\\)'s), randomly drawn with replacement. We then sample 64 rationales (\\(Z\\)'s) for every \\(X\\), before using \\(p_{\\mathrm{LM}}(ZY \\mid X)\\) as the reward. We also use the replay buffer described in §3.3 and seed it with potential rationales generated from \\(p_{\\mathrm{LM}}\\). For supervised fine-tuning, both on its own and on top of GFlowNet fine-tuning, we use a batch size of 256 with 8 queries, randomly drawn with replacement. We train for 100 steps with a linear warm-up over 20 steps and a constant learning rate. We sweep the learning rate in [0.001, 0.0001] and report the best performance. The reward temperature is annealed from 1.2 down to 1 over the first 150 steps of training."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.404,
+                0.828,
+                0.475
+            ],
+            "angle": 0,
+            "content": "Additional results. To encourage exploration in the space of \\( Z \\), we inverse-prompt the model to generate potential chains of thoughts and store them in a replay buffer at the beginning of training. The replay buffer is a priority queue indexed by the reward. Under-performing chains of thoughts are evicted as we collect more trajectories. We ablate the effect of seeding the replay buffer with inverse-prompted chains of thoughts and aggregating multiple chains at test time in Table D.4."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.484,
+                0.398,
+                0.499
+            ],
+            "angle": 0,
+            "content": "E INTEGER ARITHMETIC"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.508,
+                0.825,
+                0.537
+            ],
+            "angle": 0,
+            "content": "Additional details. See Table E.1 for examples from synthetically generated training data for the integer arithmetic task."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.542,
+                0.827,
+                0.64
+            ],
+            "angle": 0,
+            "content": "As with the infilling task, for the integer arithmetic task, the pretrained base models we tested were imperfect knowledge models, i.e. incorrect rationales are sometimes assigned very high likelihood. To assign the correct rationales higher rewards, we prepend some demonstrations with hand-crafted rationales to the query when computing the reward, i.e. \\( p_{\\mathrm{LM}}(XZY|(X_iZ_iY_i)^k) \\) where \\( k = 3 \\) in our experiments. This improves the calibration to some extent. Note that these \\( (X_i,Z_i,Y_i) \\) are taken from the dataset and used only in the reward. The GFlowNet policy here is conditioned only on \\( X \\), i.e., \\( q_{\\mathrm{GFN}}(Z|X) \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.644,
+                0.827,
+                0.837
+            ],
+            "angle": 0,
+            "content": "We detail the hyperparameters used for training GFlowNets in our experiments in Table E.2. During training, we sample \\((X,Y)\\) from the dataset and then sample (batch size) \\(Zs\\) for every \\((X,Y)\\), before using \\(p_{\\mathrm{LM}}(XZY|(X_iZ_iY_i)^k_{i=1})\\) as the reward. During the generation of \\(Z\\), as mentioned in §4.4, whenever a “=” is generated we extract the preceding expression and evaluate the last two terms using eval in Python. We use the replay buffer described in §3.3 and seed it with rationales from the dataset \\(p_{\\mathrm{LM}}\\). We linearly anneal the reward temperature, and the learning rate during warmup. During evaluation, for all methods, we aggregate the response over multiple samples drawn from the model at some fixed temperature. For the zero-shot baseline, we observe the best performance with the “Let us think step by step.” prompt appended at the end of the question. For supervised fine-tuning and PPO, we use the implementation from trl. For supervised fine-tuning we use a batch size of 128 with 8 gradient accumulation steps and train for 10 epochs with a learning rate of 0.0001 using LoRA. For PPO we use a minibatch size of 64 with 16 gradient accumulation steps, a learning rate of 0.0001, 4 epochs (on the minibatch), a clip range of 0.2, and an adaptive KL coefficient initialized at 0.2."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.842,
+                0.827,
+                0.926
+            ],
+            "angle": 0,
+            "content": "Effect of the number of rationales. As mentioned in §4.4, we seed the buffer with rationales from a dataset. Table E.3 summarizes the results of ablation on the number of examples used to seed the buffer. We observe, as expected, that the performance generally improves as we the number of examples used to seed the buffer grows. When no buffers are used to seed the buffer, the performance is quite poor. We hypothesize that without the good rationales used to seed the buffer, the exploration problem of searching for good rationales becomes very challenging."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.51,
+                0.96
+            ],
+            "angle": 0,
+            "content": "22"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.172,
+                0.133,
+                0.825,
+                0.159
+            ],
+            "angle": 0,
+            "content": "Table C.3: Examples of infills generated by the prompting baseline along with reference infills for the stories infilling task."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.174,
+                0.16,
+                0.822,
+                0.891
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Beginning (X)</td><td>Middle (Z)</td><td>End (Y)</td></tr><tr><td>Randy had recently separated from his wife. He felt very lonely and sad all the time. He considered trying to patch things up with Vera, his ex-wife.</td><td>He decided to get a puppy instead\nHe now happily had a new wife who he really liked.\nHis wife missed him, but for the wrong reasons. Now\nHe finally found someone he could be happy with. It was\nHe began spending more time with his new girlfriend and friends.\nRandy remained single and mostly satis-fied with that. He found his\nWhen Randy was happy he went to the aquarium, a place\nHe started going out more with a display of male affection.\nHe bought Vera a moonstone and gave it to her as\nHe and Vera got back together, even though they had stayed\nHe introduced Paige to his new employ-\nees and they were great.\nRandy is happily married to Vera and they have a healthy relationship\nHe found a new lady with a lot of the same interests</td><td>His decision made him happy and he no longer felt sad.</td></tr><tr><td>Ken put a bottle of beer in the freezer. He heard a popping noise. He looked in the freezer and saw the bottle had burst.</td><td>He didn&#x27;t want to wait for another beer to get cold\nHe went to sleep and came awake with an angry stomach.\nKen had another beer as a treat. After the third beer\nHe stored the beer in the freezer and for-\ngot about it.\nHe walked around the bar a bit. He thinks he&#x27;ll\nWeeks later, Ken preferred his cold beer.\nHowever, he\nHe poured a fresh bottle with a sigh of relief. It\nHe went to the bar for cold beer. No one knew\nKen hung up the phone on his friend. He didn&#x27;t\nHe bought some veggies from the gro-\ncery store. It was warm\nKen watched the movie Goalkeeper. He was glad the cold\nHe laid down for a rest. His belly was full!\nHe laid down for the night and finally sleep again. The</td><td>He drank a warm beer instead.</td></tr></table>"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "23"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.172,
+                0.14,
+                0.825,
+                0.166
+            ],
+            "angle": 0,
+            "content": "Table C.4: Additional examples of infills generated by the prompting baseline along with reference infills for the stories infilling task."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.174,
+                0.167,
+                0.822,
+                0.884
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Beginning (X)</td><td>Middle (Z)</td><td>End (Y)</td></tr><tr><td>Jasmine had homework to do. She did not have a pencil. She looked around the room for one.</td><td>The room did not have a pencil Jasmine made her pencil and done her work. She looked Her mom brought her a pencil. Jasmine wrote her homework Jasmine took her pencil out of the room. She now Jasmine pretended not to have done her homework. She got Her brother finally brought a pencil. They could finally start! Her brother brought a pencil and solved her problem. Jasmine Her brother brought a pencil and solved her problem. Jasmine Her friend offered her a pencil. Jasmine used the pencil Her teacher saw her looking at her phone. Then, Jas Her mom came in and handed her the extra pencil! Jas Her mom brought her a pencil. Jasmine then got to Jasmine sent her friend money to do it. She was</td><td>Jasmine decided not to do her homework.</td></tr><tr><td>Jane had recently gotten a new job. She was nervous about her first day of work. On the first day of work, Jane overslept.</td><td>Jane arrived at work an hour late Despite arriving late, Jane was greeted warmly by her coworkers Jane learned that you never can predict the future. It was Jane learned not to rush into anything. Jane made a good Many of her peers were not pleased with her. Jane got Jane ended up quitting her new job early. She was happy More time with the wrong people gave her anxiety. She was Jane learned that you do not mess up at a job well Jane realized she needed to find new work contacts. At After her first day of work, Jane was too tired to Secretly, Jane hated her new job. It made her Jane realized she worked at a mid-sized company with long Jane&#x27;s boss realized she did not show up to work on</td><td>Jane did not make a good impression at her new job.</td></tr></table>"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "24"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.172,
+                0.126,
+                0.825,
+                0.152
+            ],
+            "angle": 0,
+            "content": "Table C.5: Examples of infills generated by the supervised fine-tuned model along with reference infills for the stories infilling task."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.174,
+                0.153,
+                0.822,
+                0.898
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Beginning (X)</td><td>Middle (Z)</td><td>End (Y)</td></tr><tr><td>Randy had recently separated from his wife. He felt very lonely and sad all the time. He considered trying to patch things up with Vera, his ex-wife.</td><td>He decided to get a puppy instead\nHe started enjoying his time with Vera even more. Now, \nEventually Randy missed Vera very much and developedWhile making memories with \nHe finally decided he had enough pain for no good reason.\nHe began spending more time with his new girlfriend. Now he \nFor the first time since the divorce he started to make new \nSoon Randy and Vera were rekindling their romance. They \nHe started seeing someone new a day.\nNow, he feels \nHe signed up for a class in Same Sex \nLoving Uniqueness \nFor the first time in months, Randy and \nVera celebrated a \nHe introduced Paige to his new employ-\nees. They are now good \nAfter some contemplating, he decided to patch things up with Vera \nHe now looks for another partner to enjoy his life with.</td><td>His decision made him happy and he no longer felt sad.</td></tr><tr><td>Ken put a bottle of beer in the freezer. He heard a popping noise. He looked in the freezer and saw the bottle had burst.</td><td>He didn&#x27;t want to wait for another beer to get cold\nThe popping noise was a freezer needle getting caught. It fell \nKen had a cold for thirty minutes. After the popping noise \nHe stored the beer in the freezer. The pouring sound was \nHe didn&#x27;t know why it didn&#x27;t stop the ringing.\nWeeks later, Ken knew the reason for the freezing. It \nLater Ken learned the gas is out of gas and will end \nHe left the freezer for the beer. It stayed in there \nBefore he knew it, the popping noise was gone. He \nThe popping noise that Ken heard indicated it should have freezer shut \nThe popping noise became louder and more frequent. Now Ken&#x27;s \nHe froze the beer and drank it from a mason jar \nHe went to the store to buy another. Now, he</td><td>He drank a warm beer instead.</td></tr></table>"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "25"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.172,
+                0.14,
+                0.825,
+                0.166
+            ],
+            "angle": 0,
+            "content": "Table C.6: Additional examples of infills generated by the supervised fine-tuned model along with reference infills for the stories infilling task."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.174,
+                0.167,
+                0.822,
+                0.884
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Beginning (X)</td><td>Middle (Z)</td><td>End (Y)</td></tr><tr><td>Jasmine had homework to do. She did not have a pencil. She looked around the room for one.</td><td>The room did not have a pencil Jasmine decided to talk to her mom about it. They Her dad now has to teach her math! Pick- ing up Jasmine took her math test later in the day. She Jasmine got an A on her assignment. She continued to Her brother finally brought a pencil. They could finally start! Her brother brought a pencil and solved her problem. Jasmine Jasmine used her pink and brown high-lighter instead. Jasmine went to school. She was able to finish her Jasmine made her own pencil. At school, everyone was Her brotherly earrings helped her stay up more. While Jasmine made her own pencil. She ended up getting good Jasmine sent her sister to do it. Now it was</td><td>Jasmine decided not to do her homework.</td></tr><tr><td>Jane had recently gotten a new job. She was nervous about her first day of work. On the first day of work, Jane overslept.</td><td>Jane arrived at work an hour late After arriving home, Jane realized her apartment was not ready. After work, Jane returned home and ate a lousy sandwich. After her first day of work, Jane learned that mistakes happen Due to her bad first day, Jane was fired from her Ending: Jane ended up late for work. Although she On her 2nd day of work, Jane came home late On the second day of work, Jane arrived late and unprepared By the end of the day, Jane was tired and weary After her first day of work, Jane&#x27;s boss recommended her Secretly, Jane hated her new job. Se-cretly, After work, Jane felt worn out and tired. The end By the end of the day Jane felt confident and like she</td><td>Jane did not make a good impression at her new job.</td></tr></table>"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.51,
+                0.961
+            ],
+            "angle": 0,
+            "content": "26"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.172,
+                0.262,
+                0.825,
+                0.289
+            ],
+            "angle": 0,
+            "content": "Table C.7: Examples of infills generated by the GFlowNet fine-tuned model along with reference infills for the stories infilling task."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.174,
+                0.289,
+                0.822,
+                0.761
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Beginning (X)</td><td>Middle (Z)</td><td>End (Y)</td></tr><tr><td>Randy had recently separated from his wife. He felt very lonely and sad all the time. He considered trying to patch things up with Vera, his ex-wife.</td><td>He decided to get a puppy instead\nHe went to a friend&#x27;s house to talk.\nHe took Vera to the park for a picnic.\nHe finally decided to call Vera on the phone.\nHe asked Vera to meet him and they did.\nHe decided to get a dog for his family.\nHe went to Vera&#x27;s house.\nHe told Vera that he was ready to love again.\nHe bought Vera a nice gift.\nHe had two dogs and felt like he missed Vera.\nHe had a good couple of years.\nHe wanted to get Vera and he eventually did.\nHe wanted to sit in a room alone.</td><td>His decision made him happy and he no longer felt sad.</td></tr><tr><td>Ken put a bottle of beer in the freezer. He heard a popping noise. He looked in the freezer and saw the bottle had burst.</td><td>He didn&#x27;t want to wait for another beer to get cold\nHe decided to stop drinking beer.\nKen suddenly smelled a bottle of beer.\nHe thought it was something\nHe walked around the freezer to see the bottle.\nHe tried to clean up the freezer.\nHe was angry when he had to drink it.\nHe stepped down from the freezer\nHe made a big joke that night.\nHe looked in a jug and see it was, out.\nHe then looked for a beer the freezer.\nHe tried to enjoy the rest.\nHe watered the bottle,</td><td>He drank a warm beer instead.</td></tr></table>"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "27"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.172,
+                0.168,
+                0.825,
+                0.193
+            ],
+            "angle": 0,
+            "content": "Table C.8: Additional examples of infills generated by the GFlowNet fine-tuned model along with reference infills for the stories infilling task."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.174,
+                0.194,
+                0.822,
+                0.639
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Beginning (X)</td><td>Middle (Z)</td><td>End (Y)</td></tr><tr><td>Jasmine had homework to do. She did not have a pencil. She looked around the room for one.</td><td>The room did not have a pencil\nShe was in a panic.\nShe had to place the pencil in her pocket.\nShe had several her classmates.\nShe looked for a neat pencil.\nShe thought she looked everywhere.\nHer brother walked in and borrowed her pencil.\nShe only had a few minutes to do it.\nShe never had her pencil.\nShe saw a pen in her closet.\nShe searched everywhere for a pencil.\nShe went to her room.\nShe didn’t find one.</td><td>Jasmine decided not to do her homework.</td></tr><tr><td>Jane had recently gotten a new job. She was nervous about her first day of work. On the first day of work, Jane overslept.</td><td>Jane arrived at work an hour late\nShe lost her job after the first day of work.\nShe was egged on by her boss.\nShe decided to over hyp her first day.\nShe arrived late and missed her first day of work.\nShe was nervous about going to work.\nShe ended up getting a good job.\nJane waited almost a day to get to work.\nHer first day was very difficult.\nShe made a couple of phone calls.\nShe arrived at the office and was fired.\nShe was all out of coffee.\nJane was surprised she did not make a good impression.</td><td>Jane did not make a good impression at her new job.</td></tr></table>"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.327,
+                0.778,
+                0.671,
+                0.792
+            ],
+            "angle": 0,
+            "content": "Table D.1: Two training examples from the SUBJ dataset."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.177,
+                0.793,
+                0.821,
+                0.843
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Text (X)</td><td>Label (Y)</td></tr><tr><td>another story follows the relationship between a stepfather ( neeson ) and his young stepson .</td><td>objective</td></tr><tr><td>hoffman ’s performance is authentic to the core of his being .</td><td>subjective</td></tr></table>"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "28"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.335,
+                0.101,
+                0.663,
+                0.114
+            ],
+            "angle": 0,
+            "content": "Table D.2: Prompts used for subjectivity classification."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.175,
+                0.115,
+                0.825,
+                0.172
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Few-shot learning / Supervised fine-tuning</td><td>GFlowNet fine-tuning</td></tr><tr><td>Classify this movie review as objective or subjective: “[X]” This review is [Y].</td><td>Classify this movie review as objective or subjective: “[X]” This review is [Z], so it is [Y].</td></tr></table>"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.248,
+                0.207,
+                0.748,
+                0.221
+            ],
+            "angle": 0,
+            "content": "Table D.3: Hyperparameters for GFlowNet fine-tuning on subjectivity classification."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.361,
+                0.222,
+                0.64,
+                0.461
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>LoRA rank</td><td>256</td></tr><tr><td>LoRA scaling factor</td><td>16</td></tr><tr><td>LoRA dropout</td><td>0.</td></tr><tr><td>Batch size</td><td>16</td></tr><tr><td>Gradient accumulation steps</td><td>32</td></tr><tr><td>Learning rate</td><td>0.0005</td></tr><tr><td>Optimizer</td><td>AdamW</td></tr><tr><td>PFtemperature max</td><td>2</td></tr><tr><td>PFtemperature min</td><td>0.5</td></tr><tr><td>Reward temperature start</td><td>1.2</td></tr><tr><td>Reward temperature end</td><td>1.0</td></tr><tr><td>Reward temperature horizon</td><td>150</td></tr><tr><td>Buffer capacity</td><td>50</td></tr><tr><td>Number of steps</td><td>100</td></tr><tr><td>Number of samples</td><td>10</td></tr><tr><td>Maximum length</td><td>5</td></tr><tr><td>Minimum length</td><td>1</td></tr></table>"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.487,
+                0.827,
+                0.571
+            ],
+            "angle": 0,
+            "content": "Analysis. In Table E.4 we show some examples generated by PPO and GFlowNet fine-tuned models. We observe that PPO generates sequences with high rewards under the base model. These sequences, however, are not valid expressions, and, in fact, do not even call the tool. Instead, the model learns to simply repeat the expression. Repetitions being assigned high likelihood is an issue that has also been noted by Welleck et al. (2020). On the other hand, GFlowNets are able to generate the correct rationale to evaluate the expression."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.575,
+                0.825,
+                0.659
+            ],
+            "angle": 0,
+            "content": "There are still cases where the GFlowNet fails to produce the correct reasoning chain. We illustrate some of these examples in Table E.5. The errors fall mainly into 3 categories: 1) missing operands in longer OOD problems 2) incorrect operand being copied 3) incorrect operator being copied. The potential source for such errors is that the reward model assigns equally high rewards to rationales with these minor mistakes and thus the model generates them with likelihood. These errors can potentially be reduced by using better and potentially larger reward models."
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.274,
+                0.799,
+                0.722,
+                0.812
+            ],
+            "angle": 0,
+            "content": "Table D.4: Ablation studies on subjectivity classification using GPT-J 6.8B."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.253,
+                0.813,
+                0.745,
+                0.923
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Method</td><td colspan=\"3\">Test accuracy (%) ↑</td></tr><tr><td></td><td colspan=\"3\">Training samples</td></tr><tr><td></td><td>10</td><td>20</td><td>50</td></tr><tr><td>GFlowNet fine-tuning</td><td>71.4 ± 1.8</td><td>81.1 ± 0.4</td><td>87.7 ± 2.2</td></tr><tr><td>(-) Seed buffer</td><td>64.7 ± 8.9</td><td>68.7 ± 1.7</td><td>77.0 ± 5.5</td></tr><tr><td>(-) Seed buffer (-) Aggregating</td><td>63.9 ± 7.1</td><td>65.9 ± 2.4</td><td>75.4 ± 3.3</td></tr></table>"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "29"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.171,
+                0.107,
+                0.825,
+                0.133
+            ],
+            "angle": 0,
+            "content": "Table D.5: Top sample rationales for the SUBJ test set using the instruct-fine-tuned GPT-J 6B as both the reward model and the base model for the GFlowNet, which is trained on 50 labeled examples."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.184,
+                0.134,
+                0.812,
+                0.361
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>True label</td><td colspan=\"2\">Rationale</td><td>Frequency</td></tr><tr><td rowspan=\"7\">Objective</td><td rowspan=\"7\">This review is</td><td>about a factual event</td><td>11.45%</td></tr><tr><td>describing a factual event</td><td>10.23%</td></tr><tr><td>based on a factual statement</td><td>9.47%</td></tr><tr><td>about a historical event</td><td>8.77%</td></tr><tr><td>about a movie review</td><td>7.41%</td></tr><tr><td>based on facts</td><td>5.64%</td></tr><tr><td>about a factual statement</td><td>4.61%</td></tr><tr><td rowspan=\"7\">Subjective</td><td rowspan=\"7\">This review is</td><td>about a movie review</td><td>26.21%</td></tr><tr><td>about the movie experience</td><td>26.17%</td></tr><tr><td>about the movie</td><td>7.60%</td></tr><tr><td>describing a movie review</td><td>3.44%</td></tr><tr><td>about the movie review</td><td>2.31%</td></tr><tr><td>based on a fictional story</td><td>2.30%</td></tr><tr><td>describing the movie experience</td><td>2.01%</td></tr></table>"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.171,
+                0.378,
+                0.825,
+                0.404
+            ],
+            "angle": 0,
+            "content": "Table E.1: Examples from the integer arithmetic dataset. Note that the results of the two-term expressions are evaluated by the calculator."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.22,
+                0.405,
+                0.776,
+                0.461
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Question (X)</td><td>Rationale (Z)</td><td>Answer (Y)</td></tr><tr><td>Question: 6 - 0 - 4 - 8 = ? Answer:</td><td>6 - 0 =, 6 - 4 =, 2 - 8 =</td><td>. The answer is -6.</td></tr><tr><td>Question: 9 + 4 - 8 = ? Answer:</td><td>9 + 4 =, 13 - 8 =</td><td>. The answer is 5.</td></tr></table>"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.321,
+                0.48,
+                0.676,
+                0.494
+            ],
+            "angle": 0,
+            "content": "Table E.2: Hyperparameters for the Integer Arithmetic Task"
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.354,
+                0.495,
+                0.641,
+                0.749
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>LoRA rank</td><td>64</td></tr><tr><td>LoRA scaling factor</td><td>16</td></tr><tr><td>LoRA dropout</td><td>0.1</td></tr><tr><td>Batch size</td><td>16</td></tr><tr><td>Gradient accumulation steps</td><td>32</td></tr><tr><td>Learning rate</td><td>0.0001</td></tr><tr><td>Optimizer</td><td>AdamW</td></tr><tr><td>PFtemperature max</td><td>2</td></tr><tr><td>PFtemperature min</td><td>0.5</td></tr><tr><td>Reward temperature start</td><td>1.1</td></tr><tr><td>Reward temperature end</td><td>0.5</td></tr><tr><td>Reward Temperature horizon</td><td>150</td></tr><tr><td>Buffer capacity</td><td>50</td></tr><tr><td>Number of training steps</td><td>1000</td></tr><tr><td>Evaluation temperature</td><td>0.1</td></tr><tr><td>Number of samples</td><td>10</td></tr><tr><td>Maximum length</td><td>20</td></tr><tr><td>Minimum length</td><td>5</td></tr></table>"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.171,
+                0.767,
+                0.825,
+                0.793
+            ],
+            "angle": 0,
+            "content": "Table E.3: Ablation study to understand the effect of the number of rationales used to seed the buffer on the test accuracy \\((\\%)\\) for the integer arithmetic task."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.32,
+                0.793,
+                0.674,
+                0.917
+            ],
+            "angle": 0,
+            "content": "<table><tr><td></td><td colspan=\"3\">Number of operands</td></tr><tr><td></td><td colspan=\"2\">In-distribution</td><td>OOD</td></tr><tr><td>Number of seed rationales</td><td>3</td><td>4</td><td>5</td></tr><tr><td>0</td><td>22.6</td><td>5.8</td><td>3.4</td></tr><tr><td>10</td><td>58.6</td><td>52.3</td><td>20.2</td></tr><tr><td>25</td><td>48.8</td><td>56.8</td><td>24.6</td></tr><tr><td>50</td><td>95.2</td><td>75.4</td><td>40.7</td></tr></table>"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.51,
+                0.96
+            ],
+            "angle": 0,
+            "content": "30"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.251,
+                0.226,
+                0.744,
+                0.24
+            ],
+            "angle": 0,
+            "content": "Table E.4: Samples generated by PPO fine-tuned and GFlowNet fine-tuned models."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.207,
+                0.24,
+                0.789,
+                0.419
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Question (X)</td><td>Generated rationale (Z)</td><td>log R</td></tr><tr><td rowspan=\"2\">Question: 1 - 9 + 8 = ? Answer:</td><td>1 - 9 - 8</td><td>-13.17</td></tr><tr><td>1 - 9 = -8, -8 + 8 = 0</td><td>-27.75</td></tr><tr><td rowspan=\"2\">Question: 8 + 7 + 2 + 7 = ? Answer:</td><td>8 + 7 + 2 + 7</td><td>-2.39</td></tr><tr><td>8 + 7 = 15, 15 + 2 = 17, 17 + 7 = 24</td><td>-11.72</td></tr><tr><td rowspan=\"2\">Question: 7 - 5 + 8 - 0 - 6=? Answer:</td><td>7 - 5 +</td><td>-1.22</td></tr><tr><td>7 - 5 = 2, 2 + 8 = 10, 10 - 0 = 10, 10 - 6 = 4</td><td>-7.99</td></tr></table>"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.283,
+                0.674,
+                0.713,
+                0.687
+            ],
+            "angle": 0,
+            "content": "Table E.5: Examples of errors made by the GFlowNet-fine-tuned model."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.216,
+                0.687,
+                0.779,
+                0.798
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Question (X)</td><td>Generated rationale (Z)</td></tr><tr><td>Question: 7 + 3 + 4 - 9 + 9 =? Answer:</td><td>7 + 3 = 10, 10 - 9 = 1, 1 + 4 = 5</td></tr><tr><td>Question: 8 + 5 - 1 - 4 + 5 =? Answer:</td><td>8 + 5 = 13, 13 - 1 = 12, 12 + 4 = 16, 16 - 4 = 12</td></tr><tr><td>Question: 0 + 5 - 6 - 1 - 6 =? Answer:</td><td>0 + 5 = 5, 5 - 6 = -1, -1 - 6 = -7, -7 + 1 = -6</td></tr></table>"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
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+            "angle": 0,
+            "content": "31"
+        }
+    ]
+]
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+# AMORTIZING INTRACTABLE INFERENCE IN LARGE LANGUAGE MODELS
+
+Edward J. Hu*, Moksh Jain*, Eric Elmoznino
+
+Mila - Quebec AI Institute, Université de Montréal
+
+{edward.hu,moksh.jain,eric.elmoznino,...
+
+Younesse Kaddar
+
+University of Oxford
+
+younesse.kaddar@chch.ox.ac.uk
+
+Guillaume Lajoie†, Yoshua Bengio°, Nikolay Malkin
+
+Mila - Quebec AI Institute, Université de Montréal
+
+...,guillaume.lajoie,yoshua.bengio,nikolay.malkin}@mila.quebec
+
+# ABSTRACT
+
+Autoregressive large language models (LLMs) compress knowledge from their training data through next-token conditional distributions. This limits tractable querying of this knowledge to start-to-end autoregressive sampling. However, many tasks of interest—including sequence continuation, infilling, and other forms of constrained generation—involve sampling from intractable posterior distributions. We address this limitation by using amortized Bayesian inference to sample from these intractable posteriors. Such amortization is algorithmically achieved by fin-tuning LLMs via diversity-seeking reinforcement learning algorithms: generative flow networks (GFlowNets). We empirically demonstrate that this distribution-matching paradigm of LLM fine-tuning can serve as an effective alternative to maximum-likelihood training and reward-maximizing policy optimization. As an important application, we interpret chain-of-thought reasoning as a latent variable modeling problem and demonstrate that our approach enables data-efficient adaptation of LLMs to tasks that require multi-step rationalization and tool use.
+
+Code: https://github.com/GFNOrg/gfn-lm-tuning.
+
+# 1 INTRODUCTION
+
+Autoregressive large language models (LLMs) trained on general-domain data are vast stores of world knowledge (Petroni et al., 2019). They are typically optimized by predicting a token given its preceding context; therefore, tractable inference over this knowledge is limited to sampling conditioned on a prefix. Many useful tasks, such as infilling (Zhu et al., 2019; Liu et al., 2019), generating text conditioned on length or lexical constraints (Hokamp & Liu, 2017; Hu et al., 2019), and finding the most likely sequence continuation, involve intractable inference in LLMs.
+
+Such tasks are related to the problem of reasoning, which has been framed as one of probabilistic inference (Gershman & Goodman, 2014). Correspondingly, the linguistic expression of reasoning can be seen as inference over language. For example, we can interpret chain-of-thought reasoning (Wei et al., 2022; Kojima et al., 2022), a paradigm of reasoning in language models, as a problem of intractable posterior inference. Given a question-answer pair  $(X,Y)$ , we are interested in finding latent chains of thought - token sequences  $Z$  that contribute the most to the conditional likelihood
+
+$$
+p (Y \mid X) = \sum_ {Z} p _ {\mathrm {L M}} (Z Y \mid X) = \sum_ {Z} p _ {\mathrm {L M}} (Y \mid X Z) p _ {\mathrm {L M}} (Z \mid X), \tag {1}
+$$
+
+where  $p_{\mathrm{LM}}$  denotes the likelihood assigned to a sequence by a language model and apposition of variables (e.g., XZY) denotes the concatenation of the token sequences.
+
+While past work has relied on prompting and in-context learning to produce  $Z$ 's that lead to the correct  $Y$ , treating  $Z$  as a hidden variable in a latent variable model (LVM) renders chain-of-thought reasoning a Bayesian inference problem (Fig. 1). For this LVM, the distribution we must sample from is the posterior  $p_{\mathrm{LM}}(Z \mid X, Y) = \frac{p_{\mathrm{LM}}(XZY)}{\sum_{Z'} p_{\mathrm{LM}}(XZ'Y)}$ . Such sampling is intractable: while it is easy to evaluate  $p_{\mathrm{LM}}(XYZ)$ , the conditional distributions needed to sample  $Z$  from  $p_{\mathrm{LM}}(Z \mid X, Y)$  one token at a time are not easy to compute.
+
+![](images/c6cca71215175d04cf3778d8e4c9df6f17270eb4fa3f17f04fe57c211809874f.jpg)  
+Figure 1: Left: Three problems of reasoning in language - sentence infilling, chain-of-thought reasoning, and problem-solving with external tool use - can all be seen as instances of the latent variable model at the top left, where an input  $(X)$  generates the output  $(Y)$  via a latent variable  $(Z)$ . Right: We fine-tune an LLM to sample from the Bayesian posterior over  $Z$ , conditioning on  $X$  and optionally on  $Y$ . If conditioned on  $Y$ , the trained policy can be used to sample diverse latent sequences (e.g., for infilling, §4.2). If not conditioned on  $Y$ , the policy can sample  $Z$ , and thus predict  $Y$ , for inputs  $X$  not seen during training (e.g., for classification and multi-step reasoning, §4.3, 4.4). As shown in §4.4, modeling the full diversity of the posterior aids generalization.
+
+A standard method to sample approximately from intractable posterior distributions is Markov chain Monte Carlo (MCMC), but it is difficult to craft good proposal distributions for multi-modal distributions over language data (Miao et al., 2019; Zhang et al., 2020a; Lew et al., 2023), and inference on a new input may be prohibitively slow. Alternatively, one can turn to reinforcement learning (RL) approaches such as proximal policy optimization (PPO; Schulman et al., 2017), where the language model is treated as a policy to be fine-tuned. However, these do not aim to model the full diversity of the distribution; instead, learned policies settle around a small number of modes. In both cases, issues with this mode collapse are exacerbated when the target distribution is misspecified, leading to the undesirable behavior of overoptimized samplers (Gao et al., 2023).
+
+Amortized probabilistic inference – that is, training a model to approximate a distribution of interest – provides a principled, efficient, and potentially scalable way to draw samples from the distribution (Beal, 2003). One way to implement amortized inference for high-dimensional discrete data such as text is using generative flow networks (GFlowNets; Bengio et al., 2021), which are diversity-seeking reinforcement learning algorithms that train policies to sample objects (such as a token sequence  $Z$ ) with probability proportional to a given reward function, such as the joint  $p_{\mathrm{LM}}(XZY)$ .
+
+In this work, we present a method that initializes the GFlowNet policy with a pretrained LLM and continues to train it with a reward objective that can be evaluated with the same LLM. The result is a different type of fine-tuning (FT) procedure for text generation that has a number of advantages, including improved sample diversity, data efficiency, and out-of-distribution generalization. GFlowNet fine-tuning makes the language model sample from the target distribution, enabling amortized inference in a number of applications (Fig. 1).
+
+Leveraging this approach, we empirically demonstrate the possibilities and benefits of learning to sample from intractable distributions over text continuations, latent reasoning chains, and tool use sequences using GFlowNet fine-tuning. Notably, the diversity of samples from the models trained with GFlowNet fine-tuning is beneficial in Bayesian model averaging settings, such as when aggregating answers to questions obtained via multiple reasoning chains. For example, using a pretrained language model with 6B parameters, our method shows an absolute improvement of  $10.9\%$  over supervised fine-tuning on subjectivity classification with only 10 labeled examples (§4.3) and outperforms supervised fine-tuning and PPO by  $63\%$  on integer arithmetic with 50 demonstrations, with notable improvements in out-of-distribution generalization (§4.4). Moreover, the benefits of amortized inference allow us to efficiently sample from the fine-tuned model at scale. Our contributions include:
+
+(1) A general algorithm for amortized sampling from intractable LLM posteriors.  
+(2) A probabilistic approach to fine-tuning LLMs to perform chain-of-thought reasoning.  
+(3) Empirical results on sequence continuation, natural language reasoning, integer arithmetic with tool use, and story infilling.
+
+# 2 MOTIVATING EXAMPLE: GENERATING RANDOM NUMBERS WITH LLMS
+
+We consider a simple task that highlights the limitations of reward-maximizing reinforcement learning (RL) methods in fine-tuning LLMs. For readers unfamiliar with RL, we refer to Sutton & Barto (2018) and include a glossary in §A.1 to define key terms used throughout this paper. The task involves generating random numbers from a given distribution when prompted 'The following is a random integer drawn uniformly between 0 and 100:'. This task is a minimal instantiation of the problem we study in the paper: sample from a target distribution given an unnormalized density.
+
+(a) Base model  
+(b) PPO fine-tuning  
+(c) GFlowNet fine-tuning  
+Figure 2: Empirical distributions of 512,000 integers from 1 to 100 generated by GPT-J fine-tuned with PPO (reward-maximizing; b) and GFlowNet fine-tuning (distribution-matching; c). Note the logarithmic  $y$ -scale.  
+![](images/b8bd60645f42132e5ddecdc5a51756e06e79e6081a470c31c756c6e0df7016a4.jpg)  
+$50.5\%$  of samples are valid numbers.  
+$95.8\%$  of samples are valid numbers.  
+$100\%$  of samples are valid numbers.
+
+Although the target distribution is tractable, making the task seemingly straightforward, it serves as a useful illustration of the behaviors of different fine-tuning methods.
+
+Renda et al. (2023) found that pretrained LLMs perform quite poorly on this task: the distribution of numbers generated with the above prompt will be far from uniform (Fig. 2a shows an example using an instruction fine-tuned GPT-J 6B (Wang & Komatsuzaki, 2021) $^{1}$ ). There may be many reasons for this, among them the effects of instruction fine-tuning and the LLM's possible bias towards numbers that are more frequent in the training data (e.g., numbers starting with '1' are more frequent due to the properties of many natural data-generating processes (Benford, 1938)).
+
+While reward-maximizing RL can teach the model to generate valid numbers (by penalizing outputs that are not numbers from 1 to 100), it would not resolve the distribution skew introduced during pretraining. Indeed, rewarding all valid integers equally leads to an expected gradient of zero for policy gradient methods. Fig. 2b shows that while most samples are valid numbers after PPO training, the distribution remains highly skewed.
+
+Instead, we can take a principled approach by training the LLM to match the target distribution with a GFlowNet learning objective. Such an objective directly optimizes the likelihood of the model generating a number to be proportional to the reward for that number, which is the number's (potentially unnormalized) probability under the target distribution. When the policy is initialized as the pretrained LLM, the resulting distribution after GFlowNet fine-tuning is shown in Fig. 2c. Quantitatively, the KL divergence from the sampling distribution to the target (uniform) distribution decreases from 3.37 for the original LLM (on the support [0, 100]) to  $9.75 \cdot 10^{-5}$  for the GFlowNet-fine-tuned model.
+
+This example illustrates a general point: GFlowNet objectives provide a principled and flexible approach to fine-tuning LLMs to match a target distribution where reward-maximizing RL fails to. On this simple task, this distribution matching could also be achieved through supervised fine-tuning; however, this would require access to samples from the target distribution, which are unavailable in general (though not in this simple example). In the following sections, we further illustrate this point in non-trivial problems involving intractable inference, reasoning with latent variables, and tool use.
+
+# 3 FINE-TUNING LLMS TO SAMPLE FROM INTRACTABLE DISTRIBUTIONS
+
+We first describe how intractable inference emerges from interesting applications of LLMs, one of which is chain-of-thought reasoning seen through the lens of latent variable models, where the posterior distribution over the latent variable is intractable. We then discuss how GFlowNet objectives can be used to train amortized samplers to perform such intractable inference.
+
+# 3.1 PROBLEM: INTRACTABLE INFERENCE IN LARGE LANGUAGE MODELS
+
+Autoregressive language models decompose the distribution over sequences of tokens as a product of ordered conditionals:  $p(w_{1:N}) = p(w_1)p(w_2 \mid w_1) \cdots p(w_N \mid w_{1:N-1})$ . While this decomposition makes left-to-right sampling from the distribution tractable, sampling from other conditional distribu
+
+Table 1: Objects in language posterior inference. Given a pretrained 'teacher' LM  $p_{\mathrm{LM}}$ , we train a GFlowNet  $q_{\mathrm{GFN}}$  to sample the posterior  $p(Z \mid X, Y)$ . Amortization and generalization are achieved by making  $X$ , and optionally  $Y$ , an input to  $q_{\mathrm{GFN}}$ .  
+
+<table><tr><td>Object</td><td>Meaning</td><td>Example 1 (infilling)</td><td>Example 2 (subjectivity classification)</td></tr><tr><td>X</td><td>cause / condition / question</td><td>The cat was hungry.</td><td>A deeply moving storyline.</td></tr><tr><td>Z</td><td>mechanism / reasoning chain</td><td>She ate a mouse.</td><td>This review expresses personal feelings.</td></tr><tr><td>Y</td><td>effect / answer</td><td>Now the cat is sleepy, not hungry.</td><td>Answer: Subjective</td></tr><tr><td>p(Z | X)</td><td>conditional prior</td><td></td><td>pLM(Z | X)</td></tr><tr><td>p(Y | X, Z)</td><td>likelihood of effect given</td><td></td><td>pLM(Y | XZ)</td></tr><tr><td></td><td>cause and mechanism</td><td></td><td></td></tr><tr><td>p(Z, Y | X)</td><td>conditional joint, reward for Z</td><td></td><td>pLM(ZY | X)</td></tr><tr><td>p(Z | X, Y)</td><td>posterior (intractable!)</td><td colspan="2">approximated and amortized by GFlowNet qGFN(Z | X), Y)</td></tr><tr><td>q(Y | X)</td><td>posterior predictive /</td><td colspan="2">approximated as ∑Z qGFN(Z | X)pLM(Y | XZ),</td></tr><tr><td></td><td>Bayesian model average</td><td colspan="2">sampled as Z ~ qGFN(Z | X), Y ~ pLM(Y | XZ)</td></tr></table>
+
+tions is intractable. Various problems of language modeling can be viewed as sampling from such intractable conditionals in the distribution over sequences of an LLM; we give two such examples and related terminologies in Table 1. Some tasks we study in §4 are instances of these examples.
+
+Tempered and contrastive sampling. In many applications (e.g., translation, summarization, dialogue systems), one wishes to sample from a low-temperature distribution over sequences  $Z$  conditioned on a prefix  $X$ , i.e.,  $q(Z \mid X) \propto p_{\mathrm{LM}}(XZ)^{1 / T}$  for some temperature  $T < 1$ , as high-likelihood samples are more likely to be fluent or accurate continuations of  $X$  (Tillmann & Ney, 2003). The limit of  $T \to 0$  gives a distribution that is peaky on the most likely continuation. However, sampling from  $q$ , or finding its mode, is intractable, and it is common to resort to approximations, such as tempering the tokenwise conditional distributions or using beam search to search for a mode. A related problem is sampling a continuation with a correction for its unconditional likelihood, e.g.,  $q(Z \mid X) \propto p_{\mathrm{LM}}(XZ)^{\alpha}p_{\mathrm{LM}}(Z)^{\beta}$  with  $\beta < 0$  and  $\alpha > 0$ , where applications again resort to approximating the next-token conditionals of  $q$  by tempering (Malkin et al., 2022b; Li et al., 2023).
+
+Infilling and reverse generation. Infilling is the task of sampling a sequence of tokens conditioned on both its prior and subsequent context, which can be understood as sampling from the distribution  $q(Z \mid X, Y) \propto p_{\mathrm{LM}}(XZY)$ , where  $X$  and  $Y$  are fixed. Reverse generation is a special case, where  $X$  is an empty sequence. Besides being a meaningful task in its own right (Liu et al., 2019; Zhu et al., 2019; Donahue et al., 2020; Susanto et al., 2020; Lu et al., 2022a), infilling and reverse generation are key components of newly emerging methods of LLM prompting, such as when LLMs are tasked with optimizing their own instruction sequences or reasoning steps (Zhou et al., 2023; Sordoni et al., 2023; Xu et al., 2023). Current applications achieve this by resorting to hand-engineered instructions and inverted prompts.
+
+Constrained generation. Sampling of text with constraints and penalties – for example, those on the presence or the absence of certain words or on the score of an auxiliary classifier evaluated on the text – can be understood as sampling from a distribution  $q(Z) \propto p_{\mathrm{LM}}(Z)c(Z)$ , where  $c$  is an externally specified constraint. Current approaches to the problem use tokenwise approximations (Liu et al., 2021), various problem-specific beam search and local search techniques (e.g., Schmaltz et al., 2016; Hokamp & Liu, 2017; Hu et al., 2019; Sha, 2020; Lu et al., 2022b) or classifier-guided conditional generation approaches (e.g., Yang & Klein, 2021; Meng et al., 2022).
+
+# 3.2 REASONING THROUGH LATENT VARIABLES
+
+Chain-of-thought reasoning (Wei et al., 2022; Kojima et al., 2022) helps LLMs solve complex problems by producing a reasoning chain before giving the final answer. LLMs pretrained on general domain data can learn to produce useful chains of thoughts given demonstrations, which are usually handcrafted or generated by prompting the LM. Interestingly, although the capacity for chain-of-thought reasoning only emerges in large language models, knowledge can also be extracted from smaller language models when they are carefully fine-tuned (Schick & Schütze, 2021).
+
+Motivated by this, we connect chain-of-thought reasoning to the general problem of inference in latent variable models illustrated in Fig. 1. Here, reasoning can be seen as posterior inference: sampling from the posterior distribution over a string of tokens  $Z$  conditioned on a prefix  $X$  and a suffix  $Y$ , given an autoregressive language model  $p_{\mathrm{LM}}$ . The posterior is defined as
+
+$$
+p _ {\mathrm {L M}} (Z \mid X, Y) = \frac {p _ {\mathrm {L M}} (X Z Y)}{\sum_ {Z ^ {\prime}} p _ {\mathrm {L M}} \left(X Z ^ {\prime} Y\right)} \propto p _ {\mathrm {L M}} (X Z Y). \tag {2}
+$$
+
+Our goal is to train models to sample  $Z$  from this posterior distribution. Intuitively, this allows us to sample likely reasoning chains that lead to the desired outcome  $Y$ . Although we take  $Z$  to be a string
+
+of tokens, the same formalism and the GFlowNet objectives apply to other structured latent objects, such as trees or sets of natural language statements, as long as one has access to a likelihood model  $p(Y \mid XZ)$ . While not investigated in this work, these generalizations could be important for formal reasoning and multi-step chains of inference. See, e.g., Yao et al. (2023); Hao et al. (2023); Besta et al. (2024) for approaches to reasoning in language using tree- or list-structured state spaces.
+
+A latent variable model of this form is useful when the marginal distribution  $p_{\mathrm{LM}}(Y \mid X)$  is harder to model than  $p_{\mathrm{LM}}(Z \mid X)$  and  $p_{\mathrm{LM}}(Y \mid XZ)$ , i.e., a difficult inference is broken down into a chain of easier ones. By training a model to match the Bayesian posterior  $p_{\mathrm{LM}}(Z \mid X, Y)$ , we can learn to sample latent reasoning chains that increase the likelihood of producing  $Y$  from  $X$  via the sampled  $Z$ .
+
+However, we can also fine-tune the language model  $p_{\mathrm{LM}}(Z \mid XY)$  itself to maximize the likelihood of data pairs  $(X,Y)$  under the LVM. While it is generally intractable to directly maximize the data likelihood  $p_{\mathrm{LM}}(X,Y) = \sum_Z p_{\mathrm{LM}}(XZY)$  because of the summation over  $Z$ , the (variational) expectation-maximization (EM) algorithm (Dempster et al., 1977; Beal, 2003; Koller & Friedman, 2009) can be used for this purpose. In the expectation step (E-step), we draw samples from the posterior over the latent variable  $p_{\mathrm{LM}}(Z \mid X,Y)$ , which could come from an amortized sampler of  $Z$ . In the maximization step (M-step), we maximize the log-likelihood of the joint probability of the sampled latent variables  $\mathbb{E}_{Z \sim p_{\mathrm{LM}}(Z|X,Y)} \log p_{\mathrm{LM}}(XZY)$  with respect to the parameters of the language model  $p_{\mathrm{LM}}$ . This combination of amortized inference (learning to sample the chain of thought) and supervised fine-tuning (optimizing the language model with the 'supervision' involving  $Z$  sampled from the amortized posterior) will be illustrated in one of our experiments (§4.3, Table 3).
+
+# 3.3 AMORTIZED INFERENCE WITH GFLOWNET OBJECTIVES
+
+For inference in the latent variable model, we leverage the probabilistic framework of generative flow networks (GFlowNets; Bengio et al., 2021; 2023). Using notation from Malkin et al. (2022a), we briefly introduce relevant GFlowNet concepts pertaining to autoregressive sequence generation. Here, GFlowNets learn policies to sample sequences  $Z = z_{1}z_{2}\dots z_{n}\top \in \mathcal{Z}$  (where  $\top$  denotes a stop symbol) from a distribution over the space of sequences  $\mathcal{Z}$ , given an unnormalized density (reward)  $R: \mathcal{Z} \to \mathbb{R}_{>0}$ . The generative process is the same as in autoregressive language models: generation begins with an empty string, and at the  $i$ -th step a token  $z_{i}$  is sampled from a policy  $q_{\mathrm{GFN}}(z_i \mid z_{1:i-1})$  which is then appended to the sequence. This process continues until a stop symbol  $\top$  is generated.
+
+The marginal likelihood  $q_{\mathrm{GFN}}^{\top}(Z)$  of sampling a terminal state  $Z = z_{1:n}\top$  is given by  $\prod_{i = 1}^{n}q_{\mathrm{GFN}}(z_i|z_{1:i - 1})q_{\mathrm{GFN}}(\top \mid z)$ , where  $z_{1:0}$  is understood to be the empty string. The goal of GFlowNet training is to fit a parametric policy  $q_{\mathrm{GFN}}(\cdot |\cdot ;\theta)$  such that  $q_{\mathrm{GFN}}^{\top}(Z)\propto R(Z)$ , i.e., the likelihood of generating a complete sequence is proportional to its reward.
+
+Learning objective. We use a modified version of the subtrajectory balance (SubTB; Madan et al., 2023) objective to account for trajectories being terminable at all states (Deleu et al., 2022). The objective for a sequence  $Z = z_{1:n} \top$  is
+
+$$
+\mathcal {L} (Z; \theta) = \sum_ {0 \leq i <   j \leq n} \left(\log \frac {R \left(z _ {1 : i} \top\right) \prod_ {k = i + 1} ^ {j} q _ {\mathrm {G F N}} \left(z _ {k} \mid z _ {1 : k - 1}\right) q _ {\mathrm {G F N}} (\top \mid z _ {1 : j})}{R \left(z _ {1 : j} \top\right) q _ {\mathrm {G F N}} (\top \mid z _ {1 : i})}\right) ^ {2}, \tag {3}
+$$
+
+For sequence generation tasks, the SubTB objective is equivalent to the path consistency objective (Nachum et al., 2017) in max-entropy RL (Haarnoja et al., 2017), which has been previously used in the context of text generation (Guo et al., 2021). See Appendix A.2 for further discussion.
+
+Training policy. As the objective in Eq. 3 can be minimized to 0 for all trajectories  $\tau$  simultaneously given enough model capacity, we can use trajectories sampled from any full-support distribution (training policy) to perform gradient descent on  $\mathcal{L}(\tau; \theta)$  with respect to  $\theta$ . As the space we are sampling from is combinatorially large, it is important to have a training policy that can efficiently explore  $\mathcal{Z}$ . To this end, we compose the mini-batch during training using trajectories from three sources: (1) the policy  $q_{\mathrm{GFN}}$ , (2) a tempered version of the current policy  $q_{\mathrm{GFN}}$  and (3) a replay buffer storing past trajectories. Replay buffers have been shown to be quite effective in improving GFlowNet training (Jain et al., 2022; Deleu et al., 2022; Shen et al., 2023).
+
+Parametrization, amortization, and generalization. To sample the latent sequence  $Z$  from the posterior defined in Eq. 2, we parametrize the GFlowNet policy as an autoregressive language model that samples the latent  $Z$  one token at a time from left to right. By setting the reward  $R(Z) = p_{\mathrm{LM}}(XZY) \propto p_{\mathrm{LM}}(Z \mid X,Y)$ , we learn a sampler for the posterior at convergence.
+
+As illustrated in Fig. 1, depending on the task, we can condition the GFlowNet policy on either  $X$  or  $X, Y$ . In cases such as reasoning (§3.2), where there is only a single correct  $Y$  for each  $X$  and we interested in predicting  $Y$  for unseen  $X$  at test time, we can simply condition on  $X$ . In this case, the GFlowNet policy is simply a language model that generates  $Z$  as a continuation of  $X$ . To be precise, we initialize  $q_{\mathrm{GFN}}$  as a copy of  $p_{\mathrm{LM}}$  that is conditioned on the prefix  $X$ , and then fine-tune<sup>2</sup> it with a GFlowNet objective. With this view, sampling  $Z$  is an inverse problem: we need to infer  $Z$  given a (conditional) prior  $p_{\mathrm{LM}}(Z \mid X)$  and an observation  $Y$  under likelihood model  $p_{\mathrm{LM}}(Y \mid XZ)$ .
+
+Allowing the GFlowNet policy to explicitly take  $X$  as input amortizes the sampling procedure and allows generalization to unseen  $X$ . In this sense, the GFlowNet is a Bayesian model (akin to a LM cascade (Dohan et al., 2022) or deep language network (Sordoni et al., 2023)), in which  $Z$  are conditionally sampled 'parameters' that transform  $X$  into  $Y$ . To predict the  $Y$  for an unseen  $X$ , one performs Bayesian model averaging by drawing samples of  $Z$  from  $q_{\mathrm{GFN}}(Z \mid X)$  followed by sampling from  $p_{\mathrm{LM}}(Y \mid XZ)$ .
+
+In tasks such as infilling (§4.2), however, the mapping from  $X$  to  $Y$  is one-to-many and  $Y$  is available at test-time. Here, we are interested in  $Z$  itself, rather than using it as an intermediate variable en route to generating  $Y$ . The GFlowNet policy thus has to be conditioned on both  $X$  and  $Y$ . To achieve this, the policy is conditioned on a prompt that contains both  $X$  and  $Y$  (for example, see Appendix C).
+
+# 4 EMPIRICAL RESULTS
+
+We first validate GFlowNet fine-tuning on text generation, where we seek to find likely sentence continuation given a prompt (§4.1) or fill in a missing sentence in a story (§4.2). Then, we study reasoning tasks that benefit from chain-of-thought reasoning (§4.3) and external tool use (§4.4).
+
+# 4.1 SENTENCE CONTINUATION
+
+Task description. A natural application for autoregressive language models is that of sequence continuation: given a prompt, the model should generate a high-likelihood completion. In applications such as creative writing, we would like the continuations to be semantically diverse while still having a high likelihood under the language model. To demonstrate the benefits of GFlowNet fine-tuning, we consider the task of sampling the next sentence following a prompt.
+
+Sampling autoregressively from the LM until a “.” token is reached is unlikely to produce samples that have a high likelihood because the distribution over sentences has a fat tail. Existing approaches to generate sequence continuations include beam search and its variations (Vijayakumar et al., 2018; Shao et al., 2017), top-  $k$  sampling (Fan et al., 2018), nucleus sampling
+
+(Holtzman et al., 2019), tempered autoregressive sampling, and fine-tuning using importance sampling (Shih et al., 2023), among others. While useful, most of these methods are ultimately hand-crafted heuristics that leave room for improvement. Furthermore, some of these methods (e.g., beam search) involve a computationally expensive search procedure, compared to a single pass of a learned inference model that amortizes over prompts. Our GFlowNet policy autoregressively samples the sequence until a period is sampled, indicating the end of the sentence. Given prompts  $X$ , the LM is fine-tuned to generate the continuations  $Z$  from the tempered posterior by being trained with reward  $R(Z) = p_{\mathrm{LM}}(Z|X)^{\frac{1}{T}}$ . When  $T = 1$ , the GFlowNet will trivially sample proportional to  $p_{\mathrm{LM}}$  without any fine-tuning, so we consider  $0 < T < 1$  to focus on the likely continuations.
+
+We consider a dataset of prompts from OpenWebText (Gokaslan et al., 2019) with a 1.5B param GPT-2 XL (Radford et al., 2019) as the base model. We draw 8 samples from the fine-tuned model conditioned on a fixed prompt, consider the maximum-likelihood sample under the LM, and report the average over the dataset of prompts. To measure the semantic diversity of the
+
+![](images/247978d45b5e4c8128a402696a7af7131f16b50fa7345f269a631361bcfee7f0.jpg)  
+Figure 3: Maximum log-likelihood and diversity of continuations sampled for fixed prompts. GFlowNet finetuning  $(\star)$  samples higher log-likelihood sentences while maintaining more sample diversity than the baselines (and ---), even when they are given  $5\times$  the compute.
+
+samples, we compute the mean pairwise cosine distance between the embeddings (from a pretrained encoder (Reimers & Gurevych, 2019)) of the generated samples and average it over the dataset. We compare to baselines that are commonly used for producing continuations from LMs at inference time (beam search, diverse beam search, nucleus sampling, autoregressive sampling, tempered autoregressive sampling, and greedy generation).
+
+Results. Quantitative results are reported in Fig. 3 and empirical samples are shown in Appendix B. At lower temperatures, our method excels in generating high-likelihood sentences, outperforming the leading baseline, diverse beam search. If we increase the number of beams (and therefore compute) to  $5 \times$  the number of samples produced by the GFlowNet, our performance remains comparable. Nevertheless, even in this scenario, the GFlowNet's generated samples exhibit notably higher diversity compared to diverse beam search and are on par with the best diversity-scoring benchmarks.
+
+# 4.2 INFILLING STORIES
+
+Task description. Next, we consider the story infilling task, a special case of the general infilling problem (§3.1), where given the beginning  $X$  and end  $Y$  of a story, the goal is to generate the middle of the story  $Z$  (Zhu et al., 2019). This is challenging for a language model sampled left to right since continuations  $Z$  conditioned only on  $X$  might be incompatible with the ending  $Y$ . We use the ROCStories corpus (Mostafazadeh et al., 2016), a dataset of short stories containing exactly 5 sentences each. Given the first 3 sentences and the last sentence, the goal is to generate the fourth sentence, which often involves a turning point in the story and is thus challenging to fill in.
+
+As we expect the base model to contain the required knowledge, for this task we use a GPT-2 Large model (Radford et al., 2019) fine-tuned on the entire ROCStories training set as the base model. For evaluating the approach, we consider 900 samples from the dataset as training data to learn  $q_{\mathrm{GFN}}(Z|X,Y)$  and evaluate the similarity of the generated infills on a dataset of 100 unseen stories. Along with the GFlowNet-fine-tuned model, we also consider two baselines: prompting the model to infill the story and supervised fine-tuning on the same data. Further details are in Appendix C.
+
+Results. To measure the similarity of the generated infills with the reference infills available in the dataset, we compute BERTScore (Zhang et al., 2020b), with DeBERTa (He et al., 2021) - which is correlated with human judgments
+
+along with BLEU-4 (Papineni et al., 2002) and GLEU-4 (better suited for sentences; Wu et al., 2016) metrics. Additionally, we also evaluate each method using GPT-4 as a judge. From our results summarized in Table 2, we observe that the infills generated by the model with GFlowNet fine-tuning are closer to the reference infills in the dataset than the baselines. By sampling from  $p_{\mathrm{LM}}(Z|X,Y)$ , the GFlowNet is able to account for the ending while generating the infill, resulting in infills that link the beginning and the end of the story coherently. For further analysis and details see Appendix C.
+
+Table 2: Evaluation of the generated infills.  
+
+<table><tr><td>Method</td><td>BERTScore</td><td>BLEU-4</td><td>GLEU-4</td><td>GPT4Eval</td></tr><tr><td>Prompting</td><td>0.081 ± 0.009</td><td>1.3 ± 0.5</td><td>3.2 ± 0.1</td><td>2.4</td></tr><tr><td>Supervised fine-tuning</td><td>0.094 ± 0.007</td><td>1.6 ± 0.8</td><td>3.7 ± 0.4</td><td>2.7</td></tr><tr><td>GFlowNet fine-tuning</td><td>0.184 ± 0.004</td><td>2.1 ± 0.2</td><td>4.2 ± 0.7</td><td>3.4</td></tr></table>
+
+# 4.3 SUBJECTIVITY CLASSIFICATION
+
+Task description. SUBJ (Pang & Lee, 2004) is a binary classification dataset for natural language understanding. It is a collection of movie reviews in which each review is labeled as objective, meaning that it references facts about the movie, or subjective, meaning that it expresses an opinion of the reviewer (see Table D.1 for examples). Given an unlabeled review, the model must predict whether it is objective or subjective. While supervised fine-tuning on the full dataset
+
+Table 3: Test accuracy  $(\%)$  on SUBJ using an instruct-fine-tuned GPT-J 6B.  
+
+<table><tr><td>Method</td><td colspan="3">Test accuracy (%) ↑</td></tr><tr><td>Zero-shot prompting</td><td colspan="3">51.7</td></tr><tr><td></td><td colspan="3">Training samples</td></tr><tr><td></td><td>10</td><td>20</td><td>50</td></tr><tr><td>Few-shot prompting</td><td>61.3 ± 6.2</td><td>61.8 ± 5.4</td><td>65.8 ± 10.5</td></tr><tr><td>Supervised fine-tuning</td><td>64.3 ± 2.8</td><td>69.1 ± 0.8</td><td>89.7 ± 0.4</td></tr><tr><td>GFlowNet fine-tuning</td><td>71.4 ± 1.8</td><td>81.1 ± 0.4</td><td>87.7 ± 2.2</td></tr><tr><td>+ Supervised fine-tuning</td><td>75.2 ± 1.8</td><td>78.7 ± 1.6</td><td>89.9 ± 0.2</td></tr></table>
+
+can achieve high test accuracy, we are interested in the low-data regime where we only have tens of labeled examples. We use the same instruction-tuned GPT-J 6B variant as in §2 for this experiment. Without any demonstrations, the model struggles with this task using the prompt in Table D.2 and achieves only  $51.7\%$  zero-shot accuracy.
+
+This task is hard likely because it requires a latent reasoning step. A review could be considered objective because it analyzes the plot or facts about the movie, or it could be subjective because it expresses a personal opinion or makes a judgment. We denote the review  $X$ , the predicted subjectivity  $Y$ , and the latent reason  $Z$ . Then, we GFlowNet-fine-tune the LLM  $q_{\mathrm{GFN}}(Z \mid X)$ , initialized with the
+
+base model  $p_{\mathrm{LM}}$  to match the Bayesian posterior over rationales in Eq. 2. At test time,  $q_{\mathrm{GFN}}(Z \mid X)$  generates 10 latent rationales ( $Z$ 's) for an unseen  $X$ . The LLM  $p_{\mathrm{LM}}$  then autoregressively samples from  $p_{\mathrm{LM}}(Y \mid XZ)$  to produce 10 answers, the majority vote of which becomes the final prediction.
+
+This posterior inference corresponds to the E-step in the EM algorithm, where the posterior  $p_{\mathrm{LM}}(Z \mid X,Y)$  is defined in Eq. 2. Further, as described in §3.2, we can take an M-step by updating  $p_{\mathrm{LM}}$  to maximize  $\log p_{\mathrm{LM}}(XZY)$  over a collection of  $Z$ 's sampled from the amortized posterior  $q_{\mathrm{GFN}}$ . This is equivalent to applying supervised fine-tuning after GFlowNet fine-tuning.
+
+Results. We present few-shot prompting and supervised fine-tuning with LoRA as baselines. In few-shot prompting, we prepend 0, 10, 20, or 50 training examples to each test example using the prompt shown in Table D.2. We randomly shuffle few-shot demonstrations and report the mean and variance in Table 3. In supervised fine-tuning, we directly maximize  $\log p_{\mathrm{LM}}(Y \mid X)$  over the same 10, 20, or 50  $(X, Y)$  pairs. The variance is over model initialization and batch order. All entries except zero-shot prompting are aggregated over 3 random seeds. See Appendix D for experiment details. GFlowNet fine-tuning consistently outperforms supervised fine-tuning in the low-data regime, as shown in Table 3. In some cases, performing supervised fine-tuning on top, which corresponds to running one step of the EM algorithm, further improves the performance.
+
+# 4.4 SOLVING ARITHMETIC PROBLEMS STEP BY STEP
+
+Task description. Arithmetic reasoning is a fitting benchmark to evaluate reasoning abilities of large language models as it requires multi-step reasoning and correctness is easy to evaluate (Cobbe et al., 2021). While the distribution of pretraining and fine-tuning data (Magister et al., 2023; Lee et al., 2023; Luo et al., 2023) and prompting choices (Imani et al., 2023) play a critical role in their arithmetic abilities, LLMs are susceptible to poor generalization by learning 'shortcuts' to reasoning (Dziri et al., 2023). We consider a simple integer arithmetic task (Fig. 1), with a general pretrained base model, rather than a one pretrained on mathematical
+
+Table 4: Test accuracy  $(\%)$  on an integer arithmetic task with addition and subtraction using a GPT-J 6B model. Training data only include samples with 3 or 4 operands.
+
+<table><tr><td rowspan="3">Method</td><td></td><td colspan="3">Number of Operands</td></tr><tr><td></td><td colspan="2">In-distribution</td><td>OOD</td></tr><tr><td></td><td>3</td><td>4</td><td>5</td></tr><tr><td rowspan="5">k-shot CoT</td><td>k=0</td><td>10.2</td><td>6.4</td><td>3.2</td></tr><tr><td>k=3</td><td>15.8 ± 3.1</td><td>11 ± 1.7</td><td>5.4 ± 0.2</td></tr><tr><td>k=5</td><td>20.4 ± 10.4</td><td>17.6 ± 0.6</td><td>6.6 ± 1.1</td></tr><tr><td>k=10</td><td>26.5 ± 1.4</td><td>15.2 ± 1.7</td><td>8.9 ± 1.9</td></tr><tr><td>k=20</td><td>35.5 ± 1.9</td><td>21 ± 1.4</td><td>10.5 ± 0.9</td></tr><tr><td colspan="2">Supervised fine-tuning</td><td>72.1 ± 1.3</td><td>19.6 ± 2.2</td><td>12.8 ± 5.7</td></tr><tr><td colspan="2">PPO</td><td>30.6 ± 4.1</td><td>13.7 ± 4.1</td><td>5.6 ± 3.1</td></tr><tr><td colspan="2">GFlowNet fine-tuning</td><td>95.2 ± 1.3</td><td>75.4 ± 2.9</td><td>40.7 ± 9.1</td></tr></table>
+
+tasks (Jelassi et al., 2023). To avoid the pitfalls of symbolic calculations with language models, we adopt the tool use setting (Schick et al., 2023), where the model is equipped with a calculator that can perform parts of the computation, implemented as in Cobbe et al. (2021): when the model outputs  $=$  the expression preceding it is evaluated and appended to the sequence. To prevent the model from "cheating", we limit the calculator to evaluate only two-term expressions. Consequently, reasoning here involves learning to plan using a tool with limited capabilities (Hao et al., 2023).
+
+For training, we use a synthetic dataset of arithmetic expressions, limited to addition and subtraction. Following Zelikman et al. (2022), we use a small set of 50 demonstrations  $(X,Z,Y)$  to seed the replay buffer in addition to 1000 examples  $(X,Y)$ . We use the same instruction-tuned GPT-J as in §4.3 as the base model. Further details are in Appendix E. We report the accuracy on two types of examples: (1) unseen in-distribution expressions (3 or 4 operands) and (2) longer out-of-distribution expressions (5 operands). As baselines, we consider zero-shot chain-of-thought prompting,  $k$ -shot prompting, supervised fine-tuning on the tool use sequences, and fine-tuning with PPO (Schulman et al., 2017). For all methods, we enable tool use and limit the model to generate only numbers and operators.
+
+Results. From the results summarized in Table 4, the base model performs poorly even with chain-of-thought prompts. Including examples in context improves the performance considerably, with monotonic improvements as the number of examples increases. Supervised fine-tuning improves the performance significantly on the in-distribution examples, but the model still struggles to generalize on the out-of-distribution examples. Fine-tuning with PPO results also yields poor performance, caused in part by the poor calibration of the base reward model, i.e. it cannot distinguish good rationales from bad ones. Even though the sequences generated with PPO (illustrated in Appendix E) have high rewards, they are spurious and do not even define valid calls to the tool.
+
+Such overoptimization to a misspecified reward is a widely noted issue in LLMs trained with RL (Gao et al., 2023). On the other hand, by matching the entire distribution, GFlowNet fine-tuning avoids collapsing to a single mode of the reward, thereby being robust to the misspecification of
+
+the reward (Eysenbach & Levine, 2022) and achieving significantly better performance on in and out-of-distribution examples. See Appendix E for additional results and analysis.
+
+# 5 FURTHER RELATED WORK
+
+Sampling from intractable marginals. Beyond the approximations mentioned in §3.1, sampling from intractable posterior distributions given by pretrained models for tasks such as infilling and constrained generation has been an object of study. Miao et al. (2019); Zhang et al. (2020a) use MCMC for these problems, Malkin et al. (2021) used a variable-neighborhood ascent for finding modes, and a sequential Monte Carlo approach was recently proposed by Lew et al. (2023). Others have studied the problem with masked language models, using them to perform variants of Gibbs sampling (Wang & Cho, 2019; Goyal et al., 2022; Yamakoshi et al., 2022) and recovering marginal distributions over small sets of tokens (Torroba Hennigen & Kim, 2023).
+
+GFlowNets. GFlowNets (Bengio et al., 2021) were originally proposed to learn policies for sampling discrete compositional objects from an unnormalized reward distribution, motivated by the need to sample diverse high-reward objects in scientific discovery (Jain et al., 2023), in particular, for biological sequence generation (Jain et al., 2022). The interpretation of GFlowNets as variational inference algorithms (Malkin et al., 2023; Zimmermann et al., 2023) makes them appropriate for sampling Bayesian posterior distributions over structured objects (e.g., Deleu et al., 2022; 2023; van Krieken et al., 2023; Hu et al., 2023).
+
+Chain-of-thought reasoning in LLMs. In recent work on classification and completion with language models, the latent reasoning chain  $Z$ , in the notation of §3.1, is called a 'chain of thought' (Wei et al., 2022). The chain of thought is typically generated by conditioning the language model on  $X$  with the use of specialized demonstrations or prompts (Kojima et al., 2022), with no guarantee of sampling the posterior accurately. Related to our Bayesian formulation, Wang et al. (2023b) noted that appropriately aggregating the conclusions  $Y$  from several latent chains  $Z$  improves predictive performance. In Xu et al. (2023); Zhou et al. (2022), a posterior over latent token sequences is sampled using MCMC, while Zelikman et al. (2022) propose fine-tuning on successful (high-reward, in our language) chains of thought, which achieves reward maximization but gives no guarantee of diversity. In concurrent work, Phan et al. (2023) use MCMC to sample chains-of-thought in problems with binary feedback. We expect these methods to generalize poorly to difficult exploration problems, while GFlowNet fine-tuning takes advantage of generalizable structure in the posterior and has a goal of sampling the full posterior over latent reasoning chains.
+
+# 6 CONCLUSION
+
+The knowledge compressed in LLMs is crucial for tasks such as infilling and constrained generation, but querying this knowledge involves sampling from intractable posterior distributions. We propose to use GFlowNet objectives to train LLMs to sample from such posterior distributions. Empirical results show that GFlowNet fine-tuning finds a better fidelity-diversity trade-off for text generation and also improves sample efficiency and generalization on downstream tasks compared to maximum-likelihood training or reward-maximizing policy optimization. As an amortized inference algorithm, our method converts computation into better test-time performance without additional data.
+
+Future work should investigate transfer and generalization across tasks, in particular, building a 'universal reasoner' as a model  $q(Z \mid X)$  shared between  $X$  from different tasks, as was recently considered by Wang et al. (2023a). One should investigate the benefit of using a better knowledge model, e.g., a more capable base LLM, as a starting point for GFlowNet fine-tuning. The ability to draw multiple samples from a GFlowNet can also be used to quantify epistemic uncertainty. Finally, we adopt the GFlowNet formalisms with the perspective of generalizing to latent variables  $Z$  with a richer generative process than left-to-right sampling. We hope that the GFlowNet paradigm will enable more flexible reasoning with LLMs in the future: extending probabilistic programming with language variables (Beurer-Kellner et al., 2023), using structured chains of thought (Yao et al., 2023; Besta et al., 2024), and extending to program synthesis and planning with world models.
+
+Limitations. Due to resource constraints, our experiments use models up to 6B parameters, but we expect the conclusions to hold for larger models. In fact, our method can potentially benefit larger models more: it is harder to optimize a larger model with maximizing objectives on a small amount of data. As with any on-policy method, exploration, especially in problems with more complex latents, remains an open problem. Furthermore, our method improves inference but not the knowledge in the LM. Issues such as hallucination or miscalibration, which are closely related to the knowledge representation, are thus not addressed.
+
+# ETHICS STATEMENT
+
+While we foresee no immediate negative societal consequences of our work, we hope that future researchers who build upon it will, as we have, bear in mind the potential of LLMs – and in particular of human-like reasoning in LLMs – to be used both for good and for harm.
+
+Research areas in safe and explainable AI that can benefit from GFlowNet fine-tuning include (1) interpretability of LLMs' reasoning processes and (2) fine-tuning with human feedback or an external reward, where diverse sampling can help prevent 'reward hacking' and overfitting to a misspecified target.
+
+# REPRODUCIBILITY
+
+We discuss the details of the proposed algorithms in §3.3 and provide all the implementation details and hyperparameters for the experiments in the main paper and appendix. Code for our experiments is available at https://github.com/GFNOrg/gfn-lm-tuning.
+
+# ACKNOWLEDGMENTS
+
+The authors are grateful to Bonaventure Dossou and Salem Lahlou for their help in the early stages of this project. We also thank Robert Hawkins, Arian Hosseini, Zhen Wang, and Anirudh Goyal for valuable discussions and suggestions of related work.
+
+GL acknowledges funding from CIFAR, Samsung, and a Canada Research Chair in Neural Computation and Interfacing.
+
+YB acknowledges funding from CIFAR, NSERC, IBM, Intel, Genentech, and Samsung.
+
+The research was enabled in part by computational resources provided by the Digital Research Alliance of Canada (https://alliancecan.ca), Mila (https://mila.quebec), and NVIDIA.
+
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+Peiyi Wang, Lei Li, Liang Chen, Feifan Song, Binghuai Lin, Yunbo Cao, Tianyu Liu, and Zhifang Sui. Making large language models better reasoners with alignment. arXiv preprint arXiv:2309.02144, 2023a.
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+Xuezhi Wang, Jason Wei, Dale Schuurmans, Quoc Le, Ed Chi, Sharan Narang, Aakanksha Chowdhery, and Denny Zhou. Self-consistency improves chain of thought reasoning in language models. International Conference on Learning Representations (ICLR), 2023b.  
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+Heiko Zimmermann, Fredrik Lindsten, Jan-Willem van de Meent, and Christian A. Naesseth. A variational perspective on generative flow networks. Transactions on Machine Learning Research (TMLR), 2023.
+
+# A ADDITIONAL BACKGROUND
+
+# A.1 GLOSSARY OF RL FOR LLMS
+
+We provide definitions for key terms used throughout the paper, with a focus on their relevance to our setting of fine-tuning large language models (LLMs).
+
+Reinforcement learning. Reinforcement learning (RL) is a branch of machine learning concerned with how agents should take actions in an environment to maximize cumulative rewards. In our context, RL is used to fine-tune the decision-making process of LLMs to improve their performance on specific tasks. For a more comprehensive overview, we refer readers to Sutton & Barto (2018).
+
+Policy. In RL, a policy is a strategy that defines the behavior of an agent by mapping states of the environment to actions. In our setting, a policy dictates how the language model generates text sequences based on the current context and learned parameters.
+
+Reward. A reward is a signal that evaluates the quality of an action taken by an agent in a particular state. In the fine-tuning of LLMs, rewards are used to guide the model towards generating more desirable text, such as more accurate predictions or more coherent continuations. More specifically, in the context of GFlowNets, the reward corresponds to the unnormalized posterior probability, and the GFlowNet aims to match it by learning a policy that generates samples proportional to their reward.
+
+Matching a distribution. Matching a distribution involves training a model to approximate a target probability distribution. In our work, this concept is applied to fine-tune LLMs so that their generated text matches the desired characteristics, such as adhering to a particular style or content constraint.
+
+Policy gradient methods. Policy gradient methods (such as PPO) are a subset of RL algorithms that optimize a policy by computing gradients of the expected reward with respect to the policy parameters. In the context of LLMs, these methods are used to fine-tune the model's parameters to increase the likelihood of generating high-reward text sequences.
+
+# A.2 LEARNING OBJECTIVE
+
+We use the subtrajectory balance learning objective for training GFlowNets (Madan et al., 2023). In the notation of that paper, with a forward policy  $P_F$ , backward policy  $P_B$  and state flow function  $F$ , the objective over a partial trajectory  $\tau = s_m \rightarrow \dots \rightarrow s_n$  is defined as follows
+
+$$
+\mathcal {L} _ {S u b T B} (Z; \theta) = \left(\log \frac {F \left(s _ {m}\right) \prod_ {i = m} ^ {n - 1} P _ {F} \left(s _ {i + 1} \mid s _ {i}\right)}{F \left(s _ {n}\right) \prod_ {i = m} ^ {n - 1} P _ {B} \left(s _ {i} \mid s _ {i + 1}\right)}\right) ^ {2} \tag {4}
+$$
+
+In the case of autoregressive generation of a sequence of tokens in a fixed order (left-right), the generative process is a tree, so there is only a single path to each state, and each state has a single parent. Thus,  $P_B(s|s') = 1$  trivially. Additionally, since each state is a valid terminable state, we can incorporate the modification to account for this from Deleu et al. (2022). Specifically, note that at convergence we have  $R(s_n^\top) = F(s_n)P_F(\top \mid s_n)$ . Using this, we can simply substitute  $F(s_{n}) = R(s_{n}^{\top}) / P_{F}(\top \mid s_{n})$  in Eq. (4). This allows us to avoid parameterizing a flow function separately, reducing additional complexity. The only learned object is now the forward sampling policy  $P_F$ , which we refer to as  $q_{GFN}$ . Summing this over all partial trajectories in a trajectory with equal weight  $(\lambda = 1)$ , we get the final learning objective in Eq. (3).
+
+# B SENTENCE CONTINUATION
+
+Additional details. We choose sentences as the level of granularity for our sequence continuation task because they are a natural unit of generation in language. Moreso than individual words, sentences are analogous to whole thoughts, reasoning steps, etc. because the compositional rules of syntax operate at the level of sentences. Whereas the meaning of a word is often ambiguous and context-dependent, the meaning of a sentence tends to be more self-contained.
+
+A naive solution to the task is to simply sample autoregressively from the LM until a “.” token is reached, marking the end of the sentence. In practice, however, this is unlikely to produce samples that have high likelihood because the distribution over sentences has a long tail: a vast number of sentences have small but non-zero probability, and collectively account for a substantial portion of the total probability mass. Instead, it would be desirable to sample from a low-temperature distribution over sentences, so that the distribution becomes more sparse and highly probable sentences are
+
+sampled more often. However, this in itself is a difficult distribution to sample from, and it is different from simply sampling autoregressively from the LM with a lower temperature applied to the distributions over the next words. For instance, as the temperature approaches 0, the desired distribution over sentences should converge to an argmax and produce the single most likely next sentence. However, if we try to apply the temperature to the distribution over the next words and sample autoregressively, then as the temperature approaches 0, the resulting policy will greedily construct a sentence by sequentially picking the most likely next word, which is unlikely to produce the highest probability sentence.
+
+Our GFlowNet sampling policy parametrizes a distribution  $p_{\mathrm{GFN}}(w_{i+1}|w_{1:i})$  using the same architecture as the LM  $p_{\mathrm{LM}}(w_{i+1}|w_{1:i})$ , and at the start of training, it is initialized with the same weights. The initial state of the GFlowNet consists of a prompt  $w_{1:k}$  that conditions generation (i.e., the text whose next sentence we are trying to sample). Each subsequent state  $w_{1:k+i}$  is an extension of this text and is obtained by sampling autoregressively from  $p_{\mathrm{GFN}}$ . This process terminates when a period “.” is sampled, indicating the end of the next sentence. If a predefined maximum sentence length is reached, we force a “.” action to manually terminate the generation.
+
+The GFlowNet policy is trained to sample sentences in proportion to their tempered probability under the original LM given the prompt. Denote the length of the prompt by  $k$ , the length of the sampled sentence by  $m$ , and the temperature by  $T$ . Then, the reward is:
+
+$$
+R \left(w _ {1: k + m}\right) \stackrel {\text {d e f}} {=} p _ {\mathrm {L M}} \left(w _ {k + 1: k + m} \mid w _ {1: k}\right) ^ {\frac {1}{T}} = \left(\prod_ {i = 1} ^ {m} p _ {\mathrm {L M}} \left(w _ {k + i} \mid w _ {1: k + i - 1}\right)\right) ^ {\frac {1}{T}} \tag {5}
+$$
+
+The intuition behind this reward is that we are assuming that the LM can reliably assign a high likelihood to high-probability sentences, that we wish to preferentially sample over low-probability ones. If we were to set  $T = 1$ , then the solution for the GFlowNet would be to sample proportionally to  $p_{\mathrm{LM}}$  (which it is initialized to). The preferential sampling of high-probability sentences is therefore obtained by setting  $0 < T < 1$ .
+
+To run our experiment, we obtained a dataset of 1000 prompts from OpenWebText (Gokaslan et al., 2019) that were each 1-3 sentences long, 50 of which were used for validation. Our LM consisted of a pretrained 1.5B parameter GPT2-XL (Radford et al., 2019), and our GFlowNet was a copy of this model that was fine-tuned in a lower-dimensional space of 80M parameters using LoRA (Hu et al., 2022).
+
+Additional results. We show examples of empirical next-sentence samples generated by our model in Table B.1 compared to baselines. The model was trained using a reward temperature of 0.875, which achieves a good balance between log-likelihood and diversity.
+
+# C INFILLING STORIES
+
+Additional details. See Table C.1 for training examples from the subset of the ROC Stories dataset used for the task. To condition the model on  $X$  and  $Y$ , as well as for the prompting baseline, we use the following prompt:
+
+"Beginning: {X}\n End: {Y}\n Middle: "
+
+An assumption we make in this paper is that the base language model already contains the knowledge required for the task, and the goal is to perform inference over this knowledge. However, for this task, none of the pretrained base models were good at assigning high likelihoods to plausible stories. Thus, we fine-tuned GPT-2 Large model (Radford et al., 2019) on the entirety of the stories dataset and used this fine-tuned model as the reward model. This was done with full fine-tuning using theTRL library (von Werra et al., 2020). We trained for 20 epochs with a batch size of 64 and 32 gradient accumulation steps and a learning rate of 0.0005.
+
+We detail the hyperparameters used for training GFlowNets in our experiments in Table C.2. During training, we sample  $(X,Y)$  from the dataset and then sample (batch size)  $Zs$  for every  $(X,Y)$ , before using  $p_{\mathrm{LM}}(XZY)$  as the reward. We use the replay buffer described in §3.3 and seed it with rationales from the dataset. We linearly anneal the reward temperature, the temperature of the behavior policy during training, and the learning rate during warmup. For supervised fine-tuning, we use a batch size of 256 and train for 10 epochs with a learning rate of 0.0001 withTRL and LoRA. At test time, we sample 1024 infills for each example in the test set from all the models at temperature 0.9, and average over 10 such draws.
+
+Table B.1: Empirical examples for sequence continuation. Sentences generated from GFlowNet fine-tuning tend to be more reasonable than autoregressively generated samples from the LM with temperature 1.0 and tend to be more diverse than samples generated from diverse beam search.  
+
+<table><tr><td>Input Prompt:</td><td>The matching campaign starts on Friday and will continue through midnight on Monday.</td></tr><tr><td>Sampling w/ T = 1.0:</td><td>(1) Participate with a fairly low $1 donation so we can motivate more volunteers on Sunday afternoon.
+(2) However, the information regarding Cutler&#x27;s suspicious death may not become widely known until early in the week.</td></tr><tr><td>Diverse beam search:</td><td>(1) If you are interested in participating in the matching campaign, you can do so by clicking here.
+(2) There is no limit to the number of times you can enter.</td></tr><tr><td>GFlowNet fine-tuning:</td><td>(1) If you are interested in signing up you can do so here.
+(2) Please share.</td></tr><tr><td>Input Prompt:</td><td>I want hockey to become a huge thing in Houston.</td></tr><tr><td>Sampling w/ T = 1.0:</td><td>(1) We&#x27;ll be here in Texas from late November till end March.
+(2) To that end, I&#x27;ve been following the Houston Aeros (Houston Dynamo&#x27;s minor-league affiliate) and their AHL affiliate (the Texas Stars).</td></tr><tr><td>Diverse beam search:</td><td>(1) That&#x27;s why I&#x27;m here.
+(2) That&#x27;s why I&#x27;m doing this.</td></tr><tr><td>GFlowNet fine-tuning:</td><td>(1) This is something I&#x27;ve always wanted.
+(2) When I was a teenager in middle school, I went to the Ice Arena in University of Houston and loved it.</td></tr><tr><td>Input Prompt:</td><td>That incident got a lot of attention in part because it was captured on video. Israel said he recorded what happened at the synagogue, and made it public, to document it and leave no doubt about what transpired.</td></tr><tr><td>Sampling w/ T = 1.0:</td><td>(1) He blogged about it here as well.
+(2) Israeli TV stations broadcast the video before it aired in Israel, as the country&#x27;s rule requires.</td></tr><tr><td>Diverse beam search:</td><td>(1) However, there is no evidence that he did so.
+(2) However, there is no video of what happened inside the synagogue.</td></tr><tr><td>GFlowNet fine-tuning:</td><td>(1) He is on tape doing all of it.
+(2) It&#x27;s a message countries can use to deter future attacks.</td></tr><tr><td>Input Prompt:</td><td>The Rolling Stones in Concert has since been released solely by ABKCO Records. The band would remain incensed with Klein for decades for that act. Klein died in 2009.</td></tr><tr><td>Sampling w/ T = 1.0:</td><td>(1) Actually art has become as obscene as investment banking.
+(2) Some believe he shot himself in the chest.</td></tr><tr><td>Diverse beam search:</td><td>(1) The Rolling Stones, however, would not be deterred.
+(2) The Rolling Stones would go on to release their own version of The Rolling Stones in Concert.</td></tr><tr><td>GFlowNet fine-tuning:</td><td>(1) He received a lifetime achievement award from the Jazz Times.
+(2) Sometimes it seems like we are destined to repeat our own mistakes.</td></tr></table>
+
+Table C.1: Examples of training samples for the stories infilling task.  
+
+<table><tr><td>Beginning (X)</td><td>Middle (Z)</td><td>End (Y)</td></tr><tr><td>I was going to a Halloween party. I looked through my clothes but could not find a costume. I cut up my old clothes and constructed a costume.</td><td>I put my costume on and went to the party.</td><td>My friends loved my costume.</td></tr><tr><td>Allen thought he was a very talented poet. He attended college to study creative writing. In college, he met a boy named Carl.</td><td>Carl told him that he wasn’t very good.</td><td>Because of this, Allen swore off poetry forever.</td></tr></table>
+
+Table C.2: Hyperparameters for the story infilling task.  
+
+<table><tr><td>LoRA rank</td><td>64</td></tr><tr><td>LoRA scaling factor</td><td>16</td></tr><tr><td>LoRA dropout</td><td>0.1</td></tr><tr><td>Batch size</td><td>64</td></tr><tr><td>Gradient accumulation steps</td><td>16</td></tr><tr><td>Learning rate</td><td>0.0001</td></tr><tr><td>Optimizer</td><td>AdamW</td></tr><tr><td>PFtemperature max</td><td>2</td></tr><tr><td>PFtemperature min</td><td>0.5</td></tr><tr><td>Reward temperature start</td><td>1.1</td></tr><tr><td>Reward temperature end</td><td>0.85</td></tr><tr><td>Reward temperature horizon</td><td>100</td></tr><tr><td>Buffer capacity</td><td>25</td></tr><tr><td>Number of training steps</td><td>1000</td></tr><tr><td>Evaluation temperature</td><td>0.8</td></tr><tr><td>Maximum length</td><td>25</td></tr><tr><td>Minimum length</td><td>5</td></tr></table>
+
+Analysis. Table C.3, C.4, C.5, C.6, C.7, C.8 illustrate some examples of the infills generated with the prompting baseline, supervised fine-tuned and GFlowNet fine-tuned models on the test examples.
+
+For evaluating the infills we generate a rating for the stories with infills from each of the methods based on the coherence of the story. We average the rating over 10 infills sampled from each method and average it over all the 100 test examples. The prompt used for evaluation is the following, adapted from Liu et al. (2023). Stories with infills generated by the GFlowNet fine-tuned model on average receive a much higher rating than the baselines. The average rating for the reference infills is 4.3 which should be viewed as an upper bound, as the stories may potentially be present in the GPT-4 training data.
+
+You will be given a short story.
+
+Your task is to rate the story on one metric.
+
+Please make sure you read and understand these instructions carefully. Please keep this document open while reviewing, and refer to it as needed.
+
+Evaluation Criteria:
+
+Coherence (1-5) - the collective quality of all sentences. The story should be well-structured and well-organized. The story should not just be a heap of related information, but should build from sentence to a coherent narrative. The story should also be realistic and all the sentences when put together should make sense.
+
+Evaluation Steps:
+
+1. Read the story carefully. Check if the story is coherent and all the sentences follow a logical order.  
+2. Assign a score for coherence on a scale of 1 to 5, where 1 is the lowest and 5 is the highest based on the Evaluation Criteria. If there are grammatical errors or logical inconsistencies, simply assign a lower score and do not elaborate on the reasoning.
+
+Example:
+
+Randy had recently separated from his wife.   
+He felt very lonely and sad all the time.   
+He considered trying to patch things up with Vera, his ex-wife. He decided to get a puppy instead.   
+His decision made him happy and he no longer felt sad.
+
+- Coherence: 5
+
+Randy had recently separated from his wife.   
+He felt very lonely and sad all the time.   
+He considered trying to patch things up with Vera, his ex-wife.   
+Eventually Randy missed Vera very much and developedWhile making memories with   
+His decision made him happy and he no longer felt sad.
+
+- Coherence 1
+
+Story:
+
+{{Story}}
+
+Evaluation Form (scores ONLY):
+
+- Coherence:
+
+# D SUBJECTIVITY CLASSIFICATION
+
+Additional details. See Table D.1 for some training examples in the SUBJ dataset. We use the prompts in Table D.2 for all baselines.
+
+We run GFlowNet fine-tuning for 1000 steps with a linear warmup over 200 steps, a fixed learning rate of 0.0005, and a batch size of 512 samples; see Table D.3 for all the hyperparameters used. Each batch consists of 8 queries ( $X$ 's), randomly drawn with replacement. We then sample 64 rationales ( $Z$ 's) for every  $X$ , before using  $p_{\mathrm{LM}}(ZY \mid X)$  as the reward. We also use the replay buffer described in §3.3 and seed it with potential rationales generated from  $p_{\mathrm{LM}}$ . For supervised fine-tuning, both on its own and on top of GFlowNet fine-tuning, we use a batch size of 256 with 8 queries, randomly drawn with replacement. We train for 100 steps with a linear warm-up over 20 steps and a constant learning rate. We sweep the learning rate in [0.001, 0.0001] and report the best performance. The reward temperature is annealed from 1.2 down to 1 over the first 150 steps of training.
+
+Additional results. To encourage exploration in the space of  $Z$ , we inverse-prompt the model to generate potential chains of thoughts and store them in a replay buffer at the beginning of training. The replay buffer is a priority queue indexed by the reward. Under-performing chains of thoughts are evicted as we collect more trajectories. We ablate the effect of seeding the replay buffer with inverse-prompted chains of thoughts and aggregating multiple chains at test time in Table D.4.
+
+# E INTEGER ARITHMETIC
+
+Additional details. See Table E.1 for examples from synthetically generated training data for the integer arithmetic task.
+
+As with the infilling task, for the integer arithmetic task, the pretrained base models we tested were imperfect knowledge models, i.e. incorrect rationales are sometimes assigned very high likelihood. To assign the correct rationales higher rewards, we prepend some demonstrations with hand-crafted rationales to the query when computing the reward, i.e.  $p_{\mathrm{LM}}(XZY|(X_iZ_iY_i)^k)$  where  $k = 3$  in our experiments. This improves the calibration to some extent. Note that these  $(X_i,Z_i,Y_i)$  are taken from the dataset and used only in the reward. The GFlowNet policy here is conditioned only on  $X$ , i.e.,  $q_{\mathrm{GFN}}(Z|X)$ .
+
+We detail the hyperparameters used for training GFlowNets in our experiments in Table E.2. During training, we sample  $(X,Y)$  from the dataset and then sample (batch size)  $Zs$  for every  $(X,Y)$ , before using  $p_{\mathrm{LM}}(XZY|(X_iZ_iY_i)^k_{i=1})$  as the reward. During the generation of  $Z$ , as mentioned in §4.4, whenever a “=” is generated we extract the preceding expression and evaluate the last two terms using eval in Python. We use the replay buffer described in §3.3 and seed it with rationales from the dataset  $p_{\mathrm{LM}}$ . We linearly anneal the reward temperature, and the learning rate during warmup. During evaluation, for all methods, we aggregate the response over multiple samples drawn from the model at some fixed temperature. For the zero-shot baseline, we observe the best performance with the “Let us think step by step.” prompt appended at the end of the question. For supervised fine-tuning and PPO, we use the implementation from trl. For supervised fine-tuning we use a batch size of 128 with 8 gradient accumulation steps and train for 10 epochs with a learning rate of 0.0001 using LoRA. For PPO we use a minibatch size of 64 with 16 gradient accumulation steps, a learning rate of 0.0001, 4 epochs (on the minibatch), a clip range of 0.2, and an adaptive KL coefficient initialized at 0.2.
+
+Effect of the number of rationales. As mentioned in §4.4, we seed the buffer with rationales from a dataset. Table E.3 summarizes the results of ablation on the number of examples used to seed the buffer. We observe, as expected, that the performance generally improves as we the number of examples used to seed the buffer grows. When no buffers are used to seed the buffer, the performance is quite poor. We hypothesize that without the good rationales used to seed the buffer, the exploration problem of searching for good rationales becomes very challenging.
+
+Table C.3: Examples of infills generated by the prompting baseline along with reference infills for the stories infilling task.  
+
+<table><tr><td>Beginning (X)</td><td>Middle (Z)</td><td>End (Y)</td></tr><tr><td>Randy had recently separated from his wife. He felt very lonely and sad all the time. He considered trying to patch things up with Vera, his ex-wife.</td><td>He decided to get a puppy instead
+He now happily had a new wife who he really liked.
+His wife missed him, but for the wrong reasons. Now
+He finally found someone he could be happy with. It was
+He began spending more time with his new girlfriend and friends.
+Randy remained single and mostly satis-fied with that. He found his
+When Randy was happy he went to the aquarium, a place
+He started going out more with a display of male affection.
+He bought Vera a moonstone and gave it to her as
+He and Vera got back together, even though they had stayed
+He introduced Paige to his new employ-
+ees and they were great.
+Randy is happily married to Vera and they have a healthy relationship
+He found a new lady with a lot of the same interests</td><td>His decision made him happy and he no longer felt sad.</td></tr><tr><td>Ken put a bottle of beer in the freezer. He heard a popping noise. He looked in the freezer and saw the bottle had burst.</td><td>He didn&#x27;t want to wait for another beer to get cold
+He went to sleep and came awake with an angry stomach.
+Ken had another beer as a treat. After the third beer
+He stored the beer in the freezer and for-
+got about it.
+He walked around the bar a bit. He thinks he&#x27;ll
+Weeks later, Ken preferred his cold beer.
+However, he
+He poured a fresh bottle with a sigh of relief. It
+He went to the bar for cold beer. No one knew
+Ken hung up the phone on his friend. He didn&#x27;t
+He bought some veggies from the gro-
+cery store. It was warm
+Ken watched the movie Goalkeeper. He was glad the cold
+He laid down for a rest. His belly was full!
+He laid down for the night and finally sleep again. The</td><td>He drank a warm beer instead.</td></tr></table>
+
+Table C.4: Additional examples of infills generated by the prompting baseline along with reference infills for the stories infilling task.  
+
+<table><tr><td>Beginning (X)</td><td>Middle (Z)</td><td>End (Y)</td></tr><tr><td>Jasmine had homework to do. She did not have a pencil. She looked around the room for one.</td><td>The room did not have a pencil Jasmine made her pencil and done her work. She looked Her mom brought her a pencil. Jasmine wrote her homework Jasmine took her pencil out of the room. She now Jasmine pretended not to have done her homework. She got Her brother finally brought a pencil. They could finally start! Her brother brought a pencil and solved her problem. Jasmine Her brother brought a pencil and solved her problem. Jasmine Her friend offered her a pencil. Jasmine used the pencil Her teacher saw her looking at her phone. Then, Jas Her mom came in and handed her the extra pencil! Jas Her mom brought her a pencil. Jasmine then got to Jasmine sent her friend money to do it. She was</td><td>Jasmine decided not to do her homework.</td></tr><tr><td>Jane had recently gotten a new job. She was nervous about her first day of work. On the first day of work, Jane overslept.</td><td>Jane arrived at work an hour late Despite arriving late, Jane was greeted warmly by her coworkers Jane learned that you never can predict the future. It was Jane learned not to rush into anything. Jane made a good Many of her peers were not pleased with her. Jane got Jane ended up quitting her new job early. She was happy More time with the wrong people gave her anxiety. She was Jane learned that you do not mess up at a job well Jane realized she needed to find new work contacts. At After her first day of work, Jane was too tired to Secretly, Jane hated her new job. It made her Jane realized she worked at a mid-sized company with long Jane&#x27;s boss realized she did not show up to work on</td><td>Jane did not make a good impression at her new job.</td></tr></table>
+
+Table C.5: Examples of infills generated by the supervised fine-tuned model along with reference infills for the stories infilling task.  
+
+<table><tr><td>Beginning (X)</td><td>Middle (Z)</td><td>End (Y)</td></tr><tr><td>Randy had recently separated from his wife. He felt very lonely and sad all the time. He considered trying to patch things up with Vera, his ex-wife.</td><td>He decided to get a puppy instead
+He started enjoying his time with Vera even more. Now, 
+Eventually Randy missed Vera very much and developedWhile making memories with 
+He finally decided he had enough pain for no good reason.
+He began spending more time with his new girlfriend. Now he 
+For the first time since the divorce he started to make new 
+Soon Randy and Vera were rekindling their romance. They 
+He started seeing someone new a day.
+Now, he feels 
+He signed up for a class in Same Sex 
+Loving Uniqueness 
+For the first time in months, Randy and 
+Vera celebrated a 
+He introduced Paige to his new employ-
+ees. They are now good 
+After some contemplating, he decided to patch things up with Vera 
+He now looks for another partner to enjoy his life with.</td><td>His decision made him happy and he no longer felt sad.</td></tr><tr><td>Ken put a bottle of beer in the freezer. He heard a popping noise. He looked in the freezer and saw the bottle had burst.</td><td>He didn&#x27;t want to wait for another beer to get cold
+The popping noise was a freezer needle getting caught. It fell 
+Ken had a cold for thirty minutes. After the popping noise 
+He stored the beer in the freezer. The pouring sound was 
+He didn&#x27;t know why it didn&#x27;t stop the ringing.
+Weeks later, Ken knew the reason for the freezing. It 
+Later Ken learned the gas is out of gas and will end 
+He left the freezer for the beer. It stayed in there 
+Before he knew it, the popping noise was gone. He 
+The popping noise that Ken heard indicated it should have freezer shut 
+The popping noise became louder and more frequent. Now Ken&#x27;s 
+He froze the beer and drank it from a mason jar 
+He went to the store to buy another. Now, he</td><td>He drank a warm beer instead.</td></tr></table>
+
+Table C.6: Additional examples of infills generated by the supervised fine-tuned model along with reference infills for the stories infilling task.  
+
+<table><tr><td>Beginning (X)</td><td>Middle (Z)</td><td>End (Y)</td></tr><tr><td>Jasmine had homework to do. She did not have a pencil. She looked around the room for one.</td><td>The room did not have a pencil Jasmine decided to talk to her mom about it. They Her dad now has to teach her math! Pick- ing up Jasmine took her math test later in the day. She Jasmine got an A on her assignment. She continued to Her brother finally brought a pencil. They could finally start! Her brother brought a pencil and solved her problem. Jasmine Jasmine used her pink and brown high-lighter instead. Jasmine went to school. She was able to finish her Jasmine made her own pencil. At school, everyone was Her brotherly earrings helped her stay up more. While Jasmine made her own pencil. She ended up getting good Jasmine sent her sister to do it. Now it was</td><td>Jasmine decided not to do her homework.</td></tr><tr><td>Jane had recently gotten a new job. She was nervous about her first day of work. On the first day of work, Jane overslept.</td><td>Jane arrived at work an hour late After arriving home, Jane realized her apartment was not ready. After work, Jane returned home and ate a lousy sandwich. After her first day of work, Jane learned that mistakes happen Due to her bad first day, Jane was fired from her Ending: Jane ended up late for work. Although she On her 2nd day of work, Jane came home late On the second day of work, Jane arrived late and unprepared By the end of the day, Jane was tired and weary After her first day of work, Jane&#x27;s boss recommended her Secretly, Jane hated her new job. Se-cretly, After work, Jane felt worn out and tired. The end By the end of the day Jane felt confident and like she</td><td>Jane did not make a good impression at her new job.</td></tr></table>
+
+Table C.7: Examples of infills generated by the GFlowNet fine-tuned model along with reference infills for the stories infilling task.  
+
+<table><tr><td>Beginning (X)</td><td>Middle (Z)</td><td>End (Y)</td></tr><tr><td>Randy had recently separated from his wife. He felt very lonely and sad all the time. He considered trying to patch things up with Vera, his ex-wife.</td><td>He decided to get a puppy instead
+He went to a friend&#x27;s house to talk.
+He took Vera to the park for a picnic.
+He finally decided to call Vera on the phone.
+He asked Vera to meet him and they did.
+He decided to get a dog for his family.
+He went to Vera&#x27;s house.
+He told Vera that he was ready to love again.
+He bought Vera a nice gift.
+He had two dogs and felt like he missed Vera.
+He had a good couple of years.
+He wanted to get Vera and he eventually did.
+He wanted to sit in a room alone.</td><td>His decision made him happy and he no longer felt sad.</td></tr><tr><td>Ken put a bottle of beer in the freezer. He heard a popping noise. He looked in the freezer and saw the bottle had burst.</td><td>He didn&#x27;t want to wait for another beer to get cold
+He decided to stop drinking beer.
+Ken suddenly smelled a bottle of beer.
+He thought it was something
+He walked around the freezer to see the bottle.
+He tried to clean up the freezer.
+He was angry when he had to drink it.
+He stepped down from the freezer
+He made a big joke that night.
+He looked in a jug and see it was, out.
+He then looked for a beer the freezer.
+He tried to enjoy the rest.
+He watered the bottle,</td><td>He drank a warm beer instead.</td></tr></table>
+
+Table C.8: Additional examples of infills generated by the GFlowNet fine-tuned model along with reference infills for the stories infilling task.  
+
+<table><tr><td>Beginning (X)</td><td>Middle (Z)</td><td>End (Y)</td></tr><tr><td>Jasmine had homework to do. She did not have a pencil. She looked around the room for one.</td><td>The room did not have a pencil
+She was in a panic.
+She had to place the pencil in her pocket.
+She had several her classmates.
+She looked for a neat pencil.
+She thought she looked everywhere.
+Her brother walked in and borrowed her pencil.
+She only had a few minutes to do it.
+She never had her pencil.
+She saw a pen in her closet.
+She searched everywhere for a pencil.
+She went to her room.
+She didn’t find one.</td><td>Jasmine decided not to do her homework.</td></tr><tr><td>Jane had recently gotten a new job. She was nervous about her first day of work. On the first day of work, Jane overslept.</td><td>Jane arrived at work an hour late
+She lost her job after the first day of work.
+She was egged on by her boss.
+She decided to over hyp her first day.
+She arrived late and missed her first day of work.
+She was nervous about going to work.
+She ended up getting a good job.
+Jane waited almost a day to get to work.
+Her first day was very difficult.
+She made a couple of phone calls.
+She arrived at the office and was fired.
+She was all out of coffee.
+Jane was surprised she did not make a good impression.</td><td>Jane did not make a good impression at her new job.</td></tr></table>
+
+Table D.1: Two training examples from the SUBJ dataset.  
+
+<table><tr><td>Text (X)</td><td>Label (Y)</td></tr><tr><td>another story follows the relationship between a stepfather ( neeson ) and his young stepson .</td><td>objective</td></tr><tr><td>hoffman ’s performance is authentic to the core of his being .</td><td>subjective</td></tr></table>
+
+Table D.2: Prompts used for subjectivity classification.  
+
+<table><tr><td>Few-shot learning / Supervised fine-tuning</td><td>GFlowNet fine-tuning</td></tr><tr><td>Classify this movie review as objective or subjective: “[X]” This review is [Y].</td><td>Classify this movie review as objective or subjective: “[X]” This review is [Z], so it is [Y].</td></tr></table>
+
+Table D.3: Hyperparameters for GFlowNet fine-tuning on subjectivity classification.  
+
+<table><tr><td>LoRA rank</td><td>256</td></tr><tr><td>LoRA scaling factor</td><td>16</td></tr><tr><td>LoRA dropout</td><td>0.</td></tr><tr><td>Batch size</td><td>16</td></tr><tr><td>Gradient accumulation steps</td><td>32</td></tr><tr><td>Learning rate</td><td>0.0005</td></tr><tr><td>Optimizer</td><td>AdamW</td></tr><tr><td>PFtemperature max</td><td>2</td></tr><tr><td>PFtemperature min</td><td>0.5</td></tr><tr><td>Reward temperature start</td><td>1.2</td></tr><tr><td>Reward temperature end</td><td>1.0</td></tr><tr><td>Reward temperature horizon</td><td>150</td></tr><tr><td>Buffer capacity</td><td>50</td></tr><tr><td>Number of steps</td><td>100</td></tr><tr><td>Number of samples</td><td>10</td></tr><tr><td>Maximum length</td><td>5</td></tr><tr><td>Minimum length</td><td>1</td></tr></table>
+
+Analysis. In Table E.4 we show some examples generated by PPO and GFlowNet fine-tuned models. We observe that PPO generates sequences with high rewards under the base model. These sequences, however, are not valid expressions, and, in fact, do not even call the tool. Instead, the model learns to simply repeat the expression. Repetitions being assigned high likelihood is an issue that has also been noted by Welleck et al. (2020). On the other hand, GFlowNets are able to generate the correct rationale to evaluate the expression.
+
+There are still cases where the GFlowNet fails to produce the correct reasoning chain. We illustrate some of these examples in Table E.5. The errors fall mainly into 3 categories: 1) missing operands in longer OOD problems 2) incorrect operand being copied 3) incorrect operator being copied. The potential source for such errors is that the reward model assigns equally high rewards to rationales with these minor mistakes and thus the model generates them with likelihood. These errors can potentially be reduced by using better and potentially larger reward models.
+
+Table D.4: Ablation studies on subjectivity classification using GPT-J 6.8B.  
+
+<table><tr><td>Method</td><td colspan="3">Test accuracy (%) ↑</td></tr><tr><td></td><td colspan="3">Training samples</td></tr><tr><td></td><td>10</td><td>20</td><td>50</td></tr><tr><td>GFlowNet fine-tuning</td><td>71.4 ± 1.8</td><td>81.1 ± 0.4</td><td>87.7 ± 2.2</td></tr><tr><td>(-) Seed buffer</td><td>64.7 ± 8.9</td><td>68.7 ± 1.7</td><td>77.0 ± 5.5</td></tr><tr><td>(-) Seed buffer (-) Aggregating</td><td>63.9 ± 7.1</td><td>65.9 ± 2.4</td><td>75.4 ± 3.3</td></tr></table>
+
+Table D.5: Top sample rationales for the SUBJ test set using the instruct-fine-tuned GPT-J 6B as both the reward model and the base model for the GFlowNet, which is trained on 50 labeled examples.  
+
+<table><tr><td>True label</td><td colspan="2">Rationale</td><td>Frequency</td></tr><tr><td rowspan="7">Objective</td><td rowspan="7">This review is</td><td>about a factual event</td><td>11.45%</td></tr><tr><td>describing a factual event</td><td>10.23%</td></tr><tr><td>based on a factual statement</td><td>9.47%</td></tr><tr><td>about a historical event</td><td>8.77%</td></tr><tr><td>about a movie review</td><td>7.41%</td></tr><tr><td>based on facts</td><td>5.64%</td></tr><tr><td>about a factual statement</td><td>4.61%</td></tr><tr><td rowspan="7">Subjective</td><td rowspan="7">This review is</td><td>about a movie review</td><td>26.21%</td></tr><tr><td>about the movie experience</td><td>26.17%</td></tr><tr><td>about the movie</td><td>7.60%</td></tr><tr><td>describing a movie review</td><td>3.44%</td></tr><tr><td>about the movie review</td><td>2.31%</td></tr><tr><td>based on a fictional story</td><td>2.30%</td></tr><tr><td>describing the movie experience</td><td>2.01%</td></tr></table>
+
+Table E.1: Examples from the integer arithmetic dataset. Note that the results of the two-term expressions are evaluated by the calculator.  
+
+<table><tr><td>Question (X)</td><td>Rationale (Z)</td><td>Answer (Y)</td></tr><tr><td>Question: 6 - 0 - 4 - 8 = ? Answer:</td><td>6 - 0 =, 6 - 4 =, 2 - 8 =</td><td>. The answer is -6.</td></tr><tr><td>Question: 9 + 4 - 8 = ? Answer:</td><td>9 + 4 =, 13 - 8 =</td><td>. The answer is 5.</td></tr></table>
+
+Table E.2: Hyperparameters for the Integer Arithmetic Task  
+
+<table><tr><td>LoRA rank</td><td>64</td></tr><tr><td>LoRA scaling factor</td><td>16</td></tr><tr><td>LoRA dropout</td><td>0.1</td></tr><tr><td>Batch size</td><td>16</td></tr><tr><td>Gradient accumulation steps</td><td>32</td></tr><tr><td>Learning rate</td><td>0.0001</td></tr><tr><td>Optimizer</td><td>AdamW</td></tr><tr><td>PFtemperature max</td><td>2</td></tr><tr><td>PFtemperature min</td><td>0.5</td></tr><tr><td>Reward temperature start</td><td>1.1</td></tr><tr><td>Reward temperature end</td><td>0.5</td></tr><tr><td>Reward Temperature horizon</td><td>150</td></tr><tr><td>Buffer capacity</td><td>50</td></tr><tr><td>Number of training steps</td><td>1000</td></tr><tr><td>Evaluation temperature</td><td>0.1</td></tr><tr><td>Number of samples</td><td>10</td></tr><tr><td>Maximum length</td><td>20</td></tr><tr><td>Minimum length</td><td>5</td></tr></table>
+
+Table E.3: Ablation study to understand the effect of the number of rationales used to seed the buffer on the test accuracy  $(\%)$  for the integer arithmetic task.  
+
+<table><tr><td></td><td colspan="3">Number of operands</td></tr><tr><td></td><td colspan="2">In-distribution</td><td>OOD</td></tr><tr><td>Number of seed rationales</td><td>3</td><td>4</td><td>5</td></tr><tr><td>0</td><td>22.6</td><td>5.8</td><td>3.4</td></tr><tr><td>10</td><td>58.6</td><td>52.3</td><td>20.2</td></tr><tr><td>25</td><td>48.8</td><td>56.8</td><td>24.6</td></tr><tr><td>50</td><td>95.2</td><td>75.4</td><td>40.7</td></tr></table>
+
+Table E.4: Samples generated by PPO fine-tuned and GFlowNet fine-tuned models.  
+
+<table><tr><td>Question (X)</td><td>Generated rationale (Z)</td><td>log R</td></tr><tr><td rowspan="2">Question: 1 - 9 + 8 = ? Answer:</td><td>1 - 9 - 8</td><td>-13.17</td></tr><tr><td>1 - 9 = -8, -8 + 8 = 0</td><td>-27.75</td></tr><tr><td rowspan="2">Question: 8 + 7 + 2 + 7 = ? Answer:</td><td>8 + 7 + 2 + 7</td><td>-2.39</td></tr><tr><td>8 + 7 = 15, 15 + 2 = 17, 17 + 7 = 24</td><td>-11.72</td></tr><tr><td rowspan="2">Question: 7 - 5 + 8 - 0 - 6=? Answer:</td><td>7 - 5 +</td><td>-1.22</td></tr><tr><td>7 - 5 = 2, 2 + 8 = 10, 10 - 0 = 10, 10 - 6 = 4</td><td>-7.99</td></tr></table>
+
+Table E.5: Examples of errors made by the GFlowNet-fine-tuned model.  
+
+<table><tr><td>Question (X)</td><td>Generated rationale (Z)</td></tr><tr><td>Question: 7 + 3 + 4 - 9 + 9 =? Answer:</td><td>7 + 3 = 10, 10 - 9 = 1, 1 + 4 = 5</td></tr><tr><td>Question: 8 + 5 - 1 - 4 + 5 =? Answer:</td><td>8 + 5 = 13, 13 - 1 = 12, 12 + 4 = 16, 16 - 4 = 12</td></tr><tr><td>Question: 0 + 5 - 6 - 1 - 6 =? Answer:</td><td>0 + 5 = 5, 5 - 6 = -1, -1 - 6 = -7, -7 + 1 = -6</td></tr></table>
\ No newline at end of file
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+                                    "content": "...,guillaume.lajoie,yoshua.bengio,nikolay.malkin}@mila.quebec"
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+                                    "content": "ABSTRACT"
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+                                    "type": "text",
+                                    "content": "Autoregressive large language models (LLMs) compress knowledge from their training data through next-token conditional distributions. This limits tractable querying of this knowledge to start-to-end autoregressive sampling. However, many tasks of interest—including sequence continuation, infilling, and other forms of constrained generation—involve sampling from intractable posterior distributions. We address this limitation by using amortized Bayesian inference to sample from these intractable posteriors. Such amortization is algorithmically achieved by fin-tuning LLMs via diversity-seeking reinforcement learning algorithms: generative flow networks (GFlowNets). We empirically demonstrate that this distribution-matching paradigm of LLM fine-tuning can serve as an effective alternative to maximum-likelihood training and reward-maximizing policy optimization. As an important application, we interpret chain-of-thought reasoning as a latent variable modeling problem and demonstrate that our approach enables data-efficient adaptation of LLMs to tasks that require multi-step rationalization and tool use."
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+                                    "content": "Code: https://github.com/GFNOrg/gfn-lm-tuning."
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+                                    "type": "text",
+                                    "content": "1 INTRODUCTION"
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+                                    "content": "Autoregressive large language models (LLMs) trained on general-domain data are vast stores of world knowledge (Petroni et al., 2019). They are typically optimized by predicting a token given its preceding context; therefore, tractable inference over this knowledge is limited to sampling conditioned on a prefix. Many useful tasks, such as infilling (Zhu et al., 2019; Liu et al., 2019), generating text conditioned on length or lexical constraints (Hokamp & Liu, 2017; Hu et al., 2019), and finding the most likely sequence continuation, involve intractable inference in LLMs."
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+                                    "content": "Such tasks are related to the problem of reasoning, which has been framed as one of probabilistic inference (Gershman & Goodman, 2014). Correspondingly, the linguistic expression of reasoning can be seen as inference over language. For example, we can interpret chain-of-thought reasoning (Wei et al., 2022; Kojima et al., 2022), a paradigm of reasoning in language models, as a problem of intractable posterior inference. Given a question-answer pair "
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+                                    "content": ", we are interested in finding latent chains of thought - token sequences "
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+                                    "content": " that contribute the most to the conditional likelihood"
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+                                    "content": " denotes the likelihood assigned to a sequence by a language model and apposition of variables (e.g., XZY) denotes the concatenation of the token sequences."
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+                                    "content": " as a hidden variable in a latent variable model (LVM) renders chain-of-thought reasoning a Bayesian inference problem (Fig. 1). For this LVM, the distribution we must sample from is the posterior "
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+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "type": "text",
+                                    "content": "*Equal contribution. *Work done during internship at Mila. †CIFAR AI Chair. °CIFAR Senior Fellow."
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+                                    "content": "A standard method to sample approximately from intractable posterior distributions is Markov chain Monte Carlo (MCMC), but it is difficult to craft good proposal distributions for multi-modal distributions over language data (Miao et al., 2019; Zhang et al., 2020a; Lew et al., 2023), and inference on a new input may be prohibitively slow. Alternatively, one can turn to reinforcement learning (RL) approaches such as proximal policy optimization (PPO; Schulman et al., 2017), where the language model is treated as a policy to be fine-tuned. However, these do not aim to model the full diversity of the distribution; instead, learned policies settle around a small number of modes. In both cases, issues with this mode collapse are exacerbated when the target distribution is misspecified, leading to the undesirable behavior of overoptimized samplers (Gao et al., 2023)."
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+                                    "content": "Amortized probabilistic inference – that is, training a model to approximate a distribution of interest – provides a principled, efficient, and potentially scalable way to draw samples from the distribution (Beal, 2003). One way to implement amortized inference for high-dimensional discrete data such as text is using generative flow networks (GFlowNets; Bengio et al., 2021), which are diversity-seeking reinforcement learning algorithms that train policies to sample objects (such as a token sequence "
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+                                    "content": "In this work, we present a method that initializes the GFlowNet policy with a pretrained LLM and continues to train it with a reward objective that can be evaluated with the same LLM. The result is a different type of fine-tuning (FT) procedure for text generation that has a number of advantages, including improved sample diversity, data efficiency, and out-of-distribution generalization. GFlowNet fine-tuning makes the language model sample from the target distribution, enabling amortized inference in a number of applications (Fig. 1)."
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+                                    "content": "Leveraging this approach, we empirically demonstrate the possibilities and benefits of learning to sample from intractable distributions over text continuations, latent reasoning chains, and tool use sequences using GFlowNet fine-tuning. Notably, the diversity of samples from the models trained with GFlowNet fine-tuning is beneficial in Bayesian model averaging settings, such as when aggregating answers to questions obtained via multiple reasoning chains. For example, using a pretrained language model with 6B parameters, our method shows an absolute improvement of "
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+                                    "content": " over supervised fine-tuning on subjectivity classification with only 10 labeled examples (§4.3) and outperforms supervised fine-tuning and PPO by "
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+                                    "content": " on integer arithmetic with 50 demonstrations, with notable improvements in out-of-distribution generalization (§4.4). Moreover, the benefits of amortized inference allow us to efficiently sample from the fine-tuned model at scale. Our contributions include:"
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+                                            "content": "(1) A general algorithm for amortized sampling from intractable LLM posteriors."
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+                                            "content": "(2) A probabilistic approach to fine-tuning LLMs to perform chain-of-thought reasoning."
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+                                            "content": "(3) Empirical results on sequence continuation, natural language reasoning, integer arithmetic with tool use, and story infilling."
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+                                    "type": "text",
+                                    "content": "2 MOTIVATING EXAMPLE: GENERATING RANDOM NUMBERS WITH LLMS"
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+                                    "content": "We consider a simple task that highlights the limitations of reward-maximizing reinforcement learning (RL) methods in fine-tuning LLMs. For readers unfamiliar with RL, we refer to Sutton & Barto (2018) and include a glossary in §A.1 to define key terms used throughout this paper. The task involves generating random numbers from a given distribution when prompted 'The following is a random integer drawn uniformly between 0 and 100:'. This task is a minimal instantiation of the problem we study in the paper: sample from a target distribution given an unnormalized density."
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+                                            "content": "Figure 2: Empirical distributions of 512,000 integers from 1 to 100 generated by GPT-J fine-tuned with PPO (reward-maximizing; b) and GFlowNet fine-tuning (distribution-matching; c). Note the logarithmic "
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+                                    "content": "Although the target distribution is tractable, making the task seemingly straightforward, it serves as a useful illustration of the behaviors of different fine-tuning methods."
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+                                    "content": "Renda et al. (2023) found that pretrained LLMs perform quite poorly on this task: the distribution of numbers generated with the above prompt will be far from uniform (Fig. 2a shows an example using an instruction fine-tuned GPT-J 6B (Wang & Komatsuzaki, 2021)"
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+                                    "type": "text",
+                                    "content": "). There may be many reasons for this, among them the effects of instruction fine-tuning and the LLM's possible bias towards numbers that are more frequent in the training data (e.g., numbers starting with '1' are more frequent due to the properties of many natural data-generating processes (Benford, 1938))."
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+                                    "content": "While reward-maximizing RL can teach the model to generate valid numbers (by penalizing outputs that are not numbers from 1 to 100), it would not resolve the distribution skew introduced during pretraining. Indeed, rewarding all valid integers equally leads to an expected gradient of zero for policy gradient methods. Fig. 2b shows that while most samples are valid numbers after PPO training, the distribution remains highly skewed."
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+                                    "content": "Instead, we can take a principled approach by training the LLM to match the target distribution with a GFlowNet learning objective. Such an objective directly optimizes the likelihood of the model generating a number to be proportional to the reward for that number, which is the number's (potentially unnormalized) probability under the target distribution. When the policy is initialized as the pretrained LLM, the resulting distribution after GFlowNet fine-tuning is shown in Fig. 2c. Quantitatively, the KL divergence from the sampling distribution to the target (uniform) distribution decreases from 3.37 for the original LLM (on the support [0, 100]) to "
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+                                    "content": "9.75 \\cdot 10^{-5}"
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+                                    "type": "text",
+                                    "content": " for the GFlowNet-fine-tuned model."
+                                }
+                            ]
+                        }
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+                                    "content": "This example illustrates a general point: GFlowNet objectives provide a principled and flexible approach to fine-tuning LLMs to match a target distribution where reward-maximizing RL fails to. On this simple task, this distribution matching could also be achieved through supervised fine-tuning; however, this would require access to samples from the target distribution, which are unavailable in general (though not in this simple example). In the following sections, we further illustrate this point in non-trivial problems involving intractable inference, reasoning with latent variables, and tool use."
+                                }
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+                                    "content": "3 FINE-TUNING LLMS TO SAMPLE FROM INTRACTABLE DISTRIBUTIONS"
+                                }
+                            ]
+                        }
+                    ],
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+                                    "content": "We first describe how intractable inference emerges from interesting applications of LLMs, one of which is chain-of-thought reasoning seen through the lens of latent variable models, where the posterior distribution over the latent variable is intractable. We then discuss how GFlowNet objectives can be used to train amortized samplers to perform such intractable inference."
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+                                    "content": "3.1 PROBLEM: INTRACTABLE INFERENCE IN LARGE LANGUAGE MODELS"
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+                        }
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+                                    "content": "Autoregressive language models decompose the distribution over sequences of tokens as a product of ordered conditionals: "
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+                                    "content": "p(w_{1:N}) = p(w_1)p(w_2 \\mid w_1) \\cdots p(w_N \\mid w_{1:N-1})"
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+                                    "type": "text",
+                                    "content": ". While this decomposition makes left-to-right sampling from the distribution tractable, sampling from other conditional distribu"
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+                }
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "type": "text",
+                                    "content": "<sup>1</sup>We use the Instruct-GPT-J model available at <sup>2</sup>.co/nlpcloud/instruct-gpt-j-fp16."
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+                                    "content": "3"
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+                                            "type": "text",
+                                            "content": "Table 1: Objects in language posterior inference. Given a pretrained 'teacher' LM "
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+                                            "content": ", we train a GFlowNet "
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+                                            "content": " to sample the posterior "
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+                                            "content": "p(Z \\mid X, Y)"
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+                                            "type": "text",
+                                            "content": ". Amortization and generalization are achieved by making "
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+                                            "content": ", and optionally "
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+                                            "content": ", an input to "
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+                                            "type": "text",
+                                            "content": "."
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+                                            "type": "table",
+                                            "html": "<table><tr><td>Object</td><td>Meaning</td><td>Example 1 (infilling)</td><td>Example 2 (subjectivity classification)</td></tr><tr><td>X</td><td>cause / condition / question</td><td>The cat was hungry.</td><td>A deeply moving storyline.</td></tr><tr><td>Z</td><td>mechanism / reasoning chain</td><td>She ate a mouse.</td><td>This review expresses personal feelings.</td></tr><tr><td>Y</td><td>effect / answer</td><td>Now the cat is sleepy, not hungry.</td><td>Answer: Subjective</td></tr><tr><td>p(Z | X)</td><td>conditional prior</td><td></td><td>pLM(Z | X)</td></tr><tr><td>p(Y | X, Z)</td><td>likelihood of effect given</td><td></td><td>pLM(Y | XZ)</td></tr><tr><td></td><td>cause and mechanism</td><td></td><td></td></tr><tr><td>p(Z, Y | X)</td><td>conditional joint, reward for Z</td><td></td><td>pLM(ZY | X)</td></tr><tr><td>p(Z | X, Y)</td><td>posterior (intractable!)</td><td colspan=\"2\">approximated and amortized by GFlowNet qGFN(Z | X), Y)</td></tr><tr><td>q(Y | X)</td><td>posterior predictive /</td><td colspan=\"2\">approximated as ∑Z qGFN(Z | X)pLM(Y | XZ),</td></tr><tr><td></td><td>Bayesian model average</td><td colspan=\"2\">sampled as Z ~ qGFN(Z | X), Y ~ pLM(Y | XZ)</td></tr></table>",
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+                                    "type": "text",
+                                    "content": "tions is intractable. Various problems of language modeling can be viewed as sampling from such intractable conditionals in the distribution over sequences of an LLM; we give two such examples and related terminologies in Table 1. Some tasks we study in §4 are instances of these examples."
+                                }
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+                                    "type": "text",
+                                    "content": "Tempered and contrastive sampling. In many applications (e.g., translation, summarization, dialogue systems), one wishes to sample from a low-temperature distribution over sequences "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "Z"
+                                },
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+                                    "type": "text",
+                                    "content": " conditioned on a prefix "
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+                                    "type": "text",
+                                    "content": ", i.e., "
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+                                    "type": "inline_equation",
+                                    "content": "q(Z \\mid X) \\propto p_{\\mathrm{LM}}(XZ)^{1 / T}"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " for some temperature "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "T < 1"
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+                                    "type": "text",
+                                    "content": ", as high-likelihood samples are more likely to be fluent or accurate continuations of "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "X"
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+                                    "type": "text",
+                                    "content": " (Tillmann & Ney, 2003). The limit of "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "T \\to 0"
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+                                    "type": "text",
+                                    "content": " gives a distribution that is peaky on the most likely continuation. However, sampling from "
+                                },
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": ", or finding its mode, is intractable, and it is common to resort to approximations, such as tempering the tokenwise conditional distributions or using beam search to search for a mode. A related problem is sampling a continuation with a correction for its unconditional likelihood, e.g., "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "q(Z \\mid X) \\propto p_{\\mathrm{LM}}(XZ)^{\\alpha}p_{\\mathrm{LM}}(Z)^{\\beta}"
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": " with "
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+                                    "type": "inline_equation",
+                                    "content": "\\beta < 0"
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+                                    "type": "text",
+                                    "content": " and "
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+                                    "type": "inline_equation",
+                                    "content": "\\alpha > 0"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ", where applications again resort to approximating the next-token conditionals of "
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": " by tempering (Malkin et al., 2022b; Li et al., 2023)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
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+                                    "type": "text",
+                                    "content": "Infilling and reverse generation. Infilling is the task of sampling a sequence of tokens conditioned on both its prior and subsequent context, which can be understood as sampling from the distribution "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "q(Z \\mid X, Y) \\propto p_{\\mathrm{LM}}(XZY)"
+                                },
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+                                    "type": "text",
+                                    "content": ", where "
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+                                    "type": "text",
+                                    "content": " and "
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+                                    "type": "inline_equation",
+                                    "content": "Y"
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+                                {
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+                                    "type": "text",
+                                    "content": " are fixed. Reverse generation is a special case, where "
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": " is an empty sequence. Besides being a meaningful task in its own right (Liu et al., 2019; Zhu et al., 2019; Donahue et al., 2020; Susanto et al., 2020; Lu et al., 2022a), infilling and reverse generation are key components of newly emerging methods of LLM prompting, such as when LLMs are tasked with optimizing their own instruction sequences or reasoning steps (Zhou et al., 2023; Sordoni et al., 2023; Xu et al., 2023). Current applications achieve this by resorting to hand-engineered instructions and inverted prompts."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
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+                                    "type": "text",
+                                    "content": "Constrained generation. Sampling of text with constraints and penalties – for example, those on the presence or the absence of certain words or on the score of an auxiliary classifier evaluated on the text – can be understood as sampling from a distribution "
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+                                    "content": "q(Z) \\propto p_{\\mathrm{LM}}(Z)c(Z)"
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+                                    "type": "text",
+                                    "content": " is an externally specified constraint. Current approaches to the problem use tokenwise approximations (Liu et al., 2021), various problem-specific beam search and local search techniques (e.g., Schmaltz et al., 2016; Hokamp & Liu, 2017; Hu et al., 2019; Sha, 2020; Lu et al., 2022b) or classifier-guided conditional generation approaches (e.g., Yang & Klein, 2021; Meng et al., 2022)."
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+                                    "content": "3.2 REASONING THROUGH LATENT VARIABLES"
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+                                    "content": "Chain-of-thought reasoning (Wei et al., 2022; Kojima et al., 2022) helps LLMs solve complex problems by producing a reasoning chain before giving the final answer. LLMs pretrained on general domain data can learn to produce useful chains of thoughts given demonstrations, which are usually handcrafted or generated by prompting the LM. Interestingly, although the capacity for chain-of-thought reasoning only emerges in large language models, knowledge can also be extracted from smaller language models when they are carefully fine-tuned (Schick & Schütze, 2021)."
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+                                    "content": "Motivated by this, we connect chain-of-thought reasoning to the general problem of inference in latent variable models illustrated in Fig. 1. Here, reasoning can be seen as posterior inference: sampling from the posterior distribution over a string of tokens "
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+                                    "content": " conditioned on a prefix "
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+                                    "content": ", given an autoregressive language model "
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+                                    "type": "text",
+                                    "content": ". The posterior is defined as"
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+                                    "content": "p _ {\\mathrm {L M}} (Z \\mid X, Y) = \\frac {p _ {\\mathrm {L M}} (X Z Y)}{\\sum_ {Z ^ {\\prime}} p _ {\\mathrm {L M}} \\left(X Z ^ {\\prime} Y\\right)} \\propto p _ {\\mathrm {L M}} (X Z Y). \\tag {2}",
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+                                    "content": "of tokens, the same formalism and the GFlowNet objectives apply to other structured latent objects, such as trees or sets of natural language statements, as long as one has access to a likelihood model "
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+                                    "content": ". While not investigated in this work, these generalizations could be important for formal reasoning and multi-step chains of inference. See, e.g., Yao et al. (2023); Hao et al. (2023); Besta et al. (2024) for approaches to reasoning in language using tree- or list-structured state spaces."
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+                                    "type": "text",
+                                    "content": " under the LVM. While it is generally intractable to directly maximize the data likelihood "
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+                                    "content": "p_{\\mathrm{LM}}(X,Y) = \\sum_Z p_{\\mathrm{LM}}(XZY)"
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+                                    "content": ", the (variational) expectation-maximization (EM) algorithm (Dempster et al., 1977; Beal, 2003; Koller & Friedman, 2009) can be used for this purpose. In the expectation step (E-step), we draw samples from the posterior over the latent variable "
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+                                    "type": "text",
+                                    "content": ", which could come from an amortized sampler of "
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+                                    "type": "text",
+                                    "content": ". In the maximization step (M-step), we maximize the log-likelihood of the joint probability of the sampled latent variables "
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+                                    "content": "\\mathbb{E}_{Z \\sim p_{\\mathrm{LM}}(Z|X,Y)} \\log p_{\\mathrm{LM}}(XZY)"
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+                                    "type": "text",
+                                    "content": " with respect to the parameters of the language model "
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+                                    "content": ". This combination of amortized inference (learning to sample the chain of thought) and supervised fine-tuning (optimizing the language model with the 'supervision' involving "
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+                                    "type": "text",
+                                    "content": " sampled from the amortized posterior) will be illustrated in one of our experiments (§4.3, Table 3)."
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+                                    "content": "3.3 AMORTIZED INFERENCE WITH GFLOWNET OBJECTIVES"
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+                                    "type": "text",
+                                    "content": "For inference in the latent variable model, we leverage the probabilistic framework of generative flow networks (GFlowNets; Bengio et al., 2021; 2023). Using notation from Malkin et al. (2022a), we briefly introduce relevant GFlowNet concepts pertaining to autoregressive sequence generation. Here, GFlowNets learn policies to sample sequences "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
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+                                        421
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Z = z_{1}z_{2}\\dots z_{n}\\top \\in \\mathcal{Z}"
+                                },
+                                {
+                                    "bbox": [
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+                                        332,
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+                                    "type": "text",
+                                    "content": " (where "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\top"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        506,
+                                        421
+                                    ],
+                                    "type": "text",
+                                    "content": " denotes a stop symbol) from a distribution over the space of sequences "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        506,
+                                        421
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{Z}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        506,
+                                        421
+                                    ],
+                                    "type": "text",
+                                    "content": ", given an unnormalized density (reward) "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        506,
+                                        421
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "R: \\mathcal{Z} \\to \\mathbb{R}_{>0}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        506,
+                                        421
+                                    ],
+                                    "type": "text",
+                                    "content": ". The generative process is the same as in autoregressive language models: generation begins with an empty string, and at the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        506,
+                                        421
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        506,
+                                        421
+                                    ],
+                                    "type": "text",
+                                    "content": "-th step a token "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        506,
+                                        421
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z_{i}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        506,
+                                        421
+                                    ],
+                                    "type": "text",
+                                    "content": " is sampled from a policy "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        506,
+                                        421
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "q_{\\mathrm{GFN}}(z_i \\mid z_{1:i-1})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        506,
+                                        421
+                                    ],
+                                    "type": "text",
+                                    "content": " which is then appended to the sequence. This process continues until a stop symbol "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        506,
+                                        421
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\top"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        506,
+                                        421
+                                    ],
+                                    "type": "text",
+                                    "content": " is generated."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "The marginal likelihood "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "q_{\\mathrm{GFN}}^{\\top}(Z)"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " of sampling a terminal state "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Z = z_{1:n}\\top"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        424,
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+                                        469
+                                    ],
+                                    "type": "text",
+                                    "content": " is given by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        424,
+                                        505,
+                                        469
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\prod_{i = 1}^{n}q_{\\mathrm{GFN}}(z_i|z_{1:i - 1})q_{\\mathrm{GFN}}(\\top \\mid z)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        424,
+                                        505,
+                                        469
+                                    ],
+                                    "type": "text",
+                                    "content": ", where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        424,
+                                        505,
+                                        469
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z_{1:0}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        424,
+                                        505,
+                                        469
+                                    ],
+                                    "type": "text",
+                                    "content": " is understood to be the empty string. The goal of GFlowNet training is to fit a parametric policy "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        424,
+                                        505,
+                                        469
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "q_{\\mathrm{GFN}}(\\cdot |\\cdot ;\\theta)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        424,
+                                        505,
+                                        469
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        424,
+                                        505,
+                                        469
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "q_{\\mathrm{GFN}}^{\\top}(Z)\\propto R(Z)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        424,
+                                        505,
+                                        469
+                                    ],
+                                    "type": "text",
+                                    "content": ", i.e., the likelihood of generating a complete sequence is proportional to its reward."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Learning objective. We use a modified version of the subtrajectory balance (SubTB; Madan et al., 2023) objective to account for trajectories being terminable at all states (Deleu et al., 2022). The objective for a sequence "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Z = z_{1:n} \\top"
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": " is"
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+                            ]
+                        }
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+                    "index": 7
+                },
+                {
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathcal {L} (Z; \\theta) = \\sum_ {0 \\leq i <   j \\leq n} \\left(\\log \\frac {R \\left(z _ {1 : i} \\top\\right) \\prod_ {k = i + 1} ^ {j} q _ {\\mathrm {G F N}} \\left(z _ {k} \\mid z _ {1 : k - 1}\\right) q _ {\\mathrm {G F N}} (\\top \\mid z _ {1 : j})}{R \\left(z _ {1 : j} \\top\\right) q _ {\\mathrm {G F N}} (\\top \\mid z _ {1 : i})}\\right) ^ {2}, \\tag {3}",
+                                    "image_path": "53b310482287a33aee736da9ba395490351cf56f06f6126de3c5e37442bb7653.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                593
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "For sequence generation tasks, the SubTB objective is equivalent to the path consistency objective (Nachum et al., 2017) in max-entropy RL (Haarnoja et al., 2017), which has been previously used in the context of text generation (Guo et al., 2021). See Appendix A.2 for further discussion."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Training policy. As the objective in Eq. 3 can be minimized to 0 for all trajectories "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " simultaneously given enough model capacity, we can use trajectories sampled from any full-support distribution (training policy) to perform gradient descent on "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}(\\tau; \\theta)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        596,
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+                                    ],
+                                    "type": "text",
+                                    "content": " with respect to "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\theta"
+                                },
+                                {
+                                    "bbox": [
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+                                        684
+                                    ],
+                                    "type": "text",
+                                    "content": ". As the space we are sampling from is combinatorially large, it is important to have a training policy that can efficiently explore "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        596,
+                                        505,
+                                        684
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{Z}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        596,
+                                        505,
+                                        684
+                                    ],
+                                    "type": "text",
+                                    "content": ". To this end, we compose the mini-batch during training using trajectories from three sources: (1) the policy "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        596,
+                                        505,
+                                        684
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "q_{\\mathrm{GFN}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": ", (2) a tempered version of the current policy "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "q_{\\mathrm{GFN}}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " and (3) a replay buffer storing past trajectories. Replay buffers have been shown to be quite effective in improving GFlowNet training (Jain et al., 2022; Deleu et al., 2022; Shen et al., 2023)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                    "type": "text",
+                                    "content": "Parametrization, amortization, and generalization. To sample the latent sequence "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "Z"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " from the posterior defined in Eq. 2, we parametrize the GFlowNet policy as an autoregressive language model that samples the latent "
+                                },
+                                {
+                                    "bbox": [
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+                                        688,
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+                                    "type": "inline_equation",
+                                    "content": "Z"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " one token at a time from left to right. By setting the reward "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "R(Z) = p_{\\mathrm{LM}}(XZY) \\propto p_{\\mathrm{LM}}(Z \\mid X,Y)"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ", we learn a sampler for the posterior at convergence."
+                                }
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+                        }
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+                    "index": 11
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+            ],
+            "discarded_blocks": [
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "type": "text",
+                                    "content": "As illustrated in Fig. 1, depending on the task, we can condition the GFlowNet policy on either "
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+                                    "content": "X, Y"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ". In cases such as reasoning (§3.2), where there is only a single correct "
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": " for each "
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+                                    "type": "inline_equation",
+                                    "content": "X"
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+                                    "type": "text",
+                                    "content": " and we interested in predicting "
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+                                    "type": "inline_equation",
+                                    "content": "Y"
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+                                {
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+                                    "type": "text",
+                                    "content": " for unseen "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "X"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " at test time, we can simply condition on "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "X"
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ". In this case, the GFlowNet policy is simply a language model that generates "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "Z"
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " as a continuation of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
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+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "X"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": ". To be precise, we initialize "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "q_{\\mathrm{GFN}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": " as a copy of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p_{\\mathrm{LM}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": " that is conditioned on the prefix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "X"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": ", and then fine-tune<sup>2</sup> it with a GFlowNet objective. With this view, sampling "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Z"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": " is an inverse problem: we need to infer "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Z"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": " given a (conditional) prior "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p_{\\mathrm{LM}}(Z \\mid X)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": " and an observation "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
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+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Y"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
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+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": " under likelihood model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p_{\\mathrm{LM}}(Y \\mid XZ)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        163,
+                        506,
+                        229
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                163,
+                                506,
+                                229
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        163,
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+                                        229
+                                    ],
+                                    "type": "text",
+                                    "content": "Allowing the GFlowNet policy to explicitly take "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        163,
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+                                        229
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "X"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " as input amortizes the sampling procedure and allows generalization to unseen "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        163,
+                                        506,
+                                        229
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "X"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        163,
+                                        506,
+                                        229
+                                    ],
+                                    "type": "text",
+                                    "content": ". In this sense, the GFlowNet is a Bayesian model (akin to a LM cascade (Dohan et al., 2022) or deep language network (Sordoni et al., 2023)), in which "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        163,
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+                                        229
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Z"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        163,
+                                        506,
+                                        229
+                                    ],
+                                    "type": "text",
+                                    "content": " are conditionally sampled 'parameters' that transform "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        163,
+                                        506,
+                                        229
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "X"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        163,
+                                        506,
+                                        229
+                                    ],
+                                    "type": "text",
+                                    "content": " into "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        163,
+                                        506,
+                                        229
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Y"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        163,
+                                        506,
+                                        229
+                                    ],
+                                    "type": "text",
+                                    "content": ". To predict the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        163,
+                                        506,
+                                        229
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Y"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        163,
+                                        506,
+                                        229
+                                    ],
+                                    "type": "text",
+                                    "content": " for an unseen "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        163,
+                                        506,
+                                        229
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "X"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        163,
+                                        506,
+                                        229
+                                    ],
+                                    "type": "text",
+                                    "content": ", one performs Bayesian model averaging by drawing samples of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        163,
+                                        506,
+                                        229
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Z"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        163,
+                                        506,
+                                        229
+                                    ],
+                                    "type": "text",
+                                    "content": " from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        163,
+                                        506,
+                                        229
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "q_{\\mathrm{GFN}}(Z \\mid X)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        163,
+                                        506,
+                                        229
+                                    ],
+                                    "type": "text",
+                                    "content": " followed by sampling from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        163,
+                                        506,
+                                        229
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p_{\\mathrm{LM}}(Y \\mid XZ)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        163,
+                                        506,
+                                        229
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        233,
+                        506,
+                        278
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                233,
+                                506,
+                                278
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        506,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": "In tasks such as infilling (§4.2), however, the mapping from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
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+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "X"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        506,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Y"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        506,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": " is one-to-many and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        506,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Y"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        506,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": " is available at test-time. Here, we are interested in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        506,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Z"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        506,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": " itself, rather than using it as an intermediate variable en route to generating "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        506,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Y"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        506,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": ". The GFlowNet policy thus has to be conditioned on both "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        506,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "X"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        506,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        506,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Y"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        506,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": ". To achieve this, the policy is conditioned on a prompt that contains both "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        506,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "X"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        506,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        506,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Y"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
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+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": " (for example, see Appendix C)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        105,
+                        285,
+                        233,
+                        298
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                285,
+                                233,
+                                298
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "4 EMPIRICAL RESULTS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        304,
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+                        338
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "We first validate GFlowNet fine-tuning on text generation, where we seek to find likely sentence continuation given a prompt (§4.1) or fill in a missing sentence in a story (§4.2). Then, we study reasoning tasks that benefit from chain-of-thought reasoning (§4.3) and external tool use (§4.4)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
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+                        354
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
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+                            ],
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "4.1 SENTENCE CONTINUATION"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Task description. A natural application for autoregressive language models is that of sequence continuation: given a prompt, the model should generate a high-likelihood completion. In applications such as creative writing, we would like the continuations to be semantically diverse while still having a high likelihood under the language model. To demonstrate the benefits of GFlowNet fine-tuning, we consider the task of sampling the next sentence following a prompt."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Sampling autoregressively from the LM until a “.” token is reached is unlikely to produce samples that have a high likelihood because the distribution over sentences has a fat tail. Existing approaches to generate sequence continuations include beam search and its variations (Vijayakumar et al., 2018; Shao et al., 2017), top- "
+                                },
+                                {
+                                    "bbox": [
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+                                        474,
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+                                        561
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        474,
+                                        298,
+                                        561
+                                    ],
+                                    "type": "text",
+                                    "content": " sampling (Fan et al., 2018), nucleus sampling"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        561,
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+                                        661
+                                    ],
+                                    "type": "text",
+                                    "content": "(Holtzman et al., 2019), tempered autoregressive sampling, and fine-tuning using importance sampling (Shih et al., 2023), among others. While useful, most of these methods are ultimately hand-crafted heuristics that leave room for improvement. Furthermore, some of these methods (e.g., beam search) involve a computationally expensive search procedure, compared to a single pass of a learned inference model that amortizes over prompts. Our GFlowNet policy autoregressively samples the sequence until a period is sampled, indicating the end of the sentence. Given prompts "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        561,
+                                        506,
+                                        661
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "X"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        561,
+                                        506,
+                                        661
+                                    ],
+                                    "type": "text",
+                                    "content": ", the LM is fine-tuned to generate the continuations "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        561,
+                                        506,
+                                        661
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Z"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        561,
+                                        506,
+                                        661
+                                    ],
+                                    "type": "text",
+                                    "content": " from the tempered posterior by being trained with reward "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        561,
+                                        506,
+                                        661
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "R(Z) = p_{\\mathrm{LM}}(Z|X)^{\\frac{1}{T}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        561,
+                                        506,
+                                        661
+                                    ],
+                                    "type": "text",
+                                    "content": ". When "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        561,
+                                        506,
+                                        661
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T = 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        561,
+                                        506,
+                                        661
+                                    ],
+                                    "type": "text",
+                                    "content": ", the GFlowNet will trivially sample proportional to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        561,
+                                        506,
+                                        661
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p_{\\mathrm{LM}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        561,
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+                                    ],
+                                    "type": "text",
+                                    "content": " without any fine-tuning, so we consider "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        561,
+                                        506,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "0 < T < 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        561,
+                                        506,
+                                        661
+                                    ],
+                                    "type": "text",
+                                    "content": " to focus on the likely continuations."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
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+                                    "content": "2We use LoRA (Hu et al., 2022) instead of full fine-tuning for hardware efficiency in all experiments."
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+                                    "type": "text",
+                                    "content": "samples, we compute the mean pairwise cosine distance between the embeddings (from a pretrained encoder (Reimers & Gurevych, 2019)) of the generated samples and average it over the dataset. We compare to baselines that are commonly used for producing continuations from LMs at inference time (beam search, diverse beam search, nucleus sampling, autoregressive sampling, tempered autoregressive sampling, and greedy generation)."
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+                                    "type": "text",
+                                    "content": "Results. Quantitative results are reported in Fig. 3 and empirical samples are shown in Appendix B. At lower temperatures, our method excels in generating high-likelihood sentences, outperforming the leading baseline, diverse beam search. If we increase the number of beams (and therefore compute) to "
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+                                    "content": " the number of samples produced by the GFlowNet, our performance remains comparable. Nevertheless, even in this scenario, the GFlowNet's generated samples exhibit notably higher diversity compared to diverse beam search and are on par with the best diversity-scoring benchmarks."
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+                                    "content": "4.2 INFILLING STORIES"
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+                                    "type": "text",
+                                    "content": "Task description. Next, we consider the story infilling task, a special case of the general infilling problem (§3.1), where given the beginning "
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+                                    "content": " and end "
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+                                    "type": "text",
+                                    "content": " of a story, the goal is to generate the middle of the story "
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+                                    "type": "text",
+                                    "content": " (Zhu et al., 2019). This is challenging for a language model sampled left to right since continuations "
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+                                    "type": "text",
+                                    "content": " conditioned only on "
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+                                    "type": "text",
+                                    "content": " might be incompatible with the ending "
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+                                    "type": "text",
+                                    "content": ". We use the ROCStories corpus (Mostafazadeh et al., 2016), a dataset of short stories containing exactly 5 sentences each. Given the first 3 sentences and the last sentence, the goal is to generate the fourth sentence, which often involves a turning point in the story and is thus challenging to fill in."
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+                                    "content": "As we expect the base model to contain the required knowledge, for this task we use a GPT-2 Large model (Radford et al., 2019) fine-tuned on the entire ROCStories training set as the base model. For evaluating the approach, we consider 900 samples from the dataset as training data to learn "
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+                                    "type": "inline_equation",
+                                    "content": "q_{\\mathrm{GFN}}(Z|X,Y)"
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+                                    "type": "text",
+                                    "content": " and evaluate the similarity of the generated infills on a dataset of 100 unseen stories. Along with the GFlowNet-fine-tuned model, we also consider two baselines: prompting the model to infill the story and supervised fine-tuning on the same data. Further details are in Appendix C."
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+                                    "type": "text",
+                                    "content": "Results. To measure the similarity of the generated infills with the reference infills available in the dataset, we compute BERTScore (Zhang et al., 2020b), with DeBERTa (He et al., 2021) - which is correlated with human judgments"
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+                                    "type": "text",
+                                    "content": "along with BLEU-4 (Papineni et al., 2002) and GLEU-4 (better suited for sentences; Wu et al., 2016) metrics. Additionally, we also evaluate each method using GPT-4 as a judge. From our results summarized in Table 2, we observe that the infills generated by the model with GFlowNet fine-tuning are closer to the reference infills in the dataset than the baselines. By sampling from "
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+                                    "content": "p_{\\mathrm{LM}}(Z|X,Y)"
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+                                    "type": "text",
+                                    "content": ", the GFlowNet is able to account for the ending while generating the infill, resulting in infills that link the beginning and the end of the story coherently. For further analysis and details see Appendix C."
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+                                            "content": "Table 2: Evaluation of the generated infills."
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+                                            "html": "<table><tr><td>Method</td><td>BERTScore</td><td>BLEU-4</td><td>GLEU-4</td><td>GPT4Eval</td></tr><tr><td>Prompting</td><td>0.081 ± 0.009</td><td>1.3 ± 0.5</td><td>3.2 ± 0.1</td><td>2.4</td></tr><tr><td>Supervised fine-tuning</td><td>0.094 ± 0.007</td><td>1.6 ± 0.8</td><td>3.7 ± 0.4</td><td>2.7</td></tr><tr><td>GFlowNet fine-tuning</td><td>0.184 ± 0.004</td><td>2.1 ± 0.2</td><td>4.2 ± 0.7</td><td>3.4</td></tr></table>",
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+                                    "content": "4.3 SUBJECTIVITY CLASSIFICATION"
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+                                    "content": "Task description. SUBJ (Pang & Lee, 2004) is a binary classification dataset for natural language understanding. It is a collection of movie reviews in which each review is labeled as objective, meaning that it references facts about the movie, or subjective, meaning that it expresses an opinion of the reviewer (see Table D.1 for examples). Given an unlabeled review, the model must predict whether it is objective or subjective. While supervised fine-tuning on the full dataset"
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+                                            "content": " on SUBJ using an instruct-fine-tuned GPT-J 6B."
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+                                    "content": "can achieve high test accuracy, we are interested in the low-data regime where we only have tens of labeled examples. We use the same instruction-tuned GPT-J 6B variant as in §2 for this experiment. Without any demonstrations, the model struggles with this task using the prompt in Table D.2 and achieves only "
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+                                    "content": " zero-shot accuracy."
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+                                    "content": "This task is hard likely because it requires a latent reasoning step. A review could be considered objective because it analyzes the plot or facts about the movie, or it could be subjective because it expresses a personal opinion or makes a judgment. We denote the review "
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+                                    "content": ". Then, we GFlowNet-fine-tune the LLM "
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+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "content": " to match the Bayesian posterior over rationales in Eq. 2. At test time, "
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+                                    "content": " generates 10 latent rationales ("
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+                                    "content": " to produce 10 answers, the majority vote of which becomes the final prediction."
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+                                    "content": " is defined in Eq. 2. Further, as described in §3.2, we can take an M-step by updating "
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+                                    "type": "inline_equation",
+                                    "content": "\\log p_{\\mathrm{LM}}(XZY)"
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+                                    "content": " over a collection of "
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+                                    "content": "'s sampled from the amortized posterior "
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+                                    "type": "text",
+                                    "content": ". This is equivalent to applying supervised fine-tuning after GFlowNet fine-tuning."
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+                                    "content": "Results. We present few-shot prompting and supervised fine-tuning with LoRA as baselines. In few-shot prompting, we prepend 0, 10, 20, or 50 training examples to each test example using the prompt shown in Table D.2. We randomly shuffle few-shot demonstrations and report the mean and variance in Table 3. In supervised fine-tuning, we directly maximize "
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+                                    "content": "\\log p_{\\mathrm{LM}}(Y \\mid X)"
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+                                    "type": "text",
+                                    "content": " over the same 10, 20, or 50 "
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+                                    "type": "inline_equation",
+                                    "content": "(X, Y)"
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+                                    "type": "text",
+                                    "content": " pairs. The variance is over model initialization and batch order. All entries except zero-shot prompting are aggregated over 3 random seeds. See Appendix D for experiment details. GFlowNet fine-tuning consistently outperforms supervised fine-tuning in the low-data regime, as shown in Table 3. In some cases, performing supervised fine-tuning on top, which corresponds to running one step of the EM algorithm, further improves the performance."
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+                                    "type": "text",
+                                    "content": "4.4 SOLVING ARITHMETIC PROBLEMS STEP BY STEP"
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+                                    "content": "Task description. Arithmetic reasoning is a fitting benchmark to evaluate reasoning abilities of large language models as it requires multi-step reasoning and correctness is easy to evaluate (Cobbe et al., 2021). While the distribution of pretraining and fine-tuning data (Magister et al., 2023; Lee et al., 2023; Luo et al., 2023) and prompting choices (Imani et al., 2023) play a critical role in their arithmetic abilities, LLMs are susceptible to poor generalization by learning 'shortcuts' to reasoning (Dziri et al., 2023). We consider a simple integer arithmetic task (Fig. 1), with a general pretrained base model, rather than a one pretrained on mathematical"
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+                                    "content": "Table 4: Test accuracy "
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+                                    "type": "inline_equation",
+                                    "content": "(\\%)"
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+                                    "type": "text",
+                                    "content": " on an integer arithmetic task with addition and subtraction using a GPT-J 6B model. Training data only include samples with 3 or 4 operands."
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+                                            "html": "<table><tr><td rowspan=\"3\">Method</td><td></td><td colspan=\"3\">Number of Operands</td></tr><tr><td></td><td colspan=\"2\">In-distribution</td><td>OOD</td></tr><tr><td></td><td>3</td><td>4</td><td>5</td></tr><tr><td rowspan=\"5\">k-shot CoT</td><td>k=0</td><td>10.2</td><td>6.4</td><td>3.2</td></tr><tr><td>k=3</td><td>15.8 ± 3.1</td><td>11 ± 1.7</td><td>5.4 ± 0.2</td></tr><tr><td>k=5</td><td>20.4 ± 10.4</td><td>17.6 ± 0.6</td><td>6.6 ± 1.1</td></tr><tr><td>k=10</td><td>26.5 ± 1.4</td><td>15.2 ± 1.7</td><td>8.9 ± 1.9</td></tr><tr><td>k=20</td><td>35.5 ± 1.9</td><td>21 ± 1.4</td><td>10.5 ± 0.9</td></tr><tr><td colspan=\"2\">Supervised fine-tuning</td><td>72.1 ± 1.3</td><td>19.6 ± 2.2</td><td>12.8 ± 5.7</td></tr><tr><td colspan=\"2\">PPO</td><td>30.6 ± 4.1</td><td>13.7 ± 4.1</td><td>5.6 ± 3.1</td></tr><tr><td colspan=\"2\">GFlowNet fine-tuning</td><td>95.2 ± 1.3</td><td>75.4 ± 2.9</td><td>40.7 ± 9.1</td></tr></table>",
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+                                    "content": "tasks (Jelassi et al., 2023). To avoid the pitfalls of symbolic calculations with language models, we adopt the tool use setting (Schick et al., 2023), where the model is equipped with a calculator that can perform parts of the computation, implemented as in Cobbe et al. (2021): when the model outputs "
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+                                    "type": "text",
+                                    "content": " the expression preceding it is evaluated and appended to the sequence. To prevent the model from \"cheating\", we limit the calculator to evaluate only two-term expressions. Consequently, reasoning here involves learning to plan using a tool with limited capabilities (Hao et al., 2023)."
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+                                    "content": "For training, we use a synthetic dataset of arithmetic expressions, limited to addition and subtraction. Following Zelikman et al. (2022), we use a small set of 50 demonstrations "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "(X,Z,Y)"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " to seed the replay buffer in addition to 1000 examples "
+                                },
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+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "(X,Y)"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ". We use the same instruction-tuned GPT-J as in §4.3 as the base model. Further details are in Appendix E. We report the accuracy on two types of examples: (1) unseen in-distribution expressions (3 or 4 operands) and (2) longer out-of-distribution expressions (5 operands). As baselines, we consider zero-shot chain-of-thought prompting, "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "k"
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+                                    "type": "text",
+                                    "content": "-shot prompting, supervised fine-tuning on the tool use sequences, and fine-tuning with PPO (Schulman et al., 2017). For all methods, we enable tool use and limit the model to generate only numbers and operators."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
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+                                    "type": "text",
+                                    "content": "Results. From the results summarized in Table 4, the base model performs poorly even with chain-of-thought prompts. Including examples in context improves the performance considerably, with monotonic improvements as the number of examples increases. Supervised fine-tuning improves the performance significantly on the in-distribution examples, but the model still struggles to generalize on the out-of-distribution examples. Fine-tuning with PPO results also yields poor performance, caused in part by the poor calibration of the base reward model, i.e. it cannot distinguish good rationales from bad ones. Even though the sequences generated with PPO (illustrated in Appendix E) have high rewards, they are spurious and do not even define valid calls to the tool."
+                                }
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+                                    "content": "Such overoptimization to a misspecified reward is a widely noted issue in LLMs trained with RL (Gao et al., 2023). On the other hand, by matching the entire distribution, GFlowNet fine-tuning avoids collapsing to a single mode of the reward, thereby being robust to the misspecification of"
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+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "content": "8"
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+                                    "type": "text",
+                                    "content": "the reward (Eysenbach & Levine, 2022) and achieving significantly better performance on in and out-of-distribution examples. See Appendix E for additional results and analysis."
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+                            ]
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+                    ],
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+                                    "type": "text",
+                                    "content": "5 FURTHER RELATED WORK"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
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+                                    "type": "text",
+                                    "content": "Sampling from intractable marginals. Beyond the approximations mentioned in §3.1, sampling from intractable posterior distributions given by pretrained models for tasks such as infilling and constrained generation has been an object of study. Miao et al. (2019); Zhang et al. (2020a) use MCMC for these problems, Malkin et al. (2021) used a variable-neighborhood ascent for finding modes, and a sequential Monte Carlo approach was recently proposed by Lew et al. (2023). Others have studied the problem with masked language models, using them to perform variants of Gibbs sampling (Wang & Cho, 2019; Goyal et al., 2022; Yamakoshi et al., 2022) and recovering marginal distributions over small sets of tokens (Torroba Hennigen & Kim, 2023)."
+                                }
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+                    ],
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+                                    "type": "text",
+                                    "content": "GFlowNets. GFlowNets (Bengio et al., 2021) were originally proposed to learn policies for sampling discrete compositional objects from an unnormalized reward distribution, motivated by the need to sample diverse high-reward objects in scientific discovery (Jain et al., 2023), in particular, for biological sequence generation (Jain et al., 2022). The interpretation of GFlowNets as variational inference algorithms (Malkin et al., 2023; Zimmermann et al., 2023) makes them appropriate for sampling Bayesian posterior distributions over structured objects (e.g., Deleu et al., 2022; 2023; van Krieken et al., 2023; Hu et al., 2023)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
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+                                    "type": "text",
+                                    "content": "Chain-of-thought reasoning in LLMs. In recent work on classification and completion with language models, the latent reasoning chain "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "Z"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ", in the notation of §3.1, is called a 'chain of thought' (Wei et al., 2022). The chain of thought is typically generated by conditioning the language model on "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "X"
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+                                {
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+                                    "type": "text",
+                                    "content": " with the use of specialized demonstrations or prompts (Kojima et al., 2022), with no guarantee of sampling the posterior accurately. Related to our Bayesian formulation, Wang et al. (2023b) noted that appropriately aggregating the conclusions "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "Y"
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+                                {
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+                                    "type": "text",
+                                    "content": " from several latent chains "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "Z"
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " improves predictive performance. In Xu et al. (2023); Zhou et al. (2022), a posterior over latent token sequences is sampled using MCMC, while Zelikman et al. (2022) propose fine-tuning on successful (high-reward, in our language) chains of thought, which achieves reward maximization but gives no guarantee of diversity. In concurrent work, Phan et al. (2023) use MCMC to sample chains-of-thought in problems with binary feedback. We expect these methods to generalize poorly to difficult exploration problems, while GFlowNet fine-tuning takes advantage of generalizable structure in the posterior and has a goal of sampling the full posterior over latent reasoning chains."
+                                }
+                            ]
+                        }
+                    ],
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+                                    "type": "text",
+                                    "content": "6 CONCLUSION"
+                                }
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+                    "index": 6
+                },
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+                                    "type": "text",
+                                    "content": "The knowledge compressed in LLMs is crucial for tasks such as infilling and constrained generation, but querying this knowledge involves sampling from intractable posterior distributions. We propose to use GFlowNet objectives to train LLMs to sample from such posterior distributions. Empirical results show that GFlowNet fine-tuning finds a better fidelity-diversity trade-off for text generation and also improves sample efficiency and generalization on downstream tasks compared to maximum-likelihood training or reward-maximizing policy optimization. As an amortized inference algorithm, our method converts computation into better test-time performance without additional data."
+                                }
+                            ]
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+                    ],
+                    "index": 7
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+                                    "type": "text",
+                                    "content": "Future work should investigate transfer and generalization across tasks, in particular, building a 'universal reasoner' as a model "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "q(Z \\mid X)"
+                                },
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+                                    "type": "text",
+                                    "content": " shared between "
+                                },
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+                                    "type": "inline_equation",
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+                                },
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+                                    "type": "text",
+                                    "content": " from different tasks, as was recently considered by Wang et al. (2023a). One should investigate the benefit of using a better knowledge model, e.g., a more capable base LLM, as a starting point for GFlowNet fine-tuning. The ability to draw multiple samples from a GFlowNet can also be used to quantify epistemic uncertainty. Finally, we adopt the GFlowNet formalisms with the perspective of generalizing to latent variables "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "Z"
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+                                    "type": "text",
+                                    "content": " with a richer generative process than left-to-right sampling. We hope that the GFlowNet paradigm will enable more flexible reasoning with LLMs in the future: extending probabilistic programming with language variables (Beurer-Kellner et al., 2023), using structured chains of thought (Yao et al., 2023; Besta et al., 2024), and extending to program synthesis and planning with world models."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
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+                                    "type": "text",
+                                    "content": "Limitations. Due to resource constraints, our experiments use models up to 6B parameters, but we expect the conclusions to hold for larger models. In fact, our method can potentially benefit larger models more: it is harder to optimize a larger model with maximizing objectives on a small amount of data. As with any on-policy method, exploration, especially in problems with more complex latents, remains an open problem. Furthermore, our method improves inference but not the knowledge in the LM. Issues such as hallucination or miscalibration, which are closely related to the knowledge representation, are thus not addressed."
+                                }
+                            ]
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+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "content": "9"
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+                                    "type": "text",
+                                    "content": "ETHICS STATEMENT"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
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+                                    "content": "While we foresee no immediate negative societal consequences of our work, we hope that future researchers who build upon it will, as we have, bear in mind the potential of LLMs – and in particular of human-like reasoning in LLMs – to be used both for good and for harm."
+                                }
+                            ]
+                        }
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+                                    "content": "Research areas in safe and explainable AI that can benefit from GFlowNet fine-tuning include (1) interpretability of LLMs' reasoning processes and (2) fine-tuning with human feedback or an external reward, where diverse sampling can help prevent 'reward hacking' and overfitting to a misspecified target."
+                                }
+                            ]
+                        }
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+                                    "content": "REPRODUCIBILITY"
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+                        }
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+                    "index": 4
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+                                    "content": "We discuss the details of the proposed algorithms in §3.3 and provide all the implementation details and hyperparameters for the experiments in the main paper and appendix. Code for our experiments is available at https://github.com/GFNOrg/gfn-lm-tuning."
+                                }
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+                                    "content": "The authors are grateful to Bonaventure Dossou and Salem Lahlou for their help in the early stages of this project. We also thank Robert Hawkins, Arian Hosseini, Zhen Wang, and Anirudh Goyal for valuable discussions and suggestions of related work."
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+                                    "content": "GL acknowledges funding from CIFAR, Samsung, and a Canada Research Chair in Neural Computation and Interfacing."
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+                                        }
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+                                }
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+                        },
+                        {
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+                                            "content": "Yongchao Zhou, Andrei Ioan Muresanu, Ziwen Han, Keiran Paster, Silviu Pitis, Harris Chan, and Jimmy Ba. Large language models are human-level prompt engineers. International Conference on Learning Representations (ICLR), 2023."
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+                                }
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+                                            "type": "text",
+                                            "content": "Wanrong Zhu, Zhiting Hu, and Eric Xing. Text infilling. arXiv preprint arXiv:1901.00158, 2019."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 14
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+                                            "content": "Heiko Zimmermann, Fredrik Lindsten, Jan-Willem van de Meent, and Christian A. Naesseth. A variational perspective on generative flow networks. Transactions on Machine Learning Research (TMLR), 2023."
+                                        }
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+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "type": "text",
+                                    "content": "A ADDITIONAL BACKGROUND"
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+                },
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+                                    "content": "A.1 GLOSSARY OF RL FOR LLMS"
+                                }
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+                    "index": 2
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+                                    "type": "text",
+                                    "content": "We provide definitions for key terms used throughout the paper, with a focus on their relevance to our setting of fine-tuning large language models (LLMs)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
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+                    ],
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Reinforcement learning. Reinforcement learning (RL) is a branch of machine learning concerned with how agents should take actions in an environment to maximize cumulative rewards. In our context, RL is used to fine-tune the decision-making process of LLMs to improve their performance on specific tasks. For a more comprehensive overview, we refer readers to Sutton & Barto (2018)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Policy. In RL, a policy is a strategy that defines the behavior of an agent by mapping states of the environment to actions. In our setting, a policy dictates how the language model generates text sequences based on the current context and learned parameters."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
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+                    "angle": 0,
+                    "lines": [
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Reward. A reward is a signal that evaluates the quality of an action taken by an agent in a particular state. In the fine-tuning of LLMs, rewards are used to guide the model towards generating more desirable text, such as more accurate predictions or more coherent continuations. More specifically, in the context of GFlowNets, the reward corresponds to the unnormalized posterior probability, and the GFlowNet aims to match it by learning a policy that generates samples proportional to their reward."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
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+                    "angle": 0,
+                    "lines": [
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Matching a distribution. Matching a distribution involves training a model to approximate a target probability distribution. In our work, this concept is applied to fine-tune LLMs so that their generated text matches the desired characteristics, such as adhering to a particular style or content constraint."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
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+                        504,
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Policy gradient methods. Policy gradient methods (such as PPO) are a subset of RL algorithms that optimize a policy by computing gradients of the expected reward with respect to the policy parameters. In the context of LLMs, these methods are used to fine-tune the model's parameters to increase the likelihood of generating high-reward text sequences."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
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+                                    "type": "text",
+                                    "content": "A.2 LEARNING OBJECTIVE"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
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+                                    "type": "text",
+                                    "content": "We use the subtrajectory balance learning objective for training GFlowNets (Madan et al., 2023). In the notation of that paper, with a forward policy "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "P_F"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ", backward policy "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "P_B"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " and state flow function "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ", the objective over a partial trajectory "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\tau = s_m \\rightarrow \\dots \\rightarrow s_n"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " is defined as follows"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
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+                                    "type": "interline_equation",
+                                    "content": "\\mathcal {L} _ {S u b T B} (Z; \\theta) = \\left(\\log \\frac {F \\left(s _ {m}\\right) \\prod_ {i = m} ^ {n - 1} P _ {F} \\left(s _ {i + 1} \\mid s _ {i}\\right)}{F \\left(s _ {n}\\right) \\prod_ {i = m} ^ {n - 1} P _ {B} \\left(s _ {i} \\mid s _ {i + 1}\\right)}\\right) ^ {2} \\tag {4}",
+                                    "image_path": "3b6789605d5316c1d99144da3f9b3d859278e9973fba1f8118130832f0aab20a.jpg"
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+                                    "type": "text",
+                                    "content": "In the case of autoregressive generation of a sequence of tokens in a fixed order (left-right), the generative process is a tree, so there is only a single path to each state, and each state has a single parent. Thus, "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "P_B(s|s') = 1"
+                                },
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " trivially. Additionally, since each state is a valid terminable state, we can incorporate the modification to account for this from Deleu et al. (2022). Specifically, note that at convergence we have "
+                                },
+                                {
+                                    "bbox": [
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+                                        481,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "R(s_n^\\top) = F(s_n)P_F(\\top \\mid s_n)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        481,
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+                                    ],
+                                    "type": "text",
+                                    "content": ". Using this, we can simply substitute "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        481,
+                                        505,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F(s_{n}) = R(s_{n}^{\\top}) / P_{F}(\\top \\mid s_{n})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        481,
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+                                    ],
+                                    "type": "text",
+                                    "content": " in Eq. (4). This allows us to avoid parameterizing a flow function separately, reducing additional complexity. The only learned object is now the forward sampling policy "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        481,
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+                                    "type": "inline_equation",
+                                    "content": "P_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": ", which we refer to as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        481,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "q_{GFN}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        481,
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+                                    ],
+                                    "type": "text",
+                                    "content": ". Summing this over all partial trajectories in a trajectory with equal weight "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        481,
+                                        505,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\lambda = 1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        481,
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+                                    ],
+                                    "type": "text",
+                                    "content": ", we get the final learning objective in Eq. (3)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
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+                        600
+                    ],
+                    "type": "title",
+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "B SENTENCE CONTINUATION"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "Additional details. We choose sentences as the level of granularity for our sequence continuation task because they are a natural unit of generation in language. Moreso than individual words, sentences are analogous to whole thoughts, reasoning steps, etc. because the compositional rules of syntax operate at the level of sentences. Whereas the meaning of a word is often ambiguous and context-dependent, the meaning of a sentence tends to be more self-contained."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
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+                    "type": "text",
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+                                    "type": "text",
+                                    "content": "A naive solution to the task is to simply sample autoregressively from the LM until a “.” token is reached, marking the end of the sentence. In practice, however, this is unlikely to produce samples that have high likelihood because the distribution over sentences has a long tail: a vast number of sentences have small but non-zero probability, and collectively account for a substantial portion of the total probability mass. Instead, it would be desirable to sample from a low-temperature distribution over sentences, so that the distribution becomes more sparse and highly probable sentences are"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                }
+            ],
+            "discarded_blocks": [
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+                    "type": "header",
+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
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+                                    "type": "text",
+                                    "content": "17"
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+                            ]
+                        }
+                    ],
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+                }
+            ],
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+        },
+        {
+            "para_blocks": [
+                {
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+                                    "type": "text",
+                                    "content": "sampled more often. However, this in itself is a difficult distribution to sample from, and it is different from simply sampling autoregressively from the LM with a lower temperature applied to the distributions over the next words. For instance, as the temperature approaches 0, the desired distribution over sentences should converge to an argmax and produce the single most likely next sentence. However, if we try to apply the temperature to the distribution over the next words and sample autoregressively, then as the temperature approaches 0, the resulting policy will greedily construct a sentence by sequentially picking the most likely next word, which is unlikely to produce the highest probability sentence."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
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+                                    "type": "text",
+                                    "content": "Our GFlowNet sampling policy parametrizes a distribution "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p_{\\mathrm{GFN}}(w_{i+1}|w_{1:i})"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " using the same architecture as the LM "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "p_{\\mathrm{LM}}(w_{i+1}|w_{1:i})"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ", and at the start of training, it is initialized with the same weights. The initial state of the GFlowNet consists of a prompt "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_{1:k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " that conditions generation (i.e., the text whose next sentence we are trying to sample). Each subsequent state "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_{1:k+i}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " is an extension of this text and is obtained by sampling autoregressively from "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p_{\\mathrm{GFN}}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ". This process terminates when a period “.” is sampled, indicating the end of the next sentence. If a predefined maximum sentence length is reached, we force a “.” action to manually terminate the generation."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "The GFlowNet policy is trained to sample sentences in proportion to their tempered probability under the original LM given the prompt. Denote the length of the prompt by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ", the length of the sampled sentence by "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m"
+                                },
+                                {
+                                    "bbox": [
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+                                        255,
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+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": ", and the temperature by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        506,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, the reward is:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "R \\left(w _ {1: k + m}\\right) \\stackrel {\\text {d e f}} {=} p _ {\\mathrm {L M}} \\left(w _ {k + 1: k + m} \\mid w _ {1: k}\\right) ^ {\\frac {1}{T}} = \\left(\\prod_ {i = 1} ^ {m} p _ {\\mathrm {L M}} \\left(w _ {k + i} \\mid w _ {1: k + i - 1}\\right)\\right) ^ {\\frac {1}{T}} \\tag {5}",
+                                    "image_path": "ccd2aaeb4b28674c920b5f8a7832e67cfaafe668abfe5c624ee9823e00b1f970.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
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+                    "type": "text",
+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "The intuition behind this reward is that we are assuming that the LM can reliably assign a high likelihood to high-probability sentences, that we wish to preferentially sample over low-probability ones. If we were to set "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "T = 1"
+                                },
+                                {
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+                                    ],
+                                    "type": "text",
+                                    "content": ", then the solution for the GFlowNet would be to sample proportionally to "
+                                },
+                                {
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p_{\\mathrm{LM}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
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+                                    ],
+                                    "type": "text",
+                                    "content": " (which it is initialized to). The preferential sampling of high-probability sentences is therefore obtained by setting "
+                                },
+                                {
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+                                        331,
+                                        504,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "0 < T < 1"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
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+                        444
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                {
+                                    "bbox": [
+                                        104,
+                                        389,
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+                                    ],
+                                    "type": "text",
+                                    "content": "To run our experiment, we obtained a dataset of 1000 prompts from OpenWebText (Gokaslan et al., 2019) that were each 1-3 sentences long, 50 of which were used for validation. Our LM consisted of a pretrained 1.5B parameter GPT2-XL (Radford et al., 2019), and our GFlowNet was a copy of this model that was fine-tuned in a lower-dimensional space of 80M parameters using LoRA (Hu et al., 2022)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
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+                        483
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                        483
+                                    ],
+                                    "type": "text",
+                                    "content": "Additional results. We show examples of empirical next-sentence samples generated by our model in Table B.1 compared to baselines. The model was trained using a reward temperature of 0.875, which achieves a good balance between log-likelihood and diversity."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        105,
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+                        502
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "C INFILLING STORIES"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        510,
+                        504,
+                        544
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                                544
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        510,
+                                        504,
+                                        544
+                                    ],
+                                    "type": "text",
+                                    "content": "Additional details. See Table C.1 for training examples from the subset of the ROC Stories dataset used for the task. To condition the model on "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        510,
+                                        504,
+                                        544
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "X"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        510,
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+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        510,
+                                        504,
+                                        544
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Y"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        510,
+                                        504,
+                                        544
+                                    ],
+                                    "type": "text",
+                                    "content": ", as well as for the prompting baseline, we use the following prompt:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
+                        547,
+                        345,
+                        560
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                547,
+                                345,
+                                560
+                            ],
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+                                {
+                                    "bbox": [
+                                        104,
+                                        547,
+                                        345,
+                                        560
+                                    ],
+                                    "type": "text",
+                                    "content": "\"Beginning: {X}\\n End: {Y}\\n Middle: \""
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        563,
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+                        640
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "An assumption we make in this paper is that the base language model already contains the knowledge required for the task, and the goal is to perform inference over this knowledge. However, for this task, none of the pretrained base models were good at assigning high likelihoods to plausible stories. Thus, we fine-tuned GPT-2 Large model (Radford et al., 2019) on the entirety of the stories dataset and used this fine-tuned model as the reward model. This was done with full fine-tuning using theTRL library (von Werra et al., 2020). We trained for 20 epochs with a batch size of 64 and 32 gradient accumulation steps and a learning rate of 0.0005."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                    "content": "We detail the hyperparameters used for training GFlowNets in our experiments in Table C.2. During training, we sample "
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+                                    "content": " from the dataset and then sample (batch size) "
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+                                    "type": "text",
+                                    "content": " for every "
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+                                    "type": "text",
+                                    "content": ", before using "
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+                                    "type": "inline_equation",
+                                    "content": "p_{\\mathrm{LM}}(XZY)"
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+                                    "content": " as the reward. We use the replay buffer described in §3.3 and seed it with rationales from the dataset. We linearly anneal the reward temperature, the temperature of the behavior policy during training, and the learning rate during warmup. For supervised fine-tuning, we use a batch size of 256 and train for 10 epochs with a learning rate of 0.0001 withTRL and LoRA. At test time, we sample 1024 infills for each example in the test set from all the models at temperature 0.9, and average over 10 such draws."
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+                                            "content": "Table B.1: Empirical examples for sequence continuation. Sentences generated from GFlowNet fine-tuning tend to be more reasonable than autoregressively generated samples from the LM with temperature 1.0 and tend to be more diverse than samples generated from diverse beam search."
+                                        }
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+                                            "type": "text",
+                                            "content": "Table C.1: Examples of training samples for the stories infilling task."
+                                        }
+                                    ]
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+                                            "html": "<table><tr><td>Beginning (X)</td><td>Middle (Z)</td><td>End (Y)</td></tr><tr><td>I was going to a Halloween party. I looked through my clothes but could not find a costume. I cut up my old clothes and constructed a costume.</td><td>I put my costume on and went to the party.</td><td>My friends loved my costume.</td></tr><tr><td>Allen thought he was a very talented poet. He attended college to study creative writing. In college, he met a boy named Carl.</td><td>Carl told him that he wasn’t very good.</td><td>Because of this, Allen swore off poetry forever.</td></tr></table>",
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+                                            "content": "Table C.2: Hyperparameters for the story infilling task."
+                                        }
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+                                }
+                            ],
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+                                            "html": "<table><tr><td>LoRA rank</td><td>64</td></tr><tr><td>LoRA scaling factor</td><td>16</td></tr><tr><td>LoRA dropout</td><td>0.1</td></tr><tr><td>Batch size</td><td>64</td></tr><tr><td>Gradient accumulation steps</td><td>16</td></tr><tr><td>Learning rate</td><td>0.0001</td></tr><tr><td>Optimizer</td><td>AdamW</td></tr><tr><td>PFtemperature max</td><td>2</td></tr><tr><td>PFtemperature min</td><td>0.5</td></tr><tr><td>Reward temperature start</td><td>1.1</td></tr><tr><td>Reward temperature end</td><td>0.85</td></tr><tr><td>Reward temperature horizon</td><td>100</td></tr><tr><td>Buffer capacity</td><td>25</td></tr><tr><td>Number of training steps</td><td>1000</td></tr><tr><td>Evaluation temperature</td><td>0.8</td></tr><tr><td>Maximum length</td><td>25</td></tr><tr><td>Minimum length</td><td>5</td></tr></table>",
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+                                    "type": "text",
+                                    "content": "Analysis. Table C.3, C.4, C.5, C.6, C.7, C.8 illustrate some examples of the infills generated with the prompting baseline, supervised fine-tuned and GFlowNet fine-tuned models on the test examples."
+                                }
+                            ]
+                        }
+                    ],
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+                },
+                {
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+                                    "type": "text",
+                                    "content": "For evaluating the infills we generate a rating for the stories with infills from each of the methods based on the coherence of the story. We average the rating over 10 infills sampled from each method and average it over all the 100 test examples. The prompt used for evaluation is the following, adapted from Liu et al. (2023). Stories with infills generated by the GFlowNet fine-tuned model on average receive a much higher rating than the baselines. The average rating for the reference infills is 4.3 which should be viewed as an upper bound, as the stories may potentially be present in the GPT-4 training data."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
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+                                    "type": "text",
+                                    "content": "You will be given a short story."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Your task is to rate the story on one metric."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
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+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "Please make sure you read and understand these instructions carefully. Please keep this document open while reviewing, and refer to it as needed."
+                                }
+                            ]
+                        }
+                    ],
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+                },
+                {
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+                                    "type": "text",
+                                    "content": "Evaluation Criteria:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Coherence (1-5) - the collective quality of all sentences. The story should be well-structured and well-organized. The story should not just be a heap of related information, but should build from sentence to a coherent narrative. The story should also be realistic and all the sentences when put together should make sense."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
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+                                    "type": "text",
+                                    "content": "Evaluation Steps:"
+                                }
+                            ]
+                        }
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+                    "index": 8
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+                                    "type": "text",
+                                    "content": "1. Read the story carefully. Check if the story is coherent and all the sentences follow a logical order.  \n2. Assign a score for coherence on a scale of 1 to 5, where 1 is the lowest and 5 is the highest based on the Evaluation Criteria. If there are grammatical errors or logical inconsistencies, simply assign a lower score and do not elaborate on the reasoning."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
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+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "Example:"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Randy had recently separated from his wife.   \nHe felt very lonely and sad all the time.   \nHe considered trying to patch things up with Vera, his ex-wife. He decided to get a puppy instead.   \nHis decision made him happy and he no longer felt sad."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
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+                                    "type": "text",
+                                    "content": "- Coherence: 5"
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+                        }
+                    ],
+                    "index": 12
+                },
+                {
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Randy had recently separated from his wife.   \nHe felt very lonely and sad all the time.   \nHe considered trying to patch things up with Vera, his ex-wife.   \nEventually Randy missed Vera very much and developedWhile making memories with   \nHis decision made him happy and he no longer felt sad."
+                                }
+                            ]
+                        }
+                    ],
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+                                    "type": "text",
+                                    "content": "- Coherence 1"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                }
+            ],
+            "discarded_blocks": [
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
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+                                    "type": "text",
+                                    "content": "21"
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+                                    "type": "text",
+                                    "content": "Story:"
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+                            ]
+                        }
+                    ],
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+                                    "type": "text",
+                                    "content": "{{Story}}"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
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+                    ],
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+                                    ],
+                                    "type": "text",
+                                    "content": "Evaluation Form (scores ONLY):"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
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+                    ],
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+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "- Coherence:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
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+                        183
+                    ],
+                    "type": "title",
+                    "angle": 0,
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+                                    ],
+                                    "type": "text",
+                                    "content": "D SUBJECTIVITY CLASSIFICATION"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                    "type": "text",
+                                    "content": "Additional details. See Table D.1 for some training examples in the SUBJ dataset. We use the prompts in Table D.2 for all baselines."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
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+                    ],
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+                    "angle": 0,
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+                                {
+                                    "bbox": [
+                                        104,
+                                        217,
+                                        506,
+                                        316
+                                    ],
+                                    "type": "text",
+                                    "content": "We run GFlowNet fine-tuning for 1000 steps with a linear warmup over 200 steps, a fixed learning rate of 0.0005, and a batch size of 512 samples; see Table D.3 for all the hyperparameters used. Each batch consists of 8 queries ("
+                                },
+                                {
+                                    "bbox": [
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+                                        217,
+                                        506,
+                                        316
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "X"
+                                },
+                                {
+                                    "bbox": [
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+                                        217,
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+                                    ],
+                                    "type": "text",
+                                    "content": "'s), randomly drawn with replacement. We then sample 64 rationales ("
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        217,
+                                        506,
+                                        316
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Z"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        217,
+                                        506,
+                                        316
+                                    ],
+                                    "type": "text",
+                                    "content": "'s) for every "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        217,
+                                        506,
+                                        316
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "X"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        217,
+                                        506,
+                                        316
+                                    ],
+                                    "type": "text",
+                                    "content": ", before using "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        217,
+                                        506,
+                                        316
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p_{\\mathrm{LM}}(ZY \\mid X)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        217,
+                                        506,
+                                        316
+                                    ],
+                                    "type": "text",
+                                    "content": " as the reward. We also use the replay buffer described in §3.3 and seed it with potential rationales generated from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        217,
+                                        506,
+                                        316
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p_{\\mathrm{LM}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        217,
+                                        506,
+                                        316
+                                    ],
+                                    "type": "text",
+                                    "content": ". For supervised fine-tuning, both on its own and on top of GFlowNet fine-tuning, we use a batch size of 256 with 8 queries, randomly drawn with replacement. We train for 100 steps with a linear warm-up over 20 steps and a constant learning rate. We sweep the learning rate in [0.001, 0.0001] and report the best performance. The reward temperature is annealed from 1.2 down to 1 over the first 150 steps of training."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
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+                        506,
+                        376
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                319,
+                                506,
+                                376
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        319,
+                                        506,
+                                        376
+                                    ],
+                                    "type": "text",
+                                    "content": "Additional results. To encourage exploration in the space of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        319,
+                                        506,
+                                        376
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Z"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        319,
+                                        506,
+                                        376
+                                    ],
+                                    "type": "text",
+                                    "content": ", we inverse-prompt the model to generate potential chains of thoughts and store them in a replay buffer at the beginning of training. The replay buffer is a priority queue indexed by the reward. Under-performing chains of thoughts are evicted as we collect more trajectories. We ablate the effect of seeding the replay buffer with inverse-prompted chains of thoughts and aggregating multiple chains at test time in Table D.4."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
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+                        383,
+                        243,
+                        395
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "E INTEGER ARITHMETIC"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
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+                    "angle": 0,
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Additional details. See Table E.1 for examples from synthetically generated training data for the integer arithmetic task."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
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+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                        506,
+                                        506
+                                    ],
+                                    "type": "text",
+                                    "content": "As with the infilling task, for the integer arithmetic task, the pretrained base models we tested were imperfect knowledge models, i.e. incorrect rationales are sometimes assigned very high likelihood. To assign the correct rationales higher rewards, we prepend some demonstrations with hand-crafted rationales to the query when computing the reward, i.e. "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        506
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p_{\\mathrm{LM}}(XZY|(X_iZ_iY_i)^k)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
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+                                        506
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
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+                                        506
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k = 3"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        506
+                                    ],
+                                    "type": "text",
+                                    "content": " in our experiments. This improves the calibration to some extent. Note that these "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        506
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(X_i,Z_i,Y_i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        506
+                                    ],
+                                    "type": "text",
+                                    "content": " are taken from the dataset and used only in the reward. The GFlowNet policy here is conditioned only on "
+                                },
+                                {
+                                    "bbox": [
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+                                        506
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "X"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
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+                                        506
+                                    ],
+                                    "type": "text",
+                                    "content": ", i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        506
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "q_{\\mathrm{GFN}}(Z|X)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        506
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
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+                    "index": 11
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+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                {
+                                    "bbox": [
+                                        104,
+                                        510,
+                                        506,
+                                        662
+                                    ],
+                                    "type": "text",
+                                    "content": "We detail the hyperparameters used for training GFlowNets in our experiments in Table E.2. During training, we sample "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        510,
+                                        506,
+                                        662
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(X,Y)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        510,
+                                        506,
+                                        662
+                                    ],
+                                    "type": "text",
+                                    "content": " from the dataset and then sample (batch size) "
+                                },
+                                {
+                                    "bbox": [
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+                                        510,
+                                        506,
+                                        662
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+                                    "type": "inline_equation",
+                                    "content": "Zs"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        510,
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+                                    ],
+                                    "type": "text",
+                                    "content": " for every "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        510,
+                                        506,
+                                        662
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(X,Y)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        510,
+                                        506,
+                                        662
+                                    ],
+                                    "type": "text",
+                                    "content": ", before using "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        510,
+                                        506,
+                                        662
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p_{\\mathrm{LM}}(XZY|(X_iZ_iY_i)^k_{i=1})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        510,
+                                        506,
+                                        662
+                                    ],
+                                    "type": "text",
+                                    "content": " as the reward. During the generation of "
+                                },
+                                {
+                                    "bbox": [
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+                                        510,
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+                                        662
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Z"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        510,
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+                                        662
+                                    ],
+                                    "type": "text",
+                                    "content": ", as mentioned in §4.4, whenever a “=” is generated we extract the preceding expression and evaluate the last two terms using eval in Python. We use the replay buffer described in §3.3 and seed it with rationales from the dataset "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        510,
+                                        506,
+                                        662
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p_{\\mathrm{LM}}"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ". We linearly anneal the reward temperature, and the learning rate during warmup. During evaluation, for all methods, we aggregate the response over multiple samples drawn from the model at some fixed temperature. For the zero-shot baseline, we observe the best performance with the “Let us think step by step.” prompt appended at the end of the question. For supervised fine-tuning and PPO, we use the implementation from trl. For supervised fine-tuning we use a batch size of 128 with 8 gradient accumulation steps and train for 10 epochs with a learning rate of 0.0001 using LoRA. For PPO we use a minibatch size of 64 with 16 gradient accumulation steps, a learning rate of 0.0001, 4 epochs (on the minibatch), a clip range of 0.2, and an adaptive KL coefficient initialized at 0.2."
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+                                    "content": "Effect of the number of rationales. As mentioned in §4.4, we seed the buffer with rationales from a dataset. Table E.3 summarizes the results of ablation on the number of examples used to seed the buffer. We observe, as expected, that the performance generally improves as we the number of examples used to seed the buffer grows. When no buffers are used to seed the buffer, the performance is quite poor. We hypothesize that without the good rationales used to seed the buffer, the exploration problem of searching for good rationales becomes very challenging."
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+                                            "content": "Table D.1: Two training examples from the SUBJ dataset."
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+                                            "html": "<table><tr><td>Text (X)</td><td>Label (Y)</td></tr><tr><td>another story follows the relationship between a stepfather ( neeson ) and his young stepson .</td><td>objective</td></tr><tr><td>hoffman ’s performance is authentic to the core of his being .</td><td>subjective</td></tr></table>",
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+                                            "content": "Table D.2: Prompts used for subjectivity classification."
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+                                            "html": "<table><tr><td>Few-shot learning / Supervised fine-tuning</td><td>GFlowNet fine-tuning</td></tr><tr><td>Classify this movie review as objective or subjective: “[X]” This review is [Y].</td><td>Classify this movie review as objective or subjective: “[X]” This review is [Z], so it is [Y].</td></tr></table>",
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+                                            "content": "Table D.3: Hyperparameters for GFlowNet fine-tuning on subjectivity classification."
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+                                            "html": "<table><tr><td>LoRA rank</td><td>256</td></tr><tr><td>LoRA scaling factor</td><td>16</td></tr><tr><td>LoRA dropout</td><td>0.</td></tr><tr><td>Batch size</td><td>16</td></tr><tr><td>Gradient accumulation steps</td><td>32</td></tr><tr><td>Learning rate</td><td>0.0005</td></tr><tr><td>Optimizer</td><td>AdamW</td></tr><tr><td>PFtemperature max</td><td>2</td></tr><tr><td>PFtemperature min</td><td>0.5</td></tr><tr><td>Reward temperature start</td><td>1.2</td></tr><tr><td>Reward temperature end</td><td>1.0</td></tr><tr><td>Reward temperature horizon</td><td>150</td></tr><tr><td>Buffer capacity</td><td>50</td></tr><tr><td>Number of steps</td><td>100</td></tr><tr><td>Number of samples</td><td>10</td></tr><tr><td>Maximum length</td><td>5</td></tr><tr><td>Minimum length</td><td>1</td></tr></table>",
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+                                    "type": "text",
+                                    "content": "Analysis. In Table E.4 we show some examples generated by PPO and GFlowNet fine-tuned models. We observe that PPO generates sequences with high rewards under the base model. These sequences, however, are not valid expressions, and, in fact, do not even call the tool. Instead, the model learns to simply repeat the expression. Repetitions being assigned high likelihood is an issue that has also been noted by Welleck et al. (2020). On the other hand, GFlowNets are able to generate the correct rationale to evaluate the expression."
+                                }
+                            ]
+                        }
+                    ],
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+                },
+                {
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+                                    "type": "text",
+                                    "content": "There are still cases where the GFlowNet fails to produce the correct reasoning chain. We illustrate some of these examples in Table E.5. The errors fall mainly into 3 categories: 1) missing operands in longer OOD problems 2) incorrect operand being copied 3) incorrect operator being copied. The potential source for such errors is that the reward model assigns equally high rewards to rationales with these minor mistakes and thus the model generates them with likelihood. These errors can potentially be reduced by using better and potentially larger reward models."
+                                }
+                            ]
+                        }
+                    ],
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+                                            "type": "text",
+                                            "content": "Table D.4: Ablation studies on subjectivity classification using GPT-J 6.8B."
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+                                            "html": "<table><tr><td>Method</td><td colspan=\"3\">Test accuracy (%) ↑</td></tr><tr><td></td><td colspan=\"3\">Training samples</td></tr><tr><td></td><td>10</td><td>20</td><td>50</td></tr><tr><td>GFlowNet fine-tuning</td><td>71.4 ± 1.8</td><td>81.1 ± 0.4</td><td>87.7 ± 2.2</td></tr><tr><td>(-) Seed buffer</td><td>64.7 ± 8.9</td><td>68.7 ± 1.7</td><td>77.0 ± 5.5</td></tr><tr><td>(-) Seed buffer (-) Aggregating</td><td>63.9 ± 7.1</td><td>65.9 ± 2.4</td><td>75.4 ± 3.3</td></tr></table>",
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+                                            "content": "Table D.5: Top sample rationales for the SUBJ test set using the instruct-fine-tuned GPT-J 6B as both the reward model and the base model for the GFlowNet, which is trained on 50 labeled examples."
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+                                            "content": "Table E.1: Examples from the integer arithmetic dataset. Note that the results of the two-term expressions are evaluated by the calculator."
+                                        }
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+                                            "content": "Table E.3: Ablation study to understand the effect of the number of rationales used to seed the buffer on the test accuracy "
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+                                            "content": "Table E.4: Samples generated by PPO fine-tuned and GFlowNet fine-tuned models."
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+                                            "content": "Table E.5: Examples of errors made by the GFlowNet-fine-tuned model."
+                                        }
+                                    ]
+                                }
+                            ],
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diff --git a/2024/An Analytical Solution to Gauss-Newton Loss for Direct Image Alignment/1a6e78f9-a669-4d03-b852-34ab11b21b42_content_list.json b/2024/An Analytical Solution to Gauss-Newton Loss for Direct Image Alignment/1a6e78f9-a669-4d03-b852-34ab11b21b42_content_list.json
new file mode 100644
index 0000000000000000000000000000000000000000..be9aa21ffe66d33bb963dd54e407e30cbcd284ca
--- /dev/null
+++ b/2024/An Analytical Solution to Gauss-Newton Loss for Direct Image Alignment/1a6e78f9-a669-4d03-b852-34ab11b21b42_content_list.json	
@@ -0,0 +1,2545 @@
+[
+    {
+        "type": "text",
+        "text": "AN ANALYTICAL SOLUTION TO GAUSS-NEWTON LOSS FOR DIRECT IMAGE ALIGNMENT",
+        "text_level": 1,
+        "bbox": [
+            169,
+            99,
+            823,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Sergei Solonets $^{1,2,*}$  Daniil Sinitsyn $^{1,2,*}$",
+        "bbox": [
+            179,
+            167,
+            468,
+            184
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Lukas von Stumberg $^{3,\\dagger}$  Nikita Araslanov $^{1,2}$  Daniel Cremers $^{1,2}$",
+        "bbox": [
+            181,
+            184,
+            651,
+            199
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "<sup>1</sup> Technical University of Munich",
+            "2 Munich Center for Machine Learning",
+            "3 Valve Software"
+        ],
+        "bbox": [
+            181,
+            204,
+            442,
+            247
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/45ed605ea8fc0824ef2dac048c5d768f6e8e7ce2fe3fe94355da849bc9204e14.jpg",
+        "image_caption": [
+            "Figure 1: Direct image alignment is a technique for aligning scenes based on image intensities. Recent learning-based methods seek to improve its success rate by increasing the convergence basin. We derive an analytical solution to the core idea of such methods, the Gauss-Newton loss, enabling fine-grained control over the basin of convergence. As a result, we can successfully align two scenes despite a highly imprecise initialization. From left to right, the example above illustrates the convergence of the reprojected keypoints (red) to the ground truth (green) by optimizing the SE(3) camera pose with our analytical solution. The blue color is the re-projection in the first iteration."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            261,
+            385,
+            371
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/eae3db6aee13ba2c8ff208b0c14ec4553a4a5e31b812be8896bd3841ed9f0658.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            393,
+            261,
+            604,
+            371
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/35752841c5a829c17a1c6f48b0342e06ec16d815c6e46982b27bd0ed82a52af5.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            612,
+            261,
+            823,
+            371
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT",
+        "text_level": 1,
+        "bbox": [
+            450,
+            494,
+            545,
+            510
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Direct image alignment is a widely used technique for relative 6DoF pose estimation between two images, but its accuracy strongly depends on pose initialization. Therefore, recent end-to-end frameworks increase the convergence basin of the learned feature descriptors with special training objectives, such as the Gauss-Newton loss. However, the training data may exhibit bias toward a specific type of motion and pose initialization, thus limiting the generalization of these methods. In this work, we derive a closed-form solution to the expected optimum of the Gauss-Newton loss. The solution is agnostic to the underlying feature representation and allows us to dynamically adjust the basin of convergence according to our assumptions about the uncertainty in the current estimates. These properties allow for effective control over the convergence in the alignment process. Despite using self-supervised feature embeddings, our solution achieves compelling accuracy w.r.t. the state-of-the-art direct image alignment methods trained end-to-end with pose supervision, and demonstrates improved robustness to pose initialization. Our analytical solution exposes some inherent limitations of end-to-end learning with the Gauss-Newton loss, and establishes an intriguing connection between direct image alignment and feature-matching approaches.",
+        "bbox": [
+            228,
+            525,
+            769,
+            763
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION",
+        "text_level": 1,
+        "bbox": [
+            173,
+            787,
+            336,
+            801
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Visual localization refers to estimating the camera pose of a query image w.r.t. a reference image where the underlying 3D structure (e.g. a point cloud) is available. Traditionally, solutions to visual localization primarily relied on estimating correspondences between 2D features in the query image and 3D features in the reference point cloud (Liu et al., 2017; Sarlin et al., 2019; Sattler et al., 2017;",
+        "bbox": [
+            169,
+            818,
+            826,
+            876
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+    },
+    {
+        "type": "page_footnote",
+        "text": "*Equal contribution. Correspondence: {s.solonets,d.sinitsyn}@tum.de.",
+        "bbox": [
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+    {
+        "type": "page_footnote",
+        "text": "†Work done while at TU Munich.",
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+    },
+    {
+        "type": "page_footnote",
+        "text": "Project code: https://github.com/tum-vision/gn_loss_analytical.",
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+        "page_idx": 0
+    },
+    {
+        "type": "page_number",
+        "text": "1",
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+    },
+    {
+        "type": "text",
+        "text": "Svarm et al., 2017; Toft et al., 2018). Challenging this approach, recent deep learning frameworks implement direct image alignment by re-projecting the 3D points onto the feature map (Sarlin et al., 2021; von Stumberg et al., 2020a;b). Using end-to-end training, deep networks learn dense feature maps suitable for regressing the pose between an image and a point cloud. In analogy to photometric image alignment (Delaunoy and Pollefeys, 2014; Engel et al., 2016), this family of methods is referred to as feature metric image alignment (von Stumberg et al., 2020b).",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The feature representation learned by feature metric image alignment is not only discriminative, but is also spatially smooth for improved convergence (von Stumberg et al., 2020a). A training process enforces the smoothness either explicitly through a specific loss, such as the Gauss-Newton loss (von Stumberg et al., 2020a), or implicitly, by backpropagating through an optimization algorithm, such as Levenberg-Marquardt (Sarlin et al., 2021). In both cases, the resulting feature map embeds the bias of the initial poses in the training data. This may lead to poor generalization, since the training poses can differ substantially from the test scenario. We here take a less bias-prone approach: we use generic feature descriptors (e.g. obtained with self-supervision) and instead control the smoothness of the feature map dynamically at test time.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This work investigates the connection between feature descriptor networks and feature metric image alignment. The main contribution is the analytical (closed-form) solution to the Gauss-Newton loss. On the one hand, this leads to a novel technique, which can utilize any feature descriptor to generate a dense feature map suitable for direct image alignment. Importantly, it allows us to dynamically adjust the smoothness of the feature map, which effectively controls the trade-off between the basin of convergence and the alignment accuracy. We empirically verify our derivation using self-supervised feature descriptors, such as SuperPoint (DeTone et al., 2018), and demonstrate on-par or even superior alignment accuracy compared to supervised frameworks. On the other hand, the analysis of our closed-form solution reveals inherent limitations of feature learning with backpropagation through Gauss-Newton optimization: feature metric alignment merely learns a form of interpolation between feature descriptors in the points of interest. Although we demonstrate this in the context of direct image alignment, a similar argument extends to other methods, even beyond computer vision, which backpropagate through Gauss-Newton or Levenberg-Marquardt optimization.",
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+    {
+        "type": "text",
+        "text": "2 PRELIMINARIES",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Image Alignment. Given two images (reference  $I_r$  and query  $I_q$ ) with known camera models and an overlapping field of view, and a 3D point cloud  $\\{\\pmb{p}^{(i)}\\}$  in the coordinate system of  $I_r$ , the problem of image alignment is to estimate the relative camera pose  $T \\in \\mathbf{SE}(3)$ .",
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+    {
+        "type": "text",
+        "text": "Gauss-Newton Optimization. Given a set of functions  $\\{r^{(1)}(\\pmb {x}),\\dots,r^{(m)}(\\pmb {x})\\}$  called residuals, Gauss-Newton (GN) optimization finds the parameters minimizing the sum of squared residuals:",
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+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nf (\\boldsymbol {x}) = \\sum_ {i} \\| r ^ {(i)} (\\boldsymbol {x}) \\| _ {2} ^ {2}. \\tag {1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Each residual  $r^{(i)}(\\pmb{x})$  could either be a vector or a single-valued function. In both scenarios, the total residual  $r(\\pmb{x})$  is a stacked vector of  $m$  residuals. The Gauss-Newton method seeks to minimize  $f(\\pmb{x})$  by iteratively updating the parameter estimate  $\\tilde{\\pmb{x}}$ . Each iteration linearizes the residuals around the current estimate and computes an update step  $\\Delta_{GN}(r(\\tilde{\\pmb{x}}))$ :",
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+        "type": "equation",
+        "text": "\n$$\n\\Delta_ {\\mathrm {G N}} [ r (\\tilde {\\boldsymbol {x}}) ] = \\left(\\boldsymbol {J} ^ {\\intercal} \\boldsymbol {J}\\right) ^ {- 1} \\boldsymbol {J} ^ {\\intercal} r (\\tilde {\\boldsymbol {x}}), \\quad \\boldsymbol {J} := \\left. \\frac {\\partial r}{\\partial \\boldsymbol {x}} \\right| _ {\\boldsymbol {x} = \\tilde {\\boldsymbol {x}}}, \\tag {2}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where  $J$  is the Jacobian matrix. The estimate  $\\tilde{\\pmb{x}}$  evolves until convergence as",
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\tilde {\\boldsymbol {x}} \\leftarrow \\tilde {\\boldsymbol {x}} \\boxminus \\Delta_ {\\mathrm {G N}} [ r (\\tilde {\\boldsymbol {x}}) ]. \\tag {3}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Here, the operator  $\\square$  denotes a specific update procedure, which depends on the nature of the optimization space. For linear spaces,  $\\square$  simplifies to the standard subtraction operation; in the context of a rigid-body transformation  $\\mathbf{SE}(3)$ , operator  $\\square$  applies a tangential update.",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Photometric and feature metric image alignment. Photometric image alignment estimates the relative 6DoF pose by minimizing the difference between pixel intensities of points in  $I_{r}$  and the",
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+        "page_idx": 1
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "page_number",
+        "text": "2",
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "corresponding points in  $I_q$ . Recalling  $\\pmb{p}^{(i)}$  as a 3D point in the coordinate frame of the reference image, we seek a transformation  $\\pmb{T} \\in \\mathbf{SE}(3)$  minimizing the following residuals:",
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nr ^ {(i)} (\\boldsymbol {T}) = I _ {r} \\left(\\langle \\boldsymbol {p} ^ {(i)} \\rangle\\right) - I _ {q} \\left(\\langle \\boldsymbol {T} \\boldsymbol {p} ^ {(i)} \\rangle\\right). \\tag {4}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Here,  $\\langle \\cdot \\rangle$  is a 2D projection operator of a 3D point onto the image plane. The projection implicitly uses the corresponding camera intrinsic parameters, assumed to be available for both  $I_{r}$  and  $I_{q}$ . We omit them in the notation for clarity. Comprising rotation  $\\mathbf{R} \\in \\mathbf{SO}(3)$  and translation  $\\pmb{t} \\in \\mathbb{R}^3$ , the target transformation  $\\pmb{T}$  is typically found by minimizing Eq. (4) with non-linear least-squares optimization, such as Gauss-Newton or Levenberg-Marquardt (LM) methods (Levenberg, 1944; Marquardt, 1963; Nocedal and Wright, 1999).",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The success of photometric image alignment critically depends on a favorable initialization of the pose, especially in the conditions of varying illumination and occlusion. As a partial remedy, feature metric image alignment (Tang and Tan, 2019) uses feature maps instead of pixel intensities. Obtained with deep learning, such feature maps have increased convergence basin compared to that derived from image intensities, which improves robustness to pose initialization.",
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+    },
+    {
+        "type": "text",
+        "text": "3 RELATED WORK",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Direct and indirect image alignment. Estimating the relative camera pose from two images is a fundamental problem in computer vision, with applications in structure from motion (SfM) (Schönberger and Frahm, 2016), SLAM and relocalization. To address this problem, indirect (feature-based) approaches detect and match interest points (Bay et al., 2006; Lowe, 2004) in both images, and then estimate the pose using PnP (Persson and Nordberg, 2018) or by minimizing the reprojection errors (Triggs et al., 1999). In contrast, direct methods sidestep the matching process and minimize the photometric error instead (Horn and Jr., 1988; Irani and Anandan, 1999). This means that they can leverage the entire image (Kerl et al., 2013; Newcombe et al., 2011), or focus on pixels with sufficient gradient (Engel et al., 2014; 2016). The foundation behind these approaches is Lucas-Kanade tracking (Baker and Matthews, 2004; Lucas and Kanade, 1981). However, direct methods, which are central to this work, optimize for a 6DoF pose instead of individual pixel displacements (i.e. optical flow).",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Pose estimation with deep learning. The advent of deep learning revitalized interest in improving pose estimation with deep networks. While some approaches are holistic (Jatavallabhula et al., 2020), there are broadly three categories of learning-based methods. i) Fully end-to-end pose estimation methods (Kendall et al., 2016; Ummenhofer et al., 2017; Zhou et al., 2017) directly regress pose estimates with deep neural networks, instead of test-time optimization. ii) Learning-based indirect methods (DeTone et al., 2018; Dusmanu et al., 2019; Revaud et al., 2019; Yi et al., 2016) replace handcrafted detectors and descriptors with deep representations in an indirect pipeline. Some approaches further extend the traditional way of obtaining correspondences. SuperGlue (Sarlin et al., 2020) learns feature matching with a graph neural network. LoFTR (Sun et al., 2021) directly regresses correspondences instead of relying on separate feature detection and matching. Similar to indirect methods, iii) learning-based direct image alignment enhances classical direct methods with deep features. This category is the most similar to our work and we discuss it in more detail next.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Learned features for direct image alignment. Previous work differs in their approach to model training and in the final task. For example, Czarnowski et al. (2017) leverage off-the-shelf CNN features to improve optical flow tracking. A number of methods (Han et al., 2018; Lv et al., 2019; Sarlin et al., 2021; Tang and Tan, 2019; Xu et al., 2021) train feature descriptors end-to-end with ground-truth poses and backpropagate the gradient through a non-linear optimization process. At test time, these methods employ a feature pyramid and refine the initial camera pose in a coarse-to-fine fashion using GN or LM optimization. In addition to the feature pyramid, some methods predict additional properties for image alignment, such as uncertainty (Xu et al., 2021) (Sarlin et al., 2021), Jacobians (Han et al., 2018), or optimization parameters (e.g. damping factors (Lv et al., 2019; Sarlin et al., 2021)). In the same spirit, our formulation leads to a continuous image pyramid, where each level of the pyramid can be generated on-the-fly based on the input level of uncertainty.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Another line of work (von Stumberg et al., 2020a;b) trains a deep network directly on the ground-truth pixel correspondences. GN-Net (von Stumberg et al., 2020a) minimizes two loss functions. The first is a contrastive loss (Schmidt et al., 2017) facilitating discriminative properties of the",
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+        "page_idx": 2
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "page_number",
+        "text": "3",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "feature representation. The second loss function accounts for a likely displacement in the initial correspondence, implemented by adding random 2D offsets to the ground-truth correspondences at training time. By learning to minimize the feature discrepancy after a Gauss-Newton step, the model learns to cope with the initially noisy estimates. We revisit this work in detail in the Sec. 4.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The learning-based approaches (e.g. (Germain et al., 2021; Sarlin et al., 2021; von Stumberg et al., 2020a)) achieve impressive accuracy of pose estimation. Nevertheless, they require pose supervision for training and struggle in scenarios of large-baseline localization. Indeed, GN-Net (von Stumberg et al., 2020a) and PixLoc (Sarlin et al., 2021) tend to exhibit a strong bias toward the noise assumptions of the training process, which cannot be easily reversed. For example, PixLoc trained on the CMU dataset exhibits a strong bias toward horizontal movement as illustrated in Appendix A.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our work takes a different approach. We rely on existing feature descriptors obtained with self-supervision, which contribute no explicit motion bias to the alignment process. We next derive a closed-form solution to the Gauss-Newton loss (von Stumberg et al., 2020a) as a functional of a probability density governing the noise assumptions of the current pose estimate. This allows us to adjust the noise assumptions in the alignment process, thus effectively controlling the convergence basin at test time, much akin to the coarse-to-fine strategy, exemplified by Fig. 2.",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 THE GAUSS-NEWTON LOSS",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this section, we recap the GN-Net (von Stumberg et al., 2020a) and introduce the notation. GN-Net is a convolutional neural network  $E(\\cdot ;\\pmb {\\theta})$  trained on a sparse set of ground-truth correspondences. Given coordinates on the image plane, let  $f:\\mathbb{R}\\times \\mathbb{R}\\to \\mathbb{R}^d$  continuously map those coordinates to a descriptor from a feature grid  $\\mathbb{R}^{d\\times w\\times h}$  produced by  $E(\\cdot ;\\pmb {\\theta})$ , e.g. using bilinear interpolation. We define  $f_{r}\\coloneqq E(I_{r};\\pmb {\\theta})$  and  $f_{q}\\coloneqq E(I_{q};\\pmb{\\theta})$  to denote feature representations of reference and query images. GN-Net learns parameters  $\\pmb{\\theta}$  to minimize the expected value of the following loss function,",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} _ {\\mathrm {G N} - \\text {N e t}} \\left(I _ {r}, I _ {q}\\right) = \\mathcal {L} _ {\\text {c o n t r a s t i v e}} \\left(f _ {r}, f _ {q}\\right) + \\mathcal {L} _ {\\mathrm {G N}} \\left(f _ {r}, f _ {q}\\right), \\tag {5}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "which comprises a contrastive loss,  $\\mathcal{L}_{\\mathrm{contrastive}}(\\cdot ,\\cdot)$ , and the Gauss-Newton loss,  $\\mathcal{L}_{\\mathrm{GN}}(\\cdot ,\\cdot)$ . The contrastive loss minimizes the distance between the features of two corresponding points while maximizing the distance between non-corresponding pairs (Schmidt et al., 2017). The contrastive loss facilitates spatially discriminative features, and its particular instantiation has little significance for the following discussion (e. g. GN-Net uses the triplet loss).",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The Gauss-Newton loss  $\\mathcal{L}_{\\mathrm{GN}}$  ensures that the feature map is sufficiently smooth for direct image alignment, thus enlarging the convergence basin. GN-Net implements this by adding a random offset to the ground-truth correspondences and encouraging a single Gauss-Newton step to recover the original location. Let us formalize this process.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Given a ground-truth correspondence  $(\\pmb{x}^{(i)},\\pmb{y}^{(i)})$  between images  $I_{r}$  and  $I_{q}$ , the assumption behind the Gauss-Newton loss is that the initial estimate  $\\tilde{\\pmb{x}}^{(i)}$  at test time falls in the vicinity of the ground truth  $\\pmb{x}^{(i)}$ , i.e.  $\\tilde{\\pmb{x}}^{(i)} = \\pmb{x}^{(i)} + \\pmb{\\epsilon}$ , where  $\\pmb{\\epsilon}$  follows some predefined distribution  $p(\\pmb{\\epsilon})$ , such as a Gaussian with zero mean. At training time, the Gauss-Newton loss aims at recovering the ground-truth location  $\\pmb{x}^{(i)}$  from a noisy initial location  $\\tilde{\\pmb{x}}^{(i)}$  by minimizing the residual w.r.t.  $\\pmb{x}$ :",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nr ^ {(i)} \\left(\\tilde {\\boldsymbol {x}} ^ {(i)}\\right) := f _ {r} \\left(\\tilde {\\boldsymbol {x}} ^ {(i)}\\right) - f _ {q} \\left(\\boldsymbol {y} ^ {(i)}\\right), \\tag {6}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The original Gauss-Newton loss for one point  $\\pmb{x}^{(i)}$ , as introduced by von Stumberg et al. (2020a), is",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} _ {\\mathrm {G N} _ {o}} ^ {(i)} \\left(f _ {r}, f _ {q}, \\epsilon\\right) := \\| J (\\epsilon - \\Delta_ {\\mathrm {G N}} [ r ^ {(i)} \\left(\\boldsymbol {x} ^ {(i)} + \\epsilon\\right) ]) \\| _ {2} ^ {2} - \\log \\det  \\boldsymbol {J} ^ {\\intercal} \\boldsymbol {J}. \\tag {7}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "From a probabilistic standpoint (von Stumberg et al., 2020a), the loss balances between the accuracy of the Gauss-Newton step and the direction uncertainty. Here,  $J^{\\mathsf{T}}J$  represents the inverse covariance matrix propagated through the photometric residuals. The loss function corresponds to the negative log-likelihood of residuals distributed as  $\\mathcal{N}\\left(0,\\left(\\mathbf{J}^{\\mathsf{T}}\\mathbf{J}\\right)^{-1}\\right)$ . In this work, we consider a variant of the Gauss-Newton loss, in which the covariance matrix is assumed to be identity. Thus, Eq. (7) becomes",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} _ {\\mathrm {G N}} ^ {(i)} \\left(f _ {r}, f _ {q}, \\boldsymbol {\\epsilon}\\right) = \\| \\boldsymbol {\\epsilon} - \\Delta_ {\\mathrm {G N}} \\left[ r ^ {(i)} \\left(\\boldsymbol {x} ^ {(i)} + \\boldsymbol {\\epsilon}\\right) \\right] \\| _ {2} ^ {2}. \\tag {8}\n$$\n",
+        "text_format": "latex",
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+        "page_idx": 3
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "page_number",
+        "text": "4",
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+    },
+    {
+        "type": "text",
+        "text": "Although this simplified version does not account for the trade-off between uncertainty and accuracy of the prediction, it admits a closed-form minimizer of the expected value, as we show in Sec. 5. We also find that the simplified version does not differ from the original one empirically in a significant way (see Appendix C). Henceforth, we will refer to the simplified version as the Gauss-Newton loss.",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The training process calculates the Gauss-Newton loss stochastically by sampling  $\\epsilon$  from  $p(\\epsilon)$ . Therefore, it minimizes a Monte-Carlo approximation to the expected value of  $\\mathcal{L}_{\\mathrm{GN}}^{(i)}$  summed over all ground-truth correspondences,",
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+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} _ {\\mathrm {G N}} \\left(f _ {r}, f _ {q}; p\\right) = \\mathbb {E} _ {\\epsilon \\sim p} \\left[ \\sum_ {i} \\mathcal {L} _ {\\mathrm {G N}} ^ {(i)} \\left(f _ {r}, f _ {q}, \\epsilon\\right) \\right]. \\tag {9}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            232,
+            825,
+            261
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Hereafter, we use the notation  $\\mathcal{L}_{\\mathrm{GN}}(f_r,f_q;p)$  to emphasize the dependence of the Gauss-Newton loss on the noise distribution  $p$ , and omit this parametarization otherwise to avoid clutter.",
+        "bbox": [
+            169,
+            273,
+            823,
+            303
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note that the GN-Net's training stage addresses the problem of optical flow, not pose estimation. The underlying assumption is that accurate optical flow facilitates pose estimation, as each pixel will contribute to the final pose estimate. At test time, the pose is determined by solving the featuremetric image alignment problem (equivalent to Eq. (4)) using Gauss-Newton optimization in SE(3).",
+        "bbox": [
+            169,
+            310,
+            826,
+            368
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5 CLOSED-FORM GAUSS-NEWTON STEP",
+        "text_level": 1,
+        "bbox": [
+            171,
+            387,
+            526,
+            402
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Decoupling the contrastive and Gauss-Newton losses. The contrastive loss provides a sparse constraint on the feature embeddings, since we can only use sparse ground-truth correspondences, the interest points, for supervision. By contrast, the Gauss-Newton loss enforces a pre-defined basin of convergence around each interest point (modeled by  $p(\\epsilon)$ ) with little effect on the feature descriptors in the interest points. This is because the residual in Eq. (6) between the corresponding interest points will be negligible, if the contrastive loss for those points is minimized. It follows that the contrastive and the Gauss-Newton loss essentially optimize over a disjoint set of feature locations. Therefore, we can decouple the Gauss-Newton loss from the joint optimization objective in Eq. (5). Let us formalize this reasoning. We aim to solve:",
+        "bbox": [
+            169,
+            419,
+            823,
+            545
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nf _ {q} ^ {*} = \\underset {f _ {q}} {\\arg \\min } \\left[ \\mathcal {L} _ {\\text {c o n t r a s t i v e}} \\left(f _ {r}, f _ {q}\\right) + \\mathcal {L} _ {\\mathrm {G N}} \\left(f _ {r}, f _ {q}\\right) \\right]. \\tag {10}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            334,
+            566,
+            825,
+            593
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We denote the values of  $f_{r}$ ,  $f_{q}$  in the interest points as  $F_{r}$ ,  $F_{q}$ :  $F_{r}^{(i)} := f_{r}(\\pmb{x}^{(i)})$ ,  $F_{q}^{(i)} := f_{q}(\\pmb{y}^{(i)})$ . The contrastive loss only depends on  $F_{r}$  and  $F_{q}$ , while the Gauss-Newton loss depends on all the query feature values  $f_{q}$  and the interest points in the reference map  $F_{r}$ . By eliminating unused parts, we introduce an equivalent problem:",
+        "bbox": [
+            169,
+            607,
+            826,
+            666
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left\\{F _ {q} ^ {*}, f _ {q} ^ {*} \\right\\} = \\underset {F _ {q}, f _ {q}} {\\arg \\min } \\left[ \\mathcal {L} _ {\\text {c o n t r a s t i v e}} \\left(F _ {r}, F _ {q}\\right) + \\mathcal {L} _ {\\mathrm {G N}} \\left(F _ {r}, f _ {q}\\right) \\right]. \\tag {11}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            310,
+            686,
+            825,
+            714
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Next, we approximate  $F_{r}$  with  $F_{q}$  in the second term. This is permissible as the contrastive loss acts as a soft constraint, ensuring that  $F_{r}$  and  $F_{q}$  are close at the optimum of the joint loss function. This allows us to decouple the minimization problem as follows:",
+        "bbox": [
+            169,
+            724,
+            823,
+            768
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nF _ {q} ^ {*} = \\underset {F _ {q}} {\\arg \\min } \\left[ \\mathcal {L} _ {\\text {c o n t r a s t i v e}} \\left(F _ {r}, F _ {q}\\right) + \\underset {f _ {q}} {\\min } \\mathcal {L} _ {\\mathrm {G N}} \\left(F _ {q}, f _ {q}\\right) \\right], \\tag {12}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            312,
+            787,
+            823,
+            825
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nf_{q}^{*} = \\operatorname *{arg  min}_{f_{q}}\\mathcal{L}_{\\mathrm{GN}}(F_{q}^{*},f_{q}).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            313,
+            824,
+            501,
+            849
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We denote",
+        "bbox": [
+            171,
+            862,
+            245,
+            876
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nG(F_{q};p):= \\operatorname *{arg  min}_{f_{q}}\\mathcal{L}_{\\mathrm{GN}}(F_{q},f_{q};p),\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            875,
+            563,
+            901
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} _ {\\mathrm {G N}} ^ {*} (F _ {q}; p) := \\min  _ {f _ {q}} \\mathcal {L} _ {\\mathrm {G N}} (F _ {q}, f _ {q}; p) = \\mathcal {L} _ {\\mathrm {G N}} (F _ {q}, G (F _ {q}; p); p). \\tag {15}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            300,
+            901,
+            823,
+            926
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "page_number",
+        "text": "5",
+        "bbox": [
+            493,
+            946,
+            504,
+            959
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/826e241123540c27976e7973fa798f729216c0e1cfe8f9f04f0420c3578f2cb0.jpg",
+        "image_caption": [
+            "Figure 2: Controlling the basin of convergence. Image samples and the corresponding feature maps  $\\tilde{f}_q$  (PCA) for different values of  $\\sigma$  and isotropic Gaussian distribution  $p$ ."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            98,
+            826,
+            335
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "$G(F_{q};p)$  is the optimal reconstruction of a feature map under sparse feature descriptors  $F_{q}$  and noise density  $p(\\epsilon)$ .  $\\mathcal{L}_{\\mathrm{GN}}^{*}(F_{q};p)$  is the corresponding value of the Gauss-Newton loss under  $F_{q}$ . We will refer to  $G(F_{q};p)$  as  $f_{q}$  to denote an optimal reconstruction under an arbitrary sparse input  $F_{q}$ . Note the difference to  $f_{q}^{*}$ , which is an optimal reconstruction under the joint loss, i.e.  $G(F_{q}^{*};p)$ . Our main contribution, detailed shortly, is a closed-form solution for both  $G(F_{q};p)$  and  $\\mathcal{L}_{\\mathrm{GN}}^{*}(F_{q};p)$ . It reveals that the original problem in Eq. (10) actually depends only on the feature values in the interest points:",
+        "bbox": [
+            169,
+            388,
+            826,
+            481
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nF _ {q} ^ {*} = \\underset {F _ {q}} {\\arg \\min } \\left[ \\mathcal {L} _ {\\text {c o n t r a s t i v e}} \\left(F _ {r}, F _ {q}\\right) + \\mathcal {L} _ {\\mathrm {G N}} ^ {*} \\left(F _ {q}; p\\right) \\right]. \\tag {14}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            330,
+            494,
+            825,
+            523
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "This means that in training a feature extractor only the features of interest points matter, while the representation of other pixels approximates our analytical solution. As the empirical validation, we demonstrate compelling results in our experiments by using feature descriptors for the interest points from a self-supervised method, while employing our analytical solution for the remaining pixels.",
+        "bbox": [
+            169,
+            531,
+            823,
+            589
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "An analytical solution to the Gauss-Newton loss. We propose to calculate the expectation of the Gauss-Newton loss in Eq. (9) as a functional of  $f_{q}$ , and analytically find its closed-form minimizer.",
+        "bbox": [
+            169,
+            594,
+            823,
+            625
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Consider the query image  $I_{q}$  as a set  $\\{(\\pmb{x}^{(j)}, F_{q}^{(j)})\\}$  containing locations of interest points  $\\pmb{x}^{(j)}$  and the corresponding feature descriptors  $F_{q}^{(j)} \\coloneqq f_{q}(\\pmb{x}^{(j)})$ . The locations  $\\pmb{x}^{(j)}$  can be extracted by an off-the-shelf feature detector (e.g. SuperPoint (DeTone et al., 2018)), while a network trained with the contrastive loss in Eq. (5) can produce the corresponding descriptors  $F_{q}^{(j)}$ .",
+        "bbox": [
+            169,
+            631,
+            823,
+            699
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Let us first re-write the expectation in Eq. (9) using Eq. (8) and Eq. (2) in the decoupled formulation of the Gauss-Newton loss (cf. Eq. (12)):",
+        "bbox": [
+            169,
+            704,
+            823,
+            733
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} _ {\\mathrm {G N}} \\left(F _ {q}, f _ {q}; p\\right) = \\sum_ {j} \\int_ {\\Omega} \\left\\| \\boldsymbol {\\epsilon} - \\left(\\boldsymbol {J} ^ {\\intercal} \\boldsymbol {J}\\right) ^ {- 1} \\boldsymbol {J} ^ {\\intercal} \\left(f _ {q} \\left(\\boldsymbol {x} ^ {(j)} + \\boldsymbol {\\epsilon}\\right) - F _ {q} ^ {(j)}\\right) \\right\\| _ {2} ^ {2} p (\\boldsymbol {\\epsilon}) d \\boldsymbol {\\epsilon}. \\tag {15}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            240,
+            750,
+            825,
+            789
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Substituting  $\\pmb {\\epsilon} = \\tilde{\\pmb{x}} -\\pmb{x}^{(j)}$  , we obtain",
+        "bbox": [
+            171,
+            801,
+            419,
+            816
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} _ {\\mathrm {G N}} \\left(F _ {q}, f _ {q}; p\\right) = \\sum_ {j} \\int_ {\\Omega} \\left\\| \\tilde {\\boldsymbol {x}} - \\left(\\boldsymbol {J} ^ {\\intercal} \\boldsymbol {J}\\right) ^ {- 1} \\boldsymbol {J} ^ {\\intercal} \\left(f _ {q} (\\tilde {\\boldsymbol {x}}) - F _ {q} ^ {(j)}\\right) - \\boldsymbol {x} ^ {(j)} \\right\\| _ {2} ^ {2} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right) d \\tilde {\\boldsymbol {x}}. \\tag {16}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            197,
+            840,
+            825,
+            880
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We first consider all  $\\tilde{\\pmb{x}}\\in \\Omega$  independently and relax the problem by eliminating the constraint  $J = \\frac{\\partial f_q}{\\partial\\tilde{\\pmb{x}}}$ . This allows us to derive analytical solutions for minimizers  $\\tilde{f}_q(\\cdot)$  and  $\\tilde{J}_q(\\cdot)$  as functions",
+        "bbox": [
+            169,
+            892,
+            825,
+            928
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "page_number",
+        "text": "6",
+        "bbox": [
+            493,
+            948,
+            504,
+            959
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/51567faa20beee434f4e3537a45a5498ca3a8c44185318e8178f84be77c1745e.jpg",
+        "image_caption": [
+            "Figure 3: Image alignment with continuous probabilistic feature pyramids. Given reference image with a 3D point cloud and a query image paired with an initial coarse pose, we start by estimating interest points and their features on the query image, along with features from the projections of the 3D point cloud onto the reference image. Next, we perform feature metric alignment between the 3D point features and derived analytical continuous image pyramid."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            171,
+            94,
+            830,
+            253
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "of  $\\tilde{\\pmb{x}}$ . We then observe that  $\\frac{\\partial}{\\partial\\tilde{\\pmb{x}}}\\tilde{f}_q(\\cdot)$  coincides with  $\\tilde{J}_q(\\cdot)$  for a uniform distribution, which confirms that in this case, our derived solution is the solution to the original problem. As a side note, in the context of direct image alignment, it has been suggested that decoupling and predicting the Jacobian independently from the function value may yield superior convergence and results (Han et al., 2018). Appendix B provides the full derivation. Below, we summarize the closed-form minimizer to Eq. (16):",
+        "bbox": [
+            169,
+            357,
+            826,
+            429
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\tilde {f} _ {q} (\\tilde {\\boldsymbol {x}}) = \\tilde {J} _ {q} (\\tilde {\\boldsymbol {x}}) \\left(\\tilde {\\boldsymbol {x}} - \\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}})\\right) + \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}}), \\tag {17}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            307,
+            454,
+            823,
+            472
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\tilde {J} _ {q} (\\tilde {\\boldsymbol {x}}) = \\left(\\operatorname {C o v} _ {\\mathbf {x y}} (\\tilde {\\boldsymbol {x}}) \\operatorname {C o v} _ {\\mathbf {y}} (\\tilde {\\boldsymbol {x}}) ^ {- 1}\\right) ^ {+}, \\tag {18}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            305,
+            474,
+            823,
+            496
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}}) = \\sum_ {j} F _ {q} ^ {(j)} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right) / \\sum_ {j} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right), \\tag {19}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            297,
+            498,
+            823,
+            531
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}}) = \\sum_ {j} \\boldsymbol {x} ^ {(j)} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right) / \\sum_ {j} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right), \\tag {20}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            297,
+            532,
+            823,
+            566
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\operatorname {C o v} _ {\\mathbf {x y}} (\\tilde {\\boldsymbol {x}}) = \\sum_ {j} \\left(\\boldsymbol {x} ^ {(j)} - \\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}})\\right) \\left(F _ {q} ^ {(j)} - \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}})\\right) ^ {\\intercal} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}), \\tag {21}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            274,
+            570,
+            825,
+            604
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\operatorname {C o v} _ {\\mathbf {y}} (\\tilde {\\boldsymbol {x}}) = \\sum_ {j} \\left(F _ {q} ^ {(j)} - \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}})\\right) \\left(F _ {q} ^ {(j)} - \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}})\\right) ^ {\\intercal} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}), \\tag {22}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            281,
+            607,
+            825,
+            643
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where  $(^{+})$  denotes the Moore-Penrose pseudo-inverse operator. The optimal point is  $G(F_{q};p)\\coloneqq \\tilde{f}_{q}$  and the loss value at this point (cf. Eq. (13)) is",
+        "bbox": [
+            169,
+            657,
+            823,
+            689
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} _ {G N} ^ {*} (F _ {q}) = \\int_ {\\Omega} \\frac {1}{2} \\operatorname {T r} \\left[ \\operatorname {C o v} _ {\\mathbf {x}} (\\tilde {\\boldsymbol {x}}) - \\operatorname {C o v} _ {\\mathbf {x y}} (\\tilde {\\boldsymbol {x}}) \\operatorname {C o v} _ {\\mathbf {y}} (\\tilde {\\boldsymbol {x}}) ^ {- 1} \\operatorname {C o v} _ {\\mathbf {y x}} (\\tilde {\\boldsymbol {x}}) \\right] d \\tilde {\\boldsymbol {x}}. \\tag {23}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            258,
+            705,
+            825,
+            746
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Our derivation of the optimal embeddings  $G(F_{q};p)$  using fixed interest points  $F_{q}$  provides an interesting insight into end-to-end learning frameworks. Jointly training both losses requires inverting a high-dimensional matrix  $\\mathrm{Cov}_{\\mathbf{y}}(\\tilde{\\boldsymbol{x}})$  in Eq. (23), which poses high numerical instability and discontinuities. Since previous work (von Stumberg et al., 2020a) can be seen as stochastic approximations to our solution, this may explain the reported training instability and divergence in those works. Additionally, the form of  $\\tilde{f}_q$  in Eq. (17) suggests that end-to-end pipelines may not necessarily yield any sophisticated representation, as they merely interpolate between features in the interest points.",
+        "bbox": [
+            169,
+            756,
+            826,
+            857
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "FeatureMetric Image Alignment. As a case study, we employ the analytical form of  $\\tilde{f}_q$  and  $\\tilde{J}_q$  for feature metric image alignment. Algorithm 1 and Fig. 3 provide an overview. Fig. 1 illustrates stages of direct alignment using our probabilistically reconstructed feature maps. We first extract the interest points  $\\{\\pmb{x}^{(j)}\\}$  and the corresponding feature descriptors  $\\{F_q^{(j)}\\}$  from  $I_{q}$  to reconstruct  $\\tilde{f}_q$  and",
+        "bbox": [
+            169,
+            864,
+            825,
+            926
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "page_number",
+        "text": "7",
+        "bbox": [
+            493,
+            948,
+            503,
+            959
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Require:  $I_r, I_q, \\{\\pmb{p}^{(i)}\\}, \\pmb{T}^{(0)}$",
+        "bbox": [
+            174,
+            99,
+            367,
+            117
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1:  $\\{\\pmb{x}^{(j)}\\} \\gets$  InterestPoints  $(I_q)$",
+            "2:  $\\{F_q^{(j)}\\} \\gets E(I_q;\\pmb {\\theta})\\circ \\{\\pmb{x}^{(j)}\\}$",
+            "3:  $\\{\\pmb{o}^{(i)}\\} \\gets E(I_r;\\pmb {\\theta})\\circ \\{\\langle \\pmb{p}^{(i)}\\rangle \\}$",
+            "4:  $\\pmb{T} \\gets \\pmb{T}^{(0)}$",
+            "5: for  $n$  from 0 to  $N_{\\mathrm{max}}$  do",
+            "6:  $p\\gets \\mathrm{Schedule}(n)$",
+            "7:  $\\tilde{f}_q\\gets G(\\{F_q^{(j)}\\} ;p)$",
+            "8:  $\\{\\tilde{\\pmb{p}}^{(i)}\\} \\gets \\{\\pmb {T}\\pmb{p}^{(i)}\\}$",
+            "9:  $\\pmb{\\delta} \\gets \\Delta_{GN}(\\tilde{f}_q \\circ \\{\\langle \\tilde{\\pmb{p}}^{(i)}\\rangle\\} - \\{\\pmb{o}^{(i)}\\})$",
+            "0:  $\\pmb{T}\\gets \\pmb{T}\\boxminus \\pmb{\\delta}$"
+        ],
+        "bbox": [
+            183,
+            118,
+            457,
+            272
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "11: end for",
+            "12: return T"
+        ],
+        "bbox": [
+            176,
+            273,
+            263,
+            300
+        ],
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+    {
+        "type": "text",
+        "text": "$\\langle \\cdot \\rangle$  denotes 2D projection.",
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+    {
+        "type": "text",
+        "text": "Algorithm 1: Image alignment.",
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+    {
+        "type": "image",
+        "img_path": "images/b1d8d6315ec75466d97ddac206aa02c2a3166ccf6d617791a310cf5e9a3c44d8.jpg",
+        "image_caption": [
+            "Figure 4: Robustness to initialization. Our closed-form solution exhibits significantly greater robustness compared to PixLoc."
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "$\\tilde{J}_q$  using Eqs. (17) and (18). The sparse set of 3D points  $\\{\\pmb{p}^{(i)}\\}$  is projected onto the reference image to obtain reference descriptors. We obtain feature descriptors for these points using the same feature embedding network  $E(\\cdot ;\\theta)$ . Starting with the initial camera pose  $\\pmb{T}^{(0)}\\in \\mathbf{S}\\mathbf{E}(3)$ , at every iteration we perform Gauss-Newton steps minimizing the residuals  $r^{(i)}$ , w.r.t. the camera transform  $\\pmb{T}$ :",
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+    {
+        "type": "equation",
+        "text": "\n$$\nr ^ {(i)} (\\boldsymbol {T}) := F _ {r} ^ {(i)} - \\tilde {f} _ {q} \\left(\\left\\langle \\boldsymbol {T} \\boldsymbol {p} ^ {(i)} \\right\\rangle\\right). \\tag {24}\n$$\n",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "$p(\\epsilon)$  explicitly controls the basin of convergence. In practice, we can use any distribution with high initial variance based on the inaccuracy assumed in the initial point projections. As our estimates improve over the course of optimization, we gradually decrease the variance. Notably, the choice of  $p(\\epsilon)$  lets us define values for points outside of the image boundaries. Fig. 2 illustrates examples of the feature maps  $\\tilde{f}_q$  with  $p$  following a Gaussian distribution of increasing variance.",
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+        "type": "text",
+        "text": "Comparison to previous work. Our derivation is agnostic to the underlying feature extractor and can operate with off-the-shelf feature descriptors. An interesting property of the derived  $\\tilde{f}_q$  is that it enables continuous coarse-to-fine alignment. By adjusting the noise prior  $p$  in the scheduler (cf. Algorithm 1), we can control the basin of convergence dynamically at runtime. We can adjust the distribution  $p$  either by choosing a different parameter set of a pre-defined distribution, or by changing the distribution itself to another family. In practice, we start the optimization with a uniform distribution and then switch to the Gaussian distribution with a slowly decreasing variance.",
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+    {
+        "type": "text",
+        "text": "A uniform distribution with a wide support offers a large basin of convergence, which sacrifices accuracy for robustness. The ensuing Gaussian distribution with decreasing variance refines the pose and leads to a more accurate solution. Appendix E provides implementation details of the scheduler.",
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+        "type": "text",
+        "text": "Connection between feature matching and feature metric image alignment. Note that the computation of  $\\tilde{f}_q$  using Eq. (17) only depends on the feature descriptors of the interest points in the query image. This observation suggests an interesting interpretation of minimizing the residual in Eq. (24) using  $\\tilde{f}_q$  with Gauss-Newton optimization as feature matching. The Gauss-Newton step is fully determined by the neighboring points of interest. Consequently, it implies that the effectiveness of such optimization-based methods may be limited in comparison to alternative optimization-inspired approaches, which learn the optimization step (Teed and Deng, 2020).",
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+    {
+        "type": "text",
+        "text": "6 EXPERIMENTS",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Datasets. We evaluate our approach on two most popular datasets for large-scale image localization, namely the Aachen Day-Night dataset (Sattler et al., 2018), extended CMU seasons (Toft et al., 2022) and 7Scenes dataset (Shotton et al., 2013). Aachen Day-Night consists of 98 night and 824 day query",
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+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+        "type": "page_number",
+        "text": "8",
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+    {
+        "type": "table",
+        "img_path": "images/0f4aacf05e7b2a97238b22a7dfee74f7d6e9f9cd36c88ac4f352a30fb7ec57b2.jpg",
+        "table_caption": [
+            "Table 1: Camera localization on Aachen Day-Night and CMU Seasons. Our evaluation shows an overall improved accuracy in diverse scenarios despite using self-supervised SuperPoint descriptors. We compare to the state-of-the-art methods supervised with pose: GN-Net (von Stumberg et al., 2020a), LM-Reloc (von Stumberg et al., 2020b) and PixLoc (Sarlin et al., 2021)."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">Aachen</td><td colspan=\"3\">CMU Seasons</td></tr><tr><td>Day</td><td>Night</td><td>Urban</td><td>Suburban</td><td>Park</td></tr><tr><td>GN-Net</td><td>62.4/69.4/76.9</td><td>49.0/58.2/66.3</td><td>75.4/79.9/92.6</td><td>64.7/67.0/81.4</td><td>46.7/48.3/65.3</td></tr><tr><td>LM-Reloc</td><td>60.4/68.0/76.3</td><td>37.8/46.9/59.2</td><td>76.6/82.5/93.4</td><td>67.3/72.0/82.8</td><td>49.1/53.4/66.9</td></tr><tr><td>PixLoc</td><td>64.3/69.3/77.4</td><td>51.0/55.1/67.3</td><td>88.3/90.4/93.7</td><td>79.6/81.1/85.2</td><td>61.0/62.5/69.4</td></tr><tr><td>Ours (SuperPoint)</td><td>66.3/72.5/78.8</td><td>43.9/50.0/56.1</td><td>86.0/90.6/95.2</td><td>79.8/85.0/92.4</td><td>63.4/67.9/77.5</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "images. Extended CMU seasons consists of 14 slices, 5 for urban environment, 5 for suburban and 4 for park. Each slice contains between 3000 and 5000 query images. The 7Scenes dataset comprises seven distinct scenes, each containing multiple sequences of 500 to 1000 frames. For our ablation study in Appendix D we used Cambridge Landmarks dataset (Kendall et al., 2016).",
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+    {
+        "type": "text",
+        "text": "Results. Table 1 presents the results for camera localization on Aachen Day-Night (Sattler et al., 2018) and the extended CMU Seasons (Toft et al., 2022). We use SuperPoint (DeTone et al., 2018) as the feature descriptor in these experiments. In each setting (Day/Night for Aachen; Urban/Suburban/Park for CMU Seasons), we report three numbers, which indicate the percentage of the query images that were successfully localized within the specified translation and rotation thresholds. We adopt the standard threshold values defined by the benchmarks (translation, rotation):  $(0.25\\mathrm{m}, 2^{\\circ}) / (0.5\\mathrm{m}, 5^{\\circ}) / (5\\mathrm{m}, 10^{\\circ})$ . We adopted the implementation from PixLoc (Sarlin et al., 2021) to run these benchmarks. A comparison with PixLoc on the 7Scenes dataset is presented in Appendix F.",
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+    {
+        "type": "text",
+        "text": "We observe that our solution, despite using self-supervised descriptors from SuperPoint (DeTone et al., 2018), achieves an overall strong localization accuracy across diverse settings in comparison to supervised frameworks. Aachen Day-Night has significant occlusions, hence many outliers. Since the Gauss-Newton loss does not incorporate any outlier filtering, we do not expect high accuracy for our approach on this dataset. Nevertheless, we were surprised to find that on Aachen Day our approach even slightly surpassed previous state of the art. On Aachen Night, our accuracy is inferior to previous work, however. This is somewhat expected, since SuperPoint feature descriptors were not directly trained on day-night correspondences. By contrast, PixLoc was trained with supervision on day-night image pairs, which provides an obvious advantage. On CMU Seasons, our approach demonstrates a clear improvement over previous feature metric alignment methods.",
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+    {
+        "type": "text",
+        "text": "The proposed approach demonstrates notable robustness to initialization noise. We compared the proposed scheme and PixLoc on the Cambridge Landmarks dataset, varying the levels of random noise applied to the ground-truth pose as illustrated in Fig. 4 and Appendix G.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Overall, these results empirically confirm the validity of our closed-form derivation. Furthermore, the comparison suggests a limited benefit of end-to-end learning with the Gauss-Newton loss. By contrast, a dynamic convergence basin offered by our analytical solution provides versatility w.r.t. the underlying feature descriptors and improves robustness to pose initialization, as a result.",
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+    {
+        "type": "text",
+        "text": "7 CONCLUSION",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We derived a closed-form solution to the Gauss-Newton loss in the context of direct image alignment, which offers two main advantages. First, it allows for dynamic control of the convergence basin, which improves robustness of the alignment to pose initialization. Furthermore, despite using self-supervised descriptors, such control leads to compelling accuracy of pose estimates in comparison to supervised pipelines on established benchmarks. Second, our derivation exposes intrinsic limitations of employing the Gauss-Newton loss in deep learning, as it only leads to a form of interpolation between the feature descriptors in the interest points. This insight offers an interesting connection between direct image alignment and feature matching, and leads to a novel perspective on learning robust features end-to-end, which we will investigate in future work.",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+    },
+    {
+        "type": "page_footnote",
+        "text": "1https://github.com/cvg/pixloc",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "page_number",
+        "text": "9",
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+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "This work was supported by the ERC Advanced Grant SIMULACRON and by TUM Georg Nemetschek Institute under the project AI4TWINNING. We thank Linus Harenstam-Nielsen for helpful discussions and proofreading.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "REPRODUCIBILITY STATEMENT",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "The main contribution of this work is the derivation of the closed-form solution to the Gauss-Newton loss. We provide details of this derivation in Appendix B. To reproduce our experimental results, we elaborate on the implementation details Appendix E. To facilitate reproducibility in future research, we also publicly release our code.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
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+        "text": "A OPTICAL FLOW BIAS",
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+    {
+        "type": "text",
+        "text": "We demonstrate that PixLoc exhibits bias inherited from the training dataset. Fig. 5 illustrates the average translation error after optimization when evaluated on the Cambridge dataset, highlighting the impact of different initial displacements along the image axes. To solely evaluate the bias learned by the feature maps, we disabled dataset-specific learned movement priors denoted as  $\\lambda$  in PixLoc (Sarlin et al., 2021). Our results demonstrate that when trained on the CMU dataset, the network predicts optical flow along the X axis more accurately. This increased accuracy is directly linked to the structure of the CMU dataset. Specifically, the dataset comprises image pairs captured by a camera inside a moving car, pointed towards the side of the road. Consequently, this positioning results in the network predominantly learning horizontal optical flow patterns, reflecting the lateral movement observed in the images of the training dataset.",
+        "bbox": [
+            171,
+            321,
+            486,
+            601
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/5a280a00dfe2fc2939830a6e761e02fd0b13169e9961761b13fb7f9bb0a85a77.jpg",
+        "image_caption": [
+            "Figure 5: The final error of the PixLoc solution along  $X / Y$ -axes w.r.t. the initial error. PixLoc remains accurate only along  $X$ -axis on CMU, reflecting the dominant motion in the training set."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            519,
+            338,
+            800,
+            523
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B DERIVATION OF THE ANALYTICAL SOLUTION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            619,
+            584,
+            636
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In this section, we provide a solution to minimizing Eq. (16) from the main text. Let  $\\Omega$  be the coordinate domain of an image plane  $[0,1] \\times [0,1]$ , and let  $E \\subseteq \\mathbb{R}^d$  define the feature space. We aim to find  $\\tilde{f}: \\Omega \\to E$ :",
+        "bbox": [
+            169,
+            651,
+            823,
+            696
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\tilde {f} = \\arg \\min  _ {f} \\sum_ {j} \\int_ {\\Omega} \\left\\| \\tilde {\\boldsymbol {x}} - (J (\\tilde {\\boldsymbol {x}}) ^ {\\intercal} J (\\tilde {\\boldsymbol {x}})) ^ {- 1} J (\\tilde {\\boldsymbol {x}}) ^ {\\intercal} (f (\\tilde {\\boldsymbol {x}}) - F _ {q} ^ {(j)}) - \\boldsymbol {x} ^ {(j)} \\right\\| _ {2} ^ {2} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}) d \\tilde {\\boldsymbol {x}}. \\tag {25}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            718,
+            825,
+            760
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We start by relaxing the constraint  $\\nabla f(\\pmb{x}) = J(\\pmb{x})$ . This makes every value of  $f(\\pmb{x})$  and  $J(\\pmb{x})$  locally independent and, therefore, the minimum is achieved by minimizing Eq. (25) independently over every point in  $\\Omega$ :",
+        "bbox": [
+            169,
+            771,
+            823,
+            815
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\{\\tilde {f} (\\tilde {\\boldsymbol {x}}), \\tilde {J} (\\tilde {\\boldsymbol {x}}) \\} = \\underset {\\hat {\\boldsymbol {f}}, \\hat {\\boldsymbol {J}}} {\\arg \\min } \\sum_ {j} \\left\\| \\tilde {\\boldsymbol {x}} - \\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal} \\left(\\hat {\\boldsymbol {f}} - F _ {q} ^ {(j)}\\right) - \\boldsymbol {x} ^ {(j)} \\right\\| _ {2} ^ {2} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}). \\tag {26}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            202,
+            835,
+            825,
+            878
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Note that the only significant part of  $\\hat{\\pmb{f}}$  is a part which is spanned by columns of  $\\hat{\\pmb{J}}$ . This can be easily seen by observing that  $\\hat{\\pmb{J}}^{\\mathrm{T}}\\hat{\\pmb{f}}$  is a non-normalized projection onto  $\\hat{\\pmb{J}}$ . More formally, let us find  $\\hat{\\pmb{f}}$  as",
+        "bbox": [
+            169,
+            892,
+            825,
+            926
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "page_number",
+        "text": "12",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "$\\hat{J}a + \\hat{J}^\\perp b$ , where  $\\hat{J}^\\perp$  is a basis for an orthogonal complement for  $\\operatorname{span}(\\hat{J})$ . By substituting this into Eq. (26), we obtain:",
+        "bbox": [
+            169,
+            101,
+            823,
+            133
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\{\\tilde {a} (\\tilde {\\boldsymbol {x}}), \\tilde {J} (\\tilde {\\boldsymbol {x}}) \\} = \\underset {\\boldsymbol {a}, \\hat {\\boldsymbol {J}}} {\\arg \\min } \\sum_ {j} \\left\\| \\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)} - \\boldsymbol {a} + \\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal} F _ {q} ^ {(j)} \\right\\| _ {2} ^ {2} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}), \\tag {27}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            232,
+            150,
+            825,
+            190
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\tilde {f} (\\tilde {\\boldsymbol {x}}) \\in \\left\\{\\tilde {J} (\\tilde {\\boldsymbol {x}}) \\tilde {a} (\\tilde {\\boldsymbol {x}}) + \\tilde {J} (\\tilde {\\boldsymbol {x}}) ^ {\\perp} \\boldsymbol {b} \\mid \\boldsymbol {b} \\in \\mathbb {R} ^ {d - 2} \\right\\}.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            287,
+            193,
+            580,
+            213
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Observe that this problem is a linear least squares in  $\\pmb{a}$  and  $\\left(\\hat{\\boldsymbol{J}}^{\\intercal}\\hat{\\boldsymbol{J}}\\right)^{-1}\\hat{\\boldsymbol{J}}^{\\intercal}$  and recall that  $p(\\cdot)\\geq 0$  (by definition), hence the problem is convex. Minimality conditions for Eq. (27) for  $\\pmb{a}$  and  $\\left(\\hat{\\boldsymbol{J}}^{\\intercal}\\hat{\\boldsymbol{J}}\\right)^{-1}\\hat{\\boldsymbol{J}}^{\\intercal}$  are:",
+        "bbox": [
+            169,
+            224,
+            823,
+            286
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\frac {d}{d \\boldsymbol {a}}: \\quad \\sum_ {j} \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)} - \\boldsymbol {a} + \\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal} F _ {q} ^ {(j)}\\right) p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}) = 0, \\tag {28}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            285,
+            313,
+            825,
+            349
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\frac {d}{d \\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal}}: \\quad \\sum_ {j} \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)} - \\boldsymbol {a} + \\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal} F _ {q} ^ {(j)}\\right) F _ {q} ^ {(j) \\intercal} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right) = 0. \\tag {29}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            200,
+            354,
+            825,
+            398
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "From Eq. (28), we have:",
+        "bbox": [
+            171,
+            412,
+            334,
+            428
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {a} = \\tilde {\\boldsymbol {x}} - \\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}}) + \\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal} \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}}), \\tag {30}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            362,
+            445,
+            825,
+            472
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "where",
+        "bbox": [
+            171,
+            484,
+            217,
+            496
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}}) := \\sum_ {j} F _ {q} ^ {(j)} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right) / \\sum_ {j} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right), \\tag {31}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            333,
+            494,
+            825,
+            534
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}}) := \\sum_ {j} \\boldsymbol {x} ^ {(j)} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}) / \\sum_ {j} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            333,
+            532,
+            660,
+            561
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Substituting  $\\pmb{a}$  into Eq. (29), we obtain",
+        "bbox": [
+            171,
+            574,
+            428,
+            589
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\sum_ {j} \\left(\\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}}) - \\boldsymbol {x} ^ {(j)} + \\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal} \\left(F _ {q} ^ {(j)} - \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}})\\right)\\right) F _ {q} ^ {(j) \\intercal} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right) = 0, \\tag {32}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            215,
+            616,
+            825,
+            652
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n- \\sum_ {j} \\left(\\boldsymbol {x} ^ {(j)} - \\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}})\\right) F _ {q} ^ {(j) \\intercal} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}) +\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            421,
+            656,
+            707,
+            690
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal} \\sum_ {i} \\left(F _ {q} ^ {(j)} - \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}})\\right) F _ {q} ^ {(j) \\intercal} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right) = 0, \\tag {33}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            354,
+            694,
+            825,
+            728
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n- \\operatorname {C o v} _ {\\mathbf {x y}} (\\tilde {\\boldsymbol {x}}) + \\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal} \\operatorname {C o v} _ {\\mathbf {y}} (\\tilde {\\boldsymbol {x}}) = 0, \\tag {34}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            454,
+            732,
+            825,
+            758
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "where",
+        "bbox": [
+            171,
+            766,
+            217,
+            777
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\operatorname {C o v} _ {\\mathbf {x y}} (\\tilde {\\boldsymbol {x}}) := \\sum_ {j} \\left(\\boldsymbol {x} ^ {(j)} - \\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}})\\right) \\left(F _ {q} ^ {(j)} - \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}})\\right) ^ {\\intercal} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}), \\tag {35}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            274,
+            785,
+            825,
+            820
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\operatorname {C o v} _ {\\mathbf {y}} (\\tilde {\\boldsymbol {x}}) := \\sum_ {j} \\left(F _ {q} ^ {(j)} - \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}})\\right) \\left(F _ {q} ^ {(j)} - \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}})\\right) ^ {\\intercal} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}). \\tag {36}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            281,
+            824,
+            825,
+            858
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Solving Eq. (34) for  $J$  and substituting the solution into Eq. (27) results in",
+        "bbox": [
+            171,
+            864,
+            661,
+            880
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\tilde {J} (\\tilde {\\boldsymbol {x}}) = \\left(\\operatorname {C o v} _ {\\mathbf {x y}} (\\tilde {\\boldsymbol {x}}) \\operatorname {C o v} _ {\\mathbf {y}} (\\tilde {\\boldsymbol {x}}) ^ {- 1}\\right) ^ {\\dagger}, \\tag {37}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            241,
+            905,
+            825,
+            926
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "page_number",
+        "text": "13",
+        "bbox": [
+            488,
+            946,
+            506,
+            959
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/d412210a2a978a178e742f9fe1f64233bc326303b34641ab01d95fed24968989.jpg",
+        "image_caption": [
+            "Figure 6: Comparison of rotation and translation recalls for our simplified version of GN-loss and probabilistic one"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            99,
+            472,
+            287
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/1c7bf869c947ac562c24b1af137e85b4fbb4042c99ddf07d99863b671a717ddf.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            531,
+            101,
+            826,
+            289
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\tilde {f} (\\tilde {\\boldsymbol {x}}) = \\tilde {J} (\\tilde {\\boldsymbol {x}}) \\boldsymbol {a} = \\tilde {J} (\\tilde {\\boldsymbol {x}}) \\left(\\tilde {\\boldsymbol {x}} - \\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}})\\right) + \\tilde {J} (\\tilde {\\boldsymbol {x}}) \\left(\\tilde {J} (\\tilde {\\boldsymbol {x}}) ^ {\\intercal} \\tilde {J} (\\tilde {\\boldsymbol {x}})\\right) ^ {- 1} \\tilde {J} (\\tilde {\\boldsymbol {x}}) ^ {\\intercal} \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}}). \\tag {38}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            238,
+            367,
+            825,
+            396
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "$\\tilde{J} (\\tilde{\\pmb{x}})\\left(\\tilde{J} (\\tilde{\\pmb{x}})^{\\intercal}\\tilde{J} (\\tilde{\\pmb{x}})\\right)^{-1}\\tilde{J} (\\tilde{\\pmb{x}})^{\\intercal}{\\bf y}_m(\\tilde{\\pmb{x}})$  can be simplified. Note that in the Gauss-Newton step we are projecting  $\\tilde{f} (\\tilde{\\pmb{x}})$  onto  $\\tilde{J} (\\tilde{\\pmb{x}})$ .  $\\tilde{J} (\\tilde{\\pmb{x}})\\left(\\tilde{J} (\\tilde{\\pmb{x}})^{\\intercal}\\tilde{J} (\\tilde{\\pmb{x}})\\right)^{-1}\\tilde{J} (\\tilde{\\pmb{x}})^{\\intercal}$  is a projection operator, so we are projecting two times. Therefore, there is another equivalent solution which we will further use for the sake of simplicity:",
+        "bbox": [
+            169,
+            412,
+            823,
+            492
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\tilde {f} (\\tilde {\\boldsymbol {x}}) = \\tilde {J} (\\tilde {\\boldsymbol {x}}) \\left(\\tilde {\\boldsymbol {x}} - \\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}})\\right) + \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}}). \\tag {39}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            369,
+            508,
+            825,
+            527
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Observe that substituting  $\\tilde{f}$  from Eq. (39) and Eq. (38) into Eq. (26) yields the same loss value.",
+        "bbox": [
+            169,
+            540,
+            794,
+            556
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Notably,  $\\nabla \\tilde{f} (\\tilde{\\pmb{x}}) = \\tilde{J} (\\tilde{\\pmb{x}})$  for a uniform distribution  $p(\\cdot)$ .",
+        "bbox": [
+            169,
+            561,
+            544,
+            579
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C PROBABILISTIC vs. SIMPLIFIED GAUSS-NEWTON LOSS",
+        "text_level": 1,
+        "bbox": [
+            169,
+            599,
+            660,
+            614
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In this section, we compare our simplified Gauss-Newton loss formulation,",
+        "bbox": [
+            169,
+            630,
+            665,
+            646
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\underset {\\hat {\\boldsymbol {f}}, \\hat {\\boldsymbol {J}}} {\\arg \\min } \\sum_ {j} \\left\\| \\tilde {\\boldsymbol {x}} - \\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal} \\left(\\hat {\\boldsymbol {f}} - F _ {q} ^ {(j)}\\right) - \\boldsymbol {x} ^ {(j)} \\right\\| _ {2} ^ {2} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}), \\tag {40}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            269,
+            661,
+            825,
+            702
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "to the original loss, which was derived through the maximum likelihood estimation (MLE),",
+        "bbox": [
+            169,
+            714,
+            769,
+            729
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\underset {\\hat {\\boldsymbol {f}}, \\hat {\\boldsymbol {J}}} {\\arg \\min } \\sum_ {j} \\left(\\left\\| \\hat {\\boldsymbol {J}} \\left(\\tilde {\\boldsymbol {x}} - \\left(\\hat {\\boldsymbol {J}} ^ {\\mathrm {T}} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\mathrm {T}} \\left(\\hat {\\boldsymbol {f}} - F _ {q} ^ {(j)}\\right) - \\boldsymbol {x} ^ {(j)}\\right) \\right\\| _ {2} ^ {2} - \\log \\det  (\\hat {\\boldsymbol {J}} ^ {\\mathrm {T}} \\hat {\\boldsymbol {J}})\\right) p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right). \\tag {41}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            184,
+            753,
+            825,
+            810
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We note that the solution to the probabilistic loss is much more complicated and computationally expensive. Derivatives for this formulation are high-order polynomials in elements of  $\\hat{\\pmb{f}}$  and  $\\hat{\\pmb{J}}$ . It could be solved by general methods of algebraic geometry like Gröbner basis or Homotopy continuation, but the existence of a closed-form solution is not guaranteed, and we were not able to find one.",
+        "bbox": [
+            169,
+            816,
+            826,
+            888
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Nevertheless, in order to compare the two formulations, we approached the problem numerically. Fig. 6 presents the results. We used the Aachen dataset with hloc poses to plot the recall of query",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "page_number",
+        "text": "14",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "images as a function of rotation and translation errors. We observe that the accuracy difference between these two formulations is negligible in practice. However, the simplified version admits a closed-form solution and can be computed several orders of magnitudes more efficiently.",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "D A STUDY OF FEATURE DESCRIPTORS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            178,
+            514,
+            193
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We show that our approach is agnostic to the choice of the underlying feature descriptor. To analyze the accuracy of our algorithm in terms of the localization error, we use Cambridge Landmarks (Kendall et al., 2016), which provides ground-truth poses.",
+        "bbox": [
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+            215,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We experiment with four popular feature descriptors: SIFT (Lowe, 2004), SOLD2 (Pau-trat et al., 2021), SuperPoint (DeTone et al., 2018) and the recent SiLK (Gleize et al., 2023). We plug them in as the embeddings for interest points detected by SuperPoint. Fig. 7 plots recall (the percentage of successfully localized queries) as a function of the tolerated translation error.",
+        "bbox": [
+            169,
+            257,
+            486,
+            381
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We observe that SuperPoint outperforms all other feature descriptors. SOLD2 (Pautrat et al., 2021) is substantially more inferior in terms of accuracy of the pose estimates. Since it is designed to be a line descriptor, its representation of non-linear structures is not well-defined, thus such outcome is somewhat expected. Interestingly, SuperPoint and SIFT both surpass the more recent SiLK on this benchmark. In comparison to SuperPoint, SiLK has 128-D descrip",
+        "bbox": [
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+            388,
+            488,
+            527
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "tor, which suggests that it may be less expressive than the 256-  $D$  SuperPoint descriptors.",
+        "bbox": [
+            169,
+            527,
+            754,
+            542
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/b8e0fb26806c5bac90e295b1be48088253a34441dee7c04d809eabe9864177d8.jpg",
+        "image_caption": [
+            "Figure 7: We study our approach with different feature descriptors on Cambridge Landmarks."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            509,
+            272,
+            805,
+            460
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "E IMPLEMENTATION DETAILS.",
+        "text_level": 1,
+        "bbox": [
+            171,
+            573,
+            441,
+            588
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We implement our approach in PyTorch (Paszke et al., 2019). The primary focus of our implementation is the experimental validation of our derived results. Therefore, we did not optimize the runtime performance, which heavily depends on the number of interest points and 3D points involved in the computation. On average, the code takes approximately 6 seconds and 10 seconds per alignment on CMU and Aachen, respectively, on a single NVIDIA A4000. The scheduler in Algorithm 1 adapts the distribution  $p(\\cdot)$  as follows. We initialize  $p(\\cdot)$  with a truncated uniform distribution of a fixed radius around all interest points. The radius decreases from  $50\\%$  of the image diagonal to  $5\\%$  in the first 30 iterations. Afterward, the scheduler switches to the normal distribution around each interest point with standard deviation  $\\sigma$ . Initially, we define  $\\sigma$  such that  $99\\%$  of the distribution covers  $10\\%$  of the image around the point, and we decrease the coverage ratio to  $1\\%$ .",
+        "bbox": [
+            169,
+            609,
+            826,
+            750
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Since our formulation does not have any outlier removal, we adopt the cut-off from the coarse tracker of DSO (Engel et al., 2016). The idea is to define a threshold that discards all but  $20\\%$  of the residuals with the lowest norm. In our experiments with SuperPoint, we set the threshold value to 0.4.",
+        "bbox": [
+            169,
+            756,
+            826,
+            799
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "F EVALUATION ON 7SCENES",
+        "text_level": 1,
+        "bbox": [
+            171,
+            829,
+            428,
+            845
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Here, we complement our evaluation with the 7Scenes dataset. This dataset features substantial blur and distortions in some scenes, as it was captured with a rolling-shutter Kinect camera. Although SuperPoint features were not trained with such distortions, direct alignment with our closed-form solution remains competitive with PixLoc, both in terms of the median error and the recall Table 2.",
+        "bbox": [
+            169,
+            867,
+            825,
+            925
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "page_number",
+        "text": "15",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/973ec986b13797e8bae3c050e4748269b52bc99682fe8b2cfeb01f0c2c2b623f.jpg",
+        "table_caption": [
+            "Table 2: 7Scenes (Shotton et al., 2013) evaluation and comparison to PixLoc. We report the median rotation and translation errors, as well as recall values at the specified thresholds of translation and rotation."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Scene</td><td rowspan=\"2\">Method</td><td rowspan=\"2\">Median error</td><td colspan=\"4\">Recall</td></tr><tr><td>(1cm,1°)</td><td>(5cm,5°)</td><td>(25cm,2°)</td><td>(50cm,5°)</td></tr><tr><td rowspan=\"2\">Heads</td><td>Ours</td><td>0.013m, 1.006°</td><td>24.80%</td><td>93.0%</td><td>86.30%</td><td>92.0%</td></tr><tr><td>Pixloc</td><td>0.013m, 0.863°</td><td>36.40%</td><td>85.60 %</td><td>84.00%</td><td>85.90%</td></tr><tr><td rowspan=\"2\">Office</td><td>Ours</td><td>0.028m, 0.941°</td><td>6.12 %</td><td>80.25%</td><td>92.55%</td><td>99.00%</td></tr><tr><td>Pixloc</td><td>0.026m, 0.792°</td><td>7.95%</td><td>80.70 %</td><td>93.12 %</td><td>96.85%</td></tr><tr><td rowspan=\"2\">Redkitchen</td><td>Ours</td><td>0.037m, 1.444°</td><td>1.56 %</td><td>64.42%</td><td>71.88%</td><td>90.16%</td></tr><tr><td>Pixloc</td><td>0.034m, 1.217°</td><td>3.74%</td><td>67.78 %</td><td>76.56 %</td><td>89.48%</td></tr><tr><td rowspan=\"2\">Pumpkin</td><td>Ours</td><td>0.049m, 1.555°</td><td>1.60 %</td><td>51.65%</td><td>62.65%</td><td>85.35%</td></tr><tr><td>Pixloc</td><td>0.041m, 1.173°</td><td>2.80%</td><td>59.75 %</td><td>71.00 %</td><td>84.40%</td></tr><tr><td rowspan=\"2\">Stairs</td><td>Ours</td><td>0.154m, 3.685°</td><td>1.90 %</td><td>19.90 %</td><td>25.60 %</td><td>60.50 %</td></tr><tr><td>Pixloc</td><td>0.048m, 1.268°</td><td>2.60%</td><td>51.10 %</td><td>59.10 %</td><td>74.70%</td></tr><tr><td rowspan=\"2\">Chess</td><td>Ours</td><td>0.026m, 0.904°</td><td>5.50 %</td><td>91.95%</td><td>95.45%</td><td>98.45%</td></tr><tr><td>Pixloc</td><td>0.024m, 0.812°</td><td>7.75%</td><td>90.75 %</td><td>94.95%</td><td>96.15%</td></tr><tr><td rowspan=\"2\">Fire</td><td>Ours</td><td>0.021m, 0.978°</td><td>10.40 %</td><td>90.65%</td><td>94.05%</td><td>98.45%</td></tr><tr><td>Pixloc</td><td>0.019m, 0.781°</td><td>15.85%</td><td>87.50 %</td><td>87.20%</td><td>90.10%</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "G QUALITATIVE EXAMPLES",
+        "text_level": 1,
+        "bbox": [
+            171,
+            420,
+            421,
+            436
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Fig. 8 visualizes convergence examples of one scene with different pose initializations. The examples consistently demonstrate successful alignment despite suboptimal initialization of varying degree.",
+        "bbox": [
+            169,
+            450,
+            823,
+            483
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            173,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "page_number",
+        "text": "16",
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/c980c871e37d2586de9c72c848c094ef39349877409313a22f28b077fa752e49.jpg",
+        "image_caption": [
+            "Figure 8: Examples of convergence from random initial poses. The green points are projections of 3D points using the current pose estimate; the red points are projections of the 3D points with the initial pose, and the yellow points denote the locations of interest points. It can be seen that despite these highly inaccurate initial poses, our approach converges to the correct solution."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            99,
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+        "page_idx": 16
+    },
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diff --git a/2024/An Analytical Solution to Gauss-Newton Loss for Direct Image Alignment/1a6e78f9-a669-4d03-b852-34ab11b21b42_model.json b/2024/An Analytical Solution to Gauss-Newton Loss for Direct Image Alignment/1a6e78f9-a669-4d03-b852-34ab11b21b42_model.json
new file mode 100644
index 0000000000000000000000000000000000000000..babeec62b8692c98a88674b99bb4e59561c6d366
--- /dev/null
+++ b/2024/An Analytical Solution to Gauss-Newton Loss for Direct Image Alignment/1a6e78f9-a669-4d03-b852-34ab11b21b42_model.json	
@@ -0,0 +1,3215 @@
+[
+    [
+        {
+            "type": "header",
+            "bbox": [
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+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.171,
+                0.1,
+                0.825,
+                0.147
+            ],
+            "angle": 0,
+            "content": "AN ANALYTICAL SOLUTION TO GAUSS-NEWTON LOSS FOR DIRECT IMAGE ALIGNMENT"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.18,
+                0.169,
+                0.47,
+                0.185
+            ],
+            "angle": 0,
+            "content": "Sergei Solonets\\(^{1,2,*}\\) Daniil Sinitsyn\\(^{1,2,*}\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.182,
+                0.185,
+                0.653,
+                0.2
+            ],
+            "angle": 0,
+            "content": "Lukas von Stumberg\\(^{3,\\dagger}\\) Nikita Araslanov\\(^{1,2}\\) Daniel Cremers\\(^{1,2}\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.184,
+                0.205,
+                0.406,
+                0.221
+            ],
+            "angle": 0,
+            "content": "<sup>1</sup> Technical University of Munich"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.183,
+                0.221,
+                0.443,
+                0.235
+            ],
+            "angle": 0,
+            "content": "2 Munich Center for Machine Learning"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.183,
+                0.235,
+                0.297,
+                0.248
+            ],
+            "angle": 0,
+            "content": "3 Valve Software"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.183,
+                0.205,
+                0.443,
+                0.248
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.174,
+                0.262,
+                0.386,
+                0.372
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image",
+            "bbox": [
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+                0.262,
+                0.605,
+                0.372
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image",
+            "bbox": [
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+                0.262,
+                0.825,
+                0.372
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.171,
+                0.382,
+                0.828,
+                0.482
+            ],
+            "angle": 0,
+            "content": "Figure 1: Direct image alignment is a technique for aligning scenes based on image intensities. Recent learning-based methods seek to improve its success rate by increasing the convergence basin. We derive an analytical solution to the core idea of such methods, the Gauss-Newton loss, enabling fine-grained control over the basin of convergence. As a result, we can successfully align two scenes despite a highly imprecise initialization. From left to right, the example above illustrates the convergence of the reprojected keypoints (red) to the ground truth (green) by optimizing the SE(3) camera pose with our analytical solution. The blue color is the re-projection in the first iteration."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.451,
+                0.496,
+                0.547,
+                0.511
+            ],
+            "angle": 0,
+            "content": "ABSTRACT"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.23,
+                0.526,
+                0.77,
+                0.765
+            ],
+            "angle": 0,
+            "content": "Direct image alignment is a widely used technique for relative 6DoF pose estimation between two images, but its accuracy strongly depends on pose initialization. Therefore, recent end-to-end frameworks increase the convergence basin of the learned feature descriptors with special training objectives, such as the Gauss-Newton loss. However, the training data may exhibit bias toward a specific type of motion and pose initialization, thus limiting the generalization of these methods. In this work, we derive a closed-form solution to the expected optimum of the Gauss-Newton loss. The solution is agnostic to the underlying feature representation and allows us to dynamically adjust the basin of convergence according to our assumptions about the uncertainty in the current estimates. These properties allow for effective control over the convergence in the alignment process. Despite using self-supervised feature embeddings, our solution achieves compelling accuracy w.r.t. the state-of-the-art direct image alignment methods trained end-to-end with pose supervision, and demonstrates improved robustness to pose initialization. Our analytical solution exposes some inherent limitations of end-to-end learning with the Gauss-Newton loss, and establishes an intriguing connection between direct image alignment and feature-matching approaches."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.174,
+                0.788,
+                0.338,
+                0.803
+            ],
+            "angle": 0,
+            "content": "1 INTRODUCTION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.819,
+                0.828,
+                0.877
+            ],
+            "angle": 0,
+            "content": "Visual localization refers to estimating the camera pose of a query image w.r.t. a reference image where the underlying 3D structure (e.g. a point cloud) is available. Traditionally, solutions to visual localization primarily relied on estimating correspondences between 2D features in the query image and 3D features in the reference point cloud (Liu et al., 2017; Sarlin et al., 2019; Sattler et al., 2017;"
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.191,
+                0.884,
+                0.696,
+                0.899
+            ],
+            "angle": 0,
+            "content": "*Equal contribution. Correspondence: {s.solonets,d.sinitsyn}@tum.de."
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.195,
+                0.899,
+                0.395,
+                0.911
+            ],
+            "angle": 0,
+            "content": "†Work done while at TU Munich."
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.199,
+                0.911,
+                0.576,
+                0.924
+            ],
+            "angle": 0,
+            "content": "Project code: https://github.com/tum-vision/gn_loss_analytical."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.191,
+                0.884,
+                0.696,
+                0.924
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.504,
+                0.96
+            ],
+            "angle": 0,
+            "content": "1"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.104,
+                0.827,
+                0.189
+            ],
+            "angle": 0,
+            "content": "Svarm et al., 2017; Toft et al., 2018). Challenging this approach, recent deep learning frameworks implement direct image alignment by re-projecting the 3D points onto the feature map (Sarlin et al., 2021; von Stumberg et al., 2020a;b). Using end-to-end training, deep networks learn dense feature maps suitable for regressing the pose between an image and a point cloud. In analogy to photometric image alignment (Delaunoy and Pollefeys, 2014; Engel et al., 2016), this family of methods is referred to as feature metric image alignment (von Stumberg et al., 2020b)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.195,
+                0.827,
+                0.321
+            ],
+            "angle": 0,
+            "content": "The feature representation learned by feature metric image alignment is not only discriminative, but is also spatially smooth for improved convergence (von Stumberg et al., 2020a). A training process enforces the smoothness either explicitly through a specific loss, such as the Gauss-Newton loss (von Stumberg et al., 2020a), or implicitly, by backpropagating through an optimization algorithm, such as Levenberg-Marquardt (Sarlin et al., 2021). In both cases, the resulting feature map embeds the bias of the initial poses in the training data. This may lead to poor generalization, since the training poses can differ substantially from the test scenario. We here take a less bias-prone approach: we use generic feature descriptors (e.g. obtained with self-supervision) and instead control the smoothness of the feature map dynamically at test time."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.327,
+                0.828,
+                0.509
+            ],
+            "angle": 0,
+            "content": "This work investigates the connection between feature descriptor networks and feature metric image alignment. The main contribution is the analytical (closed-form) solution to the Gauss-Newton loss. On the one hand, this leads to a novel technique, which can utilize any feature descriptor to generate a dense feature map suitable for direct image alignment. Importantly, it allows us to dynamically adjust the smoothness of the feature map, which effectively controls the trade-off between the basin of convergence and the alignment accuracy. We empirically verify our derivation using self-supervised feature descriptors, such as SuperPoint (DeTone et al., 2018), and demonstrate on-par or even superior alignment accuracy compared to supervised frameworks. On the other hand, the analysis of our closed-form solution reveals inherent limitations of feature learning with backpropagation through Gauss-Newton optimization: feature metric alignment merely learns a form of interpolation between feature descriptors in the points of interest. Although we demonstrate this in the context of direct image alignment, a similar argument extends to other methods, even beyond computer vision, which backpropagate through Gauss-Newton or Levenberg-Marquardt optimization."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.528,
+                0.342,
+                0.543
+            ],
+            "angle": 0,
+            "content": "2 PRELIMINARIES"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.559,
+                0.825,
+                0.604
+            ],
+            "angle": 0,
+            "content": "Image Alignment. Given two images (reference \\( I_r \\) and query \\( I_q \\)) with known camera models and an overlapping field of view, and a 3D point cloud \\( \\{\\pmb{p}^{(i)}\\} \\) in the coordinate system of \\( I_r \\), the problem of image alignment is to estimate the relative camera pose \\( T \\in \\mathbf{SE}(3) \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.611,
+                0.827,
+                0.641
+            ],
+            "angle": 0,
+            "content": "Gauss-Newton Optimization. Given a set of functions \\(\\{r^{(1)}(\\pmb {x}),\\dots,r^{(m)}(\\pmb {x})\\}\\) called residuals, Gauss-Newton (GN) optimization finds the parameters minimizing the sum of squared residuals:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.418,
+                0.646,
+                0.826,
+                0.678
+            ],
+            "angle": 0,
+            "content": "\\[\nf (\\boldsymbol {x}) = \\sum_ {i} \\| r ^ {(i)} (\\boldsymbol {x}) \\| _ {2} ^ {2}. \\tag {1}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.692,
+                0.825,
+                0.75
+            ],
+            "angle": 0,
+            "content": "Each residual \\(r^{(i)}(\\pmb{x})\\) could either be a vector or a single-valued function. In both scenarios, the total residual \\(r(\\pmb{x})\\) is a stacked vector of \\(m\\) residuals. The Gauss-Newton method seeks to minimize \\(f(\\pmb{x})\\) by iteratively updating the parameter estimate \\(\\tilde{\\pmb{x}}\\). Each iteration linearizes the residuals around the current estimate and computes an update step \\(\\Delta_{GN}(r(\\tilde{\\pmb{x}}))\\):"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.329,
+                0.755,
+                0.826,
+                0.786
+            ],
+            "angle": 0,
+            "content": "\\[\n\\Delta_ {\\mathrm {G N}} [ r (\\tilde {\\boldsymbol {x}}) ] = \\left(\\boldsymbol {J} ^ {\\intercal} \\boldsymbol {J}\\right) ^ {- 1} \\boldsymbol {J} ^ {\\intercal} r (\\tilde {\\boldsymbol {x}}), \\quad \\boldsymbol {J} := \\left. \\frac {\\partial r}{\\partial \\boldsymbol {x}} \\right| _ {\\boldsymbol {x} = \\tilde {\\boldsymbol {x}}}, \\tag {2}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.797,
+                0.678,
+                0.812
+            ],
+            "angle": 0,
+            "content": "where \\( J \\) is the Jacobian matrix. The estimate \\( \\tilde{\\pmb{x}} \\) evolves until convergence as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.424,
+                0.817,
+                0.826,
+                0.835
+            ],
+            "angle": 0,
+            "content": "\\[\n\\tilde {\\boldsymbol {x}} \\leftarrow \\tilde {\\boldsymbol {x}} \\boxminus \\Delta_ {\\mathrm {G N}} [ r (\\tilde {\\boldsymbol {x}}) ]. \\tag {3}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.847,
+                0.825,
+                0.89
+            ],
+            "angle": 0,
+            "content": "Here, the operator \\(\\square\\) denotes a specific update procedure, which depends on the nature of the optimization space. For linear spaces, \\(\\square\\) simplifies to the standard subtraction operation; in the context of a rigid-body transformation \\(\\mathbf{SE}(3)\\), operator \\(\\square\\) applies a tangential update."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.896,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "Photometric and feature metric image alignment. Photometric image alignment estimates the relative 6DoF pose by minimizing the difference between pixel intensities of points in \\( I_{r} \\) and the"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.505,
+                0.96
+            ],
+            "angle": 0,
+            "content": "2"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.825,
+                0.134
+            ],
+            "angle": 0,
+            "content": "corresponding points in \\(I_q\\). Recalling \\(\\pmb{p}^{(i)}\\) as a 3D point in the coordinate frame of the reference image, we seek a transformation \\(\\pmb{T} \\in \\mathbf{SE}(3)\\) minimizing the following residuals:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.374,
+                0.151,
+                0.826,
+                0.171
+            ],
+            "angle": 0,
+            "content": "\\[\nr ^ {(i)} (\\boldsymbol {T}) = I _ {r} \\left(\\langle \\boldsymbol {p} ^ {(i)} \\rangle\\right) - I _ {q} \\left(\\langle \\boldsymbol {T} \\boldsymbol {p} ^ {(i)} \\rangle\\right). \\tag {4}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.175,
+                0.827,
+                0.261
+            ],
+            "angle": 0,
+            "content": "Here, \\(\\langle \\cdot \\rangle\\) is a 2D projection operator of a 3D point onto the image plane. The projection implicitly uses the corresponding camera intrinsic parameters, assumed to be available for both \\(I_{r}\\) and \\(I_{q}\\). We omit them in the notation for clarity. Comprising rotation \\(\\mathbf{R} \\in \\mathbf{SO}(3)\\) and translation \\(\\pmb{t} \\in \\mathbb{R}^3\\), the target transformation \\(\\pmb{T}\\) is typically found by minimizing Eq. (4) with non-linear least-squares optimization, such as Gauss-Newton or Levenberg-Marquardt (LM) methods (Levenberg, 1944; Marquardt, 1963; Nocedal and Wright, 1999)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.267,
+                0.828,
+                0.338
+            ],
+            "angle": 0,
+            "content": "The success of photometric image alignment critically depends on a favorable initialization of the pose, especially in the conditions of varying illumination and occlusion. As a partial remedy, feature metric image alignment (Tang and Tan, 2019) uses feature maps instead of pixel intensities. Obtained with deep learning, such feature maps have increased convergence basin compared to that derived from image intensities, which improves robustness to pose initialization."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.357,
+                0.347,
+                0.373
+            ],
+            "angle": 0,
+            "content": "3 RELATED WORK"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.389,
+                0.827,
+                0.543
+            ],
+            "angle": 0,
+            "content": "Direct and indirect image alignment. Estimating the relative camera pose from two images is a fundamental problem in computer vision, with applications in structure from motion (SfM) (Schönberger and Frahm, 2016), SLAM and relocalization. To address this problem, indirect (feature-based) approaches detect and match interest points (Bay et al., 2006; Lowe, 2004) in both images, and then estimate the pose using PnP (Persson and Nordberg, 2018) or by minimizing the reprojection errors (Triggs et al., 1999). In contrast, direct methods sidestep the matching process and minimize the photometric error instead (Horn and Jr., 1988; Irani and Anandan, 1999). This means that they can leverage the entire image (Kerl et al., 2013; Newcombe et al., 2011), or focus on pixels with sufficient gradient (Engel et al., 2014; 2016). The foundation behind these approaches is Lucas-Kanade tracking (Baker and Matthews, 2004; Lucas and Kanade, 1981). However, direct methods, which are central to this work, optimize for a 6DoF pose instead of individual pixel displacements (i.e. optical flow)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.549,
+                0.827,
+                0.716
+            ],
+            "angle": 0,
+            "content": "Pose estimation with deep learning. The advent of deep learning revitalized interest in improving pose estimation with deep networks. While some approaches are holistic (Jatavallabhula et al., 2020), there are broadly three categories of learning-based methods. i) Fully end-to-end pose estimation methods (Kendall et al., 2016; Ummenhofer et al., 2017; Zhou et al., 2017) directly regress pose estimates with deep neural networks, instead of test-time optimization. ii) Learning-based indirect methods (DeTone et al., 2018; Dusmanu et al., 2019; Revaud et al., 2019; Yi et al., 2016) replace handcrafted detectors and descriptors with deep representations in an indirect pipeline. Some approaches further extend the traditional way of obtaining correspondences. SuperGlue (Sarlin et al., 2020) learns feature matching with a graph neural network. LoFTR (Sun et al., 2021) directly regresses correspondences instead of relying on separate feature detection and matching. Similar to indirect methods, iii) learning-based direct image alignment enhances classical direct methods with deep features. This category is the most similar to our work and we discuss it in more detail next."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.722,
+                0.827,
+                0.876
+            ],
+            "angle": 0,
+            "content": "Learned features for direct image alignment. Previous work differs in their approach to model training and in the final task. For example, Czarnowski et al. (2017) leverage off-the-shelf CNN features to improve optical flow tracking. A number of methods (Han et al., 2018; Lv et al., 2019; Sarlin et al., 2021; Tang and Tan, 2019; Xu et al., 2021) train feature descriptors end-to-end with ground-truth poses and backpropagate the gradient through a non-linear optimization process. At test time, these methods employ a feature pyramid and refine the initial camera pose in a coarse-to-fine fashion using GN or LM optimization. In addition to the feature pyramid, some methods predict additional properties for image alignment, such as uncertainty (Xu et al., 2021) (Sarlin et al., 2021), Jacobians (Han et al., 2018), or optimization parameters (e.g. damping factors (Lv et al., 2019; Sarlin et al., 2021)). In the same spirit, our formulation leads to a continuous image pyramid, where each level of the pyramid can be generated on-the-fly based on the input level of uncertainty."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.882,
+                0.829,
+                0.926
+            ],
+            "angle": 0,
+            "content": "Another line of work (von Stumberg et al., 2020a;b) trains a deep network directly on the ground-truth pixel correspondences. GN-Net (von Stumberg et al., 2020a) minimizes two loss functions. The first is a contrastive loss (Schmidt et al., 2017) facilitating discriminative properties of the"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.505,
+                0.96
+            ],
+            "angle": 0,
+            "content": "3"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.104,
+                0.825,
+                0.162
+            ],
+            "angle": 0,
+            "content": "feature representation. The second loss function accounts for a likely displacement in the initial correspondence, implemented by adding random 2D offsets to the ground-truth correspondences at training time. By learning to minimize the feature discrepancy after a Gauss-Newton step, the model learns to cope with the initially noisy estimates. We revisit this work in detail in the Sec. 4."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.167,
+                0.827,
+                0.254
+            ],
+            "angle": 0,
+            "content": "The learning-based approaches (e.g. (Germain et al., 2021; Sarlin et al., 2021; von Stumberg et al., 2020a)) achieve impressive accuracy of pose estimation. Nevertheless, they require pose supervision for training and struggle in scenarios of large-baseline localization. Indeed, GN-Net (von Stumberg et al., 2020a) and PixLoc (Sarlin et al., 2021) tend to exhibit a strong bias toward the noise assumptions of the training process, which cannot be easily reversed. For example, PixLoc trained on the CMU dataset exhibits a strong bias toward horizontal movement as illustrated in Appendix A."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.258,
+                0.828,
+                0.344
+            ],
+            "angle": 0,
+            "content": "Our work takes a different approach. We rely on existing feature descriptors obtained with self-supervision, which contribute no explicit motion bias to the alignment process. We next derive a closed-form solution to the Gauss-Newton loss (von Stumberg et al., 2020a) as a functional of a probability density governing the noise assumptions of the current pose estimate. This allows us to adjust the noise assumptions in the alignment process, thus effectively controlling the convergence basin at test time, much akin to the coarse-to-fine strategy, exemplified by Fig. 2."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.361,
+                0.441,
+                0.377
+            ],
+            "angle": 0,
+            "content": "4 THE GAUSS-NEWTON LOSS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.392,
+                0.825,
+                0.476
+            ],
+            "angle": 0,
+            "content": "In this section, we recap the GN-Net (von Stumberg et al., 2020a) and introduce the notation. GN-Net is a convolutional neural network \\( E(\\cdot ;\\pmb {\\theta}) \\) trained on a sparse set of ground-truth correspondences. Given coordinates on the image plane, let \\( f:\\mathbb{R}\\times \\mathbb{R}\\to \\mathbb{R}^d \\) continuously map those coordinates to a descriptor from a feature grid \\( \\mathbb{R}^{d\\times w\\times h} \\) produced by \\( E(\\cdot ;\\pmb {\\theta}) \\), e.g. using bilinear interpolation. We define \\( f_{r}\\coloneqq E(I_{r};\\pmb {\\theta}) \\) and \\( f_{q}\\coloneqq E(I_{q};\\pmb{\\theta}) \\) to denote feature representations of reference and query images. GN-Net learns parameters \\( \\pmb{\\theta} \\) to minimize the expected value of the following loss function,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.328,
+                0.479,
+                0.825,
+                0.496
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathcal {L} _ {\\mathrm {G N} - \\text {N e t}} \\left(I _ {r}, I _ {q}\\right) = \\mathcal {L} _ {\\text {c o n t r a s t i v e}} \\left(f _ {r}, f _ {q}\\right) + \\mathcal {L} _ {\\mathrm {G N}} \\left(f _ {r}, f _ {q}\\right), \\tag {5}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.498,
+                0.825,
+                0.569
+            ],
+            "angle": 0,
+            "content": "which comprises a contrastive loss, \\(\\mathcal{L}_{\\mathrm{contrastive}}(\\cdot ,\\cdot)\\), and the Gauss-Newton loss, \\(\\mathcal{L}_{\\mathrm{GN}}(\\cdot ,\\cdot)\\). The contrastive loss minimizes the distance between the features of two corresponding points while maximizing the distance between non-corresponding pairs (Schmidt et al., 2017). The contrastive loss facilitates spatially discriminative features, and its particular instantiation has little significance for the following discussion (e. g. GN-Net uses the triplet loss)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.575,
+                0.825,
+                0.632
+            ],
+            "angle": 0,
+            "content": "The Gauss-Newton loss \\(\\mathcal{L}_{\\mathrm{GN}}\\) ensures that the feature map is sufficiently smooth for direct image alignment, thus enlarging the convergence basin. GN-Net implements this by adding a random offset to the ground-truth correspondences and encouraging a single Gauss-Newton step to recover the original location. Let us formalize this process."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.638,
+                0.825,
+                0.714
+            ],
+            "angle": 0,
+            "content": "Given a ground-truth correspondence \\((\\pmb{x}^{(i)},\\pmb{y}^{(i)})\\) between images \\(I_{r}\\) and \\(I_{q}\\), the assumption behind the Gauss-Newton loss is that the initial estimate \\(\\tilde{\\pmb{x}}^{(i)}\\) at test time falls in the vicinity of the ground truth \\(\\pmb{x}^{(i)}\\), i.e. \\(\\tilde{\\pmb{x}}^{(i)} = \\pmb{x}^{(i)} + \\pmb{\\epsilon}\\), where \\(\\pmb{\\epsilon}\\) follows some predefined distribution \\(p(\\pmb{\\epsilon})\\), such as a Gaussian with zero mean. At training time, the Gauss-Newton loss aims at recovering the ground-truth location \\(\\pmb{x}^{(i)}\\) from a noisy initial location \\(\\tilde{\\pmb{x}}^{(i)}\\) by minimizing the residual w.r.t. \\(\\pmb{x}\\):"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.384,
+                0.731,
+                0.825,
+                0.751
+            ],
+            "angle": 0,
+            "content": "\\[\nr ^ {(i)} \\left(\\tilde {\\boldsymbol {x}} ^ {(i)}\\right) := f _ {r} \\left(\\tilde {\\boldsymbol {x}} ^ {(i)}\\right) - f _ {q} \\left(\\boldsymbol {y} ^ {(i)}\\right), \\tag {6}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.759,
+                0.825,
+                0.777
+            ],
+            "angle": 0,
+            "content": "The original Gauss-Newton loss for one point \\( \\pmb{x}^{(i)} \\), as introduced by von Stumberg et al. (2020a), is"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.277,
+                0.792,
+                0.826,
+                0.814
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathcal {L} _ {\\mathrm {G N} _ {o}} ^ {(i)} \\left(f _ {r}, f _ {q}, \\epsilon\\right) := \\| J (\\epsilon - \\Delta_ {\\mathrm {G N}} [ r ^ {(i)} \\left(\\boldsymbol {x} ^ {(i)} + \\epsilon\\right) ]) \\| _ {2} ^ {2} - \\log \\det  \\boldsymbol {J} ^ {\\intercal} \\boldsymbol {J}. \\tag {7}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.819,
+                0.825,
+                0.89
+            ],
+            "angle": 0,
+            "content": "From a probabilistic standpoint (von Stumberg et al., 2020a), the loss balances between the accuracy of the Gauss-Newton step and the direction uncertainty. Here, \\( J^{\\mathsf{T}}J \\) represents the inverse covariance matrix propagated through the photometric residuals. The loss function corresponds to the negative log-likelihood of residuals distributed as \\( \\mathcal{N}\\left(0,\\left(\\mathbf{J}^{\\mathsf{T}}\\mathbf{J}\\right)^{-1}\\right) \\). In this work, we consider a variant of the Gauss-Newton loss, in which the covariance matrix is assumed to be identity. Thus, Eq. (7) becomes"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.345,
+                0.906,
+                0.826,
+                0.927
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathcal {L} _ {\\mathrm {G N}} ^ {(i)} \\left(f _ {r}, f _ {q}, \\boldsymbol {\\epsilon}\\right) = \\| \\boldsymbol {\\epsilon} - \\Delta_ {\\mathrm {G N}} \\left[ r ^ {(i)} \\left(\\boldsymbol {x} ^ {(i)} + \\boldsymbol {\\epsilon}\\right) \\right] \\| _ {2} ^ {2}. \\tag {8}\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.505,
+                0.96
+            ],
+            "angle": 0,
+            "content": "4"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.104,
+                0.827,
+                0.162
+            ],
+            "angle": 0,
+            "content": "Although this simplified version does not account for the trade-off between uncertainty and accuracy of the prediction, it admits a closed-form minimizer of the expected value, as we show in Sec. 5. We also find that the simplified version does not differ from the original one empirically in a significant way (see Appendix C). Henceforth, we will refer to the simplified version as the Gauss-Newton loss."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.167,
+                0.828,
+                0.214
+            ],
+            "angle": 0,
+            "content": "The training process calculates the Gauss-Newton loss stochastically by sampling \\(\\epsilon\\) from \\(p(\\epsilon)\\). Therefore, it minimizes a Monte-Carlo approximation to the expected value of \\(\\mathcal{L}_{\\mathrm{GN}}^{(i)}\\) summed over all ground-truth correspondences,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.353,
+                0.233,
+                0.826,
+                0.262
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathcal {L} _ {\\mathrm {G N}} \\left(f _ {r}, f _ {q}; p\\right) = \\mathbb {E} _ {\\epsilon \\sim p} \\left[ \\sum_ {i} \\mathcal {L} _ {\\mathrm {G N}} ^ {(i)} \\left(f _ {r}, f _ {q}, \\epsilon\\right) \\right]. \\tag {9}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.275,
+                0.825,
+                0.304
+            ],
+            "angle": 0,
+            "content": "Hereafter, we use the notation \\(\\mathcal{L}_{\\mathrm{GN}}(f_r,f_q;p)\\) to emphasize the dependence of the Gauss-Newton loss on the noise distribution \\(p\\), and omit this parametarization otherwise to avoid clutter."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.311,
+                0.827,
+                0.369
+            ],
+            "angle": 0,
+            "content": "Note that the GN-Net's training stage addresses the problem of optical flow, not pose estimation. The underlying assumption is that accurate optical flow facilitates pose estimation, as each pixel will contribute to the final pose estimate. At test time, the pose is determined by solving the featuremetric image alignment problem (equivalent to Eq. (4)) using Gauss-Newton optimization in SE(3)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.388,
+                0.527,
+                0.404
+            ],
+            "angle": 0,
+            "content": "5 CLOSED-FORM GAUSS-NEWTON STEP"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.42,
+                0.825,
+                0.546
+            ],
+            "angle": 0,
+            "content": "Decoupling the contrastive and Gauss-Newton losses. The contrastive loss provides a sparse constraint on the feature embeddings, since we can only use sparse ground-truth correspondences, the interest points, for supervision. By contrast, the Gauss-Newton loss enforces a pre-defined basin of convergence around each interest point (modeled by \\( p(\\epsilon) \\)) with little effect on the feature descriptors in the interest points. This is because the residual in Eq. (6) between the corresponding interest points will be negligible, if the contrastive loss for those points is minimized. It follows that the contrastive and the Gauss-Newton loss essentially optimize over a disjoint set of feature locations. Therefore, we can decouple the Gauss-Newton loss from the joint optimization objective in Eq. (5). Let us formalize this reasoning. We aim to solve:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.335,
+                0.567,
+                0.826,
+                0.594
+            ],
+            "angle": 0,
+            "content": "\\[\nf _ {q} ^ {*} = \\underset {f _ {q}} {\\arg \\min } \\left[ \\mathcal {L} _ {\\text {c o n t r a s t i v e}} \\left(f _ {r}, f _ {q}\\right) + \\mathcal {L} _ {\\mathrm {G N}} \\left(f _ {r}, f _ {q}\\right) \\right]. \\tag {10}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.608,
+                0.827,
+                0.667
+            ],
+            "angle": 0,
+            "content": "We denote the values of \\( f_{r} \\), \\( f_{q} \\) in the interest points as \\( F_{r} \\), \\( F_{q} \\): \\( F_{r}^{(i)} := f_{r}(\\pmb{x}^{(i)}) \\), \\( F_{q}^{(i)} := f_{q}(\\pmb{y}^{(i)}) \\). The contrastive loss only depends on \\( F_{r} \\) and \\( F_{q} \\), while the Gauss-Newton loss depends on all the query feature values \\( f_{q} \\) and the interest points in the reference map \\( F_{r} \\). By eliminating unused parts, we introduce an equivalent problem:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.311,
+                0.688,
+                0.826,
+                0.715
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left\\{F _ {q} ^ {*}, f _ {q} ^ {*} \\right\\} = \\underset {F _ {q}, f _ {q}} {\\arg \\min } \\left[ \\mathcal {L} _ {\\text {c o n t r a s t i v e}} \\left(F _ {r}, F _ {q}\\right) + \\mathcal {L} _ {\\mathrm {G N}} \\left(F _ {r}, f _ {q}\\right) \\right]. \\tag {11}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.726,
+                0.825,
+                0.769
+            ],
+            "angle": 0,
+            "content": "Next, we approximate \\( F_{r} \\) with \\( F_{q} \\) in the second term. This is permissible as the contrastive loss acts as a soft constraint, ensuring that \\( F_{r} \\) and \\( F_{q} \\) are close at the optimum of the joint loss function. This allows us to decouple the minimization problem as follows:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.313,
+                0.788,
+                0.825,
+                0.826
+            ],
+            "angle": 0,
+            "content": "\\[\nF _ {q} ^ {*} = \\underset {F _ {q}} {\\arg \\min } \\left[ \\mathcal {L} _ {\\text {c o n t r a s t i v e}} \\left(F _ {r}, F _ {q}\\right) + \\underset {f _ {q}} {\\min } \\mathcal {L} _ {\\mathrm {G N}} \\left(F _ {q}, f _ {q}\\right) \\right], \\tag {12}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.315,
+                0.825,
+                0.503,
+                0.851
+            ],
+            "angle": 0,
+            "content": "\\[\nf_{q}^{*} = \\operatorname *{arg  min}_{f_{q}}\\mathcal{L}_{\\mathrm{GN}}(F_{q}^{*},f_{q}).\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.863,
+                0.246,
+                0.877
+            ],
+            "angle": 0,
+            "content": "We denote"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.318,
+                0.876,
+                0.564,
+                0.902
+            ],
+            "angle": 0,
+            "content": "\\[\nG(F_{q};p):= \\operatorname *{arg  min}_{f_{q}}\\mathcal{L}_{\\mathrm{GN}}(F_{q},f_{q};p),\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.302,
+                0.902,
+                0.825,
+                0.927
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathcal {L} _ {\\mathrm {G N}} ^ {*} (F _ {q}; p) := \\min  _ {f _ {q}} \\mathcal {L} _ {\\mathrm {G N}} (F _ {q}, f _ {q}; p) = \\mathcal {L} _ {\\mathrm {G N}} (F _ {q}, G (F _ {q}; p); p). \\tag {15}\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.948,
+                0.505,
+                0.96
+            ],
+            "angle": 0,
+            "content": "5"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.174,
+                0.099,
+                0.827,
+                0.337
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.171,
+                0.345,
+                0.825,
+                0.379
+            ],
+            "angle": 0,
+            "content": "Figure 2: Controlling the basin of convergence. Image samples and the corresponding feature maps \\(\\tilde{f}_q\\) (PCA) for different values of \\(\\sigma\\) and isotropic Gaussian distribution \\(p\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.39,
+                0.827,
+                0.482
+            ],
+            "angle": 0,
+            "content": "\\(G(F_{q};p)\\) is the optimal reconstruction of a feature map under sparse feature descriptors \\(F_{q}\\) and noise density \\(p(\\epsilon)\\). \\(\\mathcal{L}_{\\mathrm{GN}}^{*}(F_{q};p)\\) is the corresponding value of the Gauss-Newton loss under \\(F_{q}\\). We will refer to \\(G(F_{q};p)\\) as \\(f_{q}\\) to denote an optimal reconstruction under an arbitrary sparse input \\(F_{q}\\). Note the difference to \\(f_{q}^{*}\\), which is an optimal reconstruction under the joint loss, i.e. \\(G(F_{q}^{*};p)\\). Our main contribution, detailed shortly, is a closed-form solution for both \\(G(F_{q};p)\\) and \\(\\mathcal{L}_{\\mathrm{GN}}^{*}(F_{q};p)\\). It reveals that the original problem in Eq. (10) actually depends only on the feature values in the interest points:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.331,
+                0.495,
+                0.826,
+                0.524
+            ],
+            "angle": 0,
+            "content": "\\[\nF _ {q} ^ {*} = \\underset {F _ {q}} {\\arg \\min } \\left[ \\mathcal {L} _ {\\text {c o n t r a s t i v e}} \\left(F _ {r}, F _ {q}\\right) + \\mathcal {L} _ {\\mathrm {G N}} ^ {*} \\left(F _ {q}; p\\right) \\right]. \\tag {14}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.532,
+                0.825,
+                0.59
+            ],
+            "angle": 0,
+            "content": "This means that in training a feature extractor only the features of interest points matter, while the representation of other pixels approximates our analytical solution. As the empirical validation, we demonstrate compelling results in our experiments by using feature descriptors for the interest points from a self-supervised method, while employing our analytical solution for the remaining pixels."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.595,
+                0.825,
+                0.626
+            ],
+            "angle": 0,
+            "content": "An analytical solution to the Gauss-Newton loss. We propose to calculate the expectation of the Gauss-Newton loss in Eq. (9) as a functional of \\( f_{q} \\), and analytically find its closed-form minimizer."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.632,
+                0.825,
+                0.7
+            ],
+            "angle": 0,
+            "content": "Consider the query image \\( I_{q} \\) as a set \\( \\{(\\pmb{x}^{(j)}, F_{q}^{(j)})\\} \\) containing locations of interest points \\( \\pmb{x}^{(j)} \\) and the corresponding feature descriptors \\( F_{q}^{(j)} \\coloneqq f_{q}(\\pmb{x}^{(j)}) \\). The locations \\( \\pmb{x}^{(j)} \\) can be extracted by an off-the-shelf feature detector (e.g. SuperPoint (DeTone et al., 2018)), while a network trained with the contrastive loss in Eq. (5) can produce the corresponding descriptors \\( F_{q}^{(j)} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.705,
+                0.825,
+                0.734
+            ],
+            "angle": 0,
+            "content": "Let us first re-write the expectation in Eq. (9) using Eq. (8) and Eq. (2) in the decoupled formulation of the Gauss-Newton loss (cf. Eq. (12)):"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.241,
+                0.75,
+                0.826,
+                0.79
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathcal {L} _ {\\mathrm {G N}} \\left(F _ {q}, f _ {q}; p\\right) = \\sum_ {j} \\int_ {\\Omega} \\left\\| \\boldsymbol {\\epsilon} - \\left(\\boldsymbol {J} ^ {\\intercal} \\boldsymbol {J}\\right) ^ {- 1} \\boldsymbol {J} ^ {\\intercal} \\left(f _ {q} \\left(\\boldsymbol {x} ^ {(j)} + \\boldsymbol {\\epsilon}\\right) - F _ {q} ^ {(j)}\\right) \\right\\| _ {2} ^ {2} p (\\boldsymbol {\\epsilon}) d \\boldsymbol {\\epsilon}. \\tag {15}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.802,
+                0.421,
+                0.817
+            ],
+            "angle": 0,
+            "content": "Substituting \\(\\pmb {\\epsilon} = \\tilde{\\pmb{x}} -\\pmb{x}^{(j)}\\) , we obtain"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.198,
+                0.841,
+                0.826,
+                0.881
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathcal {L} _ {\\mathrm {G N}} \\left(F _ {q}, f _ {q}; p\\right) = \\sum_ {j} \\int_ {\\Omega} \\left\\| \\tilde {\\boldsymbol {x}} - \\left(\\boldsymbol {J} ^ {\\intercal} \\boldsymbol {J}\\right) ^ {- 1} \\boldsymbol {J} ^ {\\intercal} \\left(f _ {q} (\\tilde {\\boldsymbol {x}}) - F _ {q} ^ {(j)}\\right) - \\boldsymbol {x} ^ {(j)} \\right\\| _ {2} ^ {2} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right) d \\tilde {\\boldsymbol {x}}. \\tag {16}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.893,
+                0.826,
+                0.929
+            ],
+            "angle": 0,
+            "content": "We first consider all \\(\\tilde{\\pmb{x}}\\in \\Omega\\) independently and relax the problem by eliminating the constraint \\(J = \\frac{\\partial f_q}{\\partial\\tilde{\\pmb{x}}}\\). This allows us to derive analytical solutions for minimizers \\(\\tilde{f}_q(\\cdot)\\) and \\(\\tilde{J}_q(\\cdot)\\) as functions"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.505,
+                0.96
+            ],
+            "angle": 0,
+            "content": "6"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.173,
+                0.095,
+                0.831,
+                0.254
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.171,
+                0.262,
+                0.828,
+                0.334
+            ],
+            "angle": 0,
+            "content": "Figure 3: Image alignment with continuous probabilistic feature pyramids. Given reference image with a 3D point cloud and a query image paired with an initial coarse pose, we start by estimating interest points and their features on the query image, along with features from the projections of the 3D point cloud onto the reference image. Next, we perform feature metric alignment between the 3D point features and derived analytical continuous image pyramid."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.358,
+                0.828,
+                0.43
+            ],
+            "angle": 0,
+            "content": "of \\(\\tilde{\\pmb{x}}\\). We then observe that \\(\\frac{\\partial}{\\partial\\tilde{\\pmb{x}}}\\tilde{f}_q(\\cdot)\\) coincides with \\(\\tilde{J}_q(\\cdot)\\) for a uniform distribution, which confirms that in this case, our derived solution is the solution to the original problem. As a side note, in the context of direct image alignment, it has been suggested that decoupling and predicting the Jacobian independently from the function value may yield superior convergence and results (Han et al., 2018). Appendix B provides the full derivation. Below, we summarize the closed-form minimizer to Eq. (16):"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.308,
+                0.455,
+                0.825,
+                0.473
+            ],
+            "angle": 0,
+            "content": "\\[\n\\tilde {f} _ {q} (\\tilde {\\boldsymbol {x}}) = \\tilde {J} _ {q} (\\tilde {\\boldsymbol {x}}) \\left(\\tilde {\\boldsymbol {x}} - \\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}})\\right) + \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}}), \\tag {17}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.306,
+                0.476,
+                0.825,
+                0.497
+            ],
+            "angle": 0,
+            "content": "\\[\n\\tilde {J} _ {q} (\\tilde {\\boldsymbol {x}}) = \\left(\\operatorname {C o v} _ {\\mathbf {x y}} (\\tilde {\\boldsymbol {x}}) \\operatorname {C o v} _ {\\mathbf {y}} (\\tilde {\\boldsymbol {x}}) ^ {- 1}\\right) ^ {+}, \\tag {18}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.299,
+                0.499,
+                0.825,
+                0.532
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}}) = \\sum_ {j} F _ {q} ^ {(j)} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right) / \\sum_ {j} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right), \\tag {19}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.299,
+                0.534,
+                0.825,
+                0.568
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}}) = \\sum_ {j} \\boldsymbol {x} ^ {(j)} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right) / \\sum_ {j} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right), \\tag {20}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.275,
+                0.571,
+                0.826,
+                0.606
+            ],
+            "angle": 0,
+            "content": "\\[\n\\operatorname {C o v} _ {\\mathbf {x y}} (\\tilde {\\boldsymbol {x}}) = \\sum_ {j} \\left(\\boldsymbol {x} ^ {(j)} - \\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}})\\right) \\left(F _ {q} ^ {(j)} - \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}})\\right) ^ {\\intercal} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}), \\tag {21}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.282,
+                0.608,
+                0.826,
+                0.644
+            ],
+            "angle": 0,
+            "content": "\\[\n\\operatorname {C o v} _ {\\mathbf {y}} (\\tilde {\\boldsymbol {x}}) = \\sum_ {j} \\left(F _ {q} ^ {(j)} - \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}})\\right) \\left(F _ {q} ^ {(j)} - \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}})\\right) ^ {\\intercal} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}), \\tag {22}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.659,
+                0.825,
+                0.69
+            ],
+            "angle": 0,
+            "content": "where \\((^{+})\\) denotes the Moore-Penrose pseudo-inverse operator. The optimal point is \\(G(F_{q};p)\\coloneqq \\tilde{f}_{q}\\) and the loss value at this point (cf. Eq. (13)) is"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.259,
+                0.706,
+                0.826,
+                0.747
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathcal {L} _ {G N} ^ {*} (F _ {q}) = \\int_ {\\Omega} \\frac {1}{2} \\operatorname {T r} \\left[ \\operatorname {C o v} _ {\\mathbf {x}} (\\tilde {\\boldsymbol {x}}) - \\operatorname {C o v} _ {\\mathbf {x y}} (\\tilde {\\boldsymbol {x}}) \\operatorname {C o v} _ {\\mathbf {y}} (\\tilde {\\boldsymbol {x}}) ^ {- 1} \\operatorname {C o v} _ {\\mathbf {y x}} (\\tilde {\\boldsymbol {x}}) \\right] d \\tilde {\\boldsymbol {x}}. \\tag {23}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.757,
+                0.827,
+                0.858
+            ],
+            "angle": 0,
+            "content": "Our derivation of the optimal embeddings \\( G(F_{q};p) \\) using fixed interest points \\( F_{q} \\) provides an interesting insight into end-to-end learning frameworks. Jointly training both losses requires inverting a high-dimensional matrix \\( \\mathrm{Cov}_{\\mathbf{y}}(\\tilde{\\boldsymbol{x}}) \\) in Eq. (23), which poses high numerical instability and discontinuities. Since previous work (von Stumberg et al., 2020a) can be seen as stochastic approximations to our solution, this may explain the reported training instability and divergence in those works. Additionally, the form of \\( \\tilde{f}_q \\) in Eq. (17) suggests that end-to-end pipelines may not necessarily yield any sophisticated representation, as they merely interpolate between features in the interest points."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.865,
+                0.826,
+                0.927
+            ],
+            "angle": 0,
+            "content": "FeatureMetric Image Alignment. As a case study, we employ the analytical form of \\(\\tilde{f}_q\\) and \\(\\tilde{J}_q\\) for feature metric image alignment. Algorithm 1 and Fig. 3 provide an overview. Fig. 1 illustrates stages of direct alignment using our probabilistically reconstructed feature maps. We first extract the interest points \\(\\{\\pmb{x}^{(j)}\\}\\) and the corresponding feature descriptors \\(\\{F_q^{(j)}\\}\\) from \\(I_{q}\\) to reconstruct \\(\\tilde{f}_q\\) and"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.504,
+                0.96
+            ],
+            "angle": 0,
+            "content": "7"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.175,
+                0.101,
+                0.369,
+                0.118
+            ],
+            "angle": 0,
+            "content": "Require: \\(I_r, I_q, \\{\\pmb{p}^{(i)}\\}, \\pmb{T}^{(0)}\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.184,
+                0.119,
+                0.39,
+                0.135
+            ],
+            "angle": 0,
+            "content": "1: \\(\\{\\pmb{x}^{(j)}\\} \\gets\\) InterestPoints \\((I_q)\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.184,
+                0.135,
+                0.391,
+                0.152
+            ],
+            "angle": 0,
+            "content": "2: \\(\\{F_q^{(j)}\\} \\gets E(I_q;\\pmb {\\theta})\\circ \\{\\pmb{x}^{(j)}\\}\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.184,
+                0.152,
+                0.396,
+                0.169
+            ],
+            "angle": 0,
+            "content": "3: \\(\\{\\pmb{o}^{(i)}\\} \\gets E(I_r;\\pmb {\\theta})\\circ \\{\\langle \\pmb{p}^{(i)}\\rangle \\}\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.184,
+                0.169,
+                0.272,
+                0.181
+            ],
+            "angle": 0,
+            "content": "4: \\( \\pmb{T} \\gets \\pmb{T}^{(0)} \\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.184,
+                0.183,
+                0.362,
+                0.197
+            ],
+            "angle": 0,
+            "content": "5: for \\( n \\) from 0 to \\( N_{\\mathrm{max}} \\) do"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.184,
+                0.197,
+                0.349,
+                0.211
+            ],
+            "angle": 0,
+            "content": "6: \\(p\\gets \\mathrm{Schedule}(n)\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.184,
+                0.212,
+                0.355,
+                0.229
+            ],
+            "angle": 0,
+            "content": "7: \\(\\tilde{f}_q\\gets G(\\{F_q^{(j)}\\} ;p)\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.184,
+                0.229,
+                0.35,
+                0.246
+            ],
+            "angle": 0,
+            "content": "8: \\(\\{\\tilde{\\pmb{p}}^{(i)}\\} \\gets \\{\\pmb {T}\\pmb{p}^{(i)}\\}\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.184,
+                0.246,
+                0.458,
+                0.262
+            ],
+            "angle": 0,
+            "content": "9: \\(\\pmb{\\delta} \\gets \\Delta_{GN}(\\tilde{f}_q \\circ \\{\\langle \\tilde{\\pmb{p}}^{(i)}\\rangle\\} - \\{\\pmb{o}^{(i)}\\})\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.184,
+                0.262,
+                0.312,
+                0.273
+            ],
+            "angle": 0,
+            "content": "0: \\(\\pmb{T}\\gets \\pmb{T}\\boxminus \\pmb{\\delta}\\)"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.184,
+                0.119,
+                0.458,
+                0.273
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.177,
+                0.275,
+                0.255,
+                0.286
+            ],
+            "angle": 0,
+            "content": "11: end for"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.177,
+                0.289,
+                0.264,
+                0.301
+            ],
+            "angle": 0,
+            "content": "12: return T"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.177,
+                0.275,
+                0.264,
+                0.301
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.2,
+                0.308,
+                0.356,
+                0.323
+            ],
+            "angle": 0,
+            "content": "\\(\\langle \\cdot \\rangle\\) denotes 2D projection."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.336,
+                0.411,
+                0.352
+            ],
+            "angle": 0,
+            "content": "Algorithm 1: Image alignment."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.47,
+                0.119,
+                0.778,
+                0.317
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.458,
+                0.322,
+                0.823,
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+            "angle": 0,
+            "content": "Figure 4: Robustness to initialization. Our closed-form solution exhibits significantly greater robustness compared to PixLoc."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.389,
+                0.825,
+                0.451
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+            "angle": 0,
+            "content": "\\(\\tilde{J}_q\\) using Eqs. (17) and (18). The sparse set of 3D points \\(\\{\\pmb{p}^{(i)}\\}\\) is projected onto the reference image to obtain reference descriptors. We obtain feature descriptors for these points using the same feature embedding network \\(E(\\cdot ;\\theta)\\). Starting with the initial camera pose \\(\\pmb{T}^{(0)}\\in \\mathbf{S}\\mathbf{E}(3)\\), at every iteration we perform Gauss-Newton steps minimizing the residuals \\(r^{(i)}\\), w.r.t. the camera transform \\(\\pmb{T}\\):"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.391,
+                0.467,
+                0.826,
+                0.488
+            ],
+            "angle": 0,
+            "content": "\\[\nr ^ {(i)} (\\boldsymbol {T}) := F _ {r} ^ {(i)} - \\tilde {f} _ {q} \\left(\\left\\langle \\boldsymbol {T} \\boldsymbol {p} ^ {(i)} \\right\\rangle\\right). \\tag {24}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.495,
+                0.825,
+                0.571
+            ],
+            "angle": 0,
+            "content": "\\(p(\\epsilon)\\) explicitly controls the basin of convergence. In practice, we can use any distribution with high initial variance based on the inaccuracy assumed in the initial point projections. As our estimates improve over the course of optimization, we gradually decrease the variance. Notably, the choice of \\(p(\\epsilon)\\) lets us define values for points outside of the image boundaries. Fig. 2 illustrates examples of the feature maps \\(\\tilde{f}_q\\) with \\(p\\) following a Gaussian distribution of increasing variance."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.575,
+                0.825,
+                0.676
+            ],
+            "angle": 0,
+            "content": "Comparison to previous work. Our derivation is agnostic to the underlying feature extractor and can operate with off-the-shelf feature descriptors. An interesting property of the derived \\(\\tilde{f}_q\\) is that it enables continuous coarse-to-fine alignment. By adjusting the noise prior \\(p\\) in the scheduler (cf. Algorithm 1), we can control the basin of convergence dynamically at runtime. We can adjust the distribution \\(p\\) either by choosing a different parameter set of a pre-defined distribution, or by changing the distribution itself to another family. In practice, we start the optimization with a uniform distribution and then switch to the Gaussian distribution with a slowly decreasing variance."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.681,
+                0.827,
+                0.725
+            ],
+            "angle": 0,
+            "content": "A uniform distribution with a wide support offers a large basin of convergence, which sacrifices accuracy for robustness. The ensuing Gaussian distribution with decreasing variance refines the pose and leads to a more accurate solution. Appendix E provides implementation details of the scheduler."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.73,
+                0.827,
+                0.833
+            ],
+            "angle": 0,
+            "content": "Connection between feature matching and feature metric image alignment. Note that the computation of \\(\\tilde{f}_q\\) using Eq. (17) only depends on the feature descriptors of the interest points in the query image. This observation suggests an interesting interpretation of minimizing the residual in Eq. (24) using \\(\\tilde{f}_q\\) with Gauss-Newton optimization as feature matching. The Gauss-Newton step is fully determined by the neighboring points of interest. Consequently, it implies that the effectiveness of such optimization-based methods may be limited in comparison to alternative optimization-inspired approaches, which learn the optimization step (Teed and Deng, 2020)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.851,
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+            "angle": 0,
+            "content": "6 EXPERIMENTS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.882,
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+            "angle": 0,
+            "content": "Datasets. We evaluate our approach on two most popular datasets for large-scale image localization, namely the Aachen Day-Night dataset (Sattler et al., 2018), extended CMU seasons (Toft et al., 2022) and 7Scenes dataset (Shotton et al., 2013). Aachen Day-Night consists of 98 night and 824 day query"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
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+            "angle": 0,
+            "content": "8"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.171,
+                0.101,
+                0.828,
+                0.158
+            ],
+            "angle": 0,
+            "content": "Table 1: Camera localization on Aachen Day-Night and CMU Seasons. Our evaluation shows an overall improved accuracy in diverse scenarios despite using self-supervised SuperPoint descriptors. We compare to the state-of-the-art methods supervised with pose: GN-Net (von Stumberg et al., 2020a), LM-Reloc (von Stumberg et al., 2020b) and PixLoc (Sarlin et al., 2021)."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.172,
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+            "angle": 0,
+            "content": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">Aachen</td><td colspan=\"3\">CMU Seasons</td></tr><tr><td>Day</td><td>Night</td><td>Urban</td><td>Suburban</td><td>Park</td></tr><tr><td>GN-Net</td><td>62.4/69.4/76.9</td><td>49.0/58.2/66.3</td><td>75.4/79.9/92.6</td><td>64.7/67.0/81.4</td><td>46.7/48.3/65.3</td></tr><tr><td>LM-Reloc</td><td>60.4/68.0/76.3</td><td>37.8/46.9/59.2</td><td>76.6/82.5/93.4</td><td>67.3/72.0/82.8</td><td>49.1/53.4/66.9</td></tr><tr><td>PixLoc</td><td>64.3/69.3/77.4</td><td>51.0/55.1/67.3</td><td>88.3/90.4/93.7</td><td>79.6/81.1/85.2</td><td>61.0/62.5/69.4</td></tr><tr><td>Ours (SuperPoint)</td><td>66.3/72.5/78.8</td><td>43.9/50.0/56.1</td><td>86.0/90.6/95.2</td><td>79.8/85.0/92.4</td><td>63.4/67.9/77.5</td></tr></table>"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.292,
+                0.825,
+                0.35
+            ],
+            "angle": 0,
+            "content": "images. Extended CMU seasons consists of 14 slices, 5 for urban environment, 5 for suburban and 4 for park. Each slice contains between 3000 and 5000 query images. The 7Scenes dataset comprises seven distinct scenes, each containing multiple sequences of 500 to 1000 frames. For our ablation study in Appendix D we used Cambridge Landmarks dataset (Kendall et al., 2016)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.355,
+                0.827,
+                0.468
+            ],
+            "angle": 0,
+            "content": "Results. Table 1 presents the results for camera localization on Aachen Day-Night (Sattler et al., 2018) and the extended CMU Seasons (Toft et al., 2022). We use SuperPoint (DeTone et al., 2018) as the feature descriptor in these experiments. In each setting (Day/Night for Aachen; Urban/Suburban/Park for CMU Seasons), we report three numbers, which indicate the percentage of the query images that were successfully localized within the specified translation and rotation thresholds. We adopt the standard threshold values defined by the benchmarks (translation, rotation): \\((0.25\\mathrm{m}, 2^{\\circ}) / (0.5\\mathrm{m}, 5^{\\circ}) / (5\\mathrm{m}, 10^{\\circ})\\). We adopted the implementation from PixLoc (Sarlin et al., 2021) to run these benchmarks. A comparison with PixLoc on the 7Scenes dataset is presented in Appendix F."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.474,
+                0.825,
+                0.613
+            ],
+            "angle": 0,
+            "content": "We observe that our solution, despite using self-supervised descriptors from SuperPoint (DeTone et al., 2018), achieves an overall strong localization accuracy across diverse settings in comparison to supervised frameworks. Aachen Day-Night has significant occlusions, hence many outliers. Since the Gauss-Newton loss does not incorporate any outlier filtering, we do not expect high accuracy for our approach on this dataset. Nevertheless, we were surprised to find that on Aachen Day our approach even slightly surpassed previous state of the art. On Aachen Night, our accuracy is inferior to previous work, however. This is somewhat expected, since SuperPoint feature descriptors were not directly trained on day-night correspondences. By contrast, PixLoc was trained with supervision on day-night image pairs, which provides an obvious advantage. On CMU Seasons, our approach demonstrates a clear improvement over previous feature metric alignment methods."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.62,
+                0.825,
+                0.663
+            ],
+            "angle": 0,
+            "content": "The proposed approach demonstrates notable robustness to initialization noise. We compared the proposed scheme and PixLoc on the Cambridge Landmarks dataset, varying the levels of random noise applied to the ground-truth pose as illustrated in Fig. 4 and Appendix G."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.668,
+                0.827,
+                0.726
+            ],
+            "angle": 0,
+            "content": "Overall, these results empirically confirm the validity of our closed-form derivation. Furthermore, the comparison suggests a limited benefit of end-to-end learning with the Gauss-Newton loss. By contrast, a dynamic convergence basin offered by our analytical solution provides versatility w.r.t. the underlying feature descriptors and improves robustness to pose initialization, as a result."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.745,
+                0.321,
+                0.76
+            ],
+            "angle": 0,
+            "content": "7 CONCLUSION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.776,
+                0.827,
+                0.903
+            ],
+            "angle": 0,
+            "content": "We derived a closed-form solution to the Gauss-Newton loss in the context of direct image alignment, which offers two main advantages. First, it allows for dynamic control of the convergence basin, which improves robustness of the alignment to pose initialization. Furthermore, despite using self-supervised descriptors, such control leads to compelling accuracy of pose estimates in comparison to supervised pipelines on established benchmarks. Second, our derivation exposes intrinsic limitations of employing the Gauss-Newton loss in deep learning, as it only leads to a form of interpolation between the feature descriptors in the interest points. This insight offers an interesting connection between direct image alignment and feature matching, and leads to a novel perspective on learning robust features end-to-end, which we will investigate in future work."
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.194,
+                0.91,
+                0.376,
+                0.925
+            ],
+            "angle": 0,
+            "content": "1https://github.com/cvg/pixloc"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.505,
+                0.96
+            ],
+            "angle": 0,
+            "content": "9"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.104,
+                0.357,
+                0.119
+            ],
+            "angle": 0,
+            "content": "ACKNOWLEDGMENTS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.134,
+                0.828,
+                0.178
+            ],
+            "angle": 0,
+            "content": "This work was supported by the ERC Advanced Grant SIMULACRON and by TUM Georg Nemetschek Institute under the project AI4TWINNING. We thank Linus Harenstam-Nielsen for helpful discussions and proofreading."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.197,
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+            "angle": 0,
+            "content": "REPRODUCIBILITY STATEMENT"
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+            "angle": 0,
+            "content": "The main contribution of this work is the derivation of the closed-form solution to the Gauss-Newton loss. We provide details of this derivation in Appendix B. To reproduce our experimental results, we elaborate on the implementation details Appendix E. To facilitate reproducibility in future research, we also publicly release our code."
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+            "content": "11"
+        }
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+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "ref_text",
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+            "content": "B. Xu, A. J. Davison, and S. Leutenegger. Deep probabilistic feature-metric tracking. IEEE Robotics Autom. Lett., 6(1):223-230, 2021."
+        },
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+            "angle": 0,
+            "content": "K. M. Yi, E. Trulls, V. Lepetit, and P. Fua. LIFT: Learned invariant feature transform. In ECCV, volume 9910, pages 467-483, 2016."
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
+                0.173,
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+            ],
+            "angle": 0,
+            "content": "T. Zhou, M. Brown, N. Snavely, and D. G. Lowe. Unsupervised learning of depth and ego-motion from video. In CVPR, pages 6612-6619, 2017."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.173,
+                0.105,
+                0.826,
+                0.267
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.293,
+                0.383,
+                0.308
+            ],
+            "angle": 0,
+            "content": "A OPTICAL FLOW BIAS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.323,
+                0.487,
+                0.602
+            ],
+            "angle": 0,
+            "content": "We demonstrate that PixLoc exhibits bias inherited from the training dataset. Fig. 5 illustrates the average translation error after optimization when evaluated on the Cambridge dataset, highlighting the impact of different initial displacements along the image axes. To solely evaluate the bias learned by the feature maps, we disabled dataset-specific learned movement priors denoted as \\(\\lambda\\) in PixLoc (Sarlin et al., 2021). Our results demonstrate that when trained on the CMU dataset, the network predicts optical flow along the X axis more accurately. This increased accuracy is directly linked to the structure of the CMU dataset. Specifically, the dataset comprises image pairs captured by a camera inside a moving car, pointed towards the side of the road. Consequently, this positioning results in the network predominantly learning horizontal optical flow patterns, reflecting the lateral movement observed in the images of the training dataset."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.521,
+                0.339,
+                0.801,
+                0.525
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.495,
+                0.537,
+                0.828,
+                0.595
+            ],
+            "angle": 0,
+            "content": "Figure 5: The final error of the PixLoc solution along \\( X / Y \\)-axes w.r.t. the initial error. PixLoc remains accurate only along \\( X \\)-axis on CMU, reflecting the dominant motion in the training set."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.621,
+                0.585,
+                0.637
+            ],
+            "angle": 0,
+            "content": "B DERIVATION OF THE ANALYTICAL SOLUTION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.652,
+                0.825,
+                0.698
+            ],
+            "angle": 0,
+            "content": "In this section, we provide a solution to minimizing Eq. (16) from the main text. Let \\(\\Omega\\) be the coordinate domain of an image plane \\([0,1] \\times [0,1]\\), and let \\(E \\subseteq \\mathbb{R}^d\\) define the feature space. We aim to find \\(\\tilde{f}: \\Omega \\to E\\):"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.183,
+                0.719,
+                0.826,
+                0.761
+            ],
+            "angle": 0,
+            "content": "\\[\n\\tilde {f} = \\arg \\min  _ {f} \\sum_ {j} \\int_ {\\Omega} \\left\\| \\tilde {\\boldsymbol {x}} - (J (\\tilde {\\boldsymbol {x}}) ^ {\\intercal} J (\\tilde {\\boldsymbol {x}})) ^ {- 1} J (\\tilde {\\boldsymbol {x}}) ^ {\\intercal} (f (\\tilde {\\boldsymbol {x}}) - F _ {q} ^ {(j)}) - \\boldsymbol {x} ^ {(j)} \\right\\| _ {2} ^ {2} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}) d \\tilde {\\boldsymbol {x}}. \\tag {25}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.772,
+                0.825,
+                0.816
+            ],
+            "angle": 0,
+            "content": "We start by relaxing the constraint \\(\\nabla f(\\pmb{x}) = J(\\pmb{x})\\). This makes every value of \\(f(\\pmb{x})\\) and \\(J(\\pmb{x})\\) locally independent and, therefore, the minimum is achieved by minimizing Eq. (25) independently over every point in \\(\\Omega\\):"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.203,
+                0.837,
+                0.826,
+                0.879
+            ],
+            "angle": 0,
+            "content": "\\[\n\\{\\tilde {f} (\\tilde {\\boldsymbol {x}}), \\tilde {J} (\\tilde {\\boldsymbol {x}}) \\} = \\underset {\\hat {\\boldsymbol {f}}, \\hat {\\boldsymbol {J}}} {\\arg \\min } \\sum_ {j} \\left\\| \\tilde {\\boldsymbol {x}} - \\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal} \\left(\\hat {\\boldsymbol {f}} - F _ {q} ^ {(j)}\\right) - \\boldsymbol {x} ^ {(j)} \\right\\| _ {2} ^ {2} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}). \\tag {26}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.893,
+                0.826,
+                0.927
+            ],
+            "angle": 0,
+            "content": "Note that the only significant part of \\(\\hat{\\pmb{f}}\\) is a part which is spanned by columns of \\(\\hat{\\pmb{J}}\\). This can be easily seen by observing that \\(\\hat{\\pmb{J}}^{\\mathrm{T}}\\hat{\\pmb{f}}\\) is a non-normalized projection onto \\(\\hat{\\pmb{J}}\\). More formally, let us find \\(\\hat{\\pmb{f}}\\) as"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "12"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.102,
+                0.825,
+                0.134
+            ],
+            "angle": 0,
+            "content": "\\(\\hat{J}a + \\hat{J}^\\perp b\\), where \\(\\hat{J}^\\perp\\) is a basis for an orthogonal complement for \\(\\operatorname{span}(\\hat{J})\\). By substituting this into Eq. (26), we obtain:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.233,
+                0.151,
+                0.826,
+                0.191
+            ],
+            "angle": 0,
+            "content": "\\[\n\\{\\tilde {a} (\\tilde {\\boldsymbol {x}}), \\tilde {J} (\\tilde {\\boldsymbol {x}}) \\} = \\underset {\\boldsymbol {a}, \\hat {\\boldsymbol {J}}} {\\arg \\min } \\sum_ {j} \\left\\| \\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)} - \\boldsymbol {a} + \\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal} F _ {q} ^ {(j)} \\right\\| _ {2} ^ {2} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}), \\tag {27}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.288,
+                0.194,
+                0.581,
+                0.214
+            ],
+            "angle": 0,
+            "content": "\\[\n\\tilde {f} (\\tilde {\\boldsymbol {x}}) \\in \\left\\{\\tilde {J} (\\tilde {\\boldsymbol {x}}) \\tilde {a} (\\tilde {\\boldsymbol {x}}) + \\tilde {J} (\\tilde {\\boldsymbol {x}}) ^ {\\perp} \\boldsymbol {b} \\mid \\boldsymbol {b} \\in \\mathbb {R} ^ {d - 2} \\right\\}.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.225,
+                0.825,
+                0.287
+            ],
+            "angle": 0,
+            "content": "Observe that this problem is a linear least squares in \\(\\pmb{a}\\) and \\(\\left(\\hat{\\boldsymbol{J}}^{\\intercal}\\hat{\\boldsymbol{J}}\\right)^{-1}\\hat{\\boldsymbol{J}}^{\\intercal}\\) and recall that \\(p(\\cdot)\\geq 0\\) (by definition), hence the problem is convex. Minimality conditions for Eq. (27) for \\(\\pmb{a}\\) and \\(\\left(\\hat{\\boldsymbol{J}}^{\\intercal}\\hat{\\boldsymbol{J}}\\right)^{-1}\\hat{\\boldsymbol{J}}^{\\intercal}\\) are:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.286,
+                0.314,
+                0.826,
+                0.351
+            ],
+            "angle": 0,
+            "content": "\\[\n\\frac {d}{d \\boldsymbol {a}}: \\quad \\sum_ {j} \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)} - \\boldsymbol {a} + \\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal} F _ {q} ^ {(j)}\\right) p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}) = 0, \\tag {28}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.201,
+                0.355,
+                0.826,
+                0.4
+            ],
+            "angle": 0,
+            "content": "\\[\n\\frac {d}{d \\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal}}: \\quad \\sum_ {j} \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)} - \\boldsymbol {a} + \\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal} F _ {q} ^ {(j)}\\right) F _ {q} ^ {(j) \\intercal} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right) = 0. \\tag {29}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.414,
+                0.336,
+                0.429
+            ],
+            "angle": 0,
+            "content": "From Eq. (28), we have:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.363,
+                0.446,
+                0.826,
+                0.473
+            ],
+            "angle": 0,
+            "content": "\\[\n\\boldsymbol {a} = \\tilde {\\boldsymbol {x}} - \\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}}) + \\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal} \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}}), \\tag {30}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.485,
+                0.218,
+                0.497
+            ],
+            "angle": 0,
+            "content": "where"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.334,
+                0.496,
+                0.826,
+                0.535
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}}) := \\sum_ {j} F _ {q} ^ {(j)} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right) / \\sum_ {j} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right), \\tag {31}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.334,
+                0.533,
+                0.661,
+                0.563
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}}) := \\sum_ {j} \\boldsymbol {x} ^ {(j)} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}) / \\sum_ {j} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}).\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.575,
+                0.429,
+                0.59
+            ],
+            "angle": 0,
+            "content": "Substituting \\( \\pmb{a} \\) into Eq. (29), we obtain"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.216,
+                0.617,
+                0.826,
+                0.653
+            ],
+            "angle": 0,
+            "content": "\\[\n\\sum_ {j} \\left(\\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}}) - \\boldsymbol {x} ^ {(j)} + \\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal} \\left(F _ {q} ^ {(j)} - \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}})\\right)\\right) F _ {q} ^ {(j) \\intercal} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right) = 0, \\tag {32}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.423,
+                0.657,
+                0.709,
+                0.691
+            ],
+            "angle": 0,
+            "content": "\\[\n- \\sum_ {j} \\left(\\boldsymbol {x} ^ {(j)} - \\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}})\\right) F _ {q} ^ {(j) \\intercal} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}) +\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.355,
+                0.695,
+                0.826,
+                0.729
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal} \\sum_ {i} \\left(F _ {q} ^ {(j)} - \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}})\\right) F _ {q} ^ {(j) \\intercal} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right) = 0, \\tag {33}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.455,
+                0.733,
+                0.826,
+                0.76
+            ],
+            "angle": 0,
+            "content": "\\[\n- \\operatorname {C o v} _ {\\mathbf {x y}} (\\tilde {\\boldsymbol {x}}) + \\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal} \\operatorname {C o v} _ {\\mathbf {y}} (\\tilde {\\boldsymbol {x}}) = 0, \\tag {34}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.767,
+                0.218,
+                0.779
+            ],
+            "angle": 0,
+            "content": "where"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.275,
+                0.786,
+                0.826,
+                0.821
+            ],
+            "angle": 0,
+            "content": "\\[\n\\operatorname {C o v} _ {\\mathbf {x y}} (\\tilde {\\boldsymbol {x}}) := \\sum_ {j} \\left(\\boldsymbol {x} ^ {(j)} - \\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}})\\right) \\left(F _ {q} ^ {(j)} - \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}})\\right) ^ {\\intercal} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}), \\tag {35}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.282,
+                0.825,
+                0.826,
+                0.859
+            ],
+            "angle": 0,
+            "content": "\\[\n\\operatorname {C o v} _ {\\mathbf {y}} (\\tilde {\\boldsymbol {x}}) := \\sum_ {j} \\left(F _ {q} ^ {(j)} - \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}})\\right) \\left(F _ {q} ^ {(j)} - \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}})\\right) ^ {\\intercal} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}). \\tag {36}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.866,
+                0.663,
+                0.881
+            ],
+            "angle": 0,
+            "content": "Solving Eq. (34) for \\( J \\) and substituting the solution into Eq. (27) results in"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.243,
+                0.906,
+                0.826,
+                0.927
+            ],
+            "angle": 0,
+            "content": "\\[\n\\tilde {J} (\\tilde {\\boldsymbol {x}}) = \\left(\\operatorname {C o v} _ {\\mathbf {x y}} (\\tilde {\\boldsymbol {x}}) \\operatorname {C o v} _ {\\mathbf {y}} (\\tilde {\\boldsymbol {x}}) ^ {- 1}\\right) ^ {\\dagger}, \\tag {37}\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "13"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.175,
+                0.101,
+                0.473,
+                0.288
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.532,
+                0.102,
+                0.828,
+                0.29
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.171,
+                0.315,
+                0.825,
+                0.345
+            ],
+            "angle": 0,
+            "content": "Figure 6: Comparison of rotation and translation recalls for our simplified version of GN-loss and probabilistic one"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.24,
+                0.368,
+                0.826,
+                0.397
+            ],
+            "angle": 0,
+            "content": "\\[\n\\tilde {f} (\\tilde {\\boldsymbol {x}}) = \\tilde {J} (\\tilde {\\boldsymbol {x}}) \\boldsymbol {a} = \\tilde {J} (\\tilde {\\boldsymbol {x}}) \\left(\\tilde {\\boldsymbol {x}} - \\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}})\\right) + \\tilde {J} (\\tilde {\\boldsymbol {x}}) \\left(\\tilde {J} (\\tilde {\\boldsymbol {x}}) ^ {\\intercal} \\tilde {J} (\\tilde {\\boldsymbol {x}})\\right) ^ {- 1} \\tilde {J} (\\tilde {\\boldsymbol {x}}) ^ {\\intercal} \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}}). \\tag {38}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.414,
+                0.825,
+                0.493
+            ],
+            "angle": 0,
+            "content": "\\(\\tilde{J} (\\tilde{\\pmb{x}})\\left(\\tilde{J} (\\tilde{\\pmb{x}})^{\\intercal}\\tilde{J} (\\tilde{\\pmb{x}})\\right)^{-1}\\tilde{J} (\\tilde{\\pmb{x}})^{\\intercal}{\\bf y}_m(\\tilde{\\pmb{x}})\\) can be simplified. Note that in the Gauss-Newton step we are projecting \\(\\tilde{f} (\\tilde{\\pmb{x}})\\) onto \\(\\tilde{J} (\\tilde{\\pmb{x}})\\). \\(\\tilde{J} (\\tilde{\\pmb{x}})\\left(\\tilde{J} (\\tilde{\\pmb{x}})^{\\intercal}\\tilde{J} (\\tilde{\\pmb{x}})\\right)^{-1}\\tilde{J} (\\tilde{\\pmb{x}})^{\\intercal}\\) is a projection operator, so we are projecting two times. Therefore, there is another equivalent solution which we will further use for the sake of simplicity:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.37,
+                0.509,
+                0.826,
+                0.528
+            ],
+            "angle": 0,
+            "content": "\\[\n\\tilde {f} (\\tilde {\\boldsymbol {x}}) = \\tilde {J} (\\tilde {\\boldsymbol {x}}) \\left(\\tilde {\\boldsymbol {x}} - \\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}})\\right) + \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}}). \\tag {39}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.541,
+                0.795,
+                0.558
+            ],
+            "angle": 0,
+            "content": "Observe that substituting \\(\\tilde{f}\\) from Eq. (39) and Eq. (38) into Eq. (26) yields the same loss value."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.563,
+                0.545,
+                0.58
+            ],
+            "angle": 0,
+            "content": "Notably, \\(\\nabla \\tilde{f} (\\tilde{\\pmb{x}}) = \\tilde{J} (\\tilde{\\pmb{x}})\\) for a uniform distribution \\(p(\\cdot)\\)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.171,
+                0.6,
+                0.661,
+                0.616
+            ],
+            "angle": 0,
+            "content": "C PROBABILISTIC vs. SIMPLIFIED GAUSS-NEWTON LOSS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.631,
+                0.666,
+                0.647
+            ],
+            "angle": 0,
+            "content": "In this section, we compare our simplified Gauss-Newton loss formulation,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.271,
+                0.662,
+                0.826,
+                0.703
+            ],
+            "angle": 0,
+            "content": "\\[\n\\underset {\\hat {\\boldsymbol {f}}, \\hat {\\boldsymbol {J}}} {\\arg \\min } \\sum_ {j} \\left\\| \\tilde {\\boldsymbol {x}} - \\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal} \\left(\\hat {\\boldsymbol {f}} - F _ {q} ^ {(j)}\\right) - \\boldsymbol {x} ^ {(j)} \\right\\| _ {2} ^ {2} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}), \\tag {40}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.715,
+                0.771,
+                0.73
+            ],
+            "angle": 0,
+            "content": "to the original loss, which was derived through the maximum likelihood estimation (MLE),"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.185,
+                0.755,
+                0.826,
+                0.811
+            ],
+            "angle": 0,
+            "content": "\\[\n\\underset {\\hat {\\boldsymbol {f}}, \\hat {\\boldsymbol {J}}} {\\arg \\min } \\sum_ {j} \\left(\\left\\| \\hat {\\boldsymbol {J}} \\left(\\tilde {\\boldsymbol {x}} - \\left(\\hat {\\boldsymbol {J}} ^ {\\mathrm {T}} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\mathrm {T}} \\left(\\hat {\\boldsymbol {f}} - F _ {q} ^ {(j)}\\right) - \\boldsymbol {x} ^ {(j)}\\right) \\right\\| _ {2} ^ {2} - \\log \\det  (\\hat {\\boldsymbol {J}} ^ {\\mathrm {T}} \\hat {\\boldsymbol {J}})\\right) p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right). \\tag {41}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.817,
+                0.827,
+                0.889
+            ],
+            "angle": 0,
+            "content": "We note that the solution to the probabilistic loss is much more complicated and computationally expensive. Derivatives for this formulation are high-order polynomials in elements of \\(\\hat{\\pmb{f}}\\) and \\(\\hat{\\pmb{J}}\\). It could be solved by general methods of algebraic geometry like Gröbner basis or Homotopy continuation, but the existence of a closed-form solution is not guaranteed, and we were not able to find one."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.896,
+                0.827,
+                0.926
+            ],
+            "angle": 0,
+            "content": "Nevertheless, in order to compare the two formulations, we approached the problem numerically. Fig. 6 presents the results. We used the Aachen dataset with hloc poses to plot the recall of query"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "14"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.827,
+                0.149
+            ],
+            "angle": 0,
+            "content": "images as a function of rotation and translation errors. We observe that the accuracy difference between these two formulations is negligible in practice. However, the simplified version admits a closed-form solution and can be computed several orders of magnitudes more efficiently."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.179,
+                0.516,
+                0.194
+            ],
+            "angle": 0,
+            "content": "D A STUDY OF FEATURE DESCRIPTORS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.216,
+                0.829,
+                0.258
+            ],
+            "angle": 0,
+            "content": "We show that our approach is agnostic to the choice of the underlying feature descriptor. To analyze the accuracy of our algorithm in terms of the localization error, we use Cambridge Landmarks (Kendall et al., 2016), which provides ground-truth poses."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.258,
+                0.488,
+                0.382
+            ],
+            "angle": 0,
+            "content": "We experiment with four popular feature descriptors: SIFT (Lowe, 2004), SOLD2 (Pau-trat et al., 2021), SuperPoint (DeTone et al., 2018) and the recent SiLK (Gleize et al., 2023). We plug them in as the embeddings for interest points detected by SuperPoint. Fig. 7 plots recall (the percentage of successfully localized queries) as a function of the tolerated translation error."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.389,
+                0.49,
+                0.528
+            ],
+            "angle": 0,
+            "content": "We observe that SuperPoint outperforms all other feature descriptors. SOLD2 (Pautrat et al., 2021) is substantially more inferior in terms of accuracy of the pose estimates. Since it is designed to be a line descriptor, its representation of non-linear structures is not well-defined, thus such outcome is somewhat expected. Interestingly, SuperPoint and SIFT both surpass the more recent SiLK on this benchmark. In comparison to SuperPoint, SiLK has 128-D descrip"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.528,
+                0.755,
+                0.544
+            ],
+            "angle": 0,
+            "content": "tor, which suggests that it may be less expressive than the 256- \\(D\\) SuperPoint descriptors."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.511,
+                0.273,
+                0.807,
+                0.462
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.495,
+                0.472,
+                0.825,
+                0.502
+            ],
+            "angle": 0,
+            "content": "Figure 7: We study our approach with different feature descriptors on Cambridge Landmarks."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.574,
+                0.442,
+                0.589
+            ],
+            "angle": 0,
+            "content": "E IMPLEMENTATION DETAILS."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.611,
+                0.828,
+                0.751
+            ],
+            "angle": 0,
+            "content": "We implement our approach in PyTorch (Paszke et al., 2019). The primary focus of our implementation is the experimental validation of our derived results. Therefore, we did not optimize the runtime performance, which heavily depends on the number of interest points and 3D points involved in the computation. On average, the code takes approximately 6 seconds and 10 seconds per alignment on CMU and Aachen, respectively, on a single NVIDIA A4000. The scheduler in Algorithm 1 adapts the distribution \\( p(\\cdot) \\) as follows. We initialize \\( p(\\cdot) \\) with a truncated uniform distribution of a fixed radius around all interest points. The radius decreases from \\( 50\\% \\) of the image diagonal to \\( 5\\% \\) in the first 30 iterations. Afterward, the scheduler switches to the normal distribution around each interest point with standard deviation \\( \\sigma \\). Initially, we define \\( \\sigma \\) such that \\( 99\\% \\) of the distribution covers \\( 10\\% \\) of the image around the point, and we decrease the coverage ratio to \\( 1\\% \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.757,
+                0.828,
+                0.8
+            ],
+            "angle": 0,
+            "content": "Since our formulation does not have any outlier removal, we adopt the cut-off from the coarse tracker of DSO (Engel et al., 2016). The idea is to define a threshold that discards all but \\(20\\%\\) of the residuals with the lowest norm. In our experiments with SuperPoint, we set the threshold value to 0.4."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.83,
+                0.429,
+                0.846
+            ],
+            "angle": 0,
+            "content": "F EVALUATION ON 7SCENES"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.868,
+                0.826,
+                0.926
+            ],
+            "angle": 0,
+            "content": "Here, we complement our evaluation with the 7Scenes dataset. This dataset features substantial blur and distortions in some scenes, as it was captured with a rolling-shutter Kinect camera. Although SuperPoint features were not trained with such distortions, direct alignment with our closed-form solution remains competitive with PixLoc, both in terms of the median error and the recall Table 2."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "15"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.171,
+                0.101,
+                0.825,
+                0.143
+            ],
+            "angle": 0,
+            "content": "Table 2: 7Scenes (Shotton et al., 2013) evaluation and comparison to PixLoc. We report the median rotation and translation errors, as well as recall values at the specified thresholds of translation and rotation."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.189,
+                0.156,
+                0.812,
+                0.398
+            ],
+            "angle": 0,
+            "content": "<table><tr><td rowspan=\"2\">Scene</td><td rowspan=\"2\">Method</td><td rowspan=\"2\">Median error</td><td colspan=\"4\">Recall</td></tr><tr><td>(1cm,1°)</td><td>(5cm,5°)</td><td>(25cm,2°)</td><td>(50cm,5°)</td></tr><tr><td rowspan=\"2\">Heads</td><td>Ours</td><td>0.013m, 1.006°</td><td>24.80%</td><td>93.0%</td><td>86.30%</td><td>92.0%</td></tr><tr><td>Pixloc</td><td>0.013m, 0.863°</td><td>36.40%</td><td>85.60 %</td><td>84.00%</td><td>85.90%</td></tr><tr><td rowspan=\"2\">Office</td><td>Ours</td><td>0.028m, 0.941°</td><td>6.12 %</td><td>80.25%</td><td>92.55%</td><td>99.00%</td></tr><tr><td>Pixloc</td><td>0.026m, 0.792°</td><td>7.95%</td><td>80.70 %</td><td>93.12 %</td><td>96.85%</td></tr><tr><td rowspan=\"2\">Redkitchen</td><td>Ours</td><td>0.037m, 1.444°</td><td>1.56 %</td><td>64.42%</td><td>71.88%</td><td>90.16%</td></tr><tr><td>Pixloc</td><td>0.034m, 1.217°</td><td>3.74%</td><td>67.78 %</td><td>76.56 %</td><td>89.48%</td></tr><tr><td rowspan=\"2\">Pumpkin</td><td>Ours</td><td>0.049m, 1.555°</td><td>1.60 %</td><td>51.65%</td><td>62.65%</td><td>85.35%</td></tr><tr><td>Pixloc</td><td>0.041m, 1.173°</td><td>2.80%</td><td>59.75 %</td><td>71.00 %</td><td>84.40%</td></tr><tr><td rowspan=\"2\">Stairs</td><td>Ours</td><td>0.154m, 3.685°</td><td>1.90 %</td><td>19.90 %</td><td>25.60 %</td><td>60.50 %</td></tr><tr><td>Pixloc</td><td>0.048m, 1.268°</td><td>2.60%</td><td>51.10 %</td><td>59.10 %</td><td>74.70%</td></tr><tr><td rowspan=\"2\">Chess</td><td>Ours</td><td>0.026m, 0.904°</td><td>5.50 %</td><td>91.95%</td><td>95.45%</td><td>98.45%</td></tr><tr><td>Pixloc</td><td>0.024m, 0.812°</td><td>7.75%</td><td>90.75 %</td><td>94.95%</td><td>96.15%</td></tr><tr><td rowspan=\"2\">Fire</td><td>Ours</td><td>0.021m, 0.978°</td><td>10.40 %</td><td>90.65%</td><td>94.05%</td><td>98.45%</td></tr><tr><td>Pixloc</td><td>0.019m, 0.781°</td><td>15.85%</td><td>87.50 %</td><td>87.20%</td><td>90.10%</td></tr></table>"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.421,
+                0.422,
+                0.438
+            ],
+            "angle": 0,
+            "content": "G QUALITATIVE EXAMPLES"
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.171,
+                0.452,
+                0.825,
+                0.484
+            ],
+            "angle": 0,
+            "content": "Fig. 8 visualizes convergence examples of one scene with different pose initializations. The examples consistently demonstrate successful alignment despite suboptimal initialization of varying degree."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "16"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.174,
+                0.101,
+                0.825,
+                0.86
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.172,
+                0.872,
+                0.825,
+                0.929
+            ],
+            "angle": 0,
+            "content": "Figure 8: Examples of convergence from random initial poses. The green points are projections of 3D points using the current pose estimate; the red points are projections of the 3D points with the initial pose, and the yellow points denote the locations of interest points. It can be seen that despite these highly inaccurate initial poses, our approach converges to the correct solution."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.491,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "17"
+        }
+    ]
+]
\ No newline at end of file
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+# AN ANALYTICAL SOLUTION TO GAUSS-NEWTON LOSS FOR DIRECT IMAGE ALIGNMENT
+
+Sergei Solonets $^{1,2,*}$  Daniil Sinitsyn $^{1,2,*}$
+
+Lukas von Stumberg $^{3,\dagger}$  Nikita Araslanov $^{1,2}$  Daniel Cremers $^{1,2}$
+
+<sup>1</sup> Technical University of Munich  
+2 Munich Center for Machine Learning  
+3 Valve Software
+
+![](images/45ed605ea8fc0824ef2dac048c5d768f6e8e7ce2fe3fe94355da849bc9204e14.jpg)  
+Figure 1: Direct image alignment is a technique for aligning scenes based on image intensities. Recent learning-based methods seek to improve its success rate by increasing the convergence basin. We derive an analytical solution to the core idea of such methods, the Gauss-Newton loss, enabling fine-grained control over the basin of convergence. As a result, we can successfully align two scenes despite a highly imprecise initialization. From left to right, the example above illustrates the convergence of the reprojected keypoints (red) to the ground truth (green) by optimizing the SE(3) camera pose with our analytical solution. The blue color is the re-projection in the first iteration.
+
+![](images/eae3db6aee13ba2c8ff208b0c14ec4553a4a5e31b812be8896bd3841ed9f0658.jpg)
+
+![](images/35752841c5a829c17a1c6f48b0342e06ec16d815c6e46982b27bd0ed82a52af5.jpg)
+
+# ABSTRACT
+
+Direct image alignment is a widely used technique for relative 6DoF pose estimation between two images, but its accuracy strongly depends on pose initialization. Therefore, recent end-to-end frameworks increase the convergence basin of the learned feature descriptors with special training objectives, such as the Gauss-Newton loss. However, the training data may exhibit bias toward a specific type of motion and pose initialization, thus limiting the generalization of these methods. In this work, we derive a closed-form solution to the expected optimum of the Gauss-Newton loss. The solution is agnostic to the underlying feature representation and allows us to dynamically adjust the basin of convergence according to our assumptions about the uncertainty in the current estimates. These properties allow for effective control over the convergence in the alignment process. Despite using self-supervised feature embeddings, our solution achieves compelling accuracy w.r.t. the state-of-the-art direct image alignment methods trained end-to-end with pose supervision, and demonstrates improved robustness to pose initialization. Our analytical solution exposes some inherent limitations of end-to-end learning with the Gauss-Newton loss, and establishes an intriguing connection between direct image alignment and feature-matching approaches.
+
+# 1 INTRODUCTION
+
+Visual localization refers to estimating the camera pose of a query image w.r.t. a reference image where the underlying 3D structure (e.g. a point cloud) is available. Traditionally, solutions to visual localization primarily relied on estimating correspondences between 2D features in the query image and 3D features in the reference point cloud (Liu et al., 2017; Sarlin et al., 2019; Sattler et al., 2017;
+
+Svarm et al., 2017; Toft et al., 2018). Challenging this approach, recent deep learning frameworks implement direct image alignment by re-projecting the 3D points onto the feature map (Sarlin et al., 2021; von Stumberg et al., 2020a;b). Using end-to-end training, deep networks learn dense feature maps suitable for regressing the pose between an image and a point cloud. In analogy to photometric image alignment (Delaunoy and Pollefeys, 2014; Engel et al., 2016), this family of methods is referred to as feature metric image alignment (von Stumberg et al., 2020b).
+
+The feature representation learned by feature metric image alignment is not only discriminative, but is also spatially smooth for improved convergence (von Stumberg et al., 2020a). A training process enforces the smoothness either explicitly through a specific loss, such as the Gauss-Newton loss (von Stumberg et al., 2020a), or implicitly, by backpropagating through an optimization algorithm, such as Levenberg-Marquardt (Sarlin et al., 2021). In both cases, the resulting feature map embeds the bias of the initial poses in the training data. This may lead to poor generalization, since the training poses can differ substantially from the test scenario. We here take a less bias-prone approach: we use generic feature descriptors (e.g. obtained with self-supervision) and instead control the smoothness of the feature map dynamically at test time.
+
+This work investigates the connection between feature descriptor networks and feature metric image alignment. The main contribution is the analytical (closed-form) solution to the Gauss-Newton loss. On the one hand, this leads to a novel technique, which can utilize any feature descriptor to generate a dense feature map suitable for direct image alignment. Importantly, it allows us to dynamically adjust the smoothness of the feature map, which effectively controls the trade-off between the basin of convergence and the alignment accuracy. We empirically verify our derivation using self-supervised feature descriptors, such as SuperPoint (DeTone et al., 2018), and demonstrate on-par or even superior alignment accuracy compared to supervised frameworks. On the other hand, the analysis of our closed-form solution reveals inherent limitations of feature learning with backpropagation through Gauss-Newton optimization: feature metric alignment merely learns a form of interpolation between feature descriptors in the points of interest. Although we demonstrate this in the context of direct image alignment, a similar argument extends to other methods, even beyond computer vision, which backpropagate through Gauss-Newton or Levenberg-Marquardt optimization.
+
+# 2 PRELIMINARIES
+
+Image Alignment. Given two images (reference  $I_r$  and query  $I_q$ ) with known camera models and an overlapping field of view, and a 3D point cloud  $\{\pmb{p}^{(i)}\}$  in the coordinate system of  $I_r$ , the problem of image alignment is to estimate the relative camera pose  $T \in \mathbf{SE}(3)$ .
+
+Gauss-Newton Optimization. Given a set of functions  $\{r^{(1)}(\pmb {x}),\dots,r^{(m)}(\pmb {x})\}$  called residuals, Gauss-Newton (GN) optimization finds the parameters minimizing the sum of squared residuals:
+
+$$
+f (\boldsymbol {x}) = \sum_ {i} \| r ^ {(i)} (\boldsymbol {x}) \| _ {2} ^ {2}. \tag {1}
+$$
+
+Each residual  $r^{(i)}(\pmb{x})$  could either be a vector or a single-valued function. In both scenarios, the total residual  $r(\pmb{x})$  is a stacked vector of  $m$  residuals. The Gauss-Newton method seeks to minimize  $f(\pmb{x})$  by iteratively updating the parameter estimate  $\tilde{\pmb{x}}$ . Each iteration linearizes the residuals around the current estimate and computes an update step  $\Delta_{GN}(r(\tilde{\pmb{x}}))$ :
+
+$$
+\Delta_ {\mathrm {G N}} [ r (\tilde {\boldsymbol {x}}) ] = \left(\boldsymbol {J} ^ {\intercal} \boldsymbol {J}\right) ^ {- 1} \boldsymbol {J} ^ {\intercal} r (\tilde {\boldsymbol {x}}), \quad \boldsymbol {J} := \left. \frac {\partial r}{\partial \boldsymbol {x}} \right| _ {\boldsymbol {x} = \tilde {\boldsymbol {x}}}, \tag {2}
+$$
+
+where  $J$  is the Jacobian matrix. The estimate  $\tilde{\pmb{x}}$  evolves until convergence as
+
+$$
+\tilde {\boldsymbol {x}} \leftarrow \tilde {\boldsymbol {x}} \boxminus \Delta_ {\mathrm {G N}} [ r (\tilde {\boldsymbol {x}}) ]. \tag {3}
+$$
+
+Here, the operator  $\square$  denotes a specific update procedure, which depends on the nature of the optimization space. For linear spaces,  $\square$  simplifies to the standard subtraction operation; in the context of a rigid-body transformation  $\mathbf{SE}(3)$ , operator  $\square$  applies a tangential update.
+
+Photometric and feature metric image alignment. Photometric image alignment estimates the relative 6DoF pose by minimizing the difference between pixel intensities of points in  $I_{r}$  and the
+
+corresponding points in  $I_q$ . Recalling  $\pmb{p}^{(i)}$  as a 3D point in the coordinate frame of the reference image, we seek a transformation  $\pmb{T} \in \mathbf{SE}(3)$  minimizing the following residuals:
+
+$$
+r ^ {(i)} (\boldsymbol {T}) = I _ {r} \left(\langle \boldsymbol {p} ^ {(i)} \rangle\right) - I _ {q} \left(\langle \boldsymbol {T} \boldsymbol {p} ^ {(i)} \rangle\right). \tag {4}
+$$
+
+Here,  $\langle \cdot \rangle$  is a 2D projection operator of a 3D point onto the image plane. The projection implicitly uses the corresponding camera intrinsic parameters, assumed to be available for both  $I_{r}$  and  $I_{q}$ . We omit them in the notation for clarity. Comprising rotation  $\mathbf{R} \in \mathbf{SO}(3)$  and translation  $\pmb{t} \in \mathbb{R}^3$ , the target transformation  $\pmb{T}$  is typically found by minimizing Eq. (4) with non-linear least-squares optimization, such as Gauss-Newton or Levenberg-Marquardt (LM) methods (Levenberg, 1944; Marquardt, 1963; Nocedal and Wright, 1999).
+
+The success of photometric image alignment critically depends on a favorable initialization of the pose, especially in the conditions of varying illumination and occlusion. As a partial remedy, feature metric image alignment (Tang and Tan, 2019) uses feature maps instead of pixel intensities. Obtained with deep learning, such feature maps have increased convergence basin compared to that derived from image intensities, which improves robustness to pose initialization.
+
+# 3 RELATED WORK
+
+Direct and indirect image alignment. Estimating the relative camera pose from two images is a fundamental problem in computer vision, with applications in structure from motion (SfM) (Schönberger and Frahm, 2016), SLAM and relocalization. To address this problem, indirect (feature-based) approaches detect and match interest points (Bay et al., 2006; Lowe, 2004) in both images, and then estimate the pose using PnP (Persson and Nordberg, 2018) or by minimizing the reprojection errors (Triggs et al., 1999). In contrast, direct methods sidestep the matching process and minimize the photometric error instead (Horn and Jr., 1988; Irani and Anandan, 1999). This means that they can leverage the entire image (Kerl et al., 2013; Newcombe et al., 2011), or focus on pixels with sufficient gradient (Engel et al., 2014; 2016). The foundation behind these approaches is Lucas-Kanade tracking (Baker and Matthews, 2004; Lucas and Kanade, 1981). However, direct methods, which are central to this work, optimize for a 6DoF pose instead of individual pixel displacements (i.e. optical flow).
+
+Pose estimation with deep learning. The advent of deep learning revitalized interest in improving pose estimation with deep networks. While some approaches are holistic (Jatavallabhula et al., 2020), there are broadly three categories of learning-based methods. i) Fully end-to-end pose estimation methods (Kendall et al., 2016; Ummenhofer et al., 2017; Zhou et al., 2017) directly regress pose estimates with deep neural networks, instead of test-time optimization. ii) Learning-based indirect methods (DeTone et al., 2018; Dusmanu et al., 2019; Revaud et al., 2019; Yi et al., 2016) replace handcrafted detectors and descriptors with deep representations in an indirect pipeline. Some approaches further extend the traditional way of obtaining correspondences. SuperGlue (Sarlin et al., 2020) learns feature matching with a graph neural network. LoFTR (Sun et al., 2021) directly regresses correspondences instead of relying on separate feature detection and matching. Similar to indirect methods, iii) learning-based direct image alignment enhances classical direct methods with deep features. This category is the most similar to our work and we discuss it in more detail next.
+
+Learned features for direct image alignment. Previous work differs in their approach to model training and in the final task. For example, Czarnowski et al. (2017) leverage off-the-shelf CNN features to improve optical flow tracking. A number of methods (Han et al., 2018; Lv et al., 2019; Sarlin et al., 2021; Tang and Tan, 2019; Xu et al., 2021) train feature descriptors end-to-end with ground-truth poses and backpropagate the gradient through a non-linear optimization process. At test time, these methods employ a feature pyramid and refine the initial camera pose in a coarse-to-fine fashion using GN or LM optimization. In addition to the feature pyramid, some methods predict additional properties for image alignment, such as uncertainty (Xu et al., 2021) (Sarlin et al., 2021), Jacobians (Han et al., 2018), or optimization parameters (e.g. damping factors (Lv et al., 2019; Sarlin et al., 2021)). In the same spirit, our formulation leads to a continuous image pyramid, where each level of the pyramid can be generated on-the-fly based on the input level of uncertainty.
+
+Another line of work (von Stumberg et al., 2020a;b) trains a deep network directly on the ground-truth pixel correspondences. GN-Net (von Stumberg et al., 2020a) minimizes two loss functions. The first is a contrastive loss (Schmidt et al., 2017) facilitating discriminative properties of the
+
+feature representation. The second loss function accounts for a likely displacement in the initial correspondence, implemented by adding random 2D offsets to the ground-truth correspondences at training time. By learning to minimize the feature discrepancy after a Gauss-Newton step, the model learns to cope with the initially noisy estimates. We revisit this work in detail in the Sec. 4.
+
+The learning-based approaches (e.g. (Germain et al., 2021; Sarlin et al., 2021; von Stumberg et al., 2020a)) achieve impressive accuracy of pose estimation. Nevertheless, they require pose supervision for training and struggle in scenarios of large-baseline localization. Indeed, GN-Net (von Stumberg et al., 2020a) and PixLoc (Sarlin et al., 2021) tend to exhibit a strong bias toward the noise assumptions of the training process, which cannot be easily reversed. For example, PixLoc trained on the CMU dataset exhibits a strong bias toward horizontal movement as illustrated in Appendix A.
+
+Our work takes a different approach. We rely on existing feature descriptors obtained with self-supervision, which contribute no explicit motion bias to the alignment process. We next derive a closed-form solution to the Gauss-Newton loss (von Stumberg et al., 2020a) as a functional of a probability density governing the noise assumptions of the current pose estimate. This allows us to adjust the noise assumptions in the alignment process, thus effectively controlling the convergence basin at test time, much akin to the coarse-to-fine strategy, exemplified by Fig. 2.
+
+# 4 THE GAUSS-NEWTON LOSS
+
+In this section, we recap the GN-Net (von Stumberg et al., 2020a) and introduce the notation. GN-Net is a convolutional neural network  $E(\cdot ;\pmb {\theta})$  trained on a sparse set of ground-truth correspondences. Given coordinates on the image plane, let  $f:\mathbb{R}\times \mathbb{R}\to \mathbb{R}^d$  continuously map those coordinates to a descriptor from a feature grid  $\mathbb{R}^{d\times w\times h}$  produced by  $E(\cdot ;\pmb {\theta})$ , e.g. using bilinear interpolation. We define  $f_{r}\coloneqq E(I_{r};\pmb {\theta})$  and  $f_{q}\coloneqq E(I_{q};\pmb{\theta})$  to denote feature representations of reference and query images. GN-Net learns parameters  $\pmb{\theta}$  to minimize the expected value of the following loss function,
+
+$$
+\mathcal {L} _ {\mathrm {G N} - \text {N e t}} \left(I _ {r}, I _ {q}\right) = \mathcal {L} _ {\text {c o n t r a s t i v e}} \left(f _ {r}, f _ {q}\right) + \mathcal {L} _ {\mathrm {G N}} \left(f _ {r}, f _ {q}\right), \tag {5}
+$$
+
+which comprises a contrastive loss,  $\mathcal{L}_{\mathrm{contrastive}}(\cdot ,\cdot)$ , and the Gauss-Newton loss,  $\mathcal{L}_{\mathrm{GN}}(\cdot ,\cdot)$ . The contrastive loss minimizes the distance between the features of two corresponding points while maximizing the distance between non-corresponding pairs (Schmidt et al., 2017). The contrastive loss facilitates spatially discriminative features, and its particular instantiation has little significance for the following discussion (e. g. GN-Net uses the triplet loss).
+
+The Gauss-Newton loss  $\mathcal{L}_{\mathrm{GN}}$  ensures that the feature map is sufficiently smooth for direct image alignment, thus enlarging the convergence basin. GN-Net implements this by adding a random offset to the ground-truth correspondences and encouraging a single Gauss-Newton step to recover the original location. Let us formalize this process.
+
+Given a ground-truth correspondence  $(\pmb{x}^{(i)},\pmb{y}^{(i)})$  between images  $I_{r}$  and  $I_{q}$ , the assumption behind the Gauss-Newton loss is that the initial estimate  $\tilde{\pmb{x}}^{(i)}$  at test time falls in the vicinity of the ground truth  $\pmb{x}^{(i)}$ , i.e.  $\tilde{\pmb{x}}^{(i)} = \pmb{x}^{(i)} + \pmb{\epsilon}$ , where  $\pmb{\epsilon}$  follows some predefined distribution  $p(\pmb{\epsilon})$ , such as a Gaussian with zero mean. At training time, the Gauss-Newton loss aims at recovering the ground-truth location  $\pmb{x}^{(i)}$  from a noisy initial location  $\tilde{\pmb{x}}^{(i)}$  by minimizing the residual w.r.t.  $\pmb{x}$ :
+
+$$
+r ^ {(i)} \left(\tilde {\boldsymbol {x}} ^ {(i)}\right) := f _ {r} \left(\tilde {\boldsymbol {x}} ^ {(i)}\right) - f _ {q} \left(\boldsymbol {y} ^ {(i)}\right), \tag {6}
+$$
+
+The original Gauss-Newton loss for one point  $\pmb{x}^{(i)}$ , as introduced by von Stumberg et al. (2020a), is
+
+$$
+\mathcal {L} _ {\mathrm {G N} _ {o}} ^ {(i)} \left(f _ {r}, f _ {q}, \epsilon\right) := \| J (\epsilon - \Delta_ {\mathrm {G N}} [ r ^ {(i)} \left(\boldsymbol {x} ^ {(i)} + \epsilon\right) ]) \| _ {2} ^ {2} - \log \det  \boldsymbol {J} ^ {\intercal} \boldsymbol {J}. \tag {7}
+$$
+
+From a probabilistic standpoint (von Stumberg et al., 2020a), the loss balances between the accuracy of the Gauss-Newton step and the direction uncertainty. Here,  $J^{\mathsf{T}}J$  represents the inverse covariance matrix propagated through the photometric residuals. The loss function corresponds to the negative log-likelihood of residuals distributed as  $\mathcal{N}\left(0,\left(\mathbf{J}^{\mathsf{T}}\mathbf{J}\right)^{-1}\right)$ . In this work, we consider a variant of the Gauss-Newton loss, in which the covariance matrix is assumed to be identity. Thus, Eq. (7) becomes
+
+$$
+\mathcal {L} _ {\mathrm {G N}} ^ {(i)} \left(f _ {r}, f _ {q}, \boldsymbol {\epsilon}\right) = \| \boldsymbol {\epsilon} - \Delta_ {\mathrm {G N}} \left[ r ^ {(i)} \left(\boldsymbol {x} ^ {(i)} + \boldsymbol {\epsilon}\right) \right] \| _ {2} ^ {2}. \tag {8}
+$$
+
+Although this simplified version does not account for the trade-off between uncertainty and accuracy of the prediction, it admits a closed-form minimizer of the expected value, as we show in Sec. 5. We also find that the simplified version does not differ from the original one empirically in a significant way (see Appendix C). Henceforth, we will refer to the simplified version as the Gauss-Newton loss.
+
+The training process calculates the Gauss-Newton loss stochastically by sampling  $\epsilon$  from  $p(\epsilon)$ . Therefore, it minimizes a Monte-Carlo approximation to the expected value of  $\mathcal{L}_{\mathrm{GN}}^{(i)}$  summed over all ground-truth correspondences,
+
+$$
+\mathcal {L} _ {\mathrm {G N}} \left(f _ {r}, f _ {q}; p\right) = \mathbb {E} _ {\epsilon \sim p} \left[ \sum_ {i} \mathcal {L} _ {\mathrm {G N}} ^ {(i)} \left(f _ {r}, f _ {q}, \epsilon\right) \right]. \tag {9}
+$$
+
+Hereafter, we use the notation  $\mathcal{L}_{\mathrm{GN}}(f_r,f_q;p)$  to emphasize the dependence of the Gauss-Newton loss on the noise distribution  $p$ , and omit this parametarization otherwise to avoid clutter.
+
+Note that the GN-Net's training stage addresses the problem of optical flow, not pose estimation. The underlying assumption is that accurate optical flow facilitates pose estimation, as each pixel will contribute to the final pose estimate. At test time, the pose is determined by solving the featuremetric image alignment problem (equivalent to Eq. (4)) using Gauss-Newton optimization in SE(3).
+
+# 5 CLOSED-FORM GAUSS-NEWTON STEP
+
+Decoupling the contrastive and Gauss-Newton losses. The contrastive loss provides a sparse constraint on the feature embeddings, since we can only use sparse ground-truth correspondences, the interest points, for supervision. By contrast, the Gauss-Newton loss enforces a pre-defined basin of convergence around each interest point (modeled by  $p(\epsilon)$ ) with little effect on the feature descriptors in the interest points. This is because the residual in Eq. (6) between the corresponding interest points will be negligible, if the contrastive loss for those points is minimized. It follows that the contrastive and the Gauss-Newton loss essentially optimize over a disjoint set of feature locations. Therefore, we can decouple the Gauss-Newton loss from the joint optimization objective in Eq. (5). Let us formalize this reasoning. We aim to solve:
+
+$$
+f _ {q} ^ {*} = \underset {f _ {q}} {\arg \min } \left[ \mathcal {L} _ {\text {c o n t r a s t i v e}} \left(f _ {r}, f _ {q}\right) + \mathcal {L} _ {\mathrm {G N}} \left(f _ {r}, f _ {q}\right) \right]. \tag {10}
+$$
+
+We denote the values of  $f_{r}$ ,  $f_{q}$  in the interest points as  $F_{r}$ ,  $F_{q}$ :  $F_{r}^{(i)} := f_{r}(\pmb{x}^{(i)})$ ,  $F_{q}^{(i)} := f_{q}(\pmb{y}^{(i)})$ . The contrastive loss only depends on  $F_{r}$  and  $F_{q}$ , while the Gauss-Newton loss depends on all the query feature values  $f_{q}$  and the interest points in the reference map  $F_{r}$ . By eliminating unused parts, we introduce an equivalent problem:
+
+$$
+\left\{F _ {q} ^ {*}, f _ {q} ^ {*} \right\} = \underset {F _ {q}, f _ {q}} {\arg \min } \left[ \mathcal {L} _ {\text {c o n t r a s t i v e}} \left(F _ {r}, F _ {q}\right) + \mathcal {L} _ {\mathrm {G N}} \left(F _ {r}, f _ {q}\right) \right]. \tag {11}
+$$
+
+Next, we approximate  $F_{r}$  with  $F_{q}$  in the second term. This is permissible as the contrastive loss acts as a soft constraint, ensuring that  $F_{r}$  and  $F_{q}$  are close at the optimum of the joint loss function. This allows us to decouple the minimization problem as follows:
+
+$$
+F _ {q} ^ {*} = \underset {F _ {q}} {\arg \min } \left[ \mathcal {L} _ {\text {c o n t r a s t i v e}} \left(F _ {r}, F _ {q}\right) + \underset {f _ {q}} {\min } \mathcal {L} _ {\mathrm {G N}} \left(F _ {q}, f _ {q}\right) \right], \tag {12}
+$$
+
+$$
+f_{q}^{*} = \operatorname *{arg  min}_{f_{q}}\mathcal{L}_{\mathrm{GN}}(F_{q}^{*},f_{q}).
+$$
+
+We denote
+
+$$
+G(F_{q};p):= \operatorname *{arg  min}_{f_{q}}\mathcal{L}_{\mathrm{GN}}(F_{q},f_{q};p),
+$$
+
+$$
+\mathcal {L} _ {\mathrm {G N}} ^ {*} (F _ {q}; p) := \min  _ {f _ {q}} \mathcal {L} _ {\mathrm {G N}} (F _ {q}, f _ {q}; p) = \mathcal {L} _ {\mathrm {G N}} (F _ {q}, G (F _ {q}; p); p). \tag {15}
+$$
+
+![](images/826e241123540c27976e7973fa798f729216c0e1cfe8f9f04f0420c3578f2cb0.jpg)  
+Figure 2: Controlling the basin of convergence. Image samples and the corresponding feature maps  $\tilde{f}_q$  (PCA) for different values of  $\sigma$  and isotropic Gaussian distribution  $p$ .
+
+$G(F_{q};p)$  is the optimal reconstruction of a feature map under sparse feature descriptors  $F_{q}$  and noise density  $p(\epsilon)$ .  $\mathcal{L}_{\mathrm{GN}}^{*}(F_{q};p)$  is the corresponding value of the Gauss-Newton loss under  $F_{q}$ . We will refer to  $G(F_{q};p)$  as  $f_{q}$  to denote an optimal reconstruction under an arbitrary sparse input  $F_{q}$ . Note the difference to  $f_{q}^{*}$ , which is an optimal reconstruction under the joint loss, i.e.  $G(F_{q}^{*};p)$ . Our main contribution, detailed shortly, is a closed-form solution for both  $G(F_{q};p)$  and  $\mathcal{L}_{\mathrm{GN}}^{*}(F_{q};p)$ . It reveals that the original problem in Eq. (10) actually depends only on the feature values in the interest points:
+
+$$
+F _ {q} ^ {*} = \underset {F _ {q}} {\arg \min } \left[ \mathcal {L} _ {\text {c o n t r a s t i v e}} \left(F _ {r}, F _ {q}\right) + \mathcal {L} _ {\mathrm {G N}} ^ {*} \left(F _ {q}; p\right) \right]. \tag {14}
+$$
+
+This means that in training a feature extractor only the features of interest points matter, while the representation of other pixels approximates our analytical solution. As the empirical validation, we demonstrate compelling results in our experiments by using feature descriptors for the interest points from a self-supervised method, while employing our analytical solution for the remaining pixels.
+
+An analytical solution to the Gauss-Newton loss. We propose to calculate the expectation of the Gauss-Newton loss in Eq. (9) as a functional of  $f_{q}$ , and analytically find its closed-form minimizer.
+
+Consider the query image  $I_{q}$  as a set  $\{(\pmb{x}^{(j)}, F_{q}^{(j)})\}$  containing locations of interest points  $\pmb{x}^{(j)}$  and the corresponding feature descriptors  $F_{q}^{(j)} \coloneqq f_{q}(\pmb{x}^{(j)})$ . The locations  $\pmb{x}^{(j)}$  can be extracted by an off-the-shelf feature detector (e.g. SuperPoint (DeTone et al., 2018)), while a network trained with the contrastive loss in Eq. (5) can produce the corresponding descriptors  $F_{q}^{(j)}$ .
+
+Let us first re-write the expectation in Eq. (9) using Eq. (8) and Eq. (2) in the decoupled formulation of the Gauss-Newton loss (cf. Eq. (12)):
+
+$$
+\mathcal {L} _ {\mathrm {G N}} \left(F _ {q}, f _ {q}; p\right) = \sum_ {j} \int_ {\Omega} \left\| \boldsymbol {\epsilon} - \left(\boldsymbol {J} ^ {\intercal} \boldsymbol {J}\right) ^ {- 1} \boldsymbol {J} ^ {\intercal} \left(f _ {q} \left(\boldsymbol {x} ^ {(j)} + \boldsymbol {\epsilon}\right) - F _ {q} ^ {(j)}\right) \right\| _ {2} ^ {2} p (\boldsymbol {\epsilon}) d \boldsymbol {\epsilon}. \tag {15}
+$$
+
+Substituting  $\pmb {\epsilon} = \tilde{\pmb{x}} -\pmb{x}^{(j)}$  , we obtain
+
+$$
+\mathcal {L} _ {\mathrm {G N}} \left(F _ {q}, f _ {q}; p\right) = \sum_ {j} \int_ {\Omega} \left\| \tilde {\boldsymbol {x}} - \left(\boldsymbol {J} ^ {\intercal} \boldsymbol {J}\right) ^ {- 1} \boldsymbol {J} ^ {\intercal} \left(f _ {q} (\tilde {\boldsymbol {x}}) - F _ {q} ^ {(j)}\right) - \boldsymbol {x} ^ {(j)} \right\| _ {2} ^ {2} p \left(\tilde {\boldsymbol {x}} - \boldsymbol {x} ^ {(j)}\right) d \tilde {\boldsymbol {x}}. \tag {16}
+$$
+
+We first consider all  $\tilde{\pmb{x}}\in \Omega$  independently and relax the problem by eliminating the constraint  $J = \frac{\partial f_q}{\partial\tilde{\pmb{x}}}$ . This allows us to derive analytical solutions for minimizers  $\tilde{f}_q(\cdot)$  and  $\tilde{J}_q(\cdot)$  as functions
+
+![](images/51567faa20beee434f4e3537a45a5498ca3a8c44185318e8178f84be77c1745e.jpg)  
+Figure 3: Image alignment with continuous probabilistic feature pyramids. Given reference image with a 3D point cloud and a query image paired with an initial coarse pose, we start by estimating interest points and their features on the query image, along with features from the projections of the 3D point cloud onto the reference image. Next, we perform feature metric alignment between the 3D point features and derived analytical continuous image pyramid.
+
+of  $\tilde{\pmb{x}}$ . We then observe that  $\frac{\partial}{\partial\tilde{\pmb{x}}}\tilde{f}_q(\cdot)$  coincides with  $\tilde{J}_q(\cdot)$  for a uniform distribution, which confirms that in this case, our derived solution is the solution to the original problem. As a side note, in the context of direct image alignment, it has been suggested that decoupling and predicting the Jacobian independently from the function value may yield superior convergence and results (Han et al., 2018). Appendix B provides the full derivation. Below, we summarize the closed-form minimizer to Eq. (16):
+
+$$
+\tilde {f} _ {q} (\tilde {\boldsymbol {x}}) = \tilde {J} _ {q} (\tilde {\boldsymbol {x}}) \left(\tilde {\boldsymbol {x}} - \mathbf {x} _ {m} (\tilde {\boldsymbol {x}})\right) + \mathbf {y} _ {m} (\tilde {\boldsymbol {x}}), \tag {17}
+$$
+
+$$
+\tilde {J} _ {q} (\tilde {\boldsymbol {x}}) = \left(\operatorname {C o v} _ {\mathbf {x y}} (\tilde {\boldsymbol {x}}) \operatorname {C o v} _ {\mathbf {y}} (\tilde {\boldsymbol {x}}) ^ {- 1}\right) ^ {+}, \tag {18}
+$$
+
+$$
+\mathbf {y} _ {m} (\tilde {\boldsymbol {x}}) = \sum_ {j} F _ {q} ^ {(j)} p \left(\tilde {\boldsymbol {x}} - \boldsymbol {x} ^ {(j)}\right) / \sum_ {j} p \left(\tilde {\boldsymbol {x}} - \boldsymbol {x} ^ {(j)}\right), \tag {19}
+$$
+
+$$
+\mathbf {x} _ {m} (\tilde {\boldsymbol {x}}) = \sum_ {j} \boldsymbol {x} ^ {(j)} p \left(\tilde {\boldsymbol {x}} - \boldsymbol {x} ^ {(j)}\right) / \sum_ {j} p \left(\tilde {\boldsymbol {x}} - \boldsymbol {x} ^ {(j)}\right), \tag {20}
+$$
+
+$$
+\operatorname {C o v} _ {\mathbf {x y}} (\tilde {\boldsymbol {x}}) = \sum_ {j} \left(\boldsymbol {x} ^ {(j)} - \mathbf {x} _ {m} (\tilde {\boldsymbol {x}})\right) \left(F _ {q} ^ {(j)} - \mathbf {y} _ {m} (\tilde {\boldsymbol {x}})\right) ^ {\intercal} p (\tilde {\boldsymbol {x}} - \boldsymbol {x} ^ {(j)}), \tag {21}
+$$
+
+$$
+\operatorname {C o v} _ {\mathbf {y}} (\tilde {\boldsymbol {x}}) = \sum_ {j} \left(F _ {q} ^ {(j)} - \mathbf {y} _ {m} (\tilde {\boldsymbol {x}})\right) \left(F _ {q} ^ {(j)} - \mathbf {y} _ {m} (\tilde {\boldsymbol {x}})\right) ^ {\intercal} p (\tilde {\boldsymbol {x}} - \boldsymbol {x} ^ {(j)}), \tag {22}
+$$
+
+where  $(^{+})$  denotes the Moore-Penrose pseudo-inverse operator. The optimal point is  $G(F_{q};p)\coloneqq \tilde{f}_{q}$  and the loss value at this point (cf. Eq. (13)) is
+
+$$
+\mathcal {L} _ {G N} ^ {*} (F _ {q}) = \int_ {\Omega} \frac {1}{2} \operatorname {T r} \left[ \operatorname {C o v} _ {\mathbf {x}} (\tilde {\boldsymbol {x}}) - \operatorname {C o v} _ {\mathbf {x y}} (\tilde {\boldsymbol {x}}) \operatorname {C o v} _ {\mathbf {y}} (\tilde {\boldsymbol {x}}) ^ {- 1} \operatorname {C o v} _ {\mathbf {y x}} (\tilde {\boldsymbol {x}}) \right] d \tilde {\boldsymbol {x}}. \tag {23}
+$$
+
+Our derivation of the optimal embeddings  $G(F_{q};p)$  using fixed interest points  $F_{q}$  provides an interesting insight into end-to-end learning frameworks. Jointly training both losses requires inverting a high-dimensional matrix  $\mathrm{Cov}_{\mathbf{y}}(\tilde{\boldsymbol{x}})$  in Eq. (23), which poses high numerical instability and discontinuities. Since previous work (von Stumberg et al., 2020a) can be seen as stochastic approximations to our solution, this may explain the reported training instability and divergence in those works. Additionally, the form of  $\tilde{f}_q$  in Eq. (17) suggests that end-to-end pipelines may not necessarily yield any sophisticated representation, as they merely interpolate between features in the interest points.
+
+FeatureMetric Image Alignment. As a case study, we employ the analytical form of  $\tilde{f}_q$  and  $\tilde{J}_q$  for feature metric image alignment. Algorithm 1 and Fig. 3 provide an overview. Fig. 1 illustrates stages of direct alignment using our probabilistically reconstructed feature maps. We first extract the interest points  $\{\pmb{x}^{(j)}\}$  and the corresponding feature descriptors  $\{F_q^{(j)}\}$  from  $I_{q}$  to reconstruct  $\tilde{f}_q$  and
+
+Require:  $I_r, I_q, \{\pmb{p}^{(i)}\}, \pmb{T}^{(0)}$
+
+1:  $\{\pmb{x}^{(j)}\} \gets$  InterestPoints  $(I_q)$  
+2:  $\{F_q^{(j)}\} \gets E(I_q;\pmb {\theta})\circ \{\pmb{x}^{(j)}\}$  
+3:  $\{\pmb{o}^{(i)}\} \gets E(I_r;\pmb {\theta})\circ \{\langle \pmb{p}^{(i)}\rangle \}$  
+4:  $\pmb{T} \gets \pmb{T}^{(0)}$  
+5: for  $n$  from 0 to  $N_{\mathrm{max}}$  do  
+6:  $p\gets \mathrm{Schedule}(n)$  
+7:  $\tilde{f}_q\gets G(\{F_q^{(j)}\} ;p)$  
+8:  $\{\tilde{\pmb{p}}^{(i)}\} \gets \{\pmb {T}\pmb{p}^{(i)}\}$  
+9:  $\pmb{\delta} \gets \Delta_{GN}(\tilde{f}_q \circ \{\langle \tilde{\pmb{p}}^{(i)}\rangle\} - \{\pmb{o}^{(i)}\})$  
+0:  $\pmb{T}\gets \pmb{T}\boxminus \pmb{\delta}$
+
+11: end for  
+12: return T
+
+$\langle \cdot \rangle$  denotes 2D projection.
+
+Algorithm 1: Image alignment.
+
+![](images/b1d8d6315ec75466d97ddac206aa02c2a3166ccf6d617791a310cf5e9a3c44d8.jpg)  
+Figure 4: Robustness to initialization. Our closed-form solution exhibits significantly greater robustness compared to PixLoc.
+
+$\tilde{J}_q$  using Eqs. (17) and (18). The sparse set of 3D points  $\{\pmb{p}^{(i)}\}$  is projected onto the reference image to obtain reference descriptors. We obtain feature descriptors for these points using the same feature embedding network  $E(\cdot ;\theta)$ . Starting with the initial camera pose  $\pmb{T}^{(0)}\in \mathbf{S}\mathbf{E}(3)$ , at every iteration we perform Gauss-Newton steps minimizing the residuals  $r^{(i)}$ , w.r.t. the camera transform  $\pmb{T}$ :
+
+$$
+r ^ {(i)} (\boldsymbol {T}) := F _ {r} ^ {(i)} - \tilde {f} _ {q} \left(\left\langle \boldsymbol {T} \boldsymbol {p} ^ {(i)} \right\rangle\right). \tag {24}
+$$
+
+$p(\epsilon)$  explicitly controls the basin of convergence. In practice, we can use any distribution with high initial variance based on the inaccuracy assumed in the initial point projections. As our estimates improve over the course of optimization, we gradually decrease the variance. Notably, the choice of  $p(\epsilon)$  lets us define values for points outside of the image boundaries. Fig. 2 illustrates examples of the feature maps  $\tilde{f}_q$  with  $p$  following a Gaussian distribution of increasing variance.
+
+Comparison to previous work. Our derivation is agnostic to the underlying feature extractor and can operate with off-the-shelf feature descriptors. An interesting property of the derived  $\tilde{f}_q$  is that it enables continuous coarse-to-fine alignment. By adjusting the noise prior  $p$  in the scheduler (cf. Algorithm 1), we can control the basin of convergence dynamically at runtime. We can adjust the distribution  $p$  either by choosing a different parameter set of a pre-defined distribution, or by changing the distribution itself to another family. In practice, we start the optimization with a uniform distribution and then switch to the Gaussian distribution with a slowly decreasing variance.
+
+A uniform distribution with a wide support offers a large basin of convergence, which sacrifices accuracy for robustness. The ensuing Gaussian distribution with decreasing variance refines the pose and leads to a more accurate solution. Appendix E provides implementation details of the scheduler.
+
+Connection between feature matching and feature metric image alignment. Note that the computation of  $\tilde{f}_q$  using Eq. (17) only depends on the feature descriptors of the interest points in the query image. This observation suggests an interesting interpretation of minimizing the residual in Eq. (24) using  $\tilde{f}_q$  with Gauss-Newton optimization as feature matching. The Gauss-Newton step is fully determined by the neighboring points of interest. Consequently, it implies that the effectiveness of such optimization-based methods may be limited in comparison to alternative optimization-inspired approaches, which learn the optimization step (Teed and Deng, 2020).
+
+# 6 EXPERIMENTS
+
+Datasets. We evaluate our approach on two most popular datasets for large-scale image localization, namely the Aachen Day-Night dataset (Sattler et al., 2018), extended CMU seasons (Toft et al., 2022) and 7Scenes dataset (Shotton et al., 2013). Aachen Day-Night consists of 98 night and 824 day query
+
+Table 1: Camera localization on Aachen Day-Night and CMU Seasons. Our evaluation shows an overall improved accuracy in diverse scenarios despite using self-supervised SuperPoint descriptors. We compare to the state-of-the-art methods supervised with pose: GN-Net (von Stumberg et al., 2020a), LM-Reloc (von Stumberg et al., 2020b) and PixLoc (Sarlin et al., 2021).  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Aachen</td><td colspan="3">CMU Seasons</td></tr><tr><td>Day</td><td>Night</td><td>Urban</td><td>Suburban</td><td>Park</td></tr><tr><td>GN-Net</td><td>62.4/69.4/76.9</td><td>49.0/58.2/66.3</td><td>75.4/79.9/92.6</td><td>64.7/67.0/81.4</td><td>46.7/48.3/65.3</td></tr><tr><td>LM-Reloc</td><td>60.4/68.0/76.3</td><td>37.8/46.9/59.2</td><td>76.6/82.5/93.4</td><td>67.3/72.0/82.8</td><td>49.1/53.4/66.9</td></tr><tr><td>PixLoc</td><td>64.3/69.3/77.4</td><td>51.0/55.1/67.3</td><td>88.3/90.4/93.7</td><td>79.6/81.1/85.2</td><td>61.0/62.5/69.4</td></tr><tr><td>Ours (SuperPoint)</td><td>66.3/72.5/78.8</td><td>43.9/50.0/56.1</td><td>86.0/90.6/95.2</td><td>79.8/85.0/92.4</td><td>63.4/67.9/77.5</td></tr></table>
+
+images. Extended CMU seasons consists of 14 slices, 5 for urban environment, 5 for suburban and 4 for park. Each slice contains between 3000 and 5000 query images. The 7Scenes dataset comprises seven distinct scenes, each containing multiple sequences of 500 to 1000 frames. For our ablation study in Appendix D we used Cambridge Landmarks dataset (Kendall et al., 2016).
+
+Results. Table 1 presents the results for camera localization on Aachen Day-Night (Sattler et al., 2018) and the extended CMU Seasons (Toft et al., 2022). We use SuperPoint (DeTone et al., 2018) as the feature descriptor in these experiments. In each setting (Day/Night for Aachen; Urban/Suburban/Park for CMU Seasons), we report three numbers, which indicate the percentage of the query images that were successfully localized within the specified translation and rotation thresholds. We adopt the standard threshold values defined by the benchmarks (translation, rotation):  $(0.25\mathrm{m}, 2^{\circ}) / (0.5\mathrm{m}, 5^{\circ}) / (5\mathrm{m}, 10^{\circ})$ . We adopted the implementation from PixLoc (Sarlin et al., 2021) to run these benchmarks. A comparison with PixLoc on the 7Scenes dataset is presented in Appendix F.
+
+We observe that our solution, despite using self-supervised descriptors from SuperPoint (DeTone et al., 2018), achieves an overall strong localization accuracy across diverse settings in comparison to supervised frameworks. Aachen Day-Night has significant occlusions, hence many outliers. Since the Gauss-Newton loss does not incorporate any outlier filtering, we do not expect high accuracy for our approach on this dataset. Nevertheless, we were surprised to find that on Aachen Day our approach even slightly surpassed previous state of the art. On Aachen Night, our accuracy is inferior to previous work, however. This is somewhat expected, since SuperPoint feature descriptors were not directly trained on day-night correspondences. By contrast, PixLoc was trained with supervision on day-night image pairs, which provides an obvious advantage. On CMU Seasons, our approach demonstrates a clear improvement over previous feature metric alignment methods.
+
+The proposed approach demonstrates notable robustness to initialization noise. We compared the proposed scheme and PixLoc on the Cambridge Landmarks dataset, varying the levels of random noise applied to the ground-truth pose as illustrated in Fig. 4 and Appendix G.
+
+Overall, these results empirically confirm the validity of our closed-form derivation. Furthermore, the comparison suggests a limited benefit of end-to-end learning with the Gauss-Newton loss. By contrast, a dynamic convergence basin offered by our analytical solution provides versatility w.r.t. the underlying feature descriptors and improves robustness to pose initialization, as a result.
+
+# 7 CONCLUSION
+
+We derived a closed-form solution to the Gauss-Newton loss in the context of direct image alignment, which offers two main advantages. First, it allows for dynamic control of the convergence basin, which improves robustness of the alignment to pose initialization. Furthermore, despite using self-supervised descriptors, such control leads to compelling accuracy of pose estimates in comparison to supervised pipelines on established benchmarks. Second, our derivation exposes intrinsic limitations of employing the Gauss-Newton loss in deep learning, as it only leads to a form of interpolation between the feature descriptors in the interest points. This insight offers an interesting connection between direct image alignment and feature matching, and leads to a novel perspective on learning robust features end-to-end, which we will investigate in future work.
+
+# ACKNOWLEDGMENTS
+
+This work was supported by the ERC Advanced Grant SIMULACRON and by TUM Georg Nemetschek Institute under the project AI4TWINNING. We thank Linus Harenstam-Nielsen for helpful discussions and proofreading.
+
+# REPRODUCIBILITY STATEMENT
+
+The main contribution of this work is the derivation of the closed-form solution to the Gauss-Newton loss. We provide details of this derivation in Appendix B. To reproduce our experimental results, we elaborate on the implementation details Appendix E. To facilitate reproducibility in future research, we also publicly release our code.
+
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+
+# A OPTICAL FLOW BIAS
+
+We demonstrate that PixLoc exhibits bias inherited from the training dataset. Fig. 5 illustrates the average translation error after optimization when evaluated on the Cambridge dataset, highlighting the impact of different initial displacements along the image axes. To solely evaluate the bias learned by the feature maps, we disabled dataset-specific learned movement priors denoted as  $\lambda$  in PixLoc (Sarlin et al., 2021). Our results demonstrate that when trained on the CMU dataset, the network predicts optical flow along the X axis more accurately. This increased accuracy is directly linked to the structure of the CMU dataset. Specifically, the dataset comprises image pairs captured by a camera inside a moving car, pointed towards the side of the road. Consequently, this positioning results in the network predominantly learning horizontal optical flow patterns, reflecting the lateral movement observed in the images of the training dataset.
+
+![](images/5a280a00dfe2fc2939830a6e761e02fd0b13169e9961761b13fb7f9bb0a85a77.jpg)  
+Figure 5: The final error of the PixLoc solution along  $X / Y$ -axes w.r.t. the initial error. PixLoc remains accurate only along  $X$ -axis on CMU, reflecting the dominant motion in the training set.
+
+# B DERIVATION OF THE ANALYTICAL SOLUTION
+
+In this section, we provide a solution to minimizing Eq. (16) from the main text. Let  $\Omega$  be the coordinate domain of an image plane  $[0,1] \times [0,1]$ , and let  $E \subseteq \mathbb{R}^d$  define the feature space. We aim to find  $\tilde{f}: \Omega \to E$ :
+
+$$
+\tilde {f} = \arg \min  _ {f} \sum_ {j} \int_ {\Omega} \left\| \tilde {\boldsymbol {x}} - (J (\tilde {\boldsymbol {x}}) ^ {\intercal} J (\tilde {\boldsymbol {x}})) ^ {- 1} J (\tilde {\boldsymbol {x}}) ^ {\intercal} (f (\tilde {\boldsymbol {x}}) - F _ {q} ^ {(j)}) - \boldsymbol {x} ^ {(j)} \right\| _ {2} ^ {2} p (\tilde {\boldsymbol {x}} - \boldsymbol {x} ^ {(j)}) d \tilde {\boldsymbol {x}}. \tag {25}
+$$
+
+We start by relaxing the constraint  $\nabla f(\pmb{x}) = J(\pmb{x})$ . This makes every value of  $f(\pmb{x})$  and  $J(\pmb{x})$  locally independent and, therefore, the minimum is achieved by minimizing Eq. (25) independently over every point in  $\Omega$ :
+
+$$
+\{\tilde {f} (\tilde {\boldsymbol {x}}), \tilde {J} (\tilde {\boldsymbol {x}}) \} = \underset {\hat {\boldsymbol {f}}, \hat {\boldsymbol {J}}} {\arg \min } \sum_ {j} \left\| \tilde {\boldsymbol {x}} - \left(\hat {\boldsymbol {J}} ^ {\intercal} \hat {\boldsymbol {J}}\right) ^ {- 1} \hat {\boldsymbol {J}} ^ {\intercal} \left(\hat {\boldsymbol {f}} - F _ {q} ^ {(j)}\right) - \boldsymbol {x} ^ {(j)} \right\| _ {2} ^ {2} p (\tilde {\boldsymbol {x}} - \boldsymbol {x} ^ {(j)}). \tag {26}
+$$
+
+Note that the only significant part of  $\hat{\pmb{f}}$  is a part which is spanned by columns of  $\hat{\pmb{J}}$ . This can be easily seen by observing that  $\hat{\pmb{J}}^{\mathrm{T}}\hat{\pmb{f}}$  is a non-normalized projection onto  $\hat{\pmb{J}}$ . More formally, let us find  $\hat{\pmb{f}}$  as
+
+$\hat{J}a + \hat{J}^\perp b$ , where  $\hat{J}^\perp$  is a basis for an orthogonal complement for  $\operatorname{span}(\hat{J})$ . By substituting this into Eq. (26), we obtain:
+
+$$
+\{\tilde {a} (\tilde {\boldsymbol {x}}), \tilde {J} (\tilde {\boldsymbol {x}}) \} = \underset {\boldsymbol {a}, \hat {\boldsymbol {J}}} {\arg \min } \sum_ {j} \left\| \tilde {\boldsymbol {x}} - \boldsymbol {x} ^ {(j)} - \boldsymbol {a} + \left(\hat {\boldsymbol {J}} ^ {\intercal} \hat {\boldsymbol {J}}\right) ^ {- 1} \hat {\boldsymbol {J}} ^ {\intercal} F _ {q} ^ {(j)} \right\| _ {2} ^ {2} p (\tilde {\boldsymbol {x}} - \boldsymbol {x} ^ {(j)}), \tag {27}
+$$
+
+$$
+\tilde {f} (\tilde {\boldsymbol {x}}) \in \left\{\tilde {J} (\tilde {\boldsymbol {x}}) \tilde {a} (\tilde {\boldsymbol {x}}) + \tilde {J} (\tilde {\boldsymbol {x}}) ^ {\perp} \boldsymbol {b} \mid \boldsymbol {b} \in \mathbb {R} ^ {d - 2} \right\}.
+$$
+
+Observe that this problem is a linear least squares in  $\pmb{a}$  and  $\left(\hat{\boldsymbol{J}}^{\intercal}\hat{\boldsymbol{J}}\right)^{-1}\hat{\boldsymbol{J}}^{\intercal}$  and recall that  $p(\cdot)\geq 0$  (by definition), hence the problem is convex. Minimality conditions for Eq. (27) for  $\pmb{a}$  and  $\left(\hat{\boldsymbol{J}}^{\intercal}\hat{\boldsymbol{J}}\right)^{-1}\hat{\boldsymbol{J}}^{\intercal}$  are:
+
+$$
+\frac {d}{d \boldsymbol {a}}: \quad \sum_ {j} \left(\tilde {\boldsymbol {x}} - \boldsymbol {x} ^ {(j)} - \boldsymbol {a} + \left(\hat {\boldsymbol {J}} ^ {\intercal} \hat {\boldsymbol {J}}\right) ^ {- 1} \hat {\boldsymbol {J}} ^ {\intercal} F _ {q} ^ {(j)}\right) p (\tilde {\boldsymbol {x}} - \boldsymbol {x} ^ {(j)}) = 0, \tag {28}
+$$
+
+$$
+\frac {d}{d \left(\hat {\boldsymbol {J}} ^ {\intercal} \hat {\boldsymbol {J}}\right) ^ {- 1} \hat {\boldsymbol {J}} ^ {\intercal}}: \quad \sum_ {j} \left(\tilde {\boldsymbol {x}} - \boldsymbol {x} ^ {(j)} - \boldsymbol {a} + \left(\hat {\boldsymbol {J}} ^ {\intercal} \hat {\boldsymbol {J}}\right) ^ {- 1} \hat {\boldsymbol {J}} ^ {\intercal} F _ {q} ^ {(j)}\right) F _ {q} ^ {(j) \intercal} p \left(\tilde {\boldsymbol {x}} - \boldsymbol {x} ^ {(j)}\right) = 0. \tag {29}
+$$
+
+From Eq. (28), we have:
+
+$$
+\boldsymbol {a} = \tilde {\boldsymbol {x}} - \mathbf {x} _ {m} (\tilde {\boldsymbol {x}}) + \left(\hat {\boldsymbol {J}} ^ {\intercal} \hat {\boldsymbol {J}}\right) ^ {- 1} \hat {\boldsymbol {J}} ^ {\intercal} \mathbf {y} _ {m} (\tilde {\boldsymbol {x}}), \tag {30}
+$$
+
+where
+
+$$
+\mathbf {y} _ {m} (\tilde {\boldsymbol {x}}) := \sum_ {j} F _ {q} ^ {(j)} p \left(\tilde {\boldsymbol {x}} - \boldsymbol {x} ^ {(j)}\right) / \sum_ {j} p \left(\tilde {\boldsymbol {x}} - \boldsymbol {x} ^ {(j)}\right), \tag {31}
+$$
+
+$$
+\mathbf {x} _ {m} (\tilde {\boldsymbol {x}}) := \sum_ {j} \boldsymbol {x} ^ {(j)} p (\tilde {\boldsymbol {x}} - \boldsymbol {x} ^ {(j)}) / \sum_ {j} p (\tilde {\boldsymbol {x}} - \boldsymbol {x} ^ {(j)}).
+$$
+
+Substituting  $\pmb{a}$  into Eq. (29), we obtain
+
+$$
+\sum_ {j} \left(\mathbf {x} _ {m} (\tilde {\boldsymbol {x}}) - \boldsymbol {x} ^ {(j)} + \left(\hat {\boldsymbol {J}} ^ {\intercal} \hat {\boldsymbol {J}}\right) ^ {- 1} \hat {\boldsymbol {J}} ^ {\intercal} \left(F _ {q} ^ {(j)} - \mathbf {y} _ {m} (\tilde {\boldsymbol {x}})\right)\right) F _ {q} ^ {(j) \intercal} p \left(\tilde {\boldsymbol {x}} - \boldsymbol {x} ^ {(j)}\right) = 0, \tag {32}
+$$
+
+$$
+- \sum_ {j} \left(\boldsymbol {x} ^ {(j)} - \mathbf {x} _ {m} (\tilde {\boldsymbol {x}})\right) F _ {q} ^ {(j) \intercal} p (\tilde {\boldsymbol {x}} - \boldsymbol {x} ^ {(j)}) +
+$$
+
+$$
+\left(\hat {\boldsymbol {J}} ^ {\intercal} \hat {\boldsymbol {J}}\right) ^ {- 1} \hat {\boldsymbol {J}} ^ {\intercal} \sum_ {i} \left(F _ {q} ^ {(j)} - \mathbf {y} _ {m} (\tilde {\boldsymbol {x}})\right) F _ {q} ^ {(j) \intercal} p \left(\tilde {\boldsymbol {x}} - \boldsymbol {x} ^ {(j)}\right) = 0, \tag {33}
+$$
+
+$$
+- \operatorname {C o v} _ {\mathbf {x y}} (\tilde {\boldsymbol {x}}) + \left(\hat {\boldsymbol {J}} ^ {\intercal} \hat {\boldsymbol {J}}\right) ^ {- 1} \hat {\boldsymbol {J}} ^ {\intercal} \operatorname {C o v} _ {\mathbf {y}} (\tilde {\boldsymbol {x}}) = 0, \tag {34}
+$$
+
+where
+
+$$
+\operatorname {C o v} _ {\mathbf {x y}} (\tilde {\boldsymbol {x}}) := \sum_ {j} \left(\boldsymbol {x} ^ {(j)} - \mathbf {x} _ {m} (\tilde {\boldsymbol {x}})\right) \left(F _ {q} ^ {(j)} - \mathbf {y} _ {m} (\tilde {\boldsymbol {x}})\right) ^ {\intercal} p (\tilde {\boldsymbol {x}} - \boldsymbol {x} ^ {(j)}), \tag {35}
+$$
+
+$$
+\operatorname {C o v} _ {\mathbf {y}} (\tilde {\boldsymbol {x}}) := \sum_ {j} \left(F _ {q} ^ {(j)} - \mathbf {y} _ {m} (\tilde {\boldsymbol {x}})\right) \left(F _ {q} ^ {(j)} - \mathbf {y} _ {m} (\tilde {\boldsymbol {x}})\right) ^ {\intercal} p (\tilde {\boldsymbol {x}} - \boldsymbol {x} ^ {(j)}). \tag {36}
+$$
+
+Solving Eq. (34) for  $J$  and substituting the solution into Eq. (27) results in
+
+$$
+\tilde {J} (\tilde {\boldsymbol {x}}) = \left(\operatorname {C o v} _ {\mathbf {x y}} (\tilde {\boldsymbol {x}}) \operatorname {C o v} _ {\mathbf {y}} (\tilde {\boldsymbol {x}}) ^ {- 1}\right) ^ {\dagger}, \tag {37}
+$$
+
+![](images/d412210a2a978a178e742f9fe1f64233bc326303b34641ab01d95fed24968989.jpg)  
+Figure 6: Comparison of rotation and translation recalls for our simplified version of GN-loss and probabilistic one
+
+![](images/1c7bf869c947ac562c24b1af137e85b4fbb4042c99ddf07d99863b671a717ddf.jpg)
+
+$$
+\tilde {f} (\tilde {\boldsymbol {x}}) = \tilde {J} (\tilde {\boldsymbol {x}}) \boldsymbol {a} = \tilde {J} (\tilde {\boldsymbol {x}}) \left(\tilde {\boldsymbol {x}} - \mathbf {x} _ {m} (\tilde {\boldsymbol {x}})\right) + \tilde {J} (\tilde {\boldsymbol {x}}) \left(\tilde {J} (\tilde {\boldsymbol {x}}) ^ {\intercal} \tilde {J} (\tilde {\boldsymbol {x}})\right) ^ {- 1} \tilde {J} (\tilde {\boldsymbol {x}}) ^ {\intercal} \mathbf {y} _ {m} (\tilde {\boldsymbol {x}}). \tag {38}
+$$
+
+$\tilde{J} (\tilde{\pmb{x}})\left(\tilde{J} (\tilde{\pmb{x}})^{\intercal}\tilde{J} (\tilde{\pmb{x}})\right)^{-1}\tilde{J} (\tilde{\pmb{x}})^{\intercal}{\bf y}_m(\tilde{\pmb{x}})$  can be simplified. Note that in the Gauss-Newton step we are projecting  $\tilde{f} (\tilde{\pmb{x}})$  onto  $\tilde{J} (\tilde{\pmb{x}})$ .  $\tilde{J} (\tilde{\pmb{x}})\left(\tilde{J} (\tilde{\pmb{x}})^{\intercal}\tilde{J} (\tilde{\pmb{x}})\right)^{-1}\tilde{J} (\tilde{\pmb{x}})^{\intercal}$  is a projection operator, so we are projecting two times. Therefore, there is another equivalent solution which we will further use for the sake of simplicity:
+
+$$
+\tilde {f} (\tilde {\boldsymbol {x}}) = \tilde {J} (\tilde {\boldsymbol {x}}) \left(\tilde {\boldsymbol {x}} - \mathbf {x} _ {m} (\tilde {\boldsymbol {x}})\right) + \mathbf {y} _ {m} (\tilde {\boldsymbol {x}}). \tag {39}
+$$
+
+Observe that substituting  $\tilde{f}$  from Eq. (39) and Eq. (38) into Eq. (26) yields the same loss value.
+
+Notably,  $\nabla \tilde{f} (\tilde{\pmb{x}}) = \tilde{J} (\tilde{\pmb{x}})$  for a uniform distribution  $p(\cdot)$ .
+
+# C PROBABILISTIC vs. SIMPLIFIED GAUSS-NEWTON LOSS
+
+In this section, we compare our simplified Gauss-Newton loss formulation,
+
+$$
+\underset {\hat {\boldsymbol {f}}, \hat {\boldsymbol {J}}} {\arg \min } \sum_ {j} \left\| \tilde {\boldsymbol {x}} - \left(\hat {\boldsymbol {J}} ^ {\intercal} \hat {\boldsymbol {J}}\right) ^ {- 1} \hat {\boldsymbol {J}} ^ {\intercal} \left(\hat {\boldsymbol {f}} - F _ {q} ^ {(j)}\right) - \boldsymbol {x} ^ {(j)} \right\| _ {2} ^ {2} p (\tilde {\boldsymbol {x}} - \boldsymbol {x} ^ {(j)}), \tag {40}
+$$
+
+to the original loss, which was derived through the maximum likelihood estimation (MLE),
+
+$$
+\underset {\hat {\boldsymbol {f}}, \hat {\boldsymbol {J}}} {\arg \min } \sum_ {j} \left(\left\| \hat {\boldsymbol {J}} \left(\tilde {\boldsymbol {x}} - \left(\hat {\boldsymbol {J}} ^ {\mathrm {T}} \hat {\boldsymbol {J}}\right) ^ {- 1} \hat {\boldsymbol {J}} ^ {\mathrm {T}} \left(\hat {\boldsymbol {f}} - F _ {q} ^ {(j)}\right) - \boldsymbol {x} ^ {(j)}\right) \right\| _ {2} ^ {2} - \log \det  (\hat {\boldsymbol {J}} ^ {\mathrm {T}} \hat {\boldsymbol {J}})\right) p \left(\tilde {\boldsymbol {x}} - \boldsymbol {x} ^ {(j)}\right). \tag {41}
+$$
+
+We note that the solution to the probabilistic loss is much more complicated and computationally expensive. Derivatives for this formulation are high-order polynomials in elements of  $\hat{\pmb{f}}$  and  $\hat{\pmb{J}}$ . It could be solved by general methods of algebraic geometry like Gröbner basis or Homotopy continuation, but the existence of a closed-form solution is not guaranteed, and we were not able to find one.
+
+Nevertheless, in order to compare the two formulations, we approached the problem numerically. Fig. 6 presents the results. We used the Aachen dataset with hloc poses to plot the recall of query
+
+images as a function of rotation and translation errors. We observe that the accuracy difference between these two formulations is negligible in practice. However, the simplified version admits a closed-form solution and can be computed several orders of magnitudes more efficiently.
+
+# D A STUDY OF FEATURE DESCRIPTORS
+
+We show that our approach is agnostic to the choice of the underlying feature descriptor. To analyze the accuracy of our algorithm in terms of the localization error, we use Cambridge Landmarks (Kendall et al., 2016), which provides ground-truth poses.
+
+We experiment with four popular feature descriptors: SIFT (Lowe, 2004), SOLD2 (Pau-trat et al., 2021), SuperPoint (DeTone et al., 2018) and the recent SiLK (Gleize et al., 2023). We plug them in as the embeddings for interest points detected by SuperPoint. Fig. 7 plots recall (the percentage of successfully localized queries) as a function of the tolerated translation error.
+
+We observe that SuperPoint outperforms all other feature descriptors. SOLD2 (Pautrat et al., 2021) is substantially more inferior in terms of accuracy of the pose estimates. Since it is designed to be a line descriptor, its representation of non-linear structures is not well-defined, thus such outcome is somewhat expected. Interestingly, SuperPoint and SIFT both surpass the more recent SiLK on this benchmark. In comparison to SuperPoint, SiLK has 128-D descrip
+
+tor, which suggests that it may be less expressive than the 256-  $D$  SuperPoint descriptors.
+
+![](images/b8e0fb26806c5bac90e295b1be48088253a34441dee7c04d809eabe9864177d8.jpg)  
+Figure 7: We study our approach with different feature descriptors on Cambridge Landmarks.
+
+# E IMPLEMENTATION DETAILS.
+
+We implement our approach in PyTorch (Paszke et al., 2019). The primary focus of our implementation is the experimental validation of our derived results. Therefore, we did not optimize the runtime performance, which heavily depends on the number of interest points and 3D points involved in the computation. On average, the code takes approximately 6 seconds and 10 seconds per alignment on CMU and Aachen, respectively, on a single NVIDIA A4000. The scheduler in Algorithm 1 adapts the distribution  $p(\cdot)$  as follows. We initialize  $p(\cdot)$  with a truncated uniform distribution of a fixed radius around all interest points. The radius decreases from  $50\%$  of the image diagonal to  $5\%$  in the first 30 iterations. Afterward, the scheduler switches to the normal distribution around each interest point with standard deviation  $\sigma$ . Initially, we define  $\sigma$  such that  $99\%$  of the distribution covers  $10\%$  of the image around the point, and we decrease the coverage ratio to  $1\%$ .
+
+Since our formulation does not have any outlier removal, we adopt the cut-off from the coarse tracker of DSO (Engel et al., 2016). The idea is to define a threshold that discards all but  $20\%$  of the residuals with the lowest norm. In our experiments with SuperPoint, we set the threshold value to 0.4.
+
+# F EVALUATION ON 7SCENES
+
+Here, we complement our evaluation with the 7Scenes dataset. This dataset features substantial blur and distortions in some scenes, as it was captured with a rolling-shutter Kinect camera. Although SuperPoint features were not trained with such distortions, direct alignment with our closed-form solution remains competitive with PixLoc, both in terms of the median error and the recall Table 2.
+
+Table 2: 7Scenes (Shotton et al., 2013) evaluation and comparison to PixLoc. We report the median rotation and translation errors, as well as recall values at the specified thresholds of translation and rotation.  
+
+<table><tr><td rowspan="2">Scene</td><td rowspan="2">Method</td><td rowspan="2">Median error</td><td colspan="4">Recall</td></tr><tr><td>(1cm,1°)</td><td>(5cm,5°)</td><td>(25cm,2°)</td><td>(50cm,5°)</td></tr><tr><td rowspan="2">Heads</td><td>Ours</td><td>0.013m, 1.006°</td><td>24.80%</td><td>93.0%</td><td>86.30%</td><td>92.0%</td></tr><tr><td>Pixloc</td><td>0.013m, 0.863°</td><td>36.40%</td><td>85.60 %</td><td>84.00%</td><td>85.90%</td></tr><tr><td rowspan="2">Office</td><td>Ours</td><td>0.028m, 0.941°</td><td>6.12 %</td><td>80.25%</td><td>92.55%</td><td>99.00%</td></tr><tr><td>Pixloc</td><td>0.026m, 0.792°</td><td>7.95%</td><td>80.70 %</td><td>93.12 %</td><td>96.85%</td></tr><tr><td rowspan="2">Redkitchen</td><td>Ours</td><td>0.037m, 1.444°</td><td>1.56 %</td><td>64.42%</td><td>71.88%</td><td>90.16%</td></tr><tr><td>Pixloc</td><td>0.034m, 1.217°</td><td>3.74%</td><td>67.78 %</td><td>76.56 %</td><td>89.48%</td></tr><tr><td rowspan="2">Pumpkin</td><td>Ours</td><td>0.049m, 1.555°</td><td>1.60 %</td><td>51.65%</td><td>62.65%</td><td>85.35%</td></tr><tr><td>Pixloc</td><td>0.041m, 1.173°</td><td>2.80%</td><td>59.75 %</td><td>71.00 %</td><td>84.40%</td></tr><tr><td rowspan="2">Stairs</td><td>Ours</td><td>0.154m, 3.685°</td><td>1.90 %</td><td>19.90 %</td><td>25.60 %</td><td>60.50 %</td></tr><tr><td>Pixloc</td><td>0.048m, 1.268°</td><td>2.60%</td><td>51.10 %</td><td>59.10 %</td><td>74.70%</td></tr><tr><td rowspan="2">Chess</td><td>Ours</td><td>0.026m, 0.904°</td><td>5.50 %</td><td>91.95%</td><td>95.45%</td><td>98.45%</td></tr><tr><td>Pixloc</td><td>0.024m, 0.812°</td><td>7.75%</td><td>90.75 %</td><td>94.95%</td><td>96.15%</td></tr><tr><td rowspan="2">Fire</td><td>Ours</td><td>0.021m, 0.978°</td><td>10.40 %</td><td>90.65%</td><td>94.05%</td><td>98.45%</td></tr><tr><td>Pixloc</td><td>0.019m, 0.781°</td><td>15.85%</td><td>87.50 %</td><td>87.20%</td><td>90.10%</td></tr></table>
+
+# G QUALITATIVE EXAMPLES
+
+Fig. 8 visualizes convergence examples of one scene with different pose initializations. The examples consistently demonstrate successful alignment despite suboptimal initialization of varying degree.
+
+![](images/c980c871e37d2586de9c72c848c094ef39349877409313a22f28b077fa752e49.jpg)  
+Figure 8: Examples of convergence from random initial poses. The green points are projections of 3D points using the current pose estimate; the red points are projections of the 3D points with the initial pose, and the yellow points denote the locations of interest points. It can be seen that despite these highly inaccurate initial poses, our approach converges to the correct solution.
\ No newline at end of file
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+                                    "content": "ABSTRACT"
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+                                    "content": "Direct image alignment is a widely used technique for relative 6DoF pose estimation between two images, but its accuracy strongly depends on pose initialization. Therefore, recent end-to-end frameworks increase the convergence basin of the learned feature descriptors with special training objectives, such as the Gauss-Newton loss. However, the training data may exhibit bias toward a specific type of motion and pose initialization, thus limiting the generalization of these methods. In this work, we derive a closed-form solution to the expected optimum of the Gauss-Newton loss. The solution is agnostic to the underlying feature representation and allows us to dynamically adjust the basin of convergence according to our assumptions about the uncertainty in the current estimates. These properties allow for effective control over the convergence in the alignment process. Despite using self-supervised feature embeddings, our solution achieves compelling accuracy w.r.t. the state-of-the-art direct image alignment methods trained end-to-end with pose supervision, and demonstrates improved robustness to pose initialization. Our analytical solution exposes some inherent limitations of end-to-end learning with the Gauss-Newton loss, and establishes an intriguing connection between direct image alignment and feature-matching approaches."
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+                            ]
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+                                    "type": "text",
+                                    "content": "1 INTRODUCTION"
+                                }
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+                                    "type": "text",
+                                    "content": "Visual localization refers to estimating the camera pose of a query image w.r.t. a reference image where the underlying 3D structure (e.g. a point cloud) is available. Traditionally, solutions to visual localization primarily relied on estimating correspondences between 2D features in the query image and 3D features in the reference point cloud (Liu et al., 2017; Sarlin et al., 2019; Sattler et al., 2017;"
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "type": "text",
+                                    "content": "*Equal contribution. Correspondence: {s.solonets,d.sinitsyn}@tum.de."
+                                }
+                            ]
+                        }
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+                                    "type": "text",
+                                    "content": "†Work done while at TU Munich."
+                                }
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+                        }
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+                                    "type": "text",
+                                    "content": "Project code: https://github.com/tum-vision/gn_loss_analytical."
+                                }
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+                                    "type": "text",
+                                    "content": "Svarm et al., 2017; Toft et al., 2018). Challenging this approach, recent deep learning frameworks implement direct image alignment by re-projecting the 3D points onto the feature map (Sarlin et al., 2021; von Stumberg et al., 2020a;b). Using end-to-end training, deep networks learn dense feature maps suitable for regressing the pose between an image and a point cloud. In analogy to photometric image alignment (Delaunoy and Pollefeys, 2014; Engel et al., 2016), this family of methods is referred to as feature metric image alignment (von Stumberg et al., 2020b)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
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+                {
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+                                    "type": "text",
+                                    "content": "The feature representation learned by feature metric image alignment is not only discriminative, but is also spatially smooth for improved convergence (von Stumberg et al., 2020a). A training process enforces the smoothness either explicitly through a specific loss, such as the Gauss-Newton loss (von Stumberg et al., 2020a), or implicitly, by backpropagating through an optimization algorithm, such as Levenberg-Marquardt (Sarlin et al., 2021). In both cases, the resulting feature map embeds the bias of the initial poses in the training data. This may lead to poor generalization, since the training poses can differ substantially from the test scenario. We here take a less bias-prone approach: we use generic feature descriptors (e.g. obtained with self-supervision) and instead control the smoothness of the feature map dynamically at test time."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
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+                {
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+                                    "type": "text",
+                                    "content": "This work investigates the connection between feature descriptor networks and feature metric image alignment. The main contribution is the analytical (closed-form) solution to the Gauss-Newton loss. On the one hand, this leads to a novel technique, which can utilize any feature descriptor to generate a dense feature map suitable for direct image alignment. Importantly, it allows us to dynamically adjust the smoothness of the feature map, which effectively controls the trade-off between the basin of convergence and the alignment accuracy. We empirically verify our derivation using self-supervised feature descriptors, such as SuperPoint (DeTone et al., 2018), and demonstrate on-par or even superior alignment accuracy compared to supervised frameworks. On the other hand, the analysis of our closed-form solution reveals inherent limitations of feature learning with backpropagation through Gauss-Newton optimization: feature metric alignment merely learns a form of interpolation between feature descriptors in the points of interest. Although we demonstrate this in the context of direct image alignment, a similar argument extends to other methods, even beyond computer vision, which backpropagate through Gauss-Newton or Levenberg-Marquardt optimization."
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+                                    "type": "text",
+                                    "content": "2 PRELIMINARIES"
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+                                    "content": "Image Alignment. Given two images (reference "
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+                                    "type": "text",
+                                    "content": ") with known camera models and an overlapping field of view, and a 3D point cloud "
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+                                    "content": "\\{\\pmb{p}^{(i)}\\}"
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+                                    "type": "text",
+                                    "content": " in the coordinate system of "
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+                                    "content": ", the problem of image alignment is to estimate the relative camera pose "
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+                                    "content": "T \\in \\mathbf{SE}(3)"
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+                                    "type": "text",
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+                                    "content": "Gauss-Newton Optimization. Given a set of functions "
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+                                    "content": " called residuals, Gauss-Newton (GN) optimization finds the parameters minimizing the sum of squared residuals:"
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+                                    "content": " could either be a vector or a single-valued function. In both scenarios, the total residual "
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+                                    "content": " by iteratively updating the parameter estimate "
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+                                    "type": "text",
+                                    "content": ". Each iteration linearizes the residuals around the current estimate and computes an update step "
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+                                    "content": "\\tilde {\\boldsymbol {x}} \\leftarrow \\tilde {\\boldsymbol {x}} \\boxminus \\Delta_ {\\mathrm {G N}} [ r (\\tilde {\\boldsymbol {x}}) ]. \\tag {3}",
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+                                    "content": " applies a tangential update."
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+                                    "content": "Photometric and feature metric image alignment. Photometric image alignment estimates the relative 6DoF pose by minimizing the difference between pixel intensities of points in "
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+                                    "content": "corresponding points in "
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+                                    "type": "text",
+                                    "content": ". Recalling "
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+                                    "type": "inline_equation",
+                                    "content": "\\pmb{p}^{(i)}"
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+                                    "type": "text",
+                                    "content": " as a 3D point in the coordinate frame of the reference image, we seek a transformation "
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+                                    "type": "inline_equation",
+                                    "content": "\\pmb{T} \\in \\mathbf{SE}(3)"
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+                                    "type": "text",
+                                    "content": " minimizing the following residuals:"
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+                                    "type": "interline_equation",
+                                    "content": "r ^ {(i)} (\\boldsymbol {T}) = I _ {r} \\left(\\langle \\boldsymbol {p} ^ {(i)} \\rangle\\right) - I _ {q} \\left(\\langle \\boldsymbol {T} \\boldsymbol {p} ^ {(i)} \\rangle\\right). \\tag {4}",
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+                                    "type": "text",
+                                    "content": "Here, "
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+                                    "type": "inline_equation",
+                                    "content": "\\langle \\cdot \\rangle"
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " is a 2D projection operator of a 3D point onto the image plane. The projection implicitly uses the corresponding camera intrinsic parameters, assumed to be available for both "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "I_{r}"
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+                                    "type": "text",
+                                    "content": " and "
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+                                {
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+                                    "type": "inline_equation",
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+                                        206
+                                    ],
+                                    "type": "text",
+                                    "content": ". We omit them in the notation for clarity. Comprising rotation "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{R} \\in \\mathbf{SO}(3)"
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+                                    "content": " and translation "
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+                                    "type": "inline_equation",
+                                    "content": "\\pmb{t} \\in \\mathbb{R}^3"
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+                                    "type": "text",
+                                    "content": ", the target transformation "
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+                                {
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+                                    "type": "inline_equation",
+                                    "content": "\\pmb{T}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " is typically found by minimizing Eq. (4) with non-linear least-squares optimization, such as Gauss-Newton or Levenberg-Marquardt (LM) methods (Levenberg, 1944; Marquardt, 1963; Nocedal and Wright, 1999)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
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+                                    "type": "text",
+                                    "content": "The success of photometric image alignment critically depends on a favorable initialization of the pose, especially in the conditions of varying illumination and occlusion. As a partial remedy, feature metric image alignment (Tang and Tan, 2019) uses feature maps instead of pixel intensities. Obtained with deep learning, such feature maps have increased convergence basin compared to that derived from image intensities, which improves robustness to pose initialization."
+                                }
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+                                    "type": "text",
+                                    "content": "3 RELATED WORK"
+                                }
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+                        }
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+                },
+                {
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+                                    "type": "text",
+                                    "content": "Direct and indirect image alignment. Estimating the relative camera pose from two images is a fundamental problem in computer vision, with applications in structure from motion (SfM) (Schönberger and Frahm, 2016), SLAM and relocalization. To address this problem, indirect (feature-based) approaches detect and match interest points (Bay et al., 2006; Lowe, 2004) in both images, and then estimate the pose using PnP (Persson and Nordberg, 2018) or by minimizing the reprojection errors (Triggs et al., 1999). In contrast, direct methods sidestep the matching process and minimize the photometric error instead (Horn and Jr., 1988; Irani and Anandan, 1999). This means that they can leverage the entire image (Kerl et al., 2013; Newcombe et al., 2011), or focus on pixels with sufficient gradient (Engel et al., 2014; 2016). The foundation behind these approaches is Lucas-Kanade tracking (Baker and Matthews, 2004; Lucas and Kanade, 1981). However, direct methods, which are central to this work, optimize for a 6DoF pose instead of individual pixel displacements (i.e. optical flow)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
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+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "Pose estimation with deep learning. The advent of deep learning revitalized interest in improving pose estimation with deep networks. While some approaches are holistic (Jatavallabhula et al., 2020), there are broadly three categories of learning-based methods. i) Fully end-to-end pose estimation methods (Kendall et al., 2016; Ummenhofer et al., 2017; Zhou et al., 2017) directly regress pose estimates with deep neural networks, instead of test-time optimization. ii) Learning-based indirect methods (DeTone et al., 2018; Dusmanu et al., 2019; Revaud et al., 2019; Yi et al., 2016) replace handcrafted detectors and descriptors with deep representations in an indirect pipeline. Some approaches further extend the traditional way of obtaining correspondences. SuperGlue (Sarlin et al., 2020) learns feature matching with a graph neural network. LoFTR (Sun et al., 2021) directly regresses correspondences instead of relying on separate feature detection and matching. Similar to indirect methods, iii) learning-based direct image alignment enhances classical direct methods with deep features. This category is the most similar to our work and we discuss it in more detail next."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
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+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "Learned features for direct image alignment. Previous work differs in their approach to model training and in the final task. For example, Czarnowski et al. (2017) leverage off-the-shelf CNN features to improve optical flow tracking. A number of methods (Han et al., 2018; Lv et al., 2019; Sarlin et al., 2021; Tang and Tan, 2019; Xu et al., 2021) train feature descriptors end-to-end with ground-truth poses and backpropagate the gradient through a non-linear optimization process. At test time, these methods employ a feature pyramid and refine the initial camera pose in a coarse-to-fine fashion using GN or LM optimization. In addition to the feature pyramid, some methods predict additional properties for image alignment, such as uncertainty (Xu et al., 2021) (Sarlin et al., 2021), Jacobians (Han et al., 2018), or optimization parameters (e.g. damping factors (Lv et al., 2019; Sarlin et al., 2021)). In the same spirit, our formulation leads to a continuous image pyramid, where each level of the pyramid can be generated on-the-fly based on the input level of uncertainty."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
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+                                    "type": "text",
+                                    "content": "Another line of work (von Stumberg et al., 2020a;b) trains a deep network directly on the ground-truth pixel correspondences. GN-Net (von Stumberg et al., 2020a) minimizes two loss functions. The first is a contrastive loss (Schmidt et al., 2017) facilitating discriminative properties of the"
+                                }
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+                                    "content": "feature representation. The second loss function accounts for a likely displacement in the initial correspondence, implemented by adding random 2D offsets to the ground-truth correspondences at training time. By learning to minimize the feature discrepancy after a Gauss-Newton step, the model learns to cope with the initially noisy estimates. We revisit this work in detail in the Sec. 4."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        132,
+                        506,
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        132,
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+                                    ],
+                                    "type": "text",
+                                    "content": "The learning-based approaches (e.g. (Germain et al., 2021; Sarlin et al., 2021; von Stumberg et al., 2020a)) achieve impressive accuracy of pose estimation. Nevertheless, they require pose supervision for training and struggle in scenarios of large-baseline localization. Indeed, GN-Net (von Stumberg et al., 2020a) and PixLoc (Sarlin et al., 2021) tend to exhibit a strong bias toward the noise assumptions of the training process, which cannot be easily reversed. For example, PixLoc trained on the CMU dataset exhibits a strong bias toward horizontal movement as illustrated in Appendix A."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        204,
+                        506,
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        204,
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+                                        272
+                                    ],
+                                    "type": "text",
+                                    "content": "Our work takes a different approach. We rely on existing feature descriptors obtained with self-supervision, which contribute no explicit motion bias to the alignment process. We next derive a closed-form solution to the Gauss-Newton loss (von Stumberg et al., 2020a) as a functional of a probability density governing the noise assumptions of the current pose estimate. This allows us to adjust the noise assumptions in the alignment process, thus effectively controlling the convergence basin at test time, much akin to the coarse-to-fine strategy, exemplified by Fig. 2."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
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+                        298
+                    ],
+                    "type": "title",
+                    "angle": 0,
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+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "type": "text",
+                                    "content": "4 THE GAUSS-NEWTON LOSS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        310,
+                                        504,
+                                        376
+                                    ],
+                                    "type": "text",
+                                    "content": "In this section, we recap the GN-Net (von Stumberg et al., 2020a) and introduce the notation. GN-Net is a convolutional neural network "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        310,
+                                        504,
+                                        376
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "E(\\cdot ;\\pmb {\\theta})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        310,
+                                        504,
+                                        376
+                                    ],
+                                    "type": "text",
+                                    "content": " trained on a sparse set of ground-truth correspondences. Given coordinates on the image plane, let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        310,
+                                        504,
+                                        376
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f:\\mathbb{R}\\times \\mathbb{R}\\to \\mathbb{R}^d"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        310,
+                                        504,
+                                        376
+                                    ],
+                                    "type": "text",
+                                    "content": " continuously map those coordinates to a descriptor from a feature grid "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        310,
+                                        504,
+                                        376
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{R}^{d\\times w\\times h}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        310,
+                                        504,
+                                        376
+                                    ],
+                                    "type": "text",
+                                    "content": " produced by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        310,
+                                        504,
+                                        376
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "E(\\cdot ;\\pmb {\\theta})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        310,
+                                        504,
+                                        376
+                                    ],
+                                    "type": "text",
+                                    "content": ", e.g. using bilinear interpolation. We define "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        310,
+                                        504,
+                                        376
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f_{r}\\coloneqq E(I_{r};\\pmb {\\theta})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        310,
+                                        504,
+                                        376
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        310,
+                                        504,
+                                        376
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f_{q}\\coloneqq E(I_{q};\\pmb{\\theta})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        310,
+                                        504,
+                                        376
+                                    ],
+                                    "type": "text",
+                                    "content": " to denote feature representations of reference and query images. GN-Net learns parameters "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        310,
+                                        504,
+                                        376
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\theta}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        310,
+                                        504,
+                                        376
+                                    ],
+                                    "type": "text",
+                                    "content": " to minimize the expected value of the following loss function,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
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+                        392
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        200,
+                                        379,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathcal {L} _ {\\mathrm {G N} - \\text {N e t}} \\left(I _ {r}, I _ {q}\\right) = \\mathcal {L} _ {\\text {c o n t r a s t i v e}} \\left(f _ {r}, f _ {q}\\right) + \\mathcal {L} _ {\\mathrm {G N}} \\left(f _ {r}, f _ {q}\\right), \\tag {5}",
+                                    "image_path": "f7a69a9f8038a30e6d262a6ad18bfce28a68ed98a71d0efb99d50db557567e1b.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        394,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "text",
+                                    "content": "which comprises a contrastive loss, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        394,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}_{\\mathrm{contrastive}}(\\cdot ,\\cdot)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        394,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "text",
+                                    "content": ", and the Gauss-Newton loss, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        394,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}_{\\mathrm{GN}}(\\cdot ,\\cdot)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        394,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "text",
+                                    "content": ". The contrastive loss minimizes the distance between the features of two corresponding points while maximizing the distance between non-corresponding pairs (Schmidt et al., 2017). The contrastive loss facilitates spatially discriminative features, and its particular instantiation has little significance for the following discussion (e. g. GN-Net uses the triplet loss)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
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+                        500
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                        500
+                                    ],
+                                    "type": "text",
+                                    "content": "The Gauss-Newton loss "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        455,
+                                        504,
+                                        500
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}_{\\mathrm{GN}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        455,
+                                        504,
+                                        500
+                                    ],
+                                    "type": "text",
+                                    "content": " ensures that the feature map is sufficiently smooth for direct image alignment, thus enlarging the convergence basin. GN-Net implements this by adding a random offset to the ground-truth correspondences and encouraging a single Gauss-Newton step to recover the original location. Let us formalize this process."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
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+                    "angle": 0,
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+                            "bbox": [
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+                            ],
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+                                    "bbox": [
+                                        104,
+                                        505,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "text",
+                                    "content": "Given a ground-truth correspondence "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        505,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\pmb{x}^{(i)},\\pmb{y}^{(i)})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        505,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "text",
+                                    "content": " between images "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        505,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "I_{r}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        505,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        505,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "I_{q}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        505,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "text",
+                                    "content": ", the assumption behind the Gauss-Newton loss is that the initial estimate "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        505,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\pmb{x}}^{(i)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        505,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "text",
+                                    "content": " at test time falls in the vicinity of the ground truth "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        505,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x}^{(i)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        505,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "text",
+                                    "content": ", i.e. "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        505,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\pmb{x}}^{(i)} = \\pmb{x}^{(i)} + \\pmb{\\epsilon}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        505,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "text",
+                                    "content": ", where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        505,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{\\epsilon}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        505,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "text",
+                                    "content": " follows some predefined distribution "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        505,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p(\\pmb{\\epsilon})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        505,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "text",
+                                    "content": ", such as a Gaussian with zero mean. At training time, the Gauss-Newton loss aims at recovering the ground-truth location "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        505,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x}^{(i)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        505,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "text",
+                                    "content": " from a noisy initial location "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        505,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\pmb{x}}^{(i)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        505,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "text",
+                                    "content": " by minimizing the residual w.r.t. "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        505,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        505,
+                                        504,
+                                        565
+                                    ],
+                                    "type": "text",
+                                    "content": ":"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        235,
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+                                        594
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "r ^ {(i)} \\left(\\tilde {\\boldsymbol {x}} ^ {(i)}\\right) := f _ {r} \\left(\\tilde {\\boldsymbol {x}} ^ {(i)}\\right) - f _ {q} \\left(\\boldsymbol {y} ^ {(i)}\\right), \\tag {6}",
+                                    "image_path": "ba7ae715c1513184de25dc14620c32c8c12541fdc4f020257885ca2e9eef2bc9.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        601,
+                                        504,
+                                        615
+                                    ],
+                                    "type": "text",
+                                    "content": "The original Gauss-Newton loss for one point "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        601,
+                                        504,
+                                        615
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x}^{(i)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        601,
+                                        504,
+                                        615
+                                    ],
+                                    "type": "text",
+                                    "content": ", as introduced by von Stumberg et al. (2020a), is"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        169,
+                                        627,
+                                        505,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathcal {L} _ {\\mathrm {G N} _ {o}} ^ {(i)} \\left(f _ {r}, f _ {q}, \\epsilon\\right) := \\| J (\\epsilon - \\Delta_ {\\mathrm {G N}} [ r ^ {(i)} \\left(\\boldsymbol {x} ^ {(i)} + \\epsilon\\right) ]) \\| _ {2} ^ {2} - \\log \\det  \\boldsymbol {J} ^ {\\intercal} \\boldsymbol {J}. \\tag {7}",
+                                    "image_path": "7dfa8e530a8f2bfc954d38f9af6c5737cfe0f244be89209cdfd824ed201ac579.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                        704
+                                    ],
+                                    "type": "text",
+                                    "content": "From a probabilistic standpoint (von Stumberg et al., 2020a), the loss balances between the accuracy of the Gauss-Newton step and the direction uncertainty. Here, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        648,
+                                        504,
+                                        704
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "J^{\\mathsf{T}}J"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        648,
+                                        504,
+                                        704
+                                    ],
+                                    "type": "text",
+                                    "content": " represents the inverse covariance matrix propagated through the photometric residuals. The loss function corresponds to the negative log-likelihood of residuals distributed as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        648,
+                                        504,
+                                        704
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{N}\\left(0,\\left(\\mathbf{J}^{\\mathsf{T}}\\mathbf{J}\\right)^{-1}\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        648,
+                                        504,
+                                        704
+                                    ],
+                                    "type": "text",
+                                    "content": ". In this work, we consider a variant of the Gauss-Newton loss, in which the covariance matrix is assumed to be identity. Thus, Eq. (7) becomes"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                    "type": "interline_equation",
+                                    "content": "\\mathcal {L} _ {\\mathrm {G N}} ^ {(i)} \\left(f _ {r}, f _ {q}, \\boldsymbol {\\epsilon}\\right) = \\| \\boldsymbol {\\epsilon} - \\Delta_ {\\mathrm {G N}} \\left[ r ^ {(i)} \\left(\\boldsymbol {x} ^ {(i)} + \\boldsymbol {\\epsilon}\\right) \\right] \\| _ {2} ^ {2}. \\tag {8}",
+                                    "image_path": "79ca7af316cb4a94d0ea0e348b0f6684d2027e6115c00df64c637a5ff7ae0b88.jpg"
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
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+                                    "content": "4"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                }
+            ],
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+                                    "type": "text",
+                                    "content": "Although this simplified version does not account for the trade-off between uncertainty and accuracy of the prediction, it admits a closed-form minimizer of the expected value, as we show in Sec. 5. We also find that the simplified version does not differ from the original one empirically in a significant way (see Appendix C). Henceforth, we will refer to the simplified version as the Gauss-Newton loss."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
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+                                    "type": "text",
+                                    "content": "The training process calculates the Gauss-Newton loss stochastically by sampling "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\epsilon"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " from "
+                                },
+                                {
+                                    "bbox": [
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+                                        132,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p(\\epsilon)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        132,
+                                        506,
+                                        169
+                                    ],
+                                    "type": "text",
+                                    "content": ". Therefore, it minimizes a Monte-Carlo approximation to the expected value of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        132,
+                                        506,
+                                        169
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}_{\\mathrm{GN}}^{(i)}"
+                                },
+                                {
+                                    "bbox": [
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+                                        132,
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+                                    ],
+                                    "type": "text",
+                                    "content": " summed over all ground-truth correspondences,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
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+                                    "type": "interline_equation",
+                                    "content": "\\mathcal {L} _ {\\mathrm {G N}} \\left(f _ {r}, f _ {q}; p\\right) = \\mathbb {E} _ {\\epsilon \\sim p} \\left[ \\sum_ {i} \\mathcal {L} _ {\\mathrm {G N}} ^ {(i)} \\left(f _ {r}, f _ {q}, \\epsilon\\right) \\right]. \\tag {9}",
+                                    "image_path": "5a6fcec11d86b94d6974b250ffc345d8db870935fc3462516b607a0eab632776.jpg"
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+                {
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Hereafter, we use the notation "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}_{\\mathrm{GN}}(f_r,f_q;p)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " to emphasize the dependence of the Gauss-Newton loss on the noise distribution "
+                                },
+                                {
+                                    "bbox": [
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+                                        217,
+                                        504,
+                                        240
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        217,
+                                        504,
+                                        240
+                                    ],
+                                    "type": "text",
+                                    "content": ", and omit this parametarization otherwise to avoid clutter."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Note that the GN-Net's training stage addresses the problem of optical flow, not pose estimation. The underlying assumption is that accurate optical flow facilitates pose estimation, as each pixel will contribute to the final pose estimate. At test time, the pose is determined by solving the featuremetric image alignment problem (equivalent to Eq. (4)) using Gauss-Newton optimization in SE(3)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
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+                    ],
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+                                    "type": "text",
+                                    "content": "5 CLOSED-FORM GAUSS-NEWTON STEP"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
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+                                    "type": "text",
+                                    "content": "Decoupling the contrastive and Gauss-Newton losses. The contrastive loss provides a sparse constraint on the feature embeddings, since we can only use sparse ground-truth correspondences, the interest points, for supervision. By contrast, the Gauss-Newton loss enforces a pre-defined basin of convergence around each interest point (modeled by "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p(\\epsilon)"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ") with little effect on the feature descriptors in the interest points. This is because the residual in Eq. (6) between the corresponding interest points will be negligible, if the contrastive loss for those points is minimized. It follows that the contrastive and the Gauss-Newton loss essentially optimize over a disjoint set of feature locations. Therefore, we can decouple the Gauss-Newton loss from the joint optimization objective in Eq. (5). Let us formalize this reasoning. We aim to solve:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
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+                    ],
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+                    "angle": 0,
+                    "lines": [
+                        {
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+                                    "type": "interline_equation",
+                                    "content": "f _ {q} ^ {*} = \\underset {f _ {q}} {\\arg \\min } \\left[ \\mathcal {L} _ {\\text {c o n t r a s t i v e}} \\left(f _ {r}, f _ {q}\\right) + \\mathcal {L} _ {\\mathrm {G N}} \\left(f _ {r}, f _ {q}\\right) \\right]. \\tag {10}",
+                                    "image_path": "6056aa6ba80d46a7cee5782e602eedca1f62c4d026e43fcb785e12a067ce6043.jpg"
+                                }
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+                                    "type": "text",
+                                    "content": "We denote the values of "
+                                },
+                                {
+                                    "bbox": [
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+                                        481,
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+                                        528
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f_{r}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        481,
+                                        506,
+                                        528
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        481,
+                                        506,
+                                        528
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f_{q}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        481,
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+                                        528
+                                    ],
+                                    "type": "text",
+                                    "content": " in the interest points as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        481,
+                                        506,
+                                        528
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F_{r}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        481,
+                                        506,
+                                        528
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        481,
+                                        506,
+                                        528
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F_{q}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        481,
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+                                    ],
+                                    "type": "text",
+                                    "content": ": "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        481,
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+                                        528
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F_{r}^{(i)} := f_{r}(\\pmb{x}^{(i)})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        481,
+                                        506,
+                                        528
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        481,
+                                        506,
+                                        528
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F_{q}^{(i)} := f_{q}(\\pmb{y}^{(i)})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        481,
+                                        506,
+                                        528
+                                    ],
+                                    "type": "text",
+                                    "content": ". The contrastive loss only depends on "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        481,
+                                        506,
+                                        528
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F_{r}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        481,
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+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        481,
+                                        506,
+                                        528
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F_{q}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        481,
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+                                        528
+                                    ],
+                                    "type": "text",
+                                    "content": ", while the Gauss-Newton loss depends on all the query feature values "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        481,
+                                        506,
+                                        528
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f_{q}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        481,
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+                                    ],
+                                    "type": "text",
+                                    "content": " and the interest points in the reference map "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        481,
+                                        506,
+                                        528
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F_{r}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        481,
+                                        506,
+                                        528
+                                    ],
+                                    "type": "text",
+                                    "content": ". By eliminating unused parts, we introduce an equivalent problem:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
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+                    "type": "interline_equation",
+                    "angle": 0,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left\\{F _ {q} ^ {*}, f _ {q} ^ {*} \\right\\} = \\underset {F _ {q}, f _ {q}} {\\arg \\min } \\left[ \\mathcal {L} _ {\\text {c o n t r a s t i v e}} \\left(F _ {r}, F _ {q}\\right) + \\mathcal {L} _ {\\mathrm {G N}} \\left(F _ {r}, f _ {q}\\right) \\right]. \\tag {11}",
+                                    "image_path": "f7d809c32dd1b612fbffd7682ebb3157ec703dfaf4bacf9c23f4ff64387f0bf2.jpg"
+                                }
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Next, we approximate "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
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+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " with "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "F_{q}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " in the second term. This is permissible as the contrastive loss acts as a soft constraint, ensuring that "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "F_{r}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F_{q}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " are close at the optimum of the joint loss function. This allows us to decouple the minimization problem as follows:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "F _ {q} ^ {*} = \\underset {F _ {q}} {\\arg \\min } \\left[ \\mathcal {L} _ {\\text {c o n t r a s t i v e}} \\left(F _ {r}, F _ {q}\\right) + \\underset {f _ {q}} {\\min } \\mathcal {L} _ {\\mathrm {G N}} \\left(F _ {q}, f _ {q}\\right) \\right], \\tag {12}",
+                                    "image_path": "f8fcf75e7ff02f87b49e33f0e7baec6fe1ad9e119ac700391ce607d070fd65c6.jpg"
+                                }
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+                                    "content": "f_{q}^{*} = \\operatorname *{arg  min}_{f_{q}}\\mathcal{L}_{\\mathrm{GN}}(F_{q}^{*},f_{q}).",
+                                    "image_path": "8faca25eab736f65564fc0eb2442a15c135fcb34efdead277aefeabba96991ff.jpg"
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+                                    "type": "text",
+                                    "content": "We denote"
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+                                    "type": "interline_equation",
+                                    "content": "G(F_{q};p):= \\operatorname *{arg  min}_{f_{q}}\\mathcal{L}_{\\mathrm{GN}}(F_{q},f_{q};p),",
+                                    "image_path": "8dd8114a79b0d0e7d0b0243171ab5e6fb412fd16abd2310c277a9dccdc1bbe6d.jpg"
+                                }
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+                                    "type": "interline_equation",
+                                    "content": "\\mathcal {L} _ {\\mathrm {G N}} ^ {*} (F _ {q}; p) := \\min  _ {f _ {q}} \\mathcal {L} _ {\\mathrm {G N}} (F _ {q}, f _ {q}; p) = \\mathcal {L} _ {\\mathrm {G N}} (F _ {q}, G (F _ {q}; p); p). \\tag {15}",
+                                    "image_path": "31b0b6545397937b60dd15e4a90959443076048b0ae251d7764003135fb44782.jpg"
+                                }
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
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+                                            "content": "Figure 2: Controlling the basin of convergence. Image samples and the corresponding feature maps "
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+                                            "type": "text",
+                                            "content": " (PCA) for different values of "
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+                                            "type": "text",
+                                            "content": " and isotropic Gaussian distribution "
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+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
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+                            "index": 2,
+                            "angle": 0,
+                            "type": "image_caption"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " is the optimal reconstruction of a feature map under sparse feature descriptors "
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": " and noise density "
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p(\\epsilon)"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ". "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}_{\\mathrm{GN}}^{*}(F_{q};p)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        308,
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+                                        381
+                                    ],
+                                    "type": "text",
+                                    "content": " is the corresponding value of the Gauss-Newton loss under "
+                                },
+                                {
+                                    "bbox": [
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+                                        308,
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+                                        381
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F_{q}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ". We will refer to "
+                                },
+                                {
+                                    "bbox": [
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+                                        506,
+                                        381
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G(F_{q};p)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        308,
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+                                        381
+                                    ],
+                                    "type": "text",
+                                    "content": " as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        308,
+                                        506,
+                                        381
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f_{q}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        308,
+                                        506,
+                                        381
+                                    ],
+                                    "type": "text",
+                                    "content": " to denote an optimal reconstruction under an arbitrary sparse input "
+                                },
+                                {
+                                    "bbox": [
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+                                        308,
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+                                        381
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F_{q}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        308,
+                                        506,
+                                        381
+                                    ],
+                                    "type": "text",
+                                    "content": ". Note the difference to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        308,
+                                        506,
+                                        381
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f_{q}^{*}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        308,
+                                        506,
+                                        381
+                                    ],
+                                    "type": "text",
+                                    "content": ", which is an optimal reconstruction under the joint loss, i.e. "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        308,
+                                        506,
+                                        381
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G(F_{q}^{*};p)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        308,
+                                        506,
+                                        381
+                                    ],
+                                    "type": "text",
+                                    "content": ". Our main contribution, detailed shortly, is a closed-form solution for both "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        308,
+                                        506,
+                                        381
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G(F_{q};p)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        308,
+                                        506,
+                                        381
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        308,
+                                        506,
+                                        381
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}_{\\mathrm{GN}}^{*}(F_{q};p)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        308,
+                                        506,
+                                        381
+                                    ],
+                                    "type": "text",
+                                    "content": ". It reveals that the original problem in Eq. (10) actually depends only on the feature values in the interest points:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "F _ {q} ^ {*} = \\underset {F _ {q}} {\\arg \\min } \\left[ \\mathcal {L} _ {\\text {c o n t r a s t i v e}} \\left(F _ {r}, F _ {q}\\right) + \\mathcal {L} _ {\\mathrm {G N}} ^ {*} \\left(F _ {q}; p\\right) \\right]. \\tag {14}",
+                                    "image_path": "5408cff12bda72ce6303eb9a750b9c21476bf02e3d4b644350a1e3e2b94d66dc.jpg"
+                                }
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+                    ],
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+                },
+                {
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "This means that in training a feature extractor only the features of interest points matter, while the representation of other pixels approximates our analytical solution. As the empirical validation, we demonstrate compelling results in our experiments by using feature descriptors for the interest points from a self-supervised method, while employing our analytical solution for the remaining pixels."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        471,
+                        504,
+                        495
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        471,
+                                        504,
+                                        495
+                                    ],
+                                    "type": "text",
+                                    "content": "An analytical solution to the Gauss-Newton loss. We propose to calculate the expectation of the Gauss-Newton loss in Eq. (9) as a functional of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        471,
+                                        504,
+                                        495
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f_{q}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        471,
+                                        504,
+                                        495
+                                    ],
+                                    "type": "text",
+                                    "content": ", and analytically find its closed-form minimizer."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        500,
+                        504,
+                        554
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                500,
+                                504,
+                                554
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        500,
+                                        504,
+                                        554
+                                    ],
+                                    "type": "text",
+                                    "content": "Consider the query image "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        500,
+                                        504,
+                                        554
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "I_{q}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        500,
+                                        504,
+                                        554
+                                    ],
+                                    "type": "text",
+                                    "content": " as a set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        500,
+                                        504,
+                                        554
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{(\\pmb{x}^{(j)}, F_{q}^{(j)})\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        500,
+                                        504,
+                                        554
+                                    ],
+                                    "type": "text",
+                                    "content": " containing locations of interest points "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        500,
+                                        504,
+                                        554
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x}^{(j)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        500,
+                                        504,
+                                        554
+                                    ],
+                                    "type": "text",
+                                    "content": " and the corresponding feature descriptors "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        500,
+                                        504,
+                                        554
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F_{q}^{(j)} \\coloneqq f_{q}(\\pmb{x}^{(j)})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        500,
+                                        504,
+                                        554
+                                    ],
+                                    "type": "text",
+                                    "content": ". The locations "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        500,
+                                        504,
+                                        554
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x}^{(j)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        500,
+                                        504,
+                                        554
+                                    ],
+                                    "type": "text",
+                                    "content": " can be extracted by an off-the-shelf feature detector (e.g. SuperPoint (DeTone et al., 2018)), while a network trained with the contrastive loss in Eq. (5) can produce the corresponding descriptors "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        500,
+                                        504,
+                                        554
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F_{q}^{(j)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        500,
+                                        504,
+                                        554
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        104,
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        558,
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+                                    ],
+                                    "type": "text",
+                                    "content": "Let us first re-write the expectation in Eq. (9) using Eq. (8) and Eq. (2) in the decoupled formulation of the Gauss-Newton loss (cf. Eq. (12)):"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
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+                                {
+                                    "bbox": [
+                                        147,
+                                        594,
+                                        505,
+                                        625
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathcal {L} _ {\\mathrm {G N}} \\left(F _ {q}, f _ {q}; p\\right) = \\sum_ {j} \\int_ {\\Omega} \\left\\| \\boldsymbol {\\epsilon} - \\left(\\boldsymbol {J} ^ {\\intercal} \\boldsymbol {J}\\right) ^ {- 1} \\boldsymbol {J} ^ {\\intercal} \\left(f _ {q} \\left(\\boldsymbol {x} ^ {(j)} + \\boldsymbol {\\epsilon}\\right) - F _ {q} ^ {(j)}\\right) \\right\\| _ {2} ^ {2} p (\\boldsymbol {\\epsilon}) d \\boldsymbol {\\epsilon}. \\tag {15}",
+                                    "image_path": "fac222b4f68b0fd4e9f1dd98647dd909c0100d5027a3694386c84aa35b3fd3fd.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "type": "text",
+                                    "content": "Substituting "
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+                                    "type": "inline_equation",
+                                    "content": "\\pmb {\\epsilon} = \\tilde{\\pmb{x}} -\\pmb{x}^{(j)}"
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+                                {
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+                                    "type": "text",
+                                    "content": " , we obtain"
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+                                    "type": "interline_equation",
+                                    "content": "\\mathcal {L} _ {\\mathrm {G N}} \\left(F _ {q}, f _ {q}; p\\right) = \\sum_ {j} \\int_ {\\Omega} \\left\\| \\tilde {\\boldsymbol {x}} - \\left(\\boldsymbol {J} ^ {\\intercal} \\boldsymbol {J}\\right) ^ {- 1} \\boldsymbol {J} ^ {\\intercal} \\left(f _ {q} (\\tilde {\\boldsymbol {x}}) - F _ {q} ^ {(j)}\\right) - \\boldsymbol {x} ^ {(j)} \\right\\| _ {2} ^ {2} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right) d \\tilde {\\boldsymbol {x}}. \\tag {16}",
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+                                    "content": "We first consider all "
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+                                    "content": "\\tilde{\\pmb{x}}\\in \\Omega"
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+                                    "type": "text",
+                                    "content": " independently and relax the problem by eliminating the constraint "
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+                                    "type": "inline_equation",
+                                    "content": "J = \\frac{\\partial f_q}{\\partial\\tilde{\\pmb{x}}}"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ". This allows us to derive analytical solutions for minimizers "
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+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\tilde{f}_q(\\cdot)"
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+                                    "type": "text",
+                                    "content": " and "
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+                                    "type": "inline_equation",
+                                    "content": "\\tilde{J}_q(\\cdot)"
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+                                    "type": "text",
+                                    "content": " as functions"
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
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+                                            "type": "text",
+                                            "content": "Figure 3: Image alignment with continuous probabilistic feature pyramids. Given reference image with a 3D point cloud and a query image paired with an initial coarse pose, we start by estimating interest points and their features on the query image, along with features from the projections of the 3D point cloud onto the reference image. Next, we perform feature metric alignment between the 3D point features and derived analytical continuous image pyramid."
+                                        }
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+                            "angle": 0,
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+                                    "type": "text",
+                                    "content": "of "
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+                                    "type": "inline_equation",
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ". We then observe that "
+                                },
+                                {
+                                    "bbox": [
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+                                        340
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\frac{\\partial}{\\partial\\tilde{\\pmb{x}}}\\tilde{f}_q(\\cdot)"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " coincides with "
+                                },
+                                {
+                                    "bbox": [
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+                                        283,
+                                        506,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{J}_q(\\cdot)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
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+                                        340
+                                    ],
+                                    "type": "text",
+                                    "content": " for a uniform distribution, which confirms that in this case, our derived solution is the solution to the original problem. As a side note, in the context of direct image alignment, it has been suggested that decoupling and predicting the Jacobian independently from the function value may yield superior convergence and results (Han et al., 2018). Appendix B provides the full derivation. Below, we summarize the closed-form minimizer to Eq. (16):"
+                                }
+                            ]
+                        }
+                    ],
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+                                    "type": "interline_equation",
+                                    "content": "\\tilde {f} _ {q} (\\tilde {\\boldsymbol {x}}) = \\tilde {J} _ {q} (\\tilde {\\boldsymbol {x}}) \\left(\\tilde {\\boldsymbol {x}} - \\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}})\\right) + \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}}), \\tag {17}",
+                                    "image_path": "148afab1b9c07ab975dea8b0d7373251dca57380b0b9a0a1351f2ea62fd001fe.jpg"
+                                }
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+                                    "type": "interline_equation",
+                                    "content": "\\tilde {J} _ {q} (\\tilde {\\boldsymbol {x}}) = \\left(\\operatorname {C o v} _ {\\mathbf {x y}} (\\tilde {\\boldsymbol {x}}) \\operatorname {C o v} _ {\\mathbf {y}} (\\tilde {\\boldsymbol {x}}) ^ {- 1}\\right) ^ {+}, \\tag {18}",
+                                    "image_path": "191c315e6d42a8dfb0339026df6180c8962d47959f928181b760733e94126ed0.jpg"
+                                }
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+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}}) = \\sum_ {j} F _ {q} ^ {(j)} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right) / \\sum_ {j} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right), \\tag {19}",
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+                                }
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+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}}) = \\sum_ {j} \\boldsymbol {x} ^ {(j)} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right) / \\sum_ {j} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right), \\tag {20}",
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+                                    "type": "interline_equation",
+                                    "content": "\\operatorname {C o v} _ {\\mathbf {x y}} (\\tilde {\\boldsymbol {x}}) = \\sum_ {j} \\left(\\boldsymbol {x} ^ {(j)} - \\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}})\\right) \\left(F _ {q} ^ {(j)} - \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}})\\right) ^ {\\intercal} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}), \\tag {21}",
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+                                    "type": "interline_equation",
+                                    "content": "\\operatorname {C o v} _ {\\mathbf {y}} (\\tilde {\\boldsymbol {x}}) = \\sum_ {j} \\left(F _ {q} ^ {(j)} - \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}})\\right) \\left(F _ {q} ^ {(j)} - \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}})\\right) ^ {\\intercal} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}), \\tag {22}",
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+                                    "type": "text",
+                                    "content": "where "
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+                                    "type": "inline_equation",
+                                    "content": "(^{+})"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " denotes the Moore-Penrose pseudo-inverse operator. The optimal point is "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G(F_{q};p)\\coloneqq \\tilde{f}_{q}"
+                                },
+                                {
+                                    "bbox": [
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+                                        521,
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+                                    ],
+                                    "type": "text",
+                                    "content": " and the loss value at this point (cf. Eq. (13)) is"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
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+                {
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\mathcal {L} _ {G N} ^ {*} (F _ {q}) = \\int_ {\\Omega} \\frac {1}{2} \\operatorname {T r} \\left[ \\operatorname {C o v} _ {\\mathbf {x}} (\\tilde {\\boldsymbol {x}}) - \\operatorname {C o v} _ {\\mathbf {x y}} (\\tilde {\\boldsymbol {x}}) \\operatorname {C o v} _ {\\mathbf {y}} (\\tilde {\\boldsymbol {x}}) ^ {- 1} \\operatorname {C o v} _ {\\mathbf {y x}} (\\tilde {\\boldsymbol {x}}) \\right] d \\tilde {\\boldsymbol {x}}. \\tag {23}",
+                                    "image_path": "1e3e7ad4215ef409ec790f69c1c5f0bb59341a0833a5a25bf9b3f00c345ec302.jpg"
+                                }
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+                    "index": 11
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+                {
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+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Our derivation of the optimal embeddings "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G(F_{q};p)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        599,
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+                                    ],
+                                    "type": "text",
+                                    "content": " using fixed interest points "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        599,
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+                                        679
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F_{q}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        599,
+                                        506,
+                                        679
+                                    ],
+                                    "type": "text",
+                                    "content": " provides an interesting insight into end-to-end learning frameworks. Jointly training both losses requires inverting a high-dimensional matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        599,
+                                        506,
+                                        679
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{Cov}_{\\mathbf{y}}(\\tilde{\\boldsymbol{x}})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        599,
+                                        506,
+                                        679
+                                    ],
+                                    "type": "text",
+                                    "content": " in Eq. (23), which poses high numerical instability and discontinuities. Since previous work (von Stumberg et al., 2020a) can be seen as stochastic approximations to our solution, this may explain the reported training instability and divergence in those works. Additionally, the form of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        599,
+                                        506,
+                                        679
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{f}_q"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        599,
+                                        506,
+                                        679
+                                    ],
+                                    "type": "text",
+                                    "content": " in Eq. (17) suggests that end-to-end pipelines may not necessarily yield any sophisticated representation, as they merely interpolate between features in the interest points."
+                                }
+                            ]
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+                                    "content": "FeatureMetric Image Alignment. As a case study, we employ the analytical form of "
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+                                    "type": "text",
+                                    "content": " for feature metric image alignment. Algorithm 1 and Fig. 3 provide an overview. Fig. 1 illustrates stages of direct alignment using our probabilistically reconstructed feature maps. We first extract the interest points "
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+                                            "content": "\\{F_q^{(j)}\\} \\gets E(I_q;\\pmb {\\theta})\\circ \\{\\pmb{x}^{(j)}\\}"
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+                                            "content": "\\{\\pmb{o}^{(i)}\\} \\gets E(I_r;\\pmb {\\theta})\\circ \\{\\langle \\pmb{p}^{(i)}\\rangle \\}"
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+                                            "content": "\\pmb{T} \\gets \\pmb{T}^{(0)}"
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+                                            "content": "p\\gets \\mathrm{Schedule}(n)"
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+                                            "content": "\\tilde{f}_q\\gets G(\\{F_q^{(j)}\\} ;p)"
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+                                            "content": "\\{\\tilde{\\pmb{p}}^{(i)}\\} \\gets \\{\\pmb {T}\\pmb{p}^{(i)}\\}"
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+                                            "type": "inline_equation",
+                                            "content": "\\pmb{\\delta} \\gets \\Delta_{GN}(\\tilde{f}_q \\circ \\{\\langle \\tilde{\\pmb{p}}^{(i)}\\rangle\\} - \\{\\pmb{o}^{(i)}\\})"
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+                                    ]
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+                                            "content": "\\pmb{T}\\gets \\pmb{T}\\boxminus \\pmb{\\delta}"
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+                                    "type": "text",
+                                    "content": " denotes 2D projection."
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+                                    "content": "Algorithm 1: Image alignment."
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+                                            "content": "Figure 4: Robustness to initialization. Our closed-form solution exhibits significantly greater robustness compared to PixLoc."
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+                                    "content": " using Eqs. (17) and (18). The sparse set of 3D points "
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+                                    "content": " is projected onto the reference image to obtain reference descriptors. We obtain feature descriptors for these points using the same feature embedding network "
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+                                    "content": ", at every iteration we perform Gauss-Newton steps minimizing the residuals "
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+                                    "content": "r ^ {(i)} (\\boldsymbol {T}) := F _ {r} ^ {(i)} - \\tilde {f} _ {q} \\left(\\left\\langle \\boldsymbol {T} \\boldsymbol {p} ^ {(i)} \\right\\rangle\\right). \\tag {24}",
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+                                    "content": " explicitly controls the basin of convergence. In practice, we can use any distribution with high initial variance based on the inaccuracy assumed in the initial point projections. As our estimates improve over the course of optimization, we gradually decrease the variance. Notably, the choice of "
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+                                    "content": " lets us define values for points outside of the image boundaries. Fig. 2 illustrates examples of the feature maps "
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+                                    "content": " following a Gaussian distribution of increasing variance."
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+                                    "content": "Comparison to previous work. Our derivation is agnostic to the underlying feature extractor and can operate with off-the-shelf feature descriptors. An interesting property of the derived "
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+                                    "content": " is that it enables continuous coarse-to-fine alignment. By adjusting the noise prior "
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+                                    "content": " in the scheduler (cf. Algorithm 1), we can control the basin of convergence dynamically at runtime. We can adjust the distribution "
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+                                    "content": " either by choosing a different parameter set of a pre-defined distribution, or by changing the distribution itself to another family. In practice, we start the optimization with a uniform distribution and then switch to the Gaussian distribution with a slowly decreasing variance."
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+                                    "content": "A uniform distribution with a wide support offers a large basin of convergence, which sacrifices accuracy for robustness. The ensuing Gaussian distribution with decreasing variance refines the pose and leads to a more accurate solution. Appendix E provides implementation details of the scheduler."
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+                                    "content": " using Eq. (17) only depends on the feature descriptors of the interest points in the query image. This observation suggests an interesting interpretation of minimizing the residual in Eq. (24) using "
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+                                    "content": " with Gauss-Newton optimization as feature matching. The Gauss-Newton step is fully determined by the neighboring points of interest. Consequently, it implies that the effectiveness of such optimization-based methods may be limited in comparison to alternative optimization-inspired approaches, which learn the optimization step (Teed and Deng, 2020)."
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+                                    "type": "text",
+                                    "content": "6 EXPERIMENTS"
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+                                    "content": "Datasets. We evaluate our approach on two most popular datasets for large-scale image localization, namely the Aachen Day-Night dataset (Sattler et al., 2018), extended CMU seasons (Toft et al., 2022) and 7Scenes dataset (Shotton et al., 2013). Aachen Day-Night consists of 98 night and 824 day query"
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+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                            "content": "Table 1: Camera localization on Aachen Day-Night and CMU Seasons. Our evaluation shows an overall improved accuracy in diverse scenarios despite using self-supervised SuperPoint descriptors. We compare to the state-of-the-art methods supervised with pose: GN-Net (von Stumberg et al., 2020a), LM-Reloc (von Stumberg et al., 2020b) and PixLoc (Sarlin et al., 2021)."
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+                                            "html": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">Aachen</td><td colspan=\"3\">CMU Seasons</td></tr><tr><td>Day</td><td>Night</td><td>Urban</td><td>Suburban</td><td>Park</td></tr><tr><td>GN-Net</td><td>62.4/69.4/76.9</td><td>49.0/58.2/66.3</td><td>75.4/79.9/92.6</td><td>64.7/67.0/81.4</td><td>46.7/48.3/65.3</td></tr><tr><td>LM-Reloc</td><td>60.4/68.0/76.3</td><td>37.8/46.9/59.2</td><td>76.6/82.5/93.4</td><td>67.3/72.0/82.8</td><td>49.1/53.4/66.9</td></tr><tr><td>PixLoc</td><td>64.3/69.3/77.4</td><td>51.0/55.1/67.3</td><td>88.3/90.4/93.7</td><td>79.6/81.1/85.2</td><td>61.0/62.5/69.4</td></tr><tr><td>Ours (SuperPoint)</td><td>66.3/72.5/78.8</td><td>43.9/50.0/56.1</td><td>86.0/90.6/95.2</td><td>79.8/85.0/92.4</td><td>63.4/67.9/77.5</td></tr></table>",
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+                                    "content": "images. Extended CMU seasons consists of 14 slices, 5 for urban environment, 5 for suburban and 4 for park. Each slice contains between 3000 and 5000 query images. The 7Scenes dataset comprises seven distinct scenes, each containing multiple sequences of 500 to 1000 frames. For our ablation study in Appendix D we used Cambridge Landmarks dataset (Kendall et al., 2016)."
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+                                    "content": "Results. Table 1 presents the results for camera localization on Aachen Day-Night (Sattler et al., 2018) and the extended CMU Seasons (Toft et al., 2022). We use SuperPoint (DeTone et al., 2018) as the feature descriptor in these experiments. In each setting (Day/Night for Aachen; Urban/Suburban/Park for CMU Seasons), we report three numbers, which indicate the percentage of the query images that were successfully localized within the specified translation and rotation thresholds. We adopt the standard threshold values defined by the benchmarks (translation, rotation): "
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+                                    "content": "(0.25\\mathrm{m}, 2^{\\circ}) / (0.5\\mathrm{m}, 5^{\\circ}) / (5\\mathrm{m}, 10^{\\circ})"
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+                                    "type": "text",
+                                    "content": ". We adopted the implementation from PixLoc (Sarlin et al., 2021) to run these benchmarks. A comparison with PixLoc on the 7Scenes dataset is presented in Appendix F."
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+                                    "content": "We observe that our solution, despite using self-supervised descriptors from SuperPoint (DeTone et al., 2018), achieves an overall strong localization accuracy across diverse settings in comparison to supervised frameworks. Aachen Day-Night has significant occlusions, hence many outliers. Since the Gauss-Newton loss does not incorporate any outlier filtering, we do not expect high accuracy for our approach on this dataset. Nevertheless, we were surprised to find that on Aachen Day our approach even slightly surpassed previous state of the art. On Aachen Night, our accuracy is inferior to previous work, however. This is somewhat expected, since SuperPoint feature descriptors were not directly trained on day-night correspondences. By contrast, PixLoc was trained with supervision on day-night image pairs, which provides an obvious advantage. On CMU Seasons, our approach demonstrates a clear improvement over previous feature metric alignment methods."
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+                                    "content": "The proposed approach demonstrates notable robustness to initialization noise. We compared the proposed scheme and PixLoc on the Cambridge Landmarks dataset, varying the levels of random noise applied to the ground-truth pose as illustrated in Fig. 4 and Appendix G."
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+                                    "content": "We derived a closed-form solution to the Gauss-Newton loss in the context of direct image alignment, which offers two main advantages. First, it allows for dynamic control of the convergence basin, which improves robustness of the alignment to pose initialization. Furthermore, despite using self-supervised descriptors, such control leads to compelling accuracy of pose estimates in comparison to supervised pipelines on established benchmarks. Second, our derivation exposes intrinsic limitations of employing the Gauss-Newton loss in deep learning, as it only leads to a form of interpolation between the feature descriptors in the interest points. This insight offers an interesting connection between direct image alignment and feature matching, and leads to a novel perspective on learning robust features end-to-end, which we will investigate in future work."
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+                                    "content": "1https://github.com/cvg/pixloc"
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+                                    "content": "ACKNOWLEDGMENTS"
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+                                    "content": "This work was supported by the ERC Advanced Grant SIMULACRON and by TUM Georg Nemetschek Institute under the project AI4TWINNING. We thank Linus Harenstam-Nielsen for helpful discussions and proofreading."
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+                                    "content": "REPRODUCIBILITY STATEMENT"
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+                                    "content": "The main contribution of this work is the derivation of the closed-form solution to the Gauss-Newton loss. We provide details of this derivation in Appendix B. To reproduce our experimental results, we elaborate on the implementation details Appendix E. To facilitate reproducibility in future research, we also publicly release our code."
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+                                    "content": "REFERENCES"
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+                                            "content": "B. Ummenhofer, H. Zhou, J. Uhrig, N. Mayer, E. Ilg, A. Dosovitskiy, and T. Brox. DeMon: Depth and motion network for learning monocular stereo. In CVPR, pages 5038-5047, 2017."
+                                        }
+                                    ]
+                                }
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+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                            "bbox": [
+                                                105,
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+                                            "type": "text",
+                                            "content": "L. von Stumberg, P. Wenzel, Q. Khan, and D. Cremers. GN-Net: The Gauss-Newton loss for multi-weather relocalization. IEEE Robotics Autom. Lett., 5(2):890–897, 2020a."
+                                        }
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+                                            "type": "text",
+                                            "content": "L. von Stumberg, P. Wenzel, N. Yang, and D. Cremers. LM-Reloc: Levenberg-Marquardt based direct visual relocalization. In 3DV, pages 968–977, 2020b."
+                                        }
+                                    ]
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+                                            "type": "text",
+                                            "content": "B. Xu, A. J. Davison, and S. Leutenegger. Deep probabilistic feature-metric tracking. IEEE Robotics Autom. Lett., 6(1):223-230, 2021."
+                                        }
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+                                }
+                            ],
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+                        {
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+                                            "type": "text",
+                                            "content": "K. M. Yi, E. Trulls, V. Lepetit, and P. Fua. LIFT: Learned invariant feature transform. In ECCV, volume 9910, pages 467-483, 2016."
+                                        }
+                                    ]
+                                }
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+                        },
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+                            "bbox": [
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+                                            "type": "text",
+                                            "content": "T. Zhou, M. Brown, N. Snavely, and D. G. Lowe. Unsupervised learning of depth and ego-motion from video. In CVPR, pages 6612-6619, 2017."
+                                        }
+                                    ]
+                                }
+                            ],
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+                        }
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+                                    "type": "text",
+                                    "content": "A OPTICAL FLOW BIAS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
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+                                    "type": "text",
+                                    "content": "We demonstrate that PixLoc exhibits bias inherited from the training dataset. Fig. 5 illustrates the average translation error after optimization when evaluated on the Cambridge dataset, highlighting the impact of different initial displacements along the image axes. To solely evaluate the bias learned by the feature maps, we disabled dataset-specific learned movement priors denoted as "
+                                },
+                                {
+                                    "bbox": [
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+                                        255,
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+                                    "type": "inline_equation",
+                                    "content": "\\lambda"
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+                                    "bbox": [
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+                                        255,
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+                                    "type": "text",
+                                    "content": " in PixLoc (Sarlin et al., 2021). Our results demonstrate that when trained on the CMU dataset, the network predicts optical flow along the X axis more accurately. This increased accuracy is directly linked to the structure of the CMU dataset. Specifically, the dataset comprises image pairs captured by a camera inside a moving car, pointed towards the side of the road. Consequently, this positioning results in the network predominantly learning horizontal optical flow patterns, reflecting the lateral movement observed in the images of the training dataset."
+                                }
+                            ]
+                        }
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+                    "type": "image",
+                    "bbox": [
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+                                            "type": "text",
+                                            "content": "Figure 5: The final error of the PixLoc solution along "
+                                        },
+                                        {
+                                            "bbox": [
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+                                                425,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "X / Y"
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+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": "-axes w.r.t. the initial error. PixLoc remains accurate only along "
+                                        },
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+                                            "type": "inline_equation",
+                                            "content": "X"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": "-axis on CMU, reflecting the dominant motion in the training set."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 10,
+                            "angle": 0,
+                            "type": "image_caption"
+                        }
+                    ],
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+                },
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "B DERIVATION OF THE ANALYTICAL SOLUTION"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
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+                    ],
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+                                    "type": "text",
+                                    "content": "In this section, we provide a solution to minimizing Eq. (16) from the main text. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        516,
+                                        504,
+                                        552
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Omega"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        516,
+                                        504,
+                                        552
+                                    ],
+                                    "type": "text",
+                                    "content": " be the coordinate domain of an image plane "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        516,
+                                        504,
+                                        552
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "[0,1] \\times [0,1]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        516,
+                                        504,
+                                        552
+                                    ],
+                                    "type": "text",
+                                    "content": ", and let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        516,
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+                                        552
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "E \\subseteq \\mathbb{R}^d"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        516,
+                                        504,
+                                        552
+                                    ],
+                                    "type": "text",
+                                    "content": " define the feature space. We aim to find "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        516,
+                                        504,
+                                        552
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{f}: \\Omega \\to E"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        516,
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+                                        552
+                                    ],
+                                    "type": "text",
+                                    "content": ":"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        111,
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                {
+                                    "bbox": [
+                                        111,
+                                        569,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\tilde {f} = \\arg \\min  _ {f} \\sum_ {j} \\int_ {\\Omega} \\left\\| \\tilde {\\boldsymbol {x}} - (J (\\tilde {\\boldsymbol {x}}) ^ {\\intercal} J (\\tilde {\\boldsymbol {x}})) ^ {- 1} J (\\tilde {\\boldsymbol {x}}) ^ {\\intercal} (f (\\tilde {\\boldsymbol {x}}) - F _ {q} ^ {(j)}) - \\boldsymbol {x} ^ {(j)} \\right\\| _ {2} ^ {2} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}) d \\tilde {\\boldsymbol {x}}. \\tag {25}",
+                                    "image_path": "267f9fc0ce911096aeca6eb98e1fce3ce29f0075842aabe81b00d254454a806a.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        104,
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
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+                                    ],
+                                    "type": "text",
+                                    "content": "We start by relaxing the constraint "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
+                                        504,
+                                        646
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\nabla f(\\pmb{x}) = J(\\pmb{x})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
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+                                    ],
+                                    "type": "text",
+                                    "content": ". This makes every value of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
+                                        504,
+                                        646
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f(\\pmb{x})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
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+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
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+                                        646
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "J(\\pmb{x})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
+                                        504,
+                                        646
+                                    ],
+                                    "type": "text",
+                                    "content": " locally independent and, therefore, the minimum is achieved by minimizing Eq. (25) independently over every point in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Omega"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": ":"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
+                                        124,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\{\\tilde {f} (\\tilde {\\boldsymbol {x}}), \\tilde {J} (\\tilde {\\boldsymbol {x}}) \\} = \\underset {\\hat {\\boldsymbol {f}}, \\hat {\\boldsymbol {J}}} {\\arg \\min } \\sum_ {j} \\left\\| \\tilde {\\boldsymbol {x}} - \\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal} \\left(\\hat {\\boldsymbol {f}} - F _ {q} ^ {(j)}\\right) - \\boldsymbol {x} ^ {(j)} \\right\\| _ {2} ^ {2} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}). \\tag {26}",
+                                    "image_path": "2d1180d2ed5177f33261b17fc829477691a11dbca00f838e8547b77276b248c5.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        104,
+                        707,
+                        505,
+                        734
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        707,
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+                                    ],
+                                    "type": "text",
+                                    "content": "Note that the only significant part of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        707,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\pmb{f}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        707,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": " is a part which is spanned by columns of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        707,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\pmb{J}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        707,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": ". This can be easily seen by observing that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        707,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\pmb{J}}^{\\mathrm{T}}\\hat{\\pmb{f}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        707,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": " is a non-normalized projection onto "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        707,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\pmb{J}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        707,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": ". More formally, let us find "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        707,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\pmb{f}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        707,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": " as"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                }
+            ],
+            "discarded_blocks": [
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+                    "type": "header",
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+                                    "bbox": [
+                                        105,
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+                                        38
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "12"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 11
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        80,
+                        504,
+                        106
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                80,
+                                504,
+                                106
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{J}a + \\hat{J}^\\perp b"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": ", where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{J}^\\perp"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": " is a basis for an orthogonal complement for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{span}(\\hat{J})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": ". By substituting this into Eq. (26), we obtain:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        142,
+                        119,
+                        505,
+                        151
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                505,
+                                151
+                            ],
+                            "spans": [
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+                                    "bbox": [
+                                        142,
+                                        119,
+                                        505,
+                                        151
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\{\\tilde {a} (\\tilde {\\boldsymbol {x}}), \\tilde {J} (\\tilde {\\boldsymbol {x}}) \\} = \\underset {\\boldsymbol {a}, \\hat {\\boldsymbol {J}}} {\\arg \\min } \\sum_ {j} \\left\\| \\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)} - \\boldsymbol {a} + \\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal} F _ {q} ^ {(j)} \\right\\| _ {2} ^ {2} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}), \\tag {27}",
+                                    "image_path": "7a94e72d94dbb88bf57017a64a21720f9ef5214e9c5d7accb2b587211d8ea41a.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
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+                        153,
+                        355,
+                        169
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                176,
+                                153,
+                                355,
+                                169
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        176,
+                                        153,
+                                        355,
+                                        169
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\tilde {f} (\\tilde {\\boldsymbol {x}}) \\in \\left\\{\\tilde {J} (\\tilde {\\boldsymbol {x}}) \\tilde {a} (\\tilde {\\boldsymbol {x}}) + \\tilde {J} (\\tilde {\\boldsymbol {x}}) ^ {\\perp} \\boldsymbol {b} \\mid \\boldsymbol {b} \\in \\mathbb {R} ^ {d - 2} \\right\\}.",
+                                    "image_path": "ea26adf9cabe9bc6c129c0820f998e4562554916412a03eced7632cbe5fb46f1.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        178,
+                        504,
+                        227
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                178,
+                                504,
+                                227
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        178,
+                                        504,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": "Observe that this problem is a linear least squares in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        178,
+                                        504,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{a}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        178,
+                                        504,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        178,
+                                        504,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(\\hat{\\boldsymbol{J}}^{\\intercal}\\hat{\\boldsymbol{J}}\\right)^{-1}\\hat{\\boldsymbol{J}}^{\\intercal}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        178,
+                                        504,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": " and recall that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        178,
+                                        504,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p(\\cdot)\\geq 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        178,
+                                        504,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": " (by definition), hence the problem is convex. Minimality conditions for Eq. (27) for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        178,
+                                        504,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{a}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        178,
+                                        504,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        178,
+                                        504,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(\\hat{\\boldsymbol{J}}^{\\intercal}\\hat{\\boldsymbol{J}}\\right)^{-1}\\hat{\\boldsymbol{J}}^{\\intercal}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        178,
+                                        504,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": " are:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                277
+                            ],
+                            "spans": [
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+                                    "bbox": [
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+                                        505,
+                                        277
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\frac {d}{d \\boldsymbol {a}}: \\quad \\sum_ {j} \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)} - \\boldsymbol {a} + \\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal} F _ {q} ^ {(j)}\\right) p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}) = 0, \\tag {28}",
+                                    "image_path": "6e600d8e1269b4b6869aa3409c72645f679bfb8c711813fe6be32ad67634fbb2.jpg"
+                                }
+                            ]
+                        }
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+                    "index": 5
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+                {
+                    "bbox": [
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+                        505,
+                        316
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                123,
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+                                316
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        123,
+                                        281,
+                                        505,
+                                        316
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\frac {d}{d \\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal}}: \\quad \\sum_ {j} \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)} - \\boldsymbol {a} + \\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal} F _ {q} ^ {(j)}\\right) F _ {q} ^ {(j) \\intercal} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right) = 0. \\tag {29}",
+                                    "image_path": "404742471a4dec8216a2c14378457151b55202b618537ea1d6cfa7187dd1fe94.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
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+                        339
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                205,
+                                339
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        327,
+                                        205,
+                                        339
+                                    ],
+                                    "type": "text",
+                                    "content": "From Eq. (28), we have:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
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+                    ],
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+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
+                                        222,
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+                                        374
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\boldsymbol {a} = \\tilde {\\boldsymbol {x}} - \\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}}) + \\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal} \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}}), \\tag {30}",
+                                    "image_path": "afec821b7a576170a331e217b2604ef42711737c1f44c174d30cb9c2ab65988f.jpg"
+                                }
+                            ]
+                        }
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+                    ],
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+                    "angle": 0,
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "where"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
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+                    ],
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+                    "angle": 0,
+                    "lines": [
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+                            ],
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+                                {
+                                    "bbox": [
+                                        204,
+                                        392,
+                                        505,
+                                        423
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}}) := \\sum_ {j} F _ {q} ^ {(j)} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right) / \\sum_ {j} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right), \\tag {31}",
+                                    "image_path": "2d69466351f73db5d2c0c1d5d4313f2c6b9fd3f90fa73d460bdacf2ff357a413.jpg"
+                                }
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+                {
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+                        445
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                422,
+                                404,
+                                445
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        204,
+                                        422,
+                                        404,
+                                        445
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}}) := \\sum_ {j} \\boldsymbol {x} ^ {(j)} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}) / \\sum_ {j} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}).",
+                                    "image_path": "bc7e5aedd95c6a7d68ad84f045a33d06f042b07a9bdfc454ed1731ad4594eeef.jpg"
+                                }
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+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "type": "text",
+                                    "content": "Substituting "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{a}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        455,
+                                        262,
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+                                    ],
+                                    "type": "text",
+                                    "content": " into Eq. (29), we obtain"
+                                }
+                            ]
+                        }
+                    ],
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+                },
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+                            "bbox": [
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+                                {
+                                    "bbox": [
+                                        132,
+                                        488,
+                                        505,
+                                        517
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\sum_ {j} \\left(\\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}}) - \\boldsymbol {x} ^ {(j)} + \\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal} \\left(F _ {q} ^ {(j)} - \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}})\\right)\\right) F _ {q} ^ {(j) \\intercal} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right) = 0, \\tag {32}",
+                                    "image_path": "c91f1ad6b6a385e0c4d68b185f3781458b3fff42977aa051703fd95d5fd2c3a7.jpg"
+                                }
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+                        }
+                    ],
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+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        258,
+                                        520,
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+                                        547
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "- \\sum_ {j} \\left(\\boldsymbol {x} ^ {(j)} - \\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}})\\right) F _ {q} ^ {(j) \\intercal} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}) +",
+                                    "image_path": "f3f57e21926dd8d60cbe5785eb5006e21a566829722e28765b9bac5c30b8f063.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                550,
+                                505,
+                                577
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        217,
+                                        550,
+                                        505,
+                                        577
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal} \\sum_ {i} \\left(F _ {q} ^ {(j)} - \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}})\\right) F _ {q} ^ {(j) \\intercal} p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right) = 0, \\tag {33}",
+                                    "image_path": "57404275fd759d6fe9f55c71f5f2acf23f2577fa98ef8b22072a88de7d641c43.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        278,
+                                        580,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "- \\operatorname {C o v} _ {\\mathbf {x y}} (\\tilde {\\boldsymbol {x}}) + \\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal} \\operatorname {C o v} _ {\\mathbf {y}} (\\tilde {\\boldsymbol {x}}) = 0, \\tag {34}",
+                                    "image_path": "afeb85c7c8614ba6260a91d2e9aa2deb409c38d96daf83e0859896966c23f7b0.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
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+                    ],
+                    "type": "text",
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+                                    ],
+                                    "type": "text",
+                                    "content": "where"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
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+                                    "content": "\\operatorname {C o v} _ {\\mathbf {x y}} (\\tilde {\\boldsymbol {x}}) := \\sum_ {j} \\left(\\boldsymbol {x} ^ {(j)} - \\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}})\\right) \\left(F _ {q} ^ {(j)} - \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}})\\right) ^ {\\intercal} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}), \\tag {35}",
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+                                    "content": "\\operatorname {C o v} _ {\\mathbf {y}} (\\tilde {\\boldsymbol {x}}) := \\sum_ {j} \\left(F _ {q} ^ {(j)} - \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}})\\right) \\left(F _ {q} ^ {(j)} - \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}})\\right) ^ {\\intercal} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}). \\tag {36}",
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+                                    "content": "Solving Eq. (34) for "
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": " and substituting the solution into Eq. (27) results in"
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+                            ]
+                        }
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+                                    "type": "interline_equation",
+                                    "content": "\\tilde {J} (\\tilde {\\boldsymbol {x}}) = \\left(\\operatorname {C o v} _ {\\mathbf {x y}} (\\tilde {\\boldsymbol {x}}) \\operatorname {C o v} _ {\\mathbf {y}} (\\tilde {\\boldsymbol {x}}) ^ {- 1}\\right) ^ {\\dagger}, \\tag {37}",
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                            "content": "Figure 6: Comparison of rotation and translation recalls for our simplified version of GN-loss and probabilistic one"
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+                                    "type": "interline_equation",
+                                    "content": "\\tilde {f} (\\tilde {\\boldsymbol {x}}) = \\tilde {J} (\\tilde {\\boldsymbol {x}}) \\boldsymbol {a} = \\tilde {J} (\\tilde {\\boldsymbol {x}}) \\left(\\tilde {\\boldsymbol {x}} - \\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}})\\right) + \\tilde {J} (\\tilde {\\boldsymbol {x}}) \\left(\\tilde {J} (\\tilde {\\boldsymbol {x}}) ^ {\\intercal} \\tilde {J} (\\tilde {\\boldsymbol {x}})\\right) ^ {- 1} \\tilde {J} (\\tilde {\\boldsymbol {x}}) ^ {\\intercal} \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}}). \\tag {38}",
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+                                    "type": "inline_equation",
+                                    "content": "\\tilde{J} (\\tilde{\\pmb{x}})\\left(\\tilde{J} (\\tilde{\\pmb{x}})^{\\intercal}\\tilde{J} (\\tilde{\\pmb{x}})\\right)^{-1}\\tilde{J} (\\tilde{\\pmb{x}})^{\\intercal}{\\bf y}_m(\\tilde{\\pmb{x}})"
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+                                    "type": "text",
+                                    "content": " can be simplified. Note that in the Gauss-Newton step we are projecting "
+                                },
+                                {
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+                                    "type": "inline_equation",
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+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " onto "
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+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{J} (\\tilde{\\pmb{x}})"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ". "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        327,
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+                                        390
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{J} (\\tilde{\\pmb{x}})\\left(\\tilde{J} (\\tilde{\\pmb{x}})^{\\intercal}\\tilde{J} (\\tilde{\\pmb{x}})\\right)^{-1}\\tilde{J} (\\tilde{\\pmb{x}})^{\\intercal}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        327,
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+                                    ],
+                                    "type": "text",
+                                    "content": " is a projection operator, so we are projecting two times. Therefore, there is another equivalent solution which we will further use for the sake of simplicity:"
+                                }
+                            ]
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+                                    "type": "interline_equation",
+                                    "content": "\\tilde {f} (\\tilde {\\boldsymbol {x}}) = \\tilde {J} (\\tilde {\\boldsymbol {x}}) \\left(\\tilde {\\boldsymbol {x}} - \\mathbf {x} _ {m} (\\tilde {\\boldsymbol {x}})\\right) + \\mathbf {y} _ {m} (\\tilde {\\boldsymbol {x}}). \\tag {39}",
+                                    "image_path": "299e9873294e58f6ed4161b4144ea978d445825abb47006d9258e2cdd842040d.jpg"
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+                                    "type": "text",
+                                    "content": "Observe that substituting "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "\\tilde{f}"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " from Eq. (39) and Eq. (38) into Eq. (26) yields the same loss value."
+                                }
+                            ]
+                        }
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+                    "index": 7
+                },
+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Notably, "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\nabla \\tilde{f} (\\tilde{\\pmb{x}}) = \\tilde{J} (\\tilde{\\pmb{x}})"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " for a uniform distribution "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "p(\\cdot)"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
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+                    "type": "title",
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+                                    "type": "text",
+                                    "content": "C PROBABILISTIC vs. SIMPLIFIED GAUSS-NEWTON LOSS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
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+                                    "type": "text",
+                                    "content": "In this section, we compare our simplified Gauss-Newton loss formulation,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\underset {\\hat {\\boldsymbol {f}}, \\hat {\\boldsymbol {J}}} {\\arg \\min } \\sum_ {j} \\left\\| \\tilde {\\boldsymbol {x}} - \\left(\\hat {\\boldsymbol {J}} ^ {\\intercal} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\intercal} \\left(\\hat {\\boldsymbol {f}} - F _ {q} ^ {(j)}\\right) - \\boldsymbol {x} ^ {(j)} \\right\\| _ {2} ^ {2} p (\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}), \\tag {40}",
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "to the original loss, which was derived through the maximum likelihood estimation (MLE),"
+                                }
+                            ]
+                        }
+                    ],
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+                },
+                {
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+                                    "bbox": [
+                                        113,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\underset {\\hat {\\boldsymbol {f}}, \\hat {\\boldsymbol {J}}} {\\arg \\min } \\sum_ {j} \\left(\\left\\| \\hat {\\boldsymbol {J}} \\left(\\tilde {\\boldsymbol {x}} - \\left(\\hat {\\boldsymbol {J}} ^ {\\mathrm {T}} \\hat {\\boldsymbol {J}}\\right) ^ {- 1} \\hat {\\boldsymbol {J}} ^ {\\mathrm {T}} \\left(\\hat {\\boldsymbol {f}} - F _ {q} ^ {(j)}\\right) - \\boldsymbol {x} ^ {(j)}\\right) \\right\\| _ {2} ^ {2} - \\log \\det  (\\hat {\\boldsymbol {J}} ^ {\\mathrm {T}} \\hat {\\boldsymbol {J}})\\right) p \\left(\\tilde {\\boldsymbol {x}} - \\boldsymbol {x} ^ {(j)}\\right). \\tag {41}",
+                                    "image_path": "bfee25859cc4e1cc43f3839473871e9e301e7200e11a3e0a877aedbacd75164d.jpg"
+                                }
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+                {
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "We note that the solution to the probabilistic loss is much more complicated and computationally expensive. Derivatives for this formulation are high-order polynomials in elements of "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\pmb{f}}"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\pmb{J}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": ". It could be solved by general methods of algebraic geometry like Gröbner basis or Homotopy continuation, but the existence of a closed-form solution is not guaranteed, and we were not able to find one."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
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+                                    "type": "text",
+                                    "content": "Nevertheless, in order to compare the two formulations, we approached the problem numerically. Fig. 6 presents the results. We used the Aachen dataset with hloc poses to plot the recall of query"
+                                }
+                            ]
+                        }
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+                                    "type": "text",
+                                    "content": "images as a function of rotation and translation errors. We observe that the accuracy difference between these two formulations is negligible in practice. However, the simplified version admits a closed-form solution and can be computed several orders of magnitudes more efficiently."
+                                }
+                            ]
+                        }
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+                                    "type": "text",
+                                    "content": "D A STUDY OF FEATURE DESCRIPTORS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
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+                                    "type": "text",
+                                    "content": "We show that our approach is agnostic to the choice of the underlying feature descriptor. To analyze the accuracy of our algorithm in terms of the localization error, we use Cambridge Landmarks (Kendall et al., 2016), which provides ground-truth poses."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
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+                                    "type": "text",
+                                    "content": "We experiment with four popular feature descriptors: SIFT (Lowe, 2004), SOLD2 (Pau-trat et al., 2021), SuperPoint (DeTone et al., 2018) and the recent SiLK (Gleize et al., 2023). We plug them in as the embeddings for interest points detected by SuperPoint. Fig. 7 plots recall (the percentage of successfully localized queries) as a function of the tolerated translation error."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
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+                                    "type": "text",
+                                    "content": "We observe that SuperPoint outperforms all other feature descriptors. SOLD2 (Pautrat et al., 2021) is substantially more inferior in terms of accuracy of the pose estimates. Since it is designed to be a line descriptor, its representation of non-linear structures is not well-defined, thus such outcome is somewhat expected. Interestingly, SuperPoint and SIFT both surpass the more recent SiLK on this benchmark. In comparison to SuperPoint, SiLK has 128-D descrip"
+                                }
+                            ]
+                        }
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+                                    "type": "text",
+                                    "content": "tor, which suggests that it may be less expressive than the 256- "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "D"
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+                                    "type": "text",
+                                    "content": " SuperPoint descriptors."
+                                }
+                            ]
+                        }
+                    ],
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+                },
+                {
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+                                            "type": "text",
+                                            "content": "Figure 7: We study our approach with different feature descriptors on Cambridge Landmarks."
+                                        }
+                                    ]
+                                }
+                            ],
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+                            "angle": 0,
+                            "type": "image_caption"
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+                                    "type": "text",
+                                    "content": "E IMPLEMENTATION DETAILS."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
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+                                    "content": "We implement our approach in PyTorch (Paszke et al., 2019). The primary focus of our implementation is the experimental validation of our derived results. Therefore, we did not optimize the runtime performance, which heavily depends on the number of interest points and 3D points involved in the computation. On average, the code takes approximately 6 seconds and 10 seconds per alignment on CMU and Aachen, respectively, on a single NVIDIA A4000. The scheduler in Algorithm 1 adapts the distribution "
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+                                    "type": "inline_equation",
+                                    "content": "p(\\cdot)"
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+                                    "type": "text",
+                                    "content": " as follows. We initialize "
+                                },
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": " with a truncated uniform distribution of a fixed radius around all interest points. The radius decreases from "
+                                },
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+                                    "type": "text",
+                                    "content": " of the image diagonal to "
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": " in the first 30 iterations. Afterward, the scheduler switches to the normal distribution around each interest point with standard deviation "
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": ". Initially, we define "
+                                },
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": " such that "
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+                                    "type": "text",
+                                    "content": " of the distribution covers "
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": " of the image around the point, and we decrease the coverage ratio to "
+                                },
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+                                    "type": "text",
+                                    "content": "."
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+                        }
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+                                    "type": "text",
+                                    "content": "Since our formulation does not have any outlier removal, we adopt the cut-off from the coarse tracker of DSO (Engel et al., 2016). The idea is to define a threshold that discards all but "
+                                },
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+                                    "type": "text",
+                                    "content": " of the residuals with the lowest norm. In our experiments with SuperPoint, we set the threshold value to 0.4."
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+                            ]
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+                                    "content": "F EVALUATION ON 7SCENES"
+                                }
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+                    "index": 12
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+                                    "content": "Here, we complement our evaluation with the 7Scenes dataset. This dataset features substantial blur and distortions in some scenes, as it was captured with a rolling-shutter Kinect camera. Although SuperPoint features were not trained with such distortions, direct alignment with our closed-form solution remains competitive with PixLoc, both in terms of the median error and the recall Table 2."
+                                }
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+                                            "html": "<table><tr><td rowspan=\"2\">Scene</td><td rowspan=\"2\">Method</td><td rowspan=\"2\">Median error</td><td colspan=\"4\">Recall</td></tr><tr><td>(1cm,1°)</td><td>(5cm,5°)</td><td>(25cm,2°)</td><td>(50cm,5°)</td></tr><tr><td rowspan=\"2\">Heads</td><td>Ours</td><td>0.013m, 1.006°</td><td>24.80%</td><td>93.0%</td><td>86.30%</td><td>92.0%</td></tr><tr><td>Pixloc</td><td>0.013m, 0.863°</td><td>36.40%</td><td>85.60 %</td><td>84.00%</td><td>85.90%</td></tr><tr><td rowspan=\"2\">Office</td><td>Ours</td><td>0.028m, 0.941°</td><td>6.12 %</td><td>80.25%</td><td>92.55%</td><td>99.00%</td></tr><tr><td>Pixloc</td><td>0.026m, 0.792°</td><td>7.95%</td><td>80.70 %</td><td>93.12 %</td><td>96.85%</td></tr><tr><td rowspan=\"2\">Redkitchen</td><td>Ours</td><td>0.037m, 1.444°</td><td>1.56 %</td><td>64.42%</td><td>71.88%</td><td>90.16%</td></tr><tr><td>Pixloc</td><td>0.034m, 1.217°</td><td>3.74%</td><td>67.78 %</td><td>76.56 %</td><td>89.48%</td></tr><tr><td rowspan=\"2\">Pumpkin</td><td>Ours</td><td>0.049m, 1.555°</td><td>1.60 %</td><td>51.65%</td><td>62.65%</td><td>85.35%</td></tr><tr><td>Pixloc</td><td>0.041m, 1.173°</td><td>2.80%</td><td>59.75 %</td><td>71.00 %</td><td>84.40%</td></tr><tr><td rowspan=\"2\">Stairs</td><td>Ours</td><td>0.154m, 3.685°</td><td>1.90 %</td><td>19.90 %</td><td>25.60 %</td><td>60.50 %</td></tr><tr><td>Pixloc</td><td>0.048m, 1.268°</td><td>2.60%</td><td>51.10 %</td><td>59.10 %</td><td>74.70%</td></tr><tr><td rowspan=\"2\">Chess</td><td>Ours</td><td>0.026m, 0.904°</td><td>5.50 %</td><td>91.95%</td><td>95.45%</td><td>98.45%</td></tr><tr><td>Pixloc</td><td>0.024m, 0.812°</td><td>7.75%</td><td>90.75 %</td><td>94.95%</td><td>96.15%</td></tr><tr><td rowspan=\"2\">Fire</td><td>Ours</td><td>0.021m, 0.978°</td><td>10.40 %</td><td>90.65%</td><td>94.05%</td><td>98.45%</td></tr><tr><td>Pixloc</td><td>0.019m, 0.781°</td><td>15.85%</td><td>87.50 %</td><td>87.20%</td><td>90.10%</td></tr></table>",
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+                                    "content": "G QUALITATIVE EXAMPLES"
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+                                    "content": "Fig. 8 visualizes convergence examples of one scene with different pose initializations. The examples consistently demonstrate successful alignment despite suboptimal initialization of varying degree."
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+                                            "content": "Figure 8: Examples of convergence from random initial poses. The green points are projections of 3D points using the current pose estimate; the red points are projections of the 3D points with the initial pose, and the yellow points denote the locations of interest points. It can be seen that despite these highly inaccurate initial poses, our approach converges to the correct solution."
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+[
+    {
+        "type": "text",
+        "text": "APPROXIMATING NASH EQUILIBRIA IN NORMAL-FORM GAMES VIA STOCHASTIC OPTIMIZATION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            99,
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+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ian Gemp",
+        "bbox": [
+            181,
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+            184
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "DeepMind",
+        "bbox": [
+            181,
+            185,
+            256,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "London, UK",
+        "bbox": [
+            181,
+            198,
+            267,
+            210
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "imgemp@google.com",
+        "bbox": [
+            183,
+            212,
+            352,
+            226
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Luke Marris",
+        "bbox": [
+            403,
+            170,
+            498,
+            183
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "DeepMind",
+        "bbox": [
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+    {
+        "type": "text",
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "marris@google.com",
+        "bbox": [
+            403,
+            212,
+            573,
+            226
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Georgios Piliouras",
+        "bbox": [
+            625,
+            170,
+            759,
+            184
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "DeepMind",
+        "bbox": [
+            625,
+            185,
+            700,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "London, UK",
+        "bbox": [
+            625,
+            198,
+            712,
+            210
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "gpil@google.com",
+        "bbox": [
+            625,
+            212,
+            777,
+            226
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT",
+        "text_level": 1,
+        "bbox": [
+            450,
+            263,
+            545,
+            277
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We propose the first loss function for approximate Nash equilibria of normal-form games that is amenable to unbiased Monte Carlo estimation. This construction allows us to deploy standard non-convex stochastic optimization techniques for approximating Nash equilibria, resulting in novel algorithms with provable guarantees. We complement our theoretical analysis with experiments demonstrating that stochastic gradient descent can outperform previous state-of-the-art approaches.",
+        "bbox": [
+            228,
+            294,
+            767,
+            378
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION",
+        "text_level": 1,
+        "bbox": [
+            173,
+            405,
+            336,
+            420
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Nash equilibrium (NE) famously encodes stable behavioral outcomes in multi-agent systems and is arguably the most influential solution concept in game theory. Formally speaking, if  $n$  players independently choose  $n$ , possibly mixed, strategies  $(x_{k}$  for  $k \\in [n]$ ) and their joint strategy  $(x = \\prod_{k} x_{k})$  constitutes a Nash equilibrium, then no player has any incentive to unilaterally deviate from their strategy. This concept has sparked extensive research in various fields, ranging from economics (Milgrom and Weber, 1982) to machine learning (Goodfellow et al., 2014), and has even inspired behavioral theory generalizations such as quantal response equilibria (QREs) which allow for more realistic models of boundedly rational agents (McKelvey and Palfrey, 1995).",
+        "bbox": [
+            169,
+            436,
+            826,
+            547
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Unfortunately, when considering Nash equilibria beyond the 2-player, zero-sum scenario, two significant challenges arise. First, it becomes unclear how  $n$  independent players would collectively identify a Nash equilibrium when multiple equilibria are possible, giving rise to the equilibrium selection problem (Harsanyi et al., 1988). Secondly, even approximating a single Nash equilibrium is known to be computationally intractable and specifically PPAD-complete (Daskalakis et al., 2009). Combining both problems together, e.g., testing for the existence of equilibria with welfare greater than some fixed threshold is NP-hard, and it is in fact even hard to approximate (Austrin et al., 2011).",
+        "bbox": [
+            169,
+            554,
+            826,
+            654
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "From a machine learning practitioner's perspective, such computational complexity results hardly give pause for thought as collectively we have become all too familiar with the unreasonable effectiveness of heuristics in circumventing such obstacles. Famously, non-convex optimization is NP-hard, even if the goal is to compute a local minimizer (Murty and Kabadi, 1985), however, stochastic gradient descent (SGD) and variants succeed in training billion parameter models (Brown et al., 2020).",
+        "bbox": [
+            169,
+            659,
+            826,
+            729
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Unfortunately, computational techniques for Nash equilibrium have so far not achieved anywhere near the same level of success. In contrast, most modern NE solvers for  $n$ -player,  $m$ -action, general-sum, normal-form games (NFGs) are practically restricted to a handful of players and/or actions per player except in special cases, e.g., symmetric (Wiedenbeck and Brinkman, 2023) or mean-field games (Pérolat et al., 2022). For example, when running the suite of all 7 applicable methods from the hallmark gambit library (McKelvey et al., 2016) on a 4-player Blotto game, we find only brute-force pure-NE enumeration is able to return any NE within a 1 hour time limit. Scaling solvers to large games is difficult partially due to the fact that an NFG is represented by a tensor with an exponential  $nm^n$  entries; even reading this description into memory can be computationally prohibitive. More to the point, any computational technique that presumes exact computation of the expectation of this tensor sampled according to  $x$  similarly does not have any hope of scaling beyond small instances.",
+        "bbox": [
+            169,
+            734,
+            826,
+            888
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This inefficiency arguably lies at the core of the differential success between machine learning (ML) optimization and equilibrium computation. For example, numerous techniques exist that reduce the",
+        "bbox": [
+            169,
+            895,
+            826,
+            925
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "page_number",
+        "text": "1",
+        "bbox": [
+            493,
+            948,
+            503,
+            959
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "problem of Nash computation to the minimization of the expectation of a random variable (Section 3). Unfortunately, unlike the source of randomness in ML applications where batch learning suffices to produce unbiased estimators, these techniques do not extend easily to game theory which incorporates non-linear functions such as maximum and best-response. This raises our motivating goal:",
+        "bbox": [
+            169,
+            103,
+            826,
+            161
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Can we solve for Nash equilibria via unbiased stochastic optimization?",
+        "text_level": 1,
+        "bbox": [
+            250,
+            167,
+            746,
+            184
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our results. Following in the successful steps of the interplay between ML and stochastic optimization, we reformulate the approximation of Nash equilibria in a normal-form game as a stochastic non-convex optimization problem admitting unbiased Monte-Carlo estimation. This enables the use of powerful solvers and advances in parallel computing to efficiently enumerate Nash equilibria for  $n$ -player, general-sum games. Furthermore, this re-casting allows practitioners to incorporate other desirable objectives into the problem such as \"find an approximate Nash equilibrium with welfare above  $\\omega$ \" or \"find an approximate Nash equilibrium nearest the current observed joint strategy\" resolving the equilibrium selection problem in an effectively ad-hoc and application tailored manner. Concretely, we make the following contributions by producing:",
+        "bbox": [
+            169,
+            196,
+            826,
+            324
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- A loss  $\\mathcal{L}^{\\tau}(\\pmb{x})$  1) whose global minima well approximate Nash equilibria in normal form games, 2) admits unbiased Monte-Carlo estimation, and 3) is Lipschitz and bounded.",
+            "- Efficient randomized algorithms for approximating Nash equilibria in a novel class of games. The algorithms emerge by employing the family of  $\\mathcal{X}$ -armed bandit approaches to  $\\mathcal{L}^{\\tau}(\\pmb{x})$  and connecting their global stochastic optimization guarantees to global approximate Nash guarantees.",
+            "- An empirical comparison of SGD against state-of-the-art baselines for approximating NEs in large games. In some games, vanilla SGD actually improves upon previous state-of-the-art; in others, SGD is slowed by saddle points, a familiar challenge in deep learning (Dauphin et al., 2014)."
+        ],
+        "bbox": [
+            171,
+            334,
+            823,
+            455
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Overall, this perspective showcases a promising new route to approximating equilibria at scale in practice. We conclude the paper with discussion for future work.",
+        "bbox": [
+            169,
+            467,
+            823,
+            498
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 PRELIMINARIES",
+        "text_level": 1,
+        "bbox": [
+            171,
+            516,
+            341,
+            532
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In an  $n$ -player, normal-form game, each player  $k \\in \\{1, \\dots, n\\} = [n]$  has a strategy set  $\\mathcal{A}_k = \\{a_{k1}, \\dots, a_{km_k}\\}$  consisting of  $m_k$  pure strategies. These strategies can be naturally indexed, so we redefine  $\\mathcal{A}_k = [m_k]$  as an abuse of notation. Each player  $k$  also has a utility function,  $u_k: \\mathcal{A} = \\prod_k \\mathcal{A}_k \\to [0,1]$ , (equiv. \"payoff tensor\") that maps joint actions to payoffs in the unit-interval. We denote the average cardinality of the players' action sets by  $\\bar{m} = \\frac{1}{n} \\sum_k m_k$  and maximum by  $m^* = \\max_k m_k$ . Player  $k$  may play a mixed strategy by sampling from a distribution over their pure strategies. Let player  $k$ 's mixed strategy be represented by a vector  $x_k \\in \\Delta^{m_k-1}$  where  $\\Delta^{m_k-1}$  is the  $(m_k-1)$ -dimensional probability simplex embedded in  $\\mathbb{R}^{m_k}$ . Each function  $u_k$  is then extended to this domain so that  $u_k(\\boldsymbol{x}) = \\sum_{\\boldsymbol{a} \\in \\mathcal{A}} u_k(\\boldsymbol{a}) \\prod_j x_{ja_j}$  where  $\\boldsymbol{x} = (x_1, \\dots, x_n)$  and  $a_j \\in \\mathcal{A}_j$  denotes player  $j$ 's component of the joint action  $\\boldsymbol{a} \\in \\mathcal{A}$ . For convenience, let  $x_{-k}$  denote all components of  $\\boldsymbol{x}$  belonging to players other than player  $k$ .",
+        "bbox": [
+            169,
+            547,
+            826,
+            705
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The joint strategy  $\\pmb{x} \\in \\prod_{k} \\Delta^{m_k - 1}$  is a Nash equilibrium if and only if, for all  $k \\in [n]$ ,  $u_k(z_k, x_{-k}) \\leq u_k(\\pmb{x})$  for all  $z_k \\in \\Delta^{m_k - 1}$ , i.e., no player has any incentive to unilaterally deviate from  $\\pmb{x}$ . Nash is typically relaxed with  $\\epsilon$ -Nash, our focus:  $u_k(z_k, x_{-k}) \\leq u_k(\\pmb{x}) + \\epsilon$  for all  $z_k \\in \\Delta^{m_k - 1}$ .",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "As an abuse of notation, let the atomic action  $a_{k} = e_{k}$  also denote the  $m_{k}$ -dimensional \"one-hot\" vector with all zeros aside from a 1 at index  $a_{k}$ ; its use should be clear from the context. We also introduce  $\\nabla_{x_k}^k$  as player  $k$ 's utility gradient. And for convenience, denote by  $H_{kl}^{k} = \\mathbb{E}_{x_{-kl}}[u_{k}(a_{k},a_{l},x_{-kl})]$  the bimatrix game approximation (Janovskaja, 1968) between players  $k$  and  $l$  with all other players marginalized out;  $x_{-kl}$  denotes all strategies belonging to players other than  $k$  and  $l$  and  $u_{k}(a_{k},a_{l},x_{-kl})$  separates out  $l$ 's strategy  $x_{l}$  from the rest of the players  $x_{-k}$ . Similarly, denote by  $T_{klq}^{k} = \\mathbb{E}_{x_{-klq}}[u_{k}(a_{k},a_{l},a_{q},x_{-klq})]$  the 3-player tensor approximation to the game. Note player  $k$ 's utility can now be written succinctly as  $u_{k}(x_{k},x_{-k}) = x_{k}^{\\top}\\nabla_{x_{k}}^{k} = x_{k}^{\\top}H_{kl}^{k}x_{l} = T_{klq}^{k}x_{k}x_{l}x_{q}$  for any  $l,q$  where we use Einstein notation for tensor arithmetic. For convenience, define  $\\mathrm{diag}(z)$  as the function that places a vector  $z$  on the diagonal of a square matrix, and  $\\mathrm{diag}3: z \\in \\mathbb{R}^{d} \\to \\mathbb{R}^{d \\times d \\times d}$  as a 3-tensor of shape  $(d,d,d)$  where  $\\mathrm{diag}3(z)_{lll} = z_l$ .",
+        "bbox": [
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+            761,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "page_number",
+        "text": "2",
+        "bbox": [
+            493,
+            946,
+            504,
+            959
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "table",
+        "img_path": "images/399ce5aa934ea3dd6ccee6516dd105581301c8e63ec58799b4562f9bcc185322.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Loss</td><td>Function</td><td>Obstacle</td></tr><tr><td>Exploitability (ε)</td><td>maxkεk(x)</td><td>max of r.v.</td></tr><tr><td>Nikaido-Isoda (NI)</td><td>∑kεk(x)</td><td>max of r.v.</td></tr><tr><td>Fully-Diff. Exp</td><td>∑k∑ak∈A[k][max(0,uk(ak,x-k)-uk(x))]2</td><td>max of r.v.</td></tr><tr><td>Gradient-based NI</td><td>NI w/ BRk← aBRk = ΠΔ(xk+η∇xuk(x))</td><td>ΠΔ of r.v.</td></tr><tr><td>Unconstrained</td><td>Loss + Simplex Deviation Penalty</td><td>sampling from xk∈Rmk</td></tr></table>",
+        "bbox": [
+            173,
+            99,
+            841,
+            199
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Previous loss functions for NFGs and their obstacles to unbiased estimation. Note that  $\\epsilon_{k}(\\pmb{x}) = \\max_{z}u_{k}(z,x_{-k}) - u_{k}(\\pmb{x})$  contains a max operator (see equivalent definition in equation (1)).",
+        "bbox": [
+            169,
+            207,
+            826,
+            238
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Following convention from differential geometry, let  $T_v\\mathcal{M}$  be the tangent space of a manifold  $\\mathcal{M}$  at  $v$ . For the interior of the  $d$ -action simplex  $\\Delta^{d-1}$ , the tangent space is the same at every point, so we drop the  $v$  subscript, i.e.,  $T\\Delta^{d-1}$ . We denote the projection of a vector  $z \\in \\mathbb{R}^d$  onto this tangent space as  $\\Pi_{T\\Delta^{d-1}}(z) = [I - \\frac{1}{d}\\mathbf{11}^\\top]z$  and call  $\\Pi_{T\\Delta^{d-1}}(\\nabla_{x_k}^k)$  a projected-gradient. We drop  $d-1$  when the dimensionality is clear from the context. Finally, let  $\\mathcal{U}(S)$  denote a discrete uniform distribution over elements from set  $S$ .",
+        "bbox": [
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+            260,
+            826,
+            347
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 RELATED WORK",
+        "text_level": 1,
+        "bbox": [
+            171,
+            366,
+            346,
+            382
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Representing the problem of computing a Nash equilibrium as an optimization problem is not new. A variety of loss functions and pseudo-distance functions have been proposed. Most of them measure some function of how much each player can exploit the joint strategy by unilaterally deviating:",
+        "bbox": [
+            169,
+            397,
+            823,
+            441
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\epsilon_ {k} (\\boldsymbol {x}) \\stackrel {\\text {d e f}} {=} u _ {k} \\left(\\operatorname {B R} _ {k}, x _ {- k}\\right) - u _ {k} (\\boldsymbol {x}) \\text {w h e r e} \\operatorname {B R} _ {k} \\in \\underset {z} {\\arg \\max } u _ {k} (z, x _ {- k}). \\tag {1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            272,
+            446,
+            823,
+            474
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As argued in the introduction, we believe it is important to be able to subsample payoff tensors of normal-form games in order to scale to large instances. As Nash equilibria can consist of mixed strategies, it is advantageous to be able to sample from an equilibrium to estimate its exploitability  $\\epsilon$ . However none of these losses is amenable to unbiased estimation under sampled play. Each of the functions currently explored in the literature is biased under sampled play either because 1) a random variable appears as the argument of a complex, nonlinear (non-polynomial) function or because 2) how to sample play is unclear. Exploitability, Nikaido-Isoda (NI) (Nikaidô and Isoda, 1955) (also known by NashConv (Lanctot et al., 2017) and ADI (Gemp et al., 2022)), as well as fully-differentiable options (Shoham and Leyton-Brown, 2008, p. 106, Eqn 4.31) introduce bias when a max over payoffs is estimated using samples from  $x$ . Gradient-based NI (Raghunathan et al., 2019) requires projecting the result of a gradient-ascent step onto the simplex; for the same reason as the max, this projection is prohibitive because it is a nonlinear operation which introduces bias. Lastly, unconstrained optimization approaches (Shoham and Leyton-Brown, 2008, p. 106) that instead penalize deviation from the simplex lose the ability to sample from strategies when each iterate  $x$  is no longer a distribution (i.e.,  $x_{k} \\notin \\Delta^{m_{k}-1}$ ). Table 1 summarizes these complications.",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4 NASH EQUILIBRIUM AS STOCHASTIC OPTIMIZATION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            715,
+            645,
+            729
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We will now develop our proposed loss function which is amenable to unbiased estimation. Sub-sections 4.1-4.4 provide a warm-up in which we assume an interior (fully-mixed) Nash equilibrium exists. Subsection 4.5 then shows how to relax that assumption allowing us to approximate partially mixed equilibria as well (including pure equilibria). Our key technical insight is to pay special attention to the geometry of the simplex. To our knowledge, prior works have failed to recognize the role of the tangent space  $T\\Delta$ . Proofs are in the appendix.",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4.1 STATIONARITY ON THE SIMPLEX INTERIOR",
+        "text_level": 1,
+        "bbox": [
+            171,
+            845,
+            516,
+            859
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Lemma 1. Assuming player  $k$ 's utility,  $u_{k}(x_{k}, x_{-k})$ , is concave in its own strategy  $x_{k}$ , a strategy in the interior of the simplex is a best response  $BR_{k}$  if and only if it has zero projected-gradient $^{1}$  norm.",
+        "bbox": [
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+            872,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "page_footnote",
+        "text": "<sup>1</sup>Not to be confused with the nonlinear (biased) projected gradient operator in (Hazan et al., 2017).",
+        "bbox": [
+            191,
+            909,
+            777,
+            924
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "page_number",
+        "text": "3",
+        "bbox": [
+            493,
+            948,
+            503,
+            959
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In NFGs, each player's utility is linear in  $x_{k}$ , thereby satisfying the concavity condition of Lemma 1.",
+        "bbox": [
+            169,
+            103,
+            826,
+            119
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.2 PROJECTED-GRADIENT NORM AS A LOSS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            133,
+            504,
+            148
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "An equivalent description of a Nash equilibrium is a joint strategy  $\\pmb{x}$  where every player's strategy is a best response to the equilibrium (i.e.,  $x_{k} = \\mathbb{R}_{k}$  so that  $\\epsilon_k(\\pmb {x}) = 0$ ). Lemma 1 states that any interior best response has zero projected-gradient norm, which inspires the following loss function",
+        "bbox": [
+            169,
+            160,
+            823,
+            203
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} (\\boldsymbol {x}) = \\sum_ {k} \\eta_ {k} \\left\\| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) \\right\\| ^ {2} \\tag {2}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            395,
+            205,
+            823,
+            236
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where each  $\\eta_{k} > 0$  represents a scalar weight, or equivalently, a step size to be explained next.",
+        "bbox": [
+            169,
+            239,
+            792,
+            253
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1. The loss  $\\mathcal{L}$  is equivalent to NashConv, but where player  $k$ 's best response is approximated by a single step of projected-gradient ascent with step size  $\\eta_k$ :  $aBR_k = x_k + \\eta_k\\Pi_{T\\Delta}(\\nabla_{x_k}^k)$ .",
+        "bbox": [
+            169,
+            255,
+            826,
+            286
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This connection was already pointed out in prior work for unconstrained problems (Gemp et al., 2022; Raghunathan et al., 2019), but this result is the first for strategies constrained to the simplex.",
+        "bbox": [
+            169,
+            292,
+            826,
+            323
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.3 CONNECTION TO TRUE EXPLOITABILITY",
+        "text_level": 1,
+        "bbox": [
+            171,
+            338,
+            500,
+            351
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In general, we can bound exploitability in terms of the projected-gradient norm as long as each player's utility is concave (this result extends to subgradients of non-smooth functions).",
+        "bbox": [
+            169,
+            364,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Lemma 2. The amount a player can gain by exploiting a joint strategy  $\\pmb{x}$  is upper bounded by a quantity proportional to the norm of the projected-gradient:",
+        "bbox": [
+            169,
+            395,
+            823,
+            422
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\epsilon_ {k} (\\boldsymbol {x}) \\leq \\sqrt {2} | | \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) | |. \\tag {3}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+            425,
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+            445
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This bound is not tight on the boundary of the simplex, which can be seen clearly by considering  $x_{k}$  to be part of a pure strategy equilibrium. In that case, this analysis assumes  $x_{k}$  can be improved upon by a projected-gradient ascent step (via the equivalence pointed out in Proposition 1). However, that is false because the probability of a pure strategy cannot be increased beyond 1. We mention this to provide further intuition for why our \"warm-up\" loss  $\\mathcal{L}(\\boldsymbol{x})$  is only valid for interior equilibria.",
+        "bbox": [
+            169,
+            452,
+            823,
+            523
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Note that  $||\\Pi_{T\\Delta}(\\nabla_{x_k}^k)|| \\leq ||\\nabla_{x_k}^k||$  because  $\\Pi_{T\\Delta}$  is a projection. Therefore, this improves the naive bounds on exploitability and distance to best responses given using the \"raw\" gradient  $\\nabla_{x_k}^k$ .",
+        "bbox": [
+            169,
+            529,
+            823,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Lemma 3. The exploitability of a joint strategy  $\\pmb{x}$ , is upper bounded by a function of  $\\mathcal{L}(\\pmb{x})$ :",
+        "bbox": [
+            169,
+            564,
+            772,
+            579
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\epsilon \\leq \\sqrt {\\frac {2 n}{\\min  _ {k} \\eta_ {k}}} \\sqrt {\\mathcal {L} (\\boldsymbol {x})} \\stackrel {\\text {d e f}} {=} f (\\mathcal {L}). \\tag {4}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+            582,
+            823,
+            616
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.4 UNBIASED ESTIMATION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            628,
+            383,
+            642
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As discussed in Section 3, a primary obstacle to unbiased estimation of  $\\mathcal{L}(\\pmb{x})$  is the presence of complex, nonlinear functions of random variables, with the projection of a point onto the simplex being one such example (see  $\\Pi_{\\Delta}$  in Table 1). However,  $\\Pi_{T\\Delta}$ , the projection onto the tangent space of the simplex, is linear! This is the insight that allows us to design an unbiased estimator (Lemma 5).",
+        "bbox": [
+            169,
+            655,
+            826,
+            712
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our proposed loss requires computing the squared norm of the expected value of the projected-gradient under the players' mixed strategies, i.e., the  $l$ -th entry of player  $k$ 's gradient equals  $\\nabla_{x_{kl}}^{k} = \\mathbb{E}_{a_{-k} \\sim x_{-k}} u_{k}(a_{kl}, a_{-k})$ . By analogy, consider a random variable  $Y$ . In general,  $\\mathbb{E}[Y]^2 \\neq \\mathbb{E}[Y^2]$ . This means that we cannot just sample projected-gradients and then compute their average norm to estimate our loss. However, consider taking two independent samples from two corresponding identically distributed, independent random variables  $Y^{(1)}$  and  $Y^{(2)}$ . Then  $\\mathbb{E}[Y^{(1)}]^2 = \\mathbb{E}[Y^{(1)}] \\mathbb{E}[Y^{(2)}] = \\mathbb{E}[Y^{(1)}Y^{(2)}]$  by properties of expected value over products of independent random variables. This is a common technique to construct unbiased estimates of expectations over polynomial functions of random variables. Proceeding in this way, define  $\\nabla_{x_k}^{k(1)}$  as a random, unbiased gradient estimate (see Table 2). Let  $\\nabla_{x_k}^{k(2)}$  be independent and distributed identically to  $\\nabla_{x_k}^{k(1)}$ . Then Lemma 5 shows",
+        "bbox": [
+            169,
+            718,
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+            869
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} (\\boldsymbol {x}) = \\mathbb {E} [ \\sum_ {k} \\eta_ {k} (\\underbrace {\\hat {\\nabla} _ {x _ {k}} ^ {k (1)} - \\frac {1}{m _ {k}} \\left(\\mathbf {1} ^ {\\top} \\hat {\\nabla} _ {x _ {k}} ^ {k (1)}\\right) \\mathbf {1}} _ {\\text {p r o j e c t e d - g r a d i e n t 1}}) ^ {\\top} (\\underbrace {\\hat {\\nabla} _ {x _ {k}} ^ {k (2)} - \\frac {1}{m _ {k}} \\left(\\mathbf {1} ^ {\\top} \\hat {\\nabla} _ {x _ {k}} ^ {k (2)}\\right) \\mathbf {1}} _ {\\text {p r o j e c t e d - g r a d i e n t 2}}) ] \\tag {5}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            251,
+            871,
+            823,
+            921
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "page_number",
+        "text": "4",
+        "bbox": [
+            493,
+            948,
+            504,
+            959
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/576cad80c74d2c237fdf399b3c2a2b2781eb7705b85c3e786323caf01c63c39e.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Exact</td><td>Sample Others</td><td>Sample All</td></tr><tr><td>Estimator of ∇k(xk)</td><td>[uk(akl, x-k)]l</td><td>[uk(akl, a-k ~ x-k)]l</td><td>mkuk(akl ~ U(Ak), a-k ~ x-k)el</td></tr><tr><td>ˆ∇k(xk) Bounds</td><td>[0, 1]</td><td>[0, 1]</td><td>[0, mk]</td></tr><tr><td>ˆ∇k(xk) Query Cost</td><td>Πn{k=1}mk</td><td>mk</td><td>1</td></tr><tr><td>ˆL Bounds</td><td>±1/4 Σkηkmk</td><td>±1/4 Σkηkmk</td><td>±1/4 Σkηkm3</td></tr><tr><td>ˆL Query Cost</td><td>n Πn{k=1}mk</td><td>2nm</td><td>2n</td></tr></table>",
+        "bbox": [
+            173,
+            99,
+            862,
+            204
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Examples and Properties of Unbiased Estimators of Loss and Player Gradients  $(\\hat{\\nabla}_{x_k}^{k(p)})$",
+        "bbox": [
+            181,
+            213,
+            815,
+            232
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where  $\\hat{\\nabla}_{x_k}^{k(p)}$  is an unbiased estimator of player  $k$ 's gradient. This estimator can be constructed in several ways. The most expensive, an exact estimator, is constructed by marginalizing player  $k$ 's payoff tensor over all other players' strategies. However, a cheaper estimate can be obtained at the expense of higher variance by approximating this marginalization with a Monte Carlo (MC) estimate of the expectation. Specifically, if we sample a single action for each of the remaining players, we can construct an unbiased estimate of player  $k$ 's gradient by considering the payoff of each of its actions against the sampled background strategy. Lastly, we can consider constructing an estimate of player  $k$ 's gradient by sampling only a single action from player  $k$  to represent their entire gradient. Each of these approaches is outlined in Table 2 along with the query complexity (Babichenko, 2016) of computing the estimator and bounds on the values it can take (Lemma 9).",
+        "bbox": [
+            169,
+            258,
+            826,
+            401
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We can extend Lemma 3 to one that holds under  $T$  samples with probability  $1 - \\delta$  by applying, for example, a Hoeffding bound:  $\\epsilon \\leq f\\bigl (\\hat{\\mathcal{L}} (\\pmb {x}) + \\mathcal{O}\\bigl (\\sqrt{\\frac{1}{T}\\ln(1 / \\delta)}\\bigr)\\bigr)$  where  $\\hat{\\mathcal{L}}$  is an MC estimate of  $\\mathcal{L}$ .",
+        "bbox": [
+            169,
+            406,
+            823,
+            445
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.5 INTERIOR EQUILIBRIA",
+        "text_level": 1,
+        "bbox": [
+            171,
+            465,
+            372,
+            479
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We discussed earlier that  $\\mathcal{L}(\\pmb{x})$  captures interior equilibria. But some games may only have partially mixed equilibria, i.e., equilibria that lie on the boundary of the simplex. We show how to circumvent this shortcoming by considering quantal response equilibria (QREs), specifically, logit equilibria. By adding an entropy bonus to each player's utility, we can",
+        "bbox": [
+            169,
+            492,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- guarantee all equilibria are interior,",
+            "- still obtain unbiased estimates of our loss,",
+            "- maintain an upper bound on the exploitability  $\\epsilon$  of any approximate Nash equilibrium in the original game (i.e., the game without an entropy bonus)."
+        ],
+        "bbox": [
+            215,
+            561,
+            823,
+            636
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Define  $u_{k}^{\\tau}(\\boldsymbol{x}) = u_{k}(\\boldsymbol{x}) + \\tau S(x_{k})$  where Shannon entropy  $S(x_{k}) = -\\sum_{l}x_{kl}\\ln (x_{kl})$  is 1-strongly concave with respect to the 1-norm (Beck and Teboulle, 2003). It is known that Nash equilibria of entropy-regularized games satisfy the conditions for logit equilibria (Leonardos et al., 2021), which are solutions to the fixed point equation  $x_{k} = \\mathrm{softmax}\\left(\\frac{1}{\\tau}\\nabla_{x_{k}}^{k}\\right)$ . The softmax should make it clear to the reader that all probabilities have positive mass at positive temperature.",
+        "bbox": [
+            169,
+            648,
+            823,
+            720
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Recall that in order to construct an unbiased estimate of our loss, we simply needed to construct unbiased estimates of player gradients. The introduction of the entropy term to player  $k$ 's utility is special in that it depends entirely on known quantities, i.e., the player's own mixed strategy. We can directly and deterministically compute  $\\tau \\frac{dS}{dx_k} = -\\tau (\\ln (x_k) + 1)$  and add this to our estimator of  $\\nabla_{x_k}^{k(p)}\\colon \\hat{\\nabla}_{x_k}^{k\\tau (p)} = \\hat{\\nabla}_{x_k}^{k(p)} + \\tau \\frac{dS}{dx_k}$ . Consider our loss function refined from (2) with changes in blue:",
+        "bbox": [
+            169,
+            726,
+            826,
+            806
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) = \\sum_ {k} \\eta_ {k} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right| \\right| ^ {2}. \\tag {6}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            388,
+            832,
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+            863
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As mentioned above, the utilities with entropy bonuses are still concave, therefore, a similar bound to Lemma 2 applies. We use this to prove the QRE counterpart to Lemma 3 where  $\\epsilon_{QRE}$  is the exploitability of an approximate equilibrium in a game with entropy bonuses.",
+        "bbox": [
+            169,
+            881,
+            823,
+            925
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "page_number",
+        "text": "5",
+        "bbox": [
+            493,
+            948,
+            503,
+            959
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/aa96af27b43a90f0e534e1bbca3272e2e724c9205afedc906d1592c007e9368c.jpg",
+        "image_caption": [
+            "Figure 1: Effect of Sampled Play on a Biased Loss. The first row displays the expectation of the upper bound guaranteed by our proposed loss  $\\mathcal{L}^{\\tau}$  with  $\\eta_{k} = 1$  for all  $k$ . The second row displays the expectation of NashConv under sampled play, i.e.,  $\\sum_{k}\\epsilon_{k}$  where  $\\epsilon_{k} = \\mathbb{E}_{a_{-k}\\sim x_{-k}}[\\max_{a_{k}}u_{k}^{\\tau}(\\pmb{a})] - \\mathbb{E}_{\\pmb{a}\\sim \\pmb{x}}[u_{k}^{\\tau}(\\pmb{a})]$ . To be consistent, we subtract the offset  $\\tau \\log (m^{2})$  from  $f_{\\tau}(\\mathcal{L}^{\\tau})$  per Lemma 13, which relates the exploitability at positive temperature to that at zero temperature. The resulting loss surface clearly shows NashConv fails to recognize any interior Nash equilibrium due to its inherent bias."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            99,
+            823,
+            266
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Lemma 4. The entropy regularized exploitability,  $\\epsilon_{QRE}$ , of a joint strategy  $\\pmb{x}$ , is upper bounded as:",
+        "bbox": [
+            169,
+            393,
+            826,
+            410
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\epsilon_ {Q R E} \\leq \\sqrt {\\frac {2 n}{\\min  _ {k} \\eta_ {k}}} \\sqrt {\\mathcal {L} ^ {\\tau} (\\boldsymbol {x})} \\stackrel {\\mathrm {d e f}} {=} f \\left(\\mathcal {L} ^ {\\tau}\\right). \\tag {7}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            362,
+            419,
+            823,
+            454
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Lastly, we establish a connection between quantal response equilibria and Nash equilibria that allows us to approximate Nash equilibria in the original game via minimizing our modified loss  $\\mathcal{L}^{\\tau}(\\pmb{x})$ .",
+        "bbox": [
+            169,
+            467,
+            823,
+            497
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Lemma 13 ( $\\mathcal{L}^{\\tau}$  Scores Nash Equilibria). Let  $\\mathcal{L}^{\\tau}(\\pmb{x})$  be our proposed entropy regularized loss function and  $\\pmb{x}$  be an approximate QRE. Then it holds that",
+        "bbox": [
+            168,
+            500,
+            823,
+            531
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\epsilon \\leq \\tau \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {\\frac {2 n}{\\min  _ {k} \\eta_ {k}}} \\sqrt {\\mathcal {L} ^ {\\tau} (\\boldsymbol {x})} \\stackrel {\\text {d e f}} {=} f _ {\\tau} \\left(\\mathcal {L} ^ {\\tau}\\right). \\tag {8}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            313,
+            539,
+            825,
+            575
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "This upper bound is plotted as a heatmap for a familiar Chicken game in the top row of Figure 1. First, notice how pure equilibria are not visible as minima for zero temperature, but appear for slightly warmer temperatures. Secondly, notice that NashConv in the bottom row is unable to capture the interior Nash equilibrium because of its high bias under sampled play. In contrast, our proposed loss  $\\mathcal{L}^{\\tau}$  is guaranteed to capture all equilibria at low temperature  $\\tau$ .",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 ANALYSIS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            686,
+            292,
+            700
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In the preceding section we established a loss function that upper bounds the exploitability of an approximate equilibrium. In addition, the zeros of this loss function have a one-to-one correspondence with quantal response equilibria (which approximate Nash equilibria at low temperature).",
+        "bbox": [
+            169,
+            718,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Here, we derive properties that suggest it is \"easy\" to optimize. While this function is generally non-convex and may suffer from a proliferation of saddle points (Figure 2), it is Lipschitz continuous (over the relevant subset of the interior) and bounded. These are two commonly made assumptions in the literature on non-convex optimization, which we leverage in Section 6. In addition, we can derive its gradient, its Hessian, and characterize its behavior around global minima.",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Lemma 14. The gradient of  $\\mathcal{L}^{\\tau}(\\pmb{x})$  with respect to player  $l$ 's strategy  $x_{l}$  is",
+        "bbox": [
+            169,
+            842,
+            665,
+            858
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\nabla_ {x _ {l}} \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) = 2 \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\tag {9}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            372,
+            867,
+            823,
+            898
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where  $B_{ll} = -\\tau \\left[I - \\frac{1}{m_l}\\mathbf{11}^\\top \\right]diag\\left(\\frac{1}{x_l}\\right)$  and  $B_{kl} = \\left[I - \\frac{1}{m_k}\\mathbf{11}^\\top \\right]H_{kl}^k$  for  $k\\neq l$",
+        "bbox": [
+            169,
+            906,
+            704,
+            928
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "page_number",
+        "text": "6",
+        "bbox": [
+            493,
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+            959
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/3950c7b2214361a1a08a415ef68ef1df7362ca3ac8ec729c3ae280a202991d77.jpg",
+        "image_caption": [
+            "Figure 2: Analysis of Loss Landscape. We reapply the analysis of (Dauphin et al., 2014), originally designed to understand the success of SGD in deep learning, to \"slices\" of several popular extensive form games. To construct a slice (or meta-game), we randomly sample 6 deterministic policies and then consider the corresponding  $n$ -player, 6-action normal-form game at  $\\tau = 0.1$  (with payoffs normalized to [0, 1]). The index of a critical point  $x_{c}$  ( $\\nabla_{\\pmb{x}}\\mathcal{L}^{\\tau}(\\pmb{x}_{c}) = \\mathbf{0}$ ) indicates the fraction of negative eigenvalues in the Hessian of  $\\mathcal{L}^{\\tau}$  at  $x_{c}$ ;  $\\alpha = 0$  indicates a local minimum, 1 a maximum, else a saddle point. We see a positive correlation between exploitability ( $y$ -axis), projected-gradient norm ( $x$ -axis), and  $\\alpha$  (color) indicating a lower prevalence of local minima at high exploitability."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Lemma 16. The Hessian of  $\\mathcal{L}^{\\tau}(\\pmb{x})$  can be written",
+        "bbox": [
+            171,
+            320,
+            504,
+            335
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\operatorname {H e s s} \\left(\\mathcal {L} ^ {\\tau}\\right) = 2 \\left[ \\tilde {B} ^ {\\top} \\tilde {B} + T \\Pi_ {T \\Delta} \\left(\\tilde {\\nabla} ^ {\\tau}\\right) \\right] \\tag {10}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            369,
+            338,
+            823,
+            358
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where  $\\tilde{B}_{kl} = \\sqrt{\\eta_k} B_{kl}$ ,  $\\Pi_{T\\Delta}(\\tilde{\\nabla}^{\\tau}) = [\\eta_1\\Pi_{T\\Delta}(\\nabla_{x_1}^{1\\tau}),\\dots ,\\eta_n\\Pi_{T\\Delta}(\\nabla_{x_n}^{n\\tau})]$ , and we augment  $T$  (the 3-player approximation to the game,  $T_{lqk}^{k}$ ) so that  $T_{lll}^{l} = \\tau \\text{diag} 3\\left(\\frac{1}{x_{l}^{2}}\\right)$ .",
+        "bbox": [
+            169,
+            363,
+            823,
+            400
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "At an NE, the latter term disappears because  $\\Pi_{T\\Delta}(\\nabla_{x_k}^{k\\tau}) = 0$  for all  $k$  (Lemma 1). If  $\\mathcal{X}$  was  $\\mathbb{R}^{n\\bar{m}}$ , then we could simply check if  $\\tilde{B}$  is full-rank to determine if  $Hess\\succ 0$ , i.e., if  $\\mathcal{L}^{\\tau}$  is locally strongly-convex. However,  $\\mathcal{X}$  is a simplex product, and we only care about curvature in directions toward which we can update our strategy profile  $\\pmb{x}$ . Toward that end, define  $M$  to be the  $n(\\bar{m} +1)\\times n\\bar{m}$  matrix that stacks  $\\tilde{B}$  on top of a repeated identity matrix that encodes orthogonality to the simplex:",
+        "bbox": [
+            169,
+            410,
+            826,
+            488
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nM (\\boldsymbol {x}) = \\left[ \\begin{array}{c c c c} - \\tau \\sqrt {\\eta_ {1}} \\Pi_ {T \\Delta} \\left(\\frac {1}{x _ {1}}\\right) & \\sqrt {\\eta_ {1}} \\Pi_ {T \\Delta} \\left(H _ {1 2} ^ {1}\\right) & \\dots & \\sqrt {\\eta_ {1}} \\Pi_ {T \\Delta} \\left(H _ {1 n} ^ {1}\\right) \\\\ \\vdots & \\vdots & \\vdots & \\vdots \\\\ \\sqrt {\\eta_ {n}} \\Pi_ {T \\Delta} \\left(H _ {n 1} ^ {n}\\right) & \\dots & \\sqrt {\\eta_ {n}} \\Pi_ {T \\Delta} \\left(H _ {n, n - 1} ^ {n}\\right) & - \\tau \\sqrt {\\eta_ {n}} \\Pi_ {T \\Delta} \\left(\\frac {1}{x _ {n}}\\right) \\\\ \\mathbf {1} _ {1} ^ {\\top} & 0 & \\dots & 0 \\\\ \\vdots & \\vdots & \\vdots & \\vdots \\\\ 0 & \\dots & 0 & \\mathbf {1} _ {n} ^ {\\top} \\end{array} \\right] \\tag {11}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            184,
+            489,
+            825,
+            604
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where  $\\Pi_{T\\Delta}(z\\in \\mathbb{R}^{a\\times b}) = [I_a - \\frac{1}{a}\\mathbf{1}_a\\mathbf{1}_a^\\top ]z$  subtracts the mean from each column of  $z$  and  $\\frac{1}{x_k}$  is shorthand for  $\\mathrm{diag}\\left(\\frac{1}{x_k}\\right)$ . If  $M(\\pmb {x})z = \\mathbf{0}$  for a nonzero vector  $z\\in \\mathbb{R}^{n\\bar{m}}$ , this implies there exists a  $z$  that 1) is orthogonal to the ones vectors of each simplex (i.e., is a valid equilibrium update direction) and 2) achieves zero curvature in the direction  $z$ , i.e.,  $z^{\\top}(\\tilde{B}^{\\top}\\tilde{B})z = z^{\\top}(Hess)z = 0$ , and so  $Hess$  is not positive definite. Conversely, if  $M(\\pmb {x})$  is of rank  $nm$  for a quantal response equilibrium  $\\pmb{x}$ , then the Hessian of  $\\mathcal{L}^{\\tau}$  at  $\\pmb{x}$  in the tangent space of the simplex product  $(\\mathcal{X} = \\prod_{k}\\mathcal{X}_{k})$  is positive definite. In this case, we call  $\\pmb{x}$  polymatrix-isolated: polymatrix because we only require information of the local polymatrix approximation of the game (i.e., the  $H_{kl}^{k}$  matrices) to construct  $M$  and isolated because it implies  $\\pmb{x}$  is not connected to any other equilibria.",
+        "bbox": [
+            169,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Definition 1 (Polymatrix-Isolated Equilibrium). A Nash equilibrium  $\\pmb{x}^*$  is polymatrix-isolated iff  $\\pmb{x}^*$  is isolated according to its local polymatrix game approximation.",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "By analyzing the rank of  $M$ , we can confirm that many classical matrix games including Rock-Paper-Scissors, Chicken, Matching Pennies, and Shapley's game all induce strongly convex  $\\mathcal{L}^{\\tau}$ 's at zero temperature (i.e., they have unique mixed Nash equilibria). In contrast, a game like Prisoner's Dilemma has a unique pure strategy that will not be captured by our loss at zero temperature.",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6 ALGORITHMS",
+        "text_level": 1,
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We have formally transformed the approximation of Nash equilibria in NFGs into a stochastic optimization problem. To our knowledge, this is the first such formulation that allows one-shot",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "page_number",
+        "text": "7",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/51cefaa3f14fb55640f2d9f76479108debe8b13cf26aeac4adcfc645c166d6c0.jpg",
+        "image_caption": [
+            "Figure 3: Comparison of SGD on  $\\mathcal{L}^{\\tau = 0}$  against baselines on four games evaluated in (Gemp et al., 2022). The number of samples used to estimate each update iteration (i.e., minibatch size) is indicated by  $s$ . From left to right: 2-player, 3-action, nonsymmetric; 6-player, 5-action, nonsymmetric; 4-player, 66-action, symmetric; 3-player, 286-action, symmetric. SGD struggles at saddle points in Blotto."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
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+        "img_path": "images/3355d02bf5dbe469ec7c5b88ce1e5106f509f2401cd6eebfd7612e4d6bd0d2d9.jpg",
+        "image_caption": [],
+        "image_footnote": [],
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+        ],
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+    },
+    {
+        "type": "image",
+        "img_path": "images/1da09690fe777c5fd7c163640db91b47dcbc7b6997a551d140ea8c7abe63ac0b.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/cb1645e6ab4d6219c74bec4561b3ac9c887f14a935c2b7586b7ff917bc6a95c6.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/58308c7ba08d80e379bbefdd3a191d08e19714edaf2140e2b7ad1643afdf437d.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "unbiased Monte-Carlo estimation which is critical to introduce the use of powerful algorithms capable of solving high dimensional optimization problems. We explore two off-the-shelf approaches.",
+        "bbox": [
+            169,
+            277,
+            823,
+            308
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6.1 STOCHASTIC GRADIENT DESCENT",
+        "text_level": 1,
+        "bbox": [
+            171,
+            321,
+            455,
+            335
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Stochastic gradient descent is the workhorse of high-dimensional stochastic optimization. It is guaranteed to converge to stationary points (Cutkosky et al., 2023), however, it may converge to local, rather than global minima. It also enjoys implicit gradient regularization (Barrett and Dherin, 2020), seeking \"flat\" minima and performs approximate Bayesian inference (Mandt et al., 2017). Despite the lack of global convergence guarantee, we find it performs well empirically in games previously examined by the literature: modified Shapley's (Ostrovski and van Strien, 2014), GAMUT D7 (Nudelman et al., 2004), Blotto (Arad and Rubinstein, 2012). Figure 3 shows SGD is competitive with scalable techniques to approximating NEs: FTRL (Shalev-Shwartz and Singer, 2006; Shalev-Shwartz et al., 2012), Regret Matching (Hart and Mas-Colell, 2000), ADIDAS/  $^yQRRe^{auto}$  (Gemp et al., 2022). Shapley's game induces a strongly convex  $\\mathcal{L}$  (see Section 5) leading to SGD's strong performance. Blotto reaches low, but nonzero  $\\epsilon$ , demonstrating the challenges of saddle points.",
+        "bbox": [
+            169,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6.2 HIGH PROBABILITY, GLOBAL POLYNOMIAL CONVERGENCE RATES VIA BANDITS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            517,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We explore one other algorithmic approach to non-convex optimization based on minimizing regret, which enjoys finite time global convergence rates.  $\\mathcal{X}$ -armed bandits (Bubeck et al., 2011) systematically explore the space of solutions by refining a mesh over the joint strategy space, trading off exploration versus exploitation of promising regions. Several approaches exist (Bartlett et al., 2019; Valko et al., 2013) with open source implementations, e.g., (Li et al., 2023). Applying  $\\mathcal{X}$ -armed bandits to our  $\\mathcal{L}^{\\tau}$  can be thought of as a stochastic generalization of the exclusion method and other bandit approaches for Nash equilibria (Berg and Sandholm, 2017; Zhou et al., 2017).",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Equipped with these techniques, we establish a high probability polynomial-time global convergence rate to Nash equilibria in  $n$ -player, general-sum games given all QREs( $\\tau$ ) are polymatrix-isolated. The quality of this approximation improves as  $\\tau \\to 0$ , at the same time increasing the constant on the convergence rate via the Lipschitz constant  $\\sqrt{\\hat{L}}$  defined below. For clarity, we assume users provide a temperature in the form  $\\tau = \\frac{1}{\\ln(1/p)}$  with  $p \\in (0,1)$  which ensures all equilibria have probability mass greater than  $\\frac{p}{m^*}$  for all actions (Lemma 11). Lower  $p$  corresponds with lower temperature.",
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+        "type": "text",
+        "text": "Theorem 4 (BLiN PAC Rate). Assume  $\\eta_{k} = \\eta = 2 / \\hat{L}$ ,  $\\tau = \\frac{1}{\\ln(1 / p)}$ , and a previously pulled arm is returned uniformly at random (i.e.,  $t \\sim U([T])$ ). Then for any  $w > 0$",
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+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\epsilon_ {t} \\leq w \\left[ \\frac {1}{\\ln (1 / p)} \\log \\left(\\prod_ {k} m _ {k}\\right) + 2 \\left(1 + \\left(4 c ^ {2} C _ {z}\\right) ^ {1 / 3}\\right) \\sqrt {2 n \\hat {L}} \\left(\\frac {\\ln T}{T}\\right) ^ {\\frac {1}{2 (d z + 2)}} \\right] \\tag {12}\n$$\n",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "with probability  $(1 - w^{-1})(1 - 2T^{-2})$  where  $m^{*} = \\max_{k} m_{k}$ ,  $2|\\mathcal{L}^{\\tau}| \\leq c \\leq \\frac{1}{4}\\left(\\frac{\\ln(m^{*})}{\\ln(1 / p)} + 2\\right)$  (Corollary 1),  $\\hat{L} = \\left(\\frac{\\ln(m^{*})}{\\ln(1 / p)} + 2\\right)\\left(\\frac{m^{*2}}{p\\ln(1 / p)} + n\\bar{m}\\right)$  (Corollary 2), the zooming dimension  $d_{z} = \\frac{1}{2} n\\bar{m}$ , and the zooming constant  $C_{z} = |\\mathcal{X}^{*}|^{-1}\\left(\\frac{1}{4} r_{\\eta}^{2}\\sigma_{-\\infty}\\right)^{-n\\bar{m}}$  (Corollary 30).",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The convergence rate for BLiN (Feng et al., 2022) depends on bounds on the exploitability in terms of the loss (Lemma 13), bounds on estimates of the loss (Corollary 1), Lipschitz bounds on the",
+        "bbox": [
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
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+        "page_idx": 7
+    },
+    {
+        "type": "page_number",
+        "text": "8",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/6737fba7ce2c95955cadad5307a45761a3fc43900fed5e9e197dd9926c74b122.jpg",
+        "image_caption": [
+            "Figure 4: Bandit-based (BLiN) Nash solver applied to an artificial 7-player, symmetric, 2-action game. We search for a symmetric equilibrium, which is represented succinctly as the probability of selecting action 1. The plot shows the true exploitability  $\\epsilon$  of all symmetric strategies in black and indicates there exist potentially 5 NEs (the dips in the curve). Upper bounds on our unregularized loss  $\\mathcal{L}$  capture 4 of these equilibria, missing only the pure NE on the right. By considering our regularized loss,  $\\mathcal{L}^{\\tau}$ , we are able to capture this pure NE (see zoomed inset). The bandit algorithm selects strategies to evaluate, using 10 Monte-Carlo samples for each evaluation (arm pull) of  $\\mathcal{L}^{\\tau}$ . These samples are displayed as vertical bars above with the height of the vertical bar representing additional arm pulls. The best arms throughout search are denoted by green circles (darker indicates later in the search). The boxed numbers near equilibria display the welfare of the strategy profile."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "infinity norm of the gradient (Corollary 2), and the number of distinct strategies  $(n\\bar{m} = \\sum_{k}m_{k})$ . This result further depends on the near-optimality or zooming-dimension  $d_z$  and zooming constant  $C_z$  which quantify the number of near optimal states. In particular, we assume  $\\mathcal{L}(s(z))$  is locally  $(\\sigma_{-\\infty})$ -strongly convex with respect to  $\\| \\cdot \\|_{\\infty}$  about each global optimum within a ball of radius  $r_\\eta$ . Here,  $s:[0,1]^{n(\\bar{m} -1)}\\to \\prod_i\\Delta^{m_i - 1}$  is any function that maps from the unit hypercube to a product of simplices; we analyze two such maps in Appendix B.1. Next, we present an additional convergence rate result using an alternative  $\\mathcal{X}$ -bandit approach, StoSOO (Valko et al., 2013).",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Theorem 5 (StoSOO Rate). Corollary 1 of Valko et al. (2013) implies that with probability  $(1 - w^{-1})(1 - \\delta)$  for any  $w > 0$ , a uniformly randomly drawn arm (i.e.,  $t \\sim U([T]))$  achieves",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\epsilon_ {t} \\leq w \\left[ \\frac {1}{\\ln (1 / p)} \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {n \\hat {L}} \\sqrt {\\xi_ {1} \\sqrt {\\frac {\\log_ {b} (T k / \\delta)}{2 \\log_ {b} (e) k}} + \\xi_ {2} b ^ {- \\frac {1}{d C} \\sqrt {T / k}}} \\right] \\tag {13}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+            571,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "where  $d = n(\\bar{m} -1)$ $\\xi_{1} = (2 + 2^{2 / d})$ $\\xi_{2} = \\frac{1}{4} db^{2(1 + 2 / d)}$ $k = T\\log_b(T)^{-3}$  ,  $b$  is the branching factor for partitioning cells, and the near-optimality constant  $C = |\\mathcal{X}^* |^{-1}\\sqrt{2\\pi d}\\Big(\\frac{b^2d^2}{5r_\\eta^2\\sigma_{-2}}\\Big)^{d / 2}$  (Lemma 36).",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Here we assume  $\\mathcal{L}(s(z))$  is locally  $(\\sigma_{-2})$ -strongly convex with respect to  $||\\cdot ||_2$  about each global optimum within a ball of radius  $r_\\eta$ . Theorem 5 implies a  $\\tilde{\\mathcal{O}}(T^{-1/4})$  global convergence rate (Proposition 2), however this is achieved only after an exponential number of burn-in iterations.",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "7 CONCLUSION",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this work, we proposed a stochastic loss for approximate Nash equilibria in normal-form games. An unbiased loss estimator of Nash equilibria is the \"key\" to the stochastic optimization \"door\" which holds a wealth of research innovations uncovered over several decades. Thus, it allows the development of new algorithmic techniques for computing equilibria. We consider bandit and vanilla SGD methods in this work, but these are only two of the many options now at our disposal (e.g., adaptive methods (Antonakopoulos et al., 2022), Gaussian processes (Calandriello et al., 2022), evolutionary algorithms (Hansen et al., 2003), etc.). Such approaches as well as generalizations of these techniques to extensive-form, imperfect-information games are promising directions for future work. Similarly to how deep learning research first balked at and then marched on to train neural networks via NP-hard non-convex optimization, we hope computational game theory can march ahead to make useful equilibrium predictions of large multiplayer systems.",
+        "bbox": [
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+    },
+    {
+        "type": "page_number",
+        "text": "9",
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+    },
+    {
+        "type": "text",
+        "text": "REFERENCES",
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+        ],
+        "bbox": [
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+            102,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
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+            47
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "page_number",
+        "text": "12",
+        "bbox": [
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+            946,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX: APPROXIMATING NASH EQUILIBRIA IN NORMAL-FORM GAMES VIA STOCHASTIC OPTIMIZATION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            101,
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A Loss with its Properties and Derivatives 14",
+        "text_level": 1,
+        "bbox": [
+            174,
+            155,
+            825,
+            170
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "A.1 Loss: Connection to Exploitability, Unbiased Estimation, and Upper Bounds 14",
+            "A.2 QREs Approximate NEs at Low Temperature 19",
+            "A.3 Gradient of Loss 21",
+            "A.4 Bound on Gradient / Lipschitz Property 22",
+            "A.5 Hessian of Loss 23"
+        ],
+        "bbox": [
+            194,
+            176,
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+            277
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B Global Convergence Guarantees 26",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "B.1 Maps from Hypercube to Simplex Product 26",
+            "B.2 Near Optimality & Zooming Dimension 29",
+            "B.3 D-BLiN 32",
+            "B.4 Bounded Diameters and Well-shaped Cells 37",
+            "B.5 Stochastic Simultaneous Optimistic Optimization 39",
+            "B.6 Regret to PAC Bounds 41",
+            "B.7 Complexity of Polymatrix Games 43"
+        ],
+        "bbox": [
+            194,
+            316,
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+            460
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C Experimental Setup and Details 45",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "C.1 GAMBIT 45",
+            "C.2 Loss Visualization and Rank Test 45",
+            "C.3 Saddle Point Analysis 47",
+            "C.4 SGD on Classical Games 47",
+            "C.5 BLiN on Artificial Game 47"
+        ],
+        "bbox": [
+            194,
+            500,
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+            601
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
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+            47
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "page_number",
+        "text": "13",
+        "bbox": [
+            488,
+            946,
+            506,
+            959
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A LOSS WITH ITS PROPERTIES AND DERIVATIVES",
+        "text_level": 1,
+        "bbox": [
+            171,
+            102,
+            602,
+            118
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In this section, we construct our loss function and derive many of its properties and derivatives (e.g., gradient and hessian) useful for analyzing and executing optimization algorithms.",
+        "bbox": [
+            169,
+            133,
+            826,
+            162
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.1 LOSS: CONNECTION TO EXPLOITABILITY, UNBIASED ESTIMATION, AND UPPER BOUNDS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            179,
+            823,
+            194
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.1.1 KKT CONDITIONS IMPLY FIXED POINT SUFFICIENCY",
+        "text_level": 1,
+        "bbox": [
+            171,
+            205,
+            604,
+            219
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Consider the following constrained optimization problem:",
+        "bbox": [
+            169,
+            229,
+            555,
+            244
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\max  _ {\\boldsymbol {x} \\in \\mathbb {R} ^ {d}} f (\\boldsymbol {x}) \\tag {14a}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            467,
+            253,
+            823,
+            273
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\ns. t. g _ {i} (\\boldsymbol {x}) \\leq 0 \\forall i \\tag {14b}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            437,
+            277,
+            823,
+            294
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nh _ {j} (\\boldsymbol {x}) = 0 \\forall j \\tag {14c}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            462,
+            296,
+            823,
+            311
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "where  $f$  is concave and  $g_{i}$  and  $h_j$  represent inequality and equality constraints respectively. If  $g_{i}$  and  $h_i$  are affine functions, then any maximizer  $x^{*}$  of  $f$  must satisfy the following necessary and sufficient KKT conditions (Ghofogh et al., 2021; Boyd and Vandenberghe, 2004):",
+        "bbox": [
+            169,
+            319,
+            823,
+            362
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- Stationarity:  $\\mathbf{0} \\in \\partial f(\\mathbf{x}^{*}) - \\sum_{j} \\lambda_{j} \\partial h_{j}(\\mathbf{x}^{*}) - \\sum_{i} \\mu_{i} \\partial g_{i}(\\mathbf{x}^{*})$",
+            "- Primal feasibility:  $h_j(\\pmb{x}^*) = 0$  for all  $j$  and  $g_i(\\pmb{x}^*) \\leq 0$  for all  $i$ .",
+            "- Dual feasibility:  $\\mu_{i} \\geq 0$  for all  $i$",
+            "- Complementary slackness:  $\\mu_{i}g_{i}(\\pmb{x}^{*}) = 0$  for all  $i$ ."
+        ],
+        "bbox": [
+            215,
+            372,
+            648,
+            449
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Lemma 1. Assuming player  $k$ 's utility,  $u_{k}(x_{k}, x_{-k})$ , is concave in its own strategy  $x_{k}$ , a strategy in the interior of the simplex is a best response  $BR_{k}$  if and only if it has zero projected-gradient² norm:",
+        "bbox": [
+            169,
+            453,
+            826,
+            481
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nB R _ {k} \\in \\left(i n t \\Delta \\cap \\underset {z} {\\arg \\max } u _ {k} (z, x _ {- k}) - u _ {k} (x _ {k}, x _ {- k})\\right) \\iff (B R _ {k} \\in i n t \\Delta) \\wedge (\\| \\Pi_ {T \\Delta} [ \\nabla_ {B R _ {k}} ^ {k} ] \\| = 0).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            171,
+            508,
+            828,
+            534
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Proof. Consider the problem of formally computing  $\\epsilon_{k}(\\pmb{x}) = \\max_{z\\in int\\Delta}u_{k}(z,x_{-k}) - u_{k}(x_{k},x_{-k})$",
+        "bbox": [
+            169,
+            550,
+            826,
+            566
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\max  _ {z \\in \\mathbb {R} ^ {d}} u _ {k} (z, x _ {- k}) - u _ {k} \\left(x _ {k}, x _ {- k}\\right) \\tag {15a}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            408,
+            587,
+            823,
+            609
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\ns. t. - z _ {k} \\leq 0 \\forall k \\tag {15b}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            383,
+            612,
+            823,
+            628
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n1 - \\sum_ {k} z _ {k} = 0. \\tag {15c}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            370,
+            632,
+            823,
+            662
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Note that the objective is linear (concave) in  $z$  and the constraints are affine, therefore the KKT conditions are necessary and sufficient for optimality. Recall that we assume that the solution  $z^{*}$  lies in the interior of the simplex, i.e.,  $z_{k}^{*} > 0$  for each  $k$ . Also, let  $e_{k}$  be a onehot vector, i.e., a zeros vector except with a 1 at index  $k$ . Mapping the KKT conditions onto this problem yields the following:",
+        "bbox": [
+            169,
+            670,
+            823,
+            742
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- Stationarity:  $\\mathbf{0} \\in \\partial u_{k}(z^{*}, x_{-k}) + \\lambda \\mathbf{1} + \\sum_{k} \\mu_{k} e_{k}$",
+            "- Primal feasibility:  $\\sum_{k}z_{k}^{*} = 1$  for all  $k$",
+            "- Dual feasibility:  $\\mu_{i} \\geq 0$  for all  $k$",
+            "- Complementary slackness:  $-\\mu_k z_k^* = 0$  for all  $k$ ."
+        ],
+        "bbox": [
+            215,
+            755,
+            560,
+            844
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "For any point  $z^{*} \\in int\\Delta$ , primal feasibility will be satisfied. Given our assumption that  $z_{k}^{*} > 0$  by complementary slackness and dual feasibility, each  $\\mu_{k}$  must be identically zero. This implies",
+        "bbox": [
+            169,
+            858,
+            826,
+            887
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "page_footnote",
+        "text": "$^{2}$ Not to be confused with the nonlinear (i.e., introduces bias) projected gradient operator introduced in (Hazan et al., 2017).",
+        "bbox": [
+            169,
+            897,
+            823,
+            922
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "page_number",
+        "text": "14",
+        "bbox": [
+            488,
+            946,
+            506,
+            959
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "the stationarity condition can be simplified to  $\\mathbf{0} \\in \\partial u_k(z^*, x_{-k}) + \\lambda \\mathbf{1}$ . Rearranging terms (and repurposing  $\\lambda$ ) we find that for any  $z^*$ , there exists a  $\\lambda$  such that",
+        "bbox": [
+            169,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\lambda \\mathbf {1} \\in \\partial u _ {k} \\left(z ^ {*}, x _ {- k}\\right). \\tag {16}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            429,
+            140,
+            823,
+            156
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Equivalently,  $\\partial u_{k}(z^{*},x_{-k})\\propto \\mathbf{1}$  at  $z^{*}\\in int\\Delta$  . Any vector proportional to the ones vector has zero projected-gradient norm, completing the claim:  $[I - \\frac{1}{m_k}\\mathbf{11}^\\top ](\\lambda \\mathbf{1}) = \\lambda (\\mathbf{1} - \\frac{m_k}{m_k}\\mathbf{1}) = \\mathbf{0}.$",
+        "bbox": [
+            169,
+            164,
+            825,
+            198
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.1.2 NORM OF PROJECTED-GRADIENT AND EQUIVALENCE TO NFG EXPLOITABILITY WITH APPROXIMATE BEST RESPONSES",
+        "text_level": 1,
+        "bbox": [
+            171,
+            210,
+            823,
+            239
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1. The loss  $\\mathcal{L}$  is equivalent to NashConv, but where player  $k$ 's best response is approximated by a single step of projected-gradient ascent with step size  $\\eta_k$ :  $aBR_k = x_k + \\eta_k\\Pi_{T\\Delta}(\\nabla_{x_k}^k)$ .",
+        "bbox": [
+            169,
+            250,
+            826,
+            280
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Proof. Define an approximate best response as the result of a player adjusting their strategy via a projected-gradient ascent step, i.e.,  $\\mathsf{aBR}_k = x_k + \\eta_k\\Pi_{T\\Delta}(\\nabla_{x_k}^k)$  for player  $k$ .",
+        "bbox": [
+            169,
+            296,
+            823,
+            325
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In a normal form game, player  $k$ 's utility at this new strategy is",
+        "bbox": [
+            171,
+            330,
+            589,
+            347
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nu _ {k} \\left(\\mathrm {a B R} _ {k}, x _ {- k}\\right) = \\left(\\nabla_ {x _ {k}} ^ {k}\\right) ^ {\\top} \\left(x _ {k} + \\eta_ {k} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right)\\right) = u _ {k} (\\boldsymbol {x}) + \\eta_ {k} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right). \\tag {17}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            207,
+            353,
+            825,
+            375
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Therefore, the amount player  $k$  gains by playing aBR is",
+        "bbox": [
+            171,
+            386,
+            540,
+            402
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\hat {\\epsilon} _ {k} (\\boldsymbol {x}) = u _ {k} \\left(\\mathrm {a B R} _ {k}, x _ {- k}\\right) - u _ {k} (\\boldsymbol {x}) (18a) \\\\ = \\eta_ {k} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) (18b) \\\\ = \\eta_ {k} \\left(\\nabla_ {x _ {k}} ^ {k} - \\frac {1}{m _ {k}} \\left(\\mathbf {1} ^ {\\top} \\nabla_ {x _ {k}} ^ {k}\\right) \\mathbf {1}\\right) ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) (18c) \\\\ = \\eta_ {k} \\| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) \\| ^ {2} (18d) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            333,
+            409,
+            823,
+            500
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "where the third equality follows from the fact that the projected-gradient,  $\\Pi_{T\\Delta}(\\nabla_{x_k}^k)$ , is orthogonal to the ones vector.",
+        "bbox": [
+            169,
+            508,
+            825,
+            537
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.1.3 CONNECTION TO TRUE EXPLOITABILITY",
+        "text_level": 1,
+        "bbox": [
+            171,
+            555,
+            516,
+            569
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Lemma 2. The amount a player can gain by deviating is upper bounded by a quantity proportional to the norm of the projected-gradient:",
+        "bbox": [
+            169,
+            580,
+            823,
+            609
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\epsilon_ {k} (\\boldsymbol {x}) \\leq \\sqrt {2} \\left\\| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) \\right\\|. \\tag {19}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            405,
+            617,
+            823,
+            636
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Proof. Let  $z$  be any point on the simplex. Then by concavity of  $u_{k}$  with respect to  $z$ ,",
+        "bbox": [
+            171,
+            651,
+            728,
+            667
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} u _ {k} (z, x _ {- k}) - u _ {k} (\\boldsymbol {x}) \\leq \\left(\\nabla_ {x _ {k}} ^ {k}\\right) ^ {\\top} (z - x _ {k}) (20a) \\\\ = \\left(\\nabla_ {x _ {k}} ^ {k}\\right) ^ {\\top} \\left(z - x _ {k}\\right) - \\frac {1}{m _ {k}} \\left(\\mathbf {1} ^ {\\top} \\nabla_ {x _ {k}} ^ {k}\\right) \\overbrace {\\mathbf {1} ^ {\\top} \\left(z - x _ {k}\\right)} ^ {1 - 1 = 0} (20b) \\\\ = \\left(\\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right)\\right) ^ {\\top} \\underbrace {(z - x _ {k})} _ {\\text {D i a m} (\\Delta) \\leq \\sqrt {2}} (20c) \\\\ \\leq \\sqrt {2} \\| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) \\|. \\quad (\\text {C a u c h y - S c h w a r z}) (20d) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            267,
+            672,
+            823,
+            801
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/8d04e54b845b2ea910ecf254d645b061bf5d3ae51e95829ca35d18bd08202a7e.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            809,
+            823,
+            821
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Continuing, we can prove a bound on  $\\epsilon$  in terms of the projected-gradient loss:",
+        "bbox": [
+            169,
+            840,
+            687,
+            856
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Lemma 3. The exploitability,  $\\epsilon$ , of a joint strategy  $x$ , is upper bounded as a function of our proposed loss:",
+        "bbox": [
+            169,
+            859,
+            826,
+            887
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\epsilon \\leq \\sqrt {\\frac {2 n}{\\min  _ {k} \\eta_ {k}}} \\sqrt {\\mathcal {L} (\\boldsymbol {x})}. \\tag {21}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            415,
+            893,
+            823,
+            929
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "page_number",
+        "text": "15",
+        "bbox": [
+            488,
+            946,
+            506,
+            959
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Proof.",
+        "bbox": [
+            171,
+            104,
+            217,
+            118
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\epsilon = \\max  _ {k} \\max  _ {z} u _ {k} (z, x _ {- k}) - u _ {k} (\\boldsymbol {x}) (22a) \\\\ \\leq \\sum_ {k} \\max  _ {z} u _ {k} (z, x _ {- k}) - u _ {k} (\\boldsymbol {x}) (22b) \\\\ \\leq \\sum_ {k} \\sqrt {2} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) \\right| \\right| _ {2} \\quad (\\text {L e m m a 2}) (22c) \\\\ = \\sqrt {2} \\left|\\left|\\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {1}} ^ {1}\\right)\\right|\\right| _ {2}, \\dots , \\left|\\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {n}} ^ {n}\\right)\\right|\\right| _ {2} \\right|\\left. \\right| _ {1} (22d) \\\\ \\leq \\sqrt {2 n} \\left|\\left|\\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {1}} ^ {1}\\right)\\right|\\right| _ {2}, \\dots , \\left|\\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {n}} ^ {n}\\right)\\right|\\right| _ {2} \\right|\\left. \\right| _ {2} (22e) \\\\ = \\sqrt {2 n} \\sqrt {\\sum_ {k} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) \\right| \\right| _ {2} ^ {2}} (22f) \\\\ = \\sqrt {2 n} \\sqrt {\\sum_ {k} \\left(\\frac {1}{\\eta_ {k}}\\right) \\eta_ {k} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) \\right| \\right| _ {2} ^ {2}} (22g) \\\\ \\leq \\sqrt {\\frac {2 n}{\\min _ {k} \\eta_ {k}}} \\sqrt {\\sum_ {k} \\eta_ {k} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) \\right| \\right| _ {2} ^ {2}} (22h) \\\\ = \\sqrt {\\frac {2 n}{\\min _ {k} \\eta_ {k}}} \\sqrt {\\mathcal {L} (\\boldsymbol {x})}. (22i) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            333,
+            127,
+            825,
+            441
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/7333ec7d88ec10819ec71c4882e6574cc3a1fb01b4749ba560ac97c46a1b8ead.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            449,
+            823,
+            460
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Lemma 4. The entropy regularized exploitability,  $\\epsilon_{QRE}$ , of a joint strategy  $\\pmb{x}$ , is upper bounded as a function of our proposed loss:",
+        "bbox": [
+            168,
+            474,
+            823,
+            503
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\epsilon_ {Q R E} \\leq \\sqrt {\\frac {2 n}{\\min _ {k} \\eta_ {k}}} \\sqrt {\\mathcal {L} ^ {\\tau} (\\boldsymbol {x})}. \\tag {23}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            395,
+            512,
+            825,
+            546
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Proof. Recall that  $u_{k}^{\\tau}(x_{k}, x_{-k})$  is also concave with respect to  $x_{k}$ . Then",
+        "bbox": [
+            169,
+            564,
+            648,
+            580
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\epsilon_ {Q R E} = \\max  _ {k} \\max  _ {z} u _ {k} ^ {\\tau} (z, x _ {- k}) - u _ {k} ^ {\\tau} (\\boldsymbol {x}) (24a) \\\\ \\leq \\sum_ {k} \\max  _ {z} u _ {k} ^ {\\tau} (z, x _ {- k}) - u _ {k} ^ {\\tau} (\\boldsymbol {x}) (24b) \\\\ \\leq \\sum_ {k} \\sqrt {2} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right| \\right| _ {2} \\quad (\\text {L e m m a 2}) (24c) \\\\ = \\sqrt {2} \\left| \\left| | \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {1}} ^ {1 \\tau}\\right) | | _ {2}, \\dots , | | \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {n}} ^ {n \\tau}\\right) | | _ {2} \\right| \\right| _ {1} (24d) \\\\ \\leq \\sqrt {2 n} \\left| \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {1}} ^ {1 \\tau}\\right) \\right| \\right| _ {2}, \\dots , \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {n}} ^ {n \\tau}\\right) \\right| \\right| _ {2} \\right| _ {2} (24e) \\\\ = \\sqrt {2 n} \\sqrt {\\sum_ {k} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right| \\right| _ {2} ^ {2}} (24f) \\\\ \\leq \\sqrt {2 n} \\sqrt {\\sum_ {k} \\left(\\frac {1}{\\eta_ {k}}\\right) \\eta_ {k} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right| \\right| _ {2} ^ {2}} (24g) \\\\ \\leq \\sqrt {\\frac {2 n}{\\min _ {k} \\eta_ {k}}} \\sqrt {\\sum_ {k} \\eta_ {k} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right| \\right| _ {2} ^ {2}} (24h) \\\\ = \\sqrt {\\frac {2 n}{\\min _ {k} \\eta_ {k}}} \\sqrt {\\mathcal {L} ^ {\\tau} (\\boldsymbol {x})}. (24i) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            587,
+            825,
+            902
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "page_number",
+        "text": "16",
+        "bbox": [
+            488,
+            948,
+            506,
+            959
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "A.1.4 UNBIASED ESTIMATION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            103,
+            401,
+            118
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Lemma 5. An unbiased estimate of  $\\mathcal{L}(\\pmb{x})$  can be obtained by drawing two samples (pure strategies) from each players' mixed strategy and observing payoffs.",
+        "bbox": [
+            168,
+            128,
+            826,
+            159
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Proof. Define  $\\hat{\\nabla}_{x_k}^{k(1)}$  as a random, unbiased gradient estimate (see Table 2). Let  $\\hat{\\nabla}_{x_k}^{k(2)}$  be independent and distributed identically to  $\\hat{\\nabla}_{x_k}^{k(1)}$ . Then,",
+        "bbox": [
+            169,
+            171,
+            825,
+            208
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\mathbb {E} _ {\\boldsymbol {a} ^ {(1)} \\sim \\boldsymbol {x}, \\boldsymbol {a} ^ {(2)} \\sim \\boldsymbol {x}} [ \\sum_ {k} \\eta_ {k} (\\underbrace {\\hat {\\nabla} _ {x _ {k}} ^ {k (1)} - \\frac {1}{m _ {k}} \\left(\\boldsymbol {1} ^ {\\top} \\hat {\\nabla} _ {x _ {k}} ^ {k (1)}\\right) \\boldsymbol {1}} _ {\\text {p r o j e c t e d - g r a d i e n t 1}}) ^ {\\top} (\\underbrace {\\hat {\\nabla} _ {x _ {k}} ^ {k (2)} - \\frac {1}{m _ {k}} \\left(\\boldsymbol {1} ^ {\\top} \\hat {\\nabla} _ {x _ {k}} ^ {k (2)}\\right) \\boldsymbol {1}} _ {\\text {p r o j e c t e d - g r a d i e n t 2}}) ] (25a) \\\\ = \\sum_ {k} \\eta_ {k} \\mathbb {E} _ {\\boldsymbol {a} ^ {(1)} \\sim \\boldsymbol {x}, \\boldsymbol {a} ^ {(2)} \\sim \\boldsymbol {x}} [ (\\hat {\\nabla} _ {x _ {k}} ^ {k (1)} - \\frac {1}{m _ {k}} (\\boldsymbol {1} ^ {\\top} \\hat {\\nabla} _ {x _ {k}} ^ {k (1)}) \\boldsymbol {1}) ^ {\\top} (\\hat {\\nabla} _ {x _ {k}} ^ {k (2)} - \\frac {1}{m _ {k}} (\\boldsymbol {1} ^ {\\top} \\hat {\\nabla} _ {x _ {k}} ^ {k (2)}) \\boldsymbol {1}) ] (25b) \\\\ = \\sum_ {k} \\eta_ {k} \\mathbb {E} _ {\\boldsymbol {a} ^ {(1)} \\sim \\boldsymbol {x}} \\left[ \\left(\\hat {\\nabla} _ {x _ {k}} ^ {k (1)} - \\frac {1}{m _ {k}} \\left(\\boldsymbol {1} ^ {\\top} \\hat {\\nabla} _ {x _ {k}} ^ {k (1)}\\right) \\boldsymbol {1}\\right) \\right] ^ {\\top} \\mathbb {E} _ {\\boldsymbol {a} ^ {(2)} \\sim \\boldsymbol {x}} \\left[ \\left(\\hat {\\nabla} _ {x _ {k}} ^ {k (2)} - \\frac {1}{m _ {k}} \\left(\\boldsymbol {1} ^ {\\top} \\hat {\\nabla} _ {x _ {k}} ^ {k (2)}\\right) \\boldsymbol {1}\\right) \\right] (25c) \\\\ = \\sum_ {k} \\eta_ {k} \\left(\\nabla_ {x _ {k}} ^ {k} - \\frac {1}{m _ {k}} \\left(\\mathbf {1} ^ {\\top} \\nabla_ {x _ {k}} ^ {k}\\right) \\mathbf {1}\\right) ^ {\\top} \\left(\\nabla_ {x _ {k}} ^ {k} - \\frac {1}{m _ {k}} \\left(\\mathbf {1} ^ {\\top} \\nabla_ {x _ {k}} ^ {k}\\right) \\mathbf {1}\\right) (25d) \\\\ = \\sum_ {k} \\eta_ {k} \\left\\| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) \\right\\| ^ {2} (25e) \\\\ = \\mathcal {L} (\\boldsymbol {x}) (25f) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            200,
+            210,
+            825,
+            429
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "where the first equality follows from linearity of expectation, the second from independence of random variables, and the third from  $\\mathbb{E}_{\\pmb{a}^{(p)}\\sim \\pmb{x}}[\\hat{\\nabla}_{x_k}^{k(p)}] = \\nabla_{x_k}^k$ , i.e.,  $\\hat{\\nabla}_{x_k}^{k(p)}$  is an unbiased estimator of player  $k$ 's gradient. Therefore, equation (25a) comprises an unbiased estimate of  $\\mathcal{L}(\\pmb{x})$  proving the claim.",
+        "bbox": [
+            169,
+            433,
+            826,
+            494
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/c3bd132c0bdd7fca6af0f73d73f20fb90490aaef865754c004ed18ad5e37e94d.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            501,
+            823,
+            512
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Lemma 6. The loss formed as the sum of the squared norms of the projected-gradients,  $\\mathcal{L}^{\\tau}$ , can be decomposed into three terms as follows:",
+        "bbox": [
+            169,
+            522,
+            823,
+            551
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) = \\underbrace {\\sum_ {k} \\eta_ {k} x _ {q} ^ {\\top} B _ {k q} ^ {\\top} B _ {k q} x _ {q}} _ {(A)} + \\underbrace {2 \\sum_ {k} \\eta_ {k} E _ {k} ^ {\\top} B _ {k q} x _ {q}} _ {(B)} + \\underbrace {\\sum_ {k} \\eta_ {k} E _ {k} ^ {\\top} E _ {k}} _ {(C)} \\tag {26}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            274,
+            555,
+            825,
+            609
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "where  $q$  is any player other than  $k$ .",
+        "bbox": [
+            171,
+            614,
+            403,
+            628
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Proof. Let  $S^{\\tau} = -\\tau \\sum_{l} x_{kl} \\ln(x_{kl})$  so that  $\\frac{\\partial S^{\\tau}}{\\partial x_k} = -\\tau (\\ln(x_k) + 1)$ . Note that  $\\Pi_{T\\Delta}[\\frac{\\partial S^{\\tau}}{\\partial x_k}] = -\\tau \\Pi_{T\\Delta}[\\ln(x_k)]$ .",
+        "bbox": [
+            169,
+            642,
+            823,
+            678
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) = \\sum_ {k} \\eta_ {k} \\left(\\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right)\\right) ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) (27a) \\\\ = \\sum_ {k} \\eta_ {k} \\left[ H _ {k q} ^ {k} x _ {q} + \\frac {\\partial S ^ {\\tau}}{\\partial x _ {k}} \\right] ^ {\\top} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] \\left[ H _ {k q} ^ {k} x _ {q} + \\frac {\\partial S ^ {\\tau}}{\\partial x _ {k}} \\right] (27b) \\\\ = \\sum_ {k} \\eta_ {k} \\left(x _ {q} ^ {\\top} \\left[ H _ {k q} ^ {k} \\right] ^ {\\top} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] ^ {2} \\left[ H _ {k q} ^ {k} \\right] x _ {q} + 2 \\left[ \\frac {\\partial S ^ {\\tau}}{\\partial x _ {k}} \\right] ^ {\\top} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] ^ {2} \\left[ H _ {k q} ^ {k} x _ {q} \\right] \\quad (2 7 c) \\right. \\\\ \\left. + \\left[ \\frac {\\partial S ^ {\\tau}}{\\partial x _ {k}} \\right] ^ {\\top} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] ^ {2} \\left[ \\frac {\\partial S ^ {\\tau}}{\\partial x _ {k}} \\right]\\right) (27d) \\\\ = \\underbrace {\\sum_ {k} \\eta_ {k} x _ {q} ^ {\\top} B _ {k q} ^ {\\top} B _ {k q} x _ {q}} _ {(A)} + \\underbrace {2 \\sum_ {k} \\eta_ {k} E _ {k} ^ {\\top} B _ {k q} x _ {q}} _ {(B)} + \\underbrace {\\sum_ {k} \\eta_ {k} E _ {k} ^ {\\top} E _ {k}} _ {(C)} (27e) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            191,
+            681,
+            825,
+            878
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "where  $B_{kq} = [I - \\frac{1}{m_k}\\mathbf{11}^\\top ]H_{kq}^k$  and  $E_{k} = [I - \\frac{1}{m_{k}}\\mathbf{11}^{\\top}][\\frac{\\partial S^{\\tau}}{\\partial x_{k}} ] = -\\tau [I - \\frac{1}{m_{k}}\\mathbf{11}^{\\top}]\\ln (x_{k})$",
+        "bbox": [
+            169,
+            883,
+            769,
+            906
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "page_number",
+        "text": "17",
+        "bbox": [
+            488,
+            946,
+            506,
+            959
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "A.1.5 BOUND ON LOSS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            103,
+            352,
+            118
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "By Proposition 1, Equation (18d), we can also rewrite this loss as a weighted sum of 2-norms,  $\\mathcal{L}(\\pmb{x}) = \\sum_{k}\\eta_{k}||\\nabla_{x_{k}}^{k} - \\mu_{k}||_{2}^{2}$  where  $\\mu_{k} = \\frac{1}{m_{k}} (\\mathbf{1}^{\\top}\\nabla_{x_{k}}^{k})\\in [0,1]$  for brevity. This will allow us to more easily analyze our loss.",
+        "bbox": [
+            169,
+            128,
+            826,
+            172
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Lemma 7. Assume payoffs are bounded in  $[0,1]$ , then setting  $\\eta_k \\leq \\frac{4}{nm_k}$  or  $\\eta_k \\leq \\frac{4}{nm}$  or  $\\sum_k \\eta_k \\leq \\frac{4}{m}$  ensures  $0 \\leq \\mathcal{L}(x) \\leq 1$  for all  $x \\in \\mathcal{X}$ .",
+        "bbox": [
+            169,
+            174,
+            823,
+            209
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Proof.",
+        "bbox": [
+            171,
+            224,
+            217,
+            239
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} 0 \\leq \\mathcal {L} (\\boldsymbol {x}) = \\sum_ {k} \\eta_ {k} \\| \\nabla_ {x _ {k}} ^ {k} - \\mu_ {k} \\| _ {2} ^ {2} (28a) \\\\ = \\sum_ {k} \\eta_ {k} m _ {k} \\left[ \\frac {1}{m _ {k}} \\sum_ {l} \\left(\\nabla_ {x _ {k l}} ^ {k} - \\mu_ {k}\\right) ^ {2} \\right] (28b) \\\\ = \\sum_ {k} \\eta_ {k} m _ {k} \\operatorname {V a r} \\left[ \\nabla_ {x _ {k}} ^ {k} \\right] (28c) \\\\ \\leq \\frac {1}{4} \\sum_ {k} \\eta_ {k} m _ {k} (28d) \\\\ \\leq \\frac {1}{4} \\left(\\max  _ {k} \\eta_ {k}\\right) \\left(\\sum_ {k} m _ {k}\\right) (28e) \\\\ = \\frac {1}{4} \\left(\\max  _ {k} \\eta_ {k}\\right) n \\bar {m} \\leq 1 (28f) \\\\ \\Rightarrow \\left(\\max  _ {k} \\eta_ {k}\\right) \\leq \\frac {4}{n \\bar {m}} (28g) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            328,
+            246,
+            825,
+            489
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "where the first inequality follows from Popoviciu's inequality (Popoviciu, 1935): the variance of a bounded random variable  $X$  is upper bounded by  $Var[X] \\leq \\frac{1}{4} (\\max_{X} - \\min_{X})^{2}$ .",
+        "bbox": [
+            169,
+            496,
+            826,
+            527
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Next, we establish the following useful lemmas.",
+        "bbox": [
+            171,
+            541,
+            488,
+            556
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Lemma 8. The matrix  $I - \\frac{1}{m_k} \\mathbf{11}^\\top$  is a projection matrix and therefore idempotent. It is also symmetric, which implies it is its own square root.",
+        "bbox": [
+            169,
+            559,
+            823,
+            590
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Proof.",
+        "bbox": [
+            171,
+            607,
+            217,
+            622
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] ^ {\\top} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] = I - \\frac {2}{m _ {k}} \\mathbf {1 1} ^ {\\top} + \\frac {1}{m _ {k} ^ {2}} \\mathbf {1} \\left(\\mathbf {1} ^ {\\top} \\mathbf {1}\\right) \\mathbf {1} ^ {\\top} (29a) \\\\ = I - \\frac {2}{m _ {k}} \\mathbf {1 1} ^ {\\top} + \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} (29b) \\\\ = \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right]. (29c) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            284,
+            628,
+            823,
+            726
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Lemma 9. The product  $A[I_m - \\frac{1}{m}\\mathbf{1}_m\\mathbf{1}_m^\\top ]^p B$  for any integer  $p > 0$  has entries whose absolute value is bounded by  $\\frac{m}{4} (A_{\\mathrm{max}} - A_{\\mathrm{min}})(B_{\\mathrm{max}} - B_{\\mathrm{min}})$  where  $A_{\\mathrm{min}}, A_{\\mathrm{max}}, B_{\\mathrm{min}}, B_{\\mathrm{max}}$  represent the minima and maxima of the matrices respectively.",
+        "bbox": [
+            169,
+            755,
+            826,
+            800
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Proof. The matrix  $[I - \\frac{1}{m}\\mathbf{11}^{\\top}]$  is idempotent (Lemma 8) so we can rewrite the product for any  $p$  as",
+        "bbox": [
+            169,
+            813,
+            823,
+            832
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nA \\left[ I - \\frac {1}{m} \\mathbf {1 1} ^ {\\top} \\right] \\left[ I - \\frac {1}{m} \\mathbf {1 1} ^ {\\top} \\right] B. \\tag {30}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            395,
+            838,
+            823,
+            867
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "The matrix  $[I - \\frac{1}{m}\\mathbf{11}^{\\top}]$  has the property that it removes the mean from every row of a matrix when right multiplied against it, i.e.,  $A[I - \\frac{1}{m}\\mathbf{11}^{\\top}]$  removes the means from the rows of  $A$ . Similarly, left multiplying it removes the means from the columns. Let  $\\tilde{A}$  and  $\\tilde{B}$  represent these mean-centered",
+        "bbox": [
+            169,
+            875,
+            825,
+            925
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "page_number",
+        "text": "18",
+        "bbox": [
+            488,
+            946,
+            506,
+            959
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "results respectively. The absolute value of the  $ij$ th entry in the resulting product can then be recognized as",
+        "bbox": [
+            169,
+            103,
+            823,
+            133
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\left| \\sum_ {k} \\tilde {A} _ {i k} \\tilde {B} _ {k j} \\right| = \\left| \\sum_ {k} \\left(A _ {i k} - \\frac {1}{m} \\sum_ {k ^ {\\prime}} A _ {i k ^ {\\prime}}\\right) \\left(B _ {k j} - \\frac {1}{m} \\sum_ {k ^ {\\prime}} B _ {k ^ {\\prime} j}\\right) \\right| (31a) \\\\ = \\left| m \\cdot \\operatorname {C o r r} \\left(A _ {i,}, B _ {. j}\\right) \\cdot \\sigma_ {A _ {i,}} \\sigma_ {B _ {. j}} \\right| (31b) \\\\ \\leq m \\sigma_ {A _ {i,}} \\sigma_ {B _ {\\cdot , j}}. (31c) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            277,
+            138,
+            825,
+            213
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "By Popoviciu's inequality (Popoviciu, 1935), we know the variance of a bounded random variable  $X$  is upper bounded by  $Var[X] \\leq \\frac{1}{4} (\\max_{X} - \\min_{X})^{2}$ . Hence its standard deviation is bounded by  $Std[X] \\leq \\frac{1}{2} (\\max_{X} - \\min_{X})$ . Plugging these bounds for  $A$  and  $B$  into equation (31c) completes the claim.",
+        "bbox": [
+            171,
+            217,
+            826,
+            275
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Lemma 10. Assume payoffs are bounded in  $[0,1]$ , then",
+        "bbox": [
+            169,
+            284,
+            540,
+            299
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left| \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) \\right| \\leq \\frac {1}{4} \\left(\\max  _ {k} \\eta_ {k}\\right) n \\bar {m} \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) ^ {2} \\tag {32}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            339,
+            304,
+            825,
+            335
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "for any  $\\pmb{x}$  such that  $x_{kl} \\geq x_{\\min} \\forall k, l$ .",
+        "bbox": [
+            168,
+            340,
+            419,
+            356
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Proof. First note that for payoffs in [0, 1], the entries in",
+        "bbox": [
+            169,
+            371,
+            539,
+            386
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) = \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k} - \\tau (\\ln (x _ {k}) + 1)\\right) = \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k} - \\tau \\ln (x _ {k})\\right) \\tag {33}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            264,
+            392,
+            823,
+            412
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "are bounded within  $[0, \\tau \\ln(\\frac{1}{x_{\\min}}) + 1]$  with a range of  $\\tau \\ln(\\frac{1}{x_{\\min}}) + 1$ . Then starting from the definition of  $\\mathcal{L}^{\\tau}$  and applying Lemma 9, we find",
+        "bbox": [
+            169,
+            417,
+            823,
+            449
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\left| \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) \\right| = \\left| \\sum_ {k} \\eta_ {k} \\left\\| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right\\| ^ {2} \\right| (34a) \\\\ \\leq \\frac {1}{4} \\sum_ {k} \\eta_ {k} m _ {k} \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) ^ {2} (34b) \\\\ \\leq \\frac {1}{4} \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) ^ {2} \\left(\\max  _ {k} \\eta_ {k}\\right) \\sum_ {k} m _ {k} (34c) \\\\ = \\frac {1}{4} \\left(\\max  _ {k} \\eta_ {k}\\right) n \\bar {m} \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\min }}\\right) + 1\\right) ^ {2}. (34d) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            326,
+            454,
+            825,
+            598
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/c6c001f8226b024cd6deced7aa443ebd133fd49c815f6cf05d3b80cc418bb6e8.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            603,
+            823,
+            614
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "A.2 QRES APPROXIMATE NES AT LOW TEMPERATURE",
+        "text_level": 1,
+        "bbox": [
+            171,
+            633,
+            571,
+            648
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Lemma 11. Setting  $\\tau = \\ln (1 / p)^{-1}$  with  $p\\in (0,1)$  and payoffs in [0,1] ensures that any QRE contains probabilities greater than  $\\frac{p}{\\max_k m_k}$ .",
+        "bbox": [
+            169,
+            657,
+            823,
+            691
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Proof. Let  $m^{*} = \\max_{k} m_{k}$  and  $\\nabla_{x_{k}}^{k}$  be player  $k$ 's gradient. Recall that a QRE( $\\tau$ ), specifically a logit equilibrium, satisfies the fixed point equation  $x_{k} = \\text{softmax}\\left(\\frac{\\nabla_{x_{k}}^{k}}{\\tau}\\right)$  for all  $k$ . Then the probability for any action in a QRE strategy profile,  $\\mathbf{x}$ , is lower bounded as",
+        "bbox": [
+            169,
+            705,
+            825,
+            758
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} x _ {k l} \\geq \\min  _ {k} \\min  _ {l} \\left\\{x _ {k l} \\right\\} \\geq \\min  _ {\\boldsymbol {x} ^ {\\prime} \\mid \\boldsymbol {x} ^ {\\prime} \\text {i s Q R E}} \\min  _ {k} \\min  _ {l} \\left\\{x _ {k l} ^ {\\prime} \\right\\} (35a) \\\\ = \\min  _ {\\nabla_ {x _ {k}} ^ {k}} \\min  _ {k} \\min  _ {l} \\left[ \\operatorname {s o f t m a x} \\left(\\frac {\\nabla_ {x _ {k}} ^ {k}}{\\tau}\\right) \\right] _ {l} (35b) \\\\ = \\frac {e ^ {0}}{\\left(m ^ {*} - 1\\right) e ^ {\\frac {1}{\\tau}} + e ^ {0}} (35c) \\\\ \\stackrel {\\text {d e f}} {=} x _ {\\min } (35d) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            295,
+            763,
+            825,
+            888
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "where equation (35c) follows from minimizing the numerator and maximizing the denominator of the softmax formula subject to the payoff constraints.",
+        "bbox": [
+            169,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "page_number",
+        "text": "19",
+        "bbox": [
+            488,
+            946,
+            506,
+            959
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Rearranging terms, we find",
+        "bbox": [
+            171,
+            104,
+            354,
+            118
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\ne ^ {\\frac {1}{\\tau}} = \\frac {1}{m ^ {*} - 1} \\left(\\frac {1}{x _ {\\min }} - 1\\right) \\Rightarrow \\tau = \\frac {1}{\\ln \\left(\\frac {1}{m ^ {*} - 1} \\left(\\frac {1}{x _ {\\min }} - 1\\right)\\right)}. \\tag {36}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            292,
+            121,
+            823,
+            164
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Let  $p\\in (0,1)$  such that  $x_{\\mathrm{min}} = \\frac{p}{\\max_km_k} = \\frac{p}{m^*}$ , then",
+        "bbox": [
+            169,
+            174,
+            524,
+            191
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\tau = \\frac {1}{\\ln \\left(\\frac {1}{m ^ {*} - 1} \\left(\\frac {1}{x _ {\\operatorname* {m i n}}} - 1\\right)\\right)} (37a) \\\\ = \\frac {1}{\\ln \\left(\\frac {1}{m ^ {*} - 1} \\left(\\frac {m ^ {*}}{p} - 1\\right)\\right)} (37b) \\\\ = \\frac {1}{\\ln \\left(\\frac {m ^ {*} - p}{m ^ {*} - 1} \\frac {1}{p}\\right)} (37c) \\\\ \\leq \\frac {1}{\\ln \\left(\\frac {1}{p}\\right)}. (37d) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            403,
+            196,
+            823,
+            359
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "This implies if we set  $\\tau = \\ln (1 / p)^{-1}$ , then we are guaranteed that all QREs contain probabilities greater than  $x_{\\min} = \\frac{p}{\\max_k m_k}$ .",
+        "bbox": [
+            169,
+            364,
+            823,
+            398
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Corollary 1. Assume payoffs are bounded in  $[0,1]$  and  $\\tau = \\ln (1 / p)^{-1}$ , then",
+        "bbox": [
+            169,
+            404,
+            676,
+            420
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left| \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) \\right| \\leq \\frac {1}{4} \\left(\\max  _ {k} \\eta_ {k}\\right) n \\bar {m} \\left(\\frac {\\ln \\left(m ^ {*}\\right)}{\\ln (1 / p)} + 2\\right) ^ {2}. \\tag {38}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            422,
+            823,
+            457
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Proof. Starting with Lemma 10 and applying Lemma 11, we find",
+        "bbox": [
+            169,
+            469,
+            607,
+            484
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\left| \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) \\right| \\leq \\frac {1}{4} \\left(\\max  _ {k} \\eta_ {k}\\right) n \\bar {m} \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\min }}\\right) + 1\\right) ^ {2} (39a) \\\\ = \\frac {1}{4} \\left(\\max  _ {k} \\eta_ {k}\\right) n \\bar {m} \\left(\\frac {1}{\\ln (1 / p)} \\ln \\left(\\frac {m ^ {*}}{p}\\right) + 1\\right) ^ {2} (39b) \\\\ = \\frac {1}{4} \\left(\\max  _ {k} \\eta_ {k}\\right) n \\bar {m} \\left(\\frac {1}{\\ln (1 / p)} \\left(\\ln \\left(m ^ {*}\\right) + \\ln (1 / p)\\right) + 1\\right) ^ {2} (39c) \\\\ = \\frac {1}{4} \\left(\\max  _ {k} \\eta_ {k}\\right) n \\bar {m} \\left(\\frac {\\ln \\left(m ^ {*}\\right)}{\\ln (1 / p)} + 2\\right) ^ {2}. (39d) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            287,
+            488,
+            823,
+            622
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Lemma 12 (Low Temperature Approximate QREs are Approximate Nash Equilibria). Let  $\\nabla_{x_k}^{k\\tau}$  be player  $k$  's entropy regularized gradient and  $\\pmb{x}$  be an approximate QRE. Then it holds that",
+        "bbox": [
+            169,
+            647,
+            823,
+            676
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nu _ {k} \\left(B R _ {k}, x _ {- k}\\right) - u _ {k} (\\boldsymbol {x}) \\leq \\tau \\log \\left(m _ {k}\\right) + \\sqrt {2} \\left\\| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right\\|. \\tag {40}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            303,
+            680,
+            823,
+            700
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Proof. Beginning with the definition of exploitability, we find",
+        "bbox": [
+            171,
+            712,
+            581,
+            727
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} u _ {k} \\left(\\mathbb {B R} _ {k}, x _ {- k}\\right) - u _ {k} (\\boldsymbol {x}) = \\left(u _ {k} \\left(\\mathbb {B R} _ {k}, x _ {- k}\\right) + \\tau S \\left(\\mathbb {B R} _ {k}\\right) - \\tau S \\left(\\mathbb {B R} _ {k}\\right)\\right) (41a) \\\\ - \\left(u _ {k} (\\boldsymbol {x}) + \\tau S (x _ {k}) - \\tau S (x _ {k})\\right) \\\\ = u _ {k} ^ {\\tau} \\left(\\mathbb {B R} _ {k}, x _ {- k}\\right) - u _ {k} ^ {\\tau} (\\boldsymbol {x}) + \\tau \\left(S \\left(x _ {k}\\right) - S \\left(\\mathbb {B R} _ {k}\\right)\\right) (41b) \\\\ \\leq \\max  _ {z \\in \\Delta^ {m _ {k}} ^ {- 1}} u _ {k} ^ {\\tau} (z, x _ {- k}) - u _ {k} ^ {\\tau} (\\boldsymbol {x}) + \\tau \\max  _ {z ^ {\\prime} \\in \\Delta^ {m _ {k}} ^ {- 1}} S \\left(z ^ {\\prime}\\right) (41c) \\\\ \\leq \\sqrt {2} \\| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\| + \\tau \\max  _ {z ^ {\\prime} \\in \\Delta^ {m _ {k} - 1}} S \\left(z ^ {\\prime}\\right) (41d) \\\\ \\leq \\sqrt {2} | | \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) | | + \\tau \\log (m _ {k}) (41e) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            241,
+            729,
+            823,
+            864
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "where the second equality follows from the definition of player  $k$ 's entropy regularized utility  $u_{k}^{\\tau}$ , the first inequality from nonnegativity of entropy  $S$ , the second inequality from concavity of  $u_{k}^{\\tau}$  with respect to its first argument (Lemma 2), and the last from the maximum possible value of Shannon entropy over distributions on  $m_{k}$  actions.",
+        "bbox": [
+            169,
+            868,
+            825,
+            925
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "page_number",
+        "text": "20",
+        "bbox": [
+            488,
+            946,
+            509,
+            959
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Lemma 13 ( $\\mathcal{L}^{\\tau}$  Scores Nash Equilibria). Let  $\\mathcal{L}^{\\tau}(\\pmb{x})$  be our proposed entropy regularized loss function and  $\\pmb{x}$  be an approximate QRE. Then it holds that",
+        "bbox": [
+            168,
+            103,
+            823,
+            133
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\epsilon \\leq \\tau \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {\\frac {2 n}{\\min _ {k} \\eta_ {k}}} \\sqrt {\\mathcal {L} ^ {\\tau} (\\boldsymbol {x})}. \\tag {42}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            143,
+            821,
+            180
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Proof. Beginning with the definition of exploitability and applying Lemma 12, we find",
+        "bbox": [
+            169,
+            204,
+            745,
+            220
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\epsilon = \\max  _ {k} u _ {k} (\\mathbb {B R} _ {k}, x _ {- k}) - u _ {k} (\\boldsymbol {x}) \\quad \\text {(r e c a l l e a c h} \\epsilon_ {k} \\geq 0) (43a) \\\\ \\leq \\sum_ {k} u _ {k} (\\mathbb {R} _ {k}, x _ {- k}) - u _ {k} (\\boldsymbol {x}) (43b) \\\\ \\leq \\sum_ {k} \\left[ \\tau \\log \\left(m _ {k}\\right) + \\sqrt {2} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right| \\right| \\right] (43c) \\\\ = \\tau \\sum_ {k} \\log \\left(m _ {k}\\right) + \\sum_ {k} \\sqrt {2} \\left\\| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right\\| (43d) \\\\ \\leq \\tau \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {\\frac {2 n}{\\min  _ {k} \\eta_ {k}}} \\sqrt {\\mathcal {L} ^ {\\tau} (\\boldsymbol {x})}. (43e) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            310,
+            229,
+            823,
+            396
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "where the last inequality follows from the same steps (24c)-(24i) outlined in Lemma 4, which established the relationship between  $\\mathcal{L}(\\pmb{x})$  and  $\\epsilon$ .",
+        "bbox": [
+            169,
+            406,
+            823,
+            436
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "A.3 GRADIENT OF LOSS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            455,
+            357,
+            469
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Lemma 14. The gradient of  $\\mathcal{L}^{\\tau}(\\pmb{x})$  with respect to player  $l$ 's strategy  $x_{l}$  is",
+        "bbox": [
+            169,
+            482,
+            666,
+            498
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\nabla_ {x _ {l}} \\mathcal {L} (\\boldsymbol {x}) = 2 \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\tag {44}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            377,
+            508,
+            823,
+            539
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "where  $B_{ll} = -\\tau \\left[I - \\frac{1}{m_l}\\mathbf{11}^\\top \\right]diag\\left(\\frac{1}{x_l}\\right)$  and  $B_{kl} = \\left[I - \\frac{1}{m_k}\\mathbf{11}^\\top \\right]H_{kl}^k$  for  $k\\neq l$",
+        "bbox": [
+            169,
+            550,
+            704,
+            570
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Proof. Recall from Lemma 6 that the loss can be decomposed as  $\\mathcal{L}^{\\tau}(\\pmb{x}) = (A) + (B) + (C)$ .",
+        "bbox": [
+            169,
+            593,
+            787,
+            609
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Then",
+        "bbox": [
+            171,
+            614,
+            210,
+            628
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nD _ {x _ {l}} [ (A) ] = D _ {x _ {l}} \\left[ \\sum_ {k} \\eta_ {k} x _ {q} ^ {\\top} B _ {k q} ^ {\\top} B _ {k q} x _ {q} \\right] = 2 \\sum_ {k \\neq l} \\eta_ {k} B _ {k l} ^ {\\top} B _ {k l} x _ {l} \\tag {45}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            300,
+            637,
+            823,
+            670
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "where  $q \\neq k$  and  $B_{kq} = [I - \\frac{1}{m_k}\\mathbf{11}^\\top][H_{kq}^k]$  does not depend on  $x_{k}$ .",
+        "bbox": [
+            169,
+            681,
+            620,
+            700
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Also, letting  $B_{ll} = -\\tau [I - \\frac{1}{m_l}\\mathbf{11}^\\top ]\\mathrm{diag}\\left(\\frac{1}{x_l}\\right)$",
+        "bbox": [
+            169,
+            705,
+            486,
+            726
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} D _ {x _ {l}} [ (B) ] = D _ {x _ {l}} \\left[ - 2 \\tau \\sum_ {k} \\eta_ {k} \\ln \\left(x _ {k}\\right) ^ {\\top} B _ {k q} x _ {q} \\right] (46a) \\\\ = - 2 \\tau \\left[ \\eta_ {l} D _ {x _ {l}} \\left[ \\ln \\left(x _ {l}\\right) ^ {\\top} B _ {l q} x _ {q} \\right] + \\sum_ {k \\neq l} \\eta_ {k} D _ {x _ {l}} \\left[ \\ln \\left(x _ {k}\\right) ^ {\\top} B _ {k l} x _ {l} \\right] \\right] (46b) \\\\ = - 2 \\tau \\left[ \\eta_ {l} \\operatorname {d i a g} \\left(\\frac {1}{x _ {l}}\\right) B _ {l q} x _ {q} + \\sum_ {k \\neq l} \\eta_ {k} B _ {k l} ^ {\\top} \\ln \\left(x _ {k}\\right) \\right] (46c) \\\\ = - 2 \\tau \\left[ \\eta_ {l} \\left(\\left[ I - \\frac {1}{m _ {l}} \\mathbf {1 1} ^ {\\top} \\right] \\operatorname {d i a g} \\left(\\frac {1}{x _ {l}}\\right)\\right) ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla^ {l}\\right) + \\sum_ {k \\neq l} \\eta_ {k} B _ {k l} ^ {\\top} \\ln \\left(x _ {k}\\right) \\right] (46d) \\\\ = 2 \\left[ \\eta_ {l} B _ {l l} ^ {\\top} \\Pi_ {T \\Delta} (\\nabla^ {l}) - \\tau \\sum_ {k \\neq l} \\eta_ {k} B _ {k l} ^ {\\top} \\ln (x _ {k}) \\right]. (46e) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            215,
+            736,
+            823,
+            920
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "page_number",
+        "text": "21",
+        "bbox": [
+            488,
+            946,
+            506,
+            959
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "And",
+        "bbox": [
+            171,
+            104,
+            205,
+            118
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} D _ {x _ {l}} [ (C) ] = D _ {x _ {l}} \\left[ \\sum_ {k} \\eta_ {k} \\tau^ {2} \\ln \\left(x _ {k}\\right) ^ {\\top} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] \\ln \\left(x _ {k}\\right) \\right] (47a) \\\\ = 2 \\tau^ {2} \\left[ \\eta_ {l} \\operatorname {d i a g} \\left(\\frac {1}{x _ {l}}\\right) \\left[ I - \\frac {1}{m _ {l}} \\mathbf {1 1} ^ {\\top} \\right] \\ln \\left(x _ {l}\\right) \\right] (47b) \\\\ = - 2 \\tau \\eta_ {l} \\left(\\left[ I - \\frac {1}{m _ {l}} \\mathbf {1 1} ^ {\\top} \\right] \\operatorname {d i a g} \\left(\\frac {1}{x _ {l}}\\right)\\right) ^ {\\top} \\Pi_ {T \\Delta} (- \\tau \\ln (x _ {l})) (47c) \\\\ = 2 \\eta_ {l} B _ {l l} ^ {\\top} \\Pi_ {T \\Delta} (- \\tau \\ln (x _ {l})). (47d) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            279,
+            133,
+            823,
+            255
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "Putting these together, we find",
+        "bbox": [
+            171,
+            280,
+            375,
+            295
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\nabla_ {x _ {l}} \\mathcal {L} (\\boldsymbol {x}) = 2 \\sum_ {k \\neq l} \\eta_ {k} B _ {k l} ^ {\\top} \\left(B _ {k l} x _ {l} - \\tau \\ln \\left(x _ {k}\\right)\\right) + 2 \\eta_ {l} B _ {l l} ^ {\\top} \\left[ \\Pi_ {T \\Delta} \\left(\\nabla^ {l}\\right) + \\Pi_ {T \\Delta} (- \\tau \\ln \\left(x _ {l}\\right)) \\right] (48a) \\\\ = 2 \\eta_ {l} B _ {l l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) + 2 \\sum_ {k \\neq l} \\eta_ {k} B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) (48b) \\\\ = 2 \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right). (48c) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            196,
+            313,
+            825,
+            419
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "image",
+        "img_path": "images/656915cfebd3ae35313223e7b48f6e8cb5eb9682343feb6bbf64f9d05bbaa973.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            445,
+            823,
+            457
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "A.3.1 UNBIASED ESTIMATION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            508,
+            401,
+            523
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "In order to construct an unbiased estimate for each  $B_{kl}$ , we will need to form an independent unbiased estimate of  $H_{kl}^{k}$ . Recall that  $H_{kl}^{k}$  is simply the expected bimatrix game between players  $k$  and  $l$  when all other players sample their actions according to their current strategies.",
+        "bbox": [
+            169,
+            539,
+            826,
+            583
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "A.4 BOUND ON GRADIENT / LIPSCHITZ PROPERTY",
+        "text_level": 1,
+        "bbox": [
+            171,
+            609,
+            542,
+            625
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "Lemma 15. Assume payoffs are upper bounded by 1, then the infinity norm of the gradient is bounded as",
+        "bbox": [
+            169,
+            641,
+            826,
+            669
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left| \\left| \\nabla_ {\\boldsymbol {x}} \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) \\right| \\right| _ {\\infty} \\leq \\frac {1}{2} \\left(\\max  _ {k} \\eta_ {k}\\right) \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) \\left[ \\tau m ^ {*} \\left(\\frac {1}{x _ {\\operatorname* {m i n}}} - 1\\right) + n \\bar {m} \\right]. \\tag {49}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            246,
+            684,
+            825,
+            715
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "Proof. Recall from Lemma 14 that the gradient of  $\\mathcal{L}(\\pmb{x})$  with respect to player  $l$ 's strategy  $x_{l}$  is",
+        "bbox": [
+            171,
+            765,
+            797,
+            780
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\nabla_ {x _ {l}} \\mathcal {L} (\\boldsymbol {x}) = 2 \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\tag {50}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            377,
+            797,
+            825,
+            830
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "where  $B_{ll} = -\\tau [I - \\frac{1}{m_l}\\mathbf{11}^\\top ]\\mathrm{diag}\\left(\\frac{1}{x_l}\\right)$  and  $B_{kl} = [I - \\frac{1}{m_k}\\mathbf{11}^\\top ]H_{kl}^k$  for  $k\\neq l$",
+        "bbox": [
+            169,
+            848,
+            705,
+            869
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "As noted before in the derivation of Lemma 10, for payoffs in  $[0,1]$ , the entries in  $\\nabla_{x_k}^{k\\tau} = \\nabla_{x_k}^k - \\tau \\ln(x_k)$  are bounded within  $[0, \\tau \\ln(\\frac{1}{x_{\\min}}) + 1]$  with a range  $\\tau \\ln(\\frac{1}{x_{\\min}}) + 1$ . Similarly, the entries in  $-\\tau \\mathrm{diag}\\left(\\frac{1}{x_l}\\right)$  are bounded within  $[- \\tau \\frac{1}{x_{\\min}}, -\\tau]$  with a range of  $\\tau\\left(\\frac{1}{x_{\\min}} - 1\\right)$ .",
+        "bbox": [
+            169,
+            875,
+            823,
+            929
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "page_number",
+        "text": "22",
+        "bbox": [
+            488,
+            946,
+            509,
+            960
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "The infinity norm of the gradient can then be bounded as",
+        "bbox": [
+            171,
+            103,
+            547,
+            118
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\left| \\left| \\nabla_ {\\boldsymbol {x}} \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) \\right| \\right| _ {\\infty} = \\max  _ {l} \\left| \\left| \\nabla_ {x _ {l}} \\mathcal {L} (\\boldsymbol {x}) \\right| \\right| _ {\\infty} (51a) \\\\ = \\max  _ {l} \\left\\| 2 \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right\\| _ {\\infty} (51b) \\\\ \\leq 2 \\sum_ {k} \\eta_ {k} \\max  _ {l} | | B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) | | _ {\\infty} (51c) \\\\ \\leq \\frac {1}{2} \\sum_ {k \\neq l ^ {*}} \\eta_ {k} m _ {k} \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) + \\frac {1}{2} \\eta_ {l ^ {*}} m _ {l ^ {*}} \\tau \\left(\\frac {1}{x _ {\\operatorname* {m i n}}} - 1\\right) \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) (51d) \\\\ = \\frac {1}{2} \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) \\left[ \\eta_ {l ^ {*}} m _ {l ^ {*}} \\tau \\left(\\frac {1}{x _ {\\operatorname* {m i n}}} - 1\\right) + \\sum_ {k \\neq l ^ {*}} \\eta_ {k} m _ {k} \\right] (51e) \\\\ \\leq \\frac {1}{2} \\left(\\max  _ {k} \\eta_ {k}\\right) \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) \\left[ \\tau m _ {l ^ {*}} \\left(\\frac {1}{x _ {\\operatorname* {m i n}}} - 1\\right) + \\sum_ {k \\neq l ^ {*}} m _ {k} \\right] (51f) \\\\ \\leq \\frac {1}{2} \\left(\\max  _ {k} \\eta_ {k}\\right) \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) \\left[ \\tau m ^ {*} \\left(\\frac {1}{x _ {\\operatorname* {m i n}}} - 1\\right) + n \\bar {m} \\right] (51g) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            184,
+            121,
+            823,
+            381
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "where the second inequality follows from Lemma 9.",
+        "bbox": [
+            171,
+            383,
+            517,
+            398
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "image",
+        "img_path": "images/c3cecbfa94a28df4cc8398ee9a6a5e9dcac15ae2c798d61c4f4866495a511e19.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            405,
+            823,
+            417
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "Corollary 2. If  $\\tau$  is set according to Lemma 11 as  $\\tau = \\ln (1 / p)^{-1}$  and payoffs are in [0, 1], then the infinity norm of the gradient is bounded as",
+        "bbox": [
+            169,
+            426,
+            823,
+            455
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left| \\left| \\nabla_ {\\boldsymbol {x}} \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) \\right| \\right| _ {\\infty} \\leq \\frac {1}{2} \\left(\\max  _ {k} \\eta_ {k}\\right) \\left[ \\frac {\\ln \\left(m ^ {*}\\right)}{\\ln (1 / p)} + 2 \\right] \\left[ \\frac {m ^ {* 2}}{p \\ln (1 / p)} + n \\bar {m} \\right] = \\frac {1}{2} \\left(\\max  _ {k} \\eta_ {k}\\right) \\hat {L} \\tag {52}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            233,
+            459,
+            823,
+            494
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "where  $m^{*} = \\max_{k} m_{k}$  and  $\\hat{L}$  is defined implicitly for convenience in other derivations.",
+        "bbox": [
+            171,
+            500,
+            745,
+            513
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "Proof. Starting with Lemma 15 and applying Lemma 11 (i.e.,  $\\tau = \\ln(1/p)^{-1}$  and  $x_{\\min} = \\frac{p}{m^*}$  with payoffs in [0, 1] and where  $m^* = \\max_k m_k$ ), we find",
+        "bbox": [
+            169,
+            527,
+            823,
+            558
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\left| \\left| \\nabla_ {\\boldsymbol {x}} \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) \\right| \\right| _ {\\infty} \\leq \\frac {1}{2} \\left(\\max  _ {k} \\eta_ {k}\\right) \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) \\left[ \\tau m ^ {*} \\left(\\frac {1}{x _ {\\operatorname* {m i n}}} - 1\\right) + n \\bar {m} \\right] (53a) \\\\ = \\frac {1}{2} \\left(\\max  _ {k} \\eta_ {k}\\right) \\left[ \\frac {\\ln \\left(m ^ {*} / p\\right)}{\\ln (1 / p)} + 1 \\right] \\left[ \\frac {m ^ {*}}{\\ln (1 / p)} \\left(\\frac {m ^ {*}}{p} - 1\\right) + n \\bar {m} \\right] (53b) \\\\ \\leq \\frac {1}{2} \\left(\\max  _ {k} \\eta_ {k}\\right) \\left[ \\frac {\\ln \\left(m ^ {*}\\right)}{\\ln (1 / p)} + 2 \\right] \\left[ \\frac {m ^ {* 2}}{p \\ln (1 / p)} + n \\bar {m} \\right]. (53c) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            250,
+            561,
+            823,
+            662
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "As  $p \\to 0^+$ , the norm of the gradient blows up because the gradient of Shannon entropy blows up for small probabilities. As  $p \\to 1$ , the norm of the gradient blows up because we require infinite temperature  $\\tau$  to guarantee all QREs are nearly uniform; recall  $\\tau$  is the regularization coefficient on the entropy bonus terms which means our modified utilities blow up for large  $\\tau$ .",
+        "bbox": [
+            169,
+            674,
+            826,
+            733
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "A.5 HESSIAN OF LOSS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            747,
+            346,
+            760
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "We will now derive the Hessian of our loss. This will be useful in establishing properties about global minima that enable the application of tailored minimization algorithms. Let  $D_z[f(z)]$  denote the differential operator applied to (possibly multivalued) function  $f$  with respect to  $z$ . For example,  $D_{x_q}[H_{lk}^k] = D_{x_q}[x_qT_{qlk}^k] = T_{qlk}^k$  where  $T_{qlk}^k$  is player  $k$ 's payoff tensor according to the three-way approximation between players  $k, l$ , and  $q$  to the game at  $x$ .",
+        "bbox": [
+            169,
+            773,
+            826,
+            845
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "Lemma 16. The Hessian of  $\\mathcal{L}^{\\tau}(\\pmb{x})$  can be written",
+        "bbox": [
+            171,
+            847,
+            506,
+            863
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nH e s s \\left(\\mathcal {L} ^ {\\tau}\\right) = 2 \\tilde {B} ^ {\\top} \\tilde {B} + T \\Pi_ {T \\Delta} \\left(\\tilde {\\nabla} ^ {\\tau}\\right) \\tag {54}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            375,
+            866,
+            823,
+            885
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "where  $\\tilde{B}_{kl} = \\sqrt{\\eta_k} B_{kl}$ ,  $\\Pi_{T\\Delta}(\\tilde{\\nabla}^{\\tau}) = [\\eta_1\\Pi_{T\\Delta}(\\nabla_{x_1}^{1\\tau}),\\dots ,\\eta_n\\Pi_{T\\Delta}(\\nabla_{x_n}^{n\\tau})]$ , and we augment  $T$  (the 3-player tensor approximation to the game,  $T_{lqk}^{k}$ ) so that  $T_{ll}^{l} = \\tau \\text{diag} 3\\left(\\frac{1}{x_{l}^{2}}\\right)$  and otherwise 0.",
+        "bbox": [
+            169,
+            890,
+            823,
+            928
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "page_number",
+        "text": "23",
+        "bbox": [
+            488,
+            946,
+            508,
+            960
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "Proof. Recall the gradient of our proposed loss:",
+        "bbox": [
+            171,
+            103,
+            488,
+            119
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\nabla_ {x _ {l}} \\mathcal {L} (\\boldsymbol {x}) = 2 \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\tag {55}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            377,
+            148,
+            823,
+            180
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "where  $B_{ll} = -\\tau [I - \\frac{1}{m_l}\\mathbf{11}^\\top ]\\mathrm{diag}\\left(\\frac{1}{x_l}\\right)$  and  $B_{kl} = [I - \\frac{1}{m_k}\\mathbf{11}^\\top ]H_{kl}^k$  for  $k\\neq l$",
+        "bbox": [
+            169,
+            208,
+            702,
+            229
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "Consider the following Jacobians, which will play an auxiliary role in our derivation of the Hessian:",
+        "bbox": [
+            169,
+            233,
+            818,
+            250
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nD _ {l} \\left[ B _ {l l} \\right] = \\tau \\left[ I - \\frac {1}{m _ {l}} \\mathbf {1 1} ^ {\\top} \\right] \\operatorname {d i a g} 3 \\left(\\frac {1}{x _ {l} ^ {2}}\\right) \\tag {56a}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            375,
+            277,
+            823,
+            310
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nD _ {q} \\left[ B _ {l l} \\right] = 0 \\tag {56b}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            372,
+            311,
+            823,
+            328
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nD _ {l} \\left[ B _ {k l} \\right] = 0 \\tag {56c}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            372,
+            330,
+            823,
+            345
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nD _ {q} \\left[ B _ {k l} \\right] = \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] T _ {k l q} ^ {k} \\tag {56d}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            370,
+            348,
+            823,
+            378
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} D _ {k} \\left[ \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right] = \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] D _ {k} \\left[ \\nabla_ {x _ {k}} ^ {k \\tau} \\right] (56e) \\\\ = \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] D _ {k} \\left[ \\nabla_ {x _ {k}} ^ {k} - \\tau \\ln \\left(x _ {k}\\right) \\right] (56f) \\\\ = \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] \\left[ - \\tau \\operatorname {d i a g} \\left(\\frac {1}{x _ {k}}\\right) \\right] (56g) \\\\ = B _ {k k} (56h) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            318,
+            381,
+            823,
+            493
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} D _ {l} \\left[ \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right] = \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] D _ {l} \\left[ \\nabla_ {x _ {k}} ^ {k \\tau} \\right] (56i) \\\\ = \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] D _ {l} \\left[ \\nabla_ {x _ {k}} ^ {k} - \\tau \\ln (x _ {k}) \\right] (56j) \\\\ = \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] \\left[ H _ {k l} ^ {k} \\right] (56k) \\\\ = B _ {k l}. (561) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            323,
+            496,
+            823,
+            608
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "We can derive the diagonal blocks of the Hessian as",
+        "bbox": [
+            171,
+            645,
+            514,
+            660
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} D _ {l l} [ \\mathcal {L} (\\boldsymbol {x}) ] = D _ {l} \\left[ \\nabla_ {x _ {l}} \\mathcal {L} (\\boldsymbol {x}) \\right] (57a) \\\\ = 2 D _ {l} \\left[ \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right] (57b) \\\\ = 2 \\left[ \\eta_ {l} D _ {l} \\left[ B _ {l l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {l}} ^ {l \\tau}\\right) \\right] + \\sum_ {k \\neq l} \\eta_ {k} D _ {l} \\left[ B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right] \\right] (57c) \\\\ = 2 \\left[ \\eta_ {l} \\left[ D _ {l} \\left[ B _ {l l} \\right] ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {l}} ^ {l \\tau}\\right) + B _ {l l} ^ {\\top} D _ {l} \\left[ \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {l}} ^ {l \\tau}\\right) \\right] \\right] \\right. (57d) \\\\ \\left. + \\sum_ {k \\neq l} \\eta_ {k} \\left[ D _ {t} \\left[ B _ {k l} \\right] ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) + B _ {k l} ^ {\\top} D _ {l} \\left[ \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right] \\right] \\right] (57e) \\\\ = 2 \\left[ \\eta_ {l} \\left[ \\tau \\operatorname {d i a g} 3 \\left(\\frac {1}{x _ {l} ^ {2}}\\right) [ I - \\frac {1}{m _ {l}} \\mathbf {1 1} ^ {\\top} ] \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {l}} ^ {l \\tau}\\right) + B _ {l l} ^ {\\top} B _ {l l} \\right] + \\sum_ {k \\neq l} \\eta_ {k} B _ {k l} ^ {\\top} B _ {k l} \\right] (57f) \\\\ = 2 \\left[ \\tau \\eta_ {l} \\operatorname {d i a g} \\left(\\left[ \\frac {1}{x _ {l} ^ {2}} \\right] \\odot \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {l}} ^ {l \\tau}\\right)\\right) + \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} B _ {k l} \\right] (57g) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            194,
+            689,
+            823,
+            921
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "page_number",
+        "text": "24",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "and the off-diagonal blocks as",
+        "bbox": [
+            171,
+            104,
+            374,
+            119
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} D _ {l q} [ \\mathcal {L} (\\boldsymbol {x}) ] = D _ {q} \\left[ \\nabla_ {x _ {l}} \\mathcal {L} (\\boldsymbol {x}) \\right] (58a) \\\\ = 2 D _ {q} \\left[ \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right] (58b) \\\\ = 2 \\left[ \\eta_ {l} D _ {q} \\left[ B _ {l l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {l}} ^ {l \\tau}\\right) \\right] + \\sum_ {k \\neq l} \\eta_ {k} D _ {q} \\left[ B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right] \\right] (58c) \\\\ = 2 \\left[ \\eta_ {l} \\left[ D _ {q} \\left[ B _ {\\bar {l} l} \\right] ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {l}} ^ {l \\tau}\\right) + B _ {l l} ^ {\\top} D _ {q} \\left[ \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {l}} ^ {l \\tau}\\right) \\right] \\right] \\right. (58d) \\\\ \\left. + \\sum_ {k \\neq l} \\eta_ {k} \\left[ D _ {q} \\left[ B _ {k l} \\right] ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) + B _ {k l} ^ {\\top} D _ {q} \\left[ \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right] \\right] \\right] (58e) \\\\ = 2 \\left[ \\eta_ {l} B _ {l l} ^ {\\top} B _ {l q} + \\sum_ {k \\neq l} \\eta_ {k} \\left[ T _ {l q k} ^ {k} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) + B _ {k l} ^ {\\top} B _ {k q} \\right] \\right] (58f) \\\\ = 2 \\left[ \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} B _ {k q} + \\sum_ {k \\neq l} \\eta_ {k} T _ {l q k} ^ {k} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right]. (58g) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            222,
+            126,
+            823,
+            356
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "Therefore, the Hessian can be written concisely as",
+        "bbox": [
+            171,
+            371,
+            504,
+            386
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n2 \\left[ \\tilde {B} ^ {\\top} \\tilde {B} + T \\Pi_ {T \\Delta} \\left(\\tilde {\\nabla} ^ {\\tau}\\right) \\right] \\tag {59}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            416,
+            393,
+            823,
+            412
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "where  $\\tilde{B}_{kl} = \\sqrt{\\eta_k} B_{kl}$ ,  $\\Pi_{T\\Delta}(\\tilde{\\nabla}^{\\tau}) = [\\eta_1\\Pi_{T\\Delta}(\\nabla_{x_1}^{1\\tau}),\\dots ,\\eta_n\\Pi_{T\\Delta}(\\nabla_{x_n}^{n\\tau})]$ , and we augment  $T$  (the 3-player tensor approximation to the game,  $T_{lqk}^{k}$ ) so that  $T_{lll}^{l} = \\tau \\mathrm{diag}3\\left(\\frac{1}{x_{l}^{2}}\\right)$  and otherwise 0.",
+        "bbox": [
+            169,
+            420,
+            823,
+            458
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "image",
+        "img_path": "images/6c19be3c9e2bc7550fabf8ab2030175dabaf14544af78c0bcc1ecf5dd2c3ec4a.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            464,
+            823,
+            474
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "page_number",
+        "text": "25",
+        "bbox": [
+            488,
+            946,
+            506,
+            959
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "B GLOBAL CONVERGENCE GUARANTEES",
+        "text_level": 1,
+        "bbox": [
+            171,
+            102,
+            537,
+            118
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "In this section, we analyze the application of optimization techniques such as bandit algorithms and gradient descent to our loss function.",
+        "bbox": [
+            169,
+            133,
+            823,
+            162
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "B.1 MAPS FROM HYPERCUBE TO SIMPLEX PRODUCT",
+        "text_level": 1,
+        "bbox": [
+            171,
+            180,
+            555,
+            194
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "In this subsection, we derive properties of a map  $s$  from the unit-hypercube to the simplex product. This map is necessary to adapt our proposed loss  $\\mathcal{L}^{\\tau}$  to the commonly assumed setting in the  $\\mathcal{X}$ -armed bandit literature (Bubeck et al., 2011) where the feasible set is a hypercube. We derive relevant properties of two such maps: the softmax and a mapping that interprets dimensions of the hypercube as angles on a unit-sphere that are then  $\\ell_1$ -normalized.",
+        "bbox": [
+            169,
+            207,
+            826,
+            276
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "Lemma 17. Let  $f(x) = -\\mathcal{L}(s(x))$ . Then  $||\\nabla f(x)||_{\\infty} \\leq ||J(s(x))^{\\top}||_{\\infty}||\\nabla \\mathcal{L}(s(x))||_{\\infty}$ .",
+        "bbox": [
+            169,
+            280,
+            750,
+            297
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "Proof.",
+        "bbox": [
+            171,
+            314,
+            217,
+            328
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left. \\right.\\left|\\left| \\nabla f (x) \\right|\\right| _ {\\infty} = \\left|\\left| J (s (x)) ^ {\\top} \\nabla \\mathcal {L} (s (x)) \\right|\\right| _ {\\infty} \\leq \\left|\\left| J (s (x)) ^ {\\top} \\right|\\right| _ {\\infty} \\left|\\left| \\nabla \\mathcal {L} (s (x)) \\right|\\right| _ {\\infty}. \\tag {60}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            253,
+            335,
+            823,
+            354
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "image",
+        "img_path": "images/1e08a5207ad835653f0ce7f7340fbc475fc5637f70c05eeb3542b9286c58ee43.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            363,
+            823,
+            375
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "Lemma 18. The  $\\infty$ -norm of the Jacobian-transpose of a transformation  $s(x)$  applied elementwise to a product space is bounded by the  $\\infty$ -norm of the Jacobian-transpose of a single transformation from that product space, i.e.,  $\\|J(s(\\pmb{x}))^\\top\\|_\\infty \\leq \\max_{x_i \\in \\mathcal{X}_i} \\|J(s(x_i))^\\top\\|_\\infty$  for any  $i$ .",
+        "bbox": [
+            168,
+            387,
+            823,
+            431
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "Proof. Let  $\\boldsymbol{x} \\in \\mathcal{X} = \\prod_{i=1}^{n} \\mathcal{X}_i$ ,  $\\mathcal{Z} = \\prod_{i=1}^{n} \\mathcal{Z}_i$  and  $S: \\mathcal{X} \\to \\mathcal{Z} = [s(x_1); \\dots; s(x_n)]^\\top$  where; denotes column-wise stacking,  $x_i \\in \\mathcal{X}_i$ . Also,  $\\mathcal{X}_i = \\mathcal{X}_j$  and  $\\mathcal{Z}_i = \\mathcal{Z}_j$  for all  $i$  and  $j$ . Then the Jacobian of  $S(\\boldsymbol{x})$  is",
+        "bbox": [
+            169,
+            446,
+            826,
+            491
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nJ (S (\\boldsymbol {x})) ^ {\\top} = \\left[ \\begin{array}{c c c} J (s (x _ {1})) ^ {\\top} & 0 \\dots & 0 \\\\ 0 & J (s (x _ {2})) ^ {\\top} \\dots & 0 \\\\ 0 & 0 \\ddots & 0 \\\\ 0 & 0 \\dots & J (s (x _ {n})) ^ {\\top} \\end{array} \\right]. \\tag {61}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            300,
+            497,
+            823,
+            571
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "The  $\\infty$ -norm of this matrix is the max 1-norm of any row. This matrix is diagonal, therefore, the  $\\infty$ -norm of each elementwise Jacobian-transpose represents the max 1-norm of the rows spanned by its block. Given that the domains, ranges, and transformations  $s$  for all blocks are the same, their  $\\infty$ -norms are also the same. The max  $\\infty$  over the blocks is then equal to the  $\\infty$ -norm of any individual  $J(s(x_i))^\\top$ .",
+        "bbox": [
+            169,
+            577,
+            826,
+            647
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "B.1.1 HESSIAN OF BANDIT REWARD FUNCTION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            664,
+            519,
+            679
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "Lemma 19. Let  $s(x)$  be a function that maps the unit hypercube to the simplex product (mixed strategy space). Then the objective function  $f(x) = -\\mathcal{L}(s(x))$ . The Hessian of  $-f(x)$  at an optimum  $x^{*}$  in direction  $\\Delta$  is  $\\Delta x^{\\top}[Ds(x)^{\\top}H_{\\mathcal{L}}(x)Ds(x)]\\Big|_{x^{*}}\\Delta x$  where  $H_{\\mathcal{L}}$  is the Hessian of  $\\mathcal{L}$  and  $Ds(x)$  is the Jacobian of  $s(x)$ .",
+        "bbox": [
+            169,
+            689,
+            826,
+            756
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "Proof.",
+        "bbox": [
+            171,
+            775,
+            217,
+            789
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\left. \\left(D ^ {2} (\\mathcal {L} \\circ s) \\left(x ^ {*}\\right)\\right) (\\Delta x, \\Delta x) = \\Delta x ^ {\\top} \\left[ \\sum_ {i} \\overbrace {\\partial_ {i} \\mathcal {L} (s (x))} ^ {= 0 \\text {a t} x = x ^ {*}} D ^ {2} h _ {i} (x) \\right] \\right| _ {x ^ {*}} \\Delta x (62a) \\\\ + \\Delta x ^ {\\top} [ D s (x) ^ {\\top} H _ {\\mathcal {L}} (x) D s (x) ] \\bigg | _ {x ^ {*}} \\Delta x \\\\ = \\Delta x ^ {\\top} \\left[ D s (x) ^ {\\top} H _ {\\mathcal {L}} (x) D s (x) \\right] \\Big | _ {x ^ {*}} \\Delta x. (62b) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            267,
+            797,
+            825,
+            902
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "image",
+        "img_path": "images/25a01722fe07d536f17dc9ae0bfaee5e73934a66ff0b933fce7eae71fe419569.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            910,
+            823,
+            921
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "page_number",
+        "text": "26",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "Lemma 20. Let  $s(x): \\mathcal{X} \\to \\prod_k \\Delta^{m_k - 1}$  be an injective function, i.e.,  $x \\neq y \\Rightarrow s(x) \\neq s(y)$ . Also let  $J = J(s(x))$  be the Jacobian of  $s$  with respect to  $x$  and  $\\Delta x$  be a nonzero vector in the tangent space of  $\\mathcal{X}$ . Then",
+        "bbox": [
+            168,
+            103,
+            826,
+            145
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nJ \\Delta x \\neq 0. \\tag {63}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            460,
+            148,
+            823,
+            162
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "Proof. Recall that the  $ij$ th entry of the Jacobian represents  $\\frac{\\partial s_i}{\\partial x_j}$  so that the  $i$ th entry of  $J\\Delta x$  is",
+        "bbox": [
+            169,
+            179,
+            792,
+            200
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n[ J \\Delta x ] _ {i} = \\sum_ {j} \\frac {\\partial s _ {i}}{\\partial x _ {j}} \\Delta x _ {j} = d s _ {i}. \\tag {64}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            393,
+            202,
+            823,
+            237
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "Assume  $J\\Delta x = 0$ . This would imply a change in  $x \\in \\mathcal{X}$  results in no change in  $s$  ( $ds = 0$ ), contradicting the fact that  $s$  is injective. Therefore, we must conclude the claim that  $J\\Delta x \\neq 0$ .",
+        "bbox": [
+            169,
+            239,
+            826,
+            268
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "Lemma 21. Let  $J = J(s(x))$  be the Jacobian of any composition of transformations  $s = s_t \\circ \\ldots \\circ s_1$  where  $s_t(z) = [z_i / \\sum_j z_j]_i$ . Then  $J\\Delta x$  lies in the tangent space of the simplex.",
+        "bbox": [
+            169,
+            275,
+            823,
+            306
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "Proof. We aim to show  $\\mathbf{1}^{\\top}J\\Delta x = \\mathbf{0}$  for any  $\\Delta x$  and  $x$ . By chain rule, the Jacobian of  $s$  is  $J = J(s) = \\prod_{t'=t}^{t'=1} J(s_t')$ . Therefore,  $\\mathbf{1}^{\\top}J\\Delta x = \\mathbf{1}^{\\top}(\\prod_{t'=t}^{t'=1} J(s_t'))\\Delta x$ . Consider the first product:",
+        "bbox": [
+            169,
+            321,
+            823,
+            356
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {1} ^ {\\top} J \\left(s _ {t}\\right) = \\mathbf {0} \\tag {65}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            450,
+            358,
+            823,
+            375
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "by Lemma 24. Therefore  $\\mathbf{1}^{\\top}J\\Delta x = \\mathbf{1}^{\\top}J(s_{t})(\\prod_{t^{\\prime} = t - 1}^{t^{\\prime} = 1}J(s_{t}^{\\prime}))\\Delta x = \\mathbf{0}^{\\top}(\\prod_{t^{\\prime} = t - 1}^{t^{\\prime} = 1}J(s_{t}^{\\prime}))\\Delta x = 0$ . This implies  $J\\Delta x$  is orthogonal to  $\\mathbf{1}$  for any  $x\\in \\mathcal{X}$  and  $\\Delta x$ , therefore  $J\\Delta x$  lies in the tangent space of the simplex for any  $x\\in \\mathcal{X}$  and  $\\Delta x$ .",
+        "bbox": [
+            169,
+            378,
+            826,
+            424
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "B.1.2 SOFTMAX MAP",
+        "text_level": 1,
+        "bbox": [
+            171,
+            439,
+            341,
+            452
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "Let  $s:[0,1]^{d - 1}\\to \\Delta^{d - 1}\\in \\mathbb{R}^d$  be the softmax function. See (Gao and Pavel, 2017) for an analysis of many of its properties and in the context of game theory. Note that  $s$  maps a  $(d - 1)$  dimensional variable to a  $d$  dimensional distribution. This can be practically handled by always appending a 0 to the  $(d - 1)$ -dimensional input prior to applying the standard softmax. We perform our analysis below in terms of the standard softmax, but note the norms we derive apply to our modified (invertible) softmax.",
+        "bbox": [
+            169,
+            462,
+            826,
+            546
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "Standard:",
+        "bbox": [
+            171,
+            554,
+            238,
+            566
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\ns (x) = \\frac {1}{\\sum_ {j = 1} ^ {d} e ^ {x _ {j}}} \\left[ e ^ {x _ {1}}, \\dots e ^ {x _ {d}} \\right]. \\tag {66}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            387,
+            566,
+            823,
+            603
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "Modified:",
+        "bbox": [
+            171,
+            611,
+            240,
+            625
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\ns (x) = \\frac {1}{1 + \\sum_ {j = 1} ^ {d - 1} e ^ {x _ {j}}} \\left[ e ^ {x _ {1}}, \\dots e ^ {x _ {d - 1}}, 1 \\right]. \\tag {67}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            357,
+            625,
+            823,
+            661
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "Lemma 22. Let  $J$  be the Jacobian of the softmax operator. Then  $\\| J\\|_{\\infty}\\leq 2$  and  $\\| J^{\\top}\\|_{\\infty}\\leq 2$",
+        "bbox": [
+            169,
+            662,
+            797,
+            680
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "Proof. Let  $S_{i}$  represent the  $i$ th entry of  $S = \\mathrm{softmax}(z)$  for any  $z \\in \\mathbb{R}^{m}$ . Then the 1-norm of row  $i$  is upper bounded as",
+        "bbox": [
+            169,
+            693,
+            823,
+            722
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\Rightarrow \\sum_ {j} \\left| D _ {j} S _ {i} \\right| = \\sum_ {j} \\left| S _ {i} \\left(\\delta_ {i j} - S _ {j}\\right) \\right| \\tag {68b}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            369,
+            744,
+            823,
+            775
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} D _ {j} S _ {i} = S _ {i} \\left(\\delta_ {i j} - S _ {j}\\right) (68a) \\\\ \\leq \\sum_ {j} \\left| \\delta_ {i j} S _ {i} \\right| + \\left| S _ {i} S _ {j} \\right| (68c) \\\\ = S _ {i} + \\sum_ {j} S _ {i} S _ {j} (68d) \\\\ = S _ {i} + S _ {i} \\sum_ {j} S _ {j} (68e) \\\\ = 2 S _ {i} (68f) \\\\ \\leq 2 \\forall i. (68g) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            439,
+            724,
+            823,
+            920
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "page_number",
+        "text": "27",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "Also, the 1-norm of column  $j$  is upper bounded similarly as",
+        "bbox": [
+            169,
+            103,
+            566,
+            119
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} (6 9 a) \\\\ \\sum_ {i} \\left| D _ {j} S _ {i} \\right| = \\sum_ {i} \\left| S _ {i} \\left(\\delta_ {i j} - S _ {j}\\right) \\right| (69b) \\\\ \\leq \\sum_ {i} \\left| \\delta_ {i j} S _ {i} \\right| + \\left| S _ {i} S _ {j} \\right| (69c) \\\\ = S _ {j} + \\sum_ {i} S _ {i} S _ {j} (69d) \\\\ = S _ {i} + S _ {j} \\sum_ {i} S _ {i} (69e) \\\\ = 2 S _ {j} (69f) \\\\ \\leq 2 \\forall j. (69g) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            383,
+            128,
+            825,
+            320
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "The  $\\infty$ -norm of a matrix is the maximum 1-norm of any row. Therefore,  $||J||_{\\infty}$  and  $||J^{\\top}||_{\\infty}$  are both upper bounded by 2.",
+        "bbox": [
+            169,
+            330,
+            826,
+            362
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "Corollary 3. Let  $J$  be the Jacobian of the softmax operator. Then  $\\| J\\| _2\\leq 2$  and  $\\| J^{\\top}\\|_{2}\\leq 2$",
+        "bbox": [
+            169,
+            378,
+            790,
+            396
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "Proof. The Gershgorin circle theorem states that every eigenvalue of  $J$  lies within one of the discs centered at the diagonal of  $J$  with radius equal to the 1-norm of the row (excluding the diagonal term). The Jacobian of the softmax operator contains only nonnegative diagonal terms, therefore, the 1-norm of any entire row also represents the maximum magnitude of any eigenvalue allowed by any Gershgorin disc. In addition,  $J$  is symmetric and therefore, the magnitude of its eigenvalues are equivalent to its singular values. Therefore by Lemma 22,  $||J||_2 \\leq 2$ .",
+        "bbox": [
+            169,
+            422,
+            826,
+            507
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "B.1.3 SPHERICAL MAP",
+        "text_level": 1,
+        "bbox": [
+            171,
+            534,
+            349,
+            547
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "For spherical coordinates, let  $s(x) = n(l(c(x)))$  where  $c(x) = \\pi / 2x$ ,  $l(\\psi)$  maps angles to the unit sphere, and  $n(z) = [z_i / \\sum_j z_j]_i$ .",
+        "bbox": [
+            169,
+            559,
+            823,
+            590
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "Definition 2. Define  $l(\\psi)$  as the transformation to the unit-sphere using spherical coordinates:",
+        "bbox": [
+            169,
+            597,
+            797,
+            612
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nl _ {1} (\\psi) = \\cos (\\psi_ {1}) \\tag {70a}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            372,
+            623,
+            823,
+            638
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nl _ {2} (\\psi) = \\sin \\left(\\psi_ {1}\\right) \\cos \\left(\\psi_ {2}\\right) \\tag {70b}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            375,
+            641,
+            823,
+            656
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nl _ {3} (\\psi) = \\sin (\\psi_ {1}) \\sin (\\psi_ {2}) \\cos (\\psi_ {3}) \\tag {70c}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            375,
+            657,
+            823,
+            674
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\vdots = \\vdots \\tag {70d}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            406,
+            684,
+            823,
+            700
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nl _ {m - 1} (\\psi) = \\sin \\left(\\psi_ {1}\\right) \\sin \\left(\\psi_ {2}\\right) \\dots \\cos \\left(\\psi_ {m - 1}\\right) \\tag {70e}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            704,
+            823,
+            719
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nl _ {m} (\\psi) = \\sin \\left(\\psi_ {1}\\right) \\sin \\left(\\psi_ {2}\\right) \\dots \\sin \\left(\\psi_ {m - 1}\\right). \\tag {70f}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            370,
+            720,
+            823,
+            737
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "Lemma 23. Let  $J$  be the Jacobian of the transformation to the unit-sphere using spherical coordinates, i.e.  $z = l(\\psi)$  where  $\\|l\\|^2 = 1$  and  $\\psi_i \\in [0, \\frac{\\pi}{2}]$  represents an angle for each  $i$ . Then  $\\|J\\|_F \\leq \\sqrt{m}$ .",
+        "bbox": [
+            169,
+            748,
+            826,
+            779
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "Proof. The Jacobian of the transformation is",
+        "bbox": [
+            171,
+            804,
+            470,
+            818
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nJ (l) = \\left[ \\begin{array}{c c c c} - \\sin (\\psi_ {1}) & 0 & \\dots & 0 \\\\ \\cos (\\psi_ {1}) \\cos (\\psi_ {2}) & - \\sin (\\psi_ {1}) \\sin (\\psi_ {2}) & \\dots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\cos (\\psi_ {1}) \\sin (\\psi_ {2}) \\dots \\cos (\\psi_ {m - 1}) & \\dots & \\dots & - \\sin (\\psi_ {1}) \\dots \\sin (\\psi_ {m - 2}) \\sin (\\psi_ {m - 1}) \\\\ \\cos (\\psi_ {1}) \\sin (\\psi_ {2}) \\dots \\sin (\\psi_ {m - 1}) & \\dots & \\dots & \\sin (\\psi_ {1}) \\dots \\sin (\\psi_ {m - 2}) \\cos (\\psi_ {m - 1}) \\end{array} \\right] \\tag {71}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            171,
+            828,
+            875,
+            925
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "page_number",
+        "text": "28",
+        "bbox": [
+            488,
+            946,
+            508,
+            960
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "and it square is",
+        "bbox": [
+            171,
+            104,
+            274,
+            118
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nJ (l) = \\left[ \\begin{array}{c c c c} t _ {1} & 0 & \\dots & 0 \\\\ \\cos (\\psi_ {1}) ^ {2} \\cos (\\psi_ {2}) ^ {2} & \\sin (\\psi_ {1}) ^ {2} t _ {2} & \\dots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\cos (\\psi_ {1}) ^ {2} \\sin (\\psi_ {2}) ^ {2} \\dots \\cos (\\psi_ {m - 1}) ^ {2} & \\dots & \\dots & \\sin (\\psi_ {1}) ^ {2} \\dots \\sin (\\psi_ {m - 2}) ^ {2} t _ {m - 1} \\\\ \\cos (\\psi_ {1}) ^ {2} \\sin (\\psi_ {2}) ^ {2} \\dots \\sin (\\psi_ {m - 1}) ^ {2} & \\dots & \\dots & \\sin (\\psi_ {1}) ^ {2} \\dots \\sin (\\psi_ {m - 2}) ^ {2} t _ {m} \\end{array} \\right] \\tag {72}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            176,
+            126,
+            823,
+            223
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "where",
+        "bbox": [
+            171,
+            229,
+            217,
+            243
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\delta_ {i m} = 1 \\text {i f} i = m, 0 \\text {e l s e} \\tag {73a}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            331,
+            252,
+            823,
+            268
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nt _ {i} = \\delta_ {i m} \\cos^ {2} \\left(\\psi_ {i - 1}\\right) + \\left(1 - \\delta_ {i m}\\right) \\sin^ {2} \\left(\\psi_ {i}\\right) \\leq 1. \\tag {73b}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            346,
+            271,
+            823,
+            289
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "To compute the Frobenius norm, we will need the sum of the squares of all entries. We will consider the sum of each row individually using the following auxiliary variable  $R_{i,k\\leq i}$  where  $\\sum_{j}J_{ij}^{2} = R_{i,1}$  and apply a recursive inequality.",
+        "bbox": [
+            169,
+            301,
+            823,
+            347
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "(74a)",
+        "bbox": [
+            785,
+            354,
+            823,
+            369
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} R _ {i, k \\leq i} = \\sum_ {k ^ {\\prime} = k} ^ {i - 1} \\cos^ {2} \\left(\\psi_ {k ^ {\\prime}}\\right) \\left[ \\prod_ {l = k, l \\neq k ^ {\\prime}} ^ {i - 1} \\sin^ {2} \\left(\\psi_ {l}\\right) \\right] \\cos^ {2} \\left(\\psi_ {i}\\right) + t _ {i} \\prod_ {l = k} ^ {i - 1} \\sin^ {2} \\left(\\psi_ {l}\\right) (74b) \\\\ = \\cos^ {2} \\left(\\psi_ {k}\\right) \\underbrace {\\left[ \\prod_ {l = k + 1} ^ {i - 1} \\sin^ {2} \\left(\\psi_ {l}\\right) \\right] \\cos^ {2} \\left(\\psi_ {i}\\right)} _ {\\leq 1} (74c) \\\\ + \\sin^ {2} \\left(\\psi_ {k}\\right) \\sum_ {k ^ {\\prime} = k + 1} ^ {i - 1} \\cos^ {2} \\left(\\psi_ {k ^ {\\prime}}\\right) \\left[ \\prod_ {l = k + 1, l \\neq k ^ {\\prime}} ^ {i - 1} \\sin^ {2} \\left(\\psi_ {l}\\right) \\right] \\cos^ {2} \\left(\\psi_ {i}\\right) (74d) \\\\ + \\sin^ {2} \\left(\\psi_ {k}\\right) t _ {i} \\prod_ {l = k + 1} ^ {i - 1} \\sin^ {2} \\left(\\psi_ {l}\\right) (74e) \\\\ \\leq \\cos^ {2} \\left(\\psi_ {k}\\right) (74f) \\\\ + \\sin^ {2} \\left(\\psi_ {k}\\right) \\left(\\sum_ {k ^ {\\prime} = k + 1} ^ {i - 1} \\cos^ {2} \\left(\\psi_ {k ^ {\\prime}}\\right) \\left[ \\prod_ {l = k + 1, l \\neq k ^ {\\prime}} ^ {i - 1} \\sin^ {2} \\left(\\psi_ {l}\\right) \\right] \\cos^ {2} \\left(\\psi_ {i}\\right) + t _ {i} \\prod_ {l = k + 1} ^ {i - 1} \\sin^ {2} \\left(\\psi_ {l}\\right)\\right) (7 4 g) \\\\ = \\cos^ {2} (\\psi_ {k}) + \\sin^ {2} (\\psi_ {k}) R _ {i, k + 1}. (74h) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            373,
+            825,
+            660
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "Note then that  $R_{i,k+1} \\leq 1 \\Rightarrow R_{i,k} \\leq 1$ . We know  $R_{i,i} = t_i \\leq 1$ , therefore,  $R_{i,1} \\leq 1$  by applying the inequality recursively. Finally,  $\\sum_j J_{ij}^2 = R_{i,1} \\leq 1$  implies the claim  $||J||_F^2 = \\sum_i R_{i,1} \\leq m$ .",
+        "bbox": [
+            169,
+            665,
+            826,
+            698
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "Lemma 24. Let  $J$  be the Jacobian of  $n(z) = z / Z$  where  $Z = \\sum_{k}z_{k}$ . Then  $\\mathbf{1}^{\\top}J = \\mathbf{0}^{\\top}$ .",
+        "bbox": [
+            171,
+            705,
+            753,
+            724
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "Proof. The  $i$ th entry of the Jacobian of  $n(z)$  is",
+        "bbox": [
+            171,
+            737,
+            488,
+            752
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nJ (n) _ {i j} = \\frac {1}{Z ^ {2}} \\left(- z _ {i} + \\delta_ {i j} Z\\right). \\tag {75}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            403,
+            758,
+            825,
+            789
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "Therefore  $[ \\mathbf{1}^{\\top} J ]_{j} = \\sum_{i} J(n)_{ij} = \\frac{1}{Z^{2}} (-Z + Z) = 0$  where  $z$  is a point on the unit-sphere in the positive orthant.",
+        "bbox": [
+            169,
+            795,
+            825,
+            825
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "B.2 NEAR OPTIMALITY & ZOOMING DIMENSION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            840,
+            531,
+            856
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "In this subsection, we derive bounds on the near-optimality dimension and zooming dimension of the associated bandit optimization problems we use to derive global convergence guarantees to Nash equilibria. This is written in terms of maximizing a function  $f$  rather than minimizing a loss to better match the bandit literature (Bubeck et al., 2011; Valko et al., 2013; Feng et al., 2022).",
+        "bbox": [
+            169,
+            867,
+            826,
+            925
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "page_number",
+        "text": "29",
+        "bbox": [
+            488,
+            946,
+            508,
+            960
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "Assumption 1. Locally around each interior  $x^{*} \\in \\mathcal{X}^{*}$ ,  $-f(x)$  is lower bounded by  $-f(x^{*}) + \\sigma_{-}||x - x^{*}||^{\\alpha_{hi}}$ . In other words, for all  $f(x) \\geq f(x^{*}) - \\eta$ :",
+        "bbox": [
+            169,
+            103,
+            826,
+            133
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nf \\left(x ^ {*}\\right) - f (x) \\geq \\sigma_ {-} \\| x - x ^ {*} \\| ^ {\\alpha_ {h i}} \\tag {76}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            385,
+            142,
+            823,
+            160
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "where we have left the precise norm unspecified for generality. Let  $\\ell(x, x^{*}) = \\sigma_{+}||x - x^{*}||^{\\alpha_{lo}}$ .",
+        "bbox": [
+            169,
+            169,
+            794,
+            186
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Definition 3.  $\\mathcal{X}_{\\epsilon} \\stackrel{\\mathrm{def}}{=} \\{x \\in \\mathcal{X} \\mid \\exists x^{*} \\in \\mathcal{X}^{*}$  s.t.  $f(x) \\geq f(x^{*}) - \\epsilon\\}$",
+        "bbox": [
+            169,
+            191,
+            604,
+            210
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Definition 4.  $\\mathcal{X}_{\\epsilon}^{lower} \\stackrel{\\mathrm{def}}{=} \\{x \\in \\mathcal{X} \\mid \\exists x^* \\in \\mathcal{X}^* \\text{ s.t. } f(x^*) - \\sigma_{-} \\|x - x^*\\|^{\\alpha_{hi}} \\geq f(x^*) - \\epsilon\\}$",
+        "bbox": [
+            169,
+            217,
+            767,
+            237
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Corollary 4.  $\\mathcal{X}_{\\varepsilon} \\subseteq \\mathcal{X}_{\\varepsilon}^{lower}$ .",
+        "bbox": [
+            169,
+            239,
+            359,
+            257
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Proof. By Assumption 1,  $f(x^{*}) - \\sigma_{-}||x - x^{*}||^{\\alpha_{hi}}\\geq f(x)$ . Therefore, any  $x\\in \\mathcal{X}$  that satisfies the requirement for an element of  $\\mathcal{X}_{\\epsilon}$ ,  $f(x)\\geq f(x^{*}) - \\epsilon$ , will also satisfy the requirement for an element of  $\\mathcal{X}_{\\epsilon}^{lower}$ .",
+        "bbox": [
+            169,
+            282,
+            826,
+            325
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Definition 5 ( $\\psi$ -near Optimality Dimension). The  $\\psi$ -near optimality dimension is the smallest  $d' > 0$  such that there exists  $C > 0$  such that for any  $\\epsilon > 0$ , the maximum number of disjoint  $\\ell$ -balls of radius  $\\psi \\epsilon$  and center in  $\\mathcal{X}_{\\epsilon}$  is less than  $C \\epsilon^{-d'}$ .",
+        "bbox": [
+            169,
+            344,
+            826,
+            388
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Definition 6 (Zooming Dimension). The zooming dimension is the smallest  $d_z > 0$  such that there exists  $C_z > 0$  such that for any  $r > 0$ , the maximum number of disjoint  $\\ell$ -balls of radius  $\\frac{r}{2}$  and center in  $\\mathcal{X}_{16r}$  is less than  $C_z r^{-d_z}$ .",
+        "bbox": [
+            169,
+            393,
+            825,
+            440
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Corollary 5 (Zooming Dimension from  $\\psi$ -near Optimality Dimension). The zooming dimension of  $f: x \\in [0,1]^d \\to [-1,1]$  under  $\\ell(x,y) = \\sigma_+||x - y||^{\\alpha_{lo}}$  is  $d_z = d\\left(\\frac{\\alpha_{hi} - \\alpha_{lo}}{\\alpha_{lo}\\alpha_{hi}}\\right)$  with constant  $C_z = 16^{-d'}C$  with  $\\psi = \\frac{1}{32}$  ( $\\psi$  is needed to compute  $C$ ).",
+        "bbox": [
+            169,
+            445,
+            825,
+            494
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Proof. Mapping the definition of  $\\psi$ -near optimality onto zooming dimension, we find  $\\psi \\epsilon = r / 2$  and  $\\epsilon = 16r$ . Then we can infer  $\\psi = 1 / 32$ ; this is used to compute  $C$  (see Theorem 1). Rewriting the bound from  $\\psi$ -near optimality in terms of  $r$ , we find",
+        "bbox": [
+            169,
+            520,
+            823,
+            563
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nN _ {\\epsilon} = N _ {1 6 r} \\leq C (1 6 r) ^ {- d ^ {\\prime}} = C 1 6 ^ {- d ^ {\\prime}} r ^ {- d ^ {\\prime}} = C _ {z} r ^ {- d _ {z}} \\tag {77}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            321,
+            573,
+            823,
+            592
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "where  $N_{16r}$  denotes the number of  $\\ell$ -balls of radius  $r/2$  with center in  $\\mathcal{X}_{16r}$ ,  $d_z = d'$ ,  $C_z = C16^{-d'} = C16^{-d_z}$ . Therefore, this translation only effects the constant  $C_z$ , not the zooming dimension.",
+        "bbox": [
+            169,
+            602,
+            825,
+            647
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Lemma 25 ( $N_{\\epsilon \\leq \\eta} \\leq C_{\\epsilon \\leq \\eta}\\epsilon^{-d'}$ ). The maximum number of disjoint  $l$ -balls with radius  $\\psi \\in$  and center in  $\\mathcal{X}_{\\epsilon \\leq \\eta}$ ,  $N_{\\epsilon \\leq \\eta}$ , is upper bounded by  $C_{\\epsilon \\leq \\eta}\\epsilon^{-d'}$  where  $C_{\\epsilon \\leq \\eta} = |\\mathcal{X}^*| \\left( \\frac{\\sigma_+}{\\psi \\sigma_-^{\\alpha_{lo} / \\alpha_{hi}}} \\right)^{d / \\alpha_{lo}}$  and  $d' = d \\left( \\frac{\\alpha_{hi} - \\alpha_{lo}}{\\alpha_{lo} \\alpha_{hi}} \\right)$ .",
+        "bbox": [
+            169,
+            666,
+            826,
+            733
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Proof. The number of disjoint  $\\ell$ -balls of radius  $\\psi \\epsilon$  and center in  $\\mathcal{X}_{\\epsilon \\leq \\eta}$  can be upper bounded as follows.",
+        "bbox": [
+            169,
+            757,
+            823,
+            785
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Rewrite  $\\mathcal{X}_{\\epsilon}^{lower}$  by rearranging terms as",
+        "bbox": [
+            171,
+            791,
+            437,
+            808
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {X} _ {\\epsilon} ^ {l o w e r} = \\left\\{x \\in \\mathcal {X} | \\exists x ^ {*} \\in \\mathcal {X} ^ {*} s. t. \\left| \\left| x - x ^ {*} \\right| \\right| \\leq \\left(\\frac {\\epsilon}{\\sigma_ {-}}\\right) ^ {1 / \\alpha_ {h i}} \\stackrel {\\text {d e f}} {=} r _ {\\epsilon} \\right\\} \\tag {78}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            277,
+            819,
+            823,
+            849
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "and recall that from Corollary 4 that  $\\mathcal{X}_{\\epsilon} \\subseteq \\mathcal{X}_{\\epsilon}^{lower}$ . Furthermore, an  $\\ell$ -ball of radius  $\\psi \\epsilon$  implies",
+        "bbox": [
+            169,
+            862,
+            794,
+            878
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\ell (x, y) = \\sigma_ {+} | | x - y | | ^ {\\alpha_ {l o}} \\leq \\psi \\epsilon \\Longrightarrow | | x - y | | \\leq \\left(\\frac {\\psi \\epsilon}{\\sigma_ {+}}\\right) ^ {1 / \\alpha_ {l o}} \\stackrel {\\text {d e f}} {=} r _ {\\ell}. \\tag {79}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            274,
+            888,
+            823,
+            921
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "page_number",
+        "text": "30",
+        "bbox": [
+            488,
+            946,
+            509,
+            959
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "The number of disjoint  $\\ell$ -balls that can pack into a set  $\\mathcal{X}_{\\epsilon}$ ,  $N_{\\epsilon \\leq \\eta}$ , is upper bounded by the ratio of the volumes of the two sets:",
+        "bbox": [
+            169,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} N _ {\\epsilon \\leq \\eta} \\leq \\frac {V o l \\left(\\mathcal {X} _ {\\epsilon}\\right)}{V o l \\left(\\mathcal {B} _ {\\ell}\\right)} (80a) \\\\ \\leq \\frac {\\operatorname {V o l} \\left(\\mathcal {X} _ {\\epsilon} ^ {\\text {l o w e r}}\\right)}{\\operatorname {V o l} \\left(\\mathcal {B} _ {\\ell}\\right)} (80b) \\\\ = \\frac {\\left| \\mathcal {X} ^ {*} \\right| S _ {d} r _ {\\epsilon} ^ {d}}{S _ {d} r _ {\\ell} ^ {d}} (80c) \\\\ \\leq \\frac {\\left| \\mathcal {X} ^ {*} \\right| \\left(\\frac {\\epsilon}{\\sigma_ {-}}\\right) ^ {d / \\alpha_ {h i}}}{\\left(\\frac {\\psi \\epsilon}{\\sigma_ {+}}\\right) ^ {d / \\alpha_ {l o}}} (80d) \\\\ = \\left| \\mathcal {X} ^ {*} \\right| \\left(\\frac {\\sigma_ {+} ^ {1 / \\alpha_ {l o}} \\psi^ {- 1 / \\alpha_ {l o}}}{\\sigma_ {-} ^ {1 / \\alpha_ {h i}}}\\right) ^ {d} \\epsilon^ {d \\left(1 / \\alpha_ {h i} - 1 / \\alpha_ {l o}\\right)} (80e) \\\\ = \\left| \\mathcal {X} ^ {*} \\right| \\left(\\frac {\\sigma_ {+}}{\\psi \\sigma_ {-} ^ {\\alpha_ {l o} / \\alpha_ {h i}}}\\right) ^ {d / \\alpha_ {l o}} \\epsilon^ {- d \\left(\\frac {\\alpha_ {h i} - \\alpha_ {l o}}{\\alpha_ {l o} \\alpha_ {h i}}\\right)} (80f) \\\\ = C _ {\\epsilon \\leq \\eta} \\epsilon^ {- d ^ {\\prime}} (80g) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            341,
+            138,
+            823,
+            407
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "text",
+        "text": "where  $C_{\\epsilon \\leq \\eta} = |\\mathcal{X}^{*}|\\left(\\frac{\\sigma_{+}}{\\psi\\sigma_{-}^{\\alpha_{lo} / \\alpha_{hi}}}\\right)^{d / \\alpha_{lo}}$ ,  $d' = d\\left(\\frac{\\alpha_{hi} - \\alpha_{lo}}{\\alpha_{lo}\\alpha_{hi}}\\right)$ ,  $|\\mathcal{X}^{*}|$  is the number of distinct global optima, and  $S_{d}$  is the volume constant for a  $d$ -sphere under the given norm  $||\\cdot||$ .",
+        "bbox": [
+            169,
+            416,
+            823,
+            462
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "text",
+        "text": "Recall, these results apply when  $f(x) \\geq f(x^{*}) - \\eta$ , i.e., when  $\\epsilon \\leq \\eta$ . Otherwise, we can upper bound the number of  $\\ell$ -balls by considering the entire set  $\\mathcal{X}$  which has volume 1. First, we will bound the constant associated with the volume of a  $d$ -sphere.",
+        "bbox": [
+            169,
+            474,
+            823,
+            518
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "text",
+        "text": "Lemma 26 ( $N_{\\epsilon \\geq \\eta} \\leq C_{\\epsilon \\geq \\eta}$ ). The maximum number of disjoint  $\\ell$ -balls with radius  $\\psi \\epsilon$  and center in  $\\mathcal{X}_{\\epsilon \\geq \\eta}$ ,  $N_{\\epsilon \\geq \\eta}$ , is upper bounded by  $C_{\\epsilon \\geq \\eta}$  where  $C_{\\epsilon \\geq \\eta} = S_d^{-1}\\left(\\frac{\\sigma_+}{\\psi\\eta}\\right)^{d / \\alpha_{lo}}$  and  $S_d$  is the volume constant for a  $d$ -sphere under a given norm.",
+        "bbox": [
+            169,
+            522,
+            825,
+            579
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "text",
+        "text": "Proof. We can upper bound the number of  $\\ell$ -balls needed to pack the entire space by considering the smallest possible radius  $\\psi \\eta$ :",
+        "bbox": [
+            169,
+            593,
+            823,
+            625
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} N _ {\\epsilon \\geq \\eta} \\leq \\frac {V o l (\\mathcal {X})}{V o l (\\mathcal {B} _ {\\ell})} (81a) \\\\ = \\frac {1}{S _ {d} r _ {\\ell} ^ {d}} (81b) \\\\ \\leq \\frac {1}{S _ {d} \\left(\\frac {\\psi \\eta}{\\sigma_ {+}}\\right) ^ {d / \\alpha_ {l o}}} (81c) \\\\ = S _ {d} ^ {- 1} \\left(\\frac {\\sigma_ {+}}{\\psi \\eta}\\right) ^ {d / \\alpha_ {l o}} (81d) \\\\ = C _ {\\epsilon \\geq \\eta} (81e) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            416,
+            630,
+            823,
+            795
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "text",
+        "text": "where  $r_l$  was defined in equation (79).",
+        "bbox": [
+            171,
+            801,
+            823,
+            816
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1. The  $\\psi$ -near optimality dimension of  $f: x \\in [0,1]^d \\to [-1,1]$  under  $\\ell$  is  $d' = d\\left(\\frac{\\alpha_{hi} - \\alpha_{lo}}{\\alpha_{lo}\\alpha_{hi}}\\right)$  with constant",
+        "bbox": [
+            169,
+            824,
+            825,
+            858
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nC = \\max  \\left\\{1, S _ {d} ^ {- 1} \\left(r _ {\\eta} ^ {\\frac {\\alpha_ {h i}}{\\alpha_ {l o}}} \\sigma_ {-} ^ {\\left(\\frac {\\alpha_ {h i} - \\alpha_ {l o}}{\\alpha_ {l o} \\alpha_ {h i}}\\right)}\\right) ^ {- d} \\right\\} \\left(\\frac {\\sigma_ {+}}{\\psi \\sigma_ {-} ^ {\\alpha_ {l o} / \\alpha_ {h i}}}\\right) ^ {d / \\alpha_ {l o}} \\tag {82}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            295,
+            864,
+            823,
+            901
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "text",
+        "text": "where  $S_{d}$  is the volume constant for a  $d$ -sphere under the same norm as  $\\ell$ .",
+        "bbox": [
+            171,
+            909,
+            658,
+            924
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "page_number",
+        "text": "31",
+        "bbox": [
+            488,
+            948,
+            506,
+            959
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "text",
+        "text": "Proof. First, let us define  $r_{\\eta} = \\left(\\frac{\\eta}{\\sigma_{-}}\\right)^{1 / \\alpha_{hi}}$  as in equation (78) which implies  $\\eta = \\sigma_{-}r_{\\eta}^{\\alpha_{hi}}$ . Then apply Lemmas 25 ( $N_{\\epsilon \\leq \\eta} \\leq C_{\\epsilon \\leq \\eta}\\epsilon^{-d'}$ ) and 26 ( $N_{\\epsilon \\geq \\eta} \\leq C_{\\epsilon \\geq \\eta}$ ) which bound the number of  $\\ell$ -balls required to pack  $\\mathcal{X}_{\\epsilon}$  when  $\\epsilon$  is less than and greater than  $\\eta$  respectively:",
+        "bbox": [
+            169,
+            101,
+            823,
+            160
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nC _ {\\epsilon \\leq \\eta} = \\left| \\mathcal {X} ^ {*} \\right| \\left(\\frac {\\sigma_ {+}}{\\psi \\sigma_ {-} ^ {\\alpha_ {l o} / \\alpha_ {h i}}}\\right) ^ {d / \\alpha_ {l o}} \\tag {83}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            388,
+            183,
+            823,
+            218
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nd ^ {\\prime} = d \\left(\\frac {\\alpha_ {h i} - \\alpha_ {l o}}{\\alpha_ {l o} \\alpha_ {h i}}\\right) \\tag {84}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            411,
+            223,
+            823,
+            252
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "text",
+        "text": "and",
+        "bbox": [
+            171,
+            256,
+            200,
+            270
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nC _ {\\epsilon \\geq \\eta} = S _ {d} ^ {- 1} \\left(\\frac {\\sigma_ {+}}{\\psi \\eta}\\right) ^ {d / \\alpha_ {l o}} \\tag {85a}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            302,
+            273,
+            823,
+            306
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n= \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} S _ {d} ^ {- 1} \\eta^ {- d / \\alpha_ {l o}} \\sigma_ {-} ^ {d / \\alpha_ {h i}} \\left| \\mathcal {X} ^ {*} \\right| \\left(\\frac {\\sigma_ {+}}{\\psi \\sigma_ {-} ^ {\\alpha_ {l o} / \\alpha_ {h i}}}\\right) ^ {d / \\alpha_ {l o}} \\tag {85b}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            308,
+            823,
+            343
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n= \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} S _ {d} ^ {- 1} \\eta^ {- d / \\alpha_ {l o}} \\sigma_ {-} ^ {d / \\alpha_ {h i}} C _ {\\epsilon \\leq \\eta} \\tag {85c}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            347,
+            823,
+            367
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n= \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} S _ {d} ^ {- 1} r _ {\\eta} ^ {- d \\alpha_ {h i} / \\alpha_ {l o}} \\sigma_ {-} ^ {- d / \\alpha_ {l o}} \\sigma_ {-} ^ {d / \\alpha_ {h i}} C _ {\\epsilon \\leq \\eta} \\tag {85d}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            369,
+            823,
+            391
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n= \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} S _ {d} ^ {- 1} r _ {\\eta} ^ {- d \\frac {\\alpha_ {h i}}{\\alpha_ {l o}}} \\sigma_ {-} ^ {- d \\left(\\frac {\\alpha_ {h i} - \\alpha_ {l o}}{\\alpha_ {l o} \\alpha_ {h i}}\\right)} C _ {\\epsilon \\leq \\eta} \\tag {85e}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            393,
+            823,
+            420
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n= \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} S _ {d} ^ {- 1} \\left(r _ {\\eta} ^ {\\frac {\\alpha_ {h i}}{\\alpha_ {l o}}} \\sigma_ {-} ^ {\\left(\\frac {\\alpha_ {h i} - \\alpha_ {l o}}{\\alpha_ {l o} \\alpha_ {h i}}\\right)}\\right) ^ {- d} C _ {\\epsilon \\leq \\eta} \\tag {85f}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            422,
+            823,
+            454
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "text",
+        "text": "where  $S_{d}$  is the volume constant for a  $d$ -sphere under the given norm.  $S_{d}^{-1}$  has been upper bounded for the 2-norm in Lemma 35. For the  $\\infty$ -norm,  $S_{d}^{-1} = 2^{-d}$ . We have written  $C_{\\epsilon \\geq \\eta}$  in terms of  $C_{\\epsilon \\leq \\eta}$  to clarify which is larger.",
+        "bbox": [
+            169,
+            459,
+            823,
+            506
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "text",
+        "text": "Therefore,",
+        "bbox": [
+            171,
+            511,
+            243,
+            525
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} C = \\max  \\left\\{1, | \\mathcal {X} ^ {*} | ^ {- 1} S _ {d} ^ {- 1} \\left(r \\eta^ {\\frac {\\alpha_ {h i}}{\\alpha_ {l o}}} \\sigma_ {-} ^ {\\left(\\frac {\\alpha_ {h i} - \\alpha_ {l o}}{\\alpha_ {l o} \\alpha_ {h i}}\\right)}\\right) ^ {- d} \\right\\} C _ {\\epsilon \\leq \\eta} (86a) \\\\ = \\max  \\left\\{1, \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} S _ {d} ^ {- 1} \\left(r _ {\\eta} ^ {\\frac {\\alpha_ {h i}}{\\alpha_ {l o}}} \\sigma_ {-} ^ {\\left(\\frac {\\alpha_ {h i} - \\alpha_ {l o}}{\\alpha_ {l o} \\alpha_ {h i}}\\right)}\\right) ^ {- d} \\right\\} \\left(\\frac {\\sigma_ {+}}{\\psi \\sigma_ {-} ^ {\\alpha_ {l o} / \\alpha_ {h i}}}\\right) ^ {d / \\alpha_ {l o}}. (86b) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            267,
+            532,
+            823,
+            602
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "text",
+        "text": "Intuitively, if the radius for which the polynomial bounds hold  $(r_{\\eta})$  is large and the minimum curvature constant  $\\sigma_{-}$  is also large, then the bound  $C_{\\epsilon \\leq \\eta}$  holds for large deviations from optimality  $\\eta$ . The number of  $\\eta$ -radius  $\\ell$ -balls required to cover the remaining space,  $C_{\\epsilon \\geq \\eta}$ , will be comparatively small.",
+        "bbox": [
+            169,
+            616,
+            826,
+            672
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "text",
+        "text": "B.3 D-BLIN",
+        "text_level": 1,
+        "bbox": [
+            171,
+            688,
+            277,
+            703
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "text",
+        "text": "The regret bound for Doubling BLiN (Feng et al., 2022) was originally proved assuming a standard normal distribution, however, the authors state their proof can be easily adapted to any sub-Gaussian distribution, which includes bounded random variables. This matches our setting with bounded payoffs, so we repeat their analysis here for that setting.",
+        "bbox": [
+            169,
+            714,
+            823,
+            772
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "text",
+        "text": "In what follows, in an effort to match notation in (Feng et al., 2022), let  $\\mu(x) = -\\mathcal{L}^{\\tau}(s(x))$  denote the expected reward for evaluating  $x \\in [0,1]^d$ ;  $\\hat{\\mu}(x)$  is an unbiased estimate of  $\\mu(x)$ . Here,  $s: [0,1]^{n(\\tilde{m} - 1)} \\to \\prod_i \\Delta^{m_i - 1}$  is any function that maps from the unit hypercube to a product of simplices; we analyze two such maps in Appendix B.1. Also let  $\\mathcal{A}_m$  denote the collection of hypercubes to be investigated during the  $m$ th batch of arm pulls,  $C \\in \\mathcal{A}_m$  denote a hypercube partition in  $\\mathcal{X}$ ,  $n_m$  indicates the number of times each cube in  $\\mathcal{A}_m$  is played in batch  $m$ , and  $B_{stop}$  denote the last batch.",
+        "bbox": [
+            169,
+            777,
+            826,
+            876
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "text",
+        "text": "Definition 7 (Global Arm Accuracy).  $\\mathcal{E} \\stackrel{\\mathrm{def}}{=} \\left\\{ |\\mu(x) - \\hat{\\mu}_m(C)| \\leq r_m + \\sqrt{c_1 \\frac{\\ln T}{n_m}}, \\forall 1 \\leq m \\leq B_{stop} - 1, \\forall C \\in \\mathcal{A}_m, \\forall x \\in C \\right\\}$ .",
+        "bbox": [
+            169,
+            878,
+            823,
+            928
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "page_number",
+        "text": "32",
+        "bbox": [
+            488,
+            946,
+            508,
+            960
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "text",
+        "text": "Define:  $n_m = c_2 \\frac{\\ln T}{r_m^2} \\Rightarrow r_m = \\sqrt{c_2 \\frac{\\ln T}{n_m}}$ .",
+        "bbox": [
+            169,
+            101,
+            460,
+            127
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "Definition 8 (Elimination Rule). Eliminate  $C \\in \\mathcal{A}_m$  if  $\\hat{\\mu}_m^{\\max} - \\hat{\\mu}_m(C) \\geq 2(1 + \\sqrt{c_1 / c_2})r_m = 2(\\sqrt{c_2} + \\sqrt{c_1})\\sqrt{\\frac{\\ln T}{n_m}}$  where  $\\hat{\\mu}_m^{\\max} \\stackrel{\\mathrm{def}}{=} \\max_{C \\in \\mathcal{A}_m} \\hat{\\mu}_m(C)$ .",
+        "bbox": [
+            169,
+            130,
+            823,
+            172
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "Lemma 27.  $Pr[\\mathcal{E}] \\geq 1 - 2T^{-2(c_1 / c^2 - 1)}$ .",
+        "bbox": [
+            171,
+            176,
+            452,
+            195
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "Proof. Let  $\\hat{\\mu}_m(C) = \\frac{1}{n_m}\\sum_{i=1}^{n_m}y_{C,i}$  be the unbiased estimate of  $\\mu(x\\in C)$  using  $n_m$  samples. Assume each  $y_{C,i}\\in [a,b]$  with  $c = b - a$  and  $\\hat{\\mu}(C) = \\frac{1}{n_m}\\sum_{i=1}^{n_m}y_{C,i}$ . Applying a Hoeffding inequality gives",
+        "bbox": [
+            169,
+            210,
+            826,
+            258
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} P r \\left[ | \\hat {\\mu} (C) - \\mathbb {E} [ \\hat {\\mu} (C) ] | \\geq \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m}}} \\right] \\leq 2 e ^ {- 2 c _ {1} \\ln T / c ^ {2}} (87a) \\\\ = 2 \\left(e ^ {\\ln T}\\right) ^ {- 2 c _ {1} / c ^ {2}} (87b) \\\\ = 2 T ^ {- 2 c _ {1} / c ^ {2}} \\forall C. (87c) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            315,
+            266,
+            825,
+            343
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "By Lipschitzness of  $\\mu$  we also have",
+        "bbox": [
+            171,
+            351,
+            408,
+            366
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left| \\mathbb {E} \\left[ \\hat {\\mu} (C) \\right] - \\mu (x) \\right| \\leq r _ {m}, \\forall x \\in C. \\tag {88}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            383,
+            373,
+            823,
+            390
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "Then consider",
+        "bbox": [
+            171,
+            405,
+            269,
+            419
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\sup  _ {x \\in C} | \\mu (x) - \\hat {\\mu} (C) | = \\sup  _ {x \\in C} | \\mu (x) - \\mathbb {E} [ \\hat {\\mu} (C) ] + \\mathbb {E} [ \\hat {\\mu} (C) ] - \\hat {\\mu} (C) | (89a) \\\\ \\leq \\sup  _ {x \\in C} \\left(| \\mu (x) - \\mathbb {E} [ \\hat {\\mu} (C) ] | + | \\mathbb {E} [ \\hat {\\mu} (C) ] - \\hat {\\mu} (C) |\\right) (89b) \\\\ = \\sup  _ {x \\in C} | \\mu (x) - \\mathbb {E} [ \\hat {\\mu} (C) ] | + | \\mathbb {E} [ \\hat {\\mu} (C) ] - \\hat {\\mu} (C) | (89c) \\\\ \\leq \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m}}} + r _ {m} (89d) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            269,
+            426,
+            823,
+            547
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "with probability  $1 - 2T^{-2c_1 / c^2}$ . The first inequality follows by triangle inequality and the second follows from equation (88) and considering the complement of equation (87c).",
+        "bbox": [
+            169,
+            556,
+            823,
+            587
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "The complement of this result occurs with probability",
+        "bbox": [
+            171,
+            594,
+            527,
+            609
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nP r \\left[ \\sup  _ {x \\in C} | \\mu (x) - \\hat {\\mu} (C) | \\geq r _ {m} + \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m}}} \\right] \\leq 2 T ^ {- 2 c _ {1} / c ^ {2}}. \\tag {90}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            303,
+            617,
+            823,
+            650
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "At least 1 arm is played in each cube  $C \\in \\mathcal{A}_m$  for  $1 \\leq m \\leq B_{stop} - 1$ , therefore,  $|\\mathcal{A}_m| \\leq T$  must be true given the exit condition of the algorithm. In addition, assume  $B_{stop} \\leq T$  ( $B_{stop}$  will be defined such that this is true). Then a union bound over all  $T^2$  events gives",
+        "bbox": [
+            169,
+            666,
+            823,
+            710
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} P r \\left[ \\exists m \\in [ 1, B _ {s t o p} - 1 ], C \\in \\mathcal {A} _ {m} s. t. \\sup  _ {x \\in C} | \\mu (x) - \\hat {\\mu} (C) | \\geq r _ {m} + \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m}}} \\right] (91a) \\\\ \\leq \\sum_ {m = 1} ^ {B _ {s t o p} - 1} \\sum_ {C \\in \\mathcal {A} _ {m}} P r \\left[ \\sup  _ {x \\in C} | \\mu (x) - \\hat {\\mu} (C) | \\geq r _ {m} + \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m}}} \\right] (91b) \\\\ \\leq \\sum_ {m = 1} ^ {B _ {s t o p} - 1} \\sum_ {C \\in \\mathcal {A} _ {m}} 2 T ^ {- 2 c _ {1} / c ^ {2}} (91c) \\\\ \\leq 2 T ^ {- 2 c _ {1} / c ^ {2}} T ^ {2}. (91d) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            217,
+            719,
+            825,
+            869
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "Taking the complement of this event and noting that  $\\sup_{x\\in C}|\\mu (x) - \\hat{\\mu} (C)|\\leq r_m + \\sqrt{c_1\\frac{\\ln T}{n_m}}\\Rightarrow |\\mu (x) - \\hat{\\mu} (C)|\\leq r_m + \\sqrt{c_1\\frac{\\ln T}{n_m}}\\forall x\\in C$  gives the desired result.",
+        "bbox": [
+            169,
+            878,
+            823,
+            928
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "page_number",
+        "text": "33",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "Lemma 28 (Optimal Arm Survives). Under event  $\\mathcal{E}$ , the optimal arm  $x^{*} = \\arg \\max \\mu(x)$  is not eliminated after the first  $B_{stop} - 1$  batches.",
+        "bbox": [
+            169,
+            103,
+            826,
+            133
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "Proof. Let  $C_m^*$  denote the cube containing  $x^*$  in  $\\mathcal{A}_m$ . Under event  $\\mathcal{E}$ , for any cube  $C \\in \\mathcal{A}_m$  and  $x \\in C$ , the following relation shows that  $C_m^*$  avoids the elimination rule in round  $m$ :",
+        "bbox": [
+            169,
+            146,
+            825,
+            176
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\hat {\\mu} (C) - \\hat {\\mu} \\left(C _ {m} ^ {*}\\right) \\leq \\left(\\mu (x) + r _ {m} + \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m}}}\\right) + \\left(- \\mu \\left(x ^ {*}\\right) + r _ {m} + \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m}}}\\right) (92a) \\\\ = \\underbrace {(\\mu (x) - \\mu \\left(x ^ {*}\\right))} _ {\\leq 0} + 2 r _ {m} + 2 \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m}}} (92b) \\\\ \\leq 2 \\sqrt {c _ {2} \\frac {\\ln T}{n _ {m}}} + 2 \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m}}} (92c) \\\\ = 2 \\left(\\sqrt {c _ {1}} + \\sqrt {c _ {2}}\\right) \\sqrt {\\frac {\\ln T}{n _ {m}}} (92d) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            222,
+            180,
+            823,
+            333
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "where the first inequality follows from applying Lemma 27 to upper bound  $\\hat{\\mu} (C)$  and  $\\hat{\\mu} (C_m^*)$  individually. The remaining steps use the optimality of  $x^{*}$ , the definition of  $r_m$ , and the elimination rule.",
+        "bbox": [
+            169,
+            334,
+            826,
+            376
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "Lemma 29. Under event  $\\mathcal{E}$ , for any  $1 \\leq m \\leq B_{stop}$ , any  $C \\in A_m$  and any  $x \\in C$ ,  $\\Delta_x$  satisfies",
+        "bbox": [
+            169,
+            383,
+            802,
+            400
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\Delta_ {x} \\leq 4 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {m - 1} \\tag {93}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            406,
+            402,
+            823,
+            421
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "Proof. For  $m = 1$ , recall that  $r_m$  is the side length of a cube  $C \\in \\mathcal{A}_m$ , therefore,  $\\Delta_x \\leq r_{m-1} \\leq 4(1 + \\sqrt{c_1 / c_2})r_{m-1}$  holds directly from the Lipschitzness of  $\\mu$ .",
+        "bbox": [
+            169,
+            434,
+            823,
+            467
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "For  $m > 1$ , let  $C_{m - 1}^{*}\\in \\mathcal{A}_{m - 1}$  be the cube containing  $x^{*}$ . From Lemma 28, this cube has not been eliminated under event  $\\mathcal{E}$ . For any cube  $C\\in \\mathcal{A}_m$  and  $x\\in C$ , it is clear that  $x$  is also in the parent of  $C$ , denoted  $C_{par}$  ( $x\\in C\\subset C_{par}$ ). Then for any  $x\\in C$ , it holds that",
+        "bbox": [
+            169,
+            472,
+            826,
+            516
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\Delta_ {x} = \\mu \\left(x ^ {*}\\right) - \\mu (x) \\leq \\left(\\hat {\\mu} _ {m - 1} \\left(C _ {m - 1} ^ {*}\\right) + r _ {m - 1} + \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m - 1}}}\\right) (94a) \\\\ + \\left(- \\hat {\\mu} _ {m - 1} (C _ {p a r}) + r _ {m - 1} + \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m - 1}}}\\right) \\\\ = \\left(\\hat {\\mu} _ {m - 1} \\left(C _ {m - 1} ^ {*}\\right) - \\hat {\\mu} _ {m - 1} \\left(C _ {p a r}\\right)\\right) + 2 \\left(\\sqrt {c _ {1}} + \\sqrt {c _ {2}}\\right) \\sqrt {\\frac {\\ln T}{n _ {m - 1}}} (94b) \\\\ \\leq \\left(\\hat {\\mu} _ {m - 1} ^ {\\max } - \\hat {\\mu} _ {m - 1} \\left(C _ {p a r}\\right)\\right) + 2 \\left(\\sqrt {c _ {1}} + \\sqrt {c _ {2}}\\right) \\sqrt {\\frac {\\ln T}{n _ {m - 1}}} (94c) \\\\ \\leq 2 \\left(\\sqrt {c _ {1}} + \\sqrt {c _ {2}}\\right) \\sqrt {\\frac {\\ln T}{n _ {m - 1}}} + 2 \\left(\\sqrt {c _ {1}} + \\sqrt {c _ {2}}\\right) \\sqrt {\\frac {\\ln T}{n _ {m - 1}}} (94d) \\\\ = 4 \\left(\\sqrt {c _ {1}} + \\sqrt {c _ {2}}\\right) \\sqrt {\\frac {\\ln T}{n _ {m - 1}}} (94e) \\\\ = 4 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {m - 1} (94f) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            210,
+            518,
+            823,
+            797
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "where we have applied Lemma 27 similarly as in Lemma 28 and also used the definition of  $r_{m - 1}$ . The last two inequalities use the fact that  $\\hat{\\mu}_{m - 1}(C_{m - 1}^{*})\\leq \\hat{\\mu}_{m - 1}^{\\max}$  and  $C_{par}$  was not eliminated.",
+        "bbox": [
+            169,
+            799,
+            826,
+            829
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2 (BLiN Regret Rate). With probability exceeding  $1 - 2T^{-2(c_1/c^2 - 1)}$ , the  $T$ -step total regret  $R(T)$  of BLiN with Doubling Edge-length Sequence ( $D$ -BLiN) Feng et al. (2022) satisfies",
+        "bbox": [
+            169,
+            837,
+            825,
+            868
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nR (T) \\leq 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left(2 c _ {2} + 1\\right) \\ln (T) ^ {\\frac {1}{d _ {z} + 2}} T ^ {\\frac {d _ {z} + 1}{d _ {z} + 2}} \\tag {95}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            328,
+            871,
+            823,
+            891
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "where  $d_{z}$  is the zooming dimension of the problem instance. In addition, D-BLiN only needs no more than  $B^{*} = \\frac{\\log 2(T) - \\log 2(\\ln(T))}{d_{z} + 2} + 2$  rounds of communications to achieve this regret rate.",
+        "bbox": [
+            169,
+            893,
+            823,
+            928
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "page_number",
+        "text": "34",
+        "bbox": [
+            488,
+            946,
+            509,
+            959
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "Proof. Since  $r_m = \\frac{r_{m-1}}{2} \\Rightarrow r_{m-1} = 2r_m$  for the Doubling Edge-length Sequence, Lemma 29 implies that every cube  $C \\in A_m$  is a subset of  $S(8(1 + \\sqrt{c_1 / c_2})r_m)$ . Thus from the definition of zooming dimension (Corollary 5 with appropriate condition), we have",
+        "bbox": [
+            169,
+            103,
+            826,
+            151
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left| \\mathcal {A} _ {m} \\right| \\leq N _ {r _ {m}} \\leq C _ {z} r _ {m} ^ {- d _ {z}}. \\tag {96}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            413,
+            185,
+            823,
+            202
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "text",
+        "text": "Fix any positive number  $B$ . Also by Lemma 29, we know that any arm played after batch  $B$  incurs a regret bounded by  $8(1 + \\sqrt{c_1 / c_2})r_B$ , since the cubes played after batch  $B$  have edge length no larger than  $r_B$ . Then the total regret that occurs after batch  $B$  is bounded by  $8(1 + \\sqrt{c_1 / c_2})r_BT$  (where  $T$  is an upper bound on the number of arms).",
+        "bbox": [
+            169,
+            244,
+            823,
+            306
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "text",
+        "text": "Thus the regret can be bounded as",
+        "bbox": [
+            171,
+            311,
+            398,
+            325
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nR (T) \\leq \\sum_ {m = 1} ^ {B} \\sum_ {C \\in \\mathcal {A} _ {m}} \\sum_ {i = 1} ^ {n _ {m}} \\Delta_ {x _ {C, i}} + 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {B} T \\tag {97}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            320,
+            361,
+            823,
+            405
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "text",
+        "text": "where the first term bounds the regret in the first  $B$  batches of D-BLiN, and the second term bounds the regret after the first  $B$  batches. If the algorithm stops at batch  $\\tilde{B} < B$ , we define  $\\mathcal{A}_m =$  for any  $\\tilde{B} < m \\leq B$  and inequality equation (97) still holds.",
+        "bbox": [
+            169,
+            438,
+            823,
+            484
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "text",
+        "text": "By Lemma 29, we have  $\\Delta_{x_{C,i}}\\leq 8(1 + \\sqrt{c_1 / c_2})r_m$  for all  $C\\in \\mathcal{A}_m$ . We can thus bound equation (97) by",
+        "bbox": [
+            169,
+            492,
+            826,
+            523
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} R (T) \\leq \\sum_ {m = 1} ^ {B} | \\mathcal {A} _ {m} | \\cdot n _ {m} \\cdot 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {m} + 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {B} T (98a) \\\\ \\leq \\sum_ {m = 1} ^ {B} N _ {r _ {m}} \\cdot n _ {m} \\cdot 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {m} + 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {B} T (98b) \\\\ = \\sum_ {m = 1} ^ {B} N _ {r _ {m}} \\cdot c _ {2} \\frac {\\ln T}{r _ {m} ^ {2}} \\cdot 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {m} + 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {B} T (98c) \\\\ = \\sum_ {m = 1} ^ {B} N _ {r _ {m}} \\cdot \\frac {\\ln T}{r _ {m}} \\cdot 8 c _ {2} \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) + 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {B} T (98d) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            263,
+            556,
+            823,
+            729
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "text",
+        "text": "where equation (98b) uses equation (96), and equation (98c) uses equality  $n_m = c_2 \\frac{\\ln T}{r_m^2}$ . Since  $r_m = 2^{-m + 1}$  and  $N_{r_m} \\leq C_z r_m^{-d_z} \\leq C_z 2^{(m - 1)d_z}$ , we have",
+        "bbox": [
+            169,
+            765,
+            823,
+            801
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} R (T) \\leq \\sum_ {m = 1} ^ {B} C _ {z} 2 ^ {(m - 1) d _ {z}} \\cdot \\frac {\\ln T}{2 ^ {- m + 1}} \\cdot 8 c _ {2} \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) + 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) 2 ^ {- B + 1} T (99a) \\\\ = 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ c _ {2} C _ {z} \\ln T \\sum_ {m = 1} ^ {B} 2 ^ {(m - 1) (d _ {z} + 1)} + 2 ^ {- B + 1} T \\right]. (99b) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            207,
+            835,
+            825,
+            921
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "page_number",
+        "text": "35",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "text",
+        "text": "Continuing we find",
+        "bbox": [
+            171,
+            103,
+            303,
+            118
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nR (T) \\leq 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ c _ {2} C _ {z} \\ln T \\sum_ {m = 1} ^ {B} 2 ^ {(m - 1) (d _ {z} + 1)} + 2 ^ {- B + 1} T \\right] \\tag {100a}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            192,
+            126,
+            823,
+            166
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n= 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ c _ {2} C _ {z} \\ln T \\sum_ {m = 1} ^ {B} \\left(2 ^ {d _ {z} + 1}\\right) ^ {m - 1} + 2 ^ {- B + 1} T \\right] \\tag {100b}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            233,
+            170,
+            823,
+            209
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n= 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ c _ {2} C _ {z} \\ln T \\sum_ {m = 0} ^ {B - 1} \\left(2 ^ {d _ {z} + 1}\\right) ^ {m} + 2 ^ {- B + 1} T \\right] \\tag {100c}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            233,
+            214,
+            823,
+            253
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n= 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ c _ {2} C _ {z} \\ln T \\left(\\frac {2 ^ {B (d _ {z} + 1)} - 1}{2 ^ {d _ {z} + 1} - 1}\\right) + 2 ^ {- B + 1} T \\right] \\text {v i a g e o m e t r i c s e r i e s} \\tag {100d}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            233,
+            257,
+            823,
+            287
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\leq 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ c _ {2} C _ {z} \\ln T \\left(\\frac {2 ^ {B (d _ {z} + 1)}}{2 ^ {d _ {z} + 1} - 1}\\right) + 2 ^ {- B + 1} T \\right] \\tag {100e}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            233,
+            291,
+            823,
+            321
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\leq 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ c _ {2} C _ {z} \\ln T \\left(2 \\cdot \\frac {2 ^ {B (d _ {z} + 1)}}{2 ^ {d _ {z} + 1}}\\right) + 2 ^ {- B + 1} T \\right] \\tag {100f}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            233,
+            325,
+            823,
+            356
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n= 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ 2 c _ {2} C _ {z} 2 ^ {(B - 1) (d _ {z} + 1)} \\ln T + 2 ^ {- (B - 1)} T \\right]. \\tag {100g}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            233,
+            359,
+            823,
+            383
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "This inequality holds for any positive  $B$ . By choosing  $B^{*} = 1 + \\frac{\\log_{2}\\left(\\frac{T}{\\ln T}\\right)}{d_{z} + 2}$ , we have",
+        "bbox": [
+            171,
+            390,
+            725,
+            412
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} R (T) \\leq 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ 2 c _ {2} C _ {z} \\left(\\frac {T}{\\ln T}\\right) ^ {\\frac {(d _ {z} + 1)}{(d _ {z} + 2)}} \\ln T + \\left(\\frac {\\ln T}{T}\\right) ^ {\\frac {1}{(d _ {z} + 2)}} T \\right] (101a) \\\\ = 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ 2 c _ {2} C _ {z} T ^ {\\frac {(d _ {z} + 1)}{(d _ {z} + 2)}} \\ln T ^ {1 - \\frac {(d _ {z} + 1)}{(d _ {z} + 2)}} + T ^ {1 - \\frac {1}{(d _ {z} + 2)}} \\ln T ^ {\\frac {1}{(d _ {z} + 2)}} \\right] (101b) \\\\ = 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ 2 c _ {2} C _ {z} T ^ {\\frac {(d _ {z} + 1)}{(d _ {z} + 2)}} \\ln T ^ {\\frac {1}{(d _ {z} + 2)}} + T ^ {\\frac {(d _ {z} + 1)}{(d _ {z} + 2)}} \\ln T ^ {\\frac {1}{(d _ {z} + 2)}} \\right] (101c) \\\\ = 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left(2 c _ {2} C _ {z} + 1\\right) T ^ {\\frac {(d _ {z} + 1)}{(d _ {z} + 2)}} \\ln T ^ {\\frac {1}{(d _ {z} + 2)}}. (101d) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            218,
+            419,
+            823,
+            531
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "Corollary 6 (BLiN Regret Rate Refined). Setting  $c_{1} = 2c^{2}$  and  $c_{2} = 2\\left(\\frac{c}{4C_{z}}\\right)^{2 / 3}$  simplifies Theorem 2 such that",
+        "bbox": [
+            169,
+            556,
+            826,
+            595
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nR (T) \\leq 8 \\left(1 + \\left(4 c ^ {2} C _ {z}\\right) ^ {1 / 3}\\right) ^ {2} T ^ {\\frac {(d _ {z} + 1)}{(d _ {z} + 2)}} \\ln T ^ {\\frac {1}{(d _ {z} + 2)}}. \\tag {102}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            336,
+            599,
+            823,
+            622
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "with probability  $1 - 2T^{-2}$ .",
+        "bbox": [
+            171,
+            627,
+            352,
+            643
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "Proof. If we set  $c_1 = 2c^2$  and  $c_2 = 2\\left(\\frac{c}{4C_z}\\right)^{2/3}$ , then  $\\sqrt{c_1 / c_2} = c\\left(\\frac{4C_z}{c}\\right)^{1/3} = \\left(4c^2 C_z\\right)^{1/3} = 2c_2C_z$ .",
+        "bbox": [
+            169,
+            657,
+            823,
+            700
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "Lemma 30. The zooming dimension and zooming constant under the  $\\ell (x,y) = ||x - y||_{\\infty}$  norm are",
+        "bbox": [
+            169,
+            707,
+            823,
+            722
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nd _ {z} = \\frac {1}{2} n \\bar {m} \\tag {103}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            410,
+            729,
+            823,
+            755
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nC _ {z} = \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\left(\\frac {4}{r _ {\\eta} ^ {2} \\sigma_ {- \\infty}}\\right) ^ {n \\bar {m}} \\tag {104}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            406,
+            758,
+            823,
+            792
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "where  $\\sigma_{-\\infty} = ||\\text{Hess}(-f(x))^{-1}||_{\\infty}$  is an upper bound on the infinity norm of the inverse Hessian matrix of the function at every equilibrium.",
+        "bbox": [
+            169,
+            797,
+            823,
+            827
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "Proof. Recall from Theorem 1 that  $d_{z} = d\\left(\\frac{\\alpha_{hi} - \\alpha_{lo}}{\\alpha_{lo}\\alpha_{hi}}\\right)$  with constant  $C_z = 16^{-d'}C$  where  $C$  is defined below. In addition, BLiN assumes  $\\ell (x,y) = ||x - y||_{\\infty}$ . Matching to Assumption 1, we see that  $\\sigma_{+} = \\alpha_{lo} = 1$ . Under the infinity norm, the volume constant  $S_{d} = 2^{d}$ .",
+        "bbox": [
+            169,
+            842,
+            823,
+            888
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "We will define the other constants with respect to properties of the Hessian of  $f(x)$  about each equilibrium, specifically the infinity norm of the inverse Hessian so that  $\\sigma_{-} = \\sigma_{-\\infty}$ . This means",
+        "bbox": [
+            169,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "page_number",
+        "text": "36",
+        "bbox": [
+            488,
+            948,
+            508,
+            959
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "we will bound the function locally with a quadratic, i.e.,  $\\alpha_{hi} = 2$ . Lastly, recall from Corollary 5 that  $\\psi = \\frac{1}{32}$  and the dimension of our search space (the product space of player mixed strategies) is  $d = n(\\bar{m} - 1) \\leq nm$  for simplicity.",
+        "bbox": [
+            169,
+            103,
+            823,
+            147
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "Plugging this information into Theorem 1, we find",
+        "bbox": [
+            171,
+            152,
+            506,
+            167
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nd _ {z} = d ^ {\\prime} = \\frac {1}{2} n \\bar {m} \\tag {105}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            441,
+            176,
+            823,
+            207
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "and",
+        "bbox": [
+            171,
+            215,
+            200,
+            229
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nC = \\max  \\left\\{1, | \\mathcal {X} ^ {*} | ^ {- 1} S _ {d} ^ {- 1} \\left(r _ {\\eta} ^ {\\frac {\\alpha_ {h i}}{\\alpha_ {l o}}} \\sigma_ {-} ^ {\\left(\\frac {\\alpha_ {h i} - \\alpha_ {l o}}{\\alpha_ {l o} \\alpha_ {h i}}\\right)}\\right) ^ {- d} \\right\\} \\left(\\frac {\\sigma_ {+}}{\\psi \\sigma_ {-} ^ {\\alpha_ {l o} / \\alpha_ {h i}}}\\right) ^ {d / \\alpha_ {l o}} \\tag {106a}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            269,
+            238,
+            825,
+            276
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n= \\max  \\left\\{1, \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} 2 ^ {- d} \\left(r _ {\\eta} ^ {2} \\sigma_ {- \\infty} ^ {\\frac {1}{2}}\\right) ^ {- d} \\right\\} \\left(\\frac {3 2}{\\sigma_ {- \\infty} ^ {1 / 2}}\\right) ^ {d} \\tag {106b}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            289,
+            277,
+            823,
+            315
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n= \\max  \\left\\{1, \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\left(2 r _ {\\eta} ^ {2} \\sigma_ {- \\infty} ^ {\\frac {1}{2}}\\right) ^ {- d} \\right\\} \\left(\\frac {3 2}{\\sigma_ {- \\infty} ^ {1 / 2}}\\right) ^ {d} \\tag {106c}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            289,
+            316,
+            823,
+            354
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\stackrel {h a r d} {=} \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\left(2 r _ {\\eta} ^ {2} \\sigma_ {- \\infty} ^ {\\frac {1}{2}}\\right) ^ {- d} \\left(\\frac {3 2}{\\sigma_ {- \\infty} ^ {1 / 2}}\\right) ^ {d} \\tag {106d}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            287,
+            356,
+            823,
+            392
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\stackrel {h a r d} {=} \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\left(\\frac {1 6}{r _ {\\eta} ^ {2} \\sigma_ {- \\infty}}\\right) ^ {n \\bar {m}} \\tag {106e}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            287,
+            393,
+            823,
+            429
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "where  $\\text{hard}$  indicates we are assuming  $r_{\\eta}$  and  $\\sigma_{-\\infty}$  are small enough to dominate the other operand of the max.",
+        "bbox": [
+            169,
+            438,
+            823,
+            465
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "Finally, converting the near optimality constant to a zooming constant, we find",
+        "bbox": [
+            171,
+            473,
+            689,
+            488
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} C _ {z} = 1 6 ^ {- d _ {z}} \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\left(\\frac {1 6}{r _ {\\eta} ^ {2} \\sigma_ {- \\infty}}\\right) ^ {n \\bar {m}} (107a) \\\\ = \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\left(\\frac {4}{r _ {\\eta} ^ {2} \\sigma_ {- \\infty}}\\right) ^ {n \\bar {m}}. (107b) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            383,
+            497,
+            823,
+            566
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "image",
+        "img_path": "images/da86bade918f42ff09723255caa327bbe359f0ae626b46d3e25b9cf4c3a5b407.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            575,
+            823,
+            589
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "B.4 BOUNDED DIAMETERS AND WELL-SHAPED CELLS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            611,
+            571,
+            626
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "We assume the feasible set is a unit-hypercube of dimensionality  $d$  where cells are evenly split along the longest edge to give  $b$  new partitions and  $x_{h,i}$  represents the center of each cell.",
+        "bbox": [
+            169,
+            638,
+            823,
+            667
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "There exists a decreasing sequence  $w(h) > 0$ , such that for any depth  $h \\geq 0$  and for any cell  $\\mathcal{X}_{h,i}$  of depth  $h$ , we have  $\\sup_{x \\in \\mathcal{X}_{h,i}} \\ell(x_{h,i}, x) \\leq w(h)$ . Moreover, there exists  $\\nu > 0$  such that for any depth  $h \\geq 0$ , any cell  $\\mathcal{X}_{h,i}$  contains an  $\\ell$ -ball of radius  $\\nu w(h)$  centered at  $x_{h,i}$ .",
+        "bbox": [
+            169,
+            672,
+            826,
+            720
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "table",
+        "img_path": "images/3660daab1f092afbdf411ce2db7204b4ff3562d83b7242eaa1660002b64735a6.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>l(x,y)</td><td>c</td><td>γ</td><td>ν</td></tr><tr><td>l(x,y)=||x-y||2/2</td><td>dα/2(b/2)α</td><td>b-α/d</td><td>d-α/2b-2α</td></tr><tr><td>l(x,y)=||x-y||∞/∞</td><td>(b/2)α</td><td>b-α/d</td><td>b-2α</td></tr></table>",
+        "bbox": [
+            295,
+            734,
+            699,
+            805
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "Table 3: Bounding Constants:  $\\sup_{x \\in \\mathcal{X}_{h,i}} \\ell(x_{h,i}, x) \\leq w(h) = c\\gamma^h$ .",
+        "bbox": [
+            274,
+            815,
+            718,
+            834
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "B.4.1  $L_{2}$ -NORM",
+        "text_level": 1,
+        "bbox": [
+            171,
+            862,
+            303,
+            878
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "Lemma 31 (L2-Norm Bounding Ball). Let  $\\ell(x, y) = ||x - y||_2^\\alpha$ . Then  $\\sup_{x \\in \\mathcal{X}_{h,i}} \\ell(x_{h,i}, x) \\leq w_2(h) = c\\gamma^h$  where  $c = \\left(\\frac{db^2}{4}\\right)^{\\alpha/2}$  and  $\\gamma = b^{-\\alpha/d}$ .",
+        "bbox": [
+            169,
+            887,
+            823,
+            926
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "page_number",
+        "text": "37",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "Proof.",
+        "bbox": [
+            171,
+            104,
+            217,
+            118
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nw (0) = \\left[ \\sum_ {i = 1} ^ {d} (1 / 2) ^ {2} \\right] ^ {\\alpha / 2} = \\left(\\frac {d}{4}\\right) ^ {\\alpha / 2} \\tag {108a}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            225,
+            121,
+            825,
+            162
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} w (1) = \\left[ (1 / b \\cdot 1 / 2) ^ {2} + \\sum_ {i = 2} ^ {d} (1 / 2) ^ {2} \\right] ^ {\\alpha / 2} = \\left[ (1 / b ^ {2}) (1 / 4) + (d - 1) (1 / 4) \\right] ^ {\\alpha / 2} (108b) \\\\ = \\left(\\frac {d - 1 + 1 / b ^ {2}}{4}\\right) ^ {\\alpha / 2} (108c) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            225,
+            165,
+            825,
+            239
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} w (d) = \\left[ \\sum_ {i = 1} ^ {d} (1 / b \\cdot 1 / 2) ^ {2} \\right] ^ {\\alpha / 2} = \\left(\\frac {d}{4 \\cdot b ^ {2}}\\right) ^ {\\alpha / 2} (108d) \\\\ w (h) = \\left[ r (1 / b) ^ {2 (q + 1)} (1 / 2) ^ {2} + \\sum_ {i = r} ^ {d} (1 / b) ^ {2 q} (1 / 2) ^ {2} \\right] ^ {\\alpha / 2} (108e) \\\\ = \\left[ (1 / b) ^ {2 q} (1 / 2) ^ {2} (r (1 / b) ^ {2} + (d - r) \\right] ^ {\\alpha / 2} (108f) \\\\ = \\left[ (1 / b ^ {2}) ^ {q} (1 / 4) \\left(d - r \\left(1 - \\frac {1}{b ^ {2}}\\right)\\right) \\right] ^ {\\alpha / 2} (108g) \\\\ \\leq \\left[ (1 / b ^ {2}) ^ {q} (1 / 4) d \\right] ^ {\\alpha / 2} (108h) \\\\ \\leq \\left[ \\left(1 / b ^ {2}\\right) ^ {h / d - 1} (1 / 4) d \\right] ^ {\\alpha / 2} (108i) \\\\ = \\left[ (1 / b ^ {2}) ^ {h / d} (b ^ {2} / 4) d \\right] ^ {\\alpha / 2} (108j) \\\\ = \\left(\\frac {d b ^ {2}}{4}\\right) ^ {\\alpha / 2} (1 / b) ^ {\\frac {\\alpha}{d} h} (108k) \\\\ = c \\gamma^ {h} (1081) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            225,
+            241,
+            825,
+            503
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "where  $q, r = \\text{divmod}(h, d) \\Rightarrow q \\geq h / d - 1, c = \\left(\\frac{db^2}{4}\\right)^{\\alpha / 2}$ , and  $\\gamma = (1 / b)^{\\alpha / d} = b^{-\\alpha / d}$ .",
+        "bbox": [
+            169,
+            507,
+            772,
+            529
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "Lemma 32 ( $L_2$ -Norm Inner Ball). Let  $\\ell(x, y) = ||x - y||_2^\\alpha$ . Any cell  $\\mathcal{X}_{h,i}$  contains an  $\\ell$ -ball of radius  $\\nu w_2(h)$  where  $\\nu = (db^4)^{-\\alpha/2}$ .",
+        "bbox": [
+            169,
+            534,
+            826,
+            566
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "Proof. Any cell  $\\mathcal{X}_{h,i}$  contains an  $\\ell$ -ball of radius equal to its shortest axis:",
+        "bbox": [
+            169,
+            580,
+            658,
+            595
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} r _ {\\min } = \\left[ (1 / 4) \\left(1 / b ^ {2}\\right) ^ {\\lceil h / d \\rceil} \\right] ^ {\\alpha / 2} (109a) \\\\ \\geq \\left[ (1 / 4) \\left(1 / b ^ {2}\\right) ^ {h / d + 1} \\right] ^ {\\alpha / 2} (109b) \\\\ = \\left(\\frac {1}{b ^ {2} \\cdot 4}\\right) ^ {\\alpha / 2} (1 / b) ^ {\\frac {\\alpha}{d} h} (109c) \\\\ = w (h) \\cdot \\left(\\frac {1}{d b ^ {4}}\\right) ^ {\\alpha / 2}. (109d) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            390,
+            598,
+            825,
+            704
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "B.4.2  $L_{\\infty}$ -NORM",
+        "text_level": 1,
+        "bbox": [
+            171,
+            734,
+            307,
+            750
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "Lemma 33 ( $L_{\\infty}$ -Norm Bounding Ball). Let  $\\ell(x, y) = ||x - y||_{\\infty}^{\\alpha}$ . Then  $\\sup_{x \\in \\mathcal{X}_{h,i}} \\ell(x_{h,i}, x) \\leq w_{\\infty}(h) = c\\gamma^{h}$  where  $c = \\left(\\frac{b}{2}\\right)^{\\alpha}$  and  $\\gamma = b^{-\\alpha / d}$ .",
+        "bbox": [
+            169,
+            757,
+            823,
+            792
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "Proof. Any cell  $\\mathcal{X}_{h,i}$  is contained by an  $\\ell$ -ball of radius equal to its longest axis:",
+        "bbox": [
+            169,
+            806,
+            699,
+            821
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} r _ {\\max } = \\left[ (1 / 4) \\left(1 / b ^ {2}\\right) ^ {\\lfloor h / d \\rfloor} \\right] ^ {\\alpha / 2} (110a) \\\\ \\leq \\left[ (1 / 4) \\left(1 / b ^ {2}\\right) ^ {h / d - 1} \\right] ^ {\\alpha / 2} (110b) \\\\ = \\left(\\frac {b ^ {2}}{4}\\right) ^ {\\alpha / 2} (1 / b) ^ {\\frac {\\alpha}{d} h} (110c) \\\\ = c \\gamma^ {h} (110d) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            390,
+            824,
+            825,
+            921
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "page_number",
+        "text": "38",
+        "bbox": [
+            488,
+            948,
+            506,
+            959
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "where  $c = \\left(\\frac{b^2}{4}\\right)^{\\alpha / 2}$ , and  $\\gamma = (1 / b)^{\\alpha / d} = b^{-\\alpha / d}$ .",
+        "bbox": [
+            169,
+            101,
+            493,
+            122
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "image",
+        "img_path": "images/bceee6c50ff66f24cad0648f2ad82e3b0b2853e2fd8d46f2a2a53b40be08fb15.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            106,
+            825,
+            118
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "Lemma 34 ( $L_{\\infty}$ -Norm Inner Ball). Let  $\\ell(x, y) = ||x - y||_{\\infty}^{\\alpha}$ . Any cell  $\\mathcal{X}_{h,i}$  contains an  $\\ell$ -ball of radius  $\\nu w_{\\infty}(h)$  where  $\\nu = b^{-2\\alpha}$ .",
+        "bbox": [
+            169,
+            130,
+            826,
+            161
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "Proof. Any cell  $\\mathcal{X}_{h,i}$  contains an  $\\ell$ -ball of radius equal to its shortest axis:",
+        "bbox": [
+            169,
+            178,
+            658,
+            194
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} r _ {\\min } = \\left[ (1 / 4) \\left(1 / b ^ {2}\\right) ^ {\\lceil h / d \\rceil} \\right] ^ {\\alpha / 2} (111a) \\\\ \\geq \\left[ (1 / 4) \\left(1 / b ^ {2}\\right) ^ {h / d + 1} \\right] ^ {\\alpha / 2} (111b) \\\\ = \\left(\\frac {1}{b ^ {2} \\cdot 4}\\right) ^ {\\alpha / 2} (1 / b) ^ {\\frac {\\alpha}{d} h} (111c) \\\\ = w (h) \\cdot \\left(\\frac {1}{b ^ {4}}\\right) ^ {\\alpha / 2}. (111d) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            395,
+            203,
+            823,
+            306
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "image",
+        "img_path": "images/5754ce1d5b12d10e269c95167dfbd9043c97f1055ae9a80af0106ee510413702.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            316,
+            823,
+            327
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "B.5 STOCHASTIC SIMULTANEOUS OPTIMISTIC OPTIMIZATION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            347,
+            620,
+            361
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "StoSOO is flexible in its choice of metric and partitioning structure. In StoSOO, we may choose  $\\ell(x,y) = ||x - y||_2^2$ .",
+        "bbox": [
+            169,
+            373,
+            823,
+            402
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "Lemma 35. The volume of a  $d$ -sphere with radius  $r$  and  $d$  even is given by  $S_{d}r^{d}$  where  $S_{d}^{-1} \\leq \\sqrt{2\\pi d}\\left(\\frac{d}{2\\pi e}\\right)^{d/2}$ .",
+        "bbox": [
+            169,
+            406,
+            823,
+            452
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "Proof. First, we recall Stirling's bounds on the factorial:  $\\sqrt{2\\pi n} \\left( \\frac{n}{e} \\right)^n e^{\\frac{1}{12n + 1}} < n! < \\sqrt{2\\pi n} \\left( \\frac{n}{e} \\right)^n e^{\\frac{1}{12n}}$ . This will be useful for bounding the Gamma function:  $\\Gamma(d) = (d - 1)!$  for even  $d$ .",
+        "bbox": [
+            169,
+            468,
+            826,
+            501
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "Given  $d$  is even, we start with the exact formula for  $S_{d}$ :",
+        "bbox": [
+            169,
+            506,
+            537,
+            522
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} S _ {d} ^ {- 1} = \\frac {\\Gamma (d / 2 + 1)}{\\pi^ {d / 2}} (112a) \\\\ = \\frac {(d / 2) !}{\\pi^ {d / 2}} (112b) \\\\ <   \\frac {\\sqrt {2 \\pi (d / 2)} \\left(\\frac {d / 2}{e}\\right) ^ {d / 2} e ^ {\\frac {1}{1 2 (d / 2)}}}{\\pi^ {d / 2}} (112c) \\\\ = \\frac {\\pi^ {1 / 2} d ^ {1 / 2} \\left(\\frac {d}{2 e}\\right) ^ {d / 2} e ^ {\\frac {1}{6 d}}}{\\pi^ {d / 2}} (112d) \\\\ = \\frac {\\pi^ {1 / 2} d ^ {(d + 1) / 2} e ^ {\\frac {1}{6 d}}}{(2 \\pi e) ^ {d / 2}} (112e) \\\\ \\leq \\sqrt {2 \\pi d} \\left(\\frac {d}{2 \\pi e}\\right) ^ {d / 2}. (112f) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            382,
+            529,
+            823,
+            731
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "image",
+        "img_path": "images/ad671b84f8dfee24547b9767b22bf0d9af09028040855e0cad6399d2c8f91d3a.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            739,
+            823,
+            750
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "Lemma 36. The near optimality dimension and constant under the  $\\ell(x, y) = ||x - y||_2^2$  norm are",
+        "bbox": [
+            169,
+            762,
+            815,
+            779
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nd ^ {\\prime} = 0 \\tag {113}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            370,
+            786,
+            823,
+            801
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nC = \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\sqrt {2 \\pi n \\bar {m}} \\left(\\frac {n \\bar {m}}{5 \\nu r _ {\\eta} ^ {2} \\sigma_ {- 2}}\\right) ^ {n \\bar {m} / 2} \\tag {114}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            370,
+            805,
+            823,
+            839
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "where  $\\sigma_{-2}$  is a lower bound on the singular value of the Hessian matrix of the function at every equilibrium and  $\\nu$  is defined in Table 3 for the corresponding  $\\ell(x, y)$ -ball.",
+        "bbox": [
+            169,
+            845,
+            823,
+            875
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "Proof. Recall from Theorem 1 that  $d' = d\\left(\\frac{\\alpha_{hi} - \\alpha_{lo}}{\\alpha_{lo}\\alpha_{hi}}\\right)$  with constant  $C$  defined below. In addition, we will analyze StoSOO with the choice  $\\ell(x,y) = ||x - y||_2^2$ . Matching to Assumption 1, we see",
+        "bbox": [
+            169,
+            891,
+            826,
+            925
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "page_number",
+        "text": "39",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "that  $\\sigma_{+} = 1$  and  $\\alpha_{lo} = 2$ . Under the 2-norm, the volume constant  $S_{d}^{-1} \\leq \\sqrt{2\\pi d}\\left(\\frac{d}{2\\pi e}\\right)^{d/2}$  (see Lemma 35).",
+        "bbox": [
+            169,
+            101,
+            823,
+            141
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "text",
+        "text": "We will bound the function locally with a quadratic, i.e.,  $\\alpha_{hi} = 2$ . Lastly, from (Valko et al., 2013, Corollary 1),  $\\psi = \\frac{\\nu}{3}$  with  $\\nu = \\frac{1}{db^2}$  defined in Table 3 and the dimension of our search space (the product space of player mixed strategies) is  $d = n(\\bar{m} - 1) \\leq n\\bar{m}$  for simplicity.",
+        "bbox": [
+            169,
+            147,
+            825,
+            191
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "text",
+        "text": "Plugging this information into Theorem 1, we find  $d' = 0$  and",
+        "bbox": [
+            171,
+            196,
+            581,
+            212
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nC = \\max  \\left\\{1, \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} S _ {d} ^ {- 1} \\left(r _ {\\eta} ^ {\\frac {\\alpha_ {h i}}{\\alpha_ {l o}}} \\sigma_ {-} ^ {\\left(\\frac {\\alpha_ {h i} - \\alpha_ {l o}}{\\alpha_ {l o} \\alpha_ {h i}}\\right)}\\right) ^ {- d} \\right\\} \\left(\\frac {\\sigma_ {+}}{\\psi \\sigma_ {-} ^ {\\alpha_ {l o} / \\alpha_ {h i}}}\\right) ^ {d / \\alpha_ {l o}} \\tag {115a}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            269,
+            220,
+            825,
+            258
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n= \\max  \\left\\{1, \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} S _ {d} ^ {- 1} r _ {\\eta} ^ {- d} \\right\\} \\left(\\frac {1}{\\psi \\sigma_ {- 2}}\\right) ^ {d / 2} \\tag {115b}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            289,
+            261,
+            825,
+            294
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\stackrel {h a r d} {=} \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} S _ {d} ^ {- 1} r _ {\\eta} ^ {- d} \\left(\\frac {3}{\\nu \\sigma_ {- 2}}\\right) ^ {d / 2} \\tag {115c}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            287,
+            296,
+            825,
+            327
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\leq^ {h a r d} \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\sqrt {2 \\pi d} \\left(\\frac {d}{2 \\pi e}\\right) ^ {d / 2} \\left(\\frac {3}{\\nu r _ {\\eta} ^ {2} \\sigma_ {- 2}}\\right) ^ {d / 2} \\tag {115d}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            287,
+            330,
+            825,
+            364
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\leq^ {h a r d} \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\sqrt {2 \\pi d} \\left(\\frac {b ^ {2} d ^ {2}}{5 r _ {\\eta} ^ {2} \\sigma_ {- 2}}\\right) ^ {d / 2} \\tag {115e}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            287,
+            367,
+            825,
+            402
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\stackrel {h a r d} {=} \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\sqrt {2 \\pi n \\bar {m}} \\left(\\frac {b ^ {2} n ^ {2} \\bar {m} ^ {2}}{5 r _ {\\eta} ^ {2} \\sigma_ {- 2}}\\right) ^ {n \\bar {m} / 2} \\tag {115f}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            287,
+            404,
+            825,
+            440
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "text",
+        "text": "where  $\\text{hard}$  indicates we are assuming  $r_{\\eta}$ , the radius of the ball under which the local polynomial bounds are accurate, is small enough to dominate the other operand of the max.",
+        "bbox": [
+            169,
+            446,
+            823,
+            477
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3 (StoSOO Regret Rate). Corollary 1 of Valko et al. (2013) implies that with probability  $1 - \\delta$ , the regret,  $R_{t}$ , of StoSOO after  $t$  pulls is upper bounded as",
+        "bbox": [
+            169,
+            487,
+            823,
+            517
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n(2 + b ^ {2 / d}) \\sqrt {\\frac {\\log_ {b} (t k / \\delta)}{2 \\log_ {b} (e) k}} + \\frac {1}{4} d b ^ {2 (1 + 2 / d)} b ^ {- \\frac {1}{d C} \\sqrt {t / k}} \\tag {116}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            328,
+            525,
+            825,
+            566
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "text",
+        "text": "where  $d = n(\\bar{m} - 1)$ ,  $b$  is the branching factor for partitioning cells,  $C$  is the near-optimality constant, and  $k$  is the maximum number of evaluations per node.",
+        "bbox": [
+            169,
+            574,
+            826,
+            603
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "text",
+        "text": "Proof. Plugging the constants  $c, \\gamma$ , and  $\\nu$  defined in Table 3 for the 2-norm into Corollary 1 of Valko et al. (2013), we find with probability  $1 - \\delta$ :",
+        "bbox": [
+            169,
+            622,
+            823,
+            651
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} R _ {t} \\leq (2 + 1 / \\gamma) \\epsilon + c \\gamma^ {\\frac {1}{2 C} \\sqrt {t / k} - 2} (117a) \\\\ = (2 + 1 / \\gamma) \\sqrt {\\frac {\\log (t k / \\delta)}{2 k}} + c \\gamma^ {\\frac {1}{2 C} \\sqrt {t / k} - 2} (117b) \\\\ = \\left(2 + b ^ {2 / d}\\right) \\sqrt {\\frac {\\log (t k / \\delta)}{2 k}} + \\frac {1}{4} d b ^ {2} \\left(b ^ {- 2 / d}\\right) ^ {\\frac {1}{2 C} \\sqrt {t / k} - 2} (117c) \\\\ = (2 + b ^ {2 / d}) \\sqrt {\\frac {\\log_ {b} (t k / \\delta)}{2 \\log_ {b} (e) k}} + \\frac {1}{4} d b ^ {2 (1 + 2 / d)} \\left(b ^ {- 2 / d}\\right) ^ {\\frac {1}{2 C} \\sqrt {t / k}} (117d) \\\\ = \\left(2 + b ^ {2 / d}\\right) \\sqrt {\\frac {\\log_ {b} (t k / \\delta)}{2 \\log_ {b} (e) k}} + \\frac {1}{4} d b ^ {2 (1 + 2 / d)} b ^ {- \\frac {1}{d C} \\sqrt {t / k}}. (117e) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            294,
+            659,
+            823,
+            839
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "image",
+        "img_path": "images/9ace8b6d2efdd07243cdf25690d81420b8365d871deb581108c33f389410ec51.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            845,
+            823,
+            859
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "text",
+        "text": "Proposition 2. Assume the same conditions as Theorem 3 and let  $k = t\\log_b(t)^{-\\rho}$  where  $\\rho \\geq 3$ . Then the StoSOO (Algorithm 1 of Valko et al. (2013)) bound requires  $t = b^{\\left(\\frac{dC}{2}\\right)^{\\frac{2}{\\rho - 2}}}$  pulls before exhibiting a  $\\tilde{\\mathcal{O}}(T^{-1/2})$  decay in regret.",
+        "bbox": [
+            169,
+            872,
+            826,
+            925
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "page_number",
+        "text": "40",
+        "bbox": [
+            488,
+            948,
+            508,
+            959
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "text",
+        "text": "Proof. We know from the given definition of  $k$  that  $\\sqrt{t / k} = \\log_b(t)^{\\rho /2}$ . We will analyze the second term in the regret formula of Corollary 3. Note that the second term,  $b^{-\\frac{1}{dC}\\sqrt{t / k}} = (b\\sqrt{t / k})^{-\\frac{1}{dC}} = (b^{\\log_b(t)^{\\rho /2}})^{-\\frac{1}{dC}} = (b^{\\log_b(t)\\cdot \\log_b(t)^{(\\rho -2) / 2}})^{-\\frac{1}{dC}} = (t^{\\log_b(t)^{(\\rho -2) / 2}})^{-\\frac{1}{dC}} = t^{-\\frac{\\log_b(t)^{(\\rho - 2) / 2}}{dC}}$ . To achieve a  $t^{-1 / 2}$  convergence rate (or better), we need  $t\\geq b^{(\\frac{dC}{2})^{\\frac{2}{\\rho - 2}}}$ . For convenience, we report the entire simplification of the bound below:",
+        "bbox": [
+            169,
+            102,
+            826,
+            200
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} R _ {t} \\leq (2 + b ^ {2 / d}) \\sqrt {\\frac {\\log_ {b} (t k / \\delta)}{2 \\log_ {b} (e) k}} + \\frac {1}{4} d b ^ {2 (1 + 2 / d)} b ^ {- \\frac {1}{d C}} \\sqrt {t / k} (118a) \\\\ = (2 + b ^ {2 / d}) \\sqrt {\\frac {\\log_ {b} \\left(t ^ {2} / \\left(\\delta \\log_ {b} (t) ^ {\\rho}\\right)\\right) \\log_ {b} (t) ^ {\\rho}}{2 \\log_ {b} (e) t}} + \\frac {1}{4} d b ^ {2 (1 + 2 / d)} t ^ {- \\frac {\\log_ {b} (t) ^ {(\\rho - 2) / 2}}{d C}} (118b) \\\\ \\leq \\left(2 + b ^ {2 / d}\\right) \\sqrt {\\frac {\\log_ {b} \\left(t ^ {2} / \\delta\\right) \\log_ {b} \\left(t / \\delta\\right) ^ {\\rho}}{2 \\log_ {b} (e) t}} + \\frac {1}{4} d b ^ {2 (1 + 2 / d)} t ^ {- \\frac {\\log_ {b} (t) ^ {(\\rho - 2) / 2}}{d C}} (118c) \\\\ = (2 + b ^ {2 / d}) \\sqrt {\\frac {2 \\log_ {b} (t / \\delta) ^ {\\rho + 1}}{2 \\log_ {b} (e) t}} + \\frac {1}{4} d b ^ {2 (1 + 2 / d)} t ^ {- \\frac {\\log_ {b} (t) ^ {(\\rho - 2) / 2}}{d C}} (118d) \\\\ = \\frac {\\left(2 + b ^ {2 / d}\\right)}{\\sqrt {\\log_ {b} (e)}} \\frac {\\log_ {b} (t / \\delta) ^ {(\\rho + 1) / 2}}{\\sqrt {t}} + \\frac {1}{4} d b ^ {2 (1 + 2 / d)} t ^ {- \\frac {\\log_ {b} (t) ^ {(\\rho - 2) / 2}}{d C}} (118e) \\\\ = \\frac {\\left(2 + b ^ {2 / d}\\right)}{\\sqrt {\\log_ {b} (e)}} \\frac {\\left[ \\log_ {b} (t) - \\log_ {b} (\\delta) \\right] ^ {(\\rho + 1) / 2}}{\\sqrt {t}} + \\frac {1}{4} d b ^ {2 (1 + 2 / d)} t ^ {- \\frac {\\log_ {b} (t) ^ {(\\rho - 2) / 2}}{d C}}. (118f) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            215,
+            212,
+            825,
+            460
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "image",
+        "img_path": "images/2d3ff3d62dd76cd6834c1121350eb79246f1e3f2f88373c316c4cc70635be961.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            468,
+            823,
+            481
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "text",
+        "text": "B.6 REGRET TO PAC BOUNDS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            502,
+            398,
+            516
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "text",
+        "text": "Lemma 37. [Loss Regret to Exploitability Regret] Assume exploitability of a joint strategy  $\\pmb{x}$  is upper bounded by  $f(\\mathcal{L}^{\\tau}(\\pmb{x}))$  where  $f$  is a concave function and  $\\mathcal{L}^{\\tau}$  is a loss function. Let  $\\pmb{x}_t$  be a joint strategy randomly drawn from the set of predictions made by an online learning algorithm  $\\mathcal{A}$  over  $T$  steps. Then the expected exploitability of  $\\pmb{x}_t$  is bounded by the average regret of  $\\mathcal{A}$ :",
+        "bbox": [
+            169,
+            529,
+            826,
+            585
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbb {E} _ {t} [ \\epsilon_ {t} ] \\leq f \\left(\\frac {1}{T} R (T)\\right). \\tag {119}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            424,
+            594,
+            823,
+            625
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "text",
+        "text": "Proof.",
+        "bbox": [
+            171,
+            645,
+            217,
+            660
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbb {E} _ {t} \\left[ \\epsilon_ {t} \\right] = \\mathbb {E} _ {t} \\left[ f \\left(\\mathcal {L} \\left(\\boldsymbol {x} _ {t}\\right)\\right) \\right] \\leq f \\left(\\mathbb {E} _ {t} \\left[ \\mathcal {L} \\left(\\boldsymbol {x} _ {t}\\right) \\right]\\right) = f \\left(\\frac {1}{T} \\sum_ {t} \\mathcal {L} \\left(\\boldsymbol {x} _ {t}\\right)\\right) = f \\left(\\frac {1}{T} R (T)\\right) \\tag {120}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            251,
+            669,
+            823,
+            704
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "text",
+        "text": "where the inequality follows from Jensen's inequality.",
+        "bbox": [
+            171,
+            713,
+            527,
+            729
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "image",
+        "img_path": "images/8cb08643aa045aa04bdc3c88894e7dc948749456006babe3f63950ec4680e0d7.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            714,
+            823,
+            726
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4 (BLiN PAC Rate). Assume  $\\eta_{k} = \\eta = 2 / \\hat{L}$  as defined in Lemma 2,  $\\tau = \\frac{1}{\\ln(1 / p)}$  so that all equilibria place at least  $\\frac{p}{m^*}$  mass on each strategy, and a previously pulled arm is returned uniformly at random (i.e.,  $t \\sim U(T)$ ). Then for any  $w > 0$ ,",
+        "bbox": [
+            169,
+            744,
+            826,
+            792
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\epsilon_ {t} \\leq w \\left[ \\tau \\log \\left(\\prod_ {k} m _ {k}\\right) + 2 \\left(1 + \\left(4 c ^ {2} C _ {z}\\right) ^ {1 / 3}\\right) \\sqrt {2 n \\hat {L}} \\left(\\frac {\\ln T}{T}\\right) ^ {\\frac {1}{2 \\left(d _ {z} + 2\\right)}} \\right] \\tag {121}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            269,
+            801,
+            823,
+            840
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "text",
+        "text": "with probability  $(1 - w^{-1})(1 - 2T^{-2})$  where  $m^{*} = \\max_{k} m_{k}$ , and  $c \\leq \\frac{n\\bar{m}}{\\hat{L}}\\left(\\frac{\\ln(m^{*})}{\\ln(1/p)} + 2\\right)^{2}$  is an upper bound on the range of  $\\hat{\\mathcal{L}}^{\\tau}$  (Corollary 1),  $\\hat{L} = \\left(\\frac{\\ln(m^{*})}{\\ln(1/p)} + 2\\right)\\left(\\frac{m^{*2}}{p\\ln(1/p)} + n\\bar{m}\\right)$  (Corollary 2), the zooming dimension  $d_{z} = \\frac{1}{2} n\\bar{m}$ , and the zooming constant  $C_{z} = |\\mathcal{X}^{*}|^{-1}\\left(\\frac{4}{r_{\\eta}^{2}\\sigma_{-\\infty}}\\right)^{n\\bar{m}}$  (Corollary 30).",
+        "bbox": [
+            169,
+            849,
+            826,
+            928
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "page_number",
+        "text": "41",
+        "bbox": [
+            488,
+            948,
+            506,
+            959
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "text",
+        "text": "Proof. Assume  $\\eta_{k} = \\eta = \\frac{2}{L}$  as defined in Lemma 2 so that  $\\mathcal{L}^{\\tau}$  is 1-Lipschitz with respect to  $||\\cdot ||_{\\infty}$ . Also assume a previously pulled arm is returned uniformly at random. Starting with Lemma 13 and applying Corollary 6, we find",
+        "bbox": [
+            169,
+            102,
+            826,
+            148
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\mathbb {E} \\left[ \\epsilon_ {t} \\right] \\leq \\tau \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {\\frac {2 n}{\\min  _ {k} \\eta_ {k}}} \\sqrt {\\frac {1}{T} \\sum_ {t} \\mathcal {L} ^ {\\tau} \\left(\\boldsymbol {x} _ {t}\\right)} (122a) \\\\ \\leq \\frac {1}{\\ln (1 / p)} \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {n \\hat {L}} \\sqrt {8 \\left(1 + \\left(4 c ^ {2}\\right) ^ {1 / 3}\\right) ^ {2} T ^ {\\frac {- 1}{\\left(d z + 2\\right)}} \\ln T ^ {\\frac {1}{\\left(d z + 2\\right)}}} (122b) \\\\ = \\frac {1}{\\ln (1 / p)} \\log \\left(\\prod_ {k} m _ {k}\\right) + 2 \\left(1 + \\left(4 c ^ {2} C _ {z}\\right) ^ {1 / 3}\\right) \\sqrt {2 n \\hat {L}} \\left(\\frac {\\ln T}{T}\\right) ^ {\\frac {1}{2 \\left(d _ {z} + 2\\right)}} (122c) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            217,
+            157,
+            825,
+            277
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "text",
+        "text": "with probability  $1 - 2T^{-2}$  where  $m^{*} = \\max_{k} m_{k}$ , and  $c \\leq \\frac{n\\bar{m}}{\\hat{L}} \\left( \\frac{\\ln(m^{*})}{\\ln(1/p)} + 2 \\right)^{2}$  is an upper bound on the range of sampled values from  $\\hat{\\mathcal{L}}^{\\tau}$  (see Corollary 1).",
+        "bbox": [
+            169,
+            287,
+            823,
+            330
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "text",
+        "text": "Markov's inequality then allows us to bound the pointwise exploitability of any arm returned by the algorithm as",
+        "bbox": [
+            169,
+            335,
+            825,
+            364
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\epsilon_ {t} \\leq w \\left[ \\frac {1}{\\ln (1 / p)} \\log \\left(\\prod_ {k} m _ {k}\\right) + 2 \\left(1 + \\left(4 c ^ {2} C _ {z}\\right) ^ {1 / 3}\\right) \\sqrt {2 n \\hat {L}} \\left(\\frac {\\ln T}{T}\\right) ^ {\\frac {1}{2 \\left(d _ {z} + 2\\right)}} \\right] \\tag {123}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            245,
+            375,
+            825,
+            411
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "text",
+        "text": "with probability  $(1 - w^{-1})(1 - 2T^{-2})$  for any  $w > 0$ .",
+        "bbox": [
+            169,
+            420,
+            534,
+            436
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "text",
+        "text": "Theorem 5 (StoSOO PAC Rate). Corollary 1 of Valko et al. (2013) implies that with probability  $(1 - w^{-1})(1 - \\delta)$  for any  $w > 0$ , a uniformly randomly drawn arm (i.e.,  $t \\sim U([T])$ ) achieves",
+        "bbox": [
+            169,
+            446,
+            823,
+            479
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\epsilon_ {t} \\leq w \\left[ \\frac {1}{\\ln (1 / p)} \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {n \\hat {L}} \\sqrt {\\xi_ {1} \\sqrt {\\frac {\\log_ {b} (T k / \\delta)}{2 \\log_ {b} (e) k}} + \\xi_ {2} b ^ {- \\frac {1}{d C} \\sqrt {T / k}}} \\right] \\tag {124}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            241,
+            486,
+            823,
+            535
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "text",
+        "text": "where  $d = n(\\bar{m} - 1)$ ,  $\\xi_1 = (2 + 2^{2/d})$ ,  $\\xi_2 = \\frac{1}{4} db^{2(1 + 2/d)}$ ,  $k = T \\log_b(T)^{-3}$ ,  $b$  is the branching factor for partitioning cells, and the near-optimality constant  $C = |\\mathcal{X}^*|^{-1}\\sqrt{2\\pi d}\\left(\\frac{b^2d^2}{5r_\\eta^2\\sigma_{-2}}\\right)^{d/2}$  (Lemma 36).",
+        "bbox": [
+            168,
+            542,
+            826,
+            589
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "text",
+        "text": "Proof. Assume  $\\eta_{k} = \\eta = \\frac{2}{L}$  as defined in Lemma 2. Also assume a previously pulled arm is returned uniformly at random. Starting with Lemma 13 and applying Theorem 3 and Lemma 37, we find",
+        "bbox": [
+            169,
+            608,
+            823,
+            640
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\mathbb {E} \\left[ \\epsilon_ {t} \\right] \\leq \\tau \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {\\frac {2 n}{\\min  _ {k} \\eta_ {k}}} \\sqrt {\\frac {1}{T} \\sum_ {t} \\mathcal {L} ^ {\\tau} \\left(\\boldsymbol {x} _ {t}\\right)} (125a) \\\\ \\leq \\frac {1}{\\ln (1 / p)} \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {n \\hat {L}} \\sqrt {\\xi_ {1} \\sqrt {\\frac {\\log_ {b} (T k / \\delta)}{2 \\log_ {b} (e) k}} + \\xi_ {2} b ^ {- \\frac {1}{d C} \\sqrt {T / k}}} (125b) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            223,
+            648,
+            825,
+            739
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "text",
+        "text": "with probability  $1 - \\delta$  where  $m^{*} = \\max_{k} m_{k}$ ,  $d = n(\\bar{m} - 1)$ ,  $\\xi_{1} = (2 + 2^{2 / d})$ ,  $\\xi_{2} = \\frac{1}{4} db^{2(1 + 2 / d)}$ ,  $k = T\\log_b(T)^{-3}$ ,  $b$  is the branching factor for partitioning cells, and the near-optimality constant  $C = |\\mathcal{X}^{*}|^{-1}\\sqrt{2\\pi d}\\left(\\frac{b^2d^2}{5r_\\eta^2\\sigma_{-2}}\\right)^{d / 2}$  (Lemma 36).",
+        "bbox": [
+            169,
+            748,
+            826,
+            809
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "text",
+        "text": "Markov's inequality then allows us to bound the pointwise exploitability of any arm returned by the algorithm as",
+        "bbox": [
+            169,
+            813,
+            823,
+            843
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\epsilon_ {t} \\leq w \\left[ \\frac {1}{\\ln (1 / p)} \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {n \\hat {L}} \\sqrt {\\xi_ {1} \\sqrt {\\frac {\\log_ {b} (T k / \\delta)}{2 \\log_ {b} (e) k}} + \\xi_ {2} b ^ {- \\frac {1}{d C} \\sqrt {T / k}}} \\right] \\tag {126}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            241,
+            851,
+            825,
+            900
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "text",
+        "text": "with probability  $(1 - w^{-1})(1 - \\delta)$  for any  $w > 0$ .",
+        "bbox": [
+            169,
+            907,
+            504,
+            925
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "page_number",
+        "text": "42",
+        "bbox": [
+            488,
+            948,
+            508,
+            959
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "text",
+        "text": "B.7 COMPLEXITY OF POLYMATRIX GAMES",
+        "text_level": 1,
+        "bbox": [
+            171,
+            103,
+            488,
+            118
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "text",
+        "text": "Lemma 38. For a polymatrix game defined by the set of bimatrix games with payoff matrix  $P_{kl}^{k}$  for every player  $k$  and  $l \\neq k$ , the rank of the matrix  $M(x)$  defined in equation (11) can be equivalently studied by replacing all instances of  $H_{kl}^{k}$  with  $P_{kl}^{k}$ .",
+        "bbox": [
+            169,
+            128,
+            826,
+            174
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "text",
+        "text": "Proof. Consider the polymatrix game given by the set of matrices  $P_{kl}^{k}$  for every  $k$  and  $l \\neq k$ . The polymatrix game can be equivalently written in normal form, albeit, less concisely. Note that the polymatrix approximation,  $H_{kl}^{k}$ , as we have defined it (see Section 2 Preliminaries) of this normal-form representation between players  $k$  and  $l$  with all other players' strategies marginalized out is related to the true underlying bimatrix game between them as follows:",
+        "bbox": [
+            169,
+            186,
+            826,
+            258
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} H _ {k l} ^ {k} [ a _ {k}, a _ {l} ] = \\mathbb {E} _ {x _ {- k l}} [ u _ {k} (a _ {k}, a _ {l}, x _ {- k l}) ] \\forall a _ {k}, a _ {l} (127a) \\\\ = a _ {k} ^ {\\top} \\left(P _ {k l} ^ {k} a _ {l} + \\sum_ {j \\notin \\{k, l \\}} P _ {k j} ^ {k} x _ {j}\\right) \\forall a _ {k}, a _ {l} (127b) \\\\ = a _ {k} ^ {\\top} P _ {k l} ^ {k} a _ {l} + \\underbrace {a _ {k} ^ {\\top} \\left(\\sum_ {j \\notin \\{k , l \\}} P _ {k j} ^ {k} x _ {j}\\right)} _ {p _ {k}} \\forall a _ {k}, a _ {l} (127c) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            321,
+            263,
+            823,
+            378
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "text",
+        "text": "where  $P_{kl}^{k}$  is player  $k$ 's payoff matrix for the bimatrix game between players  $k$  and  $l$  in a polymatrix game and  $p_k$  does not depend on player  $l$ 's strategy.",
+        "bbox": [
+            169,
+            383,
+            823,
+            412
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "text",
+        "text": "This implies that  $H_{kl}^{k}$  is equal to  $P_{kl}^{k}$  up to a constant offset of the rows, i.e.,",
+        "bbox": [
+            169,
+            419,
+            671,
+            435
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nH _ {k l} ^ {k} = P _ {k l} ^ {k} + C _ {k} \\tag {128}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            441,
+            440,
+            823,
+            459
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "text",
+        "text": "where  $C_k$  is a matrix with constant rows.",
+        "bbox": [
+            169,
+            463,
+            442,
+            477
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "text",
+        "text": "Consider the matrix  $M(x)$  which contains  $H_{kl}^{k}$  blocks. Recall that the bottom rows of  $M(x)$  contain rows of 1's matching each column of  $H_{kl}^{k}$  blocks. Consider multiplying the  $l$ th row of 1's (which contains 0's on all columns not in the  $l$ th block) by  $\\sqrt{\\eta_k} [I - \\frac{1}{m_k}\\mathbf{1}_k\\mathbf{1}_k^\\top ]C_k$  and subtracting it from the block containing  $\\sqrt{\\eta_k} [I - \\frac{1}{m_k}\\mathbf{1}_k\\mathbf{1}_k^\\top ]H_{kl}^k$ ,",
+        "bbox": [
+            169,
+            483,
+            823,
+            551
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\sqrt {\\eta_ {k}} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1} _ {k} \\mathbf {1} _ {k} ^ {\\top} \\right] \\left[ H _ {k l} ^ {k} - C _ {k} \\right] = \\sqrt {\\eta_ {k}} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1} _ {k} \\mathbf {1} _ {k} ^ {\\top} \\right] P _ {k l} ^ {k}. \\tag {129}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            302,
+            555,
+            823,
+            585
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "text",
+        "text": "Note that  $\\sqrt{\\eta_k} [I - \\frac{1}{m_k}\\mathbf{1}_k\\mathbf{1}_k^\\top ]C_k$  still remains a matrix with constant rows (the preconditioner effectively subtracts a constant matrix from  $C_k$ ). This multiplying and subtracting a row from another is an elementary operation on the matrix, meaning it does not change its row rank. Therefore, for a polymatrix game, we can reason about the positive definiteness of the Hessian at equilibria by examining the matrix  $M(x)$  with all  $H_{kl}^{k}$ 's swapped for  $P_{kl}^{k}$ 's.",
+        "bbox": [
+            169,
+            590,
+            826,
+            664
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "text",
+        "text": "Interestingly, at zero temperature (where  $\\mathrm{QRE} = \\mathrm{Nash}$ ),  $M$  is constant for a polymatrix game, so the rank of this matrix can be computed just once to extract information about all possible interior equilibria in the game. Furthermore, the Hessian is positive semi-definite over the entire joint strategy space, implying the loss function is convex (see Figure 5 (left) for empirical support). This indicates, by convex optimization theory, 1) all mixed Nash equilibria in polymatrix games form a convex set (i.e., they are connected) and 2) assuming mixed equilibria exist, they can be computed simply by stochastic gradient descent on  $\\mathcal{L}$ . If  $M$  is rank- $n\\bar{m}$ , then this interior equilibrium is unique.",
+        "bbox": [
+            169,
+            678,
+            826,
+            777
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "text",
+        "text": "**Complexity** Approximation of Nash equilibria in polymatrix games is known to be PPAD-hard (Deligkas et al., 2022). In contrast, if we restrict our class of polymatrix games to those with at least one interior Nash equilibrium, our analysis proves we can find an approximate Nash equilibrium in deterministic, polynomial time (Corollary 7). This follows directly from the fact that  $\\mathcal{L}$  is convex, our decision set  $\\mathcal{X} = \\prod_{k} \\mathcal{X}_{k}$  is convex, and convex optimization theory admits polynomial time approximation algorithms (e.g., gradient descent). We consider the assumption of the existence of an interior Nash equilibrium to be relatively mild $^3$ , so this positive complexity result is surprising.",
+        "bbox": [
+            169,
+            790,
+            828,
+            888
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "page_footnote",
+        "text": "3Marris et al. (2023) shows 2-player, 2-action polymatrix games with interior Nash equilibria make up a non-trivial  $1/4$  of the space of possible  $2 \\times 2$  games.",
+        "bbox": [
+            169,
+            897,
+            823,
+            925
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "page_number",
+        "text": "43",
+        "bbox": [
+            488,
+            948,
+            506,
+            959
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "text",
+        "text": "Also, note that the Hessian of the loss at Nash equilibria is encoded entirely by the polymatrix approximation at the equilibrium. Therefore, approximating the Hessian of  $\\mathcal{L}$  about the equilibrium (which amounts to observing near-equilibrium behavior (Ling et al., 2018)) allows one to recover this polymatrix approximation (up to constant offsets of the columns which equilibria are invariant to (Marris et al., 2023)).",
+        "bbox": [
+            169,
+            103,
+            823,
+            175
+        ],
+        "page_idx": 43
+    },
+    {
+        "type": "text",
+        "text": "Corollary 7 (Approximating Nash Equilibria of Polymatrix Games with Interior Equilibria). Consider the class of polymatrix games with interior Nash equilibria. This class of games admits a fully polynomial time deterministic approximation scheme (FPTAS).",
+        "bbox": [
+            169,
+            176,
+            828,
+            220
+        ],
+        "page_idx": 43
+    },
+    {
+        "type": "text",
+        "text": "Proof. Lemma 3 relates the approximation of Nash equilibria to the minimization of the loss function  $\\mathcal{L}(\\pmb{x})$ . By Lemma 1, this loss function attains its minimum value of zero if and only if  $\\pmb{x}$  is a Nash equilibrium. For polymatrix games, the Hessian of this loss function is everywhere finite and positive definite (Lemma 16), therefore, this loss function is convex. The decision set for this minimization problem is the product space of simplices, therefore it is also convex. Given that we only consider polymatrix games with interior Nash equilibria, we know that our loss function attains a global minimum within this set. By convex optimization theory, this function can be approximately minimized in a polynomial number of steps by, for example, (projected) gradient descent (Boyd and Vandenberghe, 2004). Gradient descent requires computing the gradient of the loss function at each step. From Lemma 14, we see that computing the gradient (at zero temperature) simply requires reading the polymatrix description of the game (i.e., each bi-matrix game  $H_{kl}^{k}$  between players), which is clearly polynomial in the size of the input (the polymatrix description). The remaining computational steps of gradient descent (e.g., projection onto simplices) are polynomial as well. In conclusion, gradient descent approximates a Nash equilibrium in polynomial number of steps (logarithmic if strongly-convex (Mairal, 2013)), each of which costs polynomial time, therefore the entire scheme is polynomial.",
+        "bbox": [
+            169,
+            234,
+            826,
+            459
+        ],
+        "page_idx": 43
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 43
+    },
+    {
+        "type": "page_number",
+        "text": "44",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 43
+    },
+    {
+        "type": "text",
+        "text": "C EXPERIMENTAL SETUP AND DETAILS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            102,
+            522,
+            118
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "text",
+        "text": "Here we provide further details on the experiments.",
+        "bbox": [
+            171,
+            133,
+            511,
+            148
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "text",
+        "text": "C.1 GAMBIT",
+        "text_level": 1,
+        "bbox": [
+            171,
+            167,
+            287,
+            181
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "text",
+        "text": "The seven methods from the gambit (McKelvey et al., 2016) library that we tested on the 3-player and 4-player Blotto games are listed below (with runtimes). Only gambit-enumpoly and gambit-enumpure are able to return any NE for 3-player Blotto within a 1 hour time limit (and only pure equilibria). And only gambit-enumpure returns any NE for the 4-player game.",
+        "bbox": [
+            169,
+            194,
+            826,
+            252
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- gambit-enumpoly [73 sec 3-player, timeout 4-player]",
+            "- gambit-enumpure [72 sec 3-player, 45 sec 4-player]",
+            "- gambit-gnm",
+            "- gambit-ipa",
+            "- gambit-liap",
+            "- gambit-logit",
+            "- gambit-simpdiv"
+        ],
+        "bbox": [
+            215,
+            262,
+            612,
+            398
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "text",
+        "text": "C.2 LOSS VISUALIZATION AND RANK TEST",
+        "text_level": 1,
+        "bbox": [
+            171,
+            417,
+            491,
+            431
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "text",
+        "text": "Figure 5 and claims made in Section 5 analyze several classical matrix games. We report the payoff matrices in standard row-player / column-player payoff form below. All games are then shifted and scaled so payoffs lie in  $[0,1]$  (i.e., first by subtracting the minimum and then scaling by the max).",
+        "bbox": [
+            169,
+            444,
+            826,
+            488
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "text",
+        "text": "RPS:",
+        "bbox": [
+            171,
+            493,
+            210,
+            506
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left[ \\begin{array}{c c c} 0 / 0 & - 1 / 1 & 1 / - 1 \\\\ 1 / - 1 & 0 / 0 & - 1 / 1 \\\\ - 1 / 1 & 1 / - 1 & 0 / 0 \\end{array} \\right]. \\tag {130}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            400,
+            513,
+            825,
+            579
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "text",
+        "text": "Chicken:",
+        "bbox": [
+            171,
+            594,
+            235,
+            608
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left[ \\begin{array}{c c} 0 / 0 & - 1 / 1 \\\\ 1 / - 1 & - 3 / - 3 \\end{array} \\right]. \\tag {131}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            424,
+            616,
+            825,
+            659
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "text",
+        "text": "Matching Pennies:",
+        "bbox": [
+            171,
+            675,
+            297,
+            690
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left[ \\begin{array}{l l} 1 / - 1 & - 1 / 1 \\\\ - 1 / 1 & 1 / - 1 \\end{array} \\right]. \\tag {132}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            431,
+            699,
+            825,
+            742
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "text",
+        "text": "Modified-Shapleys:",
+        "bbox": [
+            171,
+            758,
+            305,
+            773
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left[ \\begin{array}{c c c} 1 / - 0. 5 & 0 / 1 & 0. 5 / 0 \\\\ 0. 5 / 0 & 1 / - 0. 5 & 0 / 1 \\\\ 0 / 1 & 0. 5 / 0 & 1 / - 0. 5 \\end{array} \\right]. \\tag {133}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            382,
+            782,
+            825,
+            848
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "text",
+        "text": "Prisoner's Dilemma:",
+        "bbox": [
+            171,
+            862,
+            310,
+            876
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left[ \\begin{array}{c c} - 1 / - 1 & - 3 / 0 \\\\ 0 / - 3 & - 2 / - 2 \\end{array} \\right]. \\tag {134}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            418,
+            883,
+            825,
+            926
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "page_number",
+        "text": "45",
+        "bbox": [
+            488,
+            946,
+            506,
+            959
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "text",
+        "text": "C.2.1 LOSS ON FAMILIAR GAMES",
+        "text_level": 1,
+        "bbox": [
+            171,
+            103,
+            424,
+            118
+        ],
+        "page_idx": 45
+    },
+    {
+        "type": "text",
+        "text": "We visualize our proposed loss  $\\mathcal{L}^{\\tau}$  on two classic 2-player, general-sum games: Chicken, payoff matrix (131), and the Prisoner's Dilemma, payoff matrix (134). Each plot in Figure 5 visualizes the loss at various strategy profiles in probability-space; each strategy profile is represented by each player's probability of playing action 1 of 2 (top row / first column of payoff matrix). Temperature  $\\tau$  is varied across the plots. Figure 5 repeats this same visualization but in logit-space to better show the equilibria closest to the boundaries.",
+        "bbox": [
+            169,
+            132,
+            826,
+            217
+        ],
+        "page_idx": 45
+    },
+    {
+        "type": "image",
+        "img_path": "images/a911cefc70a0e4d48aa4672ba390fc60a5a5bea70b6d1b2e2c37e3e15dc48ace.jpg",
+        "image_caption": [
+            "Figure 5: Upper Bound  $(\\epsilon \\leq f_{\\tau}(\\mathcal{L}^{\\tau}))$  Heatmap Visualization. The first row examines the loss landscape for the classic anti-coordination game of Chicken (Nash equilibria:  $(0,1)$ ,  $(1,0)$ ,  $(2/3,1/3))$  while the second row examines the Prisoner's dilemma (Unique Nash equilibrium:  $(0,0))$ . For improved visibility, we subtract the offset  $\\tau \\log(m^{2})$  from  $f_{\\tau}(\\mathcal{L}^{\\tau})$  per Lemma 13, which relates the exploitability at positive temperature to that at zero temperature. Temperature increases for each plot moving to the right. For high temperatures, interior (fully-mixed) strategies are incentivized while for lower temperatures, nearly pure strategies can achieve minimum exploitability. For zero temperature, pure strategy equilibria (e.g., defect-defect) are not captured by the loss as illustrated by the bottom-left Prisoner's Dilemma plot with a constant loss surface."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            239,
+            825,
+            406
+        ],
+        "page_idx": 45
+    },
+    {
+        "type": "image",
+        "img_path": "images/7cf17c1aa14508b04dd3b35c8a089634339250d3478997e64d73489dfd45a4d9.jpg",
+        "image_caption": [
+            "Figure 6: [Figure 5 Repeated in Logit-Space  $(\\ln \\left(\\frac{p}{1 - p}\\right))$  Rather than Probability-Space  $(p)]$  Upper Bound  $(\\epsilon \\leq f_{\\tau}(\\mathcal{L}^{\\tau}))$  Heatmap Visualization. The first row examines the loss landscape for the classic anti-coordination game of Chicken (Nash equilibria:  $(0,1)$ ,  $(1,0)$ ,  $(2/3,1/3))$  while the second row examines the Prisoner's dilemma (Unique Nash equilibrium:  $(0,0)$ ). Temperature increases for each plot moving to the right. For improved visibility, we subtract the offset  $\\tau \\log(m^{2})$  from  $f_{\\tau}(\\mathcal{L}^{\\tau})$  per Lemma 13, which relates the exploitability at positive temperature to that at zero temperature. For high temperatures, interior (fully-mixed) strategies are incentivized while for lower temperatures, nearly pure strategies can achieve minimum exploitability. For zero temperature, pure strategy equilibria (e.g., defect-defect) are not captured by the loss as illustrated by the bottom-left Prisoner's Dilemma plot with a constant loss surface."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            584,
+            823,
+            748
+        ],
+        "page_idx": 45
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 45
+    },
+    {
+        "type": "page_number",
+        "text": "46",
+        "bbox": [
+            488,
+            948,
+            508,
+            959
+        ],
+        "page_idx": 45
+    },
+    {
+        "type": "text",
+        "text": "C.3 SADDLE POINT ANALYSIS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            103,
+            401,
+            118
+        ],
+        "page_idx": 46
+    },
+    {
+        "type": "text",
+        "text": "To generate Figure 2, we follow a procedure similar to the study of MNIST in (Dauphin et al., 2014) (Section 3 of Supp.). Their recommended procedure searches for critical points in two ways. The first repeats a randomized, iterative optimization process 20 times. They then sample one these 20 trials at random, select a random point along the descent trajectory, and search for a critical point (using Newton's method) nearby. They repeat this sampling process 100 times. The second approach randomly selects a feasible point in the decision set and searches for a critical point nearby (again using Newton's method). They also perform this 100 times.",
+        "bbox": [
+            169,
+            130,
+            826,
+            229
+        ],
+        "page_idx": 46
+    },
+    {
+        "type": "text",
+        "text": "Our protocol differs from theirs slightly in a few respects. One, we use SGD, rather than the saddle-free Newton algorithm to trace out an initial descent trajectory. Two, we do not add noise to strategies along the descent trajectory prior to looking for critical points. Thirdly, we minimize gradient norm rather than use Newton's method to look for critical points. Lastly, we use different experimental hyperparameters. We run SGD for 1000 iterations rather than 20 epochs and rerun SGD 100 times rather than 20. We sample 1000 points for each of the two approaches for finding critical points.",
+        "bbox": [
+            169,
+            234,
+            826,
+            321
+        ],
+        "page_idx": 46
+    },
+    {
+        "type": "text",
+        "text": "C.4 SGD ON CLASSICAL GAMES",
+        "text_level": 1,
+        "bbox": [
+            171,
+            339,
+            419,
+            353
+        ],
+        "page_idx": 46
+    },
+    {
+        "type": "text",
+        "text": "The games examined in Figure 3 were all taken from (Gemp et al., 2022). Each is available via open source implementations in OpenSpiel (Lanctot et al., 2019) or GAMUT (Nudelman et al., 2004).",
+        "bbox": [
+            169,
+            366,
+            823,
+            396
+        ],
+        "page_idx": 46
+    },
+    {
+        "type": "text",
+        "text": "We compare against several other baselines, replicating the experiments in (Gemp et al., 2022). RM indicates regret-matching and FTRL indicates follow-the-regularized-leader. These are, arguably, the two most popular scalable stochastic algorithms for approximating Nash equilibria.  ${}^{y}{\\mathrm{{QRE}}}^{\\text{auto }}$  is a stochastic algorithm developed in (Gemp et al., 2022).",
+        "bbox": [
+            169,
+            401,
+            825,
+            458
+        ],
+        "page_idx": 46
+    },
+    {
+        "type": "text",
+        "text": "For each of the experiments, we sweep over learning rates in log-space from  $10^{-3}$  to  $10^{2}$  in increments of 1. We also consider whether to run SGD with the projected-gradient and whether to constrain iterates to the simplex via Euclidean projection or entropic mirror descent (Beck and Teboulle, 2003). We then presented the results of the best performing hyperparameters. This was the same approach taken in (Gemp et al., 2022).",
+        "bbox": [
+            169,
+            463,
+            826,
+            536
+        ],
+        "page_idx": 46
+    },
+    {
+        "type": "text",
+        "text": "Saddle Points in Blotto To confirm the existence of saddle points, we computed the Hessian of  $\\mathcal{L}(\\pmb{x}_{10k})$  for SGD  $(s = \\infty)$ , deflated the matrix by removing from its eigenvectors all directions orthogonal to the simplex, and then computed its top-  $(n\\bar{m} - n)$  eigenvalues. We do this because there always exists a  $n$ -dimensional nullspace of the Hessian at zero temperature that lies outside the tangent space of the simplex, and we only care about curvature within the tangent space. Specifically, at an equilibrium  $\\pmb{x}$ , if we compute  $z^{\\top}Hess(\\mathcal{L})z$  where  $z$  is formed as a linear combination of the vectors  $\\{[x_1, 0, \\ldots, 0]^{\\top}, \\ldots, [0, \\ldots, x_n]^{\\top}\\}$ , then each block  $\\tilde{B}_{kl}$  is identically zero at an equilibrium:  $\\tilde{B}_{kl}x_l = \\sqrt{\\eta_k[I - \\frac{1}{m_k}\\mathbf{11}^{\\top}]}H_{kl}^k x_l = \\sqrt{\\eta_k\\Pi_{T\\Delta}(\\nabla_{x_k}^k)} = 0$ . By Lemma 16, this implies there is zero curvature of the loss in the direction  $z$ :  $z^{\\top}Hess(\\mathcal{L})z = 0$ .",
+        "bbox": [
+            169,
+            551,
+            826,
+            688
+        ],
+        "page_idx": 46
+    },
+    {
+        "type": "text",
+        "text": "C.5 BLIN ON ARTIFICIAL GAME",
+        "text_level": 1,
+        "bbox": [
+            171,
+            705,
+            416,
+            720
+        ],
+        "page_idx": 46
+    },
+    {
+        "type": "text",
+        "text": "To construct the 7-player, 2-action, symmetric, artificial game in Figure 4, we used the following coefficients (discovered by trial-and-error):",
+        "bbox": [
+            169,
+            732,
+            823,
+            762
+        ],
+        "page_idx": 46
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left[ \\begin{array}{c c c c c c c} 0. 0 9 9 0 6 8 7 3 & 0 & 0. 2 3 1 1 6 0 3 7 & 0 & 0. 6 2 7 4 3 5 2 8 & 0 & 0. 1 9 8 1 3 7 4 6 \\\\ 0 & 0. 3 3 0 2 2 9 0 9 & 0 & 0. 0 3 3 0 2 2 9 1 & 0 & 0. 6 2 7 4 3 5 2 8 & 0 \\end{array} \\right]. \\tag {135}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            173,
+            772,
+            844,
+            830
+        ],
+        "page_idx": 46
+    },
+    {
+        "type": "text",
+        "text": "The first row indicates the payoffs received when player  $i$  plays action 0 and the background population plays any of the possible joint actions (number of combinations with replacement). For example, the first column indicates the payoff when all background players play action 0. The second column indicates all background players play action 0 except for one which plays action 1, and so on. The last column indicates all background players play action 1. These  $2n$  scalars uniquely define the payoffs of a symmetric game.",
+        "bbox": [
+            169,
+            839,
+            828,
+            926
+        ],
+        "page_idx": 46
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 46
+    },
+    {
+        "type": "page_number",
+        "text": "47",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 46
+    },
+    {
+        "type": "text",
+        "text": "Given that this game only has two actions, we represent a mixed strategy by a single scalar  $p \\in [0,1]$ , i.e., the probability of the first action. Furthermore, this game is symmetric and we seek a symmetric equilibrium, so we can represent a full Nash equilibrium by this single scalar  $p$ . This reduces our search space from  $7 \\times 2 = 14$  variables to 1 variable (and obviates any need for a map  $s$  from the unit hypercube to the simplex—see Lemma 22).",
+        "bbox": [
+            169,
+            103,
+            826,
+            176
+        ],
+        "page_idx": 47
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 47
+    },
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+        "text": "48",
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+        ],
+        "page_idx": 47
+    }
+]
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diff --git a/2024/Approximating Nash Equilibria in Normal-Form Games via Stochastic Optimization/6057db18-b37f-44dd-ad29-cdbda9e3f494_model.json b/2024/Approximating Nash Equilibria in Normal-Form Games via Stochastic Optimization/6057db18-b37f-44dd-ad29-cdbda9e3f494_model.json
new file mode 100644
index 0000000000000000000000000000000000000000..9a66408e3ef4bc3a256d019310167ddd8106dd93
--- /dev/null
+++ b/2024/Approximating Nash Equilibria in Normal-Form Games via Stochastic Optimization/6057db18-b37f-44dd-ad29-cdbda9e3f494_model.json	
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+    [
+        {
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+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
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+            "type": "title",
+            "bbox": [
+                0.172,
+                0.1,
+                0.83,
+                0.147
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+            "angle": 0,
+            "content": "APPROXIMATING NASH EQUILIBRIA IN NORMAL-FORM GAMES VIA STOCHASTIC OPTIMIZATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.183,
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+            "angle": 0,
+            "content": "Ian Gemp"
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+            "content": "DeepMind"
+        },
+        {
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+            "content": "London, UK"
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+            "angle": 0,
+            "content": "Luke Marris"
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+            "bbox": [
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+            ],
+            "angle": 0,
+            "content": "Georgios Piliouras"
+        },
+        {
+            "type": "text",
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+                0.701,
+                0.199
+            ],
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+            "content": "DeepMind"
+        },
+        {
+            "type": "text",
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+                0.212
+            ],
+            "angle": 0,
+            "content": "London, UK"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.627,
+                0.213,
+                0.778,
+                0.227
+            ],
+            "angle": 0,
+            "content": "gpil@google.com"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.451,
+                0.264,
+                0.547,
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+            ],
+            "angle": 0,
+            "content": "ABSTRACT"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.23,
+                0.295,
+                0.768,
+                0.38
+            ],
+            "angle": 0,
+            "content": "We propose the first loss function for approximate Nash equilibria of normal-form games that is amenable to unbiased Monte Carlo estimation. This construction allows us to deploy standard non-convex stochastic optimization techniques for approximating Nash equilibria, resulting in novel algorithms with provable guarantees. We complement our theoretical analysis with experiments demonstrating that stochastic gradient descent can outperform previous state-of-the-art approaches."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.174,
+                0.406,
+                0.338,
+                0.421
+            ],
+            "angle": 0,
+            "content": "1 INTRODUCTION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.437,
+                0.827,
+                0.549
+            ],
+            "angle": 0,
+            "content": "Nash equilibrium (NE) famously encodes stable behavioral outcomes in multi-agent systems and is arguably the most influential solution concept in game theory. Formally speaking, if \\( n \\) players independently choose \\( n \\), possibly mixed, strategies \\( (x_{k} \\) for \\( k \\in [n] \\)) and their joint strategy \\( (x = \\prod_{k} x_{k}) \\) constitutes a Nash equilibrium, then no player has any incentive to unilaterally deviate from their strategy. This concept has sparked extensive research in various fields, ranging from economics (Milgrom and Weber, 1982) to machine learning (Goodfellow et al., 2014), and has even inspired behavioral theory generalizations such as quantal response equilibria (QREs) which allow for more realistic models of boundedly rational agents (McKelvey and Palfrey, 1995)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.555,
+                0.828,
+                0.655
+            ],
+            "angle": 0,
+            "content": "Unfortunately, when considering Nash equilibria beyond the 2-player, zero-sum scenario, two significant challenges arise. First, it becomes unclear how \\( n \\) independent players would collectively identify a Nash equilibrium when multiple equilibria are possible, giving rise to the equilibrium selection problem (Harsanyi et al., 1988). Secondly, even approximating a single Nash equilibrium is known to be computationally intractable and specifically PPAD-complete (Daskalakis et al., 2009). Combining both problems together, e.g., testing for the existence of equilibria with welfare greater than some fixed threshold is NP-hard, and it is in fact even hard to approximate (Austrin et al., 2011)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.66,
+                0.827,
+                0.73
+            ],
+            "angle": 0,
+            "content": "From a machine learning practitioner's perspective, such computational complexity results hardly give pause for thought as collectively we have become all too familiar with the unreasonable effectiveness of heuristics in circumventing such obstacles. Famously, non-convex optimization is NP-hard, even if the goal is to compute a local minimizer (Murty and Kabadi, 1985), however, stochastic gradient descent (SGD) and variants succeed in training billion parameter models (Brown et al., 2020)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.736,
+                0.827,
+                0.89
+            ],
+            "angle": 0,
+            "content": "Unfortunately, computational techniques for Nash equilibrium have so far not achieved anywhere near the same level of success. In contrast, most modern NE solvers for \\( n \\)-player, \\( m \\)-action, general-sum, normal-form games (NFGs) are practically restricted to a handful of players and/or actions per player except in special cases, e.g., symmetric (Wiedenbeck and Brinkman, 2023) or mean-field games (Pérolat et al., 2022). For example, when running the suite of all 7 applicable methods from the hallmark gambit library (McKelvey et al., 2016) on a 4-player Blotto game, we find only brute-force pure-NE enumeration is able to return any NE within a 1 hour time limit. Scaling solvers to large games is difficult partially due to the fact that an NFG is represented by a tensor with an exponential \\( nm^n \\) entries; even reading this description into memory can be computationally prohibitive. More to the point, any computational technique that presumes exact computation of the expectation of this tensor sampled according to \\( x \\) similarly does not have any hope of scaling beyond small instances."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.896,
+                0.827,
+                0.926
+            ],
+            "angle": 0,
+            "content": "This inefficiency arguably lies at the core of the differential success between machine learning (ML) optimization and equilibrium computation. For example, numerous techniques exist that reduce the"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.504,
+                0.96
+            ],
+            "angle": 0,
+            "content": "1"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.827,
+                0.162
+            ],
+            "angle": 0,
+            "content": "problem of Nash computation to the minimization of the expectation of a random variable (Section 3). Unfortunately, unlike the source of randomness in ML applications where batch learning suffices to produce unbiased estimators, these techniques do not extend easily to game theory which incorporates non-linear functions such as maximum and best-response. This raises our motivating goal:"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.25,
+                0.168,
+                0.747,
+                0.185
+            ],
+            "angle": 0,
+            "content": "Can we solve for Nash equilibria via unbiased stochastic optimization?"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.198,
+                0.828,
+                0.325
+            ],
+            "angle": 0,
+            "content": "Our results. Following in the successful steps of the interplay between ML and stochastic optimization, we reformulate the approximation of Nash equilibria in a normal-form game as a stochastic non-convex optimization problem admitting unbiased Monte-Carlo estimation. This enables the use of powerful solvers and advances in parallel computing to efficiently enumerate Nash equilibria for \\( n \\)-player, general-sum games. Furthermore, this re-casting allows practitioners to incorporate other desirable objectives into the problem such as \"find an approximate Nash equilibrium with welfare above \\( \\omega \\)\" or \"find an approximate Nash equilibrium nearest the current observed joint strategy\" resolving the equilibrium selection problem in an effectively ad-hoc and application tailored manner. Concretely, we make the following contributions by producing:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.335,
+                0.825,
+                0.363
+            ],
+            "angle": 0,
+            "content": "- A loss \\(\\mathcal{L}^{\\tau}(\\pmb{x})\\) 1) whose global minima well approximate Nash equilibria in normal form games, 2) admits unbiased Monte-Carlo estimation, and 3) is Lipschitz and bounded."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.368,
+                0.825,
+                0.41
+            ],
+            "angle": 0,
+            "content": "- Efficient randomized algorithms for approximating Nash equilibria in a novel class of games. The algorithms emerge by employing the family of \\(\\mathcal{X}\\)-armed bandit approaches to \\(\\mathcal{L}^{\\tau}(\\pmb{x})\\) and connecting their global stochastic optimization guarantees to global approximate Nash guarantees."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.415,
+                0.825,
+                0.456
+            ],
+            "angle": 0,
+            "content": "- An empirical comparison of SGD against state-of-the-art baselines for approximating NEs in large games. In some games, vanilla SGD actually improves upon previous state-of-the-art; in others, SGD is slowed by saddle points, a familiar challenge in deep learning (Dauphin et al., 2014)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.172,
+                0.335,
+                0.825,
+                0.456
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.468,
+                0.825,
+                0.499
+            ],
+            "angle": 0,
+            "content": "Overall, this perspective showcases a promising new route to approximating equilibria at scale in practice. We conclude the paper with discussion for future work."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.517,
+                0.342,
+                0.533
+            ],
+            "angle": 0,
+            "content": "2 PRELIMINARIES"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.548,
+                0.827,
+                0.706
+            ],
+            "angle": 0,
+            "content": "In an \\(n\\)-player, normal-form game, each player \\(k \\in \\{1, \\dots, n\\} = [n]\\) has a strategy set \\(\\mathcal{A}_k = \\{a_{k1}, \\dots, a_{km_k}\\}\\) consisting of \\(m_k\\) pure strategies. These strategies can be naturally indexed, so we redefine \\(\\mathcal{A}_k = [m_k]\\) as an abuse of notation. Each player \\(k\\) also has a utility function, \\(u_k: \\mathcal{A} = \\prod_k \\mathcal{A}_k \\to [0,1]\\), (equiv. \"payoff tensor\") that maps joint actions to payoffs in the unit-interval. We denote the average cardinality of the players' action sets by \\(\\bar{m} = \\frac{1}{n} \\sum_k m_k\\) and maximum by \\(m^* = \\max_k m_k\\). Player \\(k\\) may play a mixed strategy by sampling from a distribution over their pure strategies. Let player \\(k\\)'s mixed strategy be represented by a vector \\(x_k \\in \\Delta^{m_k-1}\\) where \\(\\Delta^{m_k-1}\\) is the \\((m_k-1)\\)-dimensional probability simplex embedded in \\(\\mathbb{R}^{m_k}\\). Each function \\(u_k\\) is then extended to this domain so that \\(u_k(\\boldsymbol{x}) = \\sum_{\\boldsymbol{a} \\in \\mathcal{A}} u_k(\\boldsymbol{a}) \\prod_j x_{ja_j}\\) where \\(\\boldsymbol{x} = (x_1, \\dots, x_n)\\) and \\(a_j \\in \\mathcal{A}_j\\) denotes player \\(j\\)'s component of the joint action \\(\\boldsymbol{a} \\in \\mathcal{A}\\). For convenience, let \\(x_{-k}\\) denote all components of \\(\\boldsymbol{x}\\) belonging to players other than player \\(k\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.711,
+                0.825,
+                0.757
+            ],
+            "angle": 0,
+            "content": "The joint strategy \\( \\pmb{x} \\in \\prod_{k} \\Delta^{m_k - 1} \\) is a Nash equilibrium if and only if, for all \\( k \\in [n] \\), \\( u_k(z_k, x_{-k}) \\leq u_k(\\pmb{x}) \\) for all \\( z_k \\in \\Delta^{m_k - 1} \\), i.e., no player has any incentive to unilaterally deviate from \\( \\pmb{x} \\). Nash is typically relaxed with \\( \\epsilon \\)-Nash, our focus: \\( u_k(z_k, x_{-k}) \\leq u_k(\\pmb{x}) + \\epsilon \\) for all \\( z_k \\in \\Delta^{m_k - 1} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.762,
+                0.829,
+                0.926
+            ],
+            "angle": 0,
+            "content": "As an abuse of notation, let the atomic action \\(a_{k} = e_{k}\\) also denote the \\(m_{k}\\)-dimensional \"one-hot\" vector with all zeros aside from a 1 at index \\(a_{k}\\); its use should be clear from the context. We also introduce \\(\\nabla_{x_k}^k\\) as player \\(k\\)'s utility gradient. And for convenience, denote by \\(H_{kl}^{k} = \\mathbb{E}_{x_{-kl}}[u_{k}(a_{k},a_{l},x_{-kl})]\\) the bimatrix game approximation (Janovskaja, 1968) between players \\(k\\) and \\(l\\) with all other players marginalized out; \\(x_{-kl}\\) denotes all strategies belonging to players other than \\(k\\) and \\(l\\) and \\(u_{k}(a_{k},a_{l},x_{-kl})\\) separates out \\(l\\)'s strategy \\(x_{l}\\) from the rest of the players \\(x_{-k}\\). Similarly, denote by \\(T_{klq}^{k} = \\mathbb{E}_{x_{-klq}}[u_{k}(a_{k},a_{l},a_{q},x_{-klq})]\\) the 3-player tensor approximation to the game. Note player \\(k\\)'s utility can now be written succinctly as \\(u_{k}(x_{k},x_{-k}) = x_{k}^{\\top}\\nabla_{x_{k}}^{k} = x_{k}^{\\top}H_{kl}^{k}x_{l} = T_{klq}^{k}x_{k}x_{l}x_{q}\\) for any \\(l,q\\) where we use Einstein notation for tensor arithmetic. For convenience, define \\(\\mathrm{diag}(z)\\) as the function that places a vector \\(z\\) on the diagonal of a square matrix, and \\(\\mathrm{diag}3: z \\in \\mathbb{R}^{d} \\to \\mathbb{R}^{d \\times d \\times d}\\) as a 3-tensor of shape \\((d,d,d)\\) where \\(\\mathrm{diag}3(z)_{lll} = z_l\\)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.948,
+                0.506,
+                0.96
+            ],
+            "angle": 0,
+            "content": "2"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.174,
+                0.1,
+                0.842,
+                0.2
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Loss</td><td>Function</td><td>Obstacle</td></tr><tr><td>Exploitability (ε)</td><td>maxkεk(x)</td><td>max of r.v.</td></tr><tr><td>Nikaido-Isoda (NI)</td><td>∑kεk(x)</td><td>max of r.v.</td></tr><tr><td>Fully-Diff. Exp</td><td>∑k∑ak∈A[k][max(0,uk(ak,x-k)-uk(x))]2</td><td>max of r.v.</td></tr><tr><td>Gradient-based NI</td><td>NI w/ BRk← aBRk = ΠΔ(xk+η∇xuk(x))</td><td>ΠΔ of r.v.</td></tr><tr><td>Unconstrained</td><td>Loss + Simplex Deviation Penalty</td><td>sampling from xk∈Rmk</td></tr></table>"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.171,
+                0.208,
+                0.828,
+                0.239
+            ],
+            "angle": 0,
+            "content": "Table 1: Previous loss functions for NFGs and their obstacles to unbiased estimation. Note that \\(\\epsilon_{k}(\\pmb{x}) = \\max_{z}u_{k}(z,x_{-k}) - u_{k}(\\pmb{x})\\) contains a max operator (see equivalent definition in equation (1))."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.261,
+                0.828,
+                0.348
+            ],
+            "angle": 0,
+            "content": "Following convention from differential geometry, let \\( T_v\\mathcal{M} \\) be the tangent space of a manifold \\( \\mathcal{M} \\) at \\( v \\). For the interior of the \\( d \\)-action simplex \\( \\Delta^{d-1} \\), the tangent space is the same at every point, so we drop the \\( v \\) subscript, i.e., \\( T\\Delta^{d-1} \\). We denote the projection of a vector \\( z \\in \\mathbb{R}^d \\) onto this tangent space as \\( \\Pi_{T\\Delta^{d-1}}(z) = [I - \\frac{1}{d}\\mathbf{11}^\\top]z \\) and call \\( \\Pi_{T\\Delta^{d-1}}(\\nabla_{x_k}^k) \\) a projected-gradient. We drop \\( d-1 \\) when the dimensionality is clear from the context. Finally, let \\( \\mathcal{U}(S) \\) denote a discrete uniform distribution over elements from set \\( S \\)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.367,
+                0.347,
+                0.383
+            ],
+            "angle": 0,
+            "content": "3 RELATED WORK"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.398,
+                0.825,
+                0.442
+            ],
+            "angle": 0,
+            "content": "Representing the problem of computing a Nash equilibrium as an optimization problem is not new. A variety of loss functions and pseudo-distance functions have been proposed. Most of them measure some function of how much each player can exploit the joint strategy by unilaterally deviating:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.273,
+                0.448,
+                0.825,
+                0.475
+            ],
+            "angle": 0,
+            "content": "\\[\n\\epsilon_ {k} (\\boldsymbol {x}) \\stackrel {\\text {d e f}} {=} u _ {k} \\left(\\operatorname {B R} _ {k}, x _ {- k}\\right) - u _ {k} (\\boldsymbol {x}) \\text {w h e r e} \\operatorname {B R} _ {k} \\in \\underset {z} {\\arg \\max } u _ {k} (z, x _ {- k}). \\tag {1}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.487,
+                0.827,
+                0.697
+            ],
+            "angle": 0,
+            "content": "As argued in the introduction, we believe it is important to be able to subsample payoff tensors of normal-form games in order to scale to large instances. As Nash equilibria can consist of mixed strategies, it is advantageous to be able to sample from an equilibrium to estimate its exploitability \\(\\epsilon\\). However none of these losses is amenable to unbiased estimation under sampled play. Each of the functions currently explored in the literature is biased under sampled play either because 1) a random variable appears as the argument of a complex, nonlinear (non-polynomial) function or because 2) how to sample play is unclear. Exploitability, Nikaido-Isoda (NI) (Nikaidô and Isoda, 1955) (also known by NashConv (Lanctot et al., 2017) and ADI (Gemp et al., 2022)), as well as fully-differentiable options (Shoham and Leyton-Brown, 2008, p. 106, Eqn 4.31) introduce bias when a max over payoffs is estimated using samples from \\(x\\). Gradient-based NI (Raghunathan et al., 2019) requires projecting the result of a gradient-ascent step onto the simplex; for the same reason as the max, this projection is prohibitive because it is a nonlinear operation which introduces bias. Lastly, unconstrained optimization approaches (Shoham and Leyton-Brown, 2008, p. 106) that instead penalize deviation from the simplex lose the ability to sample from strategies when each iterate \\(x\\) is no longer a distribution (i.e., \\(x_{k} \\notin \\Delta^{m_{k}-1}\\)). Table 1 summarizes these complications."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.716,
+                0.646,
+                0.731
+            ],
+            "angle": 0,
+            "content": "4 NASH EQUILIBRIUM AS STOCHASTIC OPTIMIZATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.746,
+                0.828,
+                0.83
+            ],
+            "angle": 0,
+            "content": "We will now develop our proposed loss function which is amenable to unbiased estimation. Sub-sections 4.1-4.4 provide a warm-up in which we assume an interior (fully-mixed) Nash equilibrium exists. Subsection 4.5 then shows how to relax that assumption allowing us to approximate partially mixed equilibria as well (including pure equilibria). Our key technical insight is to pay special attention to the geometry of the simplex. To our knowledge, prior works have failed to recognize the role of the tangent space \\( T\\Delta \\). Proofs are in the appendix."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.847,
+                0.517,
+                0.86
+            ],
+            "angle": 0,
+            "content": "4.1 STATIONARITY ON THE SIMPLEX INTERIOR"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.873,
+                0.827,
+                0.903
+            ],
+            "angle": 0,
+            "content": "Lemma 1. Assuming player \\( k \\)'s utility, \\( u_{k}(x_{k}, x_{-k}) \\), is concave in its own strategy \\( x_{k} \\), a strategy in the interior of the simplex is a best response \\( BR_{k} \\) if and only if it has zero projected-gradient\\(^{1}\\) norm."
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.192,
+                0.91,
+                0.778,
+                0.925
+            ],
+            "angle": 0,
+            "content": "<sup>1</sup>Not to be confused with the nonlinear (biased) projected gradient operator in (Hazan et al., 2017)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.504,
+                0.96
+            ],
+            "angle": 0,
+            "content": "3"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.827,
+                0.12
+            ],
+            "angle": 0,
+            "content": "In NFGs, each player's utility is linear in \\( x_{k} \\), thereby satisfying the concavity condition of Lemma 1."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.135,
+                0.506,
+                0.149
+            ],
+            "angle": 0,
+            "content": "4.2 PROJECTED-GRADIENT NORM AS A LOSS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.161,
+                0.825,
+                0.204
+            ],
+            "angle": 0,
+            "content": "An equivalent description of a Nash equilibrium is a joint strategy \\( \\pmb{x} \\) where every player's strategy is a best response to the equilibrium (i.e., \\( x_{k} = \\mathbb{R}_{k} \\) so that \\( \\epsilon_k(\\pmb {x}) = 0 \\)). Lemma 1 states that any interior best response has zero projected-gradient norm, which inspires the following loss function"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.396,
+                0.206,
+                0.825,
+                0.237
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathcal {L} (\\boldsymbol {x}) = \\sum_ {k} \\eta_ {k} \\left\\| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) \\right\\| ^ {2} \\tag {2}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.24,
+                0.794,
+                0.254
+            ],
+            "angle": 0,
+            "content": "where each \\(\\eta_{k} > 0\\) represents a scalar weight, or equivalently, a step size to be explained next."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.256,
+                0.827,
+                0.287
+            ],
+            "angle": 0,
+            "content": "Proposition 1. The loss \\(\\mathcal{L}\\) is equivalent to NashConv, but where player \\(k\\)'s best response is approximated by a single step of projected-gradient ascent with step size \\(\\eta_k\\): \\(aBR_k = x_k + \\eta_k\\Pi_{T\\Delta}(\\nabla_{x_k}^k)\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.294,
+                0.827,
+                0.324
+            ],
+            "angle": 0,
+            "content": "This connection was already pointed out in prior work for unconstrained problems (Gemp et al., 2022; Raghunathan et al., 2019), but this result is the first for strategies constrained to the simplex."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.339,
+                0.5,
+                0.352
+            ],
+            "angle": 0,
+            "content": "4.3 CONNECTION TO TRUE EXPLOITABILITY"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.365,
+                0.825,
+                0.394
+            ],
+            "angle": 0,
+            "content": "In general, we can bound exploitability in terms of the projected-gradient norm as long as each player's utility is concave (this result extends to subgradients of non-smooth functions)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.396,
+                0.825,
+                0.424
+            ],
+            "angle": 0,
+            "content": "Lemma 2. The amount a player can gain by exploiting a joint strategy \\( \\pmb{x} \\) is upper bounded by a quantity proportional to the norm of the projected-gradient:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.406,
+                0.426,
+                0.825,
+                0.446
+            ],
+            "angle": 0,
+            "content": "\\[\n\\epsilon_ {k} (\\boldsymbol {x}) \\leq \\sqrt {2} | | \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) | |. \\tag {3}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.453,
+                0.825,
+                0.525
+            ],
+            "angle": 0,
+            "content": "This bound is not tight on the boundary of the simplex, which can be seen clearly by considering \\( x_{k} \\) to be part of a pure strategy equilibrium. In that case, this analysis assumes \\( x_{k} \\) can be improved upon by a projected-gradient ascent step (via the equivalence pointed out in Proposition 1). However, that is false because the probability of a pure strategy cannot be increased beyond 1. We mention this to provide further intuition for why our \"warm-up\" loss \\( \\mathcal{L}(\\boldsymbol{x}) \\) is only valid for interior equilibria."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.53,
+                0.825,
+                0.563
+            ],
+            "angle": 0,
+            "content": "Note that \\(||\\Pi_{T\\Delta}(\\nabla_{x_k}^k)|| \\leq ||\\nabla_{x_k}^k||\\) because \\(\\Pi_{T\\Delta}\\) is a projection. Therefore, this improves the naive bounds on exploitability and distance to best responses given using the \"raw\" gradient \\(\\nabla_{x_k}^k\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.565,
+                0.774,
+                0.58
+            ],
+            "angle": 0,
+            "content": "Lemma 3. The exploitability of a joint strategy \\(\\pmb{x}\\), is upper bounded by a function of \\(\\mathcal{L}(\\pmb{x})\\):"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.388,
+                0.583,
+                0.825,
+                0.617
+            ],
+            "angle": 0,
+            "content": "\\[\n\\epsilon \\leq \\sqrt {\\frac {2 n}{\\min  _ {k} \\eta_ {k}}} \\sqrt {\\mathcal {L} (\\boldsymbol {x})} \\stackrel {\\text {d e f}} {=} f (\\mathcal {L}). \\tag {4}\n\\]"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.63,
+                0.384,
+                0.643
+            ],
+            "angle": 0,
+            "content": "4.4 UNBIASED ESTIMATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.656,
+                0.827,
+                0.713
+            ],
+            "angle": 0,
+            "content": "As discussed in Section 3, a primary obstacle to unbiased estimation of \\(\\mathcal{L}(\\pmb{x})\\) is the presence of complex, nonlinear functions of random variables, with the projection of a point onto the simplex being one such example (see \\(\\Pi_{\\Delta}\\) in Table 1). However, \\(\\Pi_{T\\Delta}\\), the projection onto the tangent space of the simplex, is linear! This is the insight that allows us to design an unbiased estimator (Lemma 5)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.719,
+                0.827,
+                0.871
+            ],
+            "angle": 0,
+            "content": "Our proposed loss requires computing the squared norm of the expected value of the projected-gradient under the players' mixed strategies, i.e., the \\(l\\)-th entry of player \\(k\\)'s gradient equals \\(\\nabla_{x_{kl}}^{k} = \\mathbb{E}_{a_{-k} \\sim x_{-k}} u_{k}(a_{kl}, a_{-k})\\). By analogy, consider a random variable \\(Y\\). In general, \\(\\mathbb{E}[Y]^2 \\neq \\mathbb{E}[Y^2]\\). This means that we cannot just sample projected-gradients and then compute their average norm to estimate our loss. However, consider taking two independent samples from two corresponding identically distributed, independent random variables \\(Y^{(1)}\\) and \\(Y^{(2)}\\). Then \\(\\mathbb{E}[Y^{(1)}]^2 = \\mathbb{E}[Y^{(1)}] \\mathbb{E}[Y^{(2)}] = \\mathbb{E}[Y^{(1)}Y^{(2)}]\\) by properties of expected value over products of independent random variables. This is a common technique to construct unbiased estimates of expectations over polynomial functions of random variables. Proceeding in this way, define \\(\\nabla_{x_k}^{k(1)}\\) as a random, unbiased gradient estimate (see Table 2). Let \\(\\nabla_{x_k}^{k(2)}\\) be independent and distributed identically to \\(\\nabla_{x_k}^{k(1)}\\). Then Lemma 5 shows"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.252,
+                0.872,
+                0.825,
+                0.922
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathcal {L} (\\boldsymbol {x}) = \\mathbb {E} [ \\sum_ {k} \\eta_ {k} (\\underbrace {\\hat {\\nabla} _ {x _ {k}} ^ {k (1)} - \\frac {1}{m _ {k}} \\left(\\mathbf {1} ^ {\\top} \\hat {\\nabla} _ {x _ {k}} ^ {k (1)}\\right) \\mathbf {1}} _ {\\text {p r o j e c t e d - g r a d i e n t 1}}) ^ {\\top} (\\underbrace {\\hat {\\nabla} _ {x _ {k}} ^ {k (2)} - \\frac {1}{m _ {k}} \\left(\\mathbf {1} ^ {\\top} \\hat {\\nabla} _ {x _ {k}} ^ {k (2)}\\right) \\mathbf {1}} _ {\\text {p r o j e c t e d - g r a d i e n t 2}}) ] \\tag {5}\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.505,
+                0.96
+            ],
+            "angle": 0,
+            "content": "4"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.174,
+                0.1,
+                0.864,
+                0.205
+            ],
+            "angle": 0,
+            "content": "<table><tr><td></td><td>Exact</td><td>Sample Others</td><td>Sample All</td></tr><tr><td>Estimator of ∇k(xk)</td><td>[uk(akl, x-k)]l</td><td>[uk(akl, a-k ~ x-k)]l</td><td>mkuk(akl ~ U(Ak), a-k ~ x-k)el</td></tr><tr><td>ˆ∇k(xk) Bounds</td><td>[0, 1]</td><td>[0, 1]</td><td>[0, mk]</td></tr><tr><td>ˆ∇k(xk) Query Cost</td><td>Πn{k=1}mk</td><td>mk</td><td>1</td></tr><tr><td>ˆL Bounds</td><td>±1/4 Σkηkmk</td><td>±1/4 Σkηkmk</td><td>±1/4 Σkηkm3</td></tr><tr><td>ˆL Query Cost</td><td>n Πn{k=1}mk</td><td>2nm</td><td>2n</td></tr></table>"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.182,
+                0.214,
+                0.816,
+                0.233
+            ],
+            "angle": 0,
+            "content": "Table 2: Examples and Properties of Unbiased Estimators of Loss and Player Gradients \\((\\hat{\\nabla}_{x_k}^{k(p)})\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.259,
+                0.827,
+                0.402
+            ],
+            "angle": 0,
+            "content": "where \\(\\hat{\\nabla}_{x_k}^{k(p)}\\) is an unbiased estimator of player \\(k\\)'s gradient. This estimator can be constructed in several ways. The most expensive, an exact estimator, is constructed by marginalizing player \\(k\\)'s payoff tensor over all other players' strategies. However, a cheaper estimate can be obtained at the expense of higher variance by approximating this marginalization with a Monte Carlo (MC) estimate of the expectation. Specifically, if we sample a single action for each of the remaining players, we can construct an unbiased estimate of player \\(k\\)'s gradient by considering the payoff of each of its actions against the sampled background strategy. Lastly, we can consider constructing an estimate of player \\(k\\)'s gradient by sampling only a single action from player \\(k\\) to represent their entire gradient. Each of these approaches is outlined in Table 2 along with the query complexity (Babichenko, 2016) of computing the estimator and bounds on the values it can take (Lemma 9)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.407,
+                0.825,
+                0.446
+            ],
+            "angle": 0,
+            "content": "We can extend Lemma 3 to one that holds under \\(T\\) samples with probability \\(1 - \\delta\\) by applying, for example, a Hoeffding bound: \\(\\epsilon \\leq f\\bigl (\\hat{\\mathcal{L}} (\\pmb {x}) + \\mathcal{O}\\bigl (\\sqrt{\\frac{1}{T}\\ln(1 / \\delta)}\\bigr)\\bigr)\\) where \\(\\hat{\\mathcal{L}}\\) is an MC estimate of \\(\\mathcal{L}\\)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.466,
+                0.373,
+                0.48
+            ],
+            "angle": 0,
+            "content": "4.5 INTERIOR EQUILIBRIA"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.493,
+                0.825,
+                0.551
+            ],
+            "angle": 0,
+            "content": "We discussed earlier that \\(\\mathcal{L}(\\pmb{x})\\) captures interior equilibria. But some games may only have partially mixed equilibria, i.e., equilibria that lie on the boundary of the simplex. We show how to circumvent this shortcoming by considering quantal response equilibria (QREs), specifically, logit equilibria. By adding an entropy bonus to each player's utility, we can"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.217,
+                0.563,
+                0.466,
+                0.578
+            ],
+            "angle": 0,
+            "content": "- guarantee all equilibria are interior,"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.217,
+                0.586,
+                0.509,
+                0.6
+            ],
+            "angle": 0,
+            "content": "- still obtain unbiased estimates of our loss,"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.217,
+                0.608,
+                0.825,
+                0.637
+            ],
+            "angle": 0,
+            "content": "- maintain an upper bound on the exploitability \\(\\epsilon\\) of any approximate Nash equilibrium in the original game (i.e., the game without an entropy bonus)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.217,
+                0.563,
+                0.825,
+                0.637
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.649,
+                0.825,
+                0.721
+            ],
+            "angle": 0,
+            "content": "Define \\( u_{k}^{\\tau}(\\boldsymbol{x}) = u_{k}(\\boldsymbol{x}) + \\tau S(x_{k}) \\) where Shannon entropy \\( S(x_{k}) = -\\sum_{l}x_{kl}\\ln (x_{kl}) \\) is 1-strongly concave with respect to the 1-norm (Beck and Teboulle, 2003). It is known that Nash equilibria of entropy-regularized games satisfy the conditions for logit equilibria (Leonardos et al., 2021), which are solutions to the fixed point equation \\( x_{k} = \\mathrm{softmax}\\left(\\frac{1}{\\tau}\\nabla_{x_{k}}^{k}\\right) \\). The softmax should make it clear to the reader that all probabilities have positive mass at positive temperature."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.727,
+                0.827,
+                0.807
+            ],
+            "angle": 0,
+            "content": "Recall that in order to construct an unbiased estimate of our loss, we simply needed to construct unbiased estimates of player gradients. The introduction of the entropy term to player \\( k \\)'s utility is special in that it depends entirely on known quantities, i.e., the player's own mixed strategy. We can directly and deterministically compute \\( \\tau \\frac{dS}{dx_k} = -\\tau (\\ln (x_k) + 1) \\) and add this to our estimator of \\( \\nabla_{x_k}^{k(p)}\\colon \\hat{\\nabla}_{x_k}^{k\\tau (p)} = \\hat{\\nabla}_{x_k}^{k(p)} + \\tau \\frac{dS}{dx_k} \\). Consider our loss function refined from (2) with changes in blue:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.389,
+                0.833,
+                0.825,
+                0.864
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) = \\sum_ {k} \\eta_ {k} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right| \\right| ^ {2}. \\tag {6}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.882,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "As mentioned above, the utilities with entropy bonuses are still concave, therefore, a similar bound to Lemma 2 applies. We use this to prove the QRE counterpart to Lemma 3 where \\(\\epsilon_{QRE}\\) is the exploitability of an approximate equilibrium in a game with entropy bonuses."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.504,
+                0.96
+            ],
+            "angle": 0,
+            "content": "5"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.175,
+                0.101,
+                0.825,
+                0.267
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.171,
+                0.281,
+                0.828,
+                0.368
+            ],
+            "angle": 0,
+            "content": "Figure 1: Effect of Sampled Play on a Biased Loss. The first row displays the expectation of the upper bound guaranteed by our proposed loss \\(\\mathcal{L}^{\\tau}\\) with \\(\\eta_{k} = 1\\) for all \\(k\\). The second row displays the expectation of NashConv under sampled play, i.e., \\(\\sum_{k}\\epsilon_{k}\\) where \\(\\epsilon_{k} = \\mathbb{E}_{a_{-k}\\sim x_{-k}}[\\max_{a_{k}}u_{k}^{\\tau}(\\pmb{a})] - \\mathbb{E}_{\\pmb{a}\\sim \\pmb{x}}[u_{k}^{\\tau}(\\pmb{a})]\\). To be consistent, we subtract the offset \\(\\tau \\log (m^{2})\\) from \\(f_{\\tau}(\\mathcal{L}^{\\tau})\\) per Lemma 13, which relates the exploitability at positive temperature to that at zero temperature. The resulting loss surface clearly shows NashConv fails to recognize any interior Nash equilibrium due to its inherent bias."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.394,
+                0.827,
+                0.411
+            ],
+            "angle": 0,
+            "content": "Lemma 4. The entropy regularized exploitability, \\(\\epsilon_{QRE}\\), of a joint strategy \\(\\pmb{x}\\), is upper bounded as:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.363,
+                0.42,
+                0.825,
+                0.455
+            ],
+            "angle": 0,
+            "content": "\\[\n\\epsilon_ {Q R E} \\leq \\sqrt {\\frac {2 n}{\\min  _ {k} \\eta_ {k}}} \\sqrt {\\mathcal {L} ^ {\\tau} (\\boldsymbol {x})} \\stackrel {\\mathrm {d e f}} {=} f \\left(\\mathcal {L} ^ {\\tau}\\right). \\tag {7}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.468,
+                0.825,
+                0.498
+            ],
+            "angle": 0,
+            "content": "Lastly, we establish a connection between quantal response equilibria and Nash equilibria that allows us to approximate Nash equilibria in the original game via minimizing our modified loss \\(\\mathcal{L}^{\\tau}(\\pmb{x})\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.169,
+                0.501,
+                0.825,
+                0.532
+            ],
+            "angle": 0,
+            "content": "Lemma 13 (\\(\\mathcal{L}^{\\tau}\\) Scores Nash Equilibria). Let \\(\\mathcal{L}^{\\tau}(\\pmb{x})\\) be our proposed entropy regularized loss function and \\(\\pmb{x}\\) be an approximate QRE. Then it holds that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.315,
+                0.54,
+                0.826,
+                0.577
+            ],
+            "angle": 0,
+            "content": "\\[\n\\epsilon \\leq \\tau \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {\\frac {2 n}{\\min  _ {k} \\eta_ {k}}} \\sqrt {\\mathcal {L} ^ {\\tau} (\\boldsymbol {x})} \\stackrel {\\text {d e f}} {=} f _ {\\tau} \\left(\\mathcal {L} ^ {\\tau}\\right). \\tag {8}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.594,
+                0.827,
+                0.665
+            ],
+            "angle": 0,
+            "content": "This upper bound is plotted as a heatmap for a familiar Chicken game in the top row of Figure 1. First, notice how pure equilibria are not visible as minima for zero temperature, but appear for slightly warmer temperatures. Secondly, notice that NashConv in the bottom row is unable to capture the interior Nash equilibrium because of its high bias under sampled play. In contrast, our proposed loss \\(\\mathcal{L}^{\\tau}\\) is guaranteed to capture all equilibria at low temperature \\(\\tau\\)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.687,
+                0.294,
+                0.701
+            ],
+            "angle": 0,
+            "content": "5 ANALYSIS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.719,
+                0.825,
+                0.763
+            ],
+            "angle": 0,
+            "content": "In the preceding section we established a loss function that upper bounds the exploitability of an approximate equilibrium. In addition, the zeros of this loss function have a one-to-one correspondence with quantal response equilibria (which approximate Nash equilibria at low temperature)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.768,
+                0.826,
+                0.839
+            ],
+            "angle": 0,
+            "content": "Here, we derive properties that suggest it is \"easy\" to optimize. While this function is generally non-convex and may suffer from a proliferation of saddle points (Figure 2), it is Lipschitz continuous (over the relevant subset of the interior) and bounded. These are two commonly made assumptions in the literature on non-convex optimization, which we leverage in Section 6. In addition, we can derive its gradient, its Hessian, and characterize its behavior around global minima."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.843,
+                0.666,
+                0.859
+            ],
+            "angle": 0,
+            "content": "Lemma 14. The gradient of \\(\\mathcal{L}^{\\tau}(\\pmb{x})\\) with respect to player \\(l\\)'s strategy \\(x_{l}\\) is"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.373,
+                0.868,
+                0.825,
+                0.9
+            ],
+            "angle": 0,
+            "content": "\\[\n\\nabla_ {x _ {l}} \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) = 2 \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\tag {9}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.907,
+                0.705,
+                0.929
+            ],
+            "angle": 0,
+            "content": "where \\(B_{ll} = -\\tau \\left[I - \\frac{1}{m_l}\\mathbf{11}^\\top \\right]diag\\left(\\frac{1}{x_l}\\right)\\) and \\(B_{kl} = \\left[I - \\frac{1}{m_k}\\mathbf{11}^\\top \\right]H_{kl}^k\\) for \\(k\\neq l\\)"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.505,
+                0.96
+            ],
+            "angle": 0,
+            "content": "6"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.175,
+                0.101,
+                0.825,
+                0.174
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.171,
+                0.187,
+                0.828,
+                0.3
+            ],
+            "angle": 0,
+            "content": "Figure 2: Analysis of Loss Landscape. We reapply the analysis of (Dauphin et al., 2014), originally designed to understand the success of SGD in deep learning, to \"slices\" of several popular extensive form games. To construct a slice (or meta-game), we randomly sample 6 deterministic policies and then consider the corresponding \\( n \\)-player, 6-action normal-form game at \\( \\tau = 0.1 \\) (with payoffs normalized to [0, 1]). The index of a critical point \\( x_{c} \\) (\\( \\nabla_{\\pmb{x}}\\mathcal{L}^{\\tau}(\\pmb{x}_{c}) = \\mathbf{0} \\)) indicates the fraction of negative eigenvalues in the Hessian of \\( \\mathcal{L}^{\\tau} \\) at \\( x_{c} \\); \\( \\alpha = 0 \\) indicates a local minimum, 1 a maximum, else a saddle point. We see a positive correlation between exploitability (\\( y \\)-axis), projected-gradient norm (\\( x \\)-axis), and \\( \\alpha \\) (color) indicating a lower prevalence of local minima at high exploitability."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.321,
+                0.506,
+                0.336
+            ],
+            "angle": 0,
+            "content": "Lemma 16. The Hessian of \\(\\mathcal{L}^{\\tau}(\\pmb{x})\\) can be written"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.37,
+                0.339,
+                0.825,
+                0.359
+            ],
+            "angle": 0,
+            "content": "\\[\n\\operatorname {H e s s} \\left(\\mathcal {L} ^ {\\tau}\\right) = 2 \\left[ \\tilde {B} ^ {\\top} \\tilde {B} + T \\Pi_ {T \\Delta} \\left(\\tilde {\\nabla} ^ {\\tau}\\right) \\right] \\tag {10}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.364,
+                0.825,
+                0.401
+            ],
+            "angle": 0,
+            "content": "where \\(\\tilde{B}_{kl} = \\sqrt{\\eta_k} B_{kl}\\), \\(\\Pi_{T\\Delta}(\\tilde{\\nabla}^{\\tau}) = [\\eta_1\\Pi_{T\\Delta}(\\nabla_{x_1}^{1\\tau}),\\dots ,\\eta_n\\Pi_{T\\Delta}(\\nabla_{x_n}^{n\\tau})]\\), and we augment \\(T\\) (the 3-player approximation to the game, \\(T_{lqk}^{k}\\)) so that \\(T_{lll}^{l} = \\tau \\text{diag} 3\\left(\\frac{1}{x_{l}^{2}}\\right)\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.411,
+                0.827,
+                0.489
+            ],
+            "angle": 0,
+            "content": "At an NE, the latter term disappears because \\(\\Pi_{T\\Delta}(\\nabla_{x_k}^{k\\tau}) = 0\\) for all \\(k\\) (Lemma 1). If \\(\\mathcal{X}\\) was \\(\\mathbb{R}^{n\\bar{m}}\\), then we could simply check if \\(\\tilde{B}\\) is full-rank to determine if \\(Hess\\succ 0\\), i.e., if \\(\\mathcal{L}^{\\tau}\\) is locally strongly-convex. However, \\(\\mathcal{X}\\) is a simplex product, and we only care about curvature in directions toward which we can update our strategy profile \\(\\pmb{x}\\). Toward that end, define \\(M\\) to be the \\(n(\\bar{m} +1)\\times n\\bar{m}\\) matrix that stacks \\(\\tilde{B}\\) on top of a repeated identity matrix that encodes orthogonality to the simplex:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.185,
+                0.491,
+                0.826,
+                0.605
+            ],
+            "angle": 0,
+            "content": "\\[\nM (\\boldsymbol {x}) = \\left[ \\begin{array}{c c c c} - \\tau \\sqrt {\\eta_ {1}} \\Pi_ {T \\Delta} \\left(\\frac {1}{x _ {1}}\\right) & \\sqrt {\\eta_ {1}} \\Pi_ {T \\Delta} \\left(H _ {1 2} ^ {1}\\right) & \\dots & \\sqrt {\\eta_ {1}} \\Pi_ {T \\Delta} \\left(H _ {1 n} ^ {1}\\right) \\\\ \\vdots & \\vdots & \\vdots & \\vdots \\\\ \\sqrt {\\eta_ {n}} \\Pi_ {T \\Delta} \\left(H _ {n 1} ^ {n}\\right) & \\dots & \\sqrt {\\eta_ {n}} \\Pi_ {T \\Delta} \\left(H _ {n, n - 1} ^ {n}\\right) & - \\tau \\sqrt {\\eta_ {n}} \\Pi_ {T \\Delta} \\left(\\frac {1}{x _ {n}}\\right) \\\\ \\mathbf {1} _ {1} ^ {\\top} & 0 & \\dots & 0 \\\\ \\vdots & \\vdots & \\vdots & \\vdots \\\\ 0 & \\dots & 0 & \\mathbf {1} _ {n} ^ {\\top} \\end{array} \\right] \\tag {11}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.614,
+                0.827,
+                0.748
+            ],
+            "angle": 0,
+            "content": "where \\(\\Pi_{T\\Delta}(z\\in \\mathbb{R}^{a\\times b}) = [I_a - \\frac{1}{a}\\mathbf{1}_a\\mathbf{1}_a^\\top ]z\\) subtracts the mean from each column of \\(z\\) and \\(\\frac{1}{x_k}\\) is shorthand for \\(\\mathrm{diag}\\left(\\frac{1}{x_k}\\right)\\). If \\(M(\\pmb {x})z = \\mathbf{0}\\) for a nonzero vector \\(z\\in \\mathbb{R}^{n\\bar{m}}\\), this implies there exists a \\(z\\) that 1) is orthogonal to the ones vectors of each simplex (i.e., is a valid equilibrium update direction) and 2) achieves zero curvature in the direction \\(z\\), i.e., \\(z^{\\top}(\\tilde{B}^{\\top}\\tilde{B})z = z^{\\top}(Hess)z = 0\\), and so \\(Hess\\) is not positive definite. Conversely, if \\(M(\\pmb {x})\\) is of rank \\(nm\\) for a quantal response equilibrium \\(\\pmb{x}\\), then the Hessian of \\(\\mathcal{L}^{\\tau}\\) at \\(\\pmb{x}\\) in the tangent space of the simplex product \\((\\mathcal{X} = \\prod_{k}\\mathcal{X}_{k})\\) is positive definite. In this case, we call \\(\\pmb{x}\\) polymatrix-isolated: polymatrix because we only require information of the local polymatrix approximation of the game (i.e., the \\(H_{kl}^{k}\\) matrices) to construct \\(M\\) and isolated because it implies \\(\\pmb{x}\\) is not connected to any other equilibria."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.75,
+                0.825,
+                0.78
+            ],
+            "angle": 0,
+            "content": "Definition 1 (Polymatrix-Isolated Equilibrium). A Nash equilibrium \\(\\pmb{x}^*\\) is polymatrix-isolated iff \\(\\pmb{x}^*\\) is isolated according to its local polymatrix game approximation."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.789,
+                0.827,
+                0.847
+            ],
+            "angle": 0,
+            "content": "By analyzing the rank of \\( M \\), we can confirm that many classical matrix games including Rock-Paper-Scissors, Chicken, Matching Pennies, and Shapley's game all induce strongly convex \\( \\mathcal{L}^{\\tau} \\)'s at zero temperature (i.e., they have unique mixed Nash equilibria). In contrast, a game like Prisoner's Dilemma has a unique pure strategy that will not be captured by our loss at zero temperature."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.866,
+                0.322,
+                0.881
+            ],
+            "angle": 0,
+            "content": "6 ALGORITHMS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.896,
+                0.826,
+                0.926
+            ],
+            "angle": 0,
+            "content": "We have formally transformed the approximation of Nash equilibria in NFGs into a stochastic optimization problem. To our knowledge, this is the first such formulation that allows one-shot"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.504,
+                0.96
+            ],
+            "angle": 0,
+            "content": "7"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.175,
+                0.101,
+                0.328,
+                0.184
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.33,
+                0.102,
+                0.464,
+                0.184
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.467,
+                0.102,
+                0.597,
+                0.184
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.598,
+                0.102,
+                0.732,
+                0.184
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.735,
+                0.102,
+                0.822,
+                0.174
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.171,
+                0.199,
+                0.828,
+                0.258
+            ],
+            "angle": 0,
+            "content": "Figure 3: Comparison of SGD on \\(\\mathcal{L}^{\\tau = 0}\\) against baselines on four games evaluated in (Gemp et al., 2022). The number of samples used to estimate each update iteration (i.e., minibatch size) is indicated by \\(s\\). From left to right: 2-player, 3-action, nonsymmetric; 6-player, 5-action, nonsymmetric; 4-player, 66-action, symmetric; 3-player, 286-action, symmetric. SGD struggles at saddle points in Blotto."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.278,
+                0.825,
+                0.309
+            ],
+            "angle": 0,
+            "content": "unbiased Monte-Carlo estimation which is critical to introduce the use of powerful algorithms capable of solving high dimensional optimization problems. We explore two off-the-shelf approaches."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.323,
+                0.457,
+                0.337
+            ],
+            "angle": 0,
+            "content": "6.1 STOCHASTIC GRADIENT DESCENT"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.349,
+                0.828,
+                0.503
+            ],
+            "angle": 0,
+            "content": "Stochastic gradient descent is the workhorse of high-dimensional stochastic optimization. It is guaranteed to converge to stationary points (Cutkosky et al., 2023), however, it may converge to local, rather than global minima. It also enjoys implicit gradient regularization (Barrett and Dherin, 2020), seeking \"flat\" minima and performs approximate Bayesian inference (Mandt et al., 2017). Despite the lack of global convergence guarantee, we find it performs well empirically in games previously examined by the literature: modified Shapley's (Ostrovski and van Strien, 2014), GAMUT D7 (Nudelman et al., 2004), Blotto (Arad and Rubinstein, 2012). Figure 3 shows SGD is competitive with scalable techniques to approximating NEs: FTRL (Shalev-Shwartz and Singer, 2006; Shalev-Shwartz et al., 2012), Regret Matching (Hart and Mas-Colell, 2000), ADIDAS/ \\( ^yQRRe^{auto} \\) (Gemp et al., 2022). Shapley's game induces a strongly convex \\( \\mathcal{L} \\) (see Section 5) leading to SGD's strong performance. Blotto reaches low, but nonzero \\( \\epsilon \\), demonstrating the challenges of saddle points."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.518,
+                0.78,
+                0.532
+            ],
+            "angle": 0,
+            "content": "6.2 HIGH PROBABILITY, GLOBAL POLYNOMIAL CONVERGENCE RATES VIA BANDITS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.544,
+                0.827,
+                0.643
+            ],
+            "angle": 0,
+            "content": "We explore one other algorithmic approach to non-convex optimization based on minimizing regret, which enjoys finite time global convergence rates. \\(\\mathcal{X}\\)-armed bandits (Bubeck et al., 2011) systematically explore the space of solutions by refining a mesh over the joint strategy space, trading off exploration versus exploitation of promising regions. Several approaches exist (Bartlett et al., 2019; Valko et al., 2013) with open source implementations, e.g., (Li et al., 2023). Applying \\(\\mathcal{X}\\)-armed bandits to our \\(\\mathcal{L}^{\\tau}\\) can be thought of as a stochastic generalization of the exclusion method and other bandit approaches for Nash equilibria (Berg and Sandholm, 2017; Zhou et al., 2017)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.648,
+                0.826,
+                0.742
+            ],
+            "angle": 0,
+            "content": "Equipped with these techniques, we establish a high probability polynomial-time global convergence rate to Nash equilibria in \\( n \\)-player, general-sum games given all QREs(\\( \\tau \\)) are polymatrix-isolated. The quality of this approximation improves as \\( \\tau \\to 0 \\), at the same time increasing the constant on the convergence rate via the Lipschitz constant \\( \\sqrt{\\hat{L}} \\) defined below. For clarity, we assume users provide a temperature in the form \\( \\tau = \\frac{1}{\\ln(1/p)} \\) with \\( p \\in (0,1) \\) which ensures all equilibria have probability mass greater than \\( \\frac{p}{m^*} \\) for all actions (Lemma 11). Lower \\( p \\) corresponds with lower temperature."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.746,
+                0.825,
+                0.779
+            ],
+            "angle": 0,
+            "content": "Theorem 4 (BLiN PAC Rate). Assume \\(\\eta_{k} = \\eta = 2 / \\hat{L}\\), \\(\\tau = \\frac{1}{\\ln(1 / p)}\\), and a previously pulled arm is returned uniformly at random (i.e., \\(t \\sim U([T])\\)). Then for any \\(w > 0\\)"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.248,
+                0.781,
+                0.825,
+                0.819
+            ],
+            "angle": 0,
+            "content": "\\[\n\\epsilon_ {t} \\leq w \\left[ \\frac {1}{\\ln (1 / p)} \\log \\left(\\prod_ {k} m _ {k}\\right) + 2 \\left(1 + \\left(4 c ^ {2} C _ {z}\\right) ^ {1 / 3}\\right) \\sqrt {2 n \\hat {L}} \\left(\\frac {\\ln T}{T}\\right) ^ {\\frac {1}{2 (d z + 2)}} \\right] \\tag {12}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.824,
+                0.825,
+                0.888
+            ],
+            "angle": 0,
+            "content": "with probability \\((1 - w^{-1})(1 - 2T^{-2})\\) where \\(m^{*} = \\max_{k} m_{k}\\), \\(2|\\mathcal{L}^{\\tau}| \\leq c \\leq \\frac{1}{4}\\left(\\frac{\\ln(m^{*})}{\\ln(1 / p)} + 2\\right)\\) (Corollary 1), \\(\\hat{L} = \\left(\\frac{\\ln(m^{*})}{\\ln(1 / p)} + 2\\right)\\left(\\frac{m^{*2}}{p\\ln(1 / p)} + n\\bar{m}\\right)\\) (Corollary 2), the zooming dimension \\(d_{z} = \\frac{1}{2} n\\bar{m}\\), and the zooming constant \\(C_{z} = |\\mathcal{X}^{*}|^{-1}\\left(\\frac{1}{4} r_{\\eta}^{2}\\sigma_{-\\infty}\\right)^{-n\\bar{m}}\\) (Corollary 30)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.896,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "The convergence rate for BLiN (Feng et al., 2022) depends on bounds on the exploitability in terms of the loss (Lemma 13), bounds on estimates of the loss (Corollary 1), Lipschitz bounds on the"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.504,
+                0.96
+            ],
+            "angle": 0,
+            "content": "8"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.198,
+                0.106,
+                0.803,
+                0.259
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.17,
+                0.277,
+                0.828,
+                0.418
+            ],
+            "angle": 0,
+            "content": "Figure 4: Bandit-based (BLiN) Nash solver applied to an artificial 7-player, symmetric, 2-action game. We search for a symmetric equilibrium, which is represented succinctly as the probability of selecting action 1. The plot shows the true exploitability \\(\\epsilon\\) of all symmetric strategies in black and indicates there exist potentially 5 NEs (the dips in the curve). Upper bounds on our unregularized loss \\(\\mathcal{L}\\) capture 4 of these equilibria, missing only the pure NE on the right. By considering our regularized loss, \\(\\mathcal{L}^{\\tau}\\), we are able to capture this pure NE (see zoomed inset). The bandit algorithm selects strategies to evaluate, using 10 Monte-Carlo samples for each evaluation (arm pull) of \\(\\mathcal{L}^{\\tau}\\). These samples are displayed as vertical bars above with the height of the vertical bar representing additional arm pulls. The best arms throughout search are denoted by green circles (darker indicates later in the search). The boxed numbers near equilibria display the welfare of the strategy profile."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.437,
+                0.825,
+                0.538
+            ],
+            "angle": 0,
+            "content": "infinity norm of the gradient (Corollary 2), and the number of distinct strategies \\((n\\bar{m} = \\sum_{k}m_{k})\\). This result further depends on the near-optimality or zooming-dimension \\(d_z\\) and zooming constant \\(C_z\\) which quantify the number of near optimal states. In particular, we assume \\(\\mathcal{L}(s(z))\\) is locally \\((\\sigma_{-\\infty})\\)-strongly convex with respect to \\(\\| \\cdot \\|_{\\infty}\\) about each global optimum within a ball of radius \\(r_\\eta\\). Here, \\(s:[0,1]^{n(\\bar{m} -1)}\\to \\prod_i\\Delta^{m_i - 1}\\) is any function that maps from the unit hypercube to a product of simplices; we analyze two such maps in Appendix B.1. Next, we present an additional convergence rate result using an alternative \\(\\mathcal{X}\\)-bandit approach, StoSOO (Valko et al., 2013)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.539,
+                0.825,
+                0.569
+            ],
+            "angle": 0,
+            "content": "Theorem 5 (StoSOO Rate). Corollary 1 of Valko et al. (2013) implies that with probability \\((1 - w^{-1})(1 - \\delta)\\) for any \\(w > 0\\), a uniformly randomly drawn arm (i.e., \\(t \\sim U([T]))\\) achieves"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.245,
+                0.573,
+                0.825,
+                0.62
+            ],
+            "angle": 0,
+            "content": "\\[\n\\epsilon_ {t} \\leq w \\left[ \\frac {1}{\\ln (1 / p)} \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {n \\hat {L}} \\sqrt {\\xi_ {1} \\sqrt {\\frac {\\log_ {b} (T k / \\delta)}{2 \\log_ {b} (e) k}} + \\xi_ {2} b ^ {- \\frac {1}{d C} \\sqrt {T / k}}} \\right] \\tag {13}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.169,
+                0.622,
+                0.825,
+                0.669
+            ],
+            "angle": 0,
+            "content": "where \\(d = n(\\bar{m} -1)\\) \\(\\xi_{1} = (2 + 2^{2 / d})\\) \\(\\xi_{2} = \\frac{1}{4} db^{2(1 + 2 / d)}\\) \\(k = T\\log_b(T)^{-3}\\) , \\(b\\) is the branching factor for partitioning cells, and the near-optimality constant \\(C = |\\mathcal{X}^* |^{-1}\\sqrt{2\\pi d}\\Big(\\frac{b^2d^2}{5r_\\eta^2\\sigma_{-2}}\\Big)^{d / 2}\\) (Lemma 36)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.677,
+                0.825,
+                0.723
+            ],
+            "angle": 0,
+            "content": "Here we assume \\(\\mathcal{L}(s(z))\\) is locally \\((\\sigma_{-2})\\)-strongly convex with respect to \\(||\\cdot ||_2\\) about each global optimum within a ball of radius \\(r_\\eta\\). Theorem 5 implies a \\(\\tilde{\\mathcal{O}}(T^{-1/4})\\) global convergence rate (Proposition 2), however this is achieved only after an exponential number of burn-in iterations."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.741,
+                0.32,
+                0.756
+            ],
+            "angle": 0,
+            "content": "7 CONCLUSION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.772,
+                0.827,
+                0.926
+            ],
+            "angle": 0,
+            "content": "In this work, we proposed a stochastic loss for approximate Nash equilibria in normal-form games. An unbiased loss estimator of Nash equilibria is the \"key\" to the stochastic optimization \"door\" which holds a wealth of research innovations uncovered over several decades. Thus, it allows the development of new algorithmic techniques for computing equilibria. We consider bandit and vanilla SGD methods in this work, but these are only two of the many options now at our disposal (e.g., adaptive methods (Antonakopoulos et al., 2022), Gaussian processes (Calandriello et al., 2022), evolutionary algorithms (Hansen et al., 2003), etc.). Such approaches as well as generalizations of these techniques to extensive-form, imperfect-information games are promising directions for future work. Similarly to how deep learning research first balked at and then marched on to train neural networks via NP-hard non-convex optimization, we hope computational game theory can march ahead to make useful equilibrium predictions of large multiplayer systems."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.504,
+                0.96
+            ],
+            "angle": 0,
+            "content": "9"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
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+                0.033,
+                0.48,
+                0.049
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+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
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+                0.103,
+                0.289,
+                0.118
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+            "angle": 0,
+            "content": "REFERENCES"
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+            "content": "B. Wiedenbeck and E. Brinkman. Data structures for deviation payoffs. In Proceedings of the 22nd International Conference on Autonomous Agents and Multiagent Systems, 2023."
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
+                0.173,
+                0.674,
+                0.825,
+                0.704
+            ],
+            "angle": 0,
+            "content": "Y. Zhou, J. Li, and J. Zhu. Identify the Nash equilibrium in static games with random payoffs. In International Conference on Machine Learning, pages 4160-4169. PMLR, 2017."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.173,
+                0.103,
+                0.828,
+                0.704
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.961
+            ],
+            "angle": 0,
+            "content": "12"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.102,
+                0.808,
+                0.136
+            ],
+            "angle": 0,
+            "content": "APPENDIX: APPROXIMATING NASH EQUILIBRIA IN NORMAL-FORM GAMES VIA STOCHASTIC OPTIMIZATION"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.175,
+                0.156,
+                0.826,
+                0.171
+            ],
+            "angle": 0,
+            "content": "A Loss with its Properties and Derivatives 14"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.196,
+                0.177,
+                0.826,
+                0.192
+            ],
+            "angle": 0,
+            "content": "A.1 Loss: Connection to Exploitability, Unbiased Estimation, and Upper Bounds 14"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.196,
+                0.198,
+                0.826,
+                0.214
+            ],
+            "angle": 0,
+            "content": "A.2 QREs Approximate NEs at Low Temperature 19"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.196,
+                0.22,
+                0.826,
+                0.235
+            ],
+            "angle": 0,
+            "content": "A.3 Gradient of Loss 21"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.196,
+                0.241,
+                0.826,
+                0.257
+            ],
+            "angle": 0,
+            "content": "A.4 Bound on Gradient / Lipschitz Property 22"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.196,
+                0.263,
+                0.826,
+                0.278
+            ],
+            "angle": 0,
+            "content": "A.5 Hessian of Loss 23"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.196,
+                0.177,
+                0.826,
+                0.278
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.174,
+                0.296,
+                0.826,
+                0.311
+            ],
+            "angle": 0,
+            "content": "B Global Convergence Guarantees 26"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.196,
+                0.318,
+                0.826,
+                0.334
+            ],
+            "angle": 0,
+            "content": "B.1 Maps from Hypercube to Simplex Product 26"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.196,
+                0.34,
+                0.826,
+                0.355
+            ],
+            "angle": 0,
+            "content": "B.2 Near Optimality & Zooming Dimension 29"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.196,
+                0.361,
+                0.826,
+                0.376
+            ],
+            "angle": 0,
+            "content": "B.3 D-BLiN 32"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.196,
+                0.382,
+                0.826,
+                0.397
+            ],
+            "angle": 0,
+            "content": "B.4 Bounded Diameters and Well-shaped Cells 37"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.196,
+                0.403,
+                0.826,
+                0.418
+            ],
+            "angle": 0,
+            "content": "B.5 Stochastic Simultaneous Optimistic Optimization 39"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.196,
+                0.425,
+                0.826,
+                0.44
+            ],
+            "angle": 0,
+            "content": "B.6 Regret to PAC Bounds 41"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.196,
+                0.446,
+                0.826,
+                0.462
+            ],
+            "angle": 0,
+            "content": "B.7 Complexity of Polymatrix Games 43"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.196,
+                0.318,
+                0.826,
+                0.462
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.174,
+                0.48,
+                0.826,
+                0.495
+            ],
+            "angle": 0,
+            "content": "C Experimental Setup and Details 45"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.196,
+                0.501,
+                0.826,
+                0.517
+            ],
+            "angle": 0,
+            "content": "C.1 GAMBIT 45"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.196,
+                0.523,
+                0.826,
+                0.538
+            ],
+            "angle": 0,
+            "content": "C.2 Loss Visualization and Rank Test 45"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.196,
+                0.544,
+                0.826,
+                0.56
+            ],
+            "angle": 0,
+            "content": "C.3 Saddle Point Analysis 47"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.196,
+                0.566,
+                0.826,
+                0.581
+            ],
+            "angle": 0,
+            "content": "C.4 SGD on Classical Games 47"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.196,
+                0.587,
+                0.826,
+                0.602
+            ],
+            "angle": 0,
+            "content": "C.5 BLiN on Artificial Game 47"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.196,
+                0.501,
+                0.826,
+                0.602
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "13"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.103,
+                0.603,
+                0.119
+            ],
+            "angle": 0,
+            "content": "A LOSS WITH ITS PROPERTIES AND DERIVATIVES"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.134,
+                0.827,
+                0.164
+            ],
+            "angle": 0,
+            "content": "In this section, we construct our loss function and derive many of its properties and derivatives (e.g., gradient and hessian) useful for analyzing and executing optimization algorithms."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.18,
+                0.825,
+                0.195
+            ],
+            "angle": 0,
+            "content": "A.1 LOSS: CONNECTION TO EXPLOITABILITY, UNBIASED ESTIMATION, AND UPPER BOUNDS"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.206,
+                0.606,
+                0.22
+            ],
+            "angle": 0,
+            "content": "A.1.1 KKT CONDITIONS IMPLY FIXED POINT SUFFICIENCY"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.231,
+                0.557,
+                0.246
+            ],
+            "angle": 0,
+            "content": "Consider the following constrained optimization problem:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.468,
+                0.254,
+                0.825,
+                0.275
+            ],
+            "angle": 0,
+            "content": "\\[\n\\max  _ {\\boldsymbol {x} \\in \\mathbb {R} ^ {d}} f (\\boldsymbol {x}) \\tag {14a}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.438,
+                0.279,
+                0.825,
+                0.295
+            ],
+            "angle": 0,
+            "content": "\\[\ns. t. g _ {i} (\\boldsymbol {x}) \\leq 0 \\forall i \\tag {14b}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.463,
+                0.297,
+                0.825,
+                0.313
+            ],
+            "angle": 0,
+            "content": "\\[\nh _ {j} (\\boldsymbol {x}) = 0 \\forall j \\tag {14c}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.32,
+                0.825,
+                0.363
+            ],
+            "angle": 0,
+            "content": "where \\( f \\) is concave and \\( g_{i} \\) and \\( h_j \\) represent inequality and equality constraints respectively. If \\( g_{i} \\) and \\( h_i \\) are affine functions, then any maximizer \\( x^{*} \\) of \\( f \\) must satisfy the following necessary and sufficient KKT conditions (Ghofogh et al., 2021; Boyd and Vandenberghe, 2004):"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.217,
+                0.373,
+                0.635,
+                0.391
+            ],
+            "angle": 0,
+            "content": "- Stationarity: \\(\\mathbf{0} \\in \\partial f(\\mathbf{x}^{*}) - \\sum_{j} \\lambda_{j} \\partial h_{j}(\\mathbf{x}^{*}) - \\sum_{i} \\mu_{i} \\partial g_{i}(\\mathbf{x}^{*})\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.217,
+                0.395,
+                0.65,
+                0.411
+            ],
+            "angle": 0,
+            "content": "- Primal feasibility: \\( h_j(\\pmb{x}^*) = 0 \\) for all \\( j \\) and \\( g_i(\\pmb{x}^*) \\leq 0 \\) for all \\( i \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.217,
+                0.415,
+                0.446,
+                0.43
+            ],
+            "angle": 0,
+            "content": "- Dual feasibility: \\(\\mu_{i} \\geq 0\\) for all \\(i\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.217,
+                0.434,
+                0.562,
+                0.45
+            ],
+            "angle": 0,
+            "content": "- Complementary slackness: \\(\\mu_{i}g_{i}(\\pmb{x}^{*}) = 0\\) for all \\(i\\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.217,
+                0.373,
+                0.65,
+                0.45
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.454,
+                0.827,
+                0.482
+            ],
+            "angle": 0,
+            "content": "Lemma 1. Assuming player \\( k \\)'s utility, \\( u_{k}(x_{k}, x_{-k}) \\), is concave in its own strategy \\( x_{k} \\), a strategy in the interior of the simplex is a best response \\( BR_{k} \\) if and only if it has zero projected-gradient² norm:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.172,
+                0.51,
+                0.83,
+                0.535
+            ],
+            "angle": 0,
+            "content": "\\[\nB R _ {k} \\in \\left(i n t \\Delta \\cap \\underset {z} {\\arg \\max } u _ {k} (z, x _ {- k}) - u _ {k} (x _ {k}, x _ {- k})\\right) \\iff (B R _ {k} \\in i n t \\Delta) \\wedge (\\| \\Pi_ {T \\Delta} [ \\nabla_ {B R _ {k}} ^ {k} ] \\| = 0).\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.551,
+                0.827,
+                0.567
+            ],
+            "angle": 0,
+            "content": "Proof. Consider the problem of formally computing \\(\\epsilon_{k}(\\pmb{x}) = \\max_{z\\in int\\Delta}u_{k}(z,x_{-k}) - u_{k}(x_{k},x_{-k})\\)"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.41,
+                0.588,
+                0.825,
+                0.61
+            ],
+            "angle": 0,
+            "content": "\\[\n\\max  _ {z \\in \\mathbb {R} ^ {d}} u _ {k} (z, x _ {- k}) - u _ {k} \\left(x _ {k}, x _ {- k}\\right) \\tag {15a}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.384,
+                0.613,
+                0.825,
+                0.629
+            ],
+            "angle": 0,
+            "content": "\\[\ns. t. - z _ {k} \\leq 0 \\forall k \\tag {15b}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.372,
+                0.633,
+                0.825,
+                0.664
+            ],
+            "angle": 0,
+            "content": "\\[\n1 - \\sum_ {k} z _ {k} = 0. \\tag {15c}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.671,
+                0.825,
+                0.743
+            ],
+            "angle": 0,
+            "content": "Note that the objective is linear (concave) in \\( z \\) and the constraints are affine, therefore the KKT conditions are necessary and sufficient for optimality. Recall that we assume that the solution \\( z^{*} \\) lies in the interior of the simplex, i.e., \\( z_{k}^{*} > 0 \\) for each \\( k \\). Also, let \\( e_{k} \\) be a onehot vector, i.e., a zeros vector except with a 1 at index \\( k \\). Mapping the KKT conditions onto this problem yields the following:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.217,
+                0.756,
+                0.561,
+                0.773
+            ],
+            "angle": 0,
+            "content": "- Stationarity: \\(\\mathbf{0} \\in \\partial u_{k}(z^{*}, x_{-k}) + \\lambda \\mathbf{1} + \\sum_{k} \\mu_{k} e_{k}\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.217,
+                0.78,
+                0.49,
+                0.797
+            ],
+            "angle": 0,
+            "content": "- Primal feasibility: \\(\\sum_{k}z_{k}^{*} = 1\\) for all \\(k\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.217,
+                0.805,
+                0.45,
+                0.82
+            ],
+            "angle": 0,
+            "content": "- Dual feasibility: \\(\\mu_{i} \\geq 0\\) for all \\(k\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.217,
+                0.829,
+                0.551,
+                0.845
+            ],
+            "angle": 0,
+            "content": "- Complementary slackness: \\(-\\mu_k z_k^* = 0\\) for all \\(k\\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.217,
+                0.756,
+                0.561,
+                0.845
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.859,
+                0.827,
+                0.888
+            ],
+            "angle": 0,
+            "content": "For any point \\( z^{*} \\in int\\Delta \\), primal feasibility will be satisfied. Given our assumption that \\( z_{k}^{*} > 0 \\) by complementary slackness and dual feasibility, each \\( \\mu_{k} \\) must be identically zero. This implies"
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.171,
+                0.898,
+                0.825,
+                0.924
+            ],
+            "angle": 0,
+            "content": "\\(^{2}\\)Not to be confused with the nonlinear (i.e., introduces bias) projected gradient operator introduced in (Hazan et al., 2017)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "14"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.825,
+                0.133
+            ],
+            "angle": 0,
+            "content": "the stationarity condition can be simplified to \\(\\mathbf{0} \\in \\partial u_k(z^*, x_{-k}) + \\lambda \\mathbf{1}\\). Rearranging terms (and repurposing \\(\\lambda\\)) we find that for any \\(z^*\\), there exists a \\(\\lambda\\) such that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.43,
+                0.141,
+                0.825,
+                0.157
+            ],
+            "angle": 0,
+            "content": "\\[\n\\lambda \\mathbf {1} \\in \\partial u _ {k} \\left(z ^ {*}, x _ {- k}\\right). \\tag {16}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.165,
+                0.826,
+                0.199
+            ],
+            "angle": 0,
+            "content": "Equivalently, \\(\\partial u_{k}(z^{*},x_{-k})\\propto \\mathbf{1}\\) at \\(z^{*}\\in int\\Delta\\) . Any vector proportional to the ones vector has zero projected-gradient norm, completing the claim: \\([I - \\frac{1}{m_k}\\mathbf{11}^\\top ](\\lambda \\mathbf{1}) = \\lambda (\\mathbf{1} - \\frac{m_k}{m_k}\\mathbf{1}) = \\mathbf{0}.\\)"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.212,
+                0.825,
+                0.24
+            ],
+            "angle": 0,
+            "content": "A.1.2 NORM OF PROJECTED-GRADIENT AND EQUIVALENCE TO NFG EXPLOITABILITY WITH APPROXIMATE BEST RESPONSES"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.251,
+                0.827,
+                0.281
+            ],
+            "angle": 0,
+            "content": "Proposition 1. The loss \\(\\mathcal{L}\\) is equivalent to NashConv, but where player \\(k\\)'s best response is approximated by a single step of projected-gradient ascent with step size \\(\\eta_k\\): \\(aBR_k = x_k + \\eta_k\\Pi_{T\\Delta}(\\nabla_{x_k}^k)\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.297,
+                0.825,
+                0.327
+            ],
+            "angle": 0,
+            "content": "Proof. Define an approximate best response as the result of a player adjusting their strategy via a projected-gradient ascent step, i.e., \\( \\mathsf{aBR}_k = x_k + \\eta_k\\Pi_{T\\Delta}(\\nabla_{x_k}^k) \\) for player \\( k \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.332,
+                0.59,
+                0.348
+            ],
+            "angle": 0,
+            "content": "In a normal form game, player \\( k \\)'s utility at this new strategy is"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.208,
+                0.354,
+                0.826,
+                0.375
+            ],
+            "angle": 0,
+            "content": "\\[\nu _ {k} \\left(\\mathrm {a B R} _ {k}, x _ {- k}\\right) = \\left(\\nabla_ {x _ {k}} ^ {k}\\right) ^ {\\top} \\left(x _ {k} + \\eta_ {k} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right)\\right) = u _ {k} (\\boldsymbol {x}) + \\eta_ {k} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right). \\tag {17}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.387,
+                0.542,
+                0.403
+            ],
+            "angle": 0,
+            "content": "Therefore, the amount player \\(k\\) gains by playing aBR is"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.334,
+                0.41,
+                0.825,
+                0.501
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\hat {\\epsilon} _ {k} (\\boldsymbol {x}) = u _ {k} \\left(\\mathrm {a B R} _ {k}, x _ {- k}\\right) - u _ {k} (\\boldsymbol {x}) (18a) \\\\ = \\eta_ {k} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) (18b) \\\\ = \\eta_ {k} \\left(\\nabla_ {x _ {k}} ^ {k} - \\frac {1}{m _ {k}} \\left(\\mathbf {1} ^ {\\top} \\nabla_ {x _ {k}} ^ {k}\\right) \\mathbf {1}\\right) ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) (18c) \\\\ = \\eta_ {k} \\| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) \\| ^ {2} (18d) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.51,
+                0.826,
+                0.539
+            ],
+            "angle": 0,
+            "content": "where the third equality follows from the fact that the projected-gradient, \\(\\Pi_{T\\Delta}(\\nabla_{x_k}^k)\\), is orthogonal to the ones vector."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.556,
+                0.517,
+                0.57
+            ],
+            "angle": 0,
+            "content": "A.1.3 CONNECTION TO TRUE EXPLOITABILITY"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.581,
+                0.825,
+                0.61
+            ],
+            "angle": 0,
+            "content": "Lemma 2. The amount a player can gain by deviating is upper bounded by a quantity proportional to the norm of the projected-gradient:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.406,
+                0.618,
+                0.825,
+                0.637
+            ],
+            "angle": 0,
+            "content": "\\[\n\\epsilon_ {k} (\\boldsymbol {x}) \\leq \\sqrt {2} \\left\\| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) \\right\\|. \\tag {19}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.652,
+                0.729,
+                0.668
+            ],
+            "angle": 0,
+            "content": "Proof. Let \\(z\\) be any point on the simplex. Then by concavity of \\(u_{k}\\) with respect to \\(z\\),"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.268,
+                0.674,
+                0.825,
+                0.803
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} u _ {k} (z, x _ {- k}) - u _ {k} (\\boldsymbol {x}) \\leq \\left(\\nabla_ {x _ {k}} ^ {k}\\right) ^ {\\top} (z - x _ {k}) (20a) \\\\ = \\left(\\nabla_ {x _ {k}} ^ {k}\\right) ^ {\\top} \\left(z - x _ {k}\\right) - \\frac {1}{m _ {k}} \\left(\\mathbf {1} ^ {\\top} \\nabla_ {x _ {k}} ^ {k}\\right) \\overbrace {\\mathbf {1} ^ {\\top} \\left(z - x _ {k}\\right)} ^ {1 - 1 = 0} (20b) \\\\ = \\left(\\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right)\\right) ^ {\\top} \\underbrace {(z - x _ {k})} _ {\\text {D i a m} (\\Delta) \\leq \\sqrt {2}} (20c) \\\\ \\leq \\sqrt {2} \\| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) \\|. \\quad (\\text {C a u c h y - S c h w a r z}) (20d) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.81,
+                0.825,
+                0.822
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.842,
+                0.688,
+                0.857
+            ],
+            "angle": 0,
+            "content": "Continuing, we can prove a bound on \\(\\epsilon\\) in terms of the projected-gradient loss:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.861,
+                0.827,
+                0.888
+            ],
+            "angle": 0,
+            "content": "Lemma 3. The exploitability, \\(\\epsilon\\), of a joint strategy \\(x\\), is upper bounded as a function of our proposed loss:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.416,
+                0.895,
+                0.825,
+                0.93
+            ],
+            "angle": 0,
+            "content": "\\[\n\\epsilon \\leq \\sqrt {\\frac {2 n}{\\min  _ {k} \\eta_ {k}}} \\sqrt {\\mathcal {L} (\\boldsymbol {x})}. \\tag {21}\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "15"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.105,
+                0.218,
+                0.119
+            ],
+            "angle": 0,
+            "content": "Proof."
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.334,
+                0.128,
+                0.826,
+                0.442
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\epsilon = \\max  _ {k} \\max  _ {z} u _ {k} (z, x _ {- k}) - u _ {k} (\\boldsymbol {x}) (22a) \\\\ \\leq \\sum_ {k} \\max  _ {z} u _ {k} (z, x _ {- k}) - u _ {k} (\\boldsymbol {x}) (22b) \\\\ \\leq \\sum_ {k} \\sqrt {2} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) \\right| \\right| _ {2} \\quad (\\text {L e m m a 2}) (22c) \\\\ = \\sqrt {2} \\left|\\left|\\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {1}} ^ {1}\\right)\\right|\\right| _ {2}, \\dots , \\left|\\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {n}} ^ {n}\\right)\\right|\\right| _ {2} \\right|\\left. \\right| _ {1} (22d) \\\\ \\leq \\sqrt {2 n} \\left|\\left|\\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {1}} ^ {1}\\right)\\right|\\right| _ {2}, \\dots , \\left|\\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {n}} ^ {n}\\right)\\right|\\right| _ {2} \\right|\\left. \\right| _ {2} (22e) \\\\ = \\sqrt {2 n} \\sqrt {\\sum_ {k} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) \\right| \\right| _ {2} ^ {2}} (22f) \\\\ = \\sqrt {2 n} \\sqrt {\\sum_ {k} \\left(\\frac {1}{\\eta_ {k}}\\right) \\eta_ {k} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) \\right| \\right| _ {2} ^ {2}} (22g) \\\\ \\leq \\sqrt {\\frac {2 n}{\\min _ {k} \\eta_ {k}}} \\sqrt {\\sum_ {k} \\eta_ {k} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) \\right| \\right| _ {2} ^ {2}} (22h) \\\\ = \\sqrt {\\frac {2 n}{\\min _ {k} \\eta_ {k}}} \\sqrt {\\mathcal {L} (\\boldsymbol {x})}. (22i) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.45,
+                0.825,
+                0.462
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.169,
+                0.476,
+                0.825,
+                0.505
+            ],
+            "angle": 0,
+            "content": "Lemma 4. The entropy regularized exploitability, \\(\\epsilon_{QRE}\\), of a joint strategy \\(\\pmb{x}\\), is upper bounded as a function of our proposed loss:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.397,
+                0.513,
+                0.826,
+                0.547
+            ],
+            "angle": 0,
+            "content": "\\[\n\\epsilon_ {Q R E} \\leq \\sqrt {\\frac {2 n}{\\min _ {k} \\eta_ {k}}} \\sqrt {\\mathcal {L} ^ {\\tau} (\\boldsymbol {x})}. \\tag {23}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.565,
+                0.649,
+                0.581
+            ],
+            "angle": 0,
+            "content": "Proof. Recall that \\( u_{k}^{\\tau}(x_{k}, x_{-k}) \\) is also concave with respect to \\( x_{k} \\). Then"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.318,
+                0.588,
+                0.826,
+                0.904
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\epsilon_ {Q R E} = \\max  _ {k} \\max  _ {z} u _ {k} ^ {\\tau} (z, x _ {- k}) - u _ {k} ^ {\\tau} (\\boldsymbol {x}) (24a) \\\\ \\leq \\sum_ {k} \\max  _ {z} u _ {k} ^ {\\tau} (z, x _ {- k}) - u _ {k} ^ {\\tau} (\\boldsymbol {x}) (24b) \\\\ \\leq \\sum_ {k} \\sqrt {2} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right| \\right| _ {2} \\quad (\\text {L e m m a 2}) (24c) \\\\ = \\sqrt {2} \\left| \\left| | \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {1}} ^ {1 \\tau}\\right) | | _ {2}, \\dots , | | \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {n}} ^ {n \\tau}\\right) | | _ {2} \\right| \\right| _ {1} (24d) \\\\ \\leq \\sqrt {2 n} \\left| \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {1}} ^ {1 \\tau}\\right) \\right| \\right| _ {2}, \\dots , \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {n}} ^ {n \\tau}\\right) \\right| \\right| _ {2} \\right| _ {2} (24e) \\\\ = \\sqrt {2 n} \\sqrt {\\sum_ {k} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right| \\right| _ {2} ^ {2}} (24f) \\\\ \\leq \\sqrt {2 n} \\sqrt {\\sum_ {k} \\left(\\frac {1}{\\eta_ {k}}\\right) \\eta_ {k} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right| \\right| _ {2} ^ {2}} (24g) \\\\ \\leq \\sqrt {\\frac {2 n}{\\min _ {k} \\eta_ {k}}} \\sqrt {\\sum_ {k} \\eta_ {k} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right| \\right| _ {2} ^ {2}} (24h) \\\\ = \\sqrt {\\frac {2 n}{\\min _ {k} \\eta_ {k}}} \\sqrt {\\mathcal {L} ^ {\\tau} (\\boldsymbol {x})}. (24i) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.949,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "16"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.104,
+                0.402,
+                0.119
+            ],
+            "angle": 0,
+            "content": "A.1.4 UNBIASED ESTIMATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.169,
+                0.129,
+                0.827,
+                0.16
+            ],
+            "angle": 0,
+            "content": "Lemma 5. An unbiased estimate of \\(\\mathcal{L}(\\pmb{x})\\) can be obtained by drawing two samples (pure strategies) from each players' mixed strategy and observing payoffs."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.172,
+                0.826,
+                0.209
+            ],
+            "angle": 0,
+            "content": "Proof. Define \\(\\hat{\\nabla}_{x_k}^{k(1)}\\) as a random, unbiased gradient estimate (see Table 2). Let \\(\\hat{\\nabla}_{x_k}^{k(2)}\\) be independent and distributed identically to \\(\\hat{\\nabla}_{x_k}^{k(1)}\\). Then,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.201,
+                0.212,
+                0.826,
+                0.43
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\mathbb {E} _ {\\boldsymbol {a} ^ {(1)} \\sim \\boldsymbol {x}, \\boldsymbol {a} ^ {(2)} \\sim \\boldsymbol {x}} [ \\sum_ {k} \\eta_ {k} (\\underbrace {\\hat {\\nabla} _ {x _ {k}} ^ {k (1)} - \\frac {1}{m _ {k}} \\left(\\boldsymbol {1} ^ {\\top} \\hat {\\nabla} _ {x _ {k}} ^ {k (1)}\\right) \\boldsymbol {1}} _ {\\text {p r o j e c t e d - g r a d i e n t 1}}) ^ {\\top} (\\underbrace {\\hat {\\nabla} _ {x _ {k}} ^ {k (2)} - \\frac {1}{m _ {k}} \\left(\\boldsymbol {1} ^ {\\top} \\hat {\\nabla} _ {x _ {k}} ^ {k (2)}\\right) \\boldsymbol {1}} _ {\\text {p r o j e c t e d - g r a d i e n t 2}}) ] (25a) \\\\ = \\sum_ {k} \\eta_ {k} \\mathbb {E} _ {\\boldsymbol {a} ^ {(1)} \\sim \\boldsymbol {x}, \\boldsymbol {a} ^ {(2)} \\sim \\boldsymbol {x}} [ (\\hat {\\nabla} _ {x _ {k}} ^ {k (1)} - \\frac {1}{m _ {k}} (\\boldsymbol {1} ^ {\\top} \\hat {\\nabla} _ {x _ {k}} ^ {k (1)}) \\boldsymbol {1}) ^ {\\top} (\\hat {\\nabla} _ {x _ {k}} ^ {k (2)} - \\frac {1}{m _ {k}} (\\boldsymbol {1} ^ {\\top} \\hat {\\nabla} _ {x _ {k}} ^ {k (2)}) \\boldsymbol {1}) ] (25b) \\\\ = \\sum_ {k} \\eta_ {k} \\mathbb {E} _ {\\boldsymbol {a} ^ {(1)} \\sim \\boldsymbol {x}} \\left[ \\left(\\hat {\\nabla} _ {x _ {k}} ^ {k (1)} - \\frac {1}{m _ {k}} \\left(\\boldsymbol {1} ^ {\\top} \\hat {\\nabla} _ {x _ {k}} ^ {k (1)}\\right) \\boldsymbol {1}\\right) \\right] ^ {\\top} \\mathbb {E} _ {\\boldsymbol {a} ^ {(2)} \\sim \\boldsymbol {x}} \\left[ \\left(\\hat {\\nabla} _ {x _ {k}} ^ {k (2)} - \\frac {1}{m _ {k}} \\left(\\boldsymbol {1} ^ {\\top} \\hat {\\nabla} _ {x _ {k}} ^ {k (2)}\\right) \\boldsymbol {1}\\right) \\right] (25c) \\\\ = \\sum_ {k} \\eta_ {k} \\left(\\nabla_ {x _ {k}} ^ {k} - \\frac {1}{m _ {k}} \\left(\\mathbf {1} ^ {\\top} \\nabla_ {x _ {k}} ^ {k}\\right) \\mathbf {1}\\right) ^ {\\top} \\left(\\nabla_ {x _ {k}} ^ {k} - \\frac {1}{m _ {k}} \\left(\\mathbf {1} ^ {\\top} \\nabla_ {x _ {k}} ^ {k}\\right) \\mathbf {1}\\right) (25d) \\\\ = \\sum_ {k} \\eta_ {k} \\left\\| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) \\right\\| ^ {2} (25e) \\\\ = \\mathcal {L} (\\boldsymbol {x}) (25f) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.434,
+                0.827,
+                0.495
+            ],
+            "angle": 0,
+            "content": "where the first equality follows from linearity of expectation, the second from independence of random variables, and the third from \\(\\mathbb{E}_{\\pmb{a}^{(p)}\\sim \\pmb{x}}[\\hat{\\nabla}_{x_k}^{k(p)}] = \\nabla_{x_k}^k\\), i.e., \\(\\hat{\\nabla}_{x_k}^{k(p)}\\) is an unbiased estimator of player \\(k\\)'s gradient. Therefore, equation (25a) comprises an unbiased estimate of \\(\\mathcal{L}(\\pmb{x})\\) proving the claim."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.502,
+                0.824,
+                0.513
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.523,
+                0.825,
+                0.553
+            ],
+            "angle": 0,
+            "content": "Lemma 6. The loss formed as the sum of the squared norms of the projected-gradients, \\(\\mathcal{L}^{\\tau}\\), can be decomposed into three terms as follows:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.275,
+                0.556,
+                0.826,
+                0.611
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) = \\underbrace {\\sum_ {k} \\eta_ {k} x _ {q} ^ {\\top} B _ {k q} ^ {\\top} B _ {k q} x _ {q}} _ {(A)} + \\underbrace {2 \\sum_ {k} \\eta_ {k} E _ {k} ^ {\\top} B _ {k q} x _ {q}} _ {(B)} + \\underbrace {\\sum_ {k} \\eta_ {k} E _ {k} ^ {\\top} E _ {k}} _ {(C)} \\tag {26}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.615,
+                0.405,
+                0.63
+            ],
+            "angle": 0,
+            "content": "where \\(q\\) is any player other than \\(k\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.643,
+                0.825,
+                0.679
+            ],
+            "angle": 0,
+            "content": "Proof. Let \\( S^{\\tau} = -\\tau \\sum_{l} x_{kl} \\ln(x_{kl}) \\) so that \\( \\frac{\\partial S^{\\tau}}{\\partial x_k} = -\\tau (\\ln(x_k) + 1) \\). Note that \\( \\Pi_{T\\Delta}[\\frac{\\partial S^{\\tau}}{\\partial x_k}] = -\\tau \\Pi_{T\\Delta}[\\ln(x_k)] \\)."
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.192,
+                0.682,
+                0.826,
+                0.88
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) = \\sum_ {k} \\eta_ {k} \\left(\\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right)\\right) ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) (27a) \\\\ = \\sum_ {k} \\eta_ {k} \\left[ H _ {k q} ^ {k} x _ {q} + \\frac {\\partial S ^ {\\tau}}{\\partial x _ {k}} \\right] ^ {\\top} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] \\left[ H _ {k q} ^ {k} x _ {q} + \\frac {\\partial S ^ {\\tau}}{\\partial x _ {k}} \\right] (27b) \\\\ = \\sum_ {k} \\eta_ {k} \\left(x _ {q} ^ {\\top} \\left[ H _ {k q} ^ {k} \\right] ^ {\\top} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] ^ {2} \\left[ H _ {k q} ^ {k} \\right] x _ {q} + 2 \\left[ \\frac {\\partial S ^ {\\tau}}{\\partial x _ {k}} \\right] ^ {\\top} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] ^ {2} \\left[ H _ {k q} ^ {k} x _ {q} \\right] \\quad (2 7 c) \\right. \\\\ \\left. + \\left[ \\frac {\\partial S ^ {\\tau}}{\\partial x _ {k}} \\right] ^ {\\top} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] ^ {2} \\left[ \\frac {\\partial S ^ {\\tau}}{\\partial x _ {k}} \\right]\\right) (27d) \\\\ = \\underbrace {\\sum_ {k} \\eta_ {k} x _ {q} ^ {\\top} B _ {k q} ^ {\\top} B _ {k q} x _ {q}} _ {(A)} + \\underbrace {2 \\sum_ {k} \\eta_ {k} E _ {k} ^ {\\top} B _ {k q} x _ {q}} _ {(B)} + \\underbrace {\\sum_ {k} \\eta_ {k} E _ {k} ^ {\\top} E _ {k}} _ {(C)} (27e) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.884,
+                0.771,
+                0.907
+            ],
+            "angle": 0,
+            "content": "where \\(B_{kq} = [I - \\frac{1}{m_k}\\mathbf{11}^\\top ]H_{kq}^k\\) and \\(E_{k} = [I - \\frac{1}{m_{k}}\\mathbf{11}^{\\top}][\\frac{\\partial S^{\\tau}}{\\partial x_{k}} ] = -\\tau [I - \\frac{1}{m_{k}}\\mathbf{11}^{\\top}]\\ln (x_{k})\\)"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "17"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.104,
+                0.353,
+                0.119
+            ],
+            "angle": 0,
+            "content": "A.1.5 BOUND ON LOSS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.129,
+                0.827,
+                0.173
+            ],
+            "angle": 0,
+            "content": "By Proposition 1, Equation (18d), we can also rewrite this loss as a weighted sum of 2-norms, \\(\\mathcal{L}(\\pmb{x}) = \\sum_{k}\\eta_{k}||\\nabla_{x_{k}}^{k} - \\mu_{k}||_{2}^{2}\\) where \\(\\mu_{k} = \\frac{1}{m_{k}} (\\mathbf{1}^{\\top}\\nabla_{x_{k}}^{k})\\in [0,1]\\) for brevity. This will allow us to more easily analyze our loss."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.175,
+                0.825,
+                0.21
+            ],
+            "angle": 0,
+            "content": "Lemma 7. Assume payoffs are bounded in \\([0,1]\\), then setting \\(\\eta_k \\leq \\frac{4}{nm_k}\\) or \\(\\eta_k \\leq \\frac{4}{nm}\\) or \\(\\sum_k \\eta_k \\leq \\frac{4}{m}\\) ensures \\(0 \\leq \\mathcal{L}(x) \\leq 1\\) for all \\(x \\in \\mathcal{X}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.225,
+                0.218,
+                0.24
+            ],
+            "angle": 0,
+            "content": "Proof."
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.329,
+                0.247,
+                0.826,
+                0.491
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} 0 \\leq \\mathcal {L} (\\boldsymbol {x}) = \\sum_ {k} \\eta_ {k} \\| \\nabla_ {x _ {k}} ^ {k} - \\mu_ {k} \\| _ {2} ^ {2} (28a) \\\\ = \\sum_ {k} \\eta_ {k} m _ {k} \\left[ \\frac {1}{m _ {k}} \\sum_ {l} \\left(\\nabla_ {x _ {k l}} ^ {k} - \\mu_ {k}\\right) ^ {2} \\right] (28b) \\\\ = \\sum_ {k} \\eta_ {k} m _ {k} \\operatorname {V a r} \\left[ \\nabla_ {x _ {k}} ^ {k} \\right] (28c) \\\\ \\leq \\frac {1}{4} \\sum_ {k} \\eta_ {k} m _ {k} (28d) \\\\ \\leq \\frac {1}{4} \\left(\\max  _ {k} \\eta_ {k}\\right) \\left(\\sum_ {k} m _ {k}\\right) (28e) \\\\ = \\frac {1}{4} \\left(\\max  _ {k} \\eta_ {k}\\right) n \\bar {m} \\leq 1 (28f) \\\\ \\Rightarrow \\left(\\max  _ {k} \\eta_ {k}\\right) \\leq \\frac {4}{n \\bar {m}} (28g) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.497,
+                0.827,
+                0.529
+            ],
+            "angle": 0,
+            "content": "where the first inequality follows from Popoviciu's inequality (Popoviciu, 1935): the variance of a bounded random variable \\( X \\) is upper bounded by \\( Var[X] \\leq \\frac{1}{4} (\\max_{X} - \\min_{X})^{2} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.542,
+                0.49,
+                0.558
+            ],
+            "angle": 0,
+            "content": "Next, we establish the following useful lemmas."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.56,
+                0.825,
+                0.592
+            ],
+            "angle": 0,
+            "content": "Lemma 8. The matrix \\( I - \\frac{1}{m_k} \\mathbf{11}^\\top \\) is a projection matrix and therefore idempotent. It is also symmetric, which implies it is its own square root."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.608,
+                0.218,
+                0.623
+            ],
+            "angle": 0,
+            "content": "Proof."
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.285,
+                0.629,
+                0.825,
+                0.727
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] ^ {\\top} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] = I - \\frac {2}{m _ {k}} \\mathbf {1 1} ^ {\\top} + \\frac {1}{m _ {k} ^ {2}} \\mathbf {1} \\left(\\mathbf {1} ^ {\\top} \\mathbf {1}\\right) \\mathbf {1} ^ {\\top} (29a) \\\\ = I - \\frac {2}{m _ {k}} \\mathbf {1 1} ^ {\\top} + \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} (29b) \\\\ = \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right]. (29c) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.756,
+                0.827,
+                0.801
+            ],
+            "angle": 0,
+            "content": "Lemma 9. The product \\( A[I_m - \\frac{1}{m}\\mathbf{1}_m\\mathbf{1}_m^\\top ]^p B \\) for any integer \\( p > 0 \\) has entries whose absolute value is bounded by \\( \\frac{m}{4} (A_{\\mathrm{max}} - A_{\\mathrm{min}})(B_{\\mathrm{max}} - B_{\\mathrm{min}}) \\) where \\( A_{\\mathrm{min}}, A_{\\mathrm{max}}, B_{\\mathrm{min}}, B_{\\mathrm{max}} \\) represent the minima and maxima of the matrices respectively."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.814,
+                0.825,
+                0.833
+            ],
+            "angle": 0,
+            "content": "Proof. The matrix \\([I - \\frac{1}{m}\\mathbf{11}^{\\top}]\\) is idempotent (Lemma 8) so we can rewrite the product for any \\(p\\) as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.396,
+                0.839,
+                0.825,
+                0.868
+            ],
+            "angle": 0,
+            "content": "\\[\nA \\left[ I - \\frac {1}{m} \\mathbf {1 1} ^ {\\top} \\right] \\left[ I - \\frac {1}{m} \\mathbf {1 1} ^ {\\top} \\right] B. \\tag {30}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.876,
+                0.826,
+                0.926
+            ],
+            "angle": 0,
+            "content": "The matrix \\([I - \\frac{1}{m}\\mathbf{11}^{\\top}]\\) has the property that it removes the mean from every row of a matrix when right multiplied against it, i.e., \\(A[I - \\frac{1}{m}\\mathbf{11}^{\\top}]\\) removes the means from the rows of \\(A\\). Similarly, left multiplying it removes the means from the columns. Let \\(\\tilde{A}\\) and \\(\\tilde{B}\\) represent these mean-centered"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "18"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.104,
+                0.825,
+                0.134
+            ],
+            "angle": 0,
+            "content": "results respectively. The absolute value of the \\( ij \\)th entry in the resulting product can then be recognized as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.278,
+                0.139,
+                0.826,
+                0.214
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\left| \\sum_ {k} \\tilde {A} _ {i k} \\tilde {B} _ {k j} \\right| = \\left| \\sum_ {k} \\left(A _ {i k} - \\frac {1}{m} \\sum_ {k ^ {\\prime}} A _ {i k ^ {\\prime}}\\right) \\left(B _ {k j} - \\frac {1}{m} \\sum_ {k ^ {\\prime}} B _ {k ^ {\\prime} j}\\right) \\right| (31a) \\\\ = \\left| m \\cdot \\operatorname {C o r r} \\left(A _ {i,}, B _ {. j}\\right) \\cdot \\sigma_ {A _ {i,}} \\sigma_ {B _ {. j}} \\right| (31b) \\\\ \\leq m \\sigma_ {A _ {i,}} \\sigma_ {B _ {\\cdot , j}}. (31c) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.218,
+                0.827,
+                0.276
+            ],
+            "angle": 0,
+            "content": "By Popoviciu's inequality (Popoviciu, 1935), we know the variance of a bounded random variable \\( X \\) is upper bounded by \\( Var[X] \\leq \\frac{1}{4} (\\max_{X} - \\min_{X})^{2} \\). Hence its standard deviation is bounded by \\( Std[X] \\leq \\frac{1}{2} (\\max_{X} - \\min_{X}) \\). Plugging these bounds for \\( A \\) and \\( B \\) into equation (31c) completes the claim."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.285,
+                0.541,
+                0.3
+            ],
+            "angle": 0,
+            "content": "Lemma 10. Assume payoffs are bounded in \\([0,1]\\), then"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.34,
+                0.305,
+                0.826,
+                0.337
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left| \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) \\right| \\leq \\frac {1}{4} \\left(\\max  _ {k} \\eta_ {k}\\right) n \\bar {m} \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) ^ {2} \\tag {32}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.169,
+                0.341,
+                0.421,
+                0.357
+            ],
+            "angle": 0,
+            "content": "for any \\( \\pmb{x} \\) such that \\( x_{kl} \\geq x_{\\min} \\forall k, l \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.372,
+                0.54,
+                0.387
+            ],
+            "angle": 0,
+            "content": "Proof. First note that for payoffs in [0, 1], the entries in"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.266,
+                0.393,
+                0.825,
+                0.413
+            ],
+            "angle": 0,
+            "content": "\\[\n\\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) = \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k} - \\tau (\\ln (x _ {k}) + 1)\\right) = \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k} - \\tau \\ln (x _ {k})\\right) \\tag {33}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.418,
+                0.825,
+                0.45
+            ],
+            "angle": 0,
+            "content": "are bounded within \\([0, \\tau \\ln(\\frac{1}{x_{\\min}}) + 1]\\) with a range of \\(\\tau \\ln(\\frac{1}{x_{\\min}}) + 1\\). Then starting from the definition of \\(\\mathcal{L}^{\\tau}\\) and applying Lemma 9, we find"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.327,
+                0.455,
+                0.826,
+                0.599
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\left| \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) \\right| = \\left| \\sum_ {k} \\eta_ {k} \\left\\| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right\\| ^ {2} \\right| (34a) \\\\ \\leq \\frac {1}{4} \\sum_ {k} \\eta_ {k} m _ {k} \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) ^ {2} (34b) \\\\ \\leq \\frac {1}{4} \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) ^ {2} \\left(\\max  _ {k} \\eta_ {k}\\right) \\sum_ {k} m _ {k} (34c) \\\\ = \\frac {1}{4} \\left(\\max  _ {k} \\eta_ {k}\\right) n \\bar {m} \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\min }}\\right) + 1\\right) ^ {2}. (34d) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.604,
+                0.825,
+                0.616
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.634,
+                0.573,
+                0.649
+            ],
+            "angle": 0,
+            "content": "A.2 QRES APPROXIMATE NES AT LOW TEMPERATURE"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.659,
+                0.825,
+                0.693
+            ],
+            "angle": 0,
+            "content": "Lemma 11. Setting \\(\\tau = \\ln (1 / p)^{-1}\\) with \\(p\\in (0,1)\\) and payoffs in [0,1] ensures that any QRE contains probabilities greater than \\(\\frac{p}{\\max_k m_k}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.707,
+                0.826,
+                0.76
+            ],
+            "angle": 0,
+            "content": "Proof. Let \\( m^{*} = \\max_{k} m_{k} \\) and \\( \\nabla_{x_{k}}^{k} \\) be player \\( k \\)'s gradient. Recall that a QRE(\\( \\tau \\)), specifically a logit equilibrium, satisfies the fixed point equation \\( x_{k} = \\text{softmax}\\left(\\frac{\\nabla_{x_{k}}^{k}}{\\tau}\\right) \\) for all \\( k \\). Then the probability for any action in a QRE strategy profile, \\( \\mathbf{x} \\), is lower bounded as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.297,
+                0.765,
+                0.826,
+                0.89
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} x _ {k l} \\geq \\min  _ {k} \\min  _ {l} \\left\\{x _ {k l} \\right\\} \\geq \\min  _ {\\boldsymbol {x} ^ {\\prime} \\mid \\boldsymbol {x} ^ {\\prime} \\text {i s Q R E}} \\min  _ {k} \\min  _ {l} \\left\\{x _ {k l} ^ {\\prime} \\right\\} (35a) \\\\ = \\min  _ {\\nabla_ {x _ {k}} ^ {k}} \\min  _ {k} \\min  _ {l} \\left[ \\operatorname {s o f t m a x} \\left(\\frac {\\nabla_ {x _ {k}} ^ {k}}{\\tau}\\right) \\right] _ {l} (35b) \\\\ = \\frac {e ^ {0}}{\\left(m ^ {*} - 1\\right) e ^ {\\frac {1}{\\tau}} + e ^ {0}} (35c) \\\\ \\stackrel {\\text {d e f}} {=} x _ {\\min } (35d) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.896,
+                0.825,
+                0.925
+            ],
+            "angle": 0,
+            "content": "where equation (35c) follows from minimizing the numerator and maximizing the denominator of the softmax formula subject to the payoff constraints."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "19"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.105,
+                0.356,
+                0.119
+            ],
+            "angle": 0,
+            "content": "Rearranging terms, we find"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.293,
+                0.122,
+                0.825,
+                0.165
+            ],
+            "angle": 0,
+            "content": "\\[\ne ^ {\\frac {1}{\\tau}} = \\frac {1}{m ^ {*} - 1} \\left(\\frac {1}{x _ {\\min }} - 1\\right) \\Rightarrow \\tau = \\frac {1}{\\ln \\left(\\frac {1}{m ^ {*} - 1} \\left(\\frac {1}{x _ {\\min }} - 1\\right)\\right)}. \\tag {36}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.175,
+                0.525,
+                0.193
+            ],
+            "angle": 0,
+            "content": "Let \\(p\\in (0,1)\\) such that \\(x_{\\mathrm{min}} = \\frac{p}{\\max_km_k} = \\frac{p}{m^*}\\), then"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.405,
+                0.197,
+                0.825,
+                0.361
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\tau = \\frac {1}{\\ln \\left(\\frac {1}{m ^ {*} - 1} \\left(\\frac {1}{x _ {\\operatorname* {m i n}}} - 1\\right)\\right)} (37a) \\\\ = \\frac {1}{\\ln \\left(\\frac {1}{m ^ {*} - 1} \\left(\\frac {m ^ {*}}{p} - 1\\right)\\right)} (37b) \\\\ = \\frac {1}{\\ln \\left(\\frac {m ^ {*} - p}{m ^ {*} - 1} \\frac {1}{p}\\right)} (37c) \\\\ \\leq \\frac {1}{\\ln \\left(\\frac {1}{p}\\right)}. (37d) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.365,
+                0.825,
+                0.399
+            ],
+            "angle": 0,
+            "content": "This implies if we set \\(\\tau = \\ln (1 / p)^{-1}\\), then we are guaranteed that all QREs contain probabilities greater than \\(x_{\\min} = \\frac{p}{\\max_k m_k}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.405,
+                0.678,
+                0.421
+            ],
+            "angle": 0,
+            "content": "Corollary 1. Assume payoffs are bounded in \\([0,1]\\) and \\(\\tau = \\ln (1 / p)^{-1}\\), then"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.353,
+                0.424,
+                0.825,
+                0.458
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left| \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) \\right| \\leq \\frac {1}{4} \\left(\\max  _ {k} \\eta_ {k}\\right) n \\bar {m} \\left(\\frac {\\ln \\left(m ^ {*}\\right)}{\\ln (1 / p)} + 2\\right) ^ {2}. \\tag {38}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.47,
+                0.609,
+                0.485
+            ],
+            "angle": 0,
+            "content": "Proof. Starting with Lemma 10 and applying Lemma 11, we find"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.288,
+                0.489,
+                0.825,
+                0.623
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\left| \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) \\right| \\leq \\frac {1}{4} \\left(\\max  _ {k} \\eta_ {k}\\right) n \\bar {m} \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\min }}\\right) + 1\\right) ^ {2} (39a) \\\\ = \\frac {1}{4} \\left(\\max  _ {k} \\eta_ {k}\\right) n \\bar {m} \\left(\\frac {1}{\\ln (1 / p)} \\ln \\left(\\frac {m ^ {*}}{p}\\right) + 1\\right) ^ {2} (39b) \\\\ = \\frac {1}{4} \\left(\\max  _ {k} \\eta_ {k}\\right) n \\bar {m} \\left(\\frac {1}{\\ln (1 / p)} \\left(\\ln \\left(m ^ {*}\\right) + \\ln (1 / p)\\right) + 1\\right) ^ {2} (39c) \\\\ = \\frac {1}{4} \\left(\\max  _ {k} \\eta_ {k}\\right) n \\bar {m} \\left(\\frac {\\ln \\left(m ^ {*}\\right)}{\\ln (1 / p)} + 2\\right) ^ {2}. (39d) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.648,
+                0.825,
+                0.677
+            ],
+            "angle": 0,
+            "content": "Lemma 12 (Low Temperature Approximate QREs are Approximate Nash Equilibria). Let \\(\\nabla_{x_k}^{k\\tau}\\) be player \\(k\\) 's entropy regularized gradient and \\(\\pmb{x}\\) be an approximate QRE. Then it holds that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.305,
+                0.681,
+                0.825,
+                0.701
+            ],
+            "angle": 0,
+            "content": "\\[\nu _ {k} \\left(B R _ {k}, x _ {- k}\\right) - u _ {k} (\\boldsymbol {x}) \\leq \\tau \\log \\left(m _ {k}\\right) + \\sqrt {2} \\left\\| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right\\|. \\tag {40}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.713,
+                0.583,
+                0.728
+            ],
+            "angle": 0,
+            "content": "Proof. Beginning with the definition of exploitability, we find"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.243,
+                0.731,
+                0.825,
+                0.866
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} u _ {k} \\left(\\mathbb {B R} _ {k}, x _ {- k}\\right) - u _ {k} (\\boldsymbol {x}) = \\left(u _ {k} \\left(\\mathbb {B R} _ {k}, x _ {- k}\\right) + \\tau S \\left(\\mathbb {B R} _ {k}\\right) - \\tau S \\left(\\mathbb {B R} _ {k}\\right)\\right) (41a) \\\\ - \\left(u _ {k} (\\boldsymbol {x}) + \\tau S (x _ {k}) - \\tau S (x _ {k})\\right) \\\\ = u _ {k} ^ {\\tau} \\left(\\mathbb {B R} _ {k}, x _ {- k}\\right) - u _ {k} ^ {\\tau} (\\boldsymbol {x}) + \\tau \\left(S \\left(x _ {k}\\right) - S \\left(\\mathbb {B R} _ {k}\\right)\\right) (41b) \\\\ \\leq \\max  _ {z \\in \\Delta^ {m _ {k}} ^ {- 1}} u _ {k} ^ {\\tau} (z, x _ {- k}) - u _ {k} ^ {\\tau} (\\boldsymbol {x}) + \\tau \\max  _ {z ^ {\\prime} \\in \\Delta^ {m _ {k}} ^ {- 1}} S \\left(z ^ {\\prime}\\right) (41c) \\\\ \\leq \\sqrt {2} \\| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\| + \\tau \\max  _ {z ^ {\\prime} \\in \\Delta^ {m _ {k} - 1}} S \\left(z ^ {\\prime}\\right) (41d) \\\\ \\leq \\sqrt {2} | | \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) | | + \\tau \\log (m _ {k}) (41e) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.869,
+                0.826,
+                0.926
+            ],
+            "angle": 0,
+            "content": "where the second equality follows from the definition of player \\(k\\)'s entropy regularized utility \\(u_{k}^{\\tau}\\), the first inequality from nonnegativity of entropy \\(S\\), the second inequality from concavity of \\(u_{k}^{\\tau}\\) with respect to its first argument (Lemma 2), and the last from the maximum possible value of Shannon entropy over distributions on \\(m_{k}\\) actions."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.51,
+                0.96
+            ],
+            "angle": 0,
+            "content": "20"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.169,
+                0.104,
+                0.825,
+                0.134
+            ],
+            "angle": 0,
+            "content": "Lemma 13 (\\(\\mathcal{L}^{\\tau}\\) Scores Nash Equilibria). Let \\(\\mathcal{L}^{\\tau}(\\pmb{x})\\) be our proposed entropy regularized loss function and \\(\\pmb{x}\\) be an approximate QRE. Then it holds that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.353,
+                0.144,
+                0.823,
+                0.181
+            ],
+            "angle": 0,
+            "content": "\\[\n\\epsilon \\leq \\tau \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {\\frac {2 n}{\\min _ {k} \\eta_ {k}}} \\sqrt {\\mathcal {L} ^ {\\tau} (\\boldsymbol {x})}. \\tag {42}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.205,
+                0.746,
+                0.221
+            ],
+            "angle": 0,
+            "content": "Proof. Beginning with the definition of exploitability and applying Lemma 12, we find"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.311,
+                0.231,
+                0.824,
+                0.397
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\epsilon = \\max  _ {k} u _ {k} (\\mathbb {B R} _ {k}, x _ {- k}) - u _ {k} (\\boldsymbol {x}) \\quad \\text {(r e c a l l e a c h} \\epsilon_ {k} \\geq 0) (43a) \\\\ \\leq \\sum_ {k} u _ {k} (\\mathbb {R} _ {k}, x _ {- k}) - u _ {k} (\\boldsymbol {x}) (43b) \\\\ \\leq \\sum_ {k} \\left[ \\tau \\log \\left(m _ {k}\\right) + \\sqrt {2} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right| \\right| \\right] (43c) \\\\ = \\tau \\sum_ {k} \\log \\left(m _ {k}\\right) + \\sum_ {k} \\sqrt {2} \\left\\| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right\\| (43d) \\\\ \\leq \\tau \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {\\frac {2 n}{\\min  _ {k} \\eta_ {k}}} \\sqrt {\\mathcal {L} ^ {\\tau} (\\boldsymbol {x})}. (43e) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.407,
+                0.825,
+                0.437
+            ],
+            "angle": 0,
+            "content": "where the last inequality follows from the same steps (24c)-(24i) outlined in Lemma 4, which established the relationship between \\(\\mathcal{L}(\\pmb{x})\\) and \\(\\epsilon\\)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.456,
+                0.359,
+                0.47
+            ],
+            "angle": 0,
+            "content": "A.3 GRADIENT OF LOSS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.483,
+                0.667,
+                0.499
+            ],
+            "angle": 0,
+            "content": "Lemma 14. The gradient of \\(\\mathcal{L}^{\\tau}(\\pmb{x})\\) with respect to player \\(l\\)'s strategy \\(x_{l}\\) is"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.378,
+                0.509,
+                0.824,
+                0.54
+            ],
+            "angle": 0,
+            "content": "\\[\n\\nabla_ {x _ {l}} \\mathcal {L} (\\boldsymbol {x}) = 2 \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\tag {44}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.551,
+                0.705,
+                0.571
+            ],
+            "angle": 0,
+            "content": "where \\(B_{ll} = -\\tau \\left[I - \\frac{1}{m_l}\\mathbf{11}^\\top \\right]diag\\left(\\frac{1}{x_l}\\right)\\) and \\(B_{kl} = \\left[I - \\frac{1}{m_k}\\mathbf{11}^\\top \\right]H_{kl}^k\\) for \\(k\\neq l\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.594,
+                0.789,
+                0.611
+            ],
+            "angle": 0,
+            "content": "Proof. Recall from Lemma 6 that the loss can be decomposed as \\(\\mathcal{L}^{\\tau}(\\pmb{x}) = (A) + (B) + (C)\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.616,
+                0.211,
+                0.629
+            ],
+            "angle": 0,
+            "content": "Then"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.301,
+                0.638,
+                0.825,
+                0.671
+            ],
+            "angle": 0,
+            "content": "\\[\nD _ {x _ {l}} [ (A) ] = D _ {x _ {l}} \\left[ \\sum_ {k} \\eta_ {k} x _ {q} ^ {\\top} B _ {k q} ^ {\\top} B _ {k q} x _ {q} \\right] = 2 \\sum_ {k \\neq l} \\eta_ {k} B _ {k l} ^ {\\top} B _ {k l} x _ {l} \\tag {45}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.682,
+                0.621,
+                0.701
+            ],
+            "angle": 0,
+            "content": "where \\(q \\neq k\\) and \\(B_{kq} = [I - \\frac{1}{m_k}\\mathbf{11}^\\top][H_{kq}^k]\\) does not depend on \\(x_{k}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.707,
+                0.488,
+                0.727
+            ],
+            "angle": 0,
+            "content": "Also, letting \\(B_{ll} = -\\tau [I - \\frac{1}{m_l}\\mathbf{11}^\\top ]\\mathrm{diag}\\left(\\frac{1}{x_l}\\right)\\)"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.217,
+                0.737,
+                0.824,
+                0.921
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} D _ {x _ {l}} [ (B) ] = D _ {x _ {l}} \\left[ - 2 \\tau \\sum_ {k} \\eta_ {k} \\ln \\left(x _ {k}\\right) ^ {\\top} B _ {k q} x _ {q} \\right] (46a) \\\\ = - 2 \\tau \\left[ \\eta_ {l} D _ {x _ {l}} \\left[ \\ln \\left(x _ {l}\\right) ^ {\\top} B _ {l q} x _ {q} \\right] + \\sum_ {k \\neq l} \\eta_ {k} D _ {x _ {l}} \\left[ \\ln \\left(x _ {k}\\right) ^ {\\top} B _ {k l} x _ {l} \\right] \\right] (46b) \\\\ = - 2 \\tau \\left[ \\eta_ {l} \\operatorname {d i a g} \\left(\\frac {1}{x _ {l}}\\right) B _ {l q} x _ {q} + \\sum_ {k \\neq l} \\eta_ {k} B _ {k l} ^ {\\top} \\ln \\left(x _ {k}\\right) \\right] (46c) \\\\ = - 2 \\tau \\left[ \\eta_ {l} \\left(\\left[ I - \\frac {1}{m _ {l}} \\mathbf {1 1} ^ {\\top} \\right] \\operatorname {d i a g} \\left(\\frac {1}{x _ {l}}\\right)\\right) ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla^ {l}\\right) + \\sum_ {k \\neq l} \\eta_ {k} B _ {k l} ^ {\\top} \\ln \\left(x _ {k}\\right) \\right] (46d) \\\\ = 2 \\left[ \\eta_ {l} B _ {l l} ^ {\\top} \\Pi_ {T \\Delta} (\\nabla^ {l}) - \\tau \\sum_ {k \\neq l} \\eta_ {k} B _ {k l} ^ {\\top} \\ln (x _ {k}) \\right]. (46e) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "21"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.105,
+                0.206,
+                0.119
+            ],
+            "angle": 0,
+            "content": "And"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.281,
+                0.134,
+                0.825,
+                0.256
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} D _ {x _ {l}} [ (C) ] = D _ {x _ {l}} \\left[ \\sum_ {k} \\eta_ {k} \\tau^ {2} \\ln \\left(x _ {k}\\right) ^ {\\top} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] \\ln \\left(x _ {k}\\right) \\right] (47a) \\\\ = 2 \\tau^ {2} \\left[ \\eta_ {l} \\operatorname {d i a g} \\left(\\frac {1}{x _ {l}}\\right) \\left[ I - \\frac {1}{m _ {l}} \\mathbf {1 1} ^ {\\top} \\right] \\ln \\left(x _ {l}\\right) \\right] (47b) \\\\ = - 2 \\tau \\eta_ {l} \\left(\\left[ I - \\frac {1}{m _ {l}} \\mathbf {1 1} ^ {\\top} \\right] \\operatorname {d i a g} \\left(\\frac {1}{x _ {l}}\\right)\\right) ^ {\\top} \\Pi_ {T \\Delta} (- \\tau \\ln (x _ {l})) (47c) \\\\ = 2 \\eta_ {l} B _ {l l} ^ {\\top} \\Pi_ {T \\Delta} (- \\tau \\ln (x _ {l})). (47d) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.281,
+                0.377,
+                0.296
+            ],
+            "angle": 0,
+            "content": "Putting these together, we find"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.197,
+                0.314,
+                0.826,
+                0.42
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\nabla_ {x _ {l}} \\mathcal {L} (\\boldsymbol {x}) = 2 \\sum_ {k \\neq l} \\eta_ {k} B _ {k l} ^ {\\top} \\left(B _ {k l} x _ {l} - \\tau \\ln \\left(x _ {k}\\right)\\right) + 2 \\eta_ {l} B _ {l l} ^ {\\top} \\left[ \\Pi_ {T \\Delta} \\left(\\nabla^ {l}\\right) + \\Pi_ {T \\Delta} (- \\tau \\ln \\left(x _ {l}\\right)) \\right] (48a) \\\\ = 2 \\eta_ {l} B _ {l l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) + 2 \\sum_ {k \\neq l} \\eta_ {k} B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) (48b) \\\\ = 2 \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right). (48c) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.446,
+                0.825,
+                0.458
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.51,
+                0.402,
+                0.525
+            ],
+            "angle": 0,
+            "content": "A.3.1 UNBIASED ESTIMATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.54,
+                0.827,
+                0.584
+            ],
+            "angle": 0,
+            "content": "In order to construct an unbiased estimate for each \\( B_{kl} \\), we will need to form an independent unbiased estimate of \\( H_{kl}^{k} \\). Recall that \\( H_{kl}^{k} \\) is simply the expected bimatrix game between players \\( k \\) and \\( l \\) when all other players sample their actions according to their current strategies."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.611,
+                0.544,
+                0.625
+            ],
+            "angle": 0,
+            "content": "A.4 BOUND ON GRADIENT / LIPSCHITZ PROPERTY"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.642,
+                0.827,
+                0.67
+            ],
+            "angle": 0,
+            "content": "Lemma 15. Assume payoffs are upper bounded by 1, then the infinity norm of the gradient is bounded as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.247,
+                0.685,
+                0.826,
+                0.716
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left| \\left| \\nabla_ {\\boldsymbol {x}} \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) \\right| \\right| _ {\\infty} \\leq \\frac {1}{2} \\left(\\max  _ {k} \\eta_ {k}\\right) \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) \\left[ \\tau m ^ {*} \\left(\\frac {1}{x _ {\\operatorname* {m i n}}} - 1\\right) + n \\bar {m} \\right]. \\tag {49}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.766,
+                0.798,
+                0.781
+            ],
+            "angle": 0,
+            "content": "Proof. Recall from Lemma 14 that the gradient of \\(\\mathcal{L}(\\pmb{x})\\) with respect to player \\(l\\)'s strategy \\(x_{l}\\) is"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.378,
+                0.799,
+                0.826,
+                0.831
+            ],
+            "angle": 0,
+            "content": "\\[\n\\nabla_ {x _ {l}} \\mathcal {L} (\\boldsymbol {x}) = 2 \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\tag {50}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.849,
+                0.706,
+                0.871
+            ],
+            "angle": 0,
+            "content": "where \\(B_{ll} = -\\tau [I - \\frac{1}{m_l}\\mathbf{11}^\\top ]\\mathrm{diag}\\left(\\frac{1}{x_l}\\right)\\) and \\(B_{kl} = [I - \\frac{1}{m_k}\\mathbf{11}^\\top ]H_{kl}^k\\) for \\(k\\neq l\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.875,
+                0.825,
+                0.93
+            ],
+            "angle": 0,
+            "content": "As noted before in the derivation of Lemma 10, for payoffs in \\([0,1]\\), the entries in \\(\\nabla_{x_k}^{k\\tau} = \\nabla_{x_k}^k - \\tau \\ln(x_k)\\) are bounded within \\([0, \\tau \\ln(\\frac{1}{x_{\\min}}) + 1]\\) with a range \\(\\tau \\ln(\\frac{1}{x_{\\min}}) + 1\\). Similarly, the entries in \\(-\\tau \\mathrm{diag}\\left(\\frac{1}{x_l}\\right)\\) are bounded within \\([- \\tau \\frac{1}{x_{\\min}}, -\\tau]\\) with a range of \\(\\tau\\left(\\frac{1}{x_{\\min}} - 1\\right)\\)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.51,
+                0.961
+            ],
+            "angle": 0,
+            "content": "22"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.104,
+                0.548,
+                0.119
+            ],
+            "angle": 0,
+            "content": "The infinity norm of the gradient can then be bounded as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.185,
+                0.122,
+                0.825,
+                0.382
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\left| \\left| \\nabla_ {\\boldsymbol {x}} \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) \\right| \\right| _ {\\infty} = \\max  _ {l} \\left| \\left| \\nabla_ {x _ {l}} \\mathcal {L} (\\boldsymbol {x}) \\right| \\right| _ {\\infty} (51a) \\\\ = \\max  _ {l} \\left\\| 2 \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right\\| _ {\\infty} (51b) \\\\ \\leq 2 \\sum_ {k} \\eta_ {k} \\max  _ {l} | | B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) | | _ {\\infty} (51c) \\\\ \\leq \\frac {1}{2} \\sum_ {k \\neq l ^ {*}} \\eta_ {k} m _ {k} \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) + \\frac {1}{2} \\eta_ {l ^ {*}} m _ {l ^ {*}} \\tau \\left(\\frac {1}{x _ {\\operatorname* {m i n}}} - 1\\right) \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) (51d) \\\\ = \\frac {1}{2} \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) \\left[ \\eta_ {l ^ {*}} m _ {l ^ {*}} \\tau \\left(\\frac {1}{x _ {\\operatorname* {m i n}}} - 1\\right) + \\sum_ {k \\neq l ^ {*}} \\eta_ {k} m _ {k} \\right] (51e) \\\\ \\leq \\frac {1}{2} \\left(\\max  _ {k} \\eta_ {k}\\right) \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) \\left[ \\tau m _ {l ^ {*}} \\left(\\frac {1}{x _ {\\operatorname* {m i n}}} - 1\\right) + \\sum_ {k \\neq l ^ {*}} m _ {k} \\right] (51f) \\\\ \\leq \\frac {1}{2} \\left(\\max  _ {k} \\eta_ {k}\\right) \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) \\left[ \\tau m ^ {*} \\left(\\frac {1}{x _ {\\operatorname* {m i n}}} - 1\\right) + n \\bar {m} \\right] (51g) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.385,
+                0.518,
+                0.399
+            ],
+            "angle": 0,
+            "content": "where the second inequality follows from Lemma 9."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.406,
+                0.824,
+                0.418
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.427,
+                0.825,
+                0.457
+            ],
+            "angle": 0,
+            "content": "Corollary 2. If \\(\\tau\\) is set according to Lemma 11 as \\(\\tau = \\ln (1 / p)^{-1}\\) and payoffs are in [0, 1], then the infinity norm of the gradient is bounded as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.235,
+                0.46,
+                0.825,
+                0.495
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left| \\left| \\nabla_ {\\boldsymbol {x}} \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) \\right| \\right| _ {\\infty} \\leq \\frac {1}{2} \\left(\\max  _ {k} \\eta_ {k}\\right) \\left[ \\frac {\\ln \\left(m ^ {*}\\right)}{\\ln (1 / p)} + 2 \\right] \\left[ \\frac {m ^ {* 2}}{p \\ln (1 / p)} + n \\bar {m} \\right] = \\frac {1}{2} \\left(\\max  _ {k} \\eta_ {k}\\right) \\hat {L} \\tag {52}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.5,
+                0.746,
+                0.515
+            ],
+            "angle": 0,
+            "content": "where \\(m^{*} = \\max_{k} m_{k}\\) and \\(\\hat{L}\\) is defined implicitly for convenience in other derivations."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.529,
+                0.825,
+                0.559
+            ],
+            "angle": 0,
+            "content": "Proof. Starting with Lemma 15 and applying Lemma 11 (i.e., \\(\\tau = \\ln(1/p)^{-1}\\) and \\(x_{\\min} = \\frac{p}{m^*}\\) with payoffs in [0, 1] and where \\(m^* = \\max_k m_k\\)), we find"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.251,
+                0.563,
+                0.825,
+                0.664
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\left| \\left| \\nabla_ {\\boldsymbol {x}} \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) \\right| \\right| _ {\\infty} \\leq \\frac {1}{2} \\left(\\max  _ {k} \\eta_ {k}\\right) \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) \\left[ \\tau m ^ {*} \\left(\\frac {1}{x _ {\\operatorname* {m i n}}} - 1\\right) + n \\bar {m} \\right] (53a) \\\\ = \\frac {1}{2} \\left(\\max  _ {k} \\eta_ {k}\\right) \\left[ \\frac {\\ln \\left(m ^ {*} / p\\right)}{\\ln (1 / p)} + 1 \\right] \\left[ \\frac {m ^ {*}}{\\ln (1 / p)} \\left(\\frac {m ^ {*}}{p} - 1\\right) + n \\bar {m} \\right] (53b) \\\\ \\leq \\frac {1}{2} \\left(\\max  _ {k} \\eta_ {k}\\right) \\left[ \\frac {\\ln \\left(m ^ {*}\\right)}{\\ln (1 / p)} + 2 \\right] \\left[ \\frac {m ^ {* 2}}{p \\ln (1 / p)} + n \\bar {m} \\right]. (53c) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.675,
+                0.827,
+                0.734
+            ],
+            "angle": 0,
+            "content": "As \\( p \\to 0^+ \\), the norm of the gradient blows up because the gradient of Shannon entropy blows up for small probabilities. As \\( p \\to 1 \\), the norm of the gradient blows up because we require infinite temperature \\( \\tau \\) to guarantee all QREs are nearly uniform; recall \\( \\tau \\) is the regularization coefficient on the entropy bonus terms which means our modified utilities blow up for large \\( \\tau \\)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.748,
+                0.347,
+                0.761
+            ],
+            "angle": 0,
+            "content": "A.5 HESSIAN OF LOSS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.774,
+                0.827,
+                0.847
+            ],
+            "angle": 0,
+            "content": "We will now derive the Hessian of our loss. This will be useful in establishing properties about global minima that enable the application of tailored minimization algorithms. Let \\( D_z[f(z)] \\) denote the differential operator applied to (possibly multivalued) function \\( f \\) with respect to \\( z \\). For example, \\( D_{x_q}[H_{lk}^k] = D_{x_q}[x_qT_{qlk}^k] = T_{qlk}^k \\) where \\( T_{qlk}^k \\) is player \\( k \\)'s payoff tensor according to the three-way approximation between players \\( k, l \\), and \\( q \\) to the game at \\( x \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.848,
+                0.507,
+                0.864
+            ],
+            "angle": 0,
+            "content": "Lemma 16. The Hessian of \\(\\mathcal{L}^{\\tau}(\\pmb{x})\\) can be written"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.377,
+                0.867,
+                0.825,
+                0.886
+            ],
+            "angle": 0,
+            "content": "\\[\nH e s s \\left(\\mathcal {L} ^ {\\tau}\\right) = 2 \\tilde {B} ^ {\\top} \\tilde {B} + T \\Pi_ {T \\Delta} \\left(\\tilde {\\nabla} ^ {\\tau}\\right) \\tag {54}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.891,
+                0.825,
+                0.929
+            ],
+            "angle": 0,
+            "content": "where \\(\\tilde{B}_{kl} = \\sqrt{\\eta_k} B_{kl}\\), \\(\\Pi_{T\\Delta}(\\tilde{\\nabla}^{\\tau}) = [\\eta_1\\Pi_{T\\Delta}(\\nabla_{x_1}^{1\\tau}),\\dots ,\\eta_n\\Pi_{T\\Delta}(\\nabla_{x_n}^{n\\tau})]\\), and we augment \\(T\\) (the 3-player tensor approximation to the game, \\(T_{lqk}^{k}\\)) so that \\(T_{ll}^{l} = \\tau \\text{diag} 3\\left(\\frac{1}{x_{l}^{2}}\\right)\\) and otherwise 0."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.961
+            ],
+            "angle": 0,
+            "content": "23"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.104,
+                0.49,
+                0.12
+            ],
+            "angle": 0,
+            "content": "Proof. Recall the gradient of our proposed loss:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.378,
+                0.149,
+                0.825,
+                0.181
+            ],
+            "angle": 0,
+            "content": "\\[\n\\nabla_ {x _ {l}} \\mathcal {L} (\\boldsymbol {x}) = 2 \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\tag {55}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.209,
+                0.704,
+                0.231
+            ],
+            "angle": 0,
+            "content": "where \\(B_{ll} = -\\tau [I - \\frac{1}{m_l}\\mathbf{11}^\\top ]\\mathrm{diag}\\left(\\frac{1}{x_l}\\right)\\) and \\(B_{kl} = [I - \\frac{1}{m_k}\\mathbf{11}^\\top ]H_{kl}^k\\) for \\(k\\neq l\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.234,
+                0.819,
+                0.25
+            ],
+            "angle": 0,
+            "content": "Consider the following Jacobians, which will play an auxiliary role in our derivation of the Hessian:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.377,
+                0.279,
+                0.825,
+                0.311
+            ],
+            "angle": 0,
+            "content": "\\[\nD _ {l} \\left[ B _ {l l} \\right] = \\tau \\left[ I - \\frac {1}{m _ {l}} \\mathbf {1 1} ^ {\\top} \\right] \\operatorname {d i a g} 3 \\left(\\frac {1}{x _ {l} ^ {2}}\\right) \\tag {56a}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.374,
+                0.313,
+                0.825,
+                0.329
+            ],
+            "angle": 0,
+            "content": "\\[\nD _ {q} \\left[ B _ {l l} \\right] = 0 \\tag {56b}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.374,
+                0.331,
+                0.825,
+                0.346
+            ],
+            "angle": 0,
+            "content": "\\[\nD _ {l} \\left[ B _ {k l} \\right] = 0 \\tag {56c}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.371,
+                0.349,
+                0.825,
+                0.38
+            ],
+            "angle": 0,
+            "content": "\\[\nD _ {q} \\left[ B _ {k l} \\right] = \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] T _ {k l q} ^ {k} \\tag {56d}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.32,
+                0.382,
+                0.825,
+                0.494
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} D _ {k} \\left[ \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right] = \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] D _ {k} \\left[ \\nabla_ {x _ {k}} ^ {k \\tau} \\right] (56e) \\\\ = \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] D _ {k} \\left[ \\nabla_ {x _ {k}} ^ {k} - \\tau \\ln \\left(x _ {k}\\right) \\right] (56f) \\\\ = \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] \\left[ - \\tau \\operatorname {d i a g} \\left(\\frac {1}{x _ {k}}\\right) \\right] (56g) \\\\ = B _ {k k} (56h) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.324,
+                0.497,
+                0.825,
+                0.609
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} D _ {l} \\left[ \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right] = \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] D _ {l} \\left[ \\nabla_ {x _ {k}} ^ {k \\tau} \\right] (56i) \\\\ = \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] D _ {l} \\left[ \\nabla_ {x _ {k}} ^ {k} - \\tau \\ln (x _ {k}) \\right] (56j) \\\\ = \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] \\left[ H _ {k l} ^ {k} \\right] (56k) \\\\ = B _ {k l}. (561) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.646,
+                0.515,
+                0.661
+            ],
+            "angle": 0,
+            "content": "We can derive the diagonal blocks of the Hessian as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.196,
+                0.69,
+                0.825,
+                0.922
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} D _ {l l} [ \\mathcal {L} (\\boldsymbol {x}) ] = D _ {l} \\left[ \\nabla_ {x _ {l}} \\mathcal {L} (\\boldsymbol {x}) \\right] (57a) \\\\ = 2 D _ {l} \\left[ \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right] (57b) \\\\ = 2 \\left[ \\eta_ {l} D _ {l} \\left[ B _ {l l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {l}} ^ {l \\tau}\\right) \\right] + \\sum_ {k \\neq l} \\eta_ {k} D _ {l} \\left[ B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right] \\right] (57c) \\\\ = 2 \\left[ \\eta_ {l} \\left[ D _ {l} \\left[ B _ {l l} \\right] ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {l}} ^ {l \\tau}\\right) + B _ {l l} ^ {\\top} D _ {l} \\left[ \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {l}} ^ {l \\tau}\\right) \\right] \\right] \\right. (57d) \\\\ \\left. + \\sum_ {k \\neq l} \\eta_ {k} \\left[ D _ {t} \\left[ B _ {k l} \\right] ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) + B _ {k l} ^ {\\top} D _ {l} \\left[ \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right] \\right] \\right] (57e) \\\\ = 2 \\left[ \\eta_ {l} \\left[ \\tau \\operatorname {d i a g} 3 \\left(\\frac {1}{x _ {l} ^ {2}}\\right) [ I - \\frac {1}{m _ {l}} \\mathbf {1 1} ^ {\\top} ] \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {l}} ^ {l \\tau}\\right) + B _ {l l} ^ {\\top} B _ {l l} \\right] + \\sum_ {k \\neq l} \\eta_ {k} B _ {k l} ^ {\\top} B _ {k l} \\right] (57f) \\\\ = 2 \\left[ \\tau \\eta_ {l} \\operatorname {d i a g} \\left(\\left[ \\frac {1}{x _ {l} ^ {2}} \\right] \\odot \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {l}} ^ {l \\tau}\\right)\\right) + \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} B _ {k l} \\right] (57g) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "24"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.105,
+                0.375,
+                0.12
+            ],
+            "angle": 0,
+            "content": "and the off-diagonal blocks as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.223,
+                0.127,
+                0.825,
+                0.357
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} D _ {l q} [ \\mathcal {L} (\\boldsymbol {x}) ] = D _ {q} \\left[ \\nabla_ {x _ {l}} \\mathcal {L} (\\boldsymbol {x}) \\right] (58a) \\\\ = 2 D _ {q} \\left[ \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right] (58b) \\\\ = 2 \\left[ \\eta_ {l} D _ {q} \\left[ B _ {l l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {l}} ^ {l \\tau}\\right) \\right] + \\sum_ {k \\neq l} \\eta_ {k} D _ {q} \\left[ B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right] \\right] (58c) \\\\ = 2 \\left[ \\eta_ {l} \\left[ D _ {q} \\left[ B _ {\\bar {l} l} \\right] ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {l}} ^ {l \\tau}\\right) + B _ {l l} ^ {\\top} D _ {q} \\left[ \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {l}} ^ {l \\tau}\\right) \\right] \\right] \\right. (58d) \\\\ \\left. + \\sum_ {k \\neq l} \\eta_ {k} \\left[ D _ {q} \\left[ B _ {k l} \\right] ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) + B _ {k l} ^ {\\top} D _ {q} \\left[ \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right] \\right] \\right] (58e) \\\\ = 2 \\left[ \\eta_ {l} B _ {l l} ^ {\\top} B _ {l q} + \\sum_ {k \\neq l} \\eta_ {k} \\left[ T _ {l q k} ^ {k} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) + B _ {k l} ^ {\\top} B _ {k q} \\right] \\right] (58f) \\\\ = 2 \\left[ \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} B _ {k q} + \\sum_ {k \\neq l} \\eta_ {k} T _ {l q k} ^ {k} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right]. (58g) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.372,
+                0.506,
+                0.387
+            ],
+            "angle": 0,
+            "content": "Therefore, the Hessian can be written concisely as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.417,
+                0.394,
+                0.825,
+                0.414
+            ],
+            "angle": 0,
+            "content": "\\[\n2 \\left[ \\tilde {B} ^ {\\top} \\tilde {B} + T \\Pi_ {T \\Delta} \\left(\\tilde {\\nabla} ^ {\\tau}\\right) \\right] \\tag {59}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.421,
+                0.825,
+                0.459
+            ],
+            "angle": 0,
+            "content": "where \\(\\tilde{B}_{kl} = \\sqrt{\\eta_k} B_{kl}\\), \\(\\Pi_{T\\Delta}(\\tilde{\\nabla}^{\\tau}) = [\\eta_1\\Pi_{T\\Delta}(\\nabla_{x_1}^{1\\tau}),\\dots ,\\eta_n\\Pi_{T\\Delta}(\\nabla_{x_n}^{n\\tau})]\\), and we augment \\(T\\) (the 3-player tensor approximation to the game, \\(T_{lqk}^{k}\\)) so that \\(T_{lll}^{l} = \\tau \\mathrm{diag}3\\left(\\frac{1}{x_{l}^{2}}\\right)\\) and otherwise 0."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.465,
+                0.824,
+                0.476
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "25"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.103,
+                0.538,
+                0.119
+            ],
+            "angle": 0,
+            "content": "B GLOBAL CONVERGENCE GUARANTEES"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.135,
+                0.825,
+                0.163
+            ],
+            "angle": 0,
+            "content": "In this section, we analyze the application of optimization techniques such as bandit algorithms and gradient descent to our loss function."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.181,
+                0.557,
+                0.195
+            ],
+            "angle": 0,
+            "content": "B.1 MAPS FROM HYPERCUBE TO SIMPLEX PRODUCT"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.208,
+                0.827,
+                0.277
+            ],
+            "angle": 0,
+            "content": "In this subsection, we derive properties of a map \\(s\\) from the unit-hypercube to the simplex product. This map is necessary to adapt our proposed loss \\(\\mathcal{L}^{\\tau}\\) to the commonly assumed setting in the \\(\\mathcal{X}\\)-armed bandit literature (Bubeck et al., 2011) where the feasible set is a hypercube. We derive relevant properties of two such maps: the softmax and a mapping that interprets dimensions of the hypercube as angles on a unit-sphere that are then \\(\\ell_1\\)-normalized."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.281,
+                0.75,
+                0.298
+            ],
+            "angle": 0,
+            "content": "Lemma 17. Let \\( f(x) = -\\mathcal{L}(s(x)) \\). Then \\( ||\\nabla f(x)||_{\\infty} \\leq ||J(s(x))^{\\top}||_{\\infty}||\\nabla \\mathcal{L}(s(x))||_{\\infty} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.315,
+                0.218,
+                0.329
+            ],
+            "angle": 0,
+            "content": "Proof."
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.254,
+                0.337,
+                0.825,
+                0.356
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left. \\right.\\left|\\left| \\nabla f (x) \\right|\\right| _ {\\infty} = \\left|\\left| J (s (x)) ^ {\\top} \\nabla \\mathcal {L} (s (x)) \\right|\\right| _ {\\infty} \\leq \\left|\\left| J (s (x)) ^ {\\top} \\right|\\right| _ {\\infty} \\left|\\left| \\nabla \\mathcal {L} (s (x)) \\right|\\right| _ {\\infty}. \\tag {60}\n\\]"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.364,
+                0.824,
+                0.375
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.169,
+                0.388,
+                0.825,
+                0.432
+            ],
+            "angle": 0,
+            "content": "Lemma 18. The \\(\\infty\\)-norm of the Jacobian-transpose of a transformation \\(s(x)\\) applied elementwise to a product space is bounded by the \\(\\infty\\)-norm of the Jacobian-transpose of a single transformation from that product space, i.e., \\(\\|J(s(\\pmb{x}))^\\top\\|_\\infty \\leq \\max_{x_i \\in \\mathcal{X}_i} \\|J(s(x_i))^\\top\\|_\\infty\\) for any \\(i\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.447,
+                0.827,
+                0.492
+            ],
+            "angle": 0,
+            "content": "Proof. Let \\( \\boldsymbol{x} \\in \\mathcal{X} = \\prod_{i=1}^{n} \\mathcal{X}_i \\), \\( \\mathcal{Z} = \\prod_{i=1}^{n} \\mathcal{Z}_i \\) and \\( S: \\mathcal{X} \\to \\mathcal{Z} = [s(x_1); \\dots; s(x_n)]^\\top \\) where; denotes column-wise stacking, \\( x_i \\in \\mathcal{X}_i \\). Also, \\( \\mathcal{X}_i = \\mathcal{X}_j \\) and \\( \\mathcal{Z}_i = \\mathcal{Z}_j \\) for all \\( i \\) and \\( j \\). Then the Jacobian of \\( S(\\boldsymbol{x}) \\) is"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.301,
+                0.498,
+                0.825,
+                0.572
+            ],
+            "angle": 0,
+            "content": "\\[\nJ (S (\\boldsymbol {x})) ^ {\\top} = \\left[ \\begin{array}{c c c} J (s (x _ {1})) ^ {\\top} & 0 \\dots & 0 \\\\ 0 & J (s (x _ {2})) ^ {\\top} \\dots & 0 \\\\ 0 & 0 \\ddots & 0 \\\\ 0 & 0 \\dots & J (s (x _ {n})) ^ {\\top} \\end{array} \\right]. \\tag {61}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.578,
+                0.827,
+                0.648
+            ],
+            "angle": 0,
+            "content": "The \\(\\infty\\)-norm of this matrix is the max 1-norm of any row. This matrix is diagonal, therefore, the \\(\\infty\\)-norm of each elementwise Jacobian-transpose represents the max 1-norm of the rows spanned by its block. Given that the domains, ranges, and transformations \\(s\\) for all blocks are the same, their \\(\\infty\\)-norms are also the same. The max \\(\\infty\\) over the blocks is then equal to the \\(\\infty\\)-norm of any individual \\(J(s(x_i))^\\top\\)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.665,
+                0.521,
+                0.68
+            ],
+            "angle": 0,
+            "content": "B.1.1 HESSIAN OF BANDIT REWARD FUNCTION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.69,
+                0.827,
+                0.757
+            ],
+            "angle": 0,
+            "content": "Lemma 19. Let \\( s(x) \\) be a function that maps the unit hypercube to the simplex product (mixed strategy space). Then the objective function \\( f(x) = -\\mathcal{L}(s(x)) \\). The Hessian of \\( -f(x) \\) at an optimum \\( x^{*} \\) in direction \\( \\Delta \\) is \\( \\Delta x^{\\top}[Ds(x)^{\\top}H_{\\mathcal{L}}(x)Ds(x)]\\Big|_{x^{*}}\\Delta x \\) where \\( H_{\\mathcal{L}} \\) is the Hessian of \\( \\mathcal{L} \\) and \\( Ds(x) \\) is the Jacobian of \\( s(x) \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.776,
+                0.218,
+                0.79
+            ],
+            "angle": 0,
+            "content": "Proof."
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.268,
+                0.798,
+                0.826,
+                0.904
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\left. \\left(D ^ {2} (\\mathcal {L} \\circ s) \\left(x ^ {*}\\right)\\right) (\\Delta x, \\Delta x) = \\Delta x ^ {\\top} \\left[ \\sum_ {i} \\overbrace {\\partial_ {i} \\mathcal {L} (s (x))} ^ {= 0 \\text {a t} x = x ^ {*}} D ^ {2} h _ {i} (x) \\right] \\right| _ {x ^ {*}} \\Delta x (62a) \\\\ + \\Delta x ^ {\\top} [ D s (x) ^ {\\top} H _ {\\mathcal {L}} (x) D s (x) ] \\bigg | _ {x ^ {*}} \\Delta x \\\\ = \\Delta x ^ {\\top} \\left[ D s (x) ^ {\\top} H _ {\\mathcal {L}} (x) D s (x) \\right] \\Big | _ {x ^ {*}} \\Delta x. (62b) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.911,
+                0.824,
+                0.922
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "26"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.169,
+                0.104,
+                0.827,
+                0.146
+            ],
+            "angle": 0,
+            "content": "Lemma 20. Let \\( s(x): \\mathcal{X} \\to \\prod_k \\Delta^{m_k - 1} \\) be an injective function, i.e., \\( x \\neq y \\Rightarrow s(x) \\neq s(y) \\). Also let \\( J = J(s(x)) \\) be the Jacobian of \\( s \\) with respect to \\( x \\) and \\( \\Delta x \\) be a nonzero vector in the tangent space of \\( \\mathcal{X} \\). Then"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.462,
+                0.149,
+                0.825,
+                0.164
+            ],
+            "angle": 0,
+            "content": "\\[\nJ \\Delta x \\neq 0. \\tag {63}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.18,
+                0.794,
+                0.201
+            ],
+            "angle": 0,
+            "content": "Proof. Recall that the \\( ij \\)th entry of the Jacobian represents \\( \\frac{\\partial s_i}{\\partial x_j} \\) so that the \\( i \\)th entry of \\( J\\Delta x \\) is"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.395,
+                0.203,
+                0.825,
+                0.238
+            ],
+            "angle": 0,
+            "content": "\\[\n[ J \\Delta x ] _ {i} = \\sum_ {j} \\frac {\\partial s _ {i}}{\\partial x _ {j}} \\Delta x _ {j} = d s _ {i}. \\tag {64}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.241,
+                0.827,
+                0.27
+            ],
+            "angle": 0,
+            "content": "Assume \\( J\\Delta x = 0 \\). This would imply a change in \\( x \\in \\mathcal{X} \\) results in no change in \\( s \\) (\\( ds = 0 \\)), contradicting the fact that \\( s \\) is injective. Therefore, we must conclude the claim that \\( J\\Delta x \\neq 0 \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.276,
+                0.825,
+                0.307
+            ],
+            "angle": 0,
+            "content": "Lemma 21. Let \\( J = J(s(x)) \\) be the Jacobian of any composition of transformations \\( s = s_t \\circ \\ldots \\circ s_1 \\) where \\( s_t(z) = [z_i / \\sum_j z_j]_i \\). Then \\( J\\Delta x \\) lies in the tangent space of the simplex."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.323,
+                0.825,
+                0.357
+            ],
+            "angle": 0,
+            "content": "Proof. We aim to show \\(\\mathbf{1}^{\\top}J\\Delta x = \\mathbf{0}\\) for any \\(\\Delta x\\) and \\(x\\). By chain rule, the Jacobian of \\(s\\) is \\(J = J(s) = \\prod_{t'=t}^{t'=1} J(s_t')\\). Therefore, \\(\\mathbf{1}^{\\top}J\\Delta x = \\mathbf{1}^{\\top}(\\prod_{t'=t}^{t'=1} J(s_t'))\\Delta x\\). Consider the first product:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.452,
+                0.359,
+                0.825,
+                0.376
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {1} ^ {\\top} J \\left(s _ {t}\\right) = \\mathbf {0} \\tag {65}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.379,
+                0.827,
+                0.425
+            ],
+            "angle": 0,
+            "content": "by Lemma 24. Therefore \\(\\mathbf{1}^{\\top}J\\Delta x = \\mathbf{1}^{\\top}J(s_{t})(\\prod_{t^{\\prime} = t - 1}^{t^{\\prime} = 1}J(s_{t}^{\\prime}))\\Delta x = \\mathbf{0}^{\\top}(\\prod_{t^{\\prime} = t - 1}^{t^{\\prime} = 1}J(s_{t}^{\\prime}))\\Delta x = 0\\). This implies \\(J\\Delta x\\) is orthogonal to \\(\\mathbf{1}\\) for any \\(x\\in \\mathcal{X}\\) and \\(\\Delta x\\), therefore \\(J\\Delta x\\) lies in the tangent space of the simplex for any \\(x\\in \\mathcal{X}\\) and \\(\\Delta x\\)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.44,
+                0.342,
+                0.453
+            ],
+            "angle": 0,
+            "content": "B.1.2 SOFTMAX MAP"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.463,
+                0.827,
+                0.547
+            ],
+            "angle": 0,
+            "content": "Let \\( s:[0,1]^{d - 1}\\to \\Delta^{d - 1}\\in \\mathbb{R}^d \\) be the softmax function. See (Gao and Pavel, 2017) for an analysis of many of its properties and in the context of game theory. Note that \\( s \\) maps a \\( (d - 1) \\) dimensional variable to a \\( d \\) dimensional distribution. This can be practically handled by always appending a 0 to the \\( (d - 1) \\)-dimensional input prior to applying the standard softmax. We perform our analysis below in terms of the standard softmax, but note the norms we derive apply to our modified (invertible) softmax."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.555,
+                0.24,
+                0.567
+            ],
+            "angle": 0,
+            "content": "Standard:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.388,
+                0.568,
+                0.825,
+                0.604
+            ],
+            "angle": 0,
+            "content": "\\[\ns (x) = \\frac {1}{\\sum_ {j = 1} ^ {d} e ^ {x _ {j}}} \\left[ e ^ {x _ {1}}, \\dots e ^ {x _ {d}} \\right]. \\tag {66}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.612,
+                0.241,
+                0.626
+            ],
+            "angle": 0,
+            "content": "Modified:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.359,
+                0.626,
+                0.825,
+                0.662
+            ],
+            "angle": 0,
+            "content": "\\[\ns (x) = \\frac {1}{1 + \\sum_ {j = 1} ^ {d - 1} e ^ {x _ {j}}} \\left[ e ^ {x _ {1}}, \\dots e ^ {x _ {d - 1}}, 1 \\right]. \\tag {67}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.664,
+                0.798,
+                0.681
+            ],
+            "angle": 0,
+            "content": "Lemma 22. Let \\(J\\) be the Jacobian of the softmax operator. Then \\(\\| J\\|_{\\infty}\\leq 2\\) and \\(\\| J^{\\top}\\|_{\\infty}\\leq 2\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.694,
+                0.825,
+                0.723
+            ],
+            "angle": 0,
+            "content": "Proof. Let \\( S_{i} \\) represent the \\( i \\)th entry of \\( S = \\mathrm{softmax}(z) \\) for any \\( z \\in \\mathbb{R}^{m} \\). Then the 1-norm of row \\( i \\) is upper bounded as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.37,
+                0.745,
+                0.825,
+                0.776
+            ],
+            "angle": 0,
+            "content": "\\[\n\\Rightarrow \\sum_ {j} \\left| D _ {j} S _ {i} \\right| = \\sum_ {j} \\left| S _ {i} \\left(\\delta_ {i j} - S _ {j}\\right) \\right| \\tag {68b}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.441,
+                0.725,
+                0.825,
+                0.921
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} D _ {j} S _ {i} = S _ {i} \\left(\\delta_ {i j} - S _ {j}\\right) (68a) \\\\ \\leq \\sum_ {j} \\left| \\delta_ {i j} S _ {i} \\right| + \\left| S _ {i} S _ {j} \\right| (68c) \\\\ = S _ {i} + \\sum_ {j} S _ {i} S _ {j} (68d) \\\\ = S _ {i} + S _ {i} \\sum_ {j} S _ {j} (68e) \\\\ = 2 S _ {i} (68f) \\\\ \\leq 2 \\forall i. (68g) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "27"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.568,
+                0.12
+            ],
+            "angle": 0,
+            "content": "Also, the 1-norm of column \\( j \\) is upper bounded similarly as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.384,
+                0.13,
+                0.826,
+                0.321
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} (6 9 a) \\\\ \\sum_ {i} \\left| D _ {j} S _ {i} \\right| = \\sum_ {i} \\left| S _ {i} \\left(\\delta_ {i j} - S _ {j}\\right) \\right| (69b) \\\\ \\leq \\sum_ {i} \\left| \\delta_ {i j} S _ {i} \\right| + \\left| S _ {i} S _ {j} \\right| (69c) \\\\ = S _ {j} + \\sum_ {i} S _ {i} S _ {j} (69d) \\\\ = S _ {i} + S _ {j} \\sum_ {i} S _ {i} (69e) \\\\ = 2 S _ {j} (69f) \\\\ \\leq 2 \\forall j. (69g) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.332,
+                0.827,
+                0.363
+            ],
+            "angle": 0,
+            "content": "The \\(\\infty\\)-norm of a matrix is the maximum 1-norm of any row. Therefore, \\(||J||_{\\infty}\\) and \\(||J^{\\top}||_{\\infty}\\) are both upper bounded by 2."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.38,
+                0.791,
+                0.397
+            ],
+            "angle": 0,
+            "content": "Corollary 3. Let \\(J\\) be the Jacobian of the softmax operator. Then \\(\\| J\\| _2\\leq 2\\) and \\(\\| J^{\\top}\\|_{2}\\leq 2\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.423,
+                0.827,
+                0.508
+            ],
+            "angle": 0,
+            "content": "Proof. The Gershgorin circle theorem states that every eigenvalue of \\( J \\) lies within one of the discs centered at the diagonal of \\( J \\) with radius equal to the 1-norm of the row (excluding the diagonal term). The Jacobian of the softmax operator contains only nonnegative diagonal terms, therefore, the 1-norm of any entire row also represents the maximum magnitude of any eigenvalue allowed by any Gershgorin disc. In addition, \\( J \\) is symmetric and therefore, the magnitude of its eigenvalues are equivalent to its singular values. Therefore by Lemma 22, \\( ||J||_2 \\leq 2 \\)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.535,
+                0.351,
+                0.548
+            ],
+            "angle": 0,
+            "content": "B.1.3 SPHERICAL MAP"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.56,
+                0.825,
+                0.592
+            ],
+            "angle": 0,
+            "content": "For spherical coordinates, let \\( s(x) = n(l(c(x))) \\) where \\( c(x) = \\pi / 2x \\), \\( l(\\psi) \\) maps angles to the unit sphere, and \\( n(z) = [z_i / \\sum_j z_j]_i \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.598,
+                0.798,
+                0.613
+            ],
+            "angle": 0,
+            "content": "Definition 2. Define \\( l(\\psi) \\) as the transformation to the unit-sphere using spherical coordinates:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.374,
+                0.624,
+                0.825,
+                0.64
+            ],
+            "angle": 0,
+            "content": "\\[\nl _ {1} (\\psi) = \\cos (\\psi_ {1}) \\tag {70a}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.377,
+                0.642,
+                0.825,
+                0.657
+            ],
+            "angle": 0,
+            "content": "\\[\nl _ {2} (\\psi) = \\sin \\left(\\psi_ {1}\\right) \\cos \\left(\\psi_ {2}\\right) \\tag {70b}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.377,
+                0.659,
+                0.825,
+                0.675
+            ],
+            "angle": 0,
+            "content": "\\[\nl _ {3} (\\psi) = \\sin (\\psi_ {1}) \\sin (\\psi_ {2}) \\cos (\\psi_ {3}) \\tag {70c}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.407,
+                0.685,
+                0.825,
+                0.701
+            ],
+            "angle": 0,
+            "content": "\\[\n\\vdots = \\vdots \\tag {70d}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.354,
+                0.705,
+                0.825,
+                0.72
+            ],
+            "angle": 0,
+            "content": "\\[\nl _ {m - 1} (\\psi) = \\sin \\left(\\psi_ {1}\\right) \\sin \\left(\\psi_ {2}\\right) \\dots \\cos \\left(\\psi_ {m - 1}\\right) \\tag {70e}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.371,
+                0.722,
+                0.825,
+                0.738
+            ],
+            "angle": 0,
+            "content": "\\[\nl _ {m} (\\psi) = \\sin \\left(\\psi_ {1}\\right) \\sin \\left(\\psi_ {2}\\right) \\dots \\sin \\left(\\psi_ {m - 1}\\right). \\tag {70f}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.749,
+                0.827,
+                0.78
+            ],
+            "angle": 0,
+            "content": "Lemma 23. Let \\(J\\) be the Jacobian of the transformation to the unit-sphere using spherical coordinates, i.e. \\(z = l(\\psi)\\) where \\(\\|l\\|^2 = 1\\) and \\(\\psi_i \\in [0, \\frac{\\pi}{2}]\\) represents an angle for each \\(i\\). Then \\(\\|J\\|_F \\leq \\sqrt{m}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.805,
+                0.471,
+                0.819
+            ],
+            "angle": 0,
+            "content": "Proof. The Jacobian of the transformation is"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.172,
+                0.829,
+                0.877,
+                0.926
+            ],
+            "angle": 0,
+            "content": "\\[\nJ (l) = \\left[ \\begin{array}{c c c c} - \\sin (\\psi_ {1}) & 0 & \\dots & 0 \\\\ \\cos (\\psi_ {1}) \\cos (\\psi_ {2}) & - \\sin (\\psi_ {1}) \\sin (\\psi_ {2}) & \\dots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\cos (\\psi_ {1}) \\sin (\\psi_ {2}) \\dots \\cos (\\psi_ {m - 1}) & \\dots & \\dots & - \\sin (\\psi_ {1}) \\dots \\sin (\\psi_ {m - 2}) \\sin (\\psi_ {m - 1}) \\\\ \\cos (\\psi_ {1}) \\sin (\\psi_ {2}) \\dots \\sin (\\psi_ {m - 1}) & \\dots & \\dots & \\sin (\\psi_ {1}) \\dots \\sin (\\psi_ {m - 2}) \\cos (\\psi_ {m - 1}) \\end{array} \\right] \\tag {71}\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.961
+            ],
+            "angle": 0,
+            "content": "28"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.105,
+                0.276,
+                0.119
+            ],
+            "angle": 0,
+            "content": "and it square is"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.178,
+                0.127,
+                0.825,
+                0.224
+            ],
+            "angle": 0,
+            "content": "\\[\nJ (l) = \\left[ \\begin{array}{c c c c} t _ {1} & 0 & \\dots & 0 \\\\ \\cos (\\psi_ {1}) ^ {2} \\cos (\\psi_ {2}) ^ {2} & \\sin (\\psi_ {1}) ^ {2} t _ {2} & \\dots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\cos (\\psi_ {1}) ^ {2} \\sin (\\psi_ {2}) ^ {2} \\dots \\cos (\\psi_ {m - 1}) ^ {2} & \\dots & \\dots & \\sin (\\psi_ {1}) ^ {2} \\dots \\sin (\\psi_ {m - 2}) ^ {2} t _ {m - 1} \\\\ \\cos (\\psi_ {1}) ^ {2} \\sin (\\psi_ {2}) ^ {2} \\dots \\sin (\\psi_ {m - 1}) ^ {2} & \\dots & \\dots & \\sin (\\psi_ {1}) ^ {2} \\dots \\sin (\\psi_ {m - 2}) ^ {2} t _ {m} \\end{array} \\right] \\tag {72}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.231,
+                0.218,
+                0.244
+            ],
+            "angle": 0,
+            "content": "where"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.333,
+                0.253,
+                0.825,
+                0.269
+            ],
+            "angle": 0,
+            "content": "\\[\n\\delta_ {i m} = 1 \\text {i f} i = m, 0 \\text {e l s e} \\tag {73a}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.347,
+                0.272,
+                0.824,
+                0.29
+            ],
+            "angle": 0,
+            "content": "\\[\nt _ {i} = \\delta_ {i m} \\cos^ {2} \\left(\\psi_ {i - 1}\\right) + \\left(1 - \\delta_ {i m}\\right) \\sin^ {2} \\left(\\psi_ {i}\\right) \\leq 1. \\tag {73b}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.303,
+                0.825,
+                0.348
+            ],
+            "angle": 0,
+            "content": "To compute the Frobenius norm, we will need the sum of the squares of all entries. We will consider the sum of each row individually using the following auxiliary variable \\( R_{i,k\\leq i} \\) where \\( \\sum_{j}J_{ij}^{2} = R_{i,1} \\) and apply a recursive inequality."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.786,
+                0.355,
+                0.825,
+                0.37
+            ],
+            "angle": 0,
+            "content": "(74a)"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.183,
+                0.374,
+                0.826,
+                0.661
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} R _ {i, k \\leq i} = \\sum_ {k ^ {\\prime} = k} ^ {i - 1} \\cos^ {2} \\left(\\psi_ {k ^ {\\prime}}\\right) \\left[ \\prod_ {l = k, l \\neq k ^ {\\prime}} ^ {i - 1} \\sin^ {2} \\left(\\psi_ {l}\\right) \\right] \\cos^ {2} \\left(\\psi_ {i}\\right) + t _ {i} \\prod_ {l = k} ^ {i - 1} \\sin^ {2} \\left(\\psi_ {l}\\right) (74b) \\\\ = \\cos^ {2} \\left(\\psi_ {k}\\right) \\underbrace {\\left[ \\prod_ {l = k + 1} ^ {i - 1} \\sin^ {2} \\left(\\psi_ {l}\\right) \\right] \\cos^ {2} \\left(\\psi_ {i}\\right)} _ {\\leq 1} (74c) \\\\ + \\sin^ {2} \\left(\\psi_ {k}\\right) \\sum_ {k ^ {\\prime} = k + 1} ^ {i - 1} \\cos^ {2} \\left(\\psi_ {k ^ {\\prime}}\\right) \\left[ \\prod_ {l = k + 1, l \\neq k ^ {\\prime}} ^ {i - 1} \\sin^ {2} \\left(\\psi_ {l}\\right) \\right] \\cos^ {2} \\left(\\psi_ {i}\\right) (74d) \\\\ + \\sin^ {2} \\left(\\psi_ {k}\\right) t _ {i} \\prod_ {l = k + 1} ^ {i - 1} \\sin^ {2} \\left(\\psi_ {l}\\right) (74e) \\\\ \\leq \\cos^ {2} \\left(\\psi_ {k}\\right) (74f) \\\\ + \\sin^ {2} \\left(\\psi_ {k}\\right) \\left(\\sum_ {k ^ {\\prime} = k + 1} ^ {i - 1} \\cos^ {2} \\left(\\psi_ {k ^ {\\prime}}\\right) \\left[ \\prod_ {l = k + 1, l \\neq k ^ {\\prime}} ^ {i - 1} \\sin^ {2} \\left(\\psi_ {l}\\right) \\right] \\cos^ {2} \\left(\\psi_ {i}\\right) + t _ {i} \\prod_ {l = k + 1} ^ {i - 1} \\sin^ {2} \\left(\\psi_ {l}\\right)\\right) (7 4 g) \\\\ = \\cos^ {2} (\\psi_ {k}) + \\sin^ {2} (\\psi_ {k}) R _ {i, k + 1}. (74h) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.666,
+                0.827,
+                0.699
+            ],
+            "angle": 0,
+            "content": "Note then that \\( R_{i,k+1} \\leq 1 \\Rightarrow R_{i,k} \\leq 1 \\). We know \\( R_{i,i} = t_i \\leq 1 \\), therefore, \\( R_{i,1} \\leq 1 \\) by applying the inequality recursively. Finally, \\( \\sum_j J_{ij}^2 = R_{i,1} \\leq 1 \\) implies the claim \\( ||J||_F^2 = \\sum_i R_{i,1} \\leq m \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.706,
+                0.754,
+                0.725
+            ],
+            "angle": 0,
+            "content": "Lemma 24. Let \\(J\\) be the Jacobian of \\(n(z) = z / Z\\) where \\(Z = \\sum_{k}z_{k}\\). Then \\(\\mathbf{1}^{\\top}J = \\mathbf{0}^{\\top}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.738,
+                0.49,
+                0.753
+            ],
+            "angle": 0,
+            "content": "Proof. The \\(i\\)th entry of the Jacobian of \\(n(z)\\) is"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.404,
+                0.76,
+                0.826,
+                0.79
+            ],
+            "angle": 0,
+            "content": "\\[\nJ (n) _ {i j} = \\frac {1}{Z ^ {2}} \\left(- z _ {i} + \\delta_ {i j} Z\\right). \\tag {75}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.796,
+                0.826,
+                0.827
+            ],
+            "angle": 0,
+            "content": "Therefore \\([ \\mathbf{1}^{\\top} J ]_{j} = \\sum_{i} J(n)_{ij} = \\frac{1}{Z^{2}} (-Z + Z) = 0\\) where \\(z\\) is a point on the unit-sphere in the positive orthant."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.842,
+                0.532,
+                0.857
+            ],
+            "angle": 0,
+            "content": "B.2 NEAR OPTIMALITY & ZOOMING DIMENSION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.868,
+                0.827,
+                0.926
+            ],
+            "angle": 0,
+            "content": "In this subsection, we derive bounds on the near-optimality dimension and zooming dimension of the associated bandit optimization problems we use to derive global convergence guarantees to Nash equilibria. This is written in terms of maximizing a function \\( f \\) rather than minimizing a loss to better match the bandit literature (Bubeck et al., 2011; Valko et al., 2013; Feng et al., 2022)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.961
+            ],
+            "angle": 0,
+            "content": "29"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.828,
+                0.135
+            ],
+            "angle": 0,
+            "content": "Assumption 1. Locally around each interior \\( x^{*} \\in \\mathcal{X}^{*} \\), \\( -f(x) \\) is lower bounded by \\( -f(x^{*}) + \\sigma_{-}||x - x^{*}||^{\\alpha_{hi}} \\). In other words, for all \\( f(x) \\geq f(x^{*}) - \\eta \\):"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.386,
+                0.143,
+                0.825,
+                0.161
+            ],
+            "angle": 0,
+            "content": "\\[\nf \\left(x ^ {*}\\right) - f (x) \\geq \\sigma_ {-} \\| x - x ^ {*} \\| ^ {\\alpha_ {h i}} \\tag {76}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.17,
+                0.795,
+                0.187
+            ],
+            "angle": 0,
+            "content": "where we have left the precise norm unspecified for generality. Let \\(\\ell(x, x^{*}) = \\sigma_{+}||x - x^{*}||^{\\alpha_{lo}}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.193,
+                0.605,
+                0.212
+            ],
+            "angle": 0,
+            "content": "Definition 3. \\(\\mathcal{X}_{\\epsilon} \\stackrel{\\mathrm{def}}{=} \\{x \\in \\mathcal{X} \\mid \\exists x^{*} \\in \\mathcal{X}^{*}\\) s.t. \\(f(x) \\geq f(x^{*}) - \\epsilon\\}\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.218,
+                0.768,
+                0.238
+            ],
+            "angle": 0,
+            "content": "Definition 4. \\(\\mathcal{X}_{\\epsilon}^{lower} \\stackrel{\\mathrm{def}}{=} \\{x \\in \\mathcal{X} \\mid \\exists x^* \\in \\mathcal{X}^* \\text{ s.t. } f(x^*) - \\sigma_{-} \\|x - x^*\\|^{\\alpha_{hi}} \\geq f(x^*) - \\epsilon\\}\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.24,
+                0.361,
+                0.258
+            ],
+            "angle": 0,
+            "content": "Corollary 4. \\(\\mathcal{X}_{\\varepsilon} \\subseteq \\mathcal{X}_{\\varepsilon}^{lower}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.283,
+                0.827,
+                0.327
+            ],
+            "angle": 0,
+            "content": "Proof. By Assumption 1, \\( f(x^{*}) - \\sigma_{-}||x - x^{*}||^{\\alpha_{hi}}\\geq f(x) \\). Therefore, any \\( x\\in \\mathcal{X} \\) that satisfies the requirement for an element of \\( \\mathcal{X}_{\\epsilon} \\), \\( f(x)\\geq f(x^{*}) - \\epsilon \\), will also satisfy the requirement for an element of \\( \\mathcal{X}_{\\epsilon}^{lower} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.345,
+                0.828,
+                0.39
+            ],
+            "angle": 0,
+            "content": "Definition 5 (\\(\\psi\\)-near Optimality Dimension). The \\(\\psi\\)-near optimality dimension is the smallest \\(d' > 0\\) such that there exists \\(C > 0\\) such that for any \\(\\epsilon > 0\\), the maximum number of disjoint \\(\\ell\\)-balls of radius \\(\\psi \\epsilon\\) and center in \\(\\mathcal{X}_{\\epsilon}\\) is less than \\(C \\epsilon^{-d'}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.395,
+                0.826,
+                0.441
+            ],
+            "angle": 0,
+            "content": "Definition 6 (Zooming Dimension). The zooming dimension is the smallest \\( d_z > 0 \\) such that there exists \\( C_z > 0 \\) such that for any \\( r > 0 \\), the maximum number of disjoint \\( \\ell \\)-balls of radius \\( \\frac{r}{2} \\) and center in \\( \\mathcal{X}_{16r} \\) is less than \\( C_z r^{-d_z} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.446,
+                0.826,
+                0.496
+            ],
+            "angle": 0,
+            "content": "Corollary 5 (Zooming Dimension from \\(\\psi\\)-near Optimality Dimension). The zooming dimension of \\(f: x \\in [0,1]^d \\to [-1,1]\\) under \\(\\ell(x,y) = \\sigma_+||x - y||^{\\alpha_{lo}}\\) is \\(d_z = d\\left(\\frac{\\alpha_{hi} - \\alpha_{lo}}{\\alpha_{lo}\\alpha_{hi}}\\right)\\) with constant \\(C_z = 16^{-d'}C\\) with \\(\\psi = \\frac{1}{32}\\) (\\(\\psi\\) is needed to compute \\(C\\))."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.521,
+                0.825,
+                0.564
+            ],
+            "angle": 0,
+            "content": "Proof. Mapping the definition of \\(\\psi\\)-near optimality onto zooming dimension, we find \\(\\psi \\epsilon = r / 2\\) and \\(\\epsilon = 16r\\). Then we can infer \\(\\psi = 1 / 32\\); this is used to compute \\(C\\) (see Theorem 1). Rewriting the bound from \\(\\psi\\)-near optimality in terms of \\(r\\), we find"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.323,
+                0.574,
+                0.825,
+                0.593
+            ],
+            "angle": 0,
+            "content": "\\[\nN _ {\\epsilon} = N _ {1 6 r} \\leq C (1 6 r) ^ {- d ^ {\\prime}} = C 1 6 ^ {- d ^ {\\prime}} r ^ {- d ^ {\\prime}} = C _ {z} r ^ {- d _ {z}} \\tag {77}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.603,
+                0.826,
+                0.648
+            ],
+            "angle": 0,
+            "content": "where \\(N_{16r}\\) denotes the number of \\(\\ell\\)-balls of radius \\(r/2\\) with center in \\(\\mathcal{X}_{16r}\\), \\(d_z = d'\\), \\(C_z = C16^{-d'} = C16^{-d_z}\\). Therefore, this translation only effects the constant \\(C_z\\), not the zooming dimension."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.667,
+                0.827,
+                0.734
+            ],
+            "angle": 0,
+            "content": "Lemma 25 (\\(N_{\\epsilon \\leq \\eta} \\leq C_{\\epsilon \\leq \\eta}\\epsilon^{-d'}\\)). The maximum number of disjoint \\(l\\)-balls with radius \\(\\psi \\in\\) and center in \\(\\mathcal{X}_{\\epsilon \\leq \\eta}\\), \\(N_{\\epsilon \\leq \\eta}\\), is upper bounded by \\(C_{\\epsilon \\leq \\eta}\\epsilon^{-d'}\\) where \\(C_{\\epsilon \\leq \\eta} = |\\mathcal{X}^*| \\left( \\frac{\\sigma_+}{\\psi \\sigma_-^{\\alpha_{lo} / \\alpha_{hi}}} \\right)^{d / \\alpha_{lo}}\\) and \\(d' = d \\left( \\frac{\\alpha_{hi} - \\alpha_{lo}}{\\alpha_{lo} \\alpha_{hi}} \\right)\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.758,
+                0.825,
+                0.786
+            ],
+            "angle": 0,
+            "content": "Proof. The number of disjoint \\( \\ell \\)-balls of radius \\( \\psi \\epsilon \\) and center in \\( \\mathcal{X}_{\\epsilon \\leq \\eta} \\) can be upper bounded as follows."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.792,
+                0.438,
+                0.809
+            ],
+            "angle": 0,
+            "content": "Rewrite \\(\\mathcal{X}_{\\epsilon}^{lower}\\) by rearranging terms as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.278,
+                0.82,
+                0.825,
+                0.85
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathcal {X} _ {\\epsilon} ^ {l o w e r} = \\left\\{x \\in \\mathcal {X} | \\exists x ^ {*} \\in \\mathcal {X} ^ {*} s. t. \\left| \\left| x - x ^ {*} \\right| \\right| \\leq \\left(\\frac {\\epsilon}{\\sigma_ {-}}\\right) ^ {1 / \\alpha_ {h i}} \\stackrel {\\text {d e f}} {=} r _ {\\epsilon} \\right\\} \\tag {78}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.863,
+                0.795,
+                0.88
+            ],
+            "angle": 0,
+            "content": "and recall that from Corollary 4 that \\(\\mathcal{X}_{\\epsilon} \\subseteq \\mathcal{X}_{\\epsilon}^{lower}\\). Furthermore, an \\(\\ell\\)-ball of radius \\(\\psi \\epsilon\\) implies"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.275,
+                0.89,
+                0.825,
+                0.922
+            ],
+            "angle": 0,
+            "content": "\\[\n\\ell (x, y) = \\sigma_ {+} | | x - y | | ^ {\\alpha_ {l o}} \\leq \\psi \\epsilon \\Longrightarrow | | x - y | | \\leq \\left(\\frac {\\psi \\epsilon}{\\sigma_ {+}}\\right) ^ {1 / \\alpha_ {l o}} \\stackrel {\\text {d e f}} {=} r _ {\\ell}. \\tag {79}\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.51,
+                0.96
+            ],
+            "angle": 0,
+            "content": "30"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.825,
+                0.133
+            ],
+            "angle": 0,
+            "content": "The number of disjoint \\( \\ell \\)-balls that can pack into a set \\( \\mathcal{X}_{\\epsilon} \\), \\( N_{\\epsilon \\leq \\eta} \\), is upper bounded by the ratio of the volumes of the two sets:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.342,
+                0.139,
+                0.825,
+                0.408
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} N _ {\\epsilon \\leq \\eta} \\leq \\frac {V o l \\left(\\mathcal {X} _ {\\epsilon}\\right)}{V o l \\left(\\mathcal {B} _ {\\ell}\\right)} (80a) \\\\ \\leq \\frac {\\operatorname {V o l} \\left(\\mathcal {X} _ {\\epsilon} ^ {\\text {l o w e r}}\\right)}{\\operatorname {V o l} \\left(\\mathcal {B} _ {\\ell}\\right)} (80b) \\\\ = \\frac {\\left| \\mathcal {X} ^ {*} \\right| S _ {d} r _ {\\epsilon} ^ {d}}{S _ {d} r _ {\\ell} ^ {d}} (80c) \\\\ \\leq \\frac {\\left| \\mathcal {X} ^ {*} \\right| \\left(\\frac {\\epsilon}{\\sigma_ {-}}\\right) ^ {d / \\alpha_ {h i}}}{\\left(\\frac {\\psi \\epsilon}{\\sigma_ {+}}\\right) ^ {d / \\alpha_ {l o}}} (80d) \\\\ = \\left| \\mathcal {X} ^ {*} \\right| \\left(\\frac {\\sigma_ {+} ^ {1 / \\alpha_ {l o}} \\psi^ {- 1 / \\alpha_ {l o}}}{\\sigma_ {-} ^ {1 / \\alpha_ {h i}}}\\right) ^ {d} \\epsilon^ {d \\left(1 / \\alpha_ {h i} - 1 / \\alpha_ {l o}\\right)} (80e) \\\\ = \\left| \\mathcal {X} ^ {*} \\right| \\left(\\frac {\\sigma_ {+}}{\\psi \\sigma_ {-} ^ {\\alpha_ {l o} / \\alpha_ {h i}}}\\right) ^ {d / \\alpha_ {l o}} \\epsilon^ {- d \\left(\\frac {\\alpha_ {h i} - \\alpha_ {l o}}{\\alpha_ {l o} \\alpha_ {h i}}\\right)} (80f) \\\\ = C _ {\\epsilon \\leq \\eta} \\epsilon^ {- d ^ {\\prime}} (80g) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.417,
+                0.825,
+                0.463
+            ],
+            "angle": 0,
+            "content": "where \\( C_{\\epsilon \\leq \\eta} = |\\mathcal{X}^{*}|\\left(\\frac{\\sigma_{+}}{\\psi\\sigma_{-}^{\\alpha_{lo} / \\alpha_{hi}}}\\right)^{d / \\alpha_{lo}} \\), \\( d' = d\\left(\\frac{\\alpha_{hi} - \\alpha_{lo}}{\\alpha_{lo}\\alpha_{hi}}\\right) \\), \\( |\\mathcal{X}^{*}| \\) is the number of distinct global optima, and \\( S_{d} \\) is the volume constant for a \\( d \\)-sphere under the given norm \\( ||\\cdot|| \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.476,
+                0.825,
+                0.519
+            ],
+            "angle": 0,
+            "content": "Recall, these results apply when \\( f(x) \\geq f(x^{*}) - \\eta \\), i.e., when \\( \\epsilon \\leq \\eta \\). Otherwise, we can upper bound the number of \\( \\ell \\)-balls by considering the entire set \\( \\mathcal{X} \\) which has volume 1. First, we will bound the constant associated with the volume of a \\( d \\)-sphere."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.523,
+                0.826,
+                0.58
+            ],
+            "angle": 0,
+            "content": "Lemma 26 (\\(N_{\\epsilon \\geq \\eta} \\leq C_{\\epsilon \\geq \\eta}\\)). The maximum number of disjoint \\(\\ell\\)-balls with radius \\(\\psi \\epsilon\\) and center in \\(\\mathcal{X}_{\\epsilon \\geq \\eta}\\), \\(N_{\\epsilon \\geq \\eta}\\), is upper bounded by \\(C_{\\epsilon \\geq \\eta}\\) where \\(C_{\\epsilon \\geq \\eta} = S_d^{-1}\\left(\\frac{\\sigma_+}{\\psi\\eta}\\right)^{d / \\alpha_{lo}}\\) and \\(S_d\\) is the volume constant for a \\(d\\)-sphere under a given norm."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.594,
+                0.825,
+                0.625
+            ],
+            "angle": 0,
+            "content": "Proof. We can upper bound the number of \\(\\ell\\)-balls needed to pack the entire space by considering the smallest possible radius \\(\\psi \\eta\\):"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.418,
+                0.631,
+                0.825,
+                0.796
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} N _ {\\epsilon \\geq \\eta} \\leq \\frac {V o l (\\mathcal {X})}{V o l (\\mathcal {B} _ {\\ell})} (81a) \\\\ = \\frac {1}{S _ {d} r _ {\\ell} ^ {d}} (81b) \\\\ \\leq \\frac {1}{S _ {d} \\left(\\frac {\\psi \\eta}{\\sigma_ {+}}\\right) ^ {d / \\alpha_ {l o}}} (81c) \\\\ = S _ {d} ^ {- 1} \\left(\\frac {\\sigma_ {+}}{\\psi \\eta}\\right) ^ {d / \\alpha_ {l o}} (81d) \\\\ = C _ {\\epsilon \\geq \\eta} (81e) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.803,
+                0.825,
+                0.817
+            ],
+            "angle": 0,
+            "content": "where \\(r_l\\) was defined in equation (79)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.825,
+                0.826,
+                0.859
+            ],
+            "angle": 0,
+            "content": "Theorem 1. The \\(\\psi\\)-near optimality dimension of \\(f: x \\in [0,1]^d \\to [-1,1]\\) under \\(\\ell\\) is \\(d' = d\\left(\\frac{\\alpha_{hi} - \\alpha_{lo}}{\\alpha_{lo}\\alpha_{hi}}\\right)\\) with constant"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.297,
+                0.865,
+                0.825,
+                0.902
+            ],
+            "angle": 0,
+            "content": "\\[\nC = \\max  \\left\\{1, S _ {d} ^ {- 1} \\left(r _ {\\eta} ^ {\\frac {\\alpha_ {h i}}{\\alpha_ {l o}}} \\sigma_ {-} ^ {\\left(\\frac {\\alpha_ {h i} - \\alpha_ {l o}}{\\alpha_ {l o} \\alpha_ {h i}}\\right)}\\right) ^ {- d} \\right\\} \\left(\\frac {\\sigma_ {+}}{\\psi \\sigma_ {-} ^ {\\alpha_ {l o} / \\alpha_ {h i}}}\\right) ^ {d / \\alpha_ {l o}} \\tag {82}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.91,
+                0.659,
+                0.925
+            ],
+            "angle": 0,
+            "content": "where \\(S_{d}\\) is the volume constant for a \\(d\\)-sphere under the same norm as \\(\\ell\\)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.949,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "31"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.102,
+                0.825,
+                0.161
+            ],
+            "angle": 0,
+            "content": "Proof. First, let us define \\(r_{\\eta} = \\left(\\frac{\\eta}{\\sigma_{-}}\\right)^{1 / \\alpha_{hi}}\\) as in equation (78) which implies \\(\\eta = \\sigma_{-}r_{\\eta}^{\\alpha_{hi}}\\). Then apply Lemmas 25 (\\(N_{\\epsilon \\leq \\eta} \\leq C_{\\epsilon \\leq \\eta}\\epsilon^{-d'}\\)) and 26 (\\(N_{\\epsilon \\geq \\eta} \\leq C_{\\epsilon \\geq \\eta}\\)) which bound the number of \\(\\ell\\)-balls required to pack \\(\\mathcal{X}_{\\epsilon}\\) when \\(\\epsilon\\) is less than and greater than \\(\\eta\\) respectively:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.39,
+                0.184,
+                0.825,
+                0.219
+            ],
+            "angle": 0,
+            "content": "\\[\nC _ {\\epsilon \\leq \\eta} = \\left| \\mathcal {X} ^ {*} \\right| \\left(\\frac {\\sigma_ {+}}{\\psi \\sigma_ {-} ^ {\\alpha_ {l o} / \\alpha_ {h i}}}\\right) ^ {d / \\alpha_ {l o}} \\tag {83}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.412,
+                0.224,
+                0.825,
+                0.253
+            ],
+            "angle": 0,
+            "content": "\\[\nd ^ {\\prime} = d \\left(\\frac {\\alpha_ {h i} - \\alpha_ {l o}}{\\alpha_ {l o} \\alpha_ {h i}}\\right) \\tag {84}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.257,
+                0.202,
+                0.271
+            ],
+            "angle": 0,
+            "content": "and"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.303,
+                0.274,
+                0.825,
+                0.307
+            ],
+            "angle": 0,
+            "content": "\\[\nC _ {\\epsilon \\geq \\eta} = S _ {d} ^ {- 1} \\left(\\frac {\\sigma_ {+}}{\\psi \\eta}\\right) ^ {d / \\alpha_ {l o}} \\tag {85a}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.344,
+                0.309,
+                0.825,
+                0.344
+            ],
+            "angle": 0,
+            "content": "\\[\n= \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} S _ {d} ^ {- 1} \\eta^ {- d / \\alpha_ {l o}} \\sigma_ {-} ^ {d / \\alpha_ {h i}} \\left| \\mathcal {X} ^ {*} \\right| \\left(\\frac {\\sigma_ {+}}{\\psi \\sigma_ {-} ^ {\\alpha_ {l o} / \\alpha_ {h i}}}\\right) ^ {d / \\alpha_ {l o}} \\tag {85b}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.344,
+                0.348,
+                0.825,
+                0.368
+            ],
+            "angle": 0,
+            "content": "\\[\n= \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} S _ {d} ^ {- 1} \\eta^ {- d / \\alpha_ {l o}} \\sigma_ {-} ^ {d / \\alpha_ {h i}} C _ {\\epsilon \\leq \\eta} \\tag {85c}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.344,
+                0.37,
+                0.825,
+                0.392
+            ],
+            "angle": 0,
+            "content": "\\[\n= \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} S _ {d} ^ {- 1} r _ {\\eta} ^ {- d \\alpha_ {h i} / \\alpha_ {l o}} \\sigma_ {-} ^ {- d / \\alpha_ {l o}} \\sigma_ {-} ^ {d / \\alpha_ {h i}} C _ {\\epsilon \\leq \\eta} \\tag {85d}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.344,
+                0.394,
+                0.825,
+                0.421
+            ],
+            "angle": 0,
+            "content": "\\[\n= \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} S _ {d} ^ {- 1} r _ {\\eta} ^ {- d \\frac {\\alpha_ {h i}}{\\alpha_ {l o}}} \\sigma_ {-} ^ {- d \\left(\\frac {\\alpha_ {h i} - \\alpha_ {l o}}{\\alpha_ {l o} \\alpha_ {h i}}\\right)} C _ {\\epsilon \\leq \\eta} \\tag {85e}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.344,
+                0.424,
+                0.825,
+                0.455
+            ],
+            "angle": 0,
+            "content": "\\[\n= \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} S _ {d} ^ {- 1} \\left(r _ {\\eta} ^ {\\frac {\\alpha_ {h i}}{\\alpha_ {l o}}} \\sigma_ {-} ^ {\\left(\\frac {\\alpha_ {h i} - \\alpha_ {l o}}{\\alpha_ {l o} \\alpha_ {h i}}\\right)}\\right) ^ {- d} C _ {\\epsilon \\leq \\eta} \\tag {85f}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.46,
+                0.825,
+                0.507
+            ],
+            "angle": 0,
+            "content": "where \\( S_{d} \\) is the volume constant for a \\( d \\)-sphere under the given norm. \\( S_{d}^{-1} \\) has been upper bounded for the 2-norm in Lemma 35. For the \\( \\infty \\)-norm, \\( S_{d}^{-1} = 2^{-d} \\). We have written \\( C_{\\epsilon \\geq \\eta} \\) in terms of \\( C_{\\epsilon \\leq \\eta} \\) to clarify which is larger."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.512,
+                0.245,
+                0.526
+            ],
+            "angle": 0,
+            "content": "Therefore,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.268,
+                0.533,
+                0.825,
+                0.603
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} C = \\max  \\left\\{1, | \\mathcal {X} ^ {*} | ^ {- 1} S _ {d} ^ {- 1} \\left(r \\eta^ {\\frac {\\alpha_ {h i}}{\\alpha_ {l o}}} \\sigma_ {-} ^ {\\left(\\frac {\\alpha_ {h i} - \\alpha_ {l o}}{\\alpha_ {l o} \\alpha_ {h i}}\\right)}\\right) ^ {- d} \\right\\} C _ {\\epsilon \\leq \\eta} (86a) \\\\ = \\max  \\left\\{1, \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} S _ {d} ^ {- 1} \\left(r _ {\\eta} ^ {\\frac {\\alpha_ {h i}}{\\alpha_ {l o}}} \\sigma_ {-} ^ {\\left(\\frac {\\alpha_ {h i} - \\alpha_ {l o}}{\\alpha_ {l o} \\alpha_ {h i}}\\right)}\\right) ^ {- d} \\right\\} \\left(\\frac {\\sigma_ {+}}{\\psi \\sigma_ {-} ^ {\\alpha_ {l o} / \\alpha_ {h i}}}\\right) ^ {d / \\alpha_ {l o}}. (86b) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.617,
+                0.827,
+                0.673
+            ],
+            "angle": 0,
+            "content": "Intuitively, if the radius for which the polynomial bounds hold \\((r_{\\eta})\\) is large and the minimum curvature constant \\(\\sigma_{-}\\) is also large, then the bound \\(C_{\\epsilon \\leq \\eta}\\) holds for large deviations from optimality \\(\\eta\\). The number of \\(\\eta\\)-radius \\(\\ell\\)-balls required to cover the remaining space, \\(C_{\\epsilon \\geq \\eta}\\), will be comparatively small."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.689,
+                0.279,
+                0.704
+            ],
+            "angle": 0,
+            "content": "B.3 D-BLIN"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.715,
+                0.825,
+                0.773
+            ],
+            "angle": 0,
+            "content": "The regret bound for Doubling BLiN (Feng et al., 2022) was originally proved assuming a standard normal distribution, however, the authors state their proof can be easily adapted to any sub-Gaussian distribution, which includes bounded random variables. This matches our setting with bounded payoffs, so we repeat their analysis here for that setting."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.778,
+                0.827,
+                0.877
+            ],
+            "angle": 0,
+            "content": "In what follows, in an effort to match notation in (Feng et al., 2022), let \\(\\mu(x) = -\\mathcal{L}^{\\tau}(s(x))\\) denote the expected reward for evaluating \\(x \\in [0,1]^d\\); \\(\\hat{\\mu}(x)\\) is an unbiased estimate of \\(\\mu(x)\\). Here, \\(s: [0,1]^{n(\\tilde{m} - 1)} \\to \\prod_i \\Delta^{m_i - 1}\\) is any function that maps from the unit hypercube to a product of simplices; we analyze two such maps in Appendix B.1. Also let \\(\\mathcal{A}_m\\) denote the collection of hypercubes to be investigated during the \\(m\\)th batch of arm pulls, \\(C \\in \\mathcal{A}_m\\) denote a hypercube partition in \\(\\mathcal{X}\\), \\(n_m\\) indicates the number of times each cube in \\(\\mathcal{A}_m\\) is played in batch \\(m\\), and \\(B_{stop}\\) denote the last batch."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.88,
+                0.825,
+                0.929
+            ],
+            "angle": 0,
+            "content": "Definition 7 (Global Arm Accuracy). \\(\\mathcal{E} \\stackrel{\\mathrm{def}}{=} \\left\\{ |\\mu(x) - \\hat{\\mu}_m(C)| \\leq r_m + \\sqrt{c_1 \\frac{\\ln T}{n_m}}, \\forall 1 \\leq m \\leq B_{stop} - 1, \\forall C \\in \\mathcal{A}_m, \\forall x \\in C \\right\\}\\)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.961
+            ],
+            "angle": 0,
+            "content": "32"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.102,
+                0.462,
+                0.128
+            ],
+            "angle": 0,
+            "content": "Define: \\(n_m = c_2 \\frac{\\ln T}{r_m^2} \\Rightarrow r_m = \\sqrt{c_2 \\frac{\\ln T}{n_m}}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.131,
+                0.825,
+                0.174
+            ],
+            "angle": 0,
+            "content": "Definition 8 (Elimination Rule). Eliminate \\( C \\in \\mathcal{A}_m \\) if \\( \\hat{\\mu}_m^{\\max} - \\hat{\\mu}_m(C) \\geq 2(1 + \\sqrt{c_1 / c_2})r_m = 2(\\sqrt{c_2} + \\sqrt{c_1})\\sqrt{\\frac{\\ln T}{n_m}} \\) where \\( \\hat{\\mu}_m^{\\max} \\stackrel{\\mathrm{def}}{=} \\max_{C \\in \\mathcal{A}_m} \\hat{\\mu}_m(C) \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.178,
+                0.453,
+                0.196
+            ],
+            "angle": 0,
+            "content": "Lemma 27. \\(Pr[\\mathcal{E}] \\geq 1 - 2T^{-2(c_1 / c^2 - 1)}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.211,
+                0.827,
+                0.26
+            ],
+            "angle": 0,
+            "content": "Proof. Let \\(\\hat{\\mu}_m(C) = \\frac{1}{n_m}\\sum_{i=1}^{n_m}y_{C,i}\\) be the unbiased estimate of \\(\\mu(x\\in C)\\) using \\(n_m\\) samples. Assume each \\(y_{C,i}\\in [a,b]\\) with \\(c = b - a\\) and \\(\\hat{\\mu}(C) = \\frac{1}{n_m}\\sum_{i=1}^{n_m}y_{C,i}\\). Applying a Hoeffding inequality gives"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.316,
+                0.267,
+                0.826,
+                0.344
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} P r \\left[ | \\hat {\\mu} (C) - \\mathbb {E} [ \\hat {\\mu} (C) ] | \\geq \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m}}} \\right] \\leq 2 e ^ {- 2 c _ {1} \\ln T / c ^ {2}} (87a) \\\\ = 2 \\left(e ^ {\\ln T}\\right) ^ {- 2 c _ {1} / c ^ {2}} (87b) \\\\ = 2 T ^ {- 2 c _ {1} / c ^ {2}} \\forall C. (87c) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.352,
+                0.41,
+                0.367
+            ],
+            "angle": 0,
+            "content": "By Lipschitzness of \\(\\mu\\) we also have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.384,
+                0.374,
+                0.825,
+                0.391
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left| \\mathbb {E} \\left[ \\hat {\\mu} (C) \\right] - \\mu (x) \\right| \\leq r _ {m}, \\forall x \\in C. \\tag {88}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.406,
+                0.271,
+                0.42
+            ],
+            "angle": 0,
+            "content": "Then consider"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.27,
+                0.427,
+                0.825,
+                0.549
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\sup  _ {x \\in C} | \\mu (x) - \\hat {\\mu} (C) | = \\sup  _ {x \\in C} | \\mu (x) - \\mathbb {E} [ \\hat {\\mu} (C) ] + \\mathbb {E} [ \\hat {\\mu} (C) ] - \\hat {\\mu} (C) | (89a) \\\\ \\leq \\sup  _ {x \\in C} \\left(| \\mu (x) - \\mathbb {E} [ \\hat {\\mu} (C) ] | + | \\mathbb {E} [ \\hat {\\mu} (C) ] - \\hat {\\mu} (C) |\\right) (89b) \\\\ = \\sup  _ {x \\in C} | \\mu (x) - \\mathbb {E} [ \\hat {\\mu} (C) ] | + | \\mathbb {E} [ \\hat {\\mu} (C) ] - \\hat {\\mu} (C) | (89c) \\\\ \\leq \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m}}} + r _ {m} (89d) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.558,
+                0.825,
+                0.588
+            ],
+            "angle": 0,
+            "content": "with probability \\(1 - 2T^{-2c_1 / c^2}\\). The first inequality follows by triangle inequality and the second follows from equation (88) and considering the complement of equation (87c)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.595,
+                0.529,
+                0.61
+            ],
+            "angle": 0,
+            "content": "The complement of this result occurs with probability"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.304,
+                0.618,
+                0.825,
+                0.651
+            ],
+            "angle": 0,
+            "content": "\\[\nP r \\left[ \\sup  _ {x \\in C} | \\mu (x) - \\hat {\\mu} (C) | \\geq r _ {m} + \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m}}} \\right] \\leq 2 T ^ {- 2 c _ {1} / c ^ {2}}. \\tag {90}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.667,
+                0.825,
+                0.711
+            ],
+            "angle": 0,
+            "content": "At least 1 arm is played in each cube \\( C \\in \\mathcal{A}_m \\) for \\( 1 \\leq m \\leq B_{stop} - 1 \\), therefore, \\( |\\mathcal{A}_m| \\leq T \\) must be true given the exit condition of the algorithm. In addition, assume \\( B_{stop} \\leq T \\) (\\( B_{stop} \\) will be defined such that this is true). Then a union bound over all \\( T^2 \\) events gives"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.218,
+                0.72,
+                0.826,
+                0.87
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} P r \\left[ \\exists m \\in [ 1, B _ {s t o p} - 1 ], C \\in \\mathcal {A} _ {m} s. t. \\sup  _ {x \\in C} | \\mu (x) - \\hat {\\mu} (C) | \\geq r _ {m} + \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m}}} \\right] (91a) \\\\ \\leq \\sum_ {m = 1} ^ {B _ {s t o p} - 1} \\sum_ {C \\in \\mathcal {A} _ {m}} P r \\left[ \\sup  _ {x \\in C} | \\mu (x) - \\hat {\\mu} (C) | \\geq r _ {m} + \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m}}} \\right] (91b) \\\\ \\leq \\sum_ {m = 1} ^ {B _ {s t o p} - 1} \\sum_ {C \\in \\mathcal {A} _ {m}} 2 T ^ {- 2 c _ {1} / c ^ {2}} (91c) \\\\ \\leq 2 T ^ {- 2 c _ {1} / c ^ {2}} T ^ {2}. (91d) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.88,
+                0.825,
+                0.929
+            ],
+            "angle": 0,
+            "content": "Taking the complement of this event and noting that \\(\\sup_{x\\in C}|\\mu (x) - \\hat{\\mu} (C)|\\leq r_m + \\sqrt{c_1\\frac{\\ln T}{n_m}}\\Rightarrow |\\mu (x) - \\hat{\\mu} (C)|\\leq r_m + \\sqrt{c_1\\frac{\\ln T}{n_m}}\\forall x\\in C\\) gives the desired result."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "33"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.827,
+                0.134
+            ],
+            "angle": 0,
+            "content": "Lemma 28 (Optimal Arm Survives). Under event \\(\\mathcal{E}\\), the optimal arm \\(x^{*} = \\arg \\max \\mu(x)\\) is not eliminated after the first \\(B_{stop} - 1\\) batches."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.147,
+                0.826,
+                0.177
+            ],
+            "angle": 0,
+            "content": "Proof. Let \\( C_m^* \\) denote the cube containing \\( x^* \\) in \\( \\mathcal{A}_m \\). Under event \\( \\mathcal{E} \\), for any cube \\( C \\in \\mathcal{A}_m \\) and \\( x \\in C \\), the following relation shows that \\( C_m^* \\) avoids the elimination rule in round \\( m \\):"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.223,
+                0.181,
+                0.825,
+                0.334
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\hat {\\mu} (C) - \\hat {\\mu} \\left(C _ {m} ^ {*}\\right) \\leq \\left(\\mu (x) + r _ {m} + \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m}}}\\right) + \\left(- \\mu \\left(x ^ {*}\\right) + r _ {m} + \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m}}}\\right) (92a) \\\\ = \\underbrace {(\\mu (x) - \\mu \\left(x ^ {*}\\right))} _ {\\leq 0} + 2 r _ {m} + 2 \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m}}} (92b) \\\\ \\leq 2 \\sqrt {c _ {2} \\frac {\\ln T}{n _ {m}}} + 2 \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m}}} (92c) \\\\ = 2 \\left(\\sqrt {c _ {1}} + \\sqrt {c _ {2}}\\right) \\sqrt {\\frac {\\ln T}{n _ {m}}} (92d) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.335,
+                0.827,
+                0.377
+            ],
+            "angle": 0,
+            "content": "where the first inequality follows from applying Lemma 27 to upper bound \\(\\hat{\\mu} (C)\\) and \\(\\hat{\\mu} (C_m^*)\\) individually. The remaining steps use the optimality of \\(x^{*}\\), the definition of \\(r_m\\), and the elimination rule."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.385,
+                0.803,
+                0.401
+            ],
+            "angle": 0,
+            "content": "Lemma 29. Under event \\(\\mathcal{E}\\), for any \\(1 \\leq m \\leq B_{stop}\\), any \\(C \\in A_m\\) and any \\(x \\in C\\), \\(\\Delta_x\\) satisfies"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.408,
+                0.404,
+                0.825,
+                0.422
+            ],
+            "angle": 0,
+            "content": "\\[\n\\Delta_ {x} \\leq 4 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {m - 1} \\tag {93}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.435,
+                0.825,
+                0.468
+            ],
+            "angle": 0,
+            "content": "Proof. For \\(m = 1\\), recall that \\(r_m\\) is the side length of a cube \\(C \\in \\mathcal{A}_m\\), therefore, \\(\\Delta_x \\leq r_{m-1} \\leq 4(1 + \\sqrt{c_1 / c_2})r_{m-1}\\) holds directly from the Lipschitzness of \\(\\mu\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.473,
+                0.827,
+                0.517
+            ],
+            "angle": 0,
+            "content": "For \\(m > 1\\), let \\(C_{m - 1}^{*}\\in \\mathcal{A}_{m - 1}\\) be the cube containing \\(x^{*}\\). From Lemma 28, this cube has not been eliminated under event \\(\\mathcal{E}\\). For any cube \\(C\\in \\mathcal{A}_m\\) and \\(x\\in C\\), it is clear that \\(x\\) is also in the parent of \\(C\\), denoted \\(C_{par}\\) (\\(x\\in C\\subset C_{par}\\)). Then for any \\(x\\in C\\), it holds that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.211,
+                0.52,
+                0.825,
+                0.798
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\Delta_ {x} = \\mu \\left(x ^ {*}\\right) - \\mu (x) \\leq \\left(\\hat {\\mu} _ {m - 1} \\left(C _ {m - 1} ^ {*}\\right) + r _ {m - 1} + \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m - 1}}}\\right) (94a) \\\\ + \\left(- \\hat {\\mu} _ {m - 1} (C _ {p a r}) + r _ {m - 1} + \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m - 1}}}\\right) \\\\ = \\left(\\hat {\\mu} _ {m - 1} \\left(C _ {m - 1} ^ {*}\\right) - \\hat {\\mu} _ {m - 1} \\left(C _ {p a r}\\right)\\right) + 2 \\left(\\sqrt {c _ {1}} + \\sqrt {c _ {2}}\\right) \\sqrt {\\frac {\\ln T}{n _ {m - 1}}} (94b) \\\\ \\leq \\left(\\hat {\\mu} _ {m - 1} ^ {\\max } - \\hat {\\mu} _ {m - 1} \\left(C _ {p a r}\\right)\\right) + 2 \\left(\\sqrt {c _ {1}} + \\sqrt {c _ {2}}\\right) \\sqrt {\\frac {\\ln T}{n _ {m - 1}}} (94c) \\\\ \\leq 2 \\left(\\sqrt {c _ {1}} + \\sqrt {c _ {2}}\\right) \\sqrt {\\frac {\\ln T}{n _ {m - 1}}} + 2 \\left(\\sqrt {c _ {1}} + \\sqrt {c _ {2}}\\right) \\sqrt {\\frac {\\ln T}{n _ {m - 1}}} (94d) \\\\ = 4 \\left(\\sqrt {c _ {1}} + \\sqrt {c _ {2}}\\right) \\sqrt {\\frac {\\ln T}{n _ {m - 1}}} (94e) \\\\ = 4 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {m - 1} (94f) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.8,
+                0.827,
+                0.83
+            ],
+            "angle": 0,
+            "content": "where we have applied Lemma 27 similarly as in Lemma 28 and also used the definition of \\(r_{m - 1}\\). The last two inequalities use the fact that \\(\\hat{\\mu}_{m - 1}(C_{m - 1}^{*})\\leq \\hat{\\mu}_{m - 1}^{\\max}\\) and \\(C_{par}\\) was not eliminated."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.838,
+                0.826,
+                0.869
+            ],
+            "angle": 0,
+            "content": "Theorem 2 (BLiN Regret Rate). With probability exceeding \\(1 - 2T^{-2(c_1/c^2 - 1)}\\), the \\(T\\)-step total regret \\(R(T)\\) of BLiN with Doubling Edge-length Sequence (\\(D\\)-BLiN) Feng et al. (2022) satisfies"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.329,
+                0.872,
+                0.825,
+                0.892
+            ],
+            "angle": 0,
+            "content": "\\[\nR (T) \\leq 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left(2 c _ {2} + 1\\right) \\ln (T) ^ {\\frac {1}{d _ {z} + 2}} T ^ {\\frac {d _ {z} + 1}{d _ {z} + 2}} \\tag {95}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.894,
+                0.825,
+                0.929
+            ],
+            "angle": 0,
+            "content": "where \\( d_{z} \\) is the zooming dimension of the problem instance. In addition, D-BLiN only needs no more than \\( B^{*} = \\frac{\\log 2(T) - \\log 2(\\ln(T))}{d_{z} + 2} + 2 \\) rounds of communications to achieve this regret rate."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.51,
+                0.96
+            ],
+            "angle": 0,
+            "content": "34"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.827,
+                0.152
+            ],
+            "angle": 0,
+            "content": "Proof. Since \\( r_m = \\frac{r_{m-1}}{2} \\Rightarrow r_{m-1} = 2r_m \\) for the Doubling Edge-length Sequence, Lemma 29 implies that every cube \\( C \\in A_m \\) is a subset of \\( S(8(1 + \\sqrt{c_1 / c_2})r_m) \\). Thus from the definition of zooming dimension (Corollary 5 with appropriate condition), we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.414,
+                0.186,
+                0.825,
+                0.203
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left| \\mathcal {A} _ {m} \\right| \\leq N _ {r _ {m}} \\leq C _ {z} r _ {m} ^ {- d _ {z}}. \\tag {96}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.245,
+                0.825,
+                0.307
+            ],
+            "angle": 0,
+            "content": "Fix any positive number \\( B \\). Also by Lemma 29, we know that any arm played after batch \\( B \\) incurs a regret bounded by \\( 8(1 + \\sqrt{c_1 / c_2})r_B \\), since the cubes played after batch \\( B \\) have edge length no larger than \\( r_B \\). Then the total regret that occurs after batch \\( B \\) is bounded by \\( 8(1 + \\sqrt{c_1 / c_2})r_BT \\) (where \\( T \\) is an upper bound on the number of arms)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.313,
+                0.4,
+                0.327
+            ],
+            "angle": 0,
+            "content": "Thus the regret can be bounded as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.321,
+                0.362,
+                0.825,
+                0.406
+            ],
+            "angle": 0,
+            "content": "\\[\nR (T) \\leq \\sum_ {m = 1} ^ {B} \\sum_ {C \\in \\mathcal {A} _ {m}} \\sum_ {i = 1} ^ {n _ {m}} \\Delta_ {x _ {C, i}} + 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {B} T \\tag {97}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.439,
+                0.825,
+                0.485
+            ],
+            "angle": 0,
+            "content": "where the first term bounds the regret in the first \\(B\\) batches of D-BLiN, and the second term bounds the regret after the first \\(B\\) batches. If the algorithm stops at batch \\(\\tilde{B} < B\\), we define \\(\\mathcal{A}_m =\\) for any \\(\\tilde{B} < m \\leq B\\) and inequality equation (97) still holds."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.493,
+                0.827,
+                0.524
+            ],
+            "angle": 0,
+            "content": "By Lemma 29, we have \\(\\Delta_{x_{C,i}}\\leq 8(1 + \\sqrt{c_1 / c_2})r_m\\) for all \\(C\\in \\mathcal{A}_m\\). We can thus bound equation (97) by"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.264,
+                0.557,
+                0.825,
+                0.731
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} R (T) \\leq \\sum_ {m = 1} ^ {B} | \\mathcal {A} _ {m} | \\cdot n _ {m} \\cdot 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {m} + 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {B} T (98a) \\\\ \\leq \\sum_ {m = 1} ^ {B} N _ {r _ {m}} \\cdot n _ {m} \\cdot 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {m} + 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {B} T (98b) \\\\ = \\sum_ {m = 1} ^ {B} N _ {r _ {m}} \\cdot c _ {2} \\frac {\\ln T}{r _ {m} ^ {2}} \\cdot 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {m} + 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {B} T (98c) \\\\ = \\sum_ {m = 1} ^ {B} N _ {r _ {m}} \\cdot \\frac {\\ln T}{r _ {m}} \\cdot 8 c _ {2} \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) + 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {B} T (98d) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.766,
+                0.825,
+                0.803
+            ],
+            "angle": 0,
+            "content": "where equation (98b) uses equation (96), and equation (98c) uses equality \\(n_m = c_2 \\frac{\\ln T}{r_m^2}\\). Since \\(r_m = 2^{-m + 1}\\) and \\(N_{r_m} \\leq C_z r_m^{-d_z} \\leq C_z 2^{(m - 1)d_z}\\), we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.208,
+                0.837,
+                0.826,
+                0.922
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} R (T) \\leq \\sum_ {m = 1} ^ {B} C _ {z} 2 ^ {(m - 1) d _ {z}} \\cdot \\frac {\\ln T}{2 ^ {- m + 1}} \\cdot 8 c _ {2} \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) + 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) 2 ^ {- B + 1} T (99a) \\\\ = 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ c _ {2} C _ {z} \\ln T \\sum_ {m = 1} ^ {B} 2 ^ {(m - 1) (d _ {z} + 1)} + 2 ^ {- B + 1} T \\right]. (99b) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "35"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.104,
+                0.304,
+                0.119
+            ],
+            "angle": 0,
+            "content": "Continuing we find"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.193,
+                0.127,
+                0.825,
+                0.167
+            ],
+            "angle": 0,
+            "content": "\\[\nR (T) \\leq 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ c _ {2} C _ {z} \\ln T \\sum_ {m = 1} ^ {B} 2 ^ {(m - 1) (d _ {z} + 1)} + 2 ^ {- B + 1} T \\right] \\tag {100a}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.234,
+                0.171,
+                0.825,
+                0.21
+            ],
+            "angle": 0,
+            "content": "\\[\n= 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ c _ {2} C _ {z} \\ln T \\sum_ {m = 1} ^ {B} \\left(2 ^ {d _ {z} + 1}\\right) ^ {m - 1} + 2 ^ {- B + 1} T \\right] \\tag {100b}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.234,
+                0.215,
+                0.825,
+                0.254
+            ],
+            "angle": 0,
+            "content": "\\[\n= 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ c _ {2} C _ {z} \\ln T \\sum_ {m = 0} ^ {B - 1} \\left(2 ^ {d _ {z} + 1}\\right) ^ {m} + 2 ^ {- B + 1} T \\right] \\tag {100c}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.234,
+                0.258,
+                0.825,
+                0.288
+            ],
+            "angle": 0,
+            "content": "\\[\n= 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ c _ {2} C _ {z} \\ln T \\left(\\frac {2 ^ {B (d _ {z} + 1)} - 1}{2 ^ {d _ {z} + 1} - 1}\\right) + 2 ^ {- B + 1} T \\right] \\text {v i a g e o m e t r i c s e r i e s} \\tag {100d}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.234,
+                0.292,
+                0.825,
+                0.323
+            ],
+            "angle": 0,
+            "content": "\\[\n\\leq 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ c _ {2} C _ {z} \\ln T \\left(\\frac {2 ^ {B (d _ {z} + 1)}}{2 ^ {d _ {z} + 1} - 1}\\right) + 2 ^ {- B + 1} T \\right] \\tag {100e}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.234,
+                0.327,
+                0.825,
+                0.357
+            ],
+            "angle": 0,
+            "content": "\\[\n\\leq 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ c _ {2} C _ {z} \\ln T \\left(2 \\cdot \\frac {2 ^ {B (d _ {z} + 1)}}{2 ^ {d _ {z} + 1}}\\right) + 2 ^ {- B + 1} T \\right] \\tag {100f}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.234,
+                0.36,
+                0.825,
+                0.385
+            ],
+            "angle": 0,
+            "content": "\\[\n= 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ 2 c _ {2} C _ {z} 2 ^ {(B - 1) (d _ {z} + 1)} \\ln T + 2 ^ {- (B - 1)} T \\right]. \\tag {100g}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.391,
+                0.726,
+                0.414
+            ],
+            "angle": 0,
+            "content": "This inequality holds for any positive \\( B \\). By choosing \\( B^{*} = 1 + \\frac{\\log_{2}\\left(\\frac{T}{\\ln T}\\right)}{d_{z} + 2} \\), we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.219,
+                0.42,
+                0.825,
+                0.532
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} R (T) \\leq 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ 2 c _ {2} C _ {z} \\left(\\frac {T}{\\ln T}\\right) ^ {\\frac {(d _ {z} + 1)}{(d _ {z} + 2)}} \\ln T + \\left(\\frac {\\ln T}{T}\\right) ^ {\\frac {1}{(d _ {z} + 2)}} T \\right] (101a) \\\\ = 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ 2 c _ {2} C _ {z} T ^ {\\frac {(d _ {z} + 1)}{(d _ {z} + 2)}} \\ln T ^ {1 - \\frac {(d _ {z} + 1)}{(d _ {z} + 2)}} + T ^ {1 - \\frac {1}{(d _ {z} + 2)}} \\ln T ^ {\\frac {1}{(d _ {z} + 2)}} \\right] (101b) \\\\ = 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ 2 c _ {2} C _ {z} T ^ {\\frac {(d _ {z} + 1)}{(d _ {z} + 2)}} \\ln T ^ {\\frac {1}{(d _ {z} + 2)}} + T ^ {\\frac {(d _ {z} + 1)}{(d _ {z} + 2)}} \\ln T ^ {\\frac {1}{(d _ {z} + 2)}} \\right] (101c) \\\\ = 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left(2 c _ {2} C _ {z} + 1\\right) T ^ {\\frac {(d _ {z} + 1)}{(d _ {z} + 2)}} \\ln T ^ {\\frac {1}{(d _ {z} + 2)}}. (101d) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.558,
+                0.827,
+                0.596
+            ],
+            "angle": 0,
+            "content": "Corollary 6 (BLiN Regret Rate Refined). Setting \\( c_{1} = 2c^{2} \\) and \\( c_{2} = 2\\left(\\frac{c}{4C_{z}}\\right)^{2 / 3} \\) simplifies Theorem 2 such that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.338,
+                0.601,
+                0.825,
+                0.623
+            ],
+            "angle": 0,
+            "content": "\\[\nR (T) \\leq 8 \\left(1 + \\left(4 c ^ {2} C _ {z}\\right) ^ {1 / 3}\\right) ^ {2} T ^ {\\frac {(d _ {z} + 1)}{(d _ {z} + 2)}} \\ln T ^ {\\frac {1}{(d _ {z} + 2)}}. \\tag {102}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.628,
+                0.354,
+                0.644
+            ],
+            "angle": 0,
+            "content": "with probability \\(1 - 2T^{-2}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.659,
+                0.825,
+                0.701
+            ],
+            "angle": 0,
+            "content": "Proof. If we set \\( c_1 = 2c^2 \\) and \\( c_2 = 2\\left(\\frac{c}{4C_z}\\right)^{2/3} \\), then \\( \\sqrt{c_1 / c_2} = c\\left(\\frac{4C_z}{c}\\right)^{1/3} = \\left(4c^2 C_z\\right)^{1/3} = 2c_2C_z \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.708,
+                0.825,
+                0.723
+            ],
+            "angle": 0,
+            "content": "Lemma 30. The zooming dimension and zooming constant under the \\(\\ell (x,y) = ||x - y||_{\\infty}\\) norm are"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.411,
+                0.73,
+                0.825,
+                0.756
+            ],
+            "angle": 0,
+            "content": "\\[\nd _ {z} = \\frac {1}{2} n \\bar {m} \\tag {103}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.407,
+                0.759,
+                0.825,
+                0.793
+            ],
+            "angle": 0,
+            "content": "\\[\nC _ {z} = \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\left(\\frac {4}{r _ {\\eta} ^ {2} \\sigma_ {- \\infty}}\\right) ^ {n \\bar {m}} \\tag {104}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.798,
+                0.825,
+                0.828
+            ],
+            "angle": 0,
+            "content": "where \\(\\sigma_{-\\infty} = ||\\text{Hess}(-f(x))^{-1}||_{\\infty}\\) is an upper bound on the infinity norm of the inverse Hessian matrix of the function at every equilibrium."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.843,
+                0.825,
+                0.89
+            ],
+            "angle": 0,
+            "content": "Proof. Recall from Theorem 1 that \\( d_{z} = d\\left(\\frac{\\alpha_{hi} - \\alpha_{lo}}{\\alpha_{lo}\\alpha_{hi}}\\right) \\) with constant \\( C_z = 16^{-d'}C \\) where \\( C \\) is defined below. In addition, BLiN assumes \\( \\ell (x,y) = ||x - y||_{\\infty} \\). Matching to Assumption 1, we see that \\( \\sigma_{+} = \\alpha_{lo} = 1 \\). Under the infinity norm, the volume constant \\( S_{d} = 2^{d} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.896,
+                0.825,
+                0.925
+            ],
+            "angle": 0,
+            "content": "We will define the other constants with respect to properties of the Hessian of \\( f(x) \\) about each equilibrium, specifically the infinity norm of the inverse Hessian so that \\( \\sigma_{-} = \\sigma_{-\\infty} \\). This means"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.949,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "36"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.825,
+                0.148
+            ],
+            "angle": 0,
+            "content": "we will bound the function locally with a quadratic, i.e., \\(\\alpha_{hi} = 2\\). Lastly, recall from Corollary 5 that \\(\\psi = \\frac{1}{32}\\) and the dimension of our search space (the product space of player mixed strategies) is \\(d = n(\\bar{m} - 1) \\leq nm\\) for simplicity."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.153,
+                0.507,
+                0.168
+            ],
+            "angle": 0,
+            "content": "Plugging this information into Theorem 1, we find"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.442,
+                0.178,
+                0.825,
+                0.208
+            ],
+            "angle": 0,
+            "content": "\\[\nd _ {z} = d ^ {\\prime} = \\frac {1}{2} n \\bar {m} \\tag {105}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.217,
+                0.202,
+                0.23
+            ],
+            "angle": 0,
+            "content": "and"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.271,
+                0.239,
+                0.826,
+                0.277
+            ],
+            "angle": 0,
+            "content": "\\[\nC = \\max  \\left\\{1, | \\mathcal {X} ^ {*} | ^ {- 1} S _ {d} ^ {- 1} \\left(r _ {\\eta} ^ {\\frac {\\alpha_ {h i}}{\\alpha_ {l o}}} \\sigma_ {-} ^ {\\left(\\frac {\\alpha_ {h i} - \\alpha_ {l o}}{\\alpha_ {l o} \\alpha_ {h i}}\\right)}\\right) ^ {- d} \\right\\} \\left(\\frac {\\sigma_ {+}}{\\psi \\sigma_ {-} ^ {\\alpha_ {l o} / \\alpha_ {h i}}}\\right) ^ {d / \\alpha_ {l o}} \\tag {106a}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.29,
+                0.279,
+                0.825,
+                0.316
+            ],
+            "angle": 0,
+            "content": "\\[\n= \\max  \\left\\{1, \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} 2 ^ {- d} \\left(r _ {\\eta} ^ {2} \\sigma_ {- \\infty} ^ {\\frac {1}{2}}\\right) ^ {- d} \\right\\} \\left(\\frac {3 2}{\\sigma_ {- \\infty} ^ {1 / 2}}\\right) ^ {d} \\tag {106b}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.29,
+                0.318,
+                0.825,
+                0.355
+            ],
+            "angle": 0,
+            "content": "\\[\n= \\max  \\left\\{1, \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\left(2 r _ {\\eta} ^ {2} \\sigma_ {- \\infty} ^ {\\frac {1}{2}}\\right) ^ {- d} \\right\\} \\left(\\frac {3 2}{\\sigma_ {- \\infty} ^ {1 / 2}}\\right) ^ {d} \\tag {106c}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.289,
+                0.357,
+                0.825,
+                0.393
+            ],
+            "angle": 0,
+            "content": "\\[\n\\stackrel {h a r d} {=} \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\left(2 r _ {\\eta} ^ {2} \\sigma_ {- \\infty} ^ {\\frac {1}{2}}\\right) ^ {- d} \\left(\\frac {3 2}{\\sigma_ {- \\infty} ^ {1 / 2}}\\right) ^ {d} \\tag {106d}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.289,
+                0.395,
+                0.825,
+                0.43
+            ],
+            "angle": 0,
+            "content": "\\[\n\\stackrel {h a r d} {=} \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\left(\\frac {1 6}{r _ {\\eta} ^ {2} \\sigma_ {- \\infty}}\\right) ^ {n \\bar {m}} \\tag {106e}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.439,
+                0.825,
+                0.467
+            ],
+            "angle": 0,
+            "content": "where \\( \\text{hard} \\) indicates we are assuming \\( r_{\\eta} \\) and \\( \\sigma_{-\\infty} \\) are small enough to dominate the other operand of the max."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.474,
+                0.69,
+                0.489
+            ],
+            "angle": 0,
+            "content": "Finally, converting the near optimality constant to a zooming constant, we find"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.384,
+                0.498,
+                0.825,
+                0.567
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} C _ {z} = 1 6 ^ {- d _ {z}} \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\left(\\frac {1 6}{r _ {\\eta} ^ {2} \\sigma_ {- \\infty}}\\right) ^ {n \\bar {m}} (107a) \\\\ = \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\left(\\frac {4}{r _ {\\eta} ^ {2} \\sigma_ {- \\infty}}\\right) ^ {n \\bar {m}}. (107b) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.577,
+                0.825,
+                0.59
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.612,
+                0.573,
+                0.627
+            ],
+            "angle": 0,
+            "content": "B.4 BOUNDED DIAMETERS AND WELL-SHAPED CELLS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.639,
+                0.825,
+                0.669
+            ],
+            "angle": 0,
+            "content": "We assume the feasible set is a unit-hypercube of dimensionality \\(d\\) where cells are evenly split along the longest edge to give \\(b\\) new partitions and \\(x_{h,i}\\) represents the center of each cell."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.674,
+                0.827,
+                0.721
+            ],
+            "angle": 0,
+            "content": "There exists a decreasing sequence \\( w(h) > 0 \\), such that for any depth \\( h \\geq 0 \\) and for any cell \\( \\mathcal{X}_{h,i} \\) of depth \\( h \\), we have \\( \\sup_{x \\in \\mathcal{X}_{h,i}} \\ell(x_{h,i}, x) \\leq w(h) \\). Moreover, there exists \\( \\nu > 0 \\) such that for any depth \\( h \\geq 0 \\), any cell \\( \\mathcal{X}_{h,i} \\) contains an \\( \\ell \\)-ball of radius \\( \\nu w(h) \\) centered at \\( x_{h,i} \\)."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.297,
+                0.736,
+                0.7,
+                0.806
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>l(x,y)</td><td>c</td><td>γ</td><td>ν</td></tr><tr><td>l(x,y)=||x-y||2/2</td><td>dα/2(b/2)α</td><td>b-α/d</td><td>d-α/2b-2α</td></tr><tr><td>l(x,y)=||x-y||∞/∞</td><td>(b/2)α</td><td>b-α/d</td><td>b-2α</td></tr></table>"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.276,
+                0.816,
+                0.72,
+                0.835
+            ],
+            "angle": 0,
+            "content": "Table 3: Bounding Constants: \\( \\sup_{x \\in \\mathcal{X}_{h,i}} \\ell(x_{h,i}, x) \\leq w(h) = c\\gamma^h \\)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.863,
+                0.304,
+                0.879
+            ],
+            "angle": 0,
+            "content": "B.4.1 \\(L_{2}\\)-NORM"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.888,
+                0.825,
+                0.927
+            ],
+            "angle": 0,
+            "content": "Lemma 31 (L2-Norm Bounding Ball). Let \\(\\ell(x, y) = ||x - y||_2^\\alpha\\). Then \\(\\sup_{x \\in \\mathcal{X}_{h,i}} \\ell(x_{h,i}, x) \\leq w_2(h) = c\\gamma^h\\) where \\(c = \\left(\\frac{db^2}{4}\\right)^{\\alpha/2}\\) and \\(\\gamma = b^{-\\alpha/d}\\)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "37"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.105,
+                0.218,
+                0.119
+            ],
+            "angle": 0,
+            "content": "Proof."
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.226,
+                0.122,
+                0.826,
+                0.164
+            ],
+            "angle": 0,
+            "content": "\\[\nw (0) = \\left[ \\sum_ {i = 1} ^ {d} (1 / 2) ^ {2} \\right] ^ {\\alpha / 2} = \\left(\\frac {d}{4}\\right) ^ {\\alpha / 2} \\tag {108a}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.226,
+                0.166,
+                0.826,
+                0.24
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} w (1) = \\left[ (1 / b \\cdot 1 / 2) ^ {2} + \\sum_ {i = 2} ^ {d} (1 / 2) ^ {2} \\right] ^ {\\alpha / 2} = \\left[ (1 / b ^ {2}) (1 / 4) + (d - 1) (1 / 4) \\right] ^ {\\alpha / 2} (108b) \\\\ = \\left(\\frac {d - 1 + 1 / b ^ {2}}{4}\\right) ^ {\\alpha / 2} (108c) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.226,
+                0.242,
+                0.826,
+                0.505
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} w (d) = \\left[ \\sum_ {i = 1} ^ {d} (1 / b \\cdot 1 / 2) ^ {2} \\right] ^ {\\alpha / 2} = \\left(\\frac {d}{4 \\cdot b ^ {2}}\\right) ^ {\\alpha / 2} (108d) \\\\ w (h) = \\left[ r (1 / b) ^ {2 (q + 1)} (1 / 2) ^ {2} + \\sum_ {i = r} ^ {d} (1 / b) ^ {2 q} (1 / 2) ^ {2} \\right] ^ {\\alpha / 2} (108e) \\\\ = \\left[ (1 / b) ^ {2 q} (1 / 2) ^ {2} (r (1 / b) ^ {2} + (d - r) \\right] ^ {\\alpha / 2} (108f) \\\\ = \\left[ (1 / b ^ {2}) ^ {q} (1 / 4) \\left(d - r \\left(1 - \\frac {1}{b ^ {2}}\\right)\\right) \\right] ^ {\\alpha / 2} (108g) \\\\ \\leq \\left[ (1 / b ^ {2}) ^ {q} (1 / 4) d \\right] ^ {\\alpha / 2} (108h) \\\\ \\leq \\left[ \\left(1 / b ^ {2}\\right) ^ {h / d - 1} (1 / 4) d \\right] ^ {\\alpha / 2} (108i) \\\\ = \\left[ (1 / b ^ {2}) ^ {h / d} (b ^ {2} / 4) d \\right] ^ {\\alpha / 2} (108j) \\\\ = \\left(\\frac {d b ^ {2}}{4}\\right) ^ {\\alpha / 2} (1 / b) ^ {\\frac {\\alpha}{d} h} (108k) \\\\ = c \\gamma^ {h} (1081) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.508,
+                0.773,
+                0.53
+            ],
+            "angle": 0,
+            "content": "where \\( q, r = \\text{divmod}(h, d) \\Rightarrow q \\geq h / d - 1, c = \\left(\\frac{db^2}{4}\\right)^{\\alpha / 2} \\), and \\( \\gamma = (1 / b)^{\\alpha / d} = b^{-\\alpha / d} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.535,
+                0.827,
+                0.567
+            ],
+            "angle": 0,
+            "content": "Lemma 32 (\\(L_2\\)-Norm Inner Ball). Let \\(\\ell(x, y) = ||x - y||_2^\\alpha\\). Any cell \\(\\mathcal{X}_{h,i}\\) contains an \\(\\ell\\)-ball of radius \\(\\nu w_2(h)\\) where \\(\\nu = (db^4)^{-\\alpha/2}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.581,
+                0.66,
+                0.597
+            ],
+            "angle": 0,
+            "content": "Proof. Any cell \\(\\mathcal{X}_{h,i}\\) contains an \\(\\ell\\)-ball of radius equal to its shortest axis:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.392,
+                0.599,
+                0.826,
+                0.705
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} r _ {\\min } = \\left[ (1 / 4) \\left(1 / b ^ {2}\\right) ^ {\\lceil h / d \\rceil} \\right] ^ {\\alpha / 2} (109a) \\\\ \\geq \\left[ (1 / 4) \\left(1 / b ^ {2}\\right) ^ {h / d + 1} \\right] ^ {\\alpha / 2} (109b) \\\\ = \\left(\\frac {1}{b ^ {2} \\cdot 4}\\right) ^ {\\alpha / 2} (1 / b) ^ {\\frac {\\alpha}{d} h} (109c) \\\\ = w (h) \\cdot \\left(\\frac {1}{d b ^ {4}}\\right) ^ {\\alpha / 2}. (109d) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.735,
+                0.308,
+                0.75
+            ],
+            "angle": 0,
+            "content": "B.4.2 \\(L_{\\infty}\\)-NORM"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.758,
+                0.825,
+                0.794
+            ],
+            "angle": 0,
+            "content": "Lemma 33 (\\(L_{\\infty}\\)-Norm Bounding Ball). Let \\(\\ell(x, y) = ||x - y||_{\\infty}^{\\alpha}\\). Then \\(\\sup_{x \\in \\mathcal{X}_{h,i}} \\ell(x_{h,i}, x) \\leq w_{\\infty}(h) = c\\gamma^{h}\\) where \\(c = \\left(\\frac{b}{2}\\right)^{\\alpha}\\) and \\(\\gamma = b^{-\\alpha / d}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.807,
+                0.7,
+                0.822
+            ],
+            "angle": 0,
+            "content": "Proof. Any cell \\(\\mathcal{X}_{h,i}\\) is contained by an \\(\\ell\\)-ball of radius equal to its longest axis:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.391,
+                0.825,
+                0.826,
+                0.922
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} r _ {\\max } = \\left[ (1 / 4) \\left(1 / b ^ {2}\\right) ^ {\\lfloor h / d \\rfloor} \\right] ^ {\\alpha / 2} (110a) \\\\ \\leq \\left[ (1 / 4) \\left(1 / b ^ {2}\\right) ^ {h / d - 1} \\right] ^ {\\alpha / 2} (110b) \\\\ = \\left(\\frac {b ^ {2}}{4}\\right) ^ {\\alpha / 2} (1 / b) ^ {\\frac {\\alpha}{d} h} (110c) \\\\ = c \\gamma^ {h} (110d) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.949,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "38"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.102,
+                0.495,
+                0.123
+            ],
+            "angle": 0,
+            "content": "where \\( c = \\left(\\frac{b^2}{4}\\right)^{\\alpha / 2} \\), and \\( \\gamma = (1 / b)^{\\alpha / d} = b^{-\\alpha / d} \\)."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.107,
+                0.826,
+                0.119
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.131,
+                0.828,
+                0.162
+            ],
+            "angle": 0,
+            "content": "Lemma 34 (\\(L_{\\infty}\\)-Norm Inner Ball). Let \\(\\ell(x, y) = ||x - y||_{\\infty}^{\\alpha}\\). Any cell \\(\\mathcal{X}_{h,i}\\) contains an \\(\\ell\\)-ball of radius \\(\\nu w_{\\infty}(h)\\) where \\(\\nu = b^{-2\\alpha}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.179,
+                0.66,
+                0.195
+            ],
+            "angle": 0,
+            "content": "Proof. Any cell \\(\\mathcal{X}_{h,i}\\) contains an \\(\\ell\\)-ball of radius equal to its shortest axis:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.397,
+                0.204,
+                0.825,
+                0.308
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} r _ {\\min } = \\left[ (1 / 4) \\left(1 / b ^ {2}\\right) ^ {\\lceil h / d \\rceil} \\right] ^ {\\alpha / 2} (111a) \\\\ \\geq \\left[ (1 / 4) \\left(1 / b ^ {2}\\right) ^ {h / d + 1} \\right] ^ {\\alpha / 2} (111b) \\\\ = \\left(\\frac {1}{b ^ {2} \\cdot 4}\\right) ^ {\\alpha / 2} (1 / b) ^ {\\frac {\\alpha}{d} h} (111c) \\\\ = w (h) \\cdot \\left(\\frac {1}{b ^ {4}}\\right) ^ {\\alpha / 2}. (111d) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.317,
+                0.825,
+                0.328
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.348,
+                0.621,
+                0.362
+            ],
+            "angle": 0,
+            "content": "B.5 STOCHASTIC SIMULTANEOUS OPTIMISTIC OPTIMIZATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.374,
+                0.825,
+                0.403
+            ],
+            "angle": 0,
+            "content": "StoSOO is flexible in its choice of metric and partitioning structure. In StoSOO, we may choose \\(\\ell(x,y) = ||x - y||_2^2\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.407,
+                0.825,
+                0.453
+            ],
+            "angle": 0,
+            "content": "Lemma 35. The volume of a \\(d\\)-sphere with radius \\(r\\) and \\(d\\) even is given by \\(S_{d}r^{d}\\) where \\(S_{d}^{-1} \\leq \\sqrt{2\\pi d}\\left(\\frac{d}{2\\pi e}\\right)^{d/2}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.469,
+                0.827,
+                0.502
+            ],
+            "angle": 0,
+            "content": "Proof. First, we recall Stirling's bounds on the factorial: \\(\\sqrt{2\\pi n} \\left( \\frac{n}{e} \\right)^n e^{\\frac{1}{12n + 1}} < n! < \\sqrt{2\\pi n} \\left( \\frac{n}{e} \\right)^n e^{\\frac{1}{12n}}\\). This will be useful for bounding the Gamma function: \\(\\Gamma(d) = (d - 1)!\\) for even \\(d\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.507,
+                0.538,
+                0.523
+            ],
+            "angle": 0,
+            "content": "Given \\(d\\) is even, we start with the exact formula for \\(S_{d}\\):"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.383,
+                0.53,
+                0.825,
+                0.732
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} S _ {d} ^ {- 1} = \\frac {\\Gamma (d / 2 + 1)}{\\pi^ {d / 2}} (112a) \\\\ = \\frac {(d / 2) !}{\\pi^ {d / 2}} (112b) \\\\ <   \\frac {\\sqrt {2 \\pi (d / 2)} \\left(\\frac {d / 2}{e}\\right) ^ {d / 2} e ^ {\\frac {1}{1 2 (d / 2)}}}{\\pi^ {d / 2}} (112c) \\\\ = \\frac {\\pi^ {1 / 2} d ^ {1 / 2} \\left(\\frac {d}{2 e}\\right) ^ {d / 2} e ^ {\\frac {1}{6 d}}}{\\pi^ {d / 2}} (112d) \\\\ = \\frac {\\pi^ {1 / 2} d ^ {(d + 1) / 2} e ^ {\\frac {1}{6 d}}}{(2 \\pi e) ^ {d / 2}} (112e) \\\\ \\leq \\sqrt {2 \\pi d} \\left(\\frac {d}{2 \\pi e}\\right) ^ {d / 2}. (112f) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.74,
+                0.825,
+                0.751
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.763,
+                0.816,
+                0.78
+            ],
+            "angle": 0,
+            "content": "Lemma 36. The near optimality dimension and constant under the \\(\\ell(x, y) = ||x - y||_2^2\\) norm are"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.371,
+                0.787,
+                0.824,
+                0.802
+            ],
+            "angle": 0,
+            "content": "\\[\nd ^ {\\prime} = 0 \\tag {113}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.371,
+                0.806,
+                0.824,
+                0.84
+            ],
+            "angle": 0,
+            "content": "\\[\nC = \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\sqrt {2 \\pi n \\bar {m}} \\left(\\frac {n \\bar {m}}{5 \\nu r _ {\\eta} ^ {2} \\sigma_ {- 2}}\\right) ^ {n \\bar {m} / 2} \\tag {114}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.847,
+                0.825,
+                0.876
+            ],
+            "angle": 0,
+            "content": "where \\(\\sigma_{-2}\\) is a lower bound on the singular value of the Hessian matrix of the function at every equilibrium and \\(\\nu\\) is defined in Table 3 for the corresponding \\(\\ell(x, y)\\)-ball."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.892,
+                0.827,
+                0.926
+            ],
+            "angle": 0,
+            "content": "Proof. Recall from Theorem 1 that \\( d' = d\\left(\\frac{\\alpha_{hi} - \\alpha_{lo}}{\\alpha_{lo}\\alpha_{hi}}\\right) \\) with constant \\( C \\) defined below. In addition, we will analyze StoSOO with the choice \\( \\ell(x,y) = ||x - y||_2^2 \\). Matching to Assumption 1, we see"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "39"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.102,
+                0.825,
+                0.142
+            ],
+            "angle": 0,
+            "content": "that \\(\\sigma_{+} = 1\\) and \\(\\alpha_{lo} = 2\\). Under the 2-norm, the volume constant \\(S_{d}^{-1} \\leq \\sqrt{2\\pi d}\\left(\\frac{d}{2\\pi e}\\right)^{d/2}\\) (see Lemma 35)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.148,
+                0.826,
+                0.193
+            ],
+            "angle": 0,
+            "content": "We will bound the function locally with a quadratic, i.e., \\(\\alpha_{hi} = 2\\). Lastly, from (Valko et al., 2013, Corollary 1), \\(\\psi = \\frac{\\nu}{3}\\) with \\(\\nu = \\frac{1}{db^2}\\) defined in Table 3 and the dimension of our search space (the product space of player mixed strategies) is \\(d = n(\\bar{m} - 1) \\leq n\\bar{m}\\) for simplicity."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.198,
+                0.583,
+                0.213
+            ],
+            "angle": 0,
+            "content": "Plugging this information into Theorem 1, we find \\( d' = 0 \\) and"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.271,
+                0.222,
+                0.826,
+                0.259
+            ],
+            "angle": 0,
+            "content": "\\[\nC = \\max  \\left\\{1, \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} S _ {d} ^ {- 1} \\left(r _ {\\eta} ^ {\\frac {\\alpha_ {h i}}{\\alpha_ {l o}}} \\sigma_ {-} ^ {\\left(\\frac {\\alpha_ {h i} - \\alpha_ {l o}}{\\alpha_ {l o} \\alpha_ {h i}}\\right)}\\right) ^ {- d} \\right\\} \\left(\\frac {\\sigma_ {+}}{\\psi \\sigma_ {-} ^ {\\alpha_ {l o} / \\alpha_ {h i}}}\\right) ^ {d / \\alpha_ {l o}} \\tag {115a}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.29,
+                0.262,
+                0.826,
+                0.295
+            ],
+            "angle": 0,
+            "content": "\\[\n= \\max  \\left\\{1, \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} S _ {d} ^ {- 1} r _ {\\eta} ^ {- d} \\right\\} \\left(\\frac {1}{\\psi \\sigma_ {- 2}}\\right) ^ {d / 2} \\tag {115b}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.289,
+                0.297,
+                0.826,
+                0.328
+            ],
+            "angle": 0,
+            "content": "\\[\n\\stackrel {h a r d} {=} \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} S _ {d} ^ {- 1} r _ {\\eta} ^ {- d} \\left(\\frac {3}{\\nu \\sigma_ {- 2}}\\right) ^ {d / 2} \\tag {115c}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.289,
+                0.331,
+                0.826,
+                0.366
+            ],
+            "angle": 0,
+            "content": "\\[\n\\leq^ {h a r d} \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\sqrt {2 \\pi d} \\left(\\frac {d}{2 \\pi e}\\right) ^ {d / 2} \\left(\\frac {3}{\\nu r _ {\\eta} ^ {2} \\sigma_ {- 2}}\\right) ^ {d / 2} \\tag {115d}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.289,
+                0.368,
+                0.826,
+                0.403
+            ],
+            "angle": 0,
+            "content": "\\[\n\\leq^ {h a r d} \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\sqrt {2 \\pi d} \\left(\\frac {b ^ {2} d ^ {2}}{5 r _ {\\eta} ^ {2} \\sigma_ {- 2}}\\right) ^ {d / 2} \\tag {115e}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.289,
+                0.405,
+                0.826,
+                0.441
+            ],
+            "angle": 0,
+            "content": "\\[\n\\stackrel {h a r d} {=} \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\sqrt {2 \\pi n \\bar {m}} \\left(\\frac {b ^ {2} n ^ {2} \\bar {m} ^ {2}}{5 r _ {\\eta} ^ {2} \\sigma_ {- 2}}\\right) ^ {n \\bar {m} / 2} \\tag {115f}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.448,
+                0.825,
+                0.478
+            ],
+            "angle": 0,
+            "content": "where \\( \\text{hard} \\) indicates we are assuming \\( r_{\\eta} \\), the radius of the ball under which the local polynomial bounds are accurate, is small enough to dominate the other operand of the max."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.488,
+                0.825,
+                0.518
+            ],
+            "angle": 0,
+            "content": "Theorem 3 (StoSOO Regret Rate). Corollary 1 of Valko et al. (2013) implies that with probability \\(1 - \\delta\\), the regret, \\(R_{t}\\), of StoSOO after \\(t\\) pulls is upper bounded as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.33,
+                0.526,
+                0.826,
+                0.567
+            ],
+            "angle": 0,
+            "content": "\\[\n(2 + b ^ {2 / d}) \\sqrt {\\frac {\\log_ {b} (t k / \\delta)}{2 \\log_ {b} (e) k}} + \\frac {1}{4} d b ^ {2 (1 + 2 / d)} b ^ {- \\frac {1}{d C} \\sqrt {t / k}} \\tag {116}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.575,
+                0.827,
+                0.604
+            ],
+            "angle": 0,
+            "content": "where \\( d = n(\\bar{m} - 1) \\), \\( b \\) is the branching factor for partitioning cells, \\( C \\) is the near-optimality constant, and \\( k \\) is the maximum number of evaluations per node."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.623,
+                0.825,
+                0.652
+            ],
+            "angle": 0,
+            "content": "Proof. Plugging the constants \\(c, \\gamma\\), and \\(\\nu\\) defined in Table 3 for the 2-norm into Corollary 1 of Valko et al. (2013), we find with probability \\(1 - \\delta\\):"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.295,
+                0.66,
+                0.825,
+                0.84
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} R _ {t} \\leq (2 + 1 / \\gamma) \\epsilon + c \\gamma^ {\\frac {1}{2 C} \\sqrt {t / k} - 2} (117a) \\\\ = (2 + 1 / \\gamma) \\sqrt {\\frac {\\log (t k / \\delta)}{2 k}} + c \\gamma^ {\\frac {1}{2 C} \\sqrt {t / k} - 2} (117b) \\\\ = \\left(2 + b ^ {2 / d}\\right) \\sqrt {\\frac {\\log (t k / \\delta)}{2 k}} + \\frac {1}{4} d b ^ {2} \\left(b ^ {- 2 / d}\\right) ^ {\\frac {1}{2 C} \\sqrt {t / k} - 2} (117c) \\\\ = (2 + b ^ {2 / d}) \\sqrt {\\frac {\\log_ {b} (t k / \\delta)}{2 \\log_ {b} (e) k}} + \\frac {1}{4} d b ^ {2 (1 + 2 / d)} \\left(b ^ {- 2 / d}\\right) ^ {\\frac {1}{2 C} \\sqrt {t / k}} (117d) \\\\ = \\left(2 + b ^ {2 / d}\\right) \\sqrt {\\frac {\\log_ {b} (t k / \\delta)}{2 \\log_ {b} (e) k}} + \\frac {1}{4} d b ^ {2 (1 + 2 / d)} b ^ {- \\frac {1}{d C} \\sqrt {t / k}}. (117e) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.847,
+                0.825,
+                0.86
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.873,
+                0.827,
+                0.926
+            ],
+            "angle": 0,
+            "content": "Proposition 2. Assume the same conditions as Theorem 3 and let \\( k = t\\log_b(t)^{-\\rho} \\) where \\( \\rho \\geq 3 \\). Then the StoSOO (Algorithm 1 of Valko et al. (2013)) bound requires \\( t = b^{\\left(\\frac{dC}{2}\\right)^{\\frac{2}{\\rho - 2}}} \\) pulls before exhibiting a \\( \\tilde{\\mathcal{O}}(T^{-1/2}) \\) decay in regret."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.489,
+                0.949,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "40"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.103,
+                0.827,
+                0.202
+            ],
+            "angle": 0,
+            "content": "Proof. We know from the given definition of \\( k \\) that \\( \\sqrt{t / k} = \\log_b(t)^{\\rho /2} \\). We will analyze the second term in the regret formula of Corollary 3. Note that the second term, \\( b^{-\\frac{1}{dC}\\sqrt{t / k}} = (b\\sqrt{t / k})^{-\\frac{1}{dC}} = (b^{\\log_b(t)^{\\rho /2}})^{-\\frac{1}{dC}} = (b^{\\log_b(t)\\cdot \\log_b(t)^{(\\rho -2) / 2}})^{-\\frac{1}{dC}} = (t^{\\log_b(t)^{(\\rho -2) / 2}})^{-\\frac{1}{dC}} = t^{-\\frac{\\log_b(t)^{(\\rho - 2) / 2}}{dC}} \\). To achieve a \\( t^{-1 / 2} \\) convergence rate (or better), we need \\( t\\geq b^{(\\frac{dC}{2})^{\\frac{2}{\\rho - 2}}} \\). For convenience, we report the entire simplification of the bound below:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.217,
+                0.213,
+                0.826,
+                0.461
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} R _ {t} \\leq (2 + b ^ {2 / d}) \\sqrt {\\frac {\\log_ {b} (t k / \\delta)}{2 \\log_ {b} (e) k}} + \\frac {1}{4} d b ^ {2 (1 + 2 / d)} b ^ {- \\frac {1}{d C}} \\sqrt {t / k} (118a) \\\\ = (2 + b ^ {2 / d}) \\sqrt {\\frac {\\log_ {b} \\left(t ^ {2} / \\left(\\delta \\log_ {b} (t) ^ {\\rho}\\right)\\right) \\log_ {b} (t) ^ {\\rho}}{2 \\log_ {b} (e) t}} + \\frac {1}{4} d b ^ {2 (1 + 2 / d)} t ^ {- \\frac {\\log_ {b} (t) ^ {(\\rho - 2) / 2}}{d C}} (118b) \\\\ \\leq \\left(2 + b ^ {2 / d}\\right) \\sqrt {\\frac {\\log_ {b} \\left(t ^ {2} / \\delta\\right) \\log_ {b} \\left(t / \\delta\\right) ^ {\\rho}}{2 \\log_ {b} (e) t}} + \\frac {1}{4} d b ^ {2 (1 + 2 / d)} t ^ {- \\frac {\\log_ {b} (t) ^ {(\\rho - 2) / 2}}{d C}} (118c) \\\\ = (2 + b ^ {2 / d}) \\sqrt {\\frac {2 \\log_ {b} (t / \\delta) ^ {\\rho + 1}}{2 \\log_ {b} (e) t}} + \\frac {1}{4} d b ^ {2 (1 + 2 / d)} t ^ {- \\frac {\\log_ {b} (t) ^ {(\\rho - 2) / 2}}{d C}} (118d) \\\\ = \\frac {\\left(2 + b ^ {2 / d}\\right)}{\\sqrt {\\log_ {b} (e)}} \\frac {\\log_ {b} (t / \\delta) ^ {(\\rho + 1) / 2}}{\\sqrt {t}} + \\frac {1}{4} d b ^ {2 (1 + 2 / d)} t ^ {- \\frac {\\log_ {b} (t) ^ {(\\rho - 2) / 2}}{d C}} (118e) \\\\ = \\frac {\\left(2 + b ^ {2 / d}\\right)}{\\sqrt {\\log_ {b} (e)}} \\frac {\\left[ \\log_ {b} (t) - \\log_ {b} (\\delta) \\right] ^ {(\\rho + 1) / 2}}{\\sqrt {t}} + \\frac {1}{4} d b ^ {2 (1 + 2 / d)} t ^ {- \\frac {\\log_ {b} (t) ^ {(\\rho - 2) / 2}}{d C}}. (118f) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.469,
+                0.825,
+                0.482
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.503,
+                0.4,
+                0.517
+            ],
+            "angle": 0,
+            "content": "B.6 REGRET TO PAC BOUNDS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.53,
+                0.827,
+                0.587
+            ],
+            "angle": 0,
+            "content": "Lemma 37. [Loss Regret to Exploitability Regret] Assume exploitability of a joint strategy \\(\\pmb{x}\\) is upper bounded by \\(f(\\mathcal{L}^{\\tau}(\\pmb{x}))\\) where \\(f\\) is a concave function and \\(\\mathcal{L}^{\\tau}\\) is a loss function. Let \\(\\pmb{x}_t\\) be a joint strategy randomly drawn from the set of predictions made by an online learning algorithm \\(\\mathcal{A}\\) over \\(T\\) steps. Then the expected exploitability of \\(\\pmb{x}_t\\) is bounded by the average regret of \\(\\mathcal{A}\\):"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.426,
+                0.595,
+                0.825,
+                0.625
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbb {E} _ {t} [ \\epsilon_ {t} ] \\leq f \\left(\\frac {1}{T} R (T)\\right). \\tag {119}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.646,
+                0.218,
+                0.661
+            ],
+            "angle": 0,
+            "content": "Proof."
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.253,
+                0.67,
+                0.825,
+                0.705
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbb {E} _ {t} \\left[ \\epsilon_ {t} \\right] = \\mathbb {E} _ {t} \\left[ f \\left(\\mathcal {L} \\left(\\boldsymbol {x} _ {t}\\right)\\right) \\right] \\leq f \\left(\\mathbb {E} _ {t} \\left[ \\mathcal {L} \\left(\\boldsymbol {x} _ {t}\\right) \\right]\\right) = f \\left(\\frac {1}{T} \\sum_ {t} \\mathcal {L} \\left(\\boldsymbol {x} _ {t}\\right)\\right) = f \\left(\\frac {1}{T} R (T)\\right) \\tag {120}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.714,
+                0.529,
+                0.73
+            ],
+            "angle": 0,
+            "content": "where the inequality follows from Jensen's inequality."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.715,
+                0.825,
+                0.727
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.746,
+                0.827,
+                0.793
+            ],
+            "angle": 0,
+            "content": "Theorem 4 (BLiN PAC Rate). Assume \\(\\eta_{k} = \\eta = 2 / \\hat{L}\\) as defined in Lemma 2, \\(\\tau = \\frac{1}{\\ln(1 / p)}\\) so that all equilibria place at least \\(\\frac{p}{m^*}\\) mass on each strategy, and a previously pulled arm is returned uniformly at random (i.e., \\(t \\sim U(T)\\)). Then for any \\(w > 0\\),"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.27,
+                0.803,
+                0.825,
+                0.841
+            ],
+            "angle": 0,
+            "content": "\\[\n\\epsilon_ {t} \\leq w \\left[ \\tau \\log \\left(\\prod_ {k} m _ {k}\\right) + 2 \\left(1 + \\left(4 c ^ {2} C _ {z}\\right) ^ {1 / 3}\\right) \\sqrt {2 n \\hat {L}} \\left(\\frac {\\ln T}{T}\\right) ^ {\\frac {1}{2 \\left(d _ {z} + 2\\right)}} \\right] \\tag {121}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.851,
+                0.828,
+                0.929
+            ],
+            "angle": 0,
+            "content": "with probability \\((1 - w^{-1})(1 - 2T^{-2})\\) where \\(m^{*} = \\max_{k} m_{k}\\), and \\(c \\leq \\frac{n\\bar{m}}{\\hat{L}}\\left(\\frac{\\ln(m^{*})}{\\ln(1/p)} + 2\\right)^{2}\\) is an upper bound on the range of \\(\\hat{\\mathcal{L}}^{\\tau}\\) (Corollary 1), \\(\\hat{L} = \\left(\\frac{\\ln(m^{*})}{\\ln(1/p)} + 2\\right)\\left(\\frac{m^{*2}}{p\\ln(1/p)} + n\\bar{m}\\right)\\) (Corollary 2), the zooming dimension \\(d_{z} = \\frac{1}{2} n\\bar{m}\\), and the zooming constant \\(C_{z} = |\\mathcal{X}^{*}|^{-1}\\left(\\frac{4}{r_{\\eta}^{2}\\sigma_{-\\infty}}\\right)^{n\\bar{m}}\\) (Corollary 30)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.489,
+                0.949,
+                0.507,
+                0.96
+            ],
+            "angle": 0,
+            "content": "41"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.103,
+                0.827,
+                0.15
+            ],
+            "angle": 0,
+            "content": "Proof. Assume \\(\\eta_{k} = \\eta = \\frac{2}{L}\\) as defined in Lemma 2 so that \\(\\mathcal{L}^{\\tau}\\) is 1-Lipschitz with respect to \\(||\\cdot ||_{\\infty}\\). Also assume a previously pulled arm is returned uniformly at random. Starting with Lemma 13 and applying Corollary 6, we find"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.218,
+                0.158,
+                0.826,
+                0.279
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\mathbb {E} \\left[ \\epsilon_ {t} \\right] \\leq \\tau \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {\\frac {2 n}{\\min  _ {k} \\eta_ {k}}} \\sqrt {\\frac {1}{T} \\sum_ {t} \\mathcal {L} ^ {\\tau} \\left(\\boldsymbol {x} _ {t}\\right)} (122a) \\\\ \\leq \\frac {1}{\\ln (1 / p)} \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {n \\hat {L}} \\sqrt {8 \\left(1 + \\left(4 c ^ {2}\\right) ^ {1 / 3}\\right) ^ {2} T ^ {\\frac {- 1}{\\left(d z + 2\\right)}} \\ln T ^ {\\frac {1}{\\left(d z + 2\\right)}}} (122b) \\\\ = \\frac {1}{\\ln (1 / p)} \\log \\left(\\prod_ {k} m _ {k}\\right) + 2 \\left(1 + \\left(4 c ^ {2} C _ {z}\\right) ^ {1 / 3}\\right) \\sqrt {2 n \\hat {L}} \\left(\\frac {\\ln T}{T}\\right) ^ {\\frac {1}{2 \\left(d _ {z} + 2\\right)}} (122c) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.288,
+                0.825,
+                0.332
+            ],
+            "angle": 0,
+            "content": "with probability \\(1 - 2T^{-2}\\) where \\(m^{*} = \\max_{k} m_{k}\\), and \\(c \\leq \\frac{n\\bar{m}}{\\hat{L}} \\left( \\frac{\\ln(m^{*})}{\\ln(1/p)} + 2 \\right)^{2}\\) is an upper bound on the range of sampled values from \\(\\hat{\\mathcal{L}}^{\\tau}\\) (see Corollary 1)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.337,
+                0.826,
+                0.366
+            ],
+            "angle": 0,
+            "content": "Markov's inequality then allows us to bound the pointwise exploitability of any arm returned by the algorithm as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.246,
+                0.375,
+                0.826,
+                0.412
+            ],
+            "angle": 0,
+            "content": "\\[\n\\epsilon_ {t} \\leq w \\left[ \\frac {1}{\\ln (1 / p)} \\log \\left(\\prod_ {k} m _ {k}\\right) + 2 \\left(1 + \\left(4 c ^ {2} C _ {z}\\right) ^ {1 / 3}\\right) \\sqrt {2 n \\hat {L}} \\left(\\frac {\\ln T}{T}\\right) ^ {\\frac {1}{2 \\left(d _ {z} + 2\\right)}} \\right] \\tag {123}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.421,
+                0.535,
+                0.438
+            ],
+            "angle": 0,
+            "content": "with probability \\((1 - w^{-1})(1 - 2T^{-2})\\) for any \\(w > 0\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.448,
+                0.825,
+                0.48
+            ],
+            "angle": 0,
+            "content": "Theorem 5 (StoSOO PAC Rate). Corollary 1 of Valko et al. (2013) implies that with probability \\((1 - w^{-1})(1 - \\delta)\\) for any \\(w > 0\\), a uniformly randomly drawn arm (i.e., \\(t \\sim U([T])\\)) achieves"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.243,
+                0.487,
+                0.825,
+                0.536
+            ],
+            "angle": 0,
+            "content": "\\[\n\\epsilon_ {t} \\leq w \\left[ \\frac {1}{\\ln (1 / p)} \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {n \\hat {L}} \\sqrt {\\xi_ {1} \\sqrt {\\frac {\\log_ {b} (T k / \\delta)}{2 \\log_ {b} (e) k}} + \\xi_ {2} b ^ {- \\frac {1}{d C} \\sqrt {T / k}}} \\right] \\tag {124}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.169,
+                0.544,
+                0.827,
+                0.59
+            ],
+            "angle": 0,
+            "content": "where \\(d = n(\\bar{m} - 1)\\), \\(\\xi_1 = (2 + 2^{2/d})\\), \\(\\xi_2 = \\frac{1}{4} db^{2(1 + 2/d)}\\), \\(k = T \\log_b(T)^{-3}\\), \\(b\\) is the branching factor for partitioning cells, and the near-optimality constant \\(C = |\\mathcal{X}^*|^{-1}\\sqrt{2\\pi d}\\left(\\frac{b^2d^2}{5r_\\eta^2\\sigma_{-2}}\\right)^{d/2}\\) (Lemma 36)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.609,
+                0.825,
+                0.641
+            ],
+            "angle": 0,
+            "content": "Proof. Assume \\(\\eta_{k} = \\eta = \\frac{2}{L}\\) as defined in Lemma 2. Also assume a previously pulled arm is returned uniformly at random. Starting with Lemma 13 and applying Theorem 3 and Lemma 37, we find"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.225,
+                0.649,
+                0.826,
+                0.741
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\mathbb {E} \\left[ \\epsilon_ {t} \\right] \\leq \\tau \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {\\frac {2 n}{\\min  _ {k} \\eta_ {k}}} \\sqrt {\\frac {1}{T} \\sum_ {t} \\mathcal {L} ^ {\\tau} \\left(\\boldsymbol {x} _ {t}\\right)} (125a) \\\\ \\leq \\frac {1}{\\ln (1 / p)} \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {n \\hat {L}} \\sqrt {\\xi_ {1} \\sqrt {\\frac {\\log_ {b} (T k / \\delta)}{2 \\log_ {b} (e) k}} + \\xi_ {2} b ^ {- \\frac {1}{d C} \\sqrt {T / k}}} (125b) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.749,
+                0.827,
+                0.81
+            ],
+            "angle": 0,
+            "content": "with probability \\(1 - \\delta\\) where \\(m^{*} = \\max_{k} m_{k}\\), \\(d = n(\\bar{m} - 1)\\), \\(\\xi_{1} = (2 + 2^{2 / d})\\), \\(\\xi_{2} = \\frac{1}{4} db^{2(1 + 2 / d)}\\), \\(k = T\\log_b(T)^{-3}\\), \\(b\\) is the branching factor for partitioning cells, and the near-optimality constant \\(C = |\\mathcal{X}^{*}|^{-1}\\sqrt{2\\pi d}\\left(\\frac{b^2d^2}{5r_\\eta^2\\sigma_{-2}}\\right)^{d / 2}\\) (Lemma 36)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.814,
+                0.825,
+                0.844
+            ],
+            "angle": 0,
+            "content": "Markov's inequality then allows us to bound the pointwise exploitability of any arm returned by the algorithm as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.243,
+                0.852,
+                0.826,
+                0.901
+            ],
+            "angle": 0,
+            "content": "\\[\n\\epsilon_ {t} \\leq w \\left[ \\frac {1}{\\ln (1 / p)} \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {n \\hat {L}} \\sqrt {\\xi_ {1} \\sqrt {\\frac {\\log_ {b} (T k / \\delta)}{2 \\log_ {b} (e) k}} + \\xi_ {2} b ^ {- \\frac {1}{d C} \\sqrt {T / k}}} \\right] \\tag {126}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.909,
+                0.505,
+                0.926
+            ],
+            "angle": 0,
+            "content": "with probability \\((1 - w^{-1})(1 - \\delta)\\) for any \\(w > 0\\)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.949,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "42"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.104,
+                0.489,
+                0.119
+            ],
+            "angle": 0,
+            "content": "B.7 COMPLEXITY OF POLYMATRIX GAMES"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.13,
+                0.827,
+                0.175
+            ],
+            "angle": 0,
+            "content": "Lemma 38. For a polymatrix game defined by the set of bimatrix games with payoff matrix \\( P_{kl}^{k} \\) for every player \\( k \\) and \\( l \\neq k \\), the rank of the matrix \\( M(x) \\) defined in equation (11) can be equivalently studied by replacing all instances of \\( H_{kl}^{k} \\) with \\( P_{kl}^{k} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.188,
+                0.828,
+                0.26
+            ],
+            "angle": 0,
+            "content": "Proof. Consider the polymatrix game given by the set of matrices \\( P_{kl}^{k} \\) for every \\( k \\) and \\( l \\neq k \\). The polymatrix game can be equivalently written in normal form, albeit, less concisely. Note that the polymatrix approximation, \\( H_{kl}^{k} \\), as we have defined it (see Section 2 Preliminaries) of this normal-form representation between players \\( k \\) and \\( l \\) with all other players' strategies marginalized out is related to the true underlying bimatrix game between them as follows:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.323,
+                0.265,
+                0.825,
+                0.379
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} H _ {k l} ^ {k} [ a _ {k}, a _ {l} ] = \\mathbb {E} _ {x _ {- k l}} [ u _ {k} (a _ {k}, a _ {l}, x _ {- k l}) ] \\forall a _ {k}, a _ {l} (127a) \\\\ = a _ {k} ^ {\\top} \\left(P _ {k l} ^ {k} a _ {l} + \\sum_ {j \\notin \\{k, l \\}} P _ {k j} ^ {k} x _ {j}\\right) \\forall a _ {k}, a _ {l} (127b) \\\\ = a _ {k} ^ {\\top} P _ {k l} ^ {k} a _ {l} + \\underbrace {a _ {k} ^ {\\top} \\left(\\sum_ {j \\notin \\{k , l \\}} P _ {k j} ^ {k} x _ {j}\\right)} _ {p _ {k}} \\forall a _ {k}, a _ {l} (127c) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.385,
+                0.825,
+                0.414
+            ],
+            "angle": 0,
+            "content": "where \\( P_{kl}^{k} \\) is player \\( k \\)'s payoff matrix for the bimatrix game between players \\( k \\) and \\( l \\) in a polymatrix game and \\( p_k \\) does not depend on player \\( l \\)'s strategy."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.42,
+                0.673,
+                0.436
+            ],
+            "angle": 0,
+            "content": "This implies that \\( H_{kl}^{k} \\) is equal to \\( P_{kl}^{k} \\) up to a constant offset of the rows, i.e.,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.442,
+                0.441,
+                0.825,
+                0.46
+            ],
+            "angle": 0,
+            "content": "\\[\nH _ {k l} ^ {k} = P _ {k l} ^ {k} + C _ {k} \\tag {128}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.464,
+                0.443,
+                0.478
+            ],
+            "angle": 0,
+            "content": "where \\(C_k\\) is a matrix with constant rows."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.484,
+                0.825,
+                0.552
+            ],
+            "angle": 0,
+            "content": "Consider the matrix \\( M(x) \\) which contains \\( H_{kl}^{k} \\) blocks. Recall that the bottom rows of \\( M(x) \\) contain rows of 1's matching each column of \\( H_{kl}^{k} \\) blocks. Consider multiplying the \\( l \\)th row of 1's (which contains 0's on all columns not in the \\( l \\)th block) by \\( \\sqrt{\\eta_k} [I - \\frac{1}{m_k}\\mathbf{1}_k\\mathbf{1}_k^\\top ]C_k \\) and subtracting it from the block containing \\( \\sqrt{\\eta_k} [I - \\frac{1}{m_k}\\mathbf{1}_k\\mathbf{1}_k^\\top ]H_{kl}^k \\),"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.303,
+                0.556,
+                0.825,
+                0.587
+            ],
+            "angle": 0,
+            "content": "\\[\n\\sqrt {\\eta_ {k}} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1} _ {k} \\mathbf {1} _ {k} ^ {\\top} \\right] \\left[ H _ {k l} ^ {k} - C _ {k} \\right] = \\sqrt {\\eta_ {k}} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1} _ {k} \\mathbf {1} _ {k} ^ {\\top} \\right] P _ {k l} ^ {k}. \\tag {129}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.591,
+                0.827,
+                0.665
+            ],
+            "angle": 0,
+            "content": "Note that \\(\\sqrt{\\eta_k} [I - \\frac{1}{m_k}\\mathbf{1}_k\\mathbf{1}_k^\\top ]C_k\\) still remains a matrix with constant rows (the preconditioner effectively subtracts a constant matrix from \\(C_k\\)). This multiplying and subtracting a row from another is an elementary operation on the matrix, meaning it does not change its row rank. Therefore, for a polymatrix game, we can reason about the positive definiteness of the Hessian at equilibria by examining the matrix \\(M(x)\\) with all \\(H_{kl}^{k}\\)'s swapped for \\(P_{kl}^{k}\\)'s."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.679,
+                0.827,
+                0.779
+            ],
+            "angle": 0,
+            "content": "Interestingly, at zero temperature (where \\(\\mathrm{QRE} = \\mathrm{Nash}\\)), \\(M\\) is constant for a polymatrix game, so the rank of this matrix can be computed just once to extract information about all possible interior equilibria in the game. Furthermore, the Hessian is positive semi-definite over the entire joint strategy space, implying the loss function is convex (see Figure 5 (left) for empirical support). This indicates, by convex optimization theory, 1) all mixed Nash equilibria in polymatrix games form a convex set (i.e., they are connected) and 2) assuming mixed equilibria exist, they can be computed simply by stochastic gradient descent on \\(\\mathcal{L}\\). If \\(M\\) is rank-\\(n\\bar{m}\\), then this interior equilibrium is unique."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.791,
+                0.829,
+                0.89
+            ],
+            "angle": 0,
+            "content": "**Complexity** Approximation of Nash equilibria in polymatrix games is known to be PPAD-hard (Deligkas et al., 2022). In contrast, if we restrict our class of polymatrix games to those with at least one interior Nash equilibrium, our analysis proves we can find an approximate Nash equilibrium in deterministic, polynomial time (Corollary 7). This follows directly from the fact that \\(\\mathcal{L}\\) is convex, our decision set \\(\\mathcal{X} = \\prod_{k} \\mathcal{X}_{k}\\) is convex, and convex optimization theory admits polynomial time approximation algorithms (e.g., gradient descent). We consider the assumption of the existence of an interior Nash equilibrium to be relatively mild\\(^3\\), so this positive complexity result is surprising."
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.171,
+                0.898,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "3Marris et al. (2023) shows 2-player, 2-action polymatrix games with interior Nash equilibria make up a non-trivial \\(1/4\\) of the space of possible \\(2 \\times 2\\) games."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.949,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "43"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.825,
+                0.176
+            ],
+            "angle": 0,
+            "content": "Also, note that the Hessian of the loss at Nash equilibria is encoded entirely by the polymatrix approximation at the equilibrium. Therefore, approximating the Hessian of \\(\\mathcal{L}\\) about the equilibrium (which amounts to observing near-equilibrium behavior (Ling et al., 2018)) allows one to recover this polymatrix approximation (up to constant offsets of the columns which equilibria are invariant to (Marris et al., 2023))."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.178,
+                0.829,
+                0.222
+            ],
+            "angle": 0,
+            "content": "Corollary 7 (Approximating Nash Equilibria of Polymatrix Games with Interior Equilibria). Consider the class of polymatrix games with interior Nash equilibria. This class of games admits a fully polynomial time deterministic approximation scheme (FPTAS)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.236,
+                0.828,
+                0.46
+            ],
+            "angle": 0,
+            "content": "Proof. Lemma 3 relates the approximation of Nash equilibria to the minimization of the loss function \\(\\mathcal{L}(\\pmb{x})\\). By Lemma 1, this loss function attains its minimum value of zero if and only if \\(\\pmb{x}\\) is a Nash equilibrium. For polymatrix games, the Hessian of this loss function is everywhere finite and positive definite (Lemma 16), therefore, this loss function is convex. The decision set for this minimization problem is the product space of simplices, therefore it is also convex. Given that we only consider polymatrix games with interior Nash equilibria, we know that our loss function attains a global minimum within this set. By convex optimization theory, this function can be approximately minimized in a polynomial number of steps by, for example, (projected) gradient descent (Boyd and Vandenberghe, 2004). Gradient descent requires computing the gradient of the loss function at each step. From Lemma 14, we see that computing the gradient (at zero temperature) simply requires reading the polymatrix description of the game (i.e., each bi-matrix game \\(H_{kl}^{k}\\) between players), which is clearly polynomial in the size of the input (the polymatrix description). The remaining computational steps of gradient descent (e.g., projection onto simplices) are polynomial as well. In conclusion, gradient descent approximates a Nash equilibrium in polynomial number of steps (logarithmic if strongly-convex (Mairal, 2013)), each of which costs polynomial time, therefore the entire scheme is polynomial."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "44"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.103,
+                0.523,
+                0.119
+            ],
+            "angle": 0,
+            "content": "C EXPERIMENTAL SETUP AND DETAILS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.135,
+                0.512,
+                0.15
+            ],
+            "angle": 0,
+            "content": "Here we provide further details on the experiments."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.168,
+                0.289,
+                0.182
+            ],
+            "angle": 0,
+            "content": "C.1 GAMBIT"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.195,
+                0.828,
+                0.253
+            ],
+            "angle": 0,
+            "content": "The seven methods from the gambit (McKelvey et al., 2016) library that we tested on the 3-player and 4-player Blotto games are listed below (with runtimes). Only gambit-enumpoly and gambit-enumpure are able to return any NE for 3-player Blotto within a 1 hour time limit (and only pure equilibria). And only gambit-enumpure returns any NE for the 4-player game."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.217,
+                0.263,
+                0.613,
+                0.278
+            ],
+            "angle": 0,
+            "content": "- gambit-enumpoly [73 sec 3-player, timeout 4-player]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.217,
+                0.284,
+                0.603,
+                0.299
+            ],
+            "angle": 0,
+            "content": "- gambit-enumpure [72 sec 3-player, 45 sec 4-player]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.217,
+                0.305,
+                0.334,
+                0.318
+            ],
+            "angle": 0,
+            "content": "- gambit-gnm"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.217,
+                0.325,
+                0.333,
+                0.339
+            ],
+            "angle": 0,
+            "content": "- gambit-ipa"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.217,
+                0.346,
+                0.343,
+                0.36
+            ],
+            "angle": 0,
+            "content": "- gambit-liap"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.217,
+                0.366,
+                0.352,
+                0.38
+            ],
+            "angle": 0,
+            "content": "- gambit-logit"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.217,
+                0.386,
+                0.372,
+                0.4
+            ],
+            "angle": 0,
+            "content": "- gambit-simpdiv"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.217,
+                0.263,
+                0.613,
+                0.4
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.419,
+                0.493,
+                0.432
+            ],
+            "angle": 0,
+            "content": "C.2 LOSS VISUALIZATION AND RANK TEST"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.445,
+                0.827,
+                0.489
+            ],
+            "angle": 0,
+            "content": "Figure 5 and claims made in Section 5 analyze several classical matrix games. We report the payoff matrices in standard row-player / column-player payoff form below. All games are then shifted and scaled so payoffs lie in \\([0,1]\\) (i.e., first by subtracting the minimum and then scaling by the max)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.494,
+                0.211,
+                0.507
+            ],
+            "angle": 0,
+            "content": "RPS:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.401,
+                0.515,
+                0.826,
+                0.58
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left[ \\begin{array}{c c c} 0 / 0 & - 1 / 1 & 1 / - 1 \\\\ 1 / - 1 & 0 / 0 & - 1 / 1 \\\\ - 1 / 1 & 1 / - 1 & 0 / 0 \\end{array} \\right]. \\tag {130}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.595,
+                0.236,
+                0.609
+            ],
+            "angle": 0,
+            "content": "Chicken:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.426,
+                0.617,
+                0.826,
+                0.66
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left[ \\begin{array}{c c} 0 / 0 & - 1 / 1 \\\\ 1 / - 1 & - 3 / - 3 \\end{array} \\right]. \\tag {131}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.676,
+                0.299,
+                0.691
+            ],
+            "angle": 0,
+            "content": "Matching Pennies:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.432,
+                0.7,
+                0.826,
+                0.743
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left[ \\begin{array}{l l} 1 / - 1 & - 1 / 1 \\\\ - 1 / 1 & 1 / - 1 \\end{array} \\right]. \\tag {132}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.759,
+                0.306,
+                0.775
+            ],
+            "angle": 0,
+            "content": "Modified-Shapleys:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.383,
+                0.783,
+                0.826,
+                0.849
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left[ \\begin{array}{c c c} 1 / - 0. 5 & 0 / 1 & 0. 5 / 0 \\\\ 0. 5 / 0 & 1 / - 0. 5 & 0 / 1 \\\\ 0 / 1 & 0. 5 / 0 & 1 / - 0. 5 \\end{array} \\right]. \\tag {133}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.863,
+                0.312,
+                0.877
+            ],
+            "angle": 0,
+            "content": "Prisoner's Dilemma:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.419,
+                0.885,
+                0.826,
+                0.928
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left[ \\begin{array}{c c} - 1 / - 1 & - 3 / 0 \\\\ 0 / - 3 & - 2 / - 2 \\end{array} \\right]. \\tag {134}\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "45"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.104,
+                0.426,
+                0.119
+            ],
+            "angle": 0,
+            "content": "C.2.1 LOSS ON FAMILIAR GAMES"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.133,
+                0.828,
+                0.218
+            ],
+            "angle": 0,
+            "content": "We visualize our proposed loss \\(\\mathcal{L}^{\\tau}\\) on two classic 2-player, general-sum games: Chicken, payoff matrix (131), and the Prisoner's Dilemma, payoff matrix (134). Each plot in Figure 5 visualizes the loss at various strategy profiles in probability-space; each strategy profile is represented by each player's probability of playing action 1 of 2 (top row / first column of payoff matrix). Temperature \\(\\tau\\) is varied across the plots. Figure 5 repeats this same visualization but in logit-space to better show the equilibria closest to the boundaries."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.174,
+                0.241,
+                0.826,
+                0.407
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.171,
+                0.417,
+                0.828,
+                0.543
+            ],
+            "angle": 0,
+            "content": "Figure 5: Upper Bound \\((\\epsilon \\leq f_{\\tau}(\\mathcal{L}^{\\tau}))\\) Heatmap Visualization. The first row examines the loss landscape for the classic anti-coordination game of Chicken (Nash equilibria: \\((0,1)\\), \\((1,0)\\), \\((2/3,1/3))\\) while the second row examines the Prisoner's dilemma (Unique Nash equilibrium: \\((0,0))\\). For improved visibility, we subtract the offset \\(\\tau \\log(m^{2})\\) from \\(f_{\\tau}(\\mathcal{L}^{\\tau})\\) per Lemma 13, which relates the exploitability at positive temperature to that at zero temperature. Temperature increases for each plot moving to the right. For high temperatures, interior (fully-mixed) strategies are incentivized while for lower temperatures, nearly pure strategies can achieve minimum exploitability. For zero temperature, pure strategy equilibria (e.g., defect-defect) are not captured by the loss as illustrated by the bottom-left Prisoner's Dilemma plot with a constant loss surface."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.174,
+                0.585,
+                0.825,
+                0.749
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.171,
+                0.761,
+                0.828,
+                0.905
+            ],
+            "angle": 0,
+            "content": "Figure 6: [Figure 5 Repeated in Logit-Space \\((\\ln \\left(\\frac{p}{1 - p}\\right))\\) Rather than Probability-Space \\((p)]\\) Upper Bound \\((\\epsilon \\leq f_{\\tau}(\\mathcal{L}^{\\tau}))\\) Heatmap Visualization. The first row examines the loss landscape for the classic anti-coordination game of Chicken (Nash equilibria: \\((0,1)\\), \\((1,0)\\), \\((2/3,1/3))\\) while the second row examines the Prisoner's dilemma (Unique Nash equilibrium: \\((0,0)\\)). Temperature increases for each plot moving to the right. For improved visibility, we subtract the offset \\(\\tau \\log(m^{2})\\) from \\(f_{\\tau}(\\mathcal{L}^{\\tau})\\) per Lemma 13, which relates the exploitability at positive temperature to that at zero temperature. For high temperatures, interior (fully-mixed) strategies are incentivized while for lower temperatures, nearly pure strategies can achieve minimum exploitability. For zero temperature, pure strategy equilibria (e.g., defect-defect) are not captured by the loss as illustrated by the bottom-left Prisoner's Dilemma plot with a constant loss surface."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.949,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "46"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.104,
+                0.402,
+                0.119
+            ],
+            "angle": 0,
+            "content": "C.3 SADDLE POINT ANALYSIS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.131,
+                0.827,
+                0.23
+            ],
+            "angle": 0,
+            "content": "To generate Figure 2, we follow a procedure similar to the study of MNIST in (Dauphin et al., 2014) (Section 3 of Supp.). Their recommended procedure searches for critical points in two ways. The first repeats a randomized, iterative optimization process 20 times. They then sample one these 20 trials at random, select a random point along the descent trajectory, and search for a critical point (using Newton's method) nearby. They repeat this sampling process 100 times. The second approach randomly selects a feasible point in the decision set and searches for a critical point nearby (again using Newton's method). They also perform this 100 times."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.235,
+                0.828,
+                0.322
+            ],
+            "angle": 0,
+            "content": "Our protocol differs from theirs slightly in a few respects. One, we use SGD, rather than the saddle-free Newton algorithm to trace out an initial descent trajectory. Two, we do not add noise to strategies along the descent trajectory prior to looking for critical points. Thirdly, we minimize gradient norm rather than use Newton's method to look for critical points. Lastly, we use different experimental hyperparameters. We run SGD for 1000 iterations rather than 20 epochs and rerun SGD 100 times rather than 20. We sample 1000 points for each of the two approaches for finding critical points."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.34,
+                0.42,
+                0.354
+            ],
+            "angle": 0,
+            "content": "C.4 SGD ON CLASSICAL GAMES"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.367,
+                0.825,
+                0.397
+            ],
+            "angle": 0,
+            "content": "The games examined in Figure 3 were all taken from (Gemp et al., 2022). Each is available via open source implementations in OpenSpiel (Lanctot et al., 2019) or GAMUT (Nudelman et al., 2004)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.402,
+                0.826,
+                0.459
+            ],
+            "angle": 0,
+            "content": "We compare against several other baselines, replicating the experiments in (Gemp et al., 2022). RM indicates regret-matching and FTRL indicates follow-the-regularized-leader. These are, arguably, the two most popular scalable stochastic algorithms for approximating Nash equilibria. \\( {}^{y}{\\mathrm{{QRE}}}^{\\text{auto }} \\) is a stochastic algorithm developed in (Gemp et al., 2022)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.464,
+                0.828,
+                0.537
+            ],
+            "angle": 0,
+            "content": "For each of the experiments, we sweep over learning rates in log-space from \\(10^{-3}\\) to \\(10^{2}\\) in increments of 1. We also consider whether to run SGD with the projected-gradient and whether to constrain iterates to the simplex via Euclidean projection or entropic mirror descent (Beck and Teboulle, 2003). We then presented the results of the best performing hyperparameters. This was the same approach taken in (Gemp et al., 2022)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.553,
+                0.828,
+                0.689
+            ],
+            "angle": 0,
+            "content": "Saddle Points in Blotto To confirm the existence of saddle points, we computed the Hessian of \\(\\mathcal{L}(\\pmb{x}_{10k})\\) for SGD \\((s = \\infty)\\), deflated the matrix by removing from its eigenvectors all directions orthogonal to the simplex, and then computed its top- \\((n\\bar{m} - n)\\) eigenvalues. We do this because there always exists a \\(n\\)-dimensional nullspace of the Hessian at zero temperature that lies outside the tangent space of the simplex, and we only care about curvature within the tangent space. Specifically, at an equilibrium \\(\\pmb{x}\\), if we compute \\(z^{\\top}Hess(\\mathcal{L})z\\) where \\(z\\) is formed as a linear combination of the vectors \\(\\{[x_1, 0, \\ldots, 0]^{\\top}, \\ldots, [0, \\ldots, x_n]^{\\top}\\}\\), then each block \\(\\tilde{B}_{kl}\\) is identically zero at an equilibrium: \\(\\tilde{B}_{kl}x_l = \\sqrt{\\eta_k[I - \\frac{1}{m_k}\\mathbf{11}^{\\top}]}H_{kl}^k x_l = \\sqrt{\\eta_k\\Pi_{T\\Delta}(\\nabla_{x_k}^k)} = 0\\). By Lemma 16, this implies there is zero curvature of the loss in the direction \\(z\\): \\(z^{\\top}Hess(\\mathcal{L})z = 0\\)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.707,
+                0.418,
+                0.721
+            ],
+            "angle": 0,
+            "content": "C.5 BLIN ON ARTIFICIAL GAME"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.733,
+                0.825,
+                0.763
+            ],
+            "angle": 0,
+            "content": "To construct the 7-player, 2-action, symmetric, artificial game in Figure 4, we used the following coefficients (discovered by trial-and-error):"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.174,
+                0.773,
+                0.845,
+                0.831
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left[ \\begin{array}{c c c c c c c} 0. 0 9 9 0 6 8 7 3 & 0 & 0. 2 3 1 1 6 0 3 7 & 0 & 0. 6 2 7 4 3 5 2 8 & 0 & 0. 1 9 8 1 3 7 4 6 \\\\ 0 & 0. 3 3 0 2 2 9 0 9 & 0 & 0. 0 3 3 0 2 2 9 1 & 0 & 0. 6 2 7 4 3 5 2 8 & 0 \\end{array} \\right]. \\tag {135}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.84,
+                0.829,
+                0.927
+            ],
+            "angle": 0,
+            "content": "The first row indicates the payoffs received when player \\( i \\) plays action 0 and the background population plays any of the possible joint actions (number of combinations with replacement). For example, the first column indicates the payoff when all background players play action 0. The second column indicates all background players play action 0 except for one which plays action 1, and so on. The last column indicates all background players play action 1. These \\( 2n \\) scalars uniquely define the payoffs of a symmetric game."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "47"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.828,
+                0.177
+            ],
+            "angle": 0,
+            "content": "Given that this game only has two actions, we represent a mixed strategy by a single scalar \\( p \\in [0,1] \\), i.e., the probability of the first action. Furthermore, this game is symmetric and we seek a symmetric equilibrium, so we can represent a full Nash equilibrium by this single scalar \\( p \\). This reduces our search space from \\( 7 \\times 2 = 14 \\) variables to 1 variable (and obviates any need for a map \\( s \\) from the unit hypercube to the simplex—see Lemma 22)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "48"
+        }
+    ]
+]
\ No newline at end of file
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+# APPROXIMATING NASH EQUILIBRIA IN NORMAL-FORM GAMES VIA STOCHASTIC OPTIMIZATION
+
+Ian Gemp
+
+DeepMind
+
+London, UK
+
+imgemp@google.com
+
+Luke Marris
+
+DeepMind
+
+London, UK
+
+marris@google.com
+
+Georgios Piliouras
+
+DeepMind
+
+London, UK
+
+gpil@google.com
+
+# ABSTRACT
+
+We propose the first loss function for approximate Nash equilibria of normal-form games that is amenable to unbiased Monte Carlo estimation. This construction allows us to deploy standard non-convex stochastic optimization techniques for approximating Nash equilibria, resulting in novel algorithms with provable guarantees. We complement our theoretical analysis with experiments demonstrating that stochastic gradient descent can outperform previous state-of-the-art approaches.
+
+# 1 INTRODUCTION
+
+Nash equilibrium (NE) famously encodes stable behavioral outcomes in multi-agent systems and is arguably the most influential solution concept in game theory. Formally speaking, if  $n$  players independently choose  $n$ , possibly mixed, strategies  $(x_{k}$  for  $k \in [n]$ ) and their joint strategy  $(x = \prod_{k} x_{k})$  constitutes a Nash equilibrium, then no player has any incentive to unilaterally deviate from their strategy. This concept has sparked extensive research in various fields, ranging from economics (Milgrom and Weber, 1982) to machine learning (Goodfellow et al., 2014), and has even inspired behavioral theory generalizations such as quantal response equilibria (QREs) which allow for more realistic models of boundedly rational agents (McKelvey and Palfrey, 1995).
+
+Unfortunately, when considering Nash equilibria beyond the 2-player, zero-sum scenario, two significant challenges arise. First, it becomes unclear how  $n$  independent players would collectively identify a Nash equilibrium when multiple equilibria are possible, giving rise to the equilibrium selection problem (Harsanyi et al., 1988). Secondly, even approximating a single Nash equilibrium is known to be computationally intractable and specifically PPAD-complete (Daskalakis et al., 2009). Combining both problems together, e.g., testing for the existence of equilibria with welfare greater than some fixed threshold is NP-hard, and it is in fact even hard to approximate (Austrin et al., 2011).
+
+From a machine learning practitioner's perspective, such computational complexity results hardly give pause for thought as collectively we have become all too familiar with the unreasonable effectiveness of heuristics in circumventing such obstacles. Famously, non-convex optimization is NP-hard, even if the goal is to compute a local minimizer (Murty and Kabadi, 1985), however, stochastic gradient descent (SGD) and variants succeed in training billion parameter models (Brown et al., 2020).
+
+Unfortunately, computational techniques for Nash equilibrium have so far not achieved anywhere near the same level of success. In contrast, most modern NE solvers for  $n$ -player,  $m$ -action, general-sum, normal-form games (NFGs) are practically restricted to a handful of players and/or actions per player except in special cases, e.g., symmetric (Wiedenbeck and Brinkman, 2023) or mean-field games (Pérolat et al., 2022). For example, when running the suite of all 7 applicable methods from the hallmark gambit library (McKelvey et al., 2016) on a 4-player Blotto game, we find only brute-force pure-NE enumeration is able to return any NE within a 1 hour time limit. Scaling solvers to large games is difficult partially due to the fact that an NFG is represented by a tensor with an exponential  $nm^n$  entries; even reading this description into memory can be computationally prohibitive. More to the point, any computational technique that presumes exact computation of the expectation of this tensor sampled according to  $x$  similarly does not have any hope of scaling beyond small instances.
+
+This inefficiency arguably lies at the core of the differential success between machine learning (ML) optimization and equilibrium computation. For example, numerous techniques exist that reduce the
+
+problem of Nash computation to the minimization of the expectation of a random variable (Section 3). Unfortunately, unlike the source of randomness in ML applications where batch learning suffices to produce unbiased estimators, these techniques do not extend easily to game theory which incorporates non-linear functions such as maximum and best-response. This raises our motivating goal:
+
+# Can we solve for Nash equilibria via unbiased stochastic optimization?
+
+Our results. Following in the successful steps of the interplay between ML and stochastic optimization, we reformulate the approximation of Nash equilibria in a normal-form game as a stochastic non-convex optimization problem admitting unbiased Monte-Carlo estimation. This enables the use of powerful solvers and advances in parallel computing to efficiently enumerate Nash equilibria for  $n$ -player, general-sum games. Furthermore, this re-casting allows practitioners to incorporate other desirable objectives into the problem such as "find an approximate Nash equilibrium with welfare above  $\omega$ " or "find an approximate Nash equilibrium nearest the current observed joint strategy" resolving the equilibrium selection problem in an effectively ad-hoc and application tailored manner. Concretely, we make the following contributions by producing:
+
+- A loss  $\mathcal{L}^{\tau}(\pmb{x})$  1) whose global minima well approximate Nash equilibria in normal form games, 2) admits unbiased Monte-Carlo estimation, and 3) is Lipschitz and bounded.  
+- Efficient randomized algorithms for approximating Nash equilibria in a novel class of games. The algorithms emerge by employing the family of  $\mathcal{X}$ -armed bandit approaches to  $\mathcal{L}^{\tau}(\pmb{x})$  and connecting their global stochastic optimization guarantees to global approximate Nash guarantees.  
+- An empirical comparison of SGD against state-of-the-art baselines for approximating NEs in large games. In some games, vanilla SGD actually improves upon previous state-of-the-art; in others, SGD is slowed by saddle points, a familiar challenge in deep learning (Dauphin et al., 2014).
+
+Overall, this perspective showcases a promising new route to approximating equilibria at scale in practice. We conclude the paper with discussion for future work.
+
+# 2 PRELIMINARIES
+
+In an  $n$ -player, normal-form game, each player  $k \in \{1, \dots, n\} = [n]$  has a strategy set  $\mathcal{A}_k = \{a_{k1}, \dots, a_{km_k}\}$  consisting of  $m_k$  pure strategies. These strategies can be naturally indexed, so we redefine  $\mathcal{A}_k = [m_k]$  as an abuse of notation. Each player  $k$  also has a utility function,  $u_k: \mathcal{A} = \prod_k \mathcal{A}_k \to [0,1]$ , (equiv. "payoff tensor") that maps joint actions to payoffs in the unit-interval. We denote the average cardinality of the players' action sets by  $\bar{m} = \frac{1}{n} \sum_k m_k$  and maximum by  $m^* = \max_k m_k$ . Player  $k$  may play a mixed strategy by sampling from a distribution over their pure strategies. Let player  $k$ 's mixed strategy be represented by a vector  $x_k \in \Delta^{m_k-1}$  where  $\Delta^{m_k-1}$  is the  $(m_k-1)$ -dimensional probability simplex embedded in  $\mathbb{R}^{m_k}$ . Each function  $u_k$  is then extended to this domain so that  $u_k(\boldsymbol{x}) = \sum_{\boldsymbol{a} \in \mathcal{A}} u_k(\boldsymbol{a}) \prod_j x_{ja_j}$  where  $\boldsymbol{x} = (x_1, \dots, x_n)$  and  $a_j \in \mathcal{A}_j$  denotes player  $j$ 's component of the joint action  $\boldsymbol{a} \in \mathcal{A}$ . For convenience, let  $x_{-k}$  denote all components of  $\boldsymbol{x}$  belonging to players other than player  $k$ .
+
+The joint strategy  $\pmb{x} \in \prod_{k} \Delta^{m_k - 1}$  is a Nash equilibrium if and only if, for all  $k \in [n]$ ,  $u_k(z_k, x_{-k}) \leq u_k(\pmb{x})$  for all  $z_k \in \Delta^{m_k - 1}$ , i.e., no player has any incentive to unilaterally deviate from  $\pmb{x}$ . Nash is typically relaxed with  $\epsilon$ -Nash, our focus:  $u_k(z_k, x_{-k}) \leq u_k(\pmb{x}) + \epsilon$  for all  $z_k \in \Delta^{m_k - 1}$ .
+
+As an abuse of notation, let the atomic action  $a_{k} = e_{k}$  also denote the  $m_{k}$ -dimensional "one-hot" vector with all zeros aside from a 1 at index  $a_{k}$ ; its use should be clear from the context. We also introduce  $\nabla_{x_k}^k$  as player  $k$ 's utility gradient. And for convenience, denote by  $H_{kl}^{k} = \mathbb{E}_{x_{-kl}}[u_{k}(a_{k},a_{l},x_{-kl})]$  the bimatrix game approximation (Janovskaja, 1968) between players  $k$  and  $l$  with all other players marginalized out;  $x_{-kl}$  denotes all strategies belonging to players other than  $k$  and  $l$  and  $u_{k}(a_{k},a_{l},x_{-kl})$  separates out  $l$ 's strategy  $x_{l}$  from the rest of the players  $x_{-k}$ . Similarly, denote by  $T_{klq}^{k} = \mathbb{E}_{x_{-klq}}[u_{k}(a_{k},a_{l},a_{q},x_{-klq})]$  the 3-player tensor approximation to the game. Note player  $k$ 's utility can now be written succinctly as  $u_{k}(x_{k},x_{-k}) = x_{k}^{\top}\nabla_{x_{k}}^{k} = x_{k}^{\top}H_{kl}^{k}x_{l} = T_{klq}^{k}x_{k}x_{l}x_{q}$  for any  $l,q$  where we use Einstein notation for tensor arithmetic. For convenience, define  $\mathrm{diag}(z)$  as the function that places a vector  $z$  on the diagonal of a square matrix, and  $\mathrm{diag}3: z \in \mathbb{R}^{d} \to \mathbb{R}^{d \times d \times d}$  as a 3-tensor of shape  $(d,d,d)$  where  $\mathrm{diag}3(z)_{lll} = z_l$ .
+
+<table><tr><td>Loss</td><td>Function</td><td>Obstacle</td></tr><tr><td>Exploitability (ε)</td><td>maxkεk(x)</td><td>max of r.v.</td></tr><tr><td>Nikaido-Isoda (NI)</td><td>∑kεk(x)</td><td>max of r.v.</td></tr><tr><td>Fully-Diff. Exp</td><td>∑k∑ak∈A[k][max(0,uk(ak,x-k)-uk(x))]2</td><td>max of r.v.</td></tr><tr><td>Gradient-based NI</td><td>NI w/ BRk← aBRk = ΠΔ(xk+η∇xuk(x))</td><td>ΠΔ of r.v.</td></tr><tr><td>Unconstrained</td><td>Loss + Simplex Deviation Penalty</td><td>sampling from xk∈Rmk</td></tr></table>
+
+Table 1: Previous loss functions for NFGs and their obstacles to unbiased estimation. Note that  $\epsilon_{k}(\pmb{x}) = \max_{z}u_{k}(z,x_{-k}) - u_{k}(\pmb{x})$  contains a max operator (see equivalent definition in equation (1)).
+
+Following convention from differential geometry, let  $T_v\mathcal{M}$  be the tangent space of a manifold  $\mathcal{M}$  at  $v$ . For the interior of the  $d$ -action simplex  $\Delta^{d-1}$ , the tangent space is the same at every point, so we drop the  $v$  subscript, i.e.,  $T\Delta^{d-1}$ . We denote the projection of a vector  $z \in \mathbb{R}^d$  onto this tangent space as  $\Pi_{T\Delta^{d-1}}(z) = [I - \frac{1}{d}\mathbf{11}^\top]z$  and call  $\Pi_{T\Delta^{d-1}}(\nabla_{x_k}^k)$  a projected-gradient. We drop  $d-1$  when the dimensionality is clear from the context. Finally, let  $\mathcal{U}(S)$  denote a discrete uniform distribution over elements from set  $S$ .
+
+# 3 RELATED WORK
+
+Representing the problem of computing a Nash equilibrium as an optimization problem is not new. A variety of loss functions and pseudo-distance functions have been proposed. Most of them measure some function of how much each player can exploit the joint strategy by unilaterally deviating:
+
+$$
+\epsilon_ {k} (\boldsymbol {x}) \stackrel {\text {d e f}} {=} u _ {k} \left(\operatorname {B R} _ {k}, x _ {- k}\right) - u _ {k} (\boldsymbol {x}) \text {w h e r e} \operatorname {B R} _ {k} \in \underset {z} {\arg \max } u _ {k} (z, x _ {- k}). \tag {1}
+$$
+
+As argued in the introduction, we believe it is important to be able to subsample payoff tensors of normal-form games in order to scale to large instances. As Nash equilibria can consist of mixed strategies, it is advantageous to be able to sample from an equilibrium to estimate its exploitability  $\epsilon$ . However none of these losses is amenable to unbiased estimation under sampled play. Each of the functions currently explored in the literature is biased under sampled play either because 1) a random variable appears as the argument of a complex, nonlinear (non-polynomial) function or because 2) how to sample play is unclear. Exploitability, Nikaido-Isoda (NI) (Nikaidô and Isoda, 1955) (also known by NashConv (Lanctot et al., 2017) and ADI (Gemp et al., 2022)), as well as fully-differentiable options (Shoham and Leyton-Brown, 2008, p. 106, Eqn 4.31) introduce bias when a max over payoffs is estimated using samples from  $x$ . Gradient-based NI (Raghunathan et al., 2019) requires projecting the result of a gradient-ascent step onto the simplex; for the same reason as the max, this projection is prohibitive because it is a nonlinear operation which introduces bias. Lastly, unconstrained optimization approaches (Shoham and Leyton-Brown, 2008, p. 106) that instead penalize deviation from the simplex lose the ability to sample from strategies when each iterate  $x$  is no longer a distribution (i.e.,  $x_{k} \notin \Delta^{m_{k}-1}$ ). Table 1 summarizes these complications.
+
+# 4 NASH EQUILIBRIUM AS STOCHASTIC OPTIMIZATION
+
+We will now develop our proposed loss function which is amenable to unbiased estimation. Sub-sections 4.1-4.4 provide a warm-up in which we assume an interior (fully-mixed) Nash equilibrium exists. Subsection 4.5 then shows how to relax that assumption allowing us to approximate partially mixed equilibria as well (including pure equilibria). Our key technical insight is to pay special attention to the geometry of the simplex. To our knowledge, prior works have failed to recognize the role of the tangent space  $T\Delta$ . Proofs are in the appendix.
+
+# 4.1 STATIONARITY ON THE SIMPLEX INTERIOR
+
+Lemma 1. Assuming player  $k$ 's utility,  $u_{k}(x_{k}, x_{-k})$ , is concave in its own strategy  $x_{k}$ , a strategy in the interior of the simplex is a best response  $BR_{k}$  if and only if it has zero projected-gradient $^{1}$  norm.
+
+In NFGs, each player's utility is linear in  $x_{k}$ , thereby satisfying the concavity condition of Lemma 1.
+
+# 4.2 PROJECTED-GRADIENT NORM AS A LOSS
+
+An equivalent description of a Nash equilibrium is a joint strategy  $\pmb{x}$  where every player's strategy is a best response to the equilibrium (i.e.,  $x_{k} = \mathbb{R}_{k}$  so that  $\epsilon_k(\pmb {x}) = 0$ ). Lemma 1 states that any interior best response has zero projected-gradient norm, which inspires the following loss function
+
+$$
+\mathcal {L} (\boldsymbol {x}) = \sum_ {k} \eta_ {k} \left\| \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k}\right) \right\| ^ {2} \tag {2}
+$$
+
+where each  $\eta_{k} > 0$  represents a scalar weight, or equivalently, a step size to be explained next.
+
+Proposition 1. The loss  $\mathcal{L}$  is equivalent to NashConv, but where player  $k$ 's best response is approximated by a single step of projected-gradient ascent with step size  $\eta_k$ :  $aBR_k = x_k + \eta_k\Pi_{T\Delta}(\nabla_{x_k}^k)$ .
+
+This connection was already pointed out in prior work for unconstrained problems (Gemp et al., 2022; Raghunathan et al., 2019), but this result is the first for strategies constrained to the simplex.
+
+# 4.3 CONNECTION TO TRUE EXPLOITABILITY
+
+In general, we can bound exploitability in terms of the projected-gradient norm as long as each player's utility is concave (this result extends to subgradients of non-smooth functions).
+
+Lemma 2. The amount a player can gain by exploiting a joint strategy  $\pmb{x}$  is upper bounded by a quantity proportional to the norm of the projected-gradient:
+
+$$
+\epsilon_ {k} (\boldsymbol {x}) \leq \sqrt {2} | | \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k}\right) | |. \tag {3}
+$$
+
+This bound is not tight on the boundary of the simplex, which can be seen clearly by considering  $x_{k}$  to be part of a pure strategy equilibrium. In that case, this analysis assumes  $x_{k}$  can be improved upon by a projected-gradient ascent step (via the equivalence pointed out in Proposition 1). However, that is false because the probability of a pure strategy cannot be increased beyond 1. We mention this to provide further intuition for why our "warm-up" loss  $\mathcal{L}(\boldsymbol{x})$  is only valid for interior equilibria.
+
+Note that  $||\Pi_{T\Delta}(\nabla_{x_k}^k)|| \leq ||\nabla_{x_k}^k||$  because  $\Pi_{T\Delta}$  is a projection. Therefore, this improves the naive bounds on exploitability and distance to best responses given using the "raw" gradient  $\nabla_{x_k}^k$ .
+
+Lemma 3. The exploitability of a joint strategy  $\pmb{x}$ , is upper bounded by a function of  $\mathcal{L}(\pmb{x})$ :
+
+$$
+\epsilon \leq \sqrt {\frac {2 n}{\min  _ {k} \eta_ {k}}} \sqrt {\mathcal {L} (\boldsymbol {x})} \stackrel {\text {d e f}} {=} f (\mathcal {L}). \tag {4}
+$$
+
+# 4.4 UNBIASED ESTIMATION
+
+As discussed in Section 3, a primary obstacle to unbiased estimation of  $\mathcal{L}(\pmb{x})$  is the presence of complex, nonlinear functions of random variables, with the projection of a point onto the simplex being one such example (see  $\Pi_{\Delta}$  in Table 1). However,  $\Pi_{T\Delta}$ , the projection onto the tangent space of the simplex, is linear! This is the insight that allows us to design an unbiased estimator (Lemma 5).
+
+Our proposed loss requires computing the squared norm of the expected value of the projected-gradient under the players' mixed strategies, i.e., the  $l$ -th entry of player  $k$ 's gradient equals  $\nabla_{x_{kl}}^{k} = \mathbb{E}_{a_{-k} \sim x_{-k}} u_{k}(a_{kl}, a_{-k})$ . By analogy, consider a random variable  $Y$ . In general,  $\mathbb{E}[Y]^2 \neq \mathbb{E}[Y^2]$ . This means that we cannot just sample projected-gradients and then compute their average norm to estimate our loss. However, consider taking two independent samples from two corresponding identically distributed, independent random variables  $Y^{(1)}$  and  $Y^{(2)}$ . Then  $\mathbb{E}[Y^{(1)}]^2 = \mathbb{E}[Y^{(1)}] \mathbb{E}[Y^{(2)}] = \mathbb{E}[Y^{(1)}Y^{(2)}]$  by properties of expected value over products of independent random variables. This is a common technique to construct unbiased estimates of expectations over polynomial functions of random variables. Proceeding in this way, define  $\nabla_{x_k}^{k(1)}$  as a random, unbiased gradient estimate (see Table 2). Let  $\nabla_{x_k}^{k(2)}$  be independent and distributed identically to  $\nabla_{x_k}^{k(1)}$ . Then Lemma 5 shows
+
+$$
+\mathcal {L} (\boldsymbol {x}) = \mathbb {E} [ \sum_ {k} \eta_ {k} (\underbrace {\hat {\nabla} _ {x _ {k}} ^ {k (1)} - \frac {1}{m _ {k}} \left(\mathbf {1} ^ {\top} \hat {\nabla} _ {x _ {k}} ^ {k (1)}\right) \mathbf {1}} _ {\text {p r o j e c t e d - g r a d i e n t 1}}) ^ {\top} (\underbrace {\hat {\nabla} _ {x _ {k}} ^ {k (2)} - \frac {1}{m _ {k}} \left(\mathbf {1} ^ {\top} \hat {\nabla} _ {x _ {k}} ^ {k (2)}\right) \mathbf {1}} _ {\text {p r o j e c t e d - g r a d i e n t 2}}) ] \tag {5}
+$$
+
+<table><tr><td></td><td>Exact</td><td>Sample Others</td><td>Sample All</td></tr><tr><td>Estimator of ∇k(xk)</td><td>[uk(akl, x-k)]l</td><td>[uk(akl, a-k ~ x-k)]l</td><td>mkuk(akl ~ U(Ak), a-k ~ x-k)el</td></tr><tr><td>ˆ∇k(xk) Bounds</td><td>[0, 1]</td><td>[0, 1]</td><td>[0, mk]</td></tr><tr><td>ˆ∇k(xk) Query Cost</td><td>Πn{k=1}mk</td><td>mk</td><td>1</td></tr><tr><td>ˆL Bounds</td><td>±1/4 Σkηkmk</td><td>±1/4 Σkηkmk</td><td>±1/4 Σkηkm3</td></tr><tr><td>ˆL Query Cost</td><td>n Πn{k=1}mk</td><td>2nm</td><td>2n</td></tr></table>
+
+Table 2: Examples and Properties of Unbiased Estimators of Loss and Player Gradients  $(\hat{\nabla}_{x_k}^{k(p)})$
+
+where  $\hat{\nabla}_{x_k}^{k(p)}$  is an unbiased estimator of player  $k$ 's gradient. This estimator can be constructed in several ways. The most expensive, an exact estimator, is constructed by marginalizing player  $k$ 's payoff tensor over all other players' strategies. However, a cheaper estimate can be obtained at the expense of higher variance by approximating this marginalization with a Monte Carlo (MC) estimate of the expectation. Specifically, if we sample a single action for each of the remaining players, we can construct an unbiased estimate of player  $k$ 's gradient by considering the payoff of each of its actions against the sampled background strategy. Lastly, we can consider constructing an estimate of player  $k$ 's gradient by sampling only a single action from player  $k$  to represent their entire gradient. Each of these approaches is outlined in Table 2 along with the query complexity (Babichenko, 2016) of computing the estimator and bounds on the values it can take (Lemma 9).
+
+We can extend Lemma 3 to one that holds under  $T$  samples with probability  $1 - \delta$  by applying, for example, a Hoeffding bound:  $\epsilon \leq f\bigl (\hat{\mathcal{L}} (\pmb {x}) + \mathcal{O}\bigl (\sqrt{\frac{1}{T}\ln(1 / \delta)}\bigr)\bigr)$  where  $\hat{\mathcal{L}}$  is an MC estimate of  $\mathcal{L}$ .
+
+# 4.5 INTERIOR EQUILIBRIA
+
+We discussed earlier that  $\mathcal{L}(\pmb{x})$  captures interior equilibria. But some games may only have partially mixed equilibria, i.e., equilibria that lie on the boundary of the simplex. We show how to circumvent this shortcoming by considering quantal response equilibria (QREs), specifically, logit equilibria. By adding an entropy bonus to each player's utility, we can
+
+- guarantee all equilibria are interior,  
+- still obtain unbiased estimates of our loss,  
+- maintain an upper bound on the exploitability  $\epsilon$  of any approximate Nash equilibrium in the original game (i.e., the game without an entropy bonus).
+
+Define  $u_{k}^{\tau}(\boldsymbol{x}) = u_{k}(\boldsymbol{x}) + \tau S(x_{k})$  where Shannon entropy  $S(x_{k}) = -\sum_{l}x_{kl}\ln (x_{kl})$  is 1-strongly concave with respect to the 1-norm (Beck and Teboulle, 2003). It is known that Nash equilibria of entropy-regularized games satisfy the conditions for logit equilibria (Leonardos et al., 2021), which are solutions to the fixed point equation  $x_{k} = \mathrm{softmax}\left(\frac{1}{\tau}\nabla_{x_{k}}^{k}\right)$ . The softmax should make it clear to the reader that all probabilities have positive mass at positive temperature.
+
+Recall that in order to construct an unbiased estimate of our loss, we simply needed to construct unbiased estimates of player gradients. The introduction of the entropy term to player  $k$ 's utility is special in that it depends entirely on known quantities, i.e., the player's own mixed strategy. We can directly and deterministically compute  $\tau \frac{dS}{dx_k} = -\tau (\ln (x_k) + 1)$  and add this to our estimator of  $\nabla_{x_k}^{k(p)}\colon \hat{\nabla}_{x_k}^{k\tau (p)} = \hat{\nabla}_{x_k}^{k(p)} + \tau \frac{dS}{dx_k}$ . Consider our loss function refined from (2) with changes in blue:
+
+$$
+\mathcal {L} ^ {\tau} (\boldsymbol {x}) = \sum_ {k} \eta_ {k} \left| \left| \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) \right| \right| ^ {2}. \tag {6}
+$$
+
+As mentioned above, the utilities with entropy bonuses are still concave, therefore, a similar bound to Lemma 2 applies. We use this to prove the QRE counterpart to Lemma 3 where  $\epsilon_{QRE}$  is the exploitability of an approximate equilibrium in a game with entropy bonuses.
+
+![](images/aa96af27b43a90f0e534e1bbca3272e2e724c9205afedc906d1592c007e9368c.jpg)  
+Figure 1: Effect of Sampled Play on a Biased Loss. The first row displays the expectation of the upper bound guaranteed by our proposed loss  $\mathcal{L}^{\tau}$  with  $\eta_{k} = 1$  for all  $k$ . The second row displays the expectation of NashConv under sampled play, i.e.,  $\sum_{k}\epsilon_{k}$  where  $\epsilon_{k} = \mathbb{E}_{a_{-k}\sim x_{-k}}[\max_{a_{k}}u_{k}^{\tau}(\pmb{a})] - \mathbb{E}_{\pmb{a}\sim \pmb{x}}[u_{k}^{\tau}(\pmb{a})]$ . To be consistent, we subtract the offset  $\tau \log (m^{2})$  from  $f_{\tau}(\mathcal{L}^{\tau})$  per Lemma 13, which relates the exploitability at positive temperature to that at zero temperature. The resulting loss surface clearly shows NashConv fails to recognize any interior Nash equilibrium due to its inherent bias.
+
+Lemma 4. The entropy regularized exploitability,  $\epsilon_{QRE}$ , of a joint strategy  $\pmb{x}$ , is upper bounded as:
+
+$$
+\epsilon_ {Q R E} \leq \sqrt {\frac {2 n}{\min  _ {k} \eta_ {k}}} \sqrt {\mathcal {L} ^ {\tau} (\boldsymbol {x})} \stackrel {\mathrm {d e f}} {=} f \left(\mathcal {L} ^ {\tau}\right). \tag {7}
+$$
+
+Lastly, we establish a connection between quantal response equilibria and Nash equilibria that allows us to approximate Nash equilibria in the original game via minimizing our modified loss  $\mathcal{L}^{\tau}(\pmb{x})$ .
+
+Lemma 13 ( $\mathcal{L}^{\tau}$  Scores Nash Equilibria). Let  $\mathcal{L}^{\tau}(\pmb{x})$  be our proposed entropy regularized loss function and  $\pmb{x}$  be an approximate QRE. Then it holds that
+
+$$
+\epsilon \leq \tau \log \left(\prod_ {k} m _ {k}\right) + \sqrt {\frac {2 n}{\min  _ {k} \eta_ {k}}} \sqrt {\mathcal {L} ^ {\tau} (\boldsymbol {x})} \stackrel {\text {d e f}} {=} f _ {\tau} \left(\mathcal {L} ^ {\tau}\right). \tag {8}
+$$
+
+This upper bound is plotted as a heatmap for a familiar Chicken game in the top row of Figure 1. First, notice how pure equilibria are not visible as minima for zero temperature, but appear for slightly warmer temperatures. Secondly, notice that NashConv in the bottom row is unable to capture the interior Nash equilibrium because of its high bias under sampled play. In contrast, our proposed loss  $\mathcal{L}^{\tau}$  is guaranteed to capture all equilibria at low temperature  $\tau$ .
+
+# 5 ANALYSIS
+
+In the preceding section we established a loss function that upper bounds the exploitability of an approximate equilibrium. In addition, the zeros of this loss function have a one-to-one correspondence with quantal response equilibria (which approximate Nash equilibria at low temperature).
+
+Here, we derive properties that suggest it is "easy" to optimize. While this function is generally non-convex and may suffer from a proliferation of saddle points (Figure 2), it is Lipschitz continuous (over the relevant subset of the interior) and bounded. These are two commonly made assumptions in the literature on non-convex optimization, which we leverage in Section 6. In addition, we can derive its gradient, its Hessian, and characterize its behavior around global minima.
+
+Lemma 14. The gradient of  $\mathcal{L}^{\tau}(\pmb{x})$  with respect to player  $l$ 's strategy  $x_{l}$  is
+
+$$
+\nabla_ {x _ {l}} \mathcal {L} ^ {\tau} (\boldsymbol {x}) = 2 \sum_ {k} \eta_ {k} B _ {k l} ^ {\top} \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) \tag {9}
+$$
+
+where  $B_{ll} = -\tau \left[I - \frac{1}{m_l}\mathbf{11}^\top \right]diag\left(\frac{1}{x_l}\right)$  and  $B_{kl} = \left[I - \frac{1}{m_k}\mathbf{11}^\top \right]H_{kl}^k$  for  $k\neq l$
+
+![](images/3950c7b2214361a1a08a415ef68ef1df7362ca3ac8ec729c3ae280a202991d77.jpg)  
+Figure 2: Analysis of Loss Landscape. We reapply the analysis of (Dauphin et al., 2014), originally designed to understand the success of SGD in deep learning, to "slices" of several popular extensive form games. To construct a slice (or meta-game), we randomly sample 6 deterministic policies and then consider the corresponding  $n$ -player, 6-action normal-form game at  $\tau = 0.1$  (with payoffs normalized to [0, 1]). The index of a critical point  $x_{c}$  ( $\nabla_{\pmb{x}}\mathcal{L}^{\tau}(\pmb{x}_{c}) = \mathbf{0}$ ) indicates the fraction of negative eigenvalues in the Hessian of  $\mathcal{L}^{\tau}$  at  $x_{c}$ ;  $\alpha = 0$  indicates a local minimum, 1 a maximum, else a saddle point. We see a positive correlation between exploitability ( $y$ -axis), projected-gradient norm ( $x$ -axis), and  $\alpha$  (color) indicating a lower prevalence of local minima at high exploitability.
+
+Lemma 16. The Hessian of  $\mathcal{L}^{\tau}(\pmb{x})$  can be written
+
+$$
+\operatorname {H e s s} \left(\mathcal {L} ^ {\tau}\right) = 2 \left[ \tilde {B} ^ {\top} \tilde {B} + T \Pi_ {T \Delta} \left(\tilde {\nabla} ^ {\tau}\right) \right] \tag {10}
+$$
+
+where  $\tilde{B}_{kl} = \sqrt{\eta_k} B_{kl}$ ,  $\Pi_{T\Delta}(\tilde{\nabla}^{\tau}) = [\eta_1\Pi_{T\Delta}(\nabla_{x_1}^{1\tau}),\dots ,\eta_n\Pi_{T\Delta}(\nabla_{x_n}^{n\tau})]$ , and we augment  $T$  (the 3-player approximation to the game,  $T_{lqk}^{k}$ ) so that  $T_{lll}^{l} = \tau \text{diag} 3\left(\frac{1}{x_{l}^{2}}\right)$ .
+
+At an NE, the latter term disappears because  $\Pi_{T\Delta}(\nabla_{x_k}^{k\tau}) = 0$  for all  $k$  (Lemma 1). If  $\mathcal{X}$  was  $\mathbb{R}^{n\bar{m}}$ , then we could simply check if  $\tilde{B}$  is full-rank to determine if  $Hess\succ 0$ , i.e., if  $\mathcal{L}^{\tau}$  is locally strongly-convex. However,  $\mathcal{X}$  is a simplex product, and we only care about curvature in directions toward which we can update our strategy profile  $\pmb{x}$ . Toward that end, define  $M$  to be the  $n(\bar{m} +1)\times n\bar{m}$  matrix that stacks  $\tilde{B}$  on top of a repeated identity matrix that encodes orthogonality to the simplex:
+
+$$
+M (\boldsymbol {x}) = \left[ \begin{array}{c c c c} - \tau \sqrt {\eta_ {1}} \Pi_ {T \Delta} \left(\frac {1}{x _ {1}}\right) & \sqrt {\eta_ {1}} \Pi_ {T \Delta} \left(H _ {1 2} ^ {1}\right) & \dots & \sqrt {\eta_ {1}} \Pi_ {T \Delta} \left(H _ {1 n} ^ {1}\right) \\ \vdots & \vdots & \vdots & \vdots \\ \sqrt {\eta_ {n}} \Pi_ {T \Delta} \left(H _ {n 1} ^ {n}\right) & \dots & \sqrt {\eta_ {n}} \Pi_ {T \Delta} \left(H _ {n, n - 1} ^ {n}\right) & - \tau \sqrt {\eta_ {n}} \Pi_ {T \Delta} \left(\frac {1}{x _ {n}}\right) \\ \mathbf {1} _ {1} ^ {\top} & 0 & \dots & 0 \\ \vdots & \vdots & \vdots & \vdots \\ 0 & \dots & 0 & \mathbf {1} _ {n} ^ {\top} \end{array} \right] \tag {11}
+$$
+
+where  $\Pi_{T\Delta}(z\in \mathbb{R}^{a\times b}) = [I_a - \frac{1}{a}\mathbf{1}_a\mathbf{1}_a^\top ]z$  subtracts the mean from each column of  $z$  and  $\frac{1}{x_k}$  is shorthand for  $\mathrm{diag}\left(\frac{1}{x_k}\right)$ . If  $M(\pmb {x})z = \mathbf{0}$  for a nonzero vector  $z\in \mathbb{R}^{n\bar{m}}$ , this implies there exists a  $z$  that 1) is orthogonal to the ones vectors of each simplex (i.e., is a valid equilibrium update direction) and 2) achieves zero curvature in the direction  $z$ , i.e.,  $z^{\top}(\tilde{B}^{\top}\tilde{B})z = z^{\top}(Hess)z = 0$ , and so  $Hess$  is not positive definite. Conversely, if  $M(\pmb {x})$  is of rank  $nm$  for a quantal response equilibrium  $\pmb{x}$ , then the Hessian of  $\mathcal{L}^{\tau}$  at  $\pmb{x}$  in the tangent space of the simplex product  $(\mathcal{X} = \prod_{k}\mathcal{X}_{k})$  is positive definite. In this case, we call  $\pmb{x}$  polymatrix-isolated: polymatrix because we only require information of the local polymatrix approximation of the game (i.e., the  $H_{kl}^{k}$  matrices) to construct  $M$  and isolated because it implies  $\pmb{x}$  is not connected to any other equilibria.
+
+Definition 1 (Polymatrix-Isolated Equilibrium). A Nash equilibrium  $\pmb{x}^*$  is polymatrix-isolated iff  $\pmb{x}^*$  is isolated according to its local polymatrix game approximation.
+
+By analyzing the rank of  $M$ , we can confirm that many classical matrix games including Rock-Paper-Scissors, Chicken, Matching Pennies, and Shapley's game all induce strongly convex  $\mathcal{L}^{\tau}$ 's at zero temperature (i.e., they have unique mixed Nash equilibria). In contrast, a game like Prisoner's Dilemma has a unique pure strategy that will not be captured by our loss at zero temperature.
+
+# 6 ALGORITHMS
+
+We have formally transformed the approximation of Nash equilibria in NFGs into a stochastic optimization problem. To our knowledge, this is the first such formulation that allows one-shot
+
+![](images/51cefaa3f14fb55640f2d9f76479108debe8b13cf26aeac4adcfc645c166d6c0.jpg)  
+Figure 3: Comparison of SGD on  $\mathcal{L}^{\tau = 0}$  against baselines on four games evaluated in (Gemp et al., 2022). The number of samples used to estimate each update iteration (i.e., minibatch size) is indicated by  $s$ . From left to right: 2-player, 3-action, nonsymmetric; 6-player, 5-action, nonsymmetric; 4-player, 66-action, symmetric; 3-player, 286-action, symmetric. SGD struggles at saddle points in Blotto.
+
+![](images/3355d02bf5dbe469ec7c5b88ce1e5106f509f2401cd6eebfd7612e4d6bd0d2d9.jpg)
+
+![](images/1da09690fe777c5fd7c163640db91b47dcbc7b6997a551d140ea8c7abe63ac0b.jpg)
+
+![](images/cb1645e6ab4d6219c74bec4561b3ac9c887f14a935c2b7586b7ff917bc6a95c6.jpg)
+
+![](images/58308c7ba08d80e379bbefdd3a191d08e19714edaf2140e2b7ad1643afdf437d.jpg)
+
+unbiased Monte-Carlo estimation which is critical to introduce the use of powerful algorithms capable of solving high dimensional optimization problems. We explore two off-the-shelf approaches.
+
+# 6.1 STOCHASTIC GRADIENT DESCENT
+
+Stochastic gradient descent is the workhorse of high-dimensional stochastic optimization. It is guaranteed to converge to stationary points (Cutkosky et al., 2023), however, it may converge to local, rather than global minima. It also enjoys implicit gradient regularization (Barrett and Dherin, 2020), seeking "flat" minima and performs approximate Bayesian inference (Mandt et al., 2017). Despite the lack of global convergence guarantee, we find it performs well empirically in games previously examined by the literature: modified Shapley's (Ostrovski and van Strien, 2014), GAMUT D7 (Nudelman et al., 2004), Blotto (Arad and Rubinstein, 2012). Figure 3 shows SGD is competitive with scalable techniques to approximating NEs: FTRL (Shalev-Shwartz and Singer, 2006; Shalev-Shwartz et al., 2012), Regret Matching (Hart and Mas-Colell, 2000), ADIDAS/  $^yQRRe^{auto}$  (Gemp et al., 2022). Shapley's game induces a strongly convex  $\mathcal{L}$  (see Section 5) leading to SGD's strong performance. Blotto reaches low, but nonzero  $\epsilon$ , demonstrating the challenges of saddle points.
+
+# 6.2 HIGH PROBABILITY, GLOBAL POLYNOMIAL CONVERGENCE RATES VIA BANDITS
+
+We explore one other algorithmic approach to non-convex optimization based on minimizing regret, which enjoys finite time global convergence rates.  $\mathcal{X}$ -armed bandits (Bubeck et al., 2011) systematically explore the space of solutions by refining a mesh over the joint strategy space, trading off exploration versus exploitation of promising regions. Several approaches exist (Bartlett et al., 2019; Valko et al., 2013) with open source implementations, e.g., (Li et al., 2023). Applying  $\mathcal{X}$ -armed bandits to our  $\mathcal{L}^{\tau}$  can be thought of as a stochastic generalization of the exclusion method and other bandit approaches for Nash equilibria (Berg and Sandholm, 2017; Zhou et al., 2017).
+
+Equipped with these techniques, we establish a high probability polynomial-time global convergence rate to Nash equilibria in  $n$ -player, general-sum games given all QREs( $\tau$ ) are polymatrix-isolated. The quality of this approximation improves as  $\tau \to 0$ , at the same time increasing the constant on the convergence rate via the Lipschitz constant  $\sqrt{\hat{L}}$  defined below. For clarity, we assume users provide a temperature in the form  $\tau = \frac{1}{\ln(1/p)}$  with  $p \in (0,1)$  which ensures all equilibria have probability mass greater than  $\frac{p}{m^*}$  for all actions (Lemma 11). Lower  $p$  corresponds with lower temperature.
+
+Theorem 4 (BLiN PAC Rate). Assume  $\eta_{k} = \eta = 2 / \hat{L}$ ,  $\tau = \frac{1}{\ln(1 / p)}$ , and a previously pulled arm is returned uniformly at random (i.e.,  $t \sim U([T])$ ). Then for any  $w > 0$
+
+$$
+\epsilon_ {t} \leq w \left[ \frac {1}{\ln (1 / p)} \log \left(\prod_ {k} m _ {k}\right) + 2 \left(1 + \left(4 c ^ {2} C _ {z}\right) ^ {1 / 3}\right) \sqrt {2 n \hat {L}} \left(\frac {\ln T}{T}\right) ^ {\frac {1}{2 (d z + 2)}} \right] \tag {12}
+$$
+
+with probability  $(1 - w^{-1})(1 - 2T^{-2})$  where  $m^{*} = \max_{k} m_{k}$ ,  $2|\mathcal{L}^{\tau}| \leq c \leq \frac{1}{4}\left(\frac{\ln(m^{*})}{\ln(1 / p)} + 2\right)$  (Corollary 1),  $\hat{L} = \left(\frac{\ln(m^{*})}{\ln(1 / p)} + 2\right)\left(\frac{m^{*2}}{p\ln(1 / p)} + n\bar{m}\right)$  (Corollary 2), the zooming dimension  $d_{z} = \frac{1}{2} n\bar{m}$ , and the zooming constant  $C_{z} = |\mathcal{X}^{*}|^{-1}\left(\frac{1}{4} r_{\eta}^{2}\sigma_{-\infty}\right)^{-n\bar{m}}$  (Corollary 30).
+
+The convergence rate for BLiN (Feng et al., 2022) depends on bounds on the exploitability in terms of the loss (Lemma 13), bounds on estimates of the loss (Corollary 1), Lipschitz bounds on the
+
+![](images/6737fba7ce2c95955cadad5307a45761a3fc43900fed5e9e197dd9926c74b122.jpg)  
+Figure 4: Bandit-based (BLiN) Nash solver applied to an artificial 7-player, symmetric, 2-action game. We search for a symmetric equilibrium, which is represented succinctly as the probability of selecting action 1. The plot shows the true exploitability  $\epsilon$  of all symmetric strategies in black and indicates there exist potentially 5 NEs (the dips in the curve). Upper bounds on our unregularized loss  $\mathcal{L}$  capture 4 of these equilibria, missing only the pure NE on the right. By considering our regularized loss,  $\mathcal{L}^{\tau}$ , we are able to capture this pure NE (see zoomed inset). The bandit algorithm selects strategies to evaluate, using 10 Monte-Carlo samples for each evaluation (arm pull) of  $\mathcal{L}^{\tau}$ . These samples are displayed as vertical bars above with the height of the vertical bar representing additional arm pulls. The best arms throughout search are denoted by green circles (darker indicates later in the search). The boxed numbers near equilibria display the welfare of the strategy profile.
+
+infinity norm of the gradient (Corollary 2), and the number of distinct strategies  $(n\bar{m} = \sum_{k}m_{k})$ . This result further depends on the near-optimality or zooming-dimension  $d_z$  and zooming constant  $C_z$  which quantify the number of near optimal states. In particular, we assume  $\mathcal{L}(s(z))$  is locally  $(\sigma_{-\infty})$ -strongly convex with respect to  $\| \cdot \|_{\infty}$  about each global optimum within a ball of radius  $r_\eta$ . Here,  $s:[0,1]^{n(\bar{m} -1)}\to \prod_i\Delta^{m_i - 1}$  is any function that maps from the unit hypercube to a product of simplices; we analyze two such maps in Appendix B.1. Next, we present an additional convergence rate result using an alternative  $\mathcal{X}$ -bandit approach, StoSOO (Valko et al., 2013).
+
+Theorem 5 (StoSOO Rate). Corollary 1 of Valko et al. (2013) implies that with probability  $(1 - w^{-1})(1 - \delta)$  for any  $w > 0$ , a uniformly randomly drawn arm (i.e.,  $t \sim U([T]))$  achieves
+
+$$
+\epsilon_ {t} \leq w \left[ \frac {1}{\ln (1 / p)} \log \left(\prod_ {k} m _ {k}\right) + \sqrt {n \hat {L}} \sqrt {\xi_ {1} \sqrt {\frac {\log_ {b} (T k / \delta)}{2 \log_ {b} (e) k}} + \xi_ {2} b ^ {- \frac {1}{d C} \sqrt {T / k}}} \right] \tag {13}
+$$
+
+where  $d = n(\bar{m} -1)$ $\xi_{1} = (2 + 2^{2 / d})$ $\xi_{2} = \frac{1}{4} db^{2(1 + 2 / d)}$ $k = T\log_b(T)^{-3}$  ,  $b$  is the branching factor for partitioning cells, and the near-optimality constant  $C = |\mathcal{X}^* |^{-1}\sqrt{2\pi d}\Big(\frac{b^2d^2}{5r_\eta^2\sigma_{-2}}\Big)^{d / 2}$  (Lemma 36).
+
+Here we assume  $\mathcal{L}(s(z))$  is locally  $(\sigma_{-2})$ -strongly convex with respect to  $||\cdot ||_2$  about each global optimum within a ball of radius  $r_\eta$ . Theorem 5 implies a  $\tilde{\mathcal{O}}(T^{-1/4})$  global convergence rate (Proposition 2), however this is achieved only after an exponential number of burn-in iterations.
+
+# 7 CONCLUSION
+
+In this work, we proposed a stochastic loss for approximate Nash equilibria in normal-form games. An unbiased loss estimator of Nash equilibria is the "key" to the stochastic optimization "door" which holds a wealth of research innovations uncovered over several decades. Thus, it allows the development of new algorithmic techniques for computing equilibria. We consider bandit and vanilla SGD methods in this work, but these are only two of the many options now at our disposal (e.g., adaptive methods (Antonakopoulos et al., 2022), Gaussian processes (Calandriello et al., 2022), evolutionary algorithms (Hansen et al., 2003), etc.). Such approaches as well as generalizations of these techniques to extensive-form, imperfect-information games are promising directions for future work. Similarly to how deep learning research first balked at and then marched on to train neural networks via NP-hard non-convex optimization, we hope computational game theory can march ahead to make useful equilibrium predictions of large multiplayer systems.
+
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+
+# APPENDIX: APPROXIMATING NASH EQUILIBRIA IN NORMAL-FORM GAMES VIA STOCHASTIC OPTIMIZATION
+
+# A Loss with its Properties and Derivatives 14
+
+A.1 Loss: Connection to Exploitability, Unbiased Estimation, and Upper Bounds 14  
+A.2 QREs Approximate NEs at Low Temperature 19  
+A.3 Gradient of Loss 21  
+A.4 Bound on Gradient / Lipschitz Property 22  
+A.5 Hessian of Loss 23
+
+# B Global Convergence Guarantees 26
+
+B.1 Maps from Hypercube to Simplex Product 26  
+B.2 Near Optimality & Zooming Dimension 29  
+B.3 D-BLiN 32  
+B.4 Bounded Diameters and Well-shaped Cells 37  
+B.5 Stochastic Simultaneous Optimistic Optimization 39  
+B.6 Regret to PAC Bounds 41  
+B.7 Complexity of Polymatrix Games 43
+
+# C Experimental Setup and Details 45
+
+C.1 GAMBIT 45  
+C.2 Loss Visualization and Rank Test 45  
+C.3 Saddle Point Analysis 47  
+C.4 SGD on Classical Games 47  
+C.5 BLiN on Artificial Game 47
+
+# A LOSS WITH ITS PROPERTIES AND DERIVATIVES
+
+In this section, we construct our loss function and derive many of its properties and derivatives (e.g., gradient and hessian) useful for analyzing and executing optimization algorithms.
+
+# A.1 LOSS: CONNECTION TO EXPLOITABILITY, UNBIASED ESTIMATION, AND UPPER BOUNDS
+
+# A.1.1 KKT CONDITIONS IMPLY FIXED POINT SUFFICIENCY
+
+Consider the following constrained optimization problem:
+
+$$
+\max  _ {\boldsymbol {x} \in \mathbb {R} ^ {d}} f (\boldsymbol {x}) \tag {14a}
+$$
+
+$$
+s. t. g _ {i} (\boldsymbol {x}) \leq 0 \forall i \tag {14b}
+$$
+
+$$
+h _ {j} (\boldsymbol {x}) = 0 \forall j \tag {14c}
+$$
+
+where  $f$  is concave and  $g_{i}$  and  $h_j$  represent inequality and equality constraints respectively. If  $g_{i}$  and  $h_i$  are affine functions, then any maximizer  $x^{*}$  of  $f$  must satisfy the following necessary and sufficient KKT conditions (Ghofogh et al., 2021; Boyd and Vandenberghe, 2004):
+
+- Stationarity:  $\mathbf{0} \in \partial f(\mathbf{x}^{*}) - \sum_{j} \lambda_{j} \partial h_{j}(\mathbf{x}^{*}) - \sum_{i} \mu_{i} \partial g_{i}(\mathbf{x}^{*})$  
+- Primal feasibility:  $h_j(\pmb{x}^*) = 0$  for all  $j$  and  $g_i(\pmb{x}^*) \leq 0$  for all  $i$ .  
+- Dual feasibility:  $\mu_{i} \geq 0$  for all  $i$  
+- Complementary slackness:  $\mu_{i}g_{i}(\pmb{x}^{*}) = 0$  for all  $i$ .
+
+Lemma 1. Assuming player  $k$ 's utility,  $u_{k}(x_{k}, x_{-k})$ , is concave in its own strategy  $x_{k}$ , a strategy in the interior of the simplex is a best response  $BR_{k}$  if and only if it has zero projected-gradient² norm:
+
+$$
+B R _ {k} \in \left(i n t \Delta \cap \underset {z} {\arg \max } u _ {k} (z, x _ {- k}) - u _ {k} (x _ {k}, x _ {- k})\right) \iff (B R _ {k} \in i n t \Delta) \wedge (\| \Pi_ {T \Delta} [ \nabla_ {B R _ {k}} ^ {k} ] \| = 0).
+$$
+
+Proof. Consider the problem of formally computing  $\epsilon_{k}(\pmb{x}) = \max_{z\in int\Delta}u_{k}(z,x_{-k}) - u_{k}(x_{k},x_{-k})$
+
+$$
+\max  _ {z \in \mathbb {R} ^ {d}} u _ {k} (z, x _ {- k}) - u _ {k} \left(x _ {k}, x _ {- k}\right) \tag {15a}
+$$
+
+$$
+s. t. - z _ {k} \leq 0 \forall k \tag {15b}
+$$
+
+$$
+1 - \sum_ {k} z _ {k} = 0. \tag {15c}
+$$
+
+Note that the objective is linear (concave) in  $z$  and the constraints are affine, therefore the KKT conditions are necessary and sufficient for optimality. Recall that we assume that the solution  $z^{*}$  lies in the interior of the simplex, i.e.,  $z_{k}^{*} > 0$  for each  $k$ . Also, let  $e_{k}$  be a onehot vector, i.e., a zeros vector except with a 1 at index  $k$ . Mapping the KKT conditions onto this problem yields the following:
+
+- Stationarity:  $\mathbf{0} \in \partial u_{k}(z^{*}, x_{-k}) + \lambda \mathbf{1} + \sum_{k} \mu_{k} e_{k}$  
+- Primal feasibility:  $\sum_{k}z_{k}^{*} = 1$  for all  $k$  
+- Dual feasibility:  $\mu_{i} \geq 0$  for all  $k$  
+- Complementary slackness:  $-\mu_k z_k^* = 0$  for all  $k$ .
+
+For any point  $z^{*} \in int\Delta$ , primal feasibility will be satisfied. Given our assumption that  $z_{k}^{*} > 0$  by complementary slackness and dual feasibility, each  $\mu_{k}$  must be identically zero. This implies
+
+the stationarity condition can be simplified to  $\mathbf{0} \in \partial u_k(z^*, x_{-k}) + \lambda \mathbf{1}$ . Rearranging terms (and repurposing  $\lambda$ ) we find that for any  $z^*$ , there exists a  $\lambda$  such that
+
+$$
+\lambda \mathbf {1} \in \partial u _ {k} \left(z ^ {*}, x _ {- k}\right). \tag {16}
+$$
+
+Equivalently,  $\partial u_{k}(z^{*},x_{-k})\propto \mathbf{1}$  at  $z^{*}\in int\Delta$  . Any vector proportional to the ones vector has zero projected-gradient norm, completing the claim:  $[I - \frac{1}{m_k}\mathbf{11}^\top ](\lambda \mathbf{1}) = \lambda (\mathbf{1} - \frac{m_k}{m_k}\mathbf{1}) = \mathbf{0}.$
+
+# A.1.2 NORM OF PROJECTED-GRADIENT AND EQUIVALENCE TO NFG EXPLOITABILITY WITH APPROXIMATE BEST RESPONSES
+
+Proposition 1. The loss  $\mathcal{L}$  is equivalent to NashConv, but where player  $k$ 's best response is approximated by a single step of projected-gradient ascent with step size  $\eta_k$ :  $aBR_k = x_k + \eta_k\Pi_{T\Delta}(\nabla_{x_k}^k)$ .
+
+Proof. Define an approximate best response as the result of a player adjusting their strategy via a projected-gradient ascent step, i.e.,  $\mathsf{aBR}_k = x_k + \eta_k\Pi_{T\Delta}(\nabla_{x_k}^k)$  for player  $k$ .
+
+In a normal form game, player  $k$ 's utility at this new strategy is
+
+$$
+u _ {k} \left(\mathrm {a B R} _ {k}, x _ {- k}\right) = \left(\nabla_ {x _ {k}} ^ {k}\right) ^ {\top} \left(x _ {k} + \eta_ {k} \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k}\right)\right) = u _ {k} (\boldsymbol {x}) + \eta_ {k} \left(\nabla_ {x _ {k}} ^ {k}\right) ^ {\top} \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k}\right). \tag {17}
+$$
+
+Therefore, the amount player  $k$  gains by playing aBR is
+
+$$
+\begin{array}{l} \hat {\epsilon} _ {k} (\boldsymbol {x}) = u _ {k} \left(\mathrm {a B R} _ {k}, x _ {- k}\right) - u _ {k} (\boldsymbol {x}) (18a) \\ = \eta_ {k} \left(\nabla_ {x _ {k}} ^ {k}\right) ^ {\top} \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k}\right) (18b) \\ = \eta_ {k} \left(\nabla_ {x _ {k}} ^ {k} - \frac {1}{m _ {k}} \left(\mathbf {1} ^ {\top} \nabla_ {x _ {k}} ^ {k}\right) \mathbf {1}\right) ^ {\top} \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k}\right) (18c) \\ = \eta_ {k} \| \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k}\right) \| ^ {2} (18d) \\ \end{array}
+$$
+
+where the third equality follows from the fact that the projected-gradient,  $\Pi_{T\Delta}(\nabla_{x_k}^k)$ , is orthogonal to the ones vector.
+
+# A.1.3 CONNECTION TO TRUE EXPLOITABILITY
+
+Lemma 2. The amount a player can gain by deviating is upper bounded by a quantity proportional to the norm of the projected-gradient:
+
+$$
+\epsilon_ {k} (\boldsymbol {x}) \leq \sqrt {2} \left\| \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k}\right) \right\|. \tag {19}
+$$
+
+Proof. Let  $z$  be any point on the simplex. Then by concavity of  $u_{k}$  with respect to  $z$ ,
+
+$$
+\begin{array}{l} u _ {k} (z, x _ {- k}) - u _ {k} (\boldsymbol {x}) \leq \left(\nabla_ {x _ {k}} ^ {k}\right) ^ {\top} (z - x _ {k}) (20a) \\ = \left(\nabla_ {x _ {k}} ^ {k}\right) ^ {\top} \left(z - x _ {k}\right) - \frac {1}{m _ {k}} \left(\mathbf {1} ^ {\top} \nabla_ {x _ {k}} ^ {k}\right) \overbrace {\mathbf {1} ^ {\top} \left(z - x _ {k}\right)} ^ {1 - 1 = 0} (20b) \\ = \left(\Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k}\right)\right) ^ {\top} \underbrace {(z - x _ {k})} _ {\text {D i a m} (\Delta) \leq \sqrt {2}} (20c) \\ \leq \sqrt {2} \| \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k}\right) \|. \quad (\text {C a u c h y - S c h w a r z}) (20d) \\ \end{array}
+$$
+
+![](images/8d04e54b845b2ea910ecf254d645b061bf5d3ae51e95829ca35d18bd08202a7e.jpg)
+
+Continuing, we can prove a bound on  $\epsilon$  in terms of the projected-gradient loss:
+
+Lemma 3. The exploitability,  $\epsilon$ , of a joint strategy  $x$ , is upper bounded as a function of our proposed loss:
+
+$$
+\epsilon \leq \sqrt {\frac {2 n}{\min  _ {k} \eta_ {k}}} \sqrt {\mathcal {L} (\boldsymbol {x})}. \tag {21}
+$$
+
+Proof.
+
+$$
+\begin{array}{l} \epsilon = \max  _ {k} \max  _ {z} u _ {k} (z, x _ {- k}) - u _ {k} (\boldsymbol {x}) (22a) \\ \leq \sum_ {k} \max  _ {z} u _ {k} (z, x _ {- k}) - u _ {k} (\boldsymbol {x}) (22b) \\ \leq \sum_ {k} \sqrt {2} \left| \left| \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k}\right) \right| \right| _ {2} \quad (\text {L e m m a 2}) (22c) \\ = \sqrt {2} \left|\left|\left| \Pi_ {T \Delta} \left(\nabla_ {x _ {1}} ^ {1}\right)\right|\right| _ {2}, \dots , \left|\left| \Pi_ {T \Delta} \left(\nabla_ {x _ {n}} ^ {n}\right)\right|\right| _ {2} \right|\left. \right| _ {1} (22d) \\ \leq \sqrt {2 n} \left|\left|\left| \Pi_ {T \Delta} \left(\nabla_ {x _ {1}} ^ {1}\right)\right|\right| _ {2}, \dots , \left|\left| \Pi_ {T \Delta} \left(\nabla_ {x _ {n}} ^ {n}\right)\right|\right| _ {2} \right|\left. \right| _ {2} (22e) \\ = \sqrt {2 n} \sqrt {\sum_ {k} \left| \left| \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k}\right) \right| \right| _ {2} ^ {2}} (22f) \\ = \sqrt {2 n} \sqrt {\sum_ {k} \left(\frac {1}{\eta_ {k}}\right) \eta_ {k} \left| \left| \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k}\right) \right| \right| _ {2} ^ {2}} (22g) \\ \leq \sqrt {\frac {2 n}{\min _ {k} \eta_ {k}}} \sqrt {\sum_ {k} \eta_ {k} \left| \left| \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k}\right) \right| \right| _ {2} ^ {2}} (22h) \\ = \sqrt {\frac {2 n}{\min _ {k} \eta_ {k}}} \sqrt {\mathcal {L} (\boldsymbol {x})}. (22i) \\ \end{array}
+$$
+
+![](images/7333ec7d88ec10819ec71c4882e6574cc3a1fb01b4749ba560ac97c46a1b8ead.jpg)
+
+Lemma 4. The entropy regularized exploitability,  $\epsilon_{QRE}$ , of a joint strategy  $\pmb{x}$ , is upper bounded as a function of our proposed loss:
+
+$$
+\epsilon_ {Q R E} \leq \sqrt {\frac {2 n}{\min _ {k} \eta_ {k}}} \sqrt {\mathcal {L} ^ {\tau} (\boldsymbol {x})}. \tag {23}
+$$
+
+Proof. Recall that  $u_{k}^{\tau}(x_{k}, x_{-k})$  is also concave with respect to  $x_{k}$ . Then
+
+$$
+\begin{array}{l} \epsilon_ {Q R E} = \max  _ {k} \max  _ {z} u _ {k} ^ {\tau} (z, x _ {- k}) - u _ {k} ^ {\tau} (\boldsymbol {x}) (24a) \\ \leq \sum_ {k} \max  _ {z} u _ {k} ^ {\tau} (z, x _ {- k}) - u _ {k} ^ {\tau} (\boldsymbol {x}) (24b) \\ \leq \sum_ {k} \sqrt {2} \left| \left| \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) \right| \right| _ {2} \quad (\text {L e m m a 2}) (24c) \\ = \sqrt {2} \left| \left| | \Pi_ {T \Delta} \left(\nabla_ {x _ {1}} ^ {1 \tau}\right) | | _ {2}, \dots , | | \Pi_ {T \Delta} \left(\nabla_ {x _ {n}} ^ {n \tau}\right) | | _ {2} \right| \right| _ {1} (24d) \\ \leq \sqrt {2 n} \left| \left| \left| \Pi_ {T \Delta} \left(\nabla_ {x _ {1}} ^ {1 \tau}\right) \right| \right| _ {2}, \dots , \left| \left| \Pi_ {T \Delta} \left(\nabla_ {x _ {n}} ^ {n \tau}\right) \right| \right| _ {2} \right| _ {2} (24e) \\ = \sqrt {2 n} \sqrt {\sum_ {k} \left| \left| \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) \right| \right| _ {2} ^ {2}} (24f) \\ \leq \sqrt {2 n} \sqrt {\sum_ {k} \left(\frac {1}{\eta_ {k}}\right) \eta_ {k} \left| \left| \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) \right| \right| _ {2} ^ {2}} (24g) \\ \leq \sqrt {\frac {2 n}{\min _ {k} \eta_ {k}}} \sqrt {\sum_ {k} \eta_ {k} \left| \left| \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) \right| \right| _ {2} ^ {2}} (24h) \\ = \sqrt {\frac {2 n}{\min _ {k} \eta_ {k}}} \sqrt {\mathcal {L} ^ {\tau} (\boldsymbol {x})}. (24i) \\ \end{array}
+$$
+
+# A.1.4 UNBIASED ESTIMATION
+
+Lemma 5. An unbiased estimate of  $\mathcal{L}(\pmb{x})$  can be obtained by drawing two samples (pure strategies) from each players' mixed strategy and observing payoffs.
+
+Proof. Define  $\hat{\nabla}_{x_k}^{k(1)}$  as a random, unbiased gradient estimate (see Table 2). Let  $\hat{\nabla}_{x_k}^{k(2)}$  be independent and distributed identically to  $\hat{\nabla}_{x_k}^{k(1)}$ . Then,
+
+$$
+\begin{array}{l} \mathbb {E} _ {\boldsymbol {a} ^ {(1)} \sim \boldsymbol {x}, \boldsymbol {a} ^ {(2)} \sim \boldsymbol {x}} [ \sum_ {k} \eta_ {k} (\underbrace {\hat {\nabla} _ {x _ {k}} ^ {k (1)} - \frac {1}{m _ {k}} \left(\boldsymbol {1} ^ {\top} \hat {\nabla} _ {x _ {k}} ^ {k (1)}\right) \boldsymbol {1}} _ {\text {p r o j e c t e d - g r a d i e n t 1}}) ^ {\top} (\underbrace {\hat {\nabla} _ {x _ {k}} ^ {k (2)} - \frac {1}{m _ {k}} \left(\boldsymbol {1} ^ {\top} \hat {\nabla} _ {x _ {k}} ^ {k (2)}\right) \boldsymbol {1}} _ {\text {p r o j e c t e d - g r a d i e n t 2}}) ] (25a) \\ = \sum_ {k} \eta_ {k} \mathbb {E} _ {\boldsymbol {a} ^ {(1)} \sim \boldsymbol {x}, \boldsymbol {a} ^ {(2)} \sim \boldsymbol {x}} [ (\hat {\nabla} _ {x _ {k}} ^ {k (1)} - \frac {1}{m _ {k}} (\boldsymbol {1} ^ {\top} \hat {\nabla} _ {x _ {k}} ^ {k (1)}) \boldsymbol {1}) ^ {\top} (\hat {\nabla} _ {x _ {k}} ^ {k (2)} - \frac {1}{m _ {k}} (\boldsymbol {1} ^ {\top} \hat {\nabla} _ {x _ {k}} ^ {k (2)}) \boldsymbol {1}) ] (25b) \\ = \sum_ {k} \eta_ {k} \mathbb {E} _ {\boldsymbol {a} ^ {(1)} \sim \boldsymbol {x}} \left[ \left(\hat {\nabla} _ {x _ {k}} ^ {k (1)} - \frac {1}{m _ {k}} \left(\boldsymbol {1} ^ {\top} \hat {\nabla} _ {x _ {k}} ^ {k (1)}\right) \boldsymbol {1}\right) \right] ^ {\top} \mathbb {E} _ {\boldsymbol {a} ^ {(2)} \sim \boldsymbol {x}} \left[ \left(\hat {\nabla} _ {x _ {k}} ^ {k (2)} - \frac {1}{m _ {k}} \left(\boldsymbol {1} ^ {\top} \hat {\nabla} _ {x _ {k}} ^ {k (2)}\right) \boldsymbol {1}\right) \right] (25c) \\ = \sum_ {k} \eta_ {k} \left(\nabla_ {x _ {k}} ^ {k} - \frac {1}{m _ {k}} \left(\mathbf {1} ^ {\top} \nabla_ {x _ {k}} ^ {k}\right) \mathbf {1}\right) ^ {\top} \left(\nabla_ {x _ {k}} ^ {k} - \frac {1}{m _ {k}} \left(\mathbf {1} ^ {\top} \nabla_ {x _ {k}} ^ {k}\right) \mathbf {1}\right) (25d) \\ = \sum_ {k} \eta_ {k} \left\| \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k}\right) \right\| ^ {2} (25e) \\ = \mathcal {L} (\boldsymbol {x}) (25f) \\ \end{array}
+$$
+
+where the first equality follows from linearity of expectation, the second from independence of random variables, and the third from  $\mathbb{E}_{\pmb{a}^{(p)}\sim \pmb{x}}[\hat{\nabla}_{x_k}^{k(p)}] = \nabla_{x_k}^k$ , i.e.,  $\hat{\nabla}_{x_k}^{k(p)}$  is an unbiased estimator of player  $k$ 's gradient. Therefore, equation (25a) comprises an unbiased estimate of  $\mathcal{L}(\pmb{x})$  proving the claim.
+
+![](images/c3bd132c0bdd7fca6af0f73d73f20fb90490aaef865754c004ed18ad5e37e94d.jpg)
+
+Lemma 6. The loss formed as the sum of the squared norms of the projected-gradients,  $\mathcal{L}^{\tau}$ , can be decomposed into three terms as follows:
+
+$$
+\mathcal {L} ^ {\tau} (\boldsymbol {x}) = \underbrace {\sum_ {k} \eta_ {k} x _ {q} ^ {\top} B _ {k q} ^ {\top} B _ {k q} x _ {q}} _ {(A)} + \underbrace {2 \sum_ {k} \eta_ {k} E _ {k} ^ {\top} B _ {k q} x _ {q}} _ {(B)} + \underbrace {\sum_ {k} \eta_ {k} E _ {k} ^ {\top} E _ {k}} _ {(C)} \tag {26}
+$$
+
+where  $q$  is any player other than  $k$ .
+
+Proof. Let  $S^{\tau} = -\tau \sum_{l} x_{kl} \ln(x_{kl})$  so that  $\frac{\partial S^{\tau}}{\partial x_k} = -\tau (\ln(x_k) + 1)$ . Note that  $\Pi_{T\Delta}[\frac{\partial S^{\tau}}{\partial x_k}] = -\tau \Pi_{T\Delta}[\ln(x_k)]$ .
+
+$$
+\begin{array}{l} \mathcal {L} ^ {\tau} (\boldsymbol {x}) = \sum_ {k} \eta_ {k} \left(\Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right)\right) ^ {\top} \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) (27a) \\ = \sum_ {k} \eta_ {k} \left[ H _ {k q} ^ {k} x _ {q} + \frac {\partial S ^ {\tau}}{\partial x _ {k}} \right] ^ {\top} \left[ I - \frac {1}{m _ {k}} \mathbf {1 1} ^ {\top} \right] \left[ I - \frac {1}{m _ {k}} \mathbf {1 1} ^ {\top} \right] \left[ H _ {k q} ^ {k} x _ {q} + \frac {\partial S ^ {\tau}}{\partial x _ {k}} \right] (27b) \\ = \sum_ {k} \eta_ {k} \left(x _ {q} ^ {\top} \left[ H _ {k q} ^ {k} \right] ^ {\top} \left[ I - \frac {1}{m _ {k}} \mathbf {1 1} ^ {\top} \right] ^ {2} \left[ H _ {k q} ^ {k} \right] x _ {q} + 2 \left[ \frac {\partial S ^ {\tau}}{\partial x _ {k}} \right] ^ {\top} \left[ I - \frac {1}{m _ {k}} \mathbf {1 1} ^ {\top} \right] ^ {2} \left[ H _ {k q} ^ {k} x _ {q} \right] \quad (2 7 c) \right. \\ \left. + \left[ \frac {\partial S ^ {\tau}}{\partial x _ {k}} \right] ^ {\top} \left[ I - \frac {1}{m _ {k}} \mathbf {1 1} ^ {\top} \right] ^ {2} \left[ \frac {\partial S ^ {\tau}}{\partial x _ {k}} \right]\right) (27d) \\ = \underbrace {\sum_ {k} \eta_ {k} x _ {q} ^ {\top} B _ {k q} ^ {\top} B _ {k q} x _ {q}} _ {(A)} + \underbrace {2 \sum_ {k} \eta_ {k} E _ {k} ^ {\top} B _ {k q} x _ {q}} _ {(B)} + \underbrace {\sum_ {k} \eta_ {k} E _ {k} ^ {\top} E _ {k}} _ {(C)} (27e) \\ \end{array}
+$$
+
+where  $B_{kq} = [I - \frac{1}{m_k}\mathbf{11}^\top ]H_{kq}^k$  and  $E_{k} = [I - \frac{1}{m_{k}}\mathbf{11}^{\top}][\frac{\partial S^{\tau}}{\partial x_{k}} ] = -\tau [I - \frac{1}{m_{k}}\mathbf{11}^{\top}]\ln (x_{k})$
+
+# A.1.5 BOUND ON LOSS
+
+By Proposition 1, Equation (18d), we can also rewrite this loss as a weighted sum of 2-norms,  $\mathcal{L}(\pmb{x}) = \sum_{k}\eta_{k}||\nabla_{x_{k}}^{k} - \mu_{k}||_{2}^{2}$  where  $\mu_{k} = \frac{1}{m_{k}} (\mathbf{1}^{\top}\nabla_{x_{k}}^{k})\in [0,1]$  for brevity. This will allow us to more easily analyze our loss.
+
+Lemma 7. Assume payoffs are bounded in  $[0,1]$ , then setting  $\eta_k \leq \frac{4}{nm_k}$  or  $\eta_k \leq \frac{4}{nm}$  or  $\sum_k \eta_k \leq \frac{4}{m}$  ensures  $0 \leq \mathcal{L}(x) \leq 1$  for all  $x \in \mathcal{X}$ .
+
+Proof.
+
+$$
+\begin{array}{l} 0 \leq \mathcal {L} (\boldsymbol {x}) = \sum_ {k} \eta_ {k} \| \nabla_ {x _ {k}} ^ {k} - \mu_ {k} \| _ {2} ^ {2} (28a) \\ = \sum_ {k} \eta_ {k} m _ {k} \left[ \frac {1}{m _ {k}} \sum_ {l} \left(\nabla_ {x _ {k l}} ^ {k} - \mu_ {k}\right) ^ {2} \right] (28b) \\ = \sum_ {k} \eta_ {k} m _ {k} \operatorname {V a r} \left[ \nabla_ {x _ {k}} ^ {k} \right] (28c) \\ \leq \frac {1}{4} \sum_ {k} \eta_ {k} m _ {k} (28d) \\ \leq \frac {1}{4} \left(\max  _ {k} \eta_ {k}\right) \left(\sum_ {k} m _ {k}\right) (28e) \\ = \frac {1}{4} \left(\max  _ {k} \eta_ {k}\right) n \bar {m} \leq 1 (28f) \\ \Rightarrow \left(\max  _ {k} \eta_ {k}\right) \leq \frac {4}{n \bar {m}} (28g) \\ \end{array}
+$$
+
+where the first inequality follows from Popoviciu's inequality (Popoviciu, 1935): the variance of a bounded random variable  $X$  is upper bounded by  $Var[X] \leq \frac{1}{4} (\max_{X} - \min_{X})^{2}$ .
+
+Next, we establish the following useful lemmas.
+
+Lemma 8. The matrix  $I - \frac{1}{m_k} \mathbf{11}^\top$  is a projection matrix and therefore idempotent. It is also symmetric, which implies it is its own square root.
+
+Proof.
+
+$$
+\begin{array}{l} \left[ I - \frac {1}{m _ {k}} \mathbf {1 1} ^ {\top} \right] ^ {\top} \left[ I - \frac {1}{m _ {k}} \mathbf {1 1} ^ {\top} \right] = I - \frac {2}{m _ {k}} \mathbf {1 1} ^ {\top} + \frac {1}{m _ {k} ^ {2}} \mathbf {1} \left(\mathbf {1} ^ {\top} \mathbf {1}\right) \mathbf {1} ^ {\top} (29a) \\ = I - \frac {2}{m _ {k}} \mathbf {1 1} ^ {\top} + \frac {1}{m _ {k}} \mathbf {1 1} ^ {\top} (29b) \\ = \left[ I - \frac {1}{m _ {k}} \mathbf {1 1} ^ {\top} \right]. (29c) \\ \end{array}
+$$
+
+Lemma 9. The product  $A[I_m - \frac{1}{m}\mathbf{1}_m\mathbf{1}_m^\top ]^p B$  for any integer  $p > 0$  has entries whose absolute value is bounded by  $\frac{m}{4} (A_{\mathrm{max}} - A_{\mathrm{min}})(B_{\mathrm{max}} - B_{\mathrm{min}})$  where  $A_{\mathrm{min}}, A_{\mathrm{max}}, B_{\mathrm{min}}, B_{\mathrm{max}}$  represent the minima and maxima of the matrices respectively.
+
+Proof. The matrix  $[I - \frac{1}{m}\mathbf{11}^{\top}]$  is idempotent (Lemma 8) so we can rewrite the product for any  $p$  as
+
+$$
+A \left[ I - \frac {1}{m} \mathbf {1 1} ^ {\top} \right] \left[ I - \frac {1}{m} \mathbf {1 1} ^ {\top} \right] B. \tag {30}
+$$
+
+The matrix  $[I - \frac{1}{m}\mathbf{11}^{\top}]$  has the property that it removes the mean from every row of a matrix when right multiplied against it, i.e.,  $A[I - \frac{1}{m}\mathbf{11}^{\top}]$  removes the means from the rows of  $A$ . Similarly, left multiplying it removes the means from the columns. Let  $\tilde{A}$  and  $\tilde{B}$  represent these mean-centered
+
+results respectively. The absolute value of the  $ij$ th entry in the resulting product can then be recognized as
+
+$$
+\begin{array}{l} \left| \sum_ {k} \tilde {A} _ {i k} \tilde {B} _ {k j} \right| = \left| \sum_ {k} \left(A _ {i k} - \frac {1}{m} \sum_ {k ^ {\prime}} A _ {i k ^ {\prime}}\right) \left(B _ {k j} - \frac {1}{m} \sum_ {k ^ {\prime}} B _ {k ^ {\prime} j}\right) \right| (31a) \\ = \left| m \cdot \operatorname {C o r r} \left(A _ {i,}, B _ {. j}\right) \cdot \sigma_ {A _ {i,}} \sigma_ {B _ {. j}} \right| (31b) \\ \leq m \sigma_ {A _ {i,}} \sigma_ {B _ {\cdot , j}}. (31c) \\ \end{array}
+$$
+
+By Popoviciu's inequality (Popoviciu, 1935), we know the variance of a bounded random variable  $X$  is upper bounded by  $Var[X] \leq \frac{1}{4} (\max_{X} - \min_{X})^{2}$ . Hence its standard deviation is bounded by  $Std[X] \leq \frac{1}{2} (\max_{X} - \min_{X})$ . Plugging these bounds for  $A$  and  $B$  into equation (31c) completes the claim.
+
+Lemma 10. Assume payoffs are bounded in  $[0,1]$ , then
+
+$$
+\left| \mathcal {L} ^ {\tau} (\boldsymbol {x}) \right| \leq \frac {1}{4} \left(\max  _ {k} \eta_ {k}\right) n \bar {m} \left(\tau \ln \left(\frac {1}{x _ {\operatorname* {m i n}}}\right) + 1\right) ^ {2} \tag {32}
+$$
+
+for any  $\pmb{x}$  such that  $x_{kl} \geq x_{\min} \forall k, l$ .
+
+Proof. First note that for payoffs in [0, 1], the entries in
+
+$$
+\Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) = \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k} - \tau (\ln (x _ {k}) + 1)\right) = \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k} - \tau \ln (x _ {k})\right) \tag {33}
+$$
+
+are bounded within  $[0, \tau \ln(\frac{1}{x_{\min}}) + 1]$  with a range of  $\tau \ln(\frac{1}{x_{\min}}) + 1$ . Then starting from the definition of  $\mathcal{L}^{\tau}$  and applying Lemma 9, we find
+
+$$
+\begin{array}{l} \left| \mathcal {L} ^ {\tau} (\boldsymbol {x}) \right| = \left| \sum_ {k} \eta_ {k} \left\| \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) \right\| ^ {2} \right| (34a) \\ \leq \frac {1}{4} \sum_ {k} \eta_ {k} m _ {k} \left(\tau \ln \left(\frac {1}{x _ {\operatorname* {m i n}}}\right) + 1\right) ^ {2} (34b) \\ \leq \frac {1}{4} \left(\tau \ln \left(\frac {1}{x _ {\operatorname* {m i n}}}\right) + 1\right) ^ {2} \left(\max  _ {k} \eta_ {k}\right) \sum_ {k} m _ {k} (34c) \\ = \frac {1}{4} \left(\max  _ {k} \eta_ {k}\right) n \bar {m} \left(\tau \ln \left(\frac {1}{x _ {\min }}\right) + 1\right) ^ {2}. (34d) \\ \end{array}
+$$
+
+![](images/c6c001f8226b024cd6deced7aa443ebd133fd49c815f6cf05d3b80cc418bb6e8.jpg)
+
+# A.2 QRES APPROXIMATE NES AT LOW TEMPERATURE
+
+Lemma 11. Setting  $\tau = \ln (1 / p)^{-1}$  with  $p\in (0,1)$  and payoffs in [0,1] ensures that any QRE contains probabilities greater than  $\frac{p}{\max_k m_k}$ .
+
+Proof. Let  $m^{*} = \max_{k} m_{k}$  and  $\nabla_{x_{k}}^{k}$  be player  $k$ 's gradient. Recall that a QRE( $\tau$ ), specifically a logit equilibrium, satisfies the fixed point equation  $x_{k} = \text{softmax}\left(\frac{\nabla_{x_{k}}^{k}}{\tau}\right)$  for all  $k$ . Then the probability for any action in a QRE strategy profile,  $\mathbf{x}$ , is lower bounded as
+
+$$
+\begin{array}{l} x _ {k l} \geq \min  _ {k} \min  _ {l} \left\{x _ {k l} \right\} \geq \min  _ {\boldsymbol {x} ^ {\prime} \mid \boldsymbol {x} ^ {\prime} \text {i s Q R E}} \min  _ {k} \min  _ {l} \left\{x _ {k l} ^ {\prime} \right\} (35a) \\ = \min  _ {\nabla_ {x _ {k}} ^ {k}} \min  _ {k} \min  _ {l} \left[ \operatorname {s o f t m a x} \left(\frac {\nabla_ {x _ {k}} ^ {k}}{\tau}\right) \right] _ {l} (35b) \\ = \frac {e ^ {0}}{\left(m ^ {*} - 1\right) e ^ {\frac {1}{\tau}} + e ^ {0}} (35c) \\ \stackrel {\text {d e f}} {=} x _ {\min } (35d) \\ \end{array}
+$$
+
+where equation (35c) follows from minimizing the numerator and maximizing the denominator of the softmax formula subject to the payoff constraints.
+
+Rearranging terms, we find
+
+$$
+e ^ {\frac {1}{\tau}} = \frac {1}{m ^ {*} - 1} \left(\frac {1}{x _ {\min }} - 1\right) \Rightarrow \tau = \frac {1}{\ln \left(\frac {1}{m ^ {*} - 1} \left(\frac {1}{x _ {\min }} - 1\right)\right)}. \tag {36}
+$$
+
+Let  $p\in (0,1)$  such that  $x_{\mathrm{min}} = \frac{p}{\max_km_k} = \frac{p}{m^*}$ , then
+
+$$
+\begin{array}{l} \tau = \frac {1}{\ln \left(\frac {1}{m ^ {*} - 1} \left(\frac {1}{x _ {\operatorname* {m i n}}} - 1\right)\right)} (37a) \\ = \frac {1}{\ln \left(\frac {1}{m ^ {*} - 1} \left(\frac {m ^ {*}}{p} - 1\right)\right)} (37b) \\ = \frac {1}{\ln \left(\frac {m ^ {*} - p}{m ^ {*} - 1} \frac {1}{p}\right)} (37c) \\ \leq \frac {1}{\ln \left(\frac {1}{p}\right)}. (37d) \\ \end{array}
+$$
+
+This implies if we set  $\tau = \ln (1 / p)^{-1}$ , then we are guaranteed that all QREs contain probabilities greater than  $x_{\min} = \frac{p}{\max_k m_k}$ .
+
+Corollary 1. Assume payoffs are bounded in  $[0,1]$  and  $\tau = \ln (1 / p)^{-1}$ , then
+
+$$
+\left| \mathcal {L} ^ {\tau} (\boldsymbol {x}) \right| \leq \frac {1}{4} \left(\max  _ {k} \eta_ {k}\right) n \bar {m} \left(\frac {\ln \left(m ^ {*}\right)}{\ln (1 / p)} + 2\right) ^ {2}. \tag {38}
+$$
+
+Proof. Starting with Lemma 10 and applying Lemma 11, we find
+
+$$
+\begin{array}{l} \left| \mathcal {L} ^ {\tau} (\boldsymbol {x}) \right| \leq \frac {1}{4} \left(\max  _ {k} \eta_ {k}\right) n \bar {m} \left(\tau \ln \left(\frac {1}{x _ {\min }}\right) + 1\right) ^ {2} (39a) \\ = \frac {1}{4} \left(\max  _ {k} \eta_ {k}\right) n \bar {m} \left(\frac {1}{\ln (1 / p)} \ln \left(\frac {m ^ {*}}{p}\right) + 1\right) ^ {2} (39b) \\ = \frac {1}{4} \left(\max  _ {k} \eta_ {k}\right) n \bar {m} \left(\frac {1}{\ln (1 / p)} \left(\ln \left(m ^ {*}\right) + \ln (1 / p)\right) + 1\right) ^ {2} (39c) \\ = \frac {1}{4} \left(\max  _ {k} \eta_ {k}\right) n \bar {m} \left(\frac {\ln \left(m ^ {*}\right)}{\ln (1 / p)} + 2\right) ^ {2}. (39d) \\ \end{array}
+$$
+
+Lemma 12 (Low Temperature Approximate QREs are Approximate Nash Equilibria). Let  $\nabla_{x_k}^{k\tau}$  be player  $k$  's entropy regularized gradient and  $\pmb{x}$  be an approximate QRE. Then it holds that
+
+$$
+u _ {k} \left(B R _ {k}, x _ {- k}\right) - u _ {k} (\boldsymbol {x}) \leq \tau \log \left(m _ {k}\right) + \sqrt {2} \left\| \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) \right\|. \tag {40}
+$$
+
+Proof. Beginning with the definition of exploitability, we find
+
+$$
+\begin{array}{l} u _ {k} \left(\mathbb {B R} _ {k}, x _ {- k}\right) - u _ {k} (\boldsymbol {x}) = \left(u _ {k} \left(\mathbb {B R} _ {k}, x _ {- k}\right) + \tau S \left(\mathbb {B R} _ {k}\right) - \tau S \left(\mathbb {B R} _ {k}\right)\right) (41a) \\ - \left(u _ {k} (\boldsymbol {x}) + \tau S (x _ {k}) - \tau S (x _ {k})\right) \\ = u _ {k} ^ {\tau} \left(\mathbb {B R} _ {k}, x _ {- k}\right) - u _ {k} ^ {\tau} (\boldsymbol {x}) + \tau \left(S \left(x _ {k}\right) - S \left(\mathbb {B R} _ {k}\right)\right) (41b) \\ \leq \max  _ {z \in \Delta^ {m _ {k}} ^ {- 1}} u _ {k} ^ {\tau} (z, x _ {- k}) - u _ {k} ^ {\tau} (\boldsymbol {x}) + \tau \max  _ {z ^ {\prime} \in \Delta^ {m _ {k}} ^ {- 1}} S \left(z ^ {\prime}\right) (41c) \\ \leq \sqrt {2} \| \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) \| + \tau \max  _ {z ^ {\prime} \in \Delta^ {m _ {k} - 1}} S \left(z ^ {\prime}\right) (41d) \\ \leq \sqrt {2} | | \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) | | + \tau \log (m _ {k}) (41e) \\ \end{array}
+$$
+
+where the second equality follows from the definition of player  $k$ 's entropy regularized utility  $u_{k}^{\tau}$ , the first inequality from nonnegativity of entropy  $S$ , the second inequality from concavity of  $u_{k}^{\tau}$  with respect to its first argument (Lemma 2), and the last from the maximum possible value of Shannon entropy over distributions on  $m_{k}$  actions.
+
+Lemma 13 ( $\mathcal{L}^{\tau}$  Scores Nash Equilibria). Let  $\mathcal{L}^{\tau}(\pmb{x})$  be our proposed entropy regularized loss function and  $\pmb{x}$  be an approximate QRE. Then it holds that
+
+$$
+\epsilon \leq \tau \log \left(\prod_ {k} m _ {k}\right) + \sqrt {\frac {2 n}{\min _ {k} \eta_ {k}}} \sqrt {\mathcal {L} ^ {\tau} (\boldsymbol {x})}. \tag {42}
+$$
+
+Proof. Beginning with the definition of exploitability and applying Lemma 12, we find
+
+$$
+\begin{array}{l} \epsilon = \max  _ {k} u _ {k} (\mathbb {B R} _ {k}, x _ {- k}) - u _ {k} (\boldsymbol {x}) \quad \text {(r e c a l l e a c h} \epsilon_ {k} \geq 0) (43a) \\ \leq \sum_ {k} u _ {k} (\mathbb {R} _ {k}, x _ {- k}) - u _ {k} (\boldsymbol {x}) (43b) \\ \leq \sum_ {k} \left[ \tau \log \left(m _ {k}\right) + \sqrt {2} \left| \left| \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) \right| \right| \right] (43c) \\ = \tau \sum_ {k} \log \left(m _ {k}\right) + \sum_ {k} \sqrt {2} \left\| \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) \right\| (43d) \\ \leq \tau \log \left(\prod_ {k} m _ {k}\right) + \sqrt {\frac {2 n}{\min  _ {k} \eta_ {k}}} \sqrt {\mathcal {L} ^ {\tau} (\boldsymbol {x})}. (43e) \\ \end{array}
+$$
+
+where the last inequality follows from the same steps (24c)-(24i) outlined in Lemma 4, which established the relationship between  $\mathcal{L}(\pmb{x})$  and  $\epsilon$ .
+
+# A.3 GRADIENT OF LOSS
+
+Lemma 14. The gradient of  $\mathcal{L}^{\tau}(\pmb{x})$  with respect to player  $l$ 's strategy  $x_{l}$  is
+
+$$
+\nabla_ {x _ {l}} \mathcal {L} (\boldsymbol {x}) = 2 \sum_ {k} \eta_ {k} B _ {k l} ^ {\top} \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) \tag {44}
+$$
+
+where  $B_{ll} = -\tau \left[I - \frac{1}{m_l}\mathbf{11}^\top \right]diag\left(\frac{1}{x_l}\right)$  and  $B_{kl} = \left[I - \frac{1}{m_k}\mathbf{11}^\top \right]H_{kl}^k$  for  $k\neq l$
+
+Proof. Recall from Lemma 6 that the loss can be decomposed as  $\mathcal{L}^{\tau}(\pmb{x}) = (A) + (B) + (C)$ .
+
+Then
+
+$$
+D _ {x _ {l}} [ (A) ] = D _ {x _ {l}} \left[ \sum_ {k} \eta_ {k} x _ {q} ^ {\top} B _ {k q} ^ {\top} B _ {k q} x _ {q} \right] = 2 \sum_ {k \neq l} \eta_ {k} B _ {k l} ^ {\top} B _ {k l} x _ {l} \tag {45}
+$$
+
+where  $q \neq k$  and  $B_{kq} = [I - \frac{1}{m_k}\mathbf{11}^\top][H_{kq}^k]$  does not depend on  $x_{k}$ .
+
+Also, letting  $B_{ll} = -\tau [I - \frac{1}{m_l}\mathbf{11}^\top ]\mathrm{diag}\left(\frac{1}{x_l}\right)$
+
+$$
+\begin{array}{l} D _ {x _ {l}} [ (B) ] = D _ {x _ {l}} \left[ - 2 \tau \sum_ {k} \eta_ {k} \ln \left(x _ {k}\right) ^ {\top} B _ {k q} x _ {q} \right] (46a) \\ = - 2 \tau \left[ \eta_ {l} D _ {x _ {l}} \left[ \ln \left(x _ {l}\right) ^ {\top} B _ {l q} x _ {q} \right] + \sum_ {k \neq l} \eta_ {k} D _ {x _ {l}} \left[ \ln \left(x _ {k}\right) ^ {\top} B _ {k l} x _ {l} \right] \right] (46b) \\ = - 2 \tau \left[ \eta_ {l} \operatorname {d i a g} \left(\frac {1}{x _ {l}}\right) B _ {l q} x _ {q} + \sum_ {k \neq l} \eta_ {k} B _ {k l} ^ {\top} \ln \left(x _ {k}\right) \right] (46c) \\ = - 2 \tau \left[ \eta_ {l} \left(\left[ I - \frac {1}{m _ {l}} \mathbf {1 1} ^ {\top} \right] \operatorname {d i a g} \left(\frac {1}{x _ {l}}\right)\right) ^ {\top} \Pi_ {T \Delta} \left(\nabla^ {l}\right) + \sum_ {k \neq l} \eta_ {k} B _ {k l} ^ {\top} \ln \left(x _ {k}\right) \right] (46d) \\ = 2 \left[ \eta_ {l} B _ {l l} ^ {\top} \Pi_ {T \Delta} (\nabla^ {l}) - \tau \sum_ {k \neq l} \eta_ {k} B _ {k l} ^ {\top} \ln (x _ {k}) \right]. (46e) \\ \end{array}
+$$
+
+And
+
+$$
+\begin{array}{l} D _ {x _ {l}} [ (C) ] = D _ {x _ {l}} \left[ \sum_ {k} \eta_ {k} \tau^ {2} \ln \left(x _ {k}\right) ^ {\top} \left[ I - \frac {1}{m _ {k}} \mathbf {1 1} ^ {\top} \right] \ln \left(x _ {k}\right) \right] (47a) \\ = 2 \tau^ {2} \left[ \eta_ {l} \operatorname {d i a g} \left(\frac {1}{x _ {l}}\right) \left[ I - \frac {1}{m _ {l}} \mathbf {1 1} ^ {\top} \right] \ln \left(x _ {l}\right) \right] (47b) \\ = - 2 \tau \eta_ {l} \left(\left[ I - \frac {1}{m _ {l}} \mathbf {1 1} ^ {\top} \right] \operatorname {d i a g} \left(\frac {1}{x _ {l}}\right)\right) ^ {\top} \Pi_ {T \Delta} (- \tau \ln (x _ {l})) (47c) \\ = 2 \eta_ {l} B _ {l l} ^ {\top} \Pi_ {T \Delta} (- \tau \ln (x _ {l})). (47d) \\ \end{array}
+$$
+
+Putting these together, we find
+
+$$
+\begin{array}{l} \nabla_ {x _ {l}} \mathcal {L} (\boldsymbol {x}) = 2 \sum_ {k \neq l} \eta_ {k} B _ {k l} ^ {\top} \left(B _ {k l} x _ {l} - \tau \ln \left(x _ {k}\right)\right) + 2 \eta_ {l} B _ {l l} ^ {\top} \left[ \Pi_ {T \Delta} \left(\nabla^ {l}\right) + \Pi_ {T \Delta} (- \tau \ln \left(x _ {l}\right)) \right] (48a) \\ = 2 \eta_ {l} B _ {l l} ^ {\top} \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) + 2 \sum_ {k \neq l} \eta_ {k} B _ {k l} ^ {\top} \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) (48b) \\ = 2 \sum_ {k} \eta_ {k} B _ {k l} ^ {\top} \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right). (48c) \\ \end{array}
+$$
+
+![](images/656915cfebd3ae35313223e7b48f6e8cb5eb9682343feb6bbf64f9d05bbaa973.jpg)
+
+# A.3.1 UNBIASED ESTIMATION
+
+In order to construct an unbiased estimate for each  $B_{kl}$ , we will need to form an independent unbiased estimate of  $H_{kl}^{k}$ . Recall that  $H_{kl}^{k}$  is simply the expected bimatrix game between players  $k$  and  $l$  when all other players sample their actions according to their current strategies.
+
+# A.4 BOUND ON GRADIENT / LIPSCHITZ PROPERTY
+
+Lemma 15. Assume payoffs are upper bounded by 1, then the infinity norm of the gradient is bounded as
+
+$$
+\left| \left| \nabla_ {\boldsymbol {x}} \mathcal {L} ^ {\tau} (\boldsymbol {x}) \right| \right| _ {\infty} \leq \frac {1}{2} \left(\max  _ {k} \eta_ {k}\right) \left(\tau \ln \left(\frac {1}{x _ {\operatorname* {m i n}}}\right) + 1\right) \left[ \tau m ^ {*} \left(\frac {1}{x _ {\operatorname* {m i n}}} - 1\right) + n \bar {m} \right]. \tag {49}
+$$
+
+Proof. Recall from Lemma 14 that the gradient of  $\mathcal{L}(\pmb{x})$  with respect to player  $l$ 's strategy  $x_{l}$  is
+
+$$
+\nabla_ {x _ {l}} \mathcal {L} (\boldsymbol {x}) = 2 \sum_ {k} \eta_ {k} B _ {k l} ^ {\top} \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) \tag {50}
+$$
+
+where  $B_{ll} = -\tau [I - \frac{1}{m_l}\mathbf{11}^\top ]\mathrm{diag}\left(\frac{1}{x_l}\right)$  and  $B_{kl} = [I - \frac{1}{m_k}\mathbf{11}^\top ]H_{kl}^k$  for  $k\neq l$
+
+As noted before in the derivation of Lemma 10, for payoffs in  $[0,1]$ , the entries in  $\nabla_{x_k}^{k\tau} = \nabla_{x_k}^k - \tau \ln(x_k)$  are bounded within  $[0, \tau \ln(\frac{1}{x_{\min}}) + 1]$  with a range  $\tau \ln(\frac{1}{x_{\min}}) + 1$ . Similarly, the entries in  $-\tau \mathrm{diag}\left(\frac{1}{x_l}\right)$  are bounded within  $[- \tau \frac{1}{x_{\min}}, -\tau]$  with a range of  $\tau\left(\frac{1}{x_{\min}} - 1\right)$ .
+
+The infinity norm of the gradient can then be bounded as
+
+$$
+\begin{array}{l} \left| \left| \nabla_ {\boldsymbol {x}} \mathcal {L} ^ {\tau} (\boldsymbol {x}) \right| \right| _ {\infty} = \max  _ {l} \left| \left| \nabla_ {x _ {l}} \mathcal {L} (\boldsymbol {x}) \right| \right| _ {\infty} (51a) \\ = \max  _ {l} \left\| 2 \sum_ {k} \eta_ {k} B _ {k l} ^ {\top} \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) \right\| _ {\infty} (51b) \\ \leq 2 \sum_ {k} \eta_ {k} \max  _ {l} | | B _ {k l} ^ {\top} \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) | | _ {\infty} (51c) \\ \leq \frac {1}{2} \sum_ {k \neq l ^ {*}} \eta_ {k} m _ {k} \left(\tau \ln \left(\frac {1}{x _ {\operatorname* {m i n}}}\right) + 1\right) + \frac {1}{2} \eta_ {l ^ {*}} m _ {l ^ {*}} \tau \left(\frac {1}{x _ {\operatorname* {m i n}}} - 1\right) \left(\tau \ln \left(\frac {1}{x _ {\operatorname* {m i n}}}\right) + 1\right) (51d) \\ = \frac {1}{2} \left(\tau \ln \left(\frac {1}{x _ {\operatorname* {m i n}}}\right) + 1\right) \left[ \eta_ {l ^ {*}} m _ {l ^ {*}} \tau \left(\frac {1}{x _ {\operatorname* {m i n}}} - 1\right) + \sum_ {k \neq l ^ {*}} \eta_ {k} m _ {k} \right] (51e) \\ \leq \frac {1}{2} \left(\max  _ {k} \eta_ {k}\right) \left(\tau \ln \left(\frac {1}{x _ {\operatorname* {m i n}}}\right) + 1\right) \left[ \tau m _ {l ^ {*}} \left(\frac {1}{x _ {\operatorname* {m i n}}} - 1\right) + \sum_ {k \neq l ^ {*}} m _ {k} \right] (51f) \\ \leq \frac {1}{2} \left(\max  _ {k} \eta_ {k}\right) \left(\tau \ln \left(\frac {1}{x _ {\operatorname* {m i n}}}\right) + 1\right) \left[ \tau m ^ {*} \left(\frac {1}{x _ {\operatorname* {m i n}}} - 1\right) + n \bar {m} \right] (51g) \\ \end{array}
+$$
+
+where the second inequality follows from Lemma 9.
+
+![](images/c3cecbfa94a28df4cc8398ee9a6a5e9dcac15ae2c798d61c4f4866495a511e19.jpg)
+
+Corollary 2. If  $\tau$  is set according to Lemma 11 as  $\tau = \ln (1 / p)^{-1}$  and payoffs are in [0, 1], then the infinity norm of the gradient is bounded as
+
+$$
+\left| \left| \nabla_ {\boldsymbol {x}} \mathcal {L} ^ {\tau} (\boldsymbol {x}) \right| \right| _ {\infty} \leq \frac {1}{2} \left(\max  _ {k} \eta_ {k}\right) \left[ \frac {\ln \left(m ^ {*}\right)}{\ln (1 / p)} + 2 \right] \left[ \frac {m ^ {* 2}}{p \ln (1 / p)} + n \bar {m} \right] = \frac {1}{2} \left(\max  _ {k} \eta_ {k}\right) \hat {L} \tag {52}
+$$
+
+where  $m^{*} = \max_{k} m_{k}$  and  $\hat{L}$  is defined implicitly for convenience in other derivations.
+
+Proof. Starting with Lemma 15 and applying Lemma 11 (i.e.,  $\tau = \ln(1/p)^{-1}$  and  $x_{\min} = \frac{p}{m^*}$  with payoffs in [0, 1] and where  $m^* = \max_k m_k$ ), we find
+
+$$
+\begin{array}{l} \left| \left| \nabla_ {\boldsymbol {x}} \mathcal {L} ^ {\tau} (\boldsymbol {x}) \right| \right| _ {\infty} \leq \frac {1}{2} \left(\max  _ {k} \eta_ {k}\right) \left(\tau \ln \left(\frac {1}{x _ {\operatorname* {m i n}}}\right) + 1\right) \left[ \tau m ^ {*} \left(\frac {1}{x _ {\operatorname* {m i n}}} - 1\right) + n \bar {m} \right] (53a) \\ = \frac {1}{2} \left(\max  _ {k} \eta_ {k}\right) \left[ \frac {\ln \left(m ^ {*} / p\right)}{\ln (1 / p)} + 1 \right] \left[ \frac {m ^ {*}}{\ln (1 / p)} \left(\frac {m ^ {*}}{p} - 1\right) + n \bar {m} \right] (53b) \\ \leq \frac {1}{2} \left(\max  _ {k} \eta_ {k}\right) \left[ \frac {\ln \left(m ^ {*}\right)}{\ln (1 / p)} + 2 \right] \left[ \frac {m ^ {* 2}}{p \ln (1 / p)} + n \bar {m} \right]. (53c) \\ \end{array}
+$$
+
+As  $p \to 0^+$ , the norm of the gradient blows up because the gradient of Shannon entropy blows up for small probabilities. As  $p \to 1$ , the norm of the gradient blows up because we require infinite temperature  $\tau$  to guarantee all QREs are nearly uniform; recall  $\tau$  is the regularization coefficient on the entropy bonus terms which means our modified utilities blow up for large  $\tau$ .
+
+# A.5 HESSIAN OF LOSS
+
+We will now derive the Hessian of our loss. This will be useful in establishing properties about global minima that enable the application of tailored minimization algorithms. Let  $D_z[f(z)]$  denote the differential operator applied to (possibly multivalued) function  $f$  with respect to  $z$ . For example,  $D_{x_q}[H_{lk}^k] = D_{x_q}[x_qT_{qlk}^k] = T_{qlk}^k$  where  $T_{qlk}^k$  is player  $k$ 's payoff tensor according to the three-way approximation between players  $k, l$ , and  $q$  to the game at  $x$ .
+
+Lemma 16. The Hessian of  $\mathcal{L}^{\tau}(\pmb{x})$  can be written
+
+$$
+H e s s \left(\mathcal {L} ^ {\tau}\right) = 2 \tilde {B} ^ {\top} \tilde {B} + T \Pi_ {T \Delta} \left(\tilde {\nabla} ^ {\tau}\right) \tag {54}
+$$
+
+where  $\tilde{B}_{kl} = \sqrt{\eta_k} B_{kl}$ ,  $\Pi_{T\Delta}(\tilde{\nabla}^{\tau}) = [\eta_1\Pi_{T\Delta}(\nabla_{x_1}^{1\tau}),\dots ,\eta_n\Pi_{T\Delta}(\nabla_{x_n}^{n\tau})]$ , and we augment  $T$  (the 3-player tensor approximation to the game,  $T_{lqk}^{k}$ ) so that  $T_{ll}^{l} = \tau \text{diag} 3\left(\frac{1}{x_{l}^{2}}\right)$  and otherwise 0.
+
+Proof. Recall the gradient of our proposed loss:
+
+$$
+\nabla_ {x _ {l}} \mathcal {L} (\boldsymbol {x}) = 2 \sum_ {k} \eta_ {k} B _ {k l} ^ {\top} \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) \tag {55}
+$$
+
+where  $B_{ll} = -\tau [I - \frac{1}{m_l}\mathbf{11}^\top ]\mathrm{diag}\left(\frac{1}{x_l}\right)$  and  $B_{kl} = [I - \frac{1}{m_k}\mathbf{11}^\top ]H_{kl}^k$  for  $k\neq l$
+
+Consider the following Jacobians, which will play an auxiliary role in our derivation of the Hessian:
+
+$$
+D _ {l} \left[ B _ {l l} \right] = \tau \left[ I - \frac {1}{m _ {l}} \mathbf {1 1} ^ {\top} \right] \operatorname {d i a g} 3 \left(\frac {1}{x _ {l} ^ {2}}\right) \tag {56a}
+$$
+
+$$
+D _ {q} \left[ B _ {l l} \right] = 0 \tag {56b}
+$$
+
+$$
+D _ {l} \left[ B _ {k l} \right] = 0 \tag {56c}
+$$
+
+$$
+D _ {q} \left[ B _ {k l} \right] = \left[ I - \frac {1}{m _ {k}} \mathbf {1 1} ^ {\top} \right] T _ {k l q} ^ {k} \tag {56d}
+$$
+
+$$
+\begin{array}{l} D _ {k} \left[ \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) \right] = \left[ I - \frac {1}{m _ {k}} \mathbf {1 1} ^ {\top} \right] D _ {k} \left[ \nabla_ {x _ {k}} ^ {k \tau} \right] (56e) \\ = \left[ I - \frac {1}{m _ {k}} \mathbf {1 1} ^ {\top} \right] D _ {k} \left[ \nabla_ {x _ {k}} ^ {k} - \tau \ln \left(x _ {k}\right) \right] (56f) \\ = \left[ I - \frac {1}{m _ {k}} \mathbf {1 1} ^ {\top} \right] \left[ - \tau \operatorname {d i a g} \left(\frac {1}{x _ {k}}\right) \right] (56g) \\ = B _ {k k} (56h) \\ \end{array}
+$$
+
+$$
+\begin{array}{l} D _ {l} \left[ \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) \right] = \left[ I - \frac {1}{m _ {k}} \mathbf {1 1} ^ {\top} \right] D _ {l} \left[ \nabla_ {x _ {k}} ^ {k \tau} \right] (56i) \\ = \left[ I - \frac {1}{m _ {k}} \mathbf {1 1} ^ {\top} \right] D _ {l} \left[ \nabla_ {x _ {k}} ^ {k} - \tau \ln (x _ {k}) \right] (56j) \\ = \left[ I - \frac {1}{m _ {k}} \mathbf {1 1} ^ {\top} \right] \left[ H _ {k l} ^ {k} \right] (56k) \\ = B _ {k l}. (561) \\ \end{array}
+$$
+
+We can derive the diagonal blocks of the Hessian as
+
+$$
+\begin{array}{l} D _ {l l} [ \mathcal {L} (\boldsymbol {x}) ] = D _ {l} \left[ \nabla_ {x _ {l}} \mathcal {L} (\boldsymbol {x}) \right] (57a) \\ = 2 D _ {l} \left[ \sum_ {k} \eta_ {k} B _ {k l} ^ {\top} \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) \right] (57b) \\ = 2 \left[ \eta_ {l} D _ {l} \left[ B _ {l l} ^ {\top} \Pi_ {T \Delta} \left(\nabla_ {x _ {l}} ^ {l \tau}\right) \right] + \sum_ {k \neq l} \eta_ {k} D _ {l} \left[ B _ {k l} ^ {\top} \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) \right] \right] (57c) \\ = 2 \left[ \eta_ {l} \left[ D _ {l} \left[ B _ {l l} \right] ^ {\top} \Pi_ {T \Delta} \left(\nabla_ {x _ {l}} ^ {l \tau}\right) + B _ {l l} ^ {\top} D _ {l} \left[ \Pi_ {T \Delta} \left(\nabla_ {x _ {l}} ^ {l \tau}\right) \right] \right] \right. (57d) \\ \left. + \sum_ {k \neq l} \eta_ {k} \left[ D _ {t} \left[ B _ {k l} \right] ^ {\top} \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) + B _ {k l} ^ {\top} D _ {l} \left[ \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) \right] \right] \right] (57e) \\ = 2 \left[ \eta_ {l} \left[ \tau \operatorname {d i a g} 3 \left(\frac {1}{x _ {l} ^ {2}}\right) [ I - \frac {1}{m _ {l}} \mathbf {1 1} ^ {\top} ] \Pi_ {T \Delta} \left(\nabla_ {x _ {l}} ^ {l \tau}\right) + B _ {l l} ^ {\top} B _ {l l} \right] + \sum_ {k \neq l} \eta_ {k} B _ {k l} ^ {\top} B _ {k l} \right] (57f) \\ = 2 \left[ \tau \eta_ {l} \operatorname {d i a g} \left(\left[ \frac {1}{x _ {l} ^ {2}} \right] \odot \Pi_ {T \Delta} \left(\nabla_ {x _ {l}} ^ {l \tau}\right)\right) + \sum_ {k} \eta_ {k} B _ {k l} ^ {\top} B _ {k l} \right] (57g) \\ \end{array}
+$$
+
+and the off-diagonal blocks as
+
+$$
+\begin{array}{l} D _ {l q} [ \mathcal {L} (\boldsymbol {x}) ] = D _ {q} \left[ \nabla_ {x _ {l}} \mathcal {L} (\boldsymbol {x}) \right] (58a) \\ = 2 D _ {q} \left[ \sum_ {k} \eta_ {k} B _ {k l} ^ {\top} \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) \right] (58b) \\ = 2 \left[ \eta_ {l} D _ {q} \left[ B _ {l l} ^ {\top} \Pi_ {T \Delta} \left(\nabla_ {x _ {l}} ^ {l \tau}\right) \right] + \sum_ {k \neq l} \eta_ {k} D _ {q} \left[ B _ {k l} ^ {\top} \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) \right] \right] (58c) \\ = 2 \left[ \eta_ {l} \left[ D _ {q} \left[ B _ {\bar {l} l} \right] ^ {\top} \Pi_ {T \Delta} \left(\nabla_ {x _ {l}} ^ {l \tau}\right) + B _ {l l} ^ {\top} D _ {q} \left[ \Pi_ {T \Delta} \left(\nabla_ {x _ {l}} ^ {l \tau}\right) \right] \right] \right. (58d) \\ \left. + \sum_ {k \neq l} \eta_ {k} \left[ D _ {q} \left[ B _ {k l} \right] ^ {\top} \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) + B _ {k l} ^ {\top} D _ {q} \left[ \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) \right] \right] \right] (58e) \\ = 2 \left[ \eta_ {l} B _ {l l} ^ {\top} B _ {l q} + \sum_ {k \neq l} \eta_ {k} \left[ T _ {l q k} ^ {k} \left[ I - \frac {1}{m _ {k}} \mathbf {1 1} ^ {\top} \right] \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) + B _ {k l} ^ {\top} B _ {k q} \right] \right] (58f) \\ = 2 \left[ \sum_ {k} \eta_ {k} B _ {k l} ^ {\top} B _ {k q} + \sum_ {k \neq l} \eta_ {k} T _ {l q k} ^ {k} \Pi_ {T \Delta} \left(\nabla_ {x _ {k}} ^ {k \tau}\right) \right]. (58g) \\ \end{array}
+$$
+
+Therefore, the Hessian can be written concisely as
+
+$$
+2 \left[ \tilde {B} ^ {\top} \tilde {B} + T \Pi_ {T \Delta} \left(\tilde {\nabla} ^ {\tau}\right) \right] \tag {59}
+$$
+
+where  $\tilde{B}_{kl} = \sqrt{\eta_k} B_{kl}$ ,  $\Pi_{T\Delta}(\tilde{\nabla}^{\tau}) = [\eta_1\Pi_{T\Delta}(\nabla_{x_1}^{1\tau}),\dots ,\eta_n\Pi_{T\Delta}(\nabla_{x_n}^{n\tau})]$ , and we augment  $T$  (the 3-player tensor approximation to the game,  $T_{lqk}^{k}$ ) so that  $T_{lll}^{l} = \tau \mathrm{diag}3\left(\frac{1}{x_{l}^{2}}\right)$  and otherwise 0.
+
+![](images/6c19be3c9e2bc7550fabf8ab2030175dabaf14544af78c0bcc1ecf5dd2c3ec4a.jpg)
+
+# B GLOBAL CONVERGENCE GUARANTEES
+
+In this section, we analyze the application of optimization techniques such as bandit algorithms and gradient descent to our loss function.
+
+# B.1 MAPS FROM HYPERCUBE TO SIMPLEX PRODUCT
+
+In this subsection, we derive properties of a map  $s$  from the unit-hypercube to the simplex product. This map is necessary to adapt our proposed loss  $\mathcal{L}^{\tau}$  to the commonly assumed setting in the  $\mathcal{X}$ -armed bandit literature (Bubeck et al., 2011) where the feasible set is a hypercube. We derive relevant properties of two such maps: the softmax and a mapping that interprets dimensions of the hypercube as angles on a unit-sphere that are then  $\ell_1$ -normalized.
+
+Lemma 17. Let  $f(x) = -\mathcal{L}(s(x))$ . Then  $||\nabla f(x)||_{\infty} \leq ||J(s(x))^{\top}||_{\infty}||\nabla \mathcal{L}(s(x))||_{\infty}$ .
+
+Proof.
+
+$$
+\left. \right.\left|\left| \nabla f (x) \right|\right| _ {\infty} = \left|\left| J (s (x)) ^ {\top} \nabla \mathcal {L} (s (x)) \right|\right| _ {\infty} \leq \left|\left| J (s (x)) ^ {\top} \right|\right| _ {\infty} \left|\left| \nabla \mathcal {L} (s (x)) \right|\right| _ {\infty}. \tag {60}
+$$
+
+![](images/1e08a5207ad835653f0ce7f7340fbc475fc5637f70c05eeb3542b9286c58ee43.jpg)
+
+Lemma 18. The  $\infty$ -norm of the Jacobian-transpose of a transformation  $s(x)$  applied elementwise to a product space is bounded by the  $\infty$ -norm of the Jacobian-transpose of a single transformation from that product space, i.e.,  $\|J(s(\pmb{x}))^\top\|_\infty \leq \max_{x_i \in \mathcal{X}_i} \|J(s(x_i))^\top\|_\infty$  for any  $i$ .
+
+Proof. Let  $\boldsymbol{x} \in \mathcal{X} = \prod_{i=1}^{n} \mathcal{X}_i$ ,  $\mathcal{Z} = \prod_{i=1}^{n} \mathcal{Z}_i$  and  $S: \mathcal{X} \to \mathcal{Z} = [s(x_1); \dots; s(x_n)]^\top$  where; denotes column-wise stacking,  $x_i \in \mathcal{X}_i$ . Also,  $\mathcal{X}_i = \mathcal{X}_j$  and  $\mathcal{Z}_i = \mathcal{Z}_j$  for all  $i$  and  $j$ . Then the Jacobian of  $S(\boldsymbol{x})$  is
+
+$$
+J (S (\boldsymbol {x})) ^ {\top} = \left[ \begin{array}{c c c} J (s (x _ {1})) ^ {\top} & 0 \dots & 0 \\ 0 & J (s (x _ {2})) ^ {\top} \dots & 0 \\ 0 & 0 \ddots & 0 \\ 0 & 0 \dots & J (s (x _ {n})) ^ {\top} \end{array} \right]. \tag {61}
+$$
+
+The  $\infty$ -norm of this matrix is the max 1-norm of any row. This matrix is diagonal, therefore, the  $\infty$ -norm of each elementwise Jacobian-transpose represents the max 1-norm of the rows spanned by its block. Given that the domains, ranges, and transformations  $s$  for all blocks are the same, their  $\infty$ -norms are also the same. The max  $\infty$  over the blocks is then equal to the  $\infty$ -norm of any individual  $J(s(x_i))^\top$ .
+
+# B.1.1 HESSIAN OF BANDIT REWARD FUNCTION
+
+Lemma 19. Let  $s(x)$  be a function that maps the unit hypercube to the simplex product (mixed strategy space). Then the objective function  $f(x) = -\mathcal{L}(s(x))$ . The Hessian of  $-f(x)$  at an optimum  $x^{*}$  in direction  $\Delta$  is  $\Delta x^{\top}[Ds(x)^{\top}H_{\mathcal{L}}(x)Ds(x)]\Big|_{x^{*}}\Delta x$  where  $H_{\mathcal{L}}$  is the Hessian of  $\mathcal{L}$  and  $Ds(x)$  is the Jacobian of  $s(x)$ .
+
+Proof.
+
+$$
+\begin{array}{l} \left. \left(D ^ {2} (\mathcal {L} \circ s) \left(x ^ {*}\right)\right) (\Delta x, \Delta x) = \Delta x ^ {\top} \left[ \sum_ {i} \overbrace {\partial_ {i} \mathcal {L} (s (x))} ^ {= 0 \text {a t} x = x ^ {*}} D ^ {2} h _ {i} (x) \right] \right| _ {x ^ {*}} \Delta x (62a) \\ + \Delta x ^ {\top} [ D s (x) ^ {\top} H _ {\mathcal {L}} (x) D s (x) ] \bigg | _ {x ^ {*}} \Delta x \\ = \Delta x ^ {\top} \left[ D s (x) ^ {\top} H _ {\mathcal {L}} (x) D s (x) \right] \Big | _ {x ^ {*}} \Delta x. (62b) \\ \end{array}
+$$
+
+![](images/25a01722fe07d536f17dc9ae0bfaee5e73934a66ff0b933fce7eae71fe419569.jpg)
+
+Lemma 20. Let  $s(x): \mathcal{X} \to \prod_k \Delta^{m_k - 1}$  be an injective function, i.e.,  $x \neq y \Rightarrow s(x) \neq s(y)$ . Also let  $J = J(s(x))$  be the Jacobian of  $s$  with respect to  $x$  and  $\Delta x$  be a nonzero vector in the tangent space of  $\mathcal{X}$ . Then
+
+$$
+J \Delta x \neq 0. \tag {63}
+$$
+
+Proof. Recall that the  $ij$ th entry of the Jacobian represents  $\frac{\partial s_i}{\partial x_j}$  so that the  $i$ th entry of  $J\Delta x$  is
+
+$$
+[ J \Delta x ] _ {i} = \sum_ {j} \frac {\partial s _ {i}}{\partial x _ {j}} \Delta x _ {j} = d s _ {i}. \tag {64}
+$$
+
+Assume  $J\Delta x = 0$ . This would imply a change in  $x \in \mathcal{X}$  results in no change in  $s$  ( $ds = 0$ ), contradicting the fact that  $s$  is injective. Therefore, we must conclude the claim that  $J\Delta x \neq 0$ .
+
+Lemma 21. Let  $J = J(s(x))$  be the Jacobian of any composition of transformations  $s = s_t \circ \ldots \circ s_1$  where  $s_t(z) = [z_i / \sum_j z_j]_i$ . Then  $J\Delta x$  lies in the tangent space of the simplex.
+
+Proof. We aim to show  $\mathbf{1}^{\top}J\Delta x = \mathbf{0}$  for any  $\Delta x$  and  $x$ . By chain rule, the Jacobian of  $s$  is  $J = J(s) = \prod_{t'=t}^{t'=1} J(s_t')$ . Therefore,  $\mathbf{1}^{\top}J\Delta x = \mathbf{1}^{\top}(\prod_{t'=t}^{t'=1} J(s_t'))\Delta x$ . Consider the first product:
+
+$$
+\mathbf {1} ^ {\top} J \left(s _ {t}\right) = \mathbf {0} \tag {65}
+$$
+
+by Lemma 24. Therefore  $\mathbf{1}^{\top}J\Delta x = \mathbf{1}^{\top}J(s_{t})(\prod_{t^{\prime} = t - 1}^{t^{\prime} = 1}J(s_{t}^{\prime}))\Delta x = \mathbf{0}^{\top}(\prod_{t^{\prime} = t - 1}^{t^{\prime} = 1}J(s_{t}^{\prime}))\Delta x = 0$ . This implies  $J\Delta x$  is orthogonal to  $\mathbf{1}$  for any  $x\in \mathcal{X}$  and  $\Delta x$ , therefore  $J\Delta x$  lies in the tangent space of the simplex for any  $x\in \mathcal{X}$  and  $\Delta x$ .
+
+# B.1.2 SOFTMAX MAP
+
+Let  $s:[0,1]^{d - 1}\to \Delta^{d - 1}\in \mathbb{R}^d$  be the softmax function. See (Gao and Pavel, 2017) for an analysis of many of its properties and in the context of game theory. Note that  $s$  maps a  $(d - 1)$  dimensional variable to a  $d$  dimensional distribution. This can be practically handled by always appending a 0 to the  $(d - 1)$ -dimensional input prior to applying the standard softmax. We perform our analysis below in terms of the standard softmax, but note the norms we derive apply to our modified (invertible) softmax.
+
+Standard:
+
+$$
+s (x) = \frac {1}{\sum_ {j = 1} ^ {d} e ^ {x _ {j}}} \left[ e ^ {x _ {1}}, \dots e ^ {x _ {d}} \right]. \tag {66}
+$$
+
+Modified:
+
+$$
+s (x) = \frac {1}{1 + \sum_ {j = 1} ^ {d - 1} e ^ {x _ {j}}} \left[ e ^ {x _ {1}}, \dots e ^ {x _ {d - 1}}, 1 \right]. \tag {67}
+$$
+
+Lemma 22. Let  $J$  be the Jacobian of the softmax operator. Then  $\| J\|_{\infty}\leq 2$  and  $\| J^{\top}\|_{\infty}\leq 2$
+
+Proof. Let  $S_{i}$  represent the  $i$ th entry of  $S = \mathrm{softmax}(z)$  for any  $z \in \mathbb{R}^{m}$ . Then the 1-norm of row  $i$  is upper bounded as
+
+$$
+\Rightarrow \sum_ {j} \left| D _ {j} S _ {i} \right| = \sum_ {j} \left| S _ {i} \left(\delta_ {i j} - S _ {j}\right) \right| \tag {68b}
+$$
+
+$$
+\begin{array}{l} D _ {j} S _ {i} = S _ {i} \left(\delta_ {i j} - S _ {j}\right) (68a) \\ \leq \sum_ {j} \left| \delta_ {i j} S _ {i} \right| + \left| S _ {i} S _ {j} \right| (68c) \\ = S _ {i} + \sum_ {j} S _ {i} S _ {j} (68d) \\ = S _ {i} + S _ {i} \sum_ {j} S _ {j} (68e) \\ = 2 S _ {i} (68f) \\ \leq 2 \forall i. (68g) \\ \end{array}
+$$
+
+Also, the 1-norm of column  $j$  is upper bounded similarly as
+
+$$
+\begin{array}{l} (6 9 a) \\ \sum_ {i} \left| D _ {j} S _ {i} \right| = \sum_ {i} \left| S _ {i} \left(\delta_ {i j} - S _ {j}\right) \right| (69b) \\ \leq \sum_ {i} \left| \delta_ {i j} S _ {i} \right| + \left| S _ {i} S _ {j} \right| (69c) \\ = S _ {j} + \sum_ {i} S _ {i} S _ {j} (69d) \\ = S _ {i} + S _ {j} \sum_ {i} S _ {i} (69e) \\ = 2 S _ {j} (69f) \\ \leq 2 \forall j. (69g) \\ \end{array}
+$$
+
+The  $\infty$ -norm of a matrix is the maximum 1-norm of any row. Therefore,  $||J||_{\infty}$  and  $||J^{\top}||_{\infty}$  are both upper bounded by 2.
+
+Corollary 3. Let  $J$  be the Jacobian of the softmax operator. Then  $\| J\| _2\leq 2$  and  $\| J^{\top}\|_{2}\leq 2$
+
+Proof. The Gershgorin circle theorem states that every eigenvalue of  $J$  lies within one of the discs centered at the diagonal of  $J$  with radius equal to the 1-norm of the row (excluding the diagonal term). The Jacobian of the softmax operator contains only nonnegative diagonal terms, therefore, the 1-norm of any entire row also represents the maximum magnitude of any eigenvalue allowed by any Gershgorin disc. In addition,  $J$  is symmetric and therefore, the magnitude of its eigenvalues are equivalent to its singular values. Therefore by Lemma 22,  $||J||_2 \leq 2$ .
+
+# B.1.3 SPHERICAL MAP
+
+For spherical coordinates, let  $s(x) = n(l(c(x)))$  where  $c(x) = \pi / 2x$ ,  $l(\psi)$  maps angles to the unit sphere, and  $n(z) = [z_i / \sum_j z_j]_i$ .
+
+Definition 2. Define  $l(\psi)$  as the transformation to the unit-sphere using spherical coordinates:
+
+$$
+l _ {1} (\psi) = \cos (\psi_ {1}) \tag {70a}
+$$
+
+$$
+l _ {2} (\psi) = \sin \left(\psi_ {1}\right) \cos \left(\psi_ {2}\right) \tag {70b}
+$$
+
+$$
+l _ {3} (\psi) = \sin (\psi_ {1}) \sin (\psi_ {2}) \cos (\psi_ {3}) \tag {70c}
+$$
+
+$$
+\vdots = \vdots \tag {70d}
+$$
+
+$$
+l _ {m - 1} (\psi) = \sin \left(\psi_ {1}\right) \sin \left(\psi_ {2}\right) \dots \cos \left(\psi_ {m - 1}\right) \tag {70e}
+$$
+
+$$
+l _ {m} (\psi) = \sin \left(\psi_ {1}\right) \sin \left(\psi_ {2}\right) \dots \sin \left(\psi_ {m - 1}\right). \tag {70f}
+$$
+
+Lemma 23. Let  $J$  be the Jacobian of the transformation to the unit-sphere using spherical coordinates, i.e.  $z = l(\psi)$  where  $\|l\|^2 = 1$  and  $\psi_i \in [0, \frac{\pi}{2}]$  represents an angle for each  $i$ . Then  $\|J\|_F \leq \sqrt{m}$ .
+
+Proof. The Jacobian of the transformation is
+
+$$
+J (l) = \left[ \begin{array}{c c c c} - \sin (\psi_ {1}) & 0 & \dots & 0 \\ \cos (\psi_ {1}) \cos (\psi_ {2}) & - \sin (\psi_ {1}) \sin (\psi_ {2}) & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ \cos (\psi_ {1}) \sin (\psi_ {2}) \dots \cos (\psi_ {m - 1}) & \dots & \dots & - \sin (\psi_ {1}) \dots \sin (\psi_ {m - 2}) \sin (\psi_ {m - 1}) \\ \cos (\psi_ {1}) \sin (\psi_ {2}) \dots \sin (\psi_ {m - 1}) & \dots & \dots & \sin (\psi_ {1}) \dots \sin (\psi_ {m - 2}) \cos (\psi_ {m - 1}) \end{array} \right] \tag {71}
+$$
+
+and it square is
+
+$$
+J (l) = \left[ \begin{array}{c c c c} t _ {1} & 0 & \dots & 0 \\ \cos (\psi_ {1}) ^ {2} \cos (\psi_ {2}) ^ {2} & \sin (\psi_ {1}) ^ {2} t _ {2} & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ \cos (\psi_ {1}) ^ {2} \sin (\psi_ {2}) ^ {2} \dots \cos (\psi_ {m - 1}) ^ {2} & \dots & \dots & \sin (\psi_ {1}) ^ {2} \dots \sin (\psi_ {m - 2}) ^ {2} t _ {m - 1} \\ \cos (\psi_ {1}) ^ {2} \sin (\psi_ {2}) ^ {2} \dots \sin (\psi_ {m - 1}) ^ {2} & \dots & \dots & \sin (\psi_ {1}) ^ {2} \dots \sin (\psi_ {m - 2}) ^ {2} t _ {m} \end{array} \right] \tag {72}
+$$
+
+where
+
+$$
+\delta_ {i m} = 1 \text {i f} i = m, 0 \text {e l s e} \tag {73a}
+$$
+
+$$
+t _ {i} = \delta_ {i m} \cos^ {2} \left(\psi_ {i - 1}\right) + \left(1 - \delta_ {i m}\right) \sin^ {2} \left(\psi_ {i}\right) \leq 1. \tag {73b}
+$$
+
+To compute the Frobenius norm, we will need the sum of the squares of all entries. We will consider the sum of each row individually using the following auxiliary variable  $R_{i,k\leq i}$  where  $\sum_{j}J_{ij}^{2} = R_{i,1}$  and apply a recursive inequality.
+
+(74a)
+
+$$
+\begin{array}{l} R _ {i, k \leq i} = \sum_ {k ^ {\prime} = k} ^ {i - 1} \cos^ {2} \left(\psi_ {k ^ {\prime}}\right) \left[ \prod_ {l = k, l \neq k ^ {\prime}} ^ {i - 1} \sin^ {2} \left(\psi_ {l}\right) \right] \cos^ {2} \left(\psi_ {i}\right) + t _ {i} \prod_ {l = k} ^ {i - 1} \sin^ {2} \left(\psi_ {l}\right) (74b) \\ = \cos^ {2} \left(\psi_ {k}\right) \underbrace {\left[ \prod_ {l = k + 1} ^ {i - 1} \sin^ {2} \left(\psi_ {l}\right) \right] \cos^ {2} \left(\psi_ {i}\right)} _ {\leq 1} (74c) \\ + \sin^ {2} \left(\psi_ {k}\right) \sum_ {k ^ {\prime} = k + 1} ^ {i - 1} \cos^ {2} \left(\psi_ {k ^ {\prime}}\right) \left[ \prod_ {l = k + 1, l \neq k ^ {\prime}} ^ {i - 1} \sin^ {2} \left(\psi_ {l}\right) \right] \cos^ {2} \left(\psi_ {i}\right) (74d) \\ + \sin^ {2} \left(\psi_ {k}\right) t _ {i} \prod_ {l = k + 1} ^ {i - 1} \sin^ {2} \left(\psi_ {l}\right) (74e) \\ \leq \cos^ {2} \left(\psi_ {k}\right) (74f) \\ + \sin^ {2} \left(\psi_ {k}\right) \left(\sum_ {k ^ {\prime} = k + 1} ^ {i - 1} \cos^ {2} \left(\psi_ {k ^ {\prime}}\right) \left[ \prod_ {l = k + 1, l \neq k ^ {\prime}} ^ {i - 1} \sin^ {2} \left(\psi_ {l}\right) \right] \cos^ {2} \left(\psi_ {i}\right) + t _ {i} \prod_ {l = k + 1} ^ {i - 1} \sin^ {2} \left(\psi_ {l}\right)\right) (7 4 g) \\ = \cos^ {2} (\psi_ {k}) + \sin^ {2} (\psi_ {k}) R _ {i, k + 1}. (74h) \\ \end{array}
+$$
+
+Note then that  $R_{i,k+1} \leq 1 \Rightarrow R_{i,k} \leq 1$ . We know  $R_{i,i} = t_i \leq 1$ , therefore,  $R_{i,1} \leq 1$  by applying the inequality recursively. Finally,  $\sum_j J_{ij}^2 = R_{i,1} \leq 1$  implies the claim  $||J||_F^2 = \sum_i R_{i,1} \leq m$ .
+
+Lemma 24. Let  $J$  be the Jacobian of  $n(z) = z / Z$  where  $Z = \sum_{k}z_{k}$ . Then  $\mathbf{1}^{\top}J = \mathbf{0}^{\top}$ .
+
+Proof. The  $i$ th entry of the Jacobian of  $n(z)$  is
+
+$$
+J (n) _ {i j} = \frac {1}{Z ^ {2}} \left(- z _ {i} + \delta_ {i j} Z\right). \tag {75}
+$$
+
+Therefore  $[ \mathbf{1}^{\top} J ]_{j} = \sum_{i} J(n)_{ij} = \frac{1}{Z^{2}} (-Z + Z) = 0$  where  $z$  is a point on the unit-sphere in the positive orthant.
+
+# B.2 NEAR OPTIMALITY & ZOOMING DIMENSION
+
+In this subsection, we derive bounds on the near-optimality dimension and zooming dimension of the associated bandit optimization problems we use to derive global convergence guarantees to Nash equilibria. This is written in terms of maximizing a function  $f$  rather than minimizing a loss to better match the bandit literature (Bubeck et al., 2011; Valko et al., 2013; Feng et al., 2022).
+
+Assumption 1. Locally around each interior  $x^{*} \in \mathcal{X}^{*}$ ,  $-f(x)$  is lower bounded by  $-f(x^{*}) + \sigma_{-}||x - x^{*}||^{\alpha_{hi}}$ . In other words, for all  $f(x) \geq f(x^{*}) - \eta$ :
+
+$$
+f \left(x ^ {*}\right) - f (x) \geq \sigma_ {-} \| x - x ^ {*} \| ^ {\alpha_ {h i}} \tag {76}
+$$
+
+where we have left the precise norm unspecified for generality. Let  $\ell(x, x^{*}) = \sigma_{+}||x - x^{*}||^{\alpha_{lo}}$ .
+
+Definition 3.  $\mathcal{X}_{\epsilon} \stackrel{\mathrm{def}}{=} \{x \in \mathcal{X} \mid \exists x^{*} \in \mathcal{X}^{*}$  s.t.  $f(x) \geq f(x^{*}) - \epsilon\}$
+
+Definition 4.  $\mathcal{X}_{\epsilon}^{lower} \stackrel{\mathrm{def}}{=} \{x \in \mathcal{X} \mid \exists x^* \in \mathcal{X}^* \text{ s.t. } f(x^*) - \sigma_{-} \|x - x^*\|^{\alpha_{hi}} \geq f(x^*) - \epsilon\}$
+
+Corollary 4.  $\mathcal{X}_{\varepsilon} \subseteq \mathcal{X}_{\varepsilon}^{lower}$ .
+
+Proof. By Assumption 1,  $f(x^{*}) - \sigma_{-}||x - x^{*}||^{\alpha_{hi}}\geq f(x)$ . Therefore, any  $x\in \mathcal{X}$  that satisfies the requirement for an element of  $\mathcal{X}_{\epsilon}$ ,  $f(x)\geq f(x^{*}) - \epsilon$ , will also satisfy the requirement for an element of  $\mathcal{X}_{\epsilon}^{lower}$ .
+
+Definition 5 ( $\psi$ -near Optimality Dimension). The  $\psi$ -near optimality dimension is the smallest  $d' > 0$  such that there exists  $C > 0$  such that for any  $\epsilon > 0$ , the maximum number of disjoint  $\ell$ -balls of radius  $\psi \epsilon$  and center in  $\mathcal{X}_{\epsilon}$  is less than  $C \epsilon^{-d'}$ .
+
+Definition 6 (Zooming Dimension). The zooming dimension is the smallest  $d_z > 0$  such that there exists  $C_z > 0$  such that for any  $r > 0$ , the maximum number of disjoint  $\ell$ -balls of radius  $\frac{r}{2}$  and center in  $\mathcal{X}_{16r}$  is less than  $C_z r^{-d_z}$ .
+
+Corollary 5 (Zooming Dimension from  $\psi$ -near Optimality Dimension). The zooming dimension of  $f: x \in [0,1]^d \to [-1,1]$  under  $\ell(x,y) = \sigma_+||x - y||^{\alpha_{lo}}$  is  $d_z = d\left(\frac{\alpha_{hi} - \alpha_{lo}}{\alpha_{lo}\alpha_{hi}}\right)$  with constant  $C_z = 16^{-d'}C$  with  $\psi = \frac{1}{32}$  ( $\psi$  is needed to compute  $C$ ).
+
+Proof. Mapping the definition of  $\psi$ -near optimality onto zooming dimension, we find  $\psi \epsilon = r / 2$  and  $\epsilon = 16r$ . Then we can infer  $\psi = 1 / 32$ ; this is used to compute  $C$  (see Theorem 1). Rewriting the bound from  $\psi$ -near optimality in terms of  $r$ , we find
+
+$$
+N _ {\epsilon} = N _ {1 6 r} \leq C (1 6 r) ^ {- d ^ {\prime}} = C 1 6 ^ {- d ^ {\prime}} r ^ {- d ^ {\prime}} = C _ {z} r ^ {- d _ {z}} \tag {77}
+$$
+
+where  $N_{16r}$  denotes the number of  $\ell$ -balls of radius  $r/2$  with center in  $\mathcal{X}_{16r}$ ,  $d_z = d'$ ,  $C_z = C16^{-d'} = C16^{-d_z}$ . Therefore, this translation only effects the constant  $C_z$ , not the zooming dimension.
+
+Lemma 25 ( $N_{\epsilon \leq \eta} \leq C_{\epsilon \leq \eta}\epsilon^{-d'}$ ). The maximum number of disjoint  $l$ -balls with radius  $\psi \in$  and center in  $\mathcal{X}_{\epsilon \leq \eta}$ ,  $N_{\epsilon \leq \eta}$ , is upper bounded by  $C_{\epsilon \leq \eta}\epsilon^{-d'}$  where  $C_{\epsilon \leq \eta} = |\mathcal{X}^*| \left( \frac{\sigma_+}{\psi \sigma_-^{\alpha_{lo} / \alpha_{hi}}} \right)^{d / \alpha_{lo}}$  and  $d' = d \left( \frac{\alpha_{hi} - \alpha_{lo}}{\alpha_{lo} \alpha_{hi}} \right)$ .
+
+Proof. The number of disjoint  $\ell$ -balls of radius  $\psi \epsilon$  and center in  $\mathcal{X}_{\epsilon \leq \eta}$  can be upper bounded as follows.
+
+Rewrite  $\mathcal{X}_{\epsilon}^{lower}$  by rearranging terms as
+
+$$
+\mathcal {X} _ {\epsilon} ^ {l o w e r} = \left\{x \in \mathcal {X} | \exists x ^ {*} \in \mathcal {X} ^ {*} s. t. \left| \left| x - x ^ {*} \right| \right| \leq \left(\frac {\epsilon}{\sigma_ {-}}\right) ^ {1 / \alpha_ {h i}} \stackrel {\text {d e f}} {=} r _ {\epsilon} \right\} \tag {78}
+$$
+
+and recall that from Corollary 4 that  $\mathcal{X}_{\epsilon} \subseteq \mathcal{X}_{\epsilon}^{lower}$ . Furthermore, an  $\ell$ -ball of radius  $\psi \epsilon$  implies
+
+$$
+\ell (x, y) = \sigma_ {+} | | x - y | | ^ {\alpha_ {l o}} \leq \psi \epsilon \Longrightarrow | | x - y | | \leq \left(\frac {\psi \epsilon}{\sigma_ {+}}\right) ^ {1 / \alpha_ {l o}} \stackrel {\text {d e f}} {=} r _ {\ell}. \tag {79}
+$$
+
+The number of disjoint  $\ell$ -balls that can pack into a set  $\mathcal{X}_{\epsilon}$ ,  $N_{\epsilon \leq \eta}$ , is upper bounded by the ratio of the volumes of the two sets:
+
+$$
+\begin{array}{l} N _ {\epsilon \leq \eta} \leq \frac {V o l \left(\mathcal {X} _ {\epsilon}\right)}{V o l \left(\mathcal {B} _ {\ell}\right)} (80a) \\ \leq \frac {\operatorname {V o l} \left(\mathcal {X} _ {\epsilon} ^ {\text {l o w e r}}\right)}{\operatorname {V o l} \left(\mathcal {B} _ {\ell}\right)} (80b) \\ = \frac {\left| \mathcal {X} ^ {*} \right| S _ {d} r _ {\epsilon} ^ {d}}{S _ {d} r _ {\ell} ^ {d}} (80c) \\ \leq \frac {\left| \mathcal {X} ^ {*} \right| \left(\frac {\epsilon}{\sigma_ {-}}\right) ^ {d / \alpha_ {h i}}}{\left(\frac {\psi \epsilon}{\sigma_ {+}}\right) ^ {d / \alpha_ {l o}}} (80d) \\ = \left| \mathcal {X} ^ {*} \right| \left(\frac {\sigma_ {+} ^ {1 / \alpha_ {l o}} \psi^ {- 1 / \alpha_ {l o}}}{\sigma_ {-} ^ {1 / \alpha_ {h i}}}\right) ^ {d} \epsilon^ {d \left(1 / \alpha_ {h i} - 1 / \alpha_ {l o}\right)} (80e) \\ = \left| \mathcal {X} ^ {*} \right| \left(\frac {\sigma_ {+}}{\psi \sigma_ {-} ^ {\alpha_ {l o} / \alpha_ {h i}}}\right) ^ {d / \alpha_ {l o}} \epsilon^ {- d \left(\frac {\alpha_ {h i} - \alpha_ {l o}}{\alpha_ {l o} \alpha_ {h i}}\right)} (80f) \\ = C _ {\epsilon \leq \eta} \epsilon^ {- d ^ {\prime}} (80g) \\ \end{array}
+$$
+
+where  $C_{\epsilon \leq \eta} = |\mathcal{X}^{*}|\left(\frac{\sigma_{+}}{\psi\sigma_{-}^{\alpha_{lo} / \alpha_{hi}}}\right)^{d / \alpha_{lo}}$ ,  $d' = d\left(\frac{\alpha_{hi} - \alpha_{lo}}{\alpha_{lo}\alpha_{hi}}\right)$ ,  $|\mathcal{X}^{*}|$  is the number of distinct global optima, and  $S_{d}$  is the volume constant for a  $d$ -sphere under the given norm  $||\cdot||$ .
+
+Recall, these results apply when  $f(x) \geq f(x^{*}) - \eta$ , i.e., when  $\epsilon \leq \eta$ . Otherwise, we can upper bound the number of  $\ell$ -balls by considering the entire set  $\mathcal{X}$  which has volume 1. First, we will bound the constant associated with the volume of a  $d$ -sphere.
+
+Lemma 26 ( $N_{\epsilon \geq \eta} \leq C_{\epsilon \geq \eta}$ ). The maximum number of disjoint  $\ell$ -balls with radius  $\psi \epsilon$  and center in  $\mathcal{X}_{\epsilon \geq \eta}$ ,  $N_{\epsilon \geq \eta}$ , is upper bounded by  $C_{\epsilon \geq \eta}$  where  $C_{\epsilon \geq \eta} = S_d^{-1}\left(\frac{\sigma_+}{\psi\eta}\right)^{d / \alpha_{lo}}$  and  $S_d$  is the volume constant for a  $d$ -sphere under a given norm.
+
+Proof. We can upper bound the number of  $\ell$ -balls needed to pack the entire space by considering the smallest possible radius  $\psi \eta$ :
+
+$$
+\begin{array}{l} N _ {\epsilon \geq \eta} \leq \frac {V o l (\mathcal {X})}{V o l (\mathcal {B} _ {\ell})} (81a) \\ = \frac {1}{S _ {d} r _ {\ell} ^ {d}} (81b) \\ \leq \frac {1}{S _ {d} \left(\frac {\psi \eta}{\sigma_ {+}}\right) ^ {d / \alpha_ {l o}}} (81c) \\ = S _ {d} ^ {- 1} \left(\frac {\sigma_ {+}}{\psi \eta}\right) ^ {d / \alpha_ {l o}} (81d) \\ = C _ {\epsilon \geq \eta} (81e) \\ \end{array}
+$$
+
+where  $r_l$  was defined in equation (79).
+
+Theorem 1. The  $\psi$ -near optimality dimension of  $f: x \in [0,1]^d \to [-1,1]$  under  $\ell$  is  $d' = d\left(\frac{\alpha_{hi} - \alpha_{lo}}{\alpha_{lo}\alpha_{hi}}\right)$  with constant
+
+$$
+C = \max  \left\{1, S _ {d} ^ {- 1} \left(r _ {\eta} ^ {\frac {\alpha_ {h i}}{\alpha_ {l o}}} \sigma_ {-} ^ {\left(\frac {\alpha_ {h i} - \alpha_ {l o}}{\alpha_ {l o} \alpha_ {h i}}\right)}\right) ^ {- d} \right\} \left(\frac {\sigma_ {+}}{\psi \sigma_ {-} ^ {\alpha_ {l o} / \alpha_ {h i}}}\right) ^ {d / \alpha_ {l o}} \tag {82}
+$$
+
+where  $S_{d}$  is the volume constant for a  $d$ -sphere under the same norm as  $\ell$ .
+
+Proof. First, let us define  $r_{\eta} = \left(\frac{\eta}{\sigma_{-}}\right)^{1 / \alpha_{hi}}$  as in equation (78) which implies  $\eta = \sigma_{-}r_{\eta}^{\alpha_{hi}}$ . Then apply Lemmas 25 ( $N_{\epsilon \leq \eta} \leq C_{\epsilon \leq \eta}\epsilon^{-d'}$ ) and 26 ( $N_{\epsilon \geq \eta} \leq C_{\epsilon \geq \eta}$ ) which bound the number of  $\ell$ -balls required to pack  $\mathcal{X}_{\epsilon}$  when  $\epsilon$  is less than and greater than  $\eta$  respectively:
+
+$$
+C _ {\epsilon \leq \eta} = \left| \mathcal {X} ^ {*} \right| \left(\frac {\sigma_ {+}}{\psi \sigma_ {-} ^ {\alpha_ {l o} / \alpha_ {h i}}}\right) ^ {d / \alpha_ {l o}} \tag {83}
+$$
+
+$$
+d ^ {\prime} = d \left(\frac {\alpha_ {h i} - \alpha_ {l o}}{\alpha_ {l o} \alpha_ {h i}}\right) \tag {84}
+$$
+
+and
+
+$$
+C _ {\epsilon \geq \eta} = S _ {d} ^ {- 1} \left(\frac {\sigma_ {+}}{\psi \eta}\right) ^ {d / \alpha_ {l o}} \tag {85a}
+$$
+
+$$
+= \left| \mathcal {X} ^ {*} \right| ^ {- 1} S _ {d} ^ {- 1} \eta^ {- d / \alpha_ {l o}} \sigma_ {-} ^ {d / \alpha_ {h i}} \left| \mathcal {X} ^ {*} \right| \left(\frac {\sigma_ {+}}{\psi \sigma_ {-} ^ {\alpha_ {l o} / \alpha_ {h i}}}\right) ^ {d / \alpha_ {l o}} \tag {85b}
+$$
+
+$$
+= \left| \mathcal {X} ^ {*} \right| ^ {- 1} S _ {d} ^ {- 1} \eta^ {- d / \alpha_ {l o}} \sigma_ {-} ^ {d / \alpha_ {h i}} C _ {\epsilon \leq \eta} \tag {85c}
+$$
+
+$$
+= \left| \mathcal {X} ^ {*} \right| ^ {- 1} S _ {d} ^ {- 1} r _ {\eta} ^ {- d \alpha_ {h i} / \alpha_ {l o}} \sigma_ {-} ^ {- d / \alpha_ {l o}} \sigma_ {-} ^ {d / \alpha_ {h i}} C _ {\epsilon \leq \eta} \tag {85d}
+$$
+
+$$
+= \left| \mathcal {X} ^ {*} \right| ^ {- 1} S _ {d} ^ {- 1} r _ {\eta} ^ {- d \frac {\alpha_ {h i}}{\alpha_ {l o}}} \sigma_ {-} ^ {- d \left(\frac {\alpha_ {h i} - \alpha_ {l o}}{\alpha_ {l o} \alpha_ {h i}}\right)} C _ {\epsilon \leq \eta} \tag {85e}
+$$
+
+$$
+= \left| \mathcal {X} ^ {*} \right| ^ {- 1} S _ {d} ^ {- 1} \left(r _ {\eta} ^ {\frac {\alpha_ {h i}}{\alpha_ {l o}}} \sigma_ {-} ^ {\left(\frac {\alpha_ {h i} - \alpha_ {l o}}{\alpha_ {l o} \alpha_ {h i}}\right)}\right) ^ {- d} C _ {\epsilon \leq \eta} \tag {85f}
+$$
+
+where  $S_{d}$  is the volume constant for a  $d$ -sphere under the given norm.  $S_{d}^{-1}$  has been upper bounded for the 2-norm in Lemma 35. For the  $\infty$ -norm,  $S_{d}^{-1} = 2^{-d}$ . We have written  $C_{\epsilon \geq \eta}$  in terms of  $C_{\epsilon \leq \eta}$  to clarify which is larger.
+
+Therefore,
+
+$$
+\begin{array}{l} C = \max  \left\{1, | \mathcal {X} ^ {*} | ^ {- 1} S _ {d} ^ {- 1} \left(r \eta^ {\frac {\alpha_ {h i}}{\alpha_ {l o}}} \sigma_ {-} ^ {\left(\frac {\alpha_ {h i} - \alpha_ {l o}}{\alpha_ {l o} \alpha_ {h i}}\right)}\right) ^ {- d} \right\} C _ {\epsilon \leq \eta} (86a) \\ = \max  \left\{1, \left| \mathcal {X} ^ {*} \right| ^ {- 1} S _ {d} ^ {- 1} \left(r _ {\eta} ^ {\frac {\alpha_ {h i}}{\alpha_ {l o}}} \sigma_ {-} ^ {\left(\frac {\alpha_ {h i} - \alpha_ {l o}}{\alpha_ {l o} \alpha_ {h i}}\right)}\right) ^ {- d} \right\} \left(\frac {\sigma_ {+}}{\psi \sigma_ {-} ^ {\alpha_ {l o} / \alpha_ {h i}}}\right) ^ {d / \alpha_ {l o}}. (86b) \\ \end{array}
+$$
+
+Intuitively, if the radius for which the polynomial bounds hold  $(r_{\eta})$  is large and the minimum curvature constant  $\sigma_{-}$  is also large, then the bound  $C_{\epsilon \leq \eta}$  holds for large deviations from optimality  $\eta$ . The number of  $\eta$ -radius  $\ell$ -balls required to cover the remaining space,  $C_{\epsilon \geq \eta}$ , will be comparatively small.
+
+# B.3 D-BLIN
+
+The regret bound for Doubling BLiN (Feng et al., 2022) was originally proved assuming a standard normal distribution, however, the authors state their proof can be easily adapted to any sub-Gaussian distribution, which includes bounded random variables. This matches our setting with bounded payoffs, so we repeat their analysis here for that setting.
+
+In what follows, in an effort to match notation in (Feng et al., 2022), let  $\mu(x) = -\mathcal{L}^{\tau}(s(x))$  denote the expected reward for evaluating  $x \in [0,1]^d$ ;  $\hat{\mu}(x)$  is an unbiased estimate of  $\mu(x)$ . Here,  $s: [0,1]^{n(\tilde{m} - 1)} \to \prod_i \Delta^{m_i - 1}$  is any function that maps from the unit hypercube to a product of simplices; we analyze two such maps in Appendix B.1. Also let  $\mathcal{A}_m$  denote the collection of hypercubes to be investigated during the  $m$ th batch of arm pulls,  $C \in \mathcal{A}_m$  denote a hypercube partition in  $\mathcal{X}$ ,  $n_m$  indicates the number of times each cube in  $\mathcal{A}_m$  is played in batch  $m$ , and  $B_{stop}$  denote the last batch.
+
+Definition 7 (Global Arm Accuracy).  $\mathcal{E} \stackrel{\mathrm{def}}{=} \left\{ |\mu(x) - \hat{\mu}_m(C)| \leq r_m + \sqrt{c_1 \frac{\ln T}{n_m}}, \forall 1 \leq m \leq B_{stop} - 1, \forall C \in \mathcal{A}_m, \forall x \in C \right\}$ .
+
+Define:  $n_m = c_2 \frac{\ln T}{r_m^2} \Rightarrow r_m = \sqrt{c_2 \frac{\ln T}{n_m}}$ .
+
+Definition 8 (Elimination Rule). Eliminate  $C \in \mathcal{A}_m$  if  $\hat{\mu}_m^{\max} - \hat{\mu}_m(C) \geq 2(1 + \sqrt{c_1 / c_2})r_m = 2(\sqrt{c_2} + \sqrt{c_1})\sqrt{\frac{\ln T}{n_m}}$  where  $\hat{\mu}_m^{\max} \stackrel{\mathrm{def}}{=} \max_{C \in \mathcal{A}_m} \hat{\mu}_m(C)$ .
+
+Lemma 27.  $Pr[\mathcal{E}] \geq 1 - 2T^{-2(c_1 / c^2 - 1)}$ .
+
+Proof. Let  $\hat{\mu}_m(C) = \frac{1}{n_m}\sum_{i=1}^{n_m}y_{C,i}$  be the unbiased estimate of  $\mu(x\in C)$  using  $n_m$  samples. Assume each  $y_{C,i}\in [a,b]$  with  $c = b - a$  and  $\hat{\mu}(C) = \frac{1}{n_m}\sum_{i=1}^{n_m}y_{C,i}$ . Applying a Hoeffding inequality gives
+
+$$
+\begin{array}{l} P r \left[ | \hat {\mu} (C) - \mathbb {E} [ \hat {\mu} (C) ] | \geq \sqrt {c _ {1} \frac {\ln T}{n _ {m}}} \right] \leq 2 e ^ {- 2 c _ {1} \ln T / c ^ {2}} (87a) \\ = 2 \left(e ^ {\ln T}\right) ^ {- 2 c _ {1} / c ^ {2}} (87b) \\ = 2 T ^ {- 2 c _ {1} / c ^ {2}} \forall C. (87c) \\ \end{array}
+$$
+
+By Lipschitzness of  $\mu$  we also have
+
+$$
+\left| \mathbb {E} \left[ \hat {\mu} (C) \right] - \mu (x) \right| \leq r _ {m}, \forall x \in C. \tag {88}
+$$
+
+Then consider
+
+$$
+\begin{array}{l} \sup  _ {x \in C} | \mu (x) - \hat {\mu} (C) | = \sup  _ {x \in C} | \mu (x) - \mathbb {E} [ \hat {\mu} (C) ] + \mathbb {E} [ \hat {\mu} (C) ] - \hat {\mu} (C) | (89a) \\ \leq \sup  _ {x \in C} \left(| \mu (x) - \mathbb {E} [ \hat {\mu} (C) ] | + | \mathbb {E} [ \hat {\mu} (C) ] - \hat {\mu} (C) |\right) (89b) \\ = \sup  _ {x \in C} | \mu (x) - \mathbb {E} [ \hat {\mu} (C) ] | + | \mathbb {E} [ \hat {\mu} (C) ] - \hat {\mu} (C) | (89c) \\ \leq \sqrt {c _ {1} \frac {\ln T}{n _ {m}}} + r _ {m} (89d) \\ \end{array}
+$$
+
+with probability  $1 - 2T^{-2c_1 / c^2}$ . The first inequality follows by triangle inequality and the second follows from equation (88) and considering the complement of equation (87c).
+
+The complement of this result occurs with probability
+
+$$
+P r \left[ \sup  _ {x \in C} | \mu (x) - \hat {\mu} (C) | \geq r _ {m} + \sqrt {c _ {1} \frac {\ln T}{n _ {m}}} \right] \leq 2 T ^ {- 2 c _ {1} / c ^ {2}}. \tag {90}
+$$
+
+At least 1 arm is played in each cube  $C \in \mathcal{A}_m$  for  $1 \leq m \leq B_{stop} - 1$ , therefore,  $|\mathcal{A}_m| \leq T$  must be true given the exit condition of the algorithm. In addition, assume  $B_{stop} \leq T$  ( $B_{stop}$  will be defined such that this is true). Then a union bound over all  $T^2$  events gives
+
+$$
+\begin{array}{l} P r \left[ \exists m \in [ 1, B _ {s t o p} - 1 ], C \in \mathcal {A} _ {m} s. t. \sup  _ {x \in C} | \mu (x) - \hat {\mu} (C) | \geq r _ {m} + \sqrt {c _ {1} \frac {\ln T}{n _ {m}}} \right] (91a) \\ \leq \sum_ {m = 1} ^ {B _ {s t o p} - 1} \sum_ {C \in \mathcal {A} _ {m}} P r \left[ \sup  _ {x \in C} | \mu (x) - \hat {\mu} (C) | \geq r _ {m} + \sqrt {c _ {1} \frac {\ln T}{n _ {m}}} \right] (91b) \\ \leq \sum_ {m = 1} ^ {B _ {s t o p} - 1} \sum_ {C \in \mathcal {A} _ {m}} 2 T ^ {- 2 c _ {1} / c ^ {2}} (91c) \\ \leq 2 T ^ {- 2 c _ {1} / c ^ {2}} T ^ {2}. (91d) \\ \end{array}
+$$
+
+Taking the complement of this event and noting that  $\sup_{x\in C}|\mu (x) - \hat{\mu} (C)|\leq r_m + \sqrt{c_1\frac{\ln T}{n_m}}\Rightarrow |\mu (x) - \hat{\mu} (C)|\leq r_m + \sqrt{c_1\frac{\ln T}{n_m}}\forall x\in C$  gives the desired result.
+
+Lemma 28 (Optimal Arm Survives). Under event  $\mathcal{E}$ , the optimal arm  $x^{*} = \arg \max \mu(x)$  is not eliminated after the first  $B_{stop} - 1$  batches.
+
+Proof. Let  $C_m^*$  denote the cube containing  $x^*$  in  $\mathcal{A}_m$ . Under event  $\mathcal{E}$ , for any cube  $C \in \mathcal{A}_m$  and  $x \in C$ , the following relation shows that  $C_m^*$  avoids the elimination rule in round  $m$ :
+
+$$
+\begin{array}{l} \hat {\mu} (C) - \hat {\mu} \left(C _ {m} ^ {*}\right) \leq \left(\mu (x) + r _ {m} + \sqrt {c _ {1} \frac {\ln T}{n _ {m}}}\right) + \left(- \mu \left(x ^ {*}\right) + r _ {m} + \sqrt {c _ {1} \frac {\ln T}{n _ {m}}}\right) (92a) \\ = \underbrace {(\mu (x) - \mu \left(x ^ {*}\right))} _ {\leq 0} + 2 r _ {m} + 2 \sqrt {c _ {1} \frac {\ln T}{n _ {m}}} (92b) \\ \leq 2 \sqrt {c _ {2} \frac {\ln T}{n _ {m}}} + 2 \sqrt {c _ {1} \frac {\ln T}{n _ {m}}} (92c) \\ = 2 \left(\sqrt {c _ {1}} + \sqrt {c _ {2}}\right) \sqrt {\frac {\ln T}{n _ {m}}} (92d) \\ \end{array}
+$$
+
+where the first inequality follows from applying Lemma 27 to upper bound  $\hat{\mu} (C)$  and  $\hat{\mu} (C_m^*)$  individually. The remaining steps use the optimality of  $x^{*}$ , the definition of  $r_m$ , and the elimination rule.
+
+Lemma 29. Under event  $\mathcal{E}$ , for any  $1 \leq m \leq B_{stop}$ , any  $C \in A_m$  and any  $x \in C$ ,  $\Delta_x$  satisfies
+
+$$
+\Delta_ {x} \leq 4 \left(1 + \sqrt {c _ {1} / c _ {2}}\right) r _ {m - 1} \tag {93}
+$$
+
+Proof. For  $m = 1$ , recall that  $r_m$  is the side length of a cube  $C \in \mathcal{A}_m$ , therefore,  $\Delta_x \leq r_{m-1} \leq 4(1 + \sqrt{c_1 / c_2})r_{m-1}$  holds directly from the Lipschitzness of  $\mu$ .
+
+For  $m > 1$ , let  $C_{m - 1}^{*}\in \mathcal{A}_{m - 1}$  be the cube containing  $x^{*}$ . From Lemma 28, this cube has not been eliminated under event  $\mathcal{E}$ . For any cube  $C\in \mathcal{A}_m$  and  $x\in C$ , it is clear that  $x$  is also in the parent of  $C$ , denoted  $C_{par}$  ( $x\in C\subset C_{par}$ ). Then for any  $x\in C$ , it holds that
+
+$$
+\begin{array}{l} \Delta_ {x} = \mu \left(x ^ {*}\right) - \mu (x) \leq \left(\hat {\mu} _ {m - 1} \left(C _ {m - 1} ^ {*}\right) + r _ {m - 1} + \sqrt {c _ {1} \frac {\ln T}{n _ {m - 1}}}\right) (94a) \\ + \left(- \hat {\mu} _ {m - 1} (C _ {p a r}) + r _ {m - 1} + \sqrt {c _ {1} \frac {\ln T}{n _ {m - 1}}}\right) \\ = \left(\hat {\mu} _ {m - 1} \left(C _ {m - 1} ^ {*}\right) - \hat {\mu} _ {m - 1} \left(C _ {p a r}\right)\right) + 2 \left(\sqrt {c _ {1}} + \sqrt {c _ {2}}\right) \sqrt {\frac {\ln T}{n _ {m - 1}}} (94b) \\ \leq \left(\hat {\mu} _ {m - 1} ^ {\max } - \hat {\mu} _ {m - 1} \left(C _ {p a r}\right)\right) + 2 \left(\sqrt {c _ {1}} + \sqrt {c _ {2}}\right) \sqrt {\frac {\ln T}{n _ {m - 1}}} (94c) \\ \leq 2 \left(\sqrt {c _ {1}} + \sqrt {c _ {2}}\right) \sqrt {\frac {\ln T}{n _ {m - 1}}} + 2 \left(\sqrt {c _ {1}} + \sqrt {c _ {2}}\right) \sqrt {\frac {\ln T}{n _ {m - 1}}} (94d) \\ = 4 \left(\sqrt {c _ {1}} + \sqrt {c _ {2}}\right) \sqrt {\frac {\ln T}{n _ {m - 1}}} (94e) \\ = 4 \left(1 + \sqrt {c _ {1} / c _ {2}}\right) r _ {m - 1} (94f) \\ \end{array}
+$$
+
+where we have applied Lemma 27 similarly as in Lemma 28 and also used the definition of  $r_{m - 1}$ . The last two inequalities use the fact that  $\hat{\mu}_{m - 1}(C_{m - 1}^{*})\leq \hat{\mu}_{m - 1}^{\max}$  and  $C_{par}$  was not eliminated.
+
+Theorem 2 (BLiN Regret Rate). With probability exceeding  $1 - 2T^{-2(c_1/c^2 - 1)}$ , the  $T$ -step total regret  $R(T)$  of BLiN with Doubling Edge-length Sequence ( $D$ -BLiN) Feng et al. (2022) satisfies
+
+$$
+R (T) \leq 8 \left(1 + \sqrt {c _ {1} / c _ {2}}\right) \left(2 c _ {2} + 1\right) \ln (T) ^ {\frac {1}{d _ {z} + 2}} T ^ {\frac {d _ {z} + 1}{d _ {z} + 2}} \tag {95}
+$$
+
+where  $d_{z}$  is the zooming dimension of the problem instance. In addition, D-BLiN only needs no more than  $B^{*} = \frac{\log 2(T) - \log 2(\ln(T))}{d_{z} + 2} + 2$  rounds of communications to achieve this regret rate.
+
+Proof. Since  $r_m = \frac{r_{m-1}}{2} \Rightarrow r_{m-1} = 2r_m$  for the Doubling Edge-length Sequence, Lemma 29 implies that every cube  $C \in A_m$  is a subset of  $S(8(1 + \sqrt{c_1 / c_2})r_m)$ . Thus from the definition of zooming dimension (Corollary 5 with appropriate condition), we have
+
+$$
+\left| \mathcal {A} _ {m} \right| \leq N _ {r _ {m}} \leq C _ {z} r _ {m} ^ {- d _ {z}}. \tag {96}
+$$
+
+Fix any positive number  $B$ . Also by Lemma 29, we know that any arm played after batch  $B$  incurs a regret bounded by  $8(1 + \sqrt{c_1 / c_2})r_B$ , since the cubes played after batch  $B$  have edge length no larger than  $r_B$ . Then the total regret that occurs after batch  $B$  is bounded by  $8(1 + \sqrt{c_1 / c_2})r_BT$  (where  $T$  is an upper bound on the number of arms).
+
+Thus the regret can be bounded as
+
+$$
+R (T) \leq \sum_ {m = 1} ^ {B} \sum_ {C \in \mathcal {A} _ {m}} \sum_ {i = 1} ^ {n _ {m}} \Delta_ {x _ {C, i}} + 8 \left(1 + \sqrt {c _ {1} / c _ {2}}\right) r _ {B} T \tag {97}
+$$
+
+where the first term bounds the regret in the first  $B$  batches of D-BLiN, and the second term bounds the regret after the first  $B$  batches. If the algorithm stops at batch  $\tilde{B} < B$ , we define  $\mathcal{A}_m =$  for any  $\tilde{B} < m \leq B$  and inequality equation (97) still holds.
+
+By Lemma 29, we have  $\Delta_{x_{C,i}}\leq 8(1 + \sqrt{c_1 / c_2})r_m$  for all  $C\in \mathcal{A}_m$ . We can thus bound equation (97) by
+
+$$
+\begin{array}{l} R (T) \leq \sum_ {m = 1} ^ {B} | \mathcal {A} _ {m} | \cdot n _ {m} \cdot 8 \left(1 + \sqrt {c _ {1} / c _ {2}}\right) r _ {m} + 8 \left(1 + \sqrt {c _ {1} / c _ {2}}\right) r _ {B} T (98a) \\ \leq \sum_ {m = 1} ^ {B} N _ {r _ {m}} \cdot n _ {m} \cdot 8 \left(1 + \sqrt {c _ {1} / c _ {2}}\right) r _ {m} + 8 \left(1 + \sqrt {c _ {1} / c _ {2}}\right) r _ {B} T (98b) \\ = \sum_ {m = 1} ^ {B} N _ {r _ {m}} \cdot c _ {2} \frac {\ln T}{r _ {m} ^ {2}} \cdot 8 \left(1 + \sqrt {c _ {1} / c _ {2}}\right) r _ {m} + 8 \left(1 + \sqrt {c _ {1} / c _ {2}}\right) r _ {B} T (98c) \\ = \sum_ {m = 1} ^ {B} N _ {r _ {m}} \cdot \frac {\ln T}{r _ {m}} \cdot 8 c _ {2} \left(1 + \sqrt {c _ {1} / c _ {2}}\right) + 8 \left(1 + \sqrt {c _ {1} / c _ {2}}\right) r _ {B} T (98d) \\ \end{array}
+$$
+
+where equation (98b) uses equation (96), and equation (98c) uses equality  $n_m = c_2 \frac{\ln T}{r_m^2}$ . Since  $r_m = 2^{-m + 1}$  and  $N_{r_m} \leq C_z r_m^{-d_z} \leq C_z 2^{(m - 1)d_z}$ , we have
+
+$$
+\begin{array}{l} R (T) \leq \sum_ {m = 1} ^ {B} C _ {z} 2 ^ {(m - 1) d _ {z}} \cdot \frac {\ln T}{2 ^ {- m + 1}} \cdot 8 c _ {2} \left(1 + \sqrt {c _ {1} / c _ {2}}\right) + 8 \left(1 + \sqrt {c _ {1} / c _ {2}}\right) 2 ^ {- B + 1} T (99a) \\ = 8 \left(1 + \sqrt {c _ {1} / c _ {2}}\right) \left[ c _ {2} C _ {z} \ln T \sum_ {m = 1} ^ {B} 2 ^ {(m - 1) (d _ {z} + 1)} + 2 ^ {- B + 1} T \right]. (99b) \\ \end{array}
+$$
+
+Continuing we find
+
+$$
+R (T) \leq 8 \left(1 + \sqrt {c _ {1} / c _ {2}}\right) \left[ c _ {2} C _ {z} \ln T \sum_ {m = 1} ^ {B} 2 ^ {(m - 1) (d _ {z} + 1)} + 2 ^ {- B + 1} T \right] \tag {100a}
+$$
+
+$$
+= 8 \left(1 + \sqrt {c _ {1} / c _ {2}}\right) \left[ c _ {2} C _ {z} \ln T \sum_ {m = 1} ^ {B} \left(2 ^ {d _ {z} + 1}\right) ^ {m - 1} + 2 ^ {- B + 1} T \right] \tag {100b}
+$$
+
+$$
+= 8 \left(1 + \sqrt {c _ {1} / c _ {2}}\right) \left[ c _ {2} C _ {z} \ln T \sum_ {m = 0} ^ {B - 1} \left(2 ^ {d _ {z} + 1}\right) ^ {m} + 2 ^ {- B + 1} T \right] \tag {100c}
+$$
+
+$$
+= 8 \left(1 + \sqrt {c _ {1} / c _ {2}}\right) \left[ c _ {2} C _ {z} \ln T \left(\frac {2 ^ {B (d _ {z} + 1)} - 1}{2 ^ {d _ {z} + 1} - 1}\right) + 2 ^ {- B + 1} T \right] \text {v i a g e o m e t r i c s e r i e s} \tag {100d}
+$$
+
+$$
+\leq 8 \left(1 + \sqrt {c _ {1} / c _ {2}}\right) \left[ c _ {2} C _ {z} \ln T \left(\frac {2 ^ {B (d _ {z} + 1)}}{2 ^ {d _ {z} + 1} - 1}\right) + 2 ^ {- B + 1} T \right] \tag {100e}
+$$
+
+$$
+\leq 8 \left(1 + \sqrt {c _ {1} / c _ {2}}\right) \left[ c _ {2} C _ {z} \ln T \left(2 \cdot \frac {2 ^ {B (d _ {z} + 1)}}{2 ^ {d _ {z} + 1}}\right) + 2 ^ {- B + 1} T \right] \tag {100f}
+$$
+
+$$
+= 8 \left(1 + \sqrt {c _ {1} / c _ {2}}\right) \left[ 2 c _ {2} C _ {z} 2 ^ {(B - 1) (d _ {z} + 1)} \ln T + 2 ^ {- (B - 1)} T \right]. \tag {100g}
+$$
+
+This inequality holds for any positive  $B$ . By choosing  $B^{*} = 1 + \frac{\log_{2}\left(\frac{T}{\ln T}\right)}{d_{z} + 2}$ , we have
+
+$$
+\begin{array}{l} R (T) \leq 8 \left(1 + \sqrt {c _ {1} / c _ {2}}\right) \left[ 2 c _ {2} C _ {z} \left(\frac {T}{\ln T}\right) ^ {\frac {(d _ {z} + 1)}{(d _ {z} + 2)}} \ln T + \left(\frac {\ln T}{T}\right) ^ {\frac {1}{(d _ {z} + 2)}} T \right] (101a) \\ = 8 \left(1 + \sqrt {c _ {1} / c _ {2}}\right) \left[ 2 c _ {2} C _ {z} T ^ {\frac {(d _ {z} + 1)}{(d _ {z} + 2)}} \ln T ^ {1 - \frac {(d _ {z} + 1)}{(d _ {z} + 2)}} + T ^ {1 - \frac {1}{(d _ {z} + 2)}} \ln T ^ {\frac {1}{(d _ {z} + 2)}} \right] (101b) \\ = 8 \left(1 + \sqrt {c _ {1} / c _ {2}}\right) \left[ 2 c _ {2} C _ {z} T ^ {\frac {(d _ {z} + 1)}{(d _ {z} + 2)}} \ln T ^ {\frac {1}{(d _ {z} + 2)}} + T ^ {\frac {(d _ {z} + 1)}{(d _ {z} + 2)}} \ln T ^ {\frac {1}{(d _ {z} + 2)}} \right] (101c) \\ = 8 \left(1 + \sqrt {c _ {1} / c _ {2}}\right) \left(2 c _ {2} C _ {z} + 1\right) T ^ {\frac {(d _ {z} + 1)}{(d _ {z} + 2)}} \ln T ^ {\frac {1}{(d _ {z} + 2)}}. (101d) \\ \end{array}
+$$
+
+Corollary 6 (BLiN Regret Rate Refined). Setting  $c_{1} = 2c^{2}$  and  $c_{2} = 2\left(\frac{c}{4C_{z}}\right)^{2 / 3}$  simplifies Theorem 2 such that
+
+$$
+R (T) \leq 8 \left(1 + \left(4 c ^ {2} C _ {z}\right) ^ {1 / 3}\right) ^ {2} T ^ {\frac {(d _ {z} + 1)}{(d _ {z} + 2)}} \ln T ^ {\frac {1}{(d _ {z} + 2)}}. \tag {102}
+$$
+
+with probability  $1 - 2T^{-2}$ .
+
+Proof. If we set  $c_1 = 2c^2$  and  $c_2 = 2\left(\frac{c}{4C_z}\right)^{2/3}$ , then  $\sqrt{c_1 / c_2} = c\left(\frac{4C_z}{c}\right)^{1/3} = \left(4c^2 C_z\right)^{1/3} = 2c_2C_z$ .
+
+Lemma 30. The zooming dimension and zooming constant under the  $\ell (x,y) = ||x - y||_{\infty}$  norm are
+
+$$
+d _ {z} = \frac {1}{2} n \bar {m} \tag {103}
+$$
+
+$$
+C _ {z} = \left| \mathcal {X} ^ {*} \right| ^ {- 1} \left(\frac {4}{r _ {\eta} ^ {2} \sigma_ {- \infty}}\right) ^ {n \bar {m}} \tag {104}
+$$
+
+where  $\sigma_{-\infty} = ||\text{Hess}(-f(x))^{-1}||_{\infty}$  is an upper bound on the infinity norm of the inverse Hessian matrix of the function at every equilibrium.
+
+Proof. Recall from Theorem 1 that  $d_{z} = d\left(\frac{\alpha_{hi} - \alpha_{lo}}{\alpha_{lo}\alpha_{hi}}\right)$  with constant  $C_z = 16^{-d'}C$  where  $C$  is defined below. In addition, BLiN assumes  $\ell (x,y) = ||x - y||_{\infty}$ . Matching to Assumption 1, we see that  $\sigma_{+} = \alpha_{lo} = 1$ . Under the infinity norm, the volume constant  $S_{d} = 2^{d}$ .
+
+We will define the other constants with respect to properties of the Hessian of  $f(x)$  about each equilibrium, specifically the infinity norm of the inverse Hessian so that  $\sigma_{-} = \sigma_{-\infty}$ . This means
+
+we will bound the function locally with a quadratic, i.e.,  $\alpha_{hi} = 2$ . Lastly, recall from Corollary 5 that  $\psi = \frac{1}{32}$  and the dimension of our search space (the product space of player mixed strategies) is  $d = n(\bar{m} - 1) \leq nm$  for simplicity.
+
+Plugging this information into Theorem 1, we find
+
+$$
+d _ {z} = d ^ {\prime} = \frac {1}{2} n \bar {m} \tag {105}
+$$
+
+and
+
+$$
+C = \max  \left\{1, | \mathcal {X} ^ {*} | ^ {- 1} S _ {d} ^ {- 1} \left(r _ {\eta} ^ {\frac {\alpha_ {h i}}{\alpha_ {l o}}} \sigma_ {-} ^ {\left(\frac {\alpha_ {h i} - \alpha_ {l o}}{\alpha_ {l o} \alpha_ {h i}}\right)}\right) ^ {- d} \right\} \left(\frac {\sigma_ {+}}{\psi \sigma_ {-} ^ {\alpha_ {l o} / \alpha_ {h i}}}\right) ^ {d / \alpha_ {l o}} \tag {106a}
+$$
+
+$$
+= \max  \left\{1, \left| \mathcal {X} ^ {*} \right| ^ {- 1} 2 ^ {- d} \left(r _ {\eta} ^ {2} \sigma_ {- \infty} ^ {\frac {1}{2}}\right) ^ {- d} \right\} \left(\frac {3 2}{\sigma_ {- \infty} ^ {1 / 2}}\right) ^ {d} \tag {106b}
+$$
+
+$$
+= \max  \left\{1, \left| \mathcal {X} ^ {*} \right| ^ {- 1} \left(2 r _ {\eta} ^ {2} \sigma_ {- \infty} ^ {\frac {1}{2}}\right) ^ {- d} \right\} \left(\frac {3 2}{\sigma_ {- \infty} ^ {1 / 2}}\right) ^ {d} \tag {106c}
+$$
+
+$$
+\stackrel {h a r d} {=} \left| \mathcal {X} ^ {*} \right| ^ {- 1} \left(2 r _ {\eta} ^ {2} \sigma_ {- \infty} ^ {\frac {1}{2}}\right) ^ {- d} \left(\frac {3 2}{\sigma_ {- \infty} ^ {1 / 2}}\right) ^ {d} \tag {106d}
+$$
+
+$$
+\stackrel {h a r d} {=} \left| \mathcal {X} ^ {*} \right| ^ {- 1} \left(\frac {1 6}{r _ {\eta} ^ {2} \sigma_ {- \infty}}\right) ^ {n \bar {m}} \tag {106e}
+$$
+
+where  $\text{hard}$  indicates we are assuming  $r_{\eta}$  and  $\sigma_{-\infty}$  are small enough to dominate the other operand of the max.
+
+Finally, converting the near optimality constant to a zooming constant, we find
+
+$$
+\begin{array}{l} C _ {z} = 1 6 ^ {- d _ {z}} \left| \mathcal {X} ^ {*} \right| ^ {- 1} \left(\frac {1 6}{r _ {\eta} ^ {2} \sigma_ {- \infty}}\right) ^ {n \bar {m}} (107a) \\ = \left| \mathcal {X} ^ {*} \right| ^ {- 1} \left(\frac {4}{r _ {\eta} ^ {2} \sigma_ {- \infty}}\right) ^ {n \bar {m}}. (107b) \\ \end{array}
+$$
+
+![](images/da86bade918f42ff09723255caa327bbe359f0ae626b46d3e25b9cf4c3a5b407.jpg)
+
+# B.4 BOUNDED DIAMETERS AND WELL-SHAPED CELLS
+
+We assume the feasible set is a unit-hypercube of dimensionality  $d$  where cells are evenly split along the longest edge to give  $b$  new partitions and  $x_{h,i}$  represents the center of each cell.
+
+There exists a decreasing sequence  $w(h) > 0$ , such that for any depth  $h \geq 0$  and for any cell  $\mathcal{X}_{h,i}$  of depth  $h$ , we have  $\sup_{x \in \mathcal{X}_{h,i}} \ell(x_{h,i}, x) \leq w(h)$ . Moreover, there exists  $\nu > 0$  such that for any depth  $h \geq 0$ , any cell  $\mathcal{X}_{h,i}$  contains an  $\ell$ -ball of radius  $\nu w(h)$  centered at  $x_{h,i}$ .
+
+<table><tr><td>l(x,y)</td><td>c</td><td>γ</td><td>ν</td></tr><tr><td>l(x,y)=||x-y||2/2</td><td>dα/2(b/2)α</td><td>b-α/d</td><td>d-α/2b-2α</td></tr><tr><td>l(x,y)=||x-y||∞/∞</td><td>(b/2)α</td><td>b-α/d</td><td>b-2α</td></tr></table>
+
+Table 3: Bounding Constants:  $\sup_{x \in \mathcal{X}_{h,i}} \ell(x_{h,i}, x) \leq w(h) = c\gamma^h$ .
+
+# B.4.1  $L_{2}$ -NORM
+
+Lemma 31 (L2-Norm Bounding Ball). Let  $\ell(x, y) = ||x - y||_2^\alpha$ . Then  $\sup_{x \in \mathcal{X}_{h,i}} \ell(x_{h,i}, x) \leq w_2(h) = c\gamma^h$  where  $c = \left(\frac{db^2}{4}\right)^{\alpha/2}$  and  $\gamma = b^{-\alpha/d}$ .
+
+Proof.
+
+$$
+w (0) = \left[ \sum_ {i = 1} ^ {d} (1 / 2) ^ {2} \right] ^ {\alpha / 2} = \left(\frac {d}{4}\right) ^ {\alpha / 2} \tag {108a}
+$$
+
+$$
+\begin{array}{l} w (1) = \left[ (1 / b \cdot 1 / 2) ^ {2} + \sum_ {i = 2} ^ {d} (1 / 2) ^ {2} \right] ^ {\alpha / 2} = \left[ (1 / b ^ {2}) (1 / 4) + (d - 1) (1 / 4) \right] ^ {\alpha / 2} (108b) \\ = \left(\frac {d - 1 + 1 / b ^ {2}}{4}\right) ^ {\alpha / 2} (108c) \\ \end{array}
+$$
+
+$$
+\begin{array}{l} w (d) = \left[ \sum_ {i = 1} ^ {d} (1 / b \cdot 1 / 2) ^ {2} \right] ^ {\alpha / 2} = \left(\frac {d}{4 \cdot b ^ {2}}\right) ^ {\alpha / 2} (108d) \\ w (h) = \left[ r (1 / b) ^ {2 (q + 1)} (1 / 2) ^ {2} + \sum_ {i = r} ^ {d} (1 / b) ^ {2 q} (1 / 2) ^ {2} \right] ^ {\alpha / 2} (108e) \\ = \left[ (1 / b) ^ {2 q} (1 / 2) ^ {2} (r (1 / b) ^ {2} + (d - r) \right] ^ {\alpha / 2} (108f) \\ = \left[ (1 / b ^ {2}) ^ {q} (1 / 4) \left(d - r \left(1 - \frac {1}{b ^ {2}}\right)\right) \right] ^ {\alpha / 2} (108g) \\ \leq \left[ (1 / b ^ {2}) ^ {q} (1 / 4) d \right] ^ {\alpha / 2} (108h) \\ \leq \left[ \left(1 / b ^ {2}\right) ^ {h / d - 1} (1 / 4) d \right] ^ {\alpha / 2} (108i) \\ = \left[ (1 / b ^ {2}) ^ {h / d} (b ^ {2} / 4) d \right] ^ {\alpha / 2} (108j) \\ = \left(\frac {d b ^ {2}}{4}\right) ^ {\alpha / 2} (1 / b) ^ {\frac {\alpha}{d} h} (108k) \\ = c \gamma^ {h} (1081) \\ \end{array}
+$$
+
+where  $q, r = \text{divmod}(h, d) \Rightarrow q \geq h / d - 1, c = \left(\frac{db^2}{4}\right)^{\alpha / 2}$ , and  $\gamma = (1 / b)^{\alpha / d} = b^{-\alpha / d}$ .
+
+Lemma 32 ( $L_2$ -Norm Inner Ball). Let  $\ell(x, y) = ||x - y||_2^\alpha$ . Any cell  $\mathcal{X}_{h,i}$  contains an  $\ell$ -ball of radius  $\nu w_2(h)$  where  $\nu = (db^4)^{-\alpha/2}$ .
+
+Proof. Any cell  $\mathcal{X}_{h,i}$  contains an  $\ell$ -ball of radius equal to its shortest axis:
+
+$$
+\begin{array}{l} r _ {\min } = \left[ (1 / 4) \left(1 / b ^ {2}\right) ^ {\lceil h / d \rceil} \right] ^ {\alpha / 2} (109a) \\ \geq \left[ (1 / 4) \left(1 / b ^ {2}\right) ^ {h / d + 1} \right] ^ {\alpha / 2} (109b) \\ = \left(\frac {1}{b ^ {2} \cdot 4}\right) ^ {\alpha / 2} (1 / b) ^ {\frac {\alpha}{d} h} (109c) \\ = w (h) \cdot \left(\frac {1}{d b ^ {4}}\right) ^ {\alpha / 2}. (109d) \\ \end{array}
+$$
+
+# B.4.2  $L_{\infty}$ -NORM
+
+Lemma 33 ( $L_{\infty}$ -Norm Bounding Ball). Let  $\ell(x, y) = ||x - y||_{\infty}^{\alpha}$ . Then  $\sup_{x \in \mathcal{X}_{h,i}} \ell(x_{h,i}, x) \leq w_{\infty}(h) = c\gamma^{h}$  where  $c = \left(\frac{b}{2}\right)^{\alpha}$  and  $\gamma = b^{-\alpha / d}$ .
+
+Proof. Any cell  $\mathcal{X}_{h,i}$  is contained by an  $\ell$ -ball of radius equal to its longest axis:
+
+$$
+\begin{array}{l} r _ {\max } = \left[ (1 / 4) \left(1 / b ^ {2}\right) ^ {\lfloor h / d \rfloor} \right] ^ {\alpha / 2} (110a) \\ \leq \left[ (1 / 4) \left(1 / b ^ {2}\right) ^ {h / d - 1} \right] ^ {\alpha / 2} (110b) \\ = \left(\frac {b ^ {2}}{4}\right) ^ {\alpha / 2} (1 / b) ^ {\frac {\alpha}{d} h} (110c) \\ = c \gamma^ {h} (110d) \\ \end{array}
+$$
+
+where  $c = \left(\frac{b^2}{4}\right)^{\alpha / 2}$ , and  $\gamma = (1 / b)^{\alpha / d} = b^{-\alpha / d}$ .
+
+![](images/bceee6c50ff66f24cad0648f2ad82e3b0b2853e2fd8d46f2a2a53b40be08fb15.jpg)
+
+Lemma 34 ( $L_{\infty}$ -Norm Inner Ball). Let  $\ell(x, y) = ||x - y||_{\infty}^{\alpha}$ . Any cell  $\mathcal{X}_{h,i}$  contains an  $\ell$ -ball of radius  $\nu w_{\infty}(h)$  where  $\nu = b^{-2\alpha}$ .
+
+Proof. Any cell  $\mathcal{X}_{h,i}$  contains an  $\ell$ -ball of radius equal to its shortest axis:
+
+$$
+\begin{array}{l} r _ {\min } = \left[ (1 / 4) \left(1 / b ^ {2}\right) ^ {\lceil h / d \rceil} \right] ^ {\alpha / 2} (111a) \\ \geq \left[ (1 / 4) \left(1 / b ^ {2}\right) ^ {h / d + 1} \right] ^ {\alpha / 2} (111b) \\ = \left(\frac {1}{b ^ {2} \cdot 4}\right) ^ {\alpha / 2} (1 / b) ^ {\frac {\alpha}{d} h} (111c) \\ = w (h) \cdot \left(\frac {1}{b ^ {4}}\right) ^ {\alpha / 2}. (111d) \\ \end{array}
+$$
+
+![](images/5754ce1d5b12d10e269c95167dfbd9043c97f1055ae9a80af0106ee510413702.jpg)
+
+# B.5 STOCHASTIC SIMULTANEOUS OPTIMISTIC OPTIMIZATION
+
+StoSOO is flexible in its choice of metric and partitioning structure. In StoSOO, we may choose  $\ell(x,y) = ||x - y||_2^2$ .
+
+Lemma 35. The volume of a  $d$ -sphere with radius  $r$  and  $d$  even is given by  $S_{d}r^{d}$  where  $S_{d}^{-1} \leq \sqrt{2\pi d}\left(\frac{d}{2\pi e}\right)^{d/2}$ .
+
+Proof. First, we recall Stirling's bounds on the factorial:  $\sqrt{2\pi n} \left( \frac{n}{e} \right)^n e^{\frac{1}{12n + 1}} < n! < \sqrt{2\pi n} \left( \frac{n}{e} \right)^n e^{\frac{1}{12n}}$ . This will be useful for bounding the Gamma function:  $\Gamma(d) = (d - 1)!$  for even  $d$ .
+
+Given  $d$  is even, we start with the exact formula for  $S_{d}$ :
+
+$$
+\begin{array}{l} S _ {d} ^ {- 1} = \frac {\Gamma (d / 2 + 1)}{\pi^ {d / 2}} (112a) \\ = \frac {(d / 2) !}{\pi^ {d / 2}} (112b) \\ <   \frac {\sqrt {2 \pi (d / 2)} \left(\frac {d / 2}{e}\right) ^ {d / 2} e ^ {\frac {1}{1 2 (d / 2)}}}{\pi^ {d / 2}} (112c) \\ = \frac {\pi^ {1 / 2} d ^ {1 / 2} \left(\frac {d}{2 e}\right) ^ {d / 2} e ^ {\frac {1}{6 d}}}{\pi^ {d / 2}} (112d) \\ = \frac {\pi^ {1 / 2} d ^ {(d + 1) / 2} e ^ {\frac {1}{6 d}}}{(2 \pi e) ^ {d / 2}} (112e) \\ \leq \sqrt {2 \pi d} \left(\frac {d}{2 \pi e}\right) ^ {d / 2}. (112f) \\ \end{array}
+$$
+
+![](images/ad671b84f8dfee24547b9767b22bf0d9af09028040855e0cad6399d2c8f91d3a.jpg)
+
+Lemma 36. The near optimality dimension and constant under the  $\ell(x, y) = ||x - y||_2^2$  norm are
+
+$$
+d ^ {\prime} = 0 \tag {113}
+$$
+
+$$
+C = \left| \mathcal {X} ^ {*} \right| ^ {- 1} \sqrt {2 \pi n \bar {m}} \left(\frac {n \bar {m}}{5 \nu r _ {\eta} ^ {2} \sigma_ {- 2}}\right) ^ {n \bar {m} / 2} \tag {114}
+$$
+
+where  $\sigma_{-2}$  is a lower bound on the singular value of the Hessian matrix of the function at every equilibrium and  $\nu$  is defined in Table 3 for the corresponding  $\ell(x, y)$ -ball.
+
+Proof. Recall from Theorem 1 that  $d' = d\left(\frac{\alpha_{hi} - \alpha_{lo}}{\alpha_{lo}\alpha_{hi}}\right)$  with constant  $C$  defined below. In addition, we will analyze StoSOO with the choice  $\ell(x,y) = ||x - y||_2^2$ . Matching to Assumption 1, we see
+
+that  $\sigma_{+} = 1$  and  $\alpha_{lo} = 2$ . Under the 2-norm, the volume constant  $S_{d}^{-1} \leq \sqrt{2\pi d}\left(\frac{d}{2\pi e}\right)^{d/2}$  (see Lemma 35).
+
+We will bound the function locally with a quadratic, i.e.,  $\alpha_{hi} = 2$ . Lastly, from (Valko et al., 2013, Corollary 1),  $\psi = \frac{\nu}{3}$  with  $\nu = \frac{1}{db^2}$  defined in Table 3 and the dimension of our search space (the product space of player mixed strategies) is  $d = n(\bar{m} - 1) \leq n\bar{m}$  for simplicity.
+
+Plugging this information into Theorem 1, we find  $d' = 0$  and
+
+$$
+C = \max  \left\{1, \left| \mathcal {X} ^ {*} \right| ^ {- 1} S _ {d} ^ {- 1} \left(r _ {\eta} ^ {\frac {\alpha_ {h i}}{\alpha_ {l o}}} \sigma_ {-} ^ {\left(\frac {\alpha_ {h i} - \alpha_ {l o}}{\alpha_ {l o} \alpha_ {h i}}\right)}\right) ^ {- d} \right\} \left(\frac {\sigma_ {+}}{\psi \sigma_ {-} ^ {\alpha_ {l o} / \alpha_ {h i}}}\right) ^ {d / \alpha_ {l o}} \tag {115a}
+$$
+
+$$
+= \max  \left\{1, \left| \mathcal {X} ^ {*} \right| ^ {- 1} S _ {d} ^ {- 1} r _ {\eta} ^ {- d} \right\} \left(\frac {1}{\psi \sigma_ {- 2}}\right) ^ {d / 2} \tag {115b}
+$$
+
+$$
+\stackrel {h a r d} {=} \left| \mathcal {X} ^ {*} \right| ^ {- 1} S _ {d} ^ {- 1} r _ {\eta} ^ {- d} \left(\frac {3}{\nu \sigma_ {- 2}}\right) ^ {d / 2} \tag {115c}
+$$
+
+$$
+\leq^ {h a r d} \left| \mathcal {X} ^ {*} \right| ^ {- 1} \sqrt {2 \pi d} \left(\frac {d}{2 \pi e}\right) ^ {d / 2} \left(\frac {3}{\nu r _ {\eta} ^ {2} \sigma_ {- 2}}\right) ^ {d / 2} \tag {115d}
+$$
+
+$$
+\leq^ {h a r d} \left| \mathcal {X} ^ {*} \right| ^ {- 1} \sqrt {2 \pi d} \left(\frac {b ^ {2} d ^ {2}}{5 r _ {\eta} ^ {2} \sigma_ {- 2}}\right) ^ {d / 2} \tag {115e}
+$$
+
+$$
+\stackrel {h a r d} {=} \left| \mathcal {X} ^ {*} \right| ^ {- 1} \sqrt {2 \pi n \bar {m}} \left(\frac {b ^ {2} n ^ {2} \bar {m} ^ {2}}{5 r _ {\eta} ^ {2} \sigma_ {- 2}}\right) ^ {n \bar {m} / 2} \tag {115f}
+$$
+
+where  $\text{hard}$  indicates we are assuming  $r_{\eta}$ , the radius of the ball under which the local polynomial bounds are accurate, is small enough to dominate the other operand of the max.
+
+Theorem 3 (StoSOO Regret Rate). Corollary 1 of Valko et al. (2013) implies that with probability  $1 - \delta$ , the regret,  $R_{t}$ , of StoSOO after  $t$  pulls is upper bounded as
+
+$$
+(2 + b ^ {2 / d}) \sqrt {\frac {\log_ {b} (t k / \delta)}{2 \log_ {b} (e) k}} + \frac {1}{4} d b ^ {2 (1 + 2 / d)} b ^ {- \frac {1}{d C} \sqrt {t / k}} \tag {116}
+$$
+
+where  $d = n(\bar{m} - 1)$ ,  $b$  is the branching factor for partitioning cells,  $C$  is the near-optimality constant, and  $k$  is the maximum number of evaluations per node.
+
+Proof. Plugging the constants  $c, \gamma$ , and  $\nu$  defined in Table 3 for the 2-norm into Corollary 1 of Valko et al. (2013), we find with probability  $1 - \delta$ :
+
+$$
+\begin{array}{l} R _ {t} \leq (2 + 1 / \gamma) \epsilon + c \gamma^ {\frac {1}{2 C} \sqrt {t / k} - 2} (117a) \\ = (2 + 1 / \gamma) \sqrt {\frac {\log (t k / \delta)}{2 k}} + c \gamma^ {\frac {1}{2 C} \sqrt {t / k} - 2} (117b) \\ = \left(2 + b ^ {2 / d}\right) \sqrt {\frac {\log (t k / \delta)}{2 k}} + \frac {1}{4} d b ^ {2} \left(b ^ {- 2 / d}\right) ^ {\frac {1}{2 C} \sqrt {t / k} - 2} (117c) \\ = (2 + b ^ {2 / d}) \sqrt {\frac {\log_ {b} (t k / \delta)}{2 \log_ {b} (e) k}} + \frac {1}{4} d b ^ {2 (1 + 2 / d)} \left(b ^ {- 2 / d}\right) ^ {\frac {1}{2 C} \sqrt {t / k}} (117d) \\ = \left(2 + b ^ {2 / d}\right) \sqrt {\frac {\log_ {b} (t k / \delta)}{2 \log_ {b} (e) k}} + \frac {1}{4} d b ^ {2 (1 + 2 / d)} b ^ {- \frac {1}{d C} \sqrt {t / k}}. (117e) \\ \end{array}
+$$
+
+![](images/9ace8b6d2efdd07243cdf25690d81420b8365d871deb581108c33f389410ec51.jpg)
+
+Proposition 2. Assume the same conditions as Theorem 3 and let  $k = t\log_b(t)^{-\rho}$  where  $\rho \geq 3$ . Then the StoSOO (Algorithm 1 of Valko et al. (2013)) bound requires  $t = b^{\left(\frac{dC}{2}\right)^{\frac{2}{\rho - 2}}}$  pulls before exhibiting a  $\tilde{\mathcal{O}}(T^{-1/2})$  decay in regret.
+
+Proof. We know from the given definition of  $k$  that  $\sqrt{t / k} = \log_b(t)^{\rho /2}$ . We will analyze the second term in the regret formula of Corollary 3. Note that the second term,  $b^{-\frac{1}{dC}\sqrt{t / k}} = (b\sqrt{t / k})^{-\frac{1}{dC}} = (b^{\log_b(t)^{\rho /2}})^{-\frac{1}{dC}} = (b^{\log_b(t)\cdot \log_b(t)^{(\rho -2) / 2}})^{-\frac{1}{dC}} = (t^{\log_b(t)^{(\rho -2) / 2}})^{-\frac{1}{dC}} = t^{-\frac{\log_b(t)^{(\rho - 2) / 2}}{dC}}$ . To achieve a  $t^{-1 / 2}$  convergence rate (or better), we need  $t\geq b^{(\frac{dC}{2})^{\frac{2}{\rho - 2}}}$ . For convenience, we report the entire simplification of the bound below:
+
+$$
+\begin{array}{l} R _ {t} \leq (2 + b ^ {2 / d}) \sqrt {\frac {\log_ {b} (t k / \delta)}{2 \log_ {b} (e) k}} + \frac {1}{4} d b ^ {2 (1 + 2 / d)} b ^ {- \frac {1}{d C}} \sqrt {t / k} (118a) \\ = (2 + b ^ {2 / d}) \sqrt {\frac {\log_ {b} \left(t ^ {2} / \left(\delta \log_ {b} (t) ^ {\rho}\right)\right) \log_ {b} (t) ^ {\rho}}{2 \log_ {b} (e) t}} + \frac {1}{4} d b ^ {2 (1 + 2 / d)} t ^ {- \frac {\log_ {b} (t) ^ {(\rho - 2) / 2}}{d C}} (118b) \\ \leq \left(2 + b ^ {2 / d}\right) \sqrt {\frac {\log_ {b} \left(t ^ {2} / \delta\right) \log_ {b} \left(t / \delta\right) ^ {\rho}}{2 \log_ {b} (e) t}} + \frac {1}{4} d b ^ {2 (1 + 2 / d)} t ^ {- \frac {\log_ {b} (t) ^ {(\rho - 2) / 2}}{d C}} (118c) \\ = (2 + b ^ {2 / d}) \sqrt {\frac {2 \log_ {b} (t / \delta) ^ {\rho + 1}}{2 \log_ {b} (e) t}} + \frac {1}{4} d b ^ {2 (1 + 2 / d)} t ^ {- \frac {\log_ {b} (t) ^ {(\rho - 2) / 2}}{d C}} (118d) \\ = \frac {\left(2 + b ^ {2 / d}\right)}{\sqrt {\log_ {b} (e)}} \frac {\log_ {b} (t / \delta) ^ {(\rho + 1) / 2}}{\sqrt {t}} + \frac {1}{4} d b ^ {2 (1 + 2 / d)} t ^ {- \frac {\log_ {b} (t) ^ {(\rho - 2) / 2}}{d C}} (118e) \\ = \frac {\left(2 + b ^ {2 / d}\right)}{\sqrt {\log_ {b} (e)}} \frac {\left[ \log_ {b} (t) - \log_ {b} (\delta) \right] ^ {(\rho + 1) / 2}}{\sqrt {t}} + \frac {1}{4} d b ^ {2 (1 + 2 / d)} t ^ {- \frac {\log_ {b} (t) ^ {(\rho - 2) / 2}}{d C}}. (118f) \\ \end{array}
+$$
+
+![](images/2d3ff3d62dd76cd6834c1121350eb79246f1e3f2f88373c316c4cc70635be961.jpg)
+
+# B.6 REGRET TO PAC BOUNDS
+
+Lemma 37. [Loss Regret to Exploitability Regret] Assume exploitability of a joint strategy  $\pmb{x}$  is upper bounded by  $f(\mathcal{L}^{\tau}(\pmb{x}))$  where  $f$  is a concave function and  $\mathcal{L}^{\tau}$  is a loss function. Let  $\pmb{x}_t$  be a joint strategy randomly drawn from the set of predictions made by an online learning algorithm  $\mathcal{A}$  over  $T$  steps. Then the expected exploitability of  $\pmb{x}_t$  is bounded by the average regret of  $\mathcal{A}$ :
+
+$$
+\mathbb {E} _ {t} [ \epsilon_ {t} ] \leq f \left(\frac {1}{T} R (T)\right). \tag {119}
+$$
+
+Proof.
+
+$$
+\mathbb {E} _ {t} \left[ \epsilon_ {t} \right] = \mathbb {E} _ {t} \left[ f \left(\mathcal {L} \left(\boldsymbol {x} _ {t}\right)\right) \right] \leq f \left(\mathbb {E} _ {t} \left[ \mathcal {L} \left(\boldsymbol {x} _ {t}\right) \right]\right) = f \left(\frac {1}{T} \sum_ {t} \mathcal {L} \left(\boldsymbol {x} _ {t}\right)\right) = f \left(\frac {1}{T} R (T)\right) \tag {120}
+$$
+
+where the inequality follows from Jensen's inequality.
+
+![](images/8cb08643aa045aa04bdc3c88894e7dc948749456006babe3f63950ec4680e0d7.jpg)
+
+Theorem 4 (BLiN PAC Rate). Assume  $\eta_{k} = \eta = 2 / \hat{L}$  as defined in Lemma 2,  $\tau = \frac{1}{\ln(1 / p)}$  so that all equilibria place at least  $\frac{p}{m^*}$  mass on each strategy, and a previously pulled arm is returned uniformly at random (i.e.,  $t \sim U(T)$ ). Then for any  $w > 0$ ,
+
+$$
+\epsilon_ {t} \leq w \left[ \tau \log \left(\prod_ {k} m _ {k}\right) + 2 \left(1 + \left(4 c ^ {2} C _ {z}\right) ^ {1 / 3}\right) \sqrt {2 n \hat {L}} \left(\frac {\ln T}{T}\right) ^ {\frac {1}{2 \left(d _ {z} + 2\right)}} \right] \tag {121}
+$$
+
+with probability  $(1 - w^{-1})(1 - 2T^{-2})$  where  $m^{*} = \max_{k} m_{k}$ , and  $c \leq \frac{n\bar{m}}{\hat{L}}\left(\frac{\ln(m^{*})}{\ln(1/p)} + 2\right)^{2}$  is an upper bound on the range of  $\hat{\mathcal{L}}^{\tau}$  (Corollary 1),  $\hat{L} = \left(\frac{\ln(m^{*})}{\ln(1/p)} + 2\right)\left(\frac{m^{*2}}{p\ln(1/p)} + n\bar{m}\right)$  (Corollary 2), the zooming dimension  $d_{z} = \frac{1}{2} n\bar{m}$ , and the zooming constant  $C_{z} = |\mathcal{X}^{*}|^{-1}\left(\frac{4}{r_{\eta}^{2}\sigma_{-\infty}}\right)^{n\bar{m}}$  (Corollary 30).
+
+Proof. Assume  $\eta_{k} = \eta = \frac{2}{L}$  as defined in Lemma 2 so that  $\mathcal{L}^{\tau}$  is 1-Lipschitz with respect to  $||\cdot ||_{\infty}$ . Also assume a previously pulled arm is returned uniformly at random. Starting with Lemma 13 and applying Corollary 6, we find
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \epsilon_ {t} \right] \leq \tau \log \left(\prod_ {k} m _ {k}\right) + \sqrt {\frac {2 n}{\min  _ {k} \eta_ {k}}} \sqrt {\frac {1}{T} \sum_ {t} \mathcal {L} ^ {\tau} \left(\boldsymbol {x} _ {t}\right)} (122a) \\ \leq \frac {1}{\ln (1 / p)} \log \left(\prod_ {k} m _ {k}\right) + \sqrt {n \hat {L}} \sqrt {8 \left(1 + \left(4 c ^ {2}\right) ^ {1 / 3}\right) ^ {2} T ^ {\frac {- 1}{\left(d z + 2\right)}} \ln T ^ {\frac {1}{\left(d z + 2\right)}}} (122b) \\ = \frac {1}{\ln (1 / p)} \log \left(\prod_ {k} m _ {k}\right) + 2 \left(1 + \left(4 c ^ {2} C _ {z}\right) ^ {1 / 3}\right) \sqrt {2 n \hat {L}} \left(\frac {\ln T}{T}\right) ^ {\frac {1}{2 \left(d _ {z} + 2\right)}} (122c) \\ \end{array}
+$$
+
+with probability  $1 - 2T^{-2}$  where  $m^{*} = \max_{k} m_{k}$ , and  $c \leq \frac{n\bar{m}}{\hat{L}} \left( \frac{\ln(m^{*})}{\ln(1/p)} + 2 \right)^{2}$  is an upper bound on the range of sampled values from  $\hat{\mathcal{L}}^{\tau}$  (see Corollary 1).
+
+Markov's inequality then allows us to bound the pointwise exploitability of any arm returned by the algorithm as
+
+$$
+\epsilon_ {t} \leq w \left[ \frac {1}{\ln (1 / p)} \log \left(\prod_ {k} m _ {k}\right) + 2 \left(1 + \left(4 c ^ {2} C _ {z}\right) ^ {1 / 3}\right) \sqrt {2 n \hat {L}} \left(\frac {\ln T}{T}\right) ^ {\frac {1}{2 \left(d _ {z} + 2\right)}} \right] \tag {123}
+$$
+
+with probability  $(1 - w^{-1})(1 - 2T^{-2})$  for any  $w > 0$ .
+
+Theorem 5 (StoSOO PAC Rate). Corollary 1 of Valko et al. (2013) implies that with probability  $(1 - w^{-1})(1 - \delta)$  for any  $w > 0$ , a uniformly randomly drawn arm (i.e.,  $t \sim U([T])$ ) achieves
+
+$$
+\epsilon_ {t} \leq w \left[ \frac {1}{\ln (1 / p)} \log \left(\prod_ {k} m _ {k}\right) + \sqrt {n \hat {L}} \sqrt {\xi_ {1} \sqrt {\frac {\log_ {b} (T k / \delta)}{2 \log_ {b} (e) k}} + \xi_ {2} b ^ {- \frac {1}{d C} \sqrt {T / k}}} \right] \tag {124}
+$$
+
+where  $d = n(\bar{m} - 1)$ ,  $\xi_1 = (2 + 2^{2/d})$ ,  $\xi_2 = \frac{1}{4} db^{2(1 + 2/d)}$ ,  $k = T \log_b(T)^{-3}$ ,  $b$  is the branching factor for partitioning cells, and the near-optimality constant  $C = |\mathcal{X}^*|^{-1}\sqrt{2\pi d}\left(\frac{b^2d^2}{5r_\eta^2\sigma_{-2}}\right)^{d/2}$  (Lemma 36).
+
+Proof. Assume  $\eta_{k} = \eta = \frac{2}{L}$  as defined in Lemma 2. Also assume a previously pulled arm is returned uniformly at random. Starting with Lemma 13 and applying Theorem 3 and Lemma 37, we find
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \epsilon_ {t} \right] \leq \tau \log \left(\prod_ {k} m _ {k}\right) + \sqrt {\frac {2 n}{\min  _ {k} \eta_ {k}}} \sqrt {\frac {1}{T} \sum_ {t} \mathcal {L} ^ {\tau} \left(\boldsymbol {x} _ {t}\right)} (125a) \\ \leq \frac {1}{\ln (1 / p)} \log \left(\prod_ {k} m _ {k}\right) + \sqrt {n \hat {L}} \sqrt {\xi_ {1} \sqrt {\frac {\log_ {b} (T k / \delta)}{2 \log_ {b} (e) k}} + \xi_ {2} b ^ {- \frac {1}{d C} \sqrt {T / k}}} (125b) \\ \end{array}
+$$
+
+with probability  $1 - \delta$  where  $m^{*} = \max_{k} m_{k}$ ,  $d = n(\bar{m} - 1)$ ,  $\xi_{1} = (2 + 2^{2 / d})$ ,  $\xi_{2} = \frac{1}{4} db^{2(1 + 2 / d)}$ ,  $k = T\log_b(T)^{-3}$ ,  $b$  is the branching factor for partitioning cells, and the near-optimality constant  $C = |\mathcal{X}^{*}|^{-1}\sqrt{2\pi d}\left(\frac{b^2d^2}{5r_\eta^2\sigma_{-2}}\right)^{d / 2}$  (Lemma 36).
+
+Markov's inequality then allows us to bound the pointwise exploitability of any arm returned by the algorithm as
+
+$$
+\epsilon_ {t} \leq w \left[ \frac {1}{\ln (1 / p)} \log \left(\prod_ {k} m _ {k}\right) + \sqrt {n \hat {L}} \sqrt {\xi_ {1} \sqrt {\frac {\log_ {b} (T k / \delta)}{2 \log_ {b} (e) k}} + \xi_ {2} b ^ {- \frac {1}{d C} \sqrt {T / k}}} \right] \tag {126}
+$$
+
+with probability  $(1 - w^{-1})(1 - \delta)$  for any  $w > 0$ .
+
+# B.7 COMPLEXITY OF POLYMATRIX GAMES
+
+Lemma 38. For a polymatrix game defined by the set of bimatrix games with payoff matrix  $P_{kl}^{k}$  for every player  $k$  and  $l \neq k$ , the rank of the matrix  $M(x)$  defined in equation (11) can be equivalently studied by replacing all instances of  $H_{kl}^{k}$  with  $P_{kl}^{k}$ .
+
+Proof. Consider the polymatrix game given by the set of matrices  $P_{kl}^{k}$  for every  $k$  and  $l \neq k$ . The polymatrix game can be equivalently written in normal form, albeit, less concisely. Note that the polymatrix approximation,  $H_{kl}^{k}$ , as we have defined it (see Section 2 Preliminaries) of this normal-form representation between players  $k$  and  $l$  with all other players' strategies marginalized out is related to the true underlying bimatrix game between them as follows:
+
+$$
+\begin{array}{l} H _ {k l} ^ {k} [ a _ {k}, a _ {l} ] = \mathbb {E} _ {x _ {- k l}} [ u _ {k} (a _ {k}, a _ {l}, x _ {- k l}) ] \forall a _ {k}, a _ {l} (127a) \\ = a _ {k} ^ {\top} \left(P _ {k l} ^ {k} a _ {l} + \sum_ {j \notin \{k, l \}} P _ {k j} ^ {k} x _ {j}\right) \forall a _ {k}, a _ {l} (127b) \\ = a _ {k} ^ {\top} P _ {k l} ^ {k} a _ {l} + \underbrace {a _ {k} ^ {\top} \left(\sum_ {j \notin \{k , l \}} P _ {k j} ^ {k} x _ {j}\right)} _ {p _ {k}} \forall a _ {k}, a _ {l} (127c) \\ \end{array}
+$$
+
+where  $P_{kl}^{k}$  is player  $k$ 's payoff matrix for the bimatrix game between players  $k$  and  $l$  in a polymatrix game and  $p_k$  does not depend on player  $l$ 's strategy.
+
+This implies that  $H_{kl}^{k}$  is equal to  $P_{kl}^{k}$  up to a constant offset of the rows, i.e.,
+
+$$
+H _ {k l} ^ {k} = P _ {k l} ^ {k} + C _ {k} \tag {128}
+$$
+
+where  $C_k$  is a matrix with constant rows.
+
+Consider the matrix  $M(x)$  which contains  $H_{kl}^{k}$  blocks. Recall that the bottom rows of  $M(x)$  contain rows of 1's matching each column of  $H_{kl}^{k}$  blocks. Consider multiplying the  $l$ th row of 1's (which contains 0's on all columns not in the  $l$ th block) by  $\sqrt{\eta_k} [I - \frac{1}{m_k}\mathbf{1}_k\mathbf{1}_k^\top ]C_k$  and subtracting it from the block containing  $\sqrt{\eta_k} [I - \frac{1}{m_k}\mathbf{1}_k\mathbf{1}_k^\top ]H_{kl}^k$ ,
+
+$$
+\sqrt {\eta_ {k}} \left[ I - \frac {1}{m _ {k}} \mathbf {1} _ {k} \mathbf {1} _ {k} ^ {\top} \right] \left[ H _ {k l} ^ {k} - C _ {k} \right] = \sqrt {\eta_ {k}} \left[ I - \frac {1}{m _ {k}} \mathbf {1} _ {k} \mathbf {1} _ {k} ^ {\top} \right] P _ {k l} ^ {k}. \tag {129}
+$$
+
+Note that  $\sqrt{\eta_k} [I - \frac{1}{m_k}\mathbf{1}_k\mathbf{1}_k^\top ]C_k$  still remains a matrix with constant rows (the preconditioner effectively subtracts a constant matrix from  $C_k$ ). This multiplying and subtracting a row from another is an elementary operation on the matrix, meaning it does not change its row rank. Therefore, for a polymatrix game, we can reason about the positive definiteness of the Hessian at equilibria by examining the matrix  $M(x)$  with all  $H_{kl}^{k}$ 's swapped for  $P_{kl}^{k}$ 's.
+
+Interestingly, at zero temperature (where  $\mathrm{QRE} = \mathrm{Nash}$ ),  $M$  is constant for a polymatrix game, so the rank of this matrix can be computed just once to extract information about all possible interior equilibria in the game. Furthermore, the Hessian is positive semi-definite over the entire joint strategy space, implying the loss function is convex (see Figure 5 (left) for empirical support). This indicates, by convex optimization theory, 1) all mixed Nash equilibria in polymatrix games form a convex set (i.e., they are connected) and 2) assuming mixed equilibria exist, they can be computed simply by stochastic gradient descent on  $\mathcal{L}$ . If  $M$  is rank- $n\bar{m}$ , then this interior equilibrium is unique.
+
+**Complexity** Approximation of Nash equilibria in polymatrix games is known to be PPAD-hard (Deligkas et al., 2022). In contrast, if we restrict our class of polymatrix games to those with at least one interior Nash equilibrium, our analysis proves we can find an approximate Nash equilibrium in deterministic, polynomial time (Corollary 7). This follows directly from the fact that  $\mathcal{L}$  is convex, our decision set  $\mathcal{X} = \prod_{k} \mathcal{X}_{k}$  is convex, and convex optimization theory admits polynomial time approximation algorithms (e.g., gradient descent). We consider the assumption of the existence of an interior Nash equilibrium to be relatively mild $^3$ , so this positive complexity result is surprising.
+
+Also, note that the Hessian of the loss at Nash equilibria is encoded entirely by the polymatrix approximation at the equilibrium. Therefore, approximating the Hessian of  $\mathcal{L}$  about the equilibrium (which amounts to observing near-equilibrium behavior (Ling et al., 2018)) allows one to recover this polymatrix approximation (up to constant offsets of the columns which equilibria are invariant to (Marris et al., 2023)).
+
+Corollary 7 (Approximating Nash Equilibria of Polymatrix Games with Interior Equilibria). Consider the class of polymatrix games with interior Nash equilibria. This class of games admits a fully polynomial time deterministic approximation scheme (FPTAS).
+
+Proof. Lemma 3 relates the approximation of Nash equilibria to the minimization of the loss function  $\mathcal{L}(\pmb{x})$ . By Lemma 1, this loss function attains its minimum value of zero if and only if  $\pmb{x}$  is a Nash equilibrium. For polymatrix games, the Hessian of this loss function is everywhere finite and positive definite (Lemma 16), therefore, this loss function is convex. The decision set for this minimization problem is the product space of simplices, therefore it is also convex. Given that we only consider polymatrix games with interior Nash equilibria, we know that our loss function attains a global minimum within this set. By convex optimization theory, this function can be approximately minimized in a polynomial number of steps by, for example, (projected) gradient descent (Boyd and Vandenberghe, 2004). Gradient descent requires computing the gradient of the loss function at each step. From Lemma 14, we see that computing the gradient (at zero temperature) simply requires reading the polymatrix description of the game (i.e., each bi-matrix game  $H_{kl}^{k}$  between players), which is clearly polynomial in the size of the input (the polymatrix description). The remaining computational steps of gradient descent (e.g., projection onto simplices) are polynomial as well. In conclusion, gradient descent approximates a Nash equilibrium in polynomial number of steps (logarithmic if strongly-convex (Mairal, 2013)), each of which costs polynomial time, therefore the entire scheme is polynomial.
+
+# C EXPERIMENTAL SETUP AND DETAILS
+
+Here we provide further details on the experiments.
+
+# C.1 GAMBIT
+
+The seven methods from the gambit (McKelvey et al., 2016) library that we tested on the 3-player and 4-player Blotto games are listed below (with runtimes). Only gambit-enumpoly and gambit-enumpure are able to return any NE for 3-player Blotto within a 1 hour time limit (and only pure equilibria). And only gambit-enumpure returns any NE for the 4-player game.
+
+- gambit-enumpoly [73 sec 3-player, timeout 4-player]  
+- gambit-enumpure [72 sec 3-player, 45 sec 4-player]  
+- gambit-gnm  
+- gambit-ipa  
+- gambit-liap  
+- gambit-logit  
+- gambit-simpdiv
+
+# C.2 LOSS VISUALIZATION AND RANK TEST
+
+Figure 5 and claims made in Section 5 analyze several classical matrix games. We report the payoff matrices in standard row-player / column-player payoff form below. All games are then shifted and scaled so payoffs lie in  $[0,1]$  (i.e., first by subtracting the minimum and then scaling by the max).
+
+RPS:
+
+$$
+\left[ \begin{array}{c c c} 0 / 0 & - 1 / 1 & 1 / - 1 \\ 1 / - 1 & 0 / 0 & - 1 / 1 \\ - 1 / 1 & 1 / - 1 & 0 / 0 \end{array} \right]. \tag {130}
+$$
+
+Chicken:
+
+$$
+\left[ \begin{array}{c c} 0 / 0 & - 1 / 1 \\ 1 / - 1 & - 3 / - 3 \end{array} \right]. \tag {131}
+$$
+
+Matching Pennies:
+
+$$
+\left[ \begin{array}{l l} 1 / - 1 & - 1 / 1 \\ - 1 / 1 & 1 / - 1 \end{array} \right]. \tag {132}
+$$
+
+Modified-Shapleys:
+
+$$
+\left[ \begin{array}{c c c} 1 / - 0. 5 & 0 / 1 & 0. 5 / 0 \\ 0. 5 / 0 & 1 / - 0. 5 & 0 / 1 \\ 0 / 1 & 0. 5 / 0 & 1 / - 0. 5 \end{array} \right]. \tag {133}
+$$
+
+Prisoner's Dilemma:
+
+$$
+\left[ \begin{array}{c c} - 1 / - 1 & - 3 / 0 \\ 0 / - 3 & - 2 / - 2 \end{array} \right]. \tag {134}
+$$
+
+# C.2.1 LOSS ON FAMILIAR GAMES
+
+We visualize our proposed loss  $\mathcal{L}^{\tau}$  on two classic 2-player, general-sum games: Chicken, payoff matrix (131), and the Prisoner's Dilemma, payoff matrix (134). Each plot in Figure 5 visualizes the loss at various strategy profiles in probability-space; each strategy profile is represented by each player's probability of playing action 1 of 2 (top row / first column of payoff matrix). Temperature  $\tau$  is varied across the plots. Figure 5 repeats this same visualization but in logit-space to better show the equilibria closest to the boundaries.
+
+![](images/a911cefc70a0e4d48aa4672ba390fc60a5a5bea70b6d1b2e2c37e3e15dc48ace.jpg)  
+Figure 5: Upper Bound  $(\epsilon \leq f_{\tau}(\mathcal{L}^{\tau}))$  Heatmap Visualization. The first row examines the loss landscape for the classic anti-coordination game of Chicken (Nash equilibria:  $(0,1)$ ,  $(1,0)$ ,  $(2/3,1/3))$  while the second row examines the Prisoner's dilemma (Unique Nash equilibrium:  $(0,0))$ . For improved visibility, we subtract the offset  $\tau \log(m^{2})$  from  $f_{\tau}(\mathcal{L}^{\tau})$  per Lemma 13, which relates the exploitability at positive temperature to that at zero temperature. Temperature increases for each plot moving to the right. For high temperatures, interior (fully-mixed) strategies are incentivized while for lower temperatures, nearly pure strategies can achieve minimum exploitability. For zero temperature, pure strategy equilibria (e.g., defect-defect) are not captured by the loss as illustrated by the bottom-left Prisoner's Dilemma plot with a constant loss surface.
+
+![](images/7cf17c1aa14508b04dd3b35c8a089634339250d3478997e64d73489dfd45a4d9.jpg)  
+Figure 6: [Figure 5 Repeated in Logit-Space  $(\ln \left(\frac{p}{1 - p}\right))$  Rather than Probability-Space  $(p)]$  Upper Bound  $(\epsilon \leq f_{\tau}(\mathcal{L}^{\tau}))$  Heatmap Visualization. The first row examines the loss landscape for the classic anti-coordination game of Chicken (Nash equilibria:  $(0,1)$ ,  $(1,0)$ ,  $(2/3,1/3))$  while the second row examines the Prisoner's dilemma (Unique Nash equilibrium:  $(0,0)$ ). Temperature increases for each plot moving to the right. For improved visibility, we subtract the offset  $\tau \log(m^{2})$  from  $f_{\tau}(\mathcal{L}^{\tau})$  per Lemma 13, which relates the exploitability at positive temperature to that at zero temperature. For high temperatures, interior (fully-mixed) strategies are incentivized while for lower temperatures, nearly pure strategies can achieve minimum exploitability. For zero temperature, pure strategy equilibria (e.g., defect-defect) are not captured by the loss as illustrated by the bottom-left Prisoner's Dilemma plot with a constant loss surface.
+
+# C.3 SADDLE POINT ANALYSIS
+
+To generate Figure 2, we follow a procedure similar to the study of MNIST in (Dauphin et al., 2014) (Section 3 of Supp.). Their recommended procedure searches for critical points in two ways. The first repeats a randomized, iterative optimization process 20 times. They then sample one these 20 trials at random, select a random point along the descent trajectory, and search for a critical point (using Newton's method) nearby. They repeat this sampling process 100 times. The second approach randomly selects a feasible point in the decision set and searches for a critical point nearby (again using Newton's method). They also perform this 100 times.
+
+Our protocol differs from theirs slightly in a few respects. One, we use SGD, rather than the saddle-free Newton algorithm to trace out an initial descent trajectory. Two, we do not add noise to strategies along the descent trajectory prior to looking for critical points. Thirdly, we minimize gradient norm rather than use Newton's method to look for critical points. Lastly, we use different experimental hyperparameters. We run SGD for 1000 iterations rather than 20 epochs and rerun SGD 100 times rather than 20. We sample 1000 points for each of the two approaches for finding critical points.
+
+# C.4 SGD ON CLASSICAL GAMES
+
+The games examined in Figure 3 were all taken from (Gemp et al., 2022). Each is available via open source implementations in OpenSpiel (Lanctot et al., 2019) or GAMUT (Nudelman et al., 2004).
+
+We compare against several other baselines, replicating the experiments in (Gemp et al., 2022). RM indicates regret-matching and FTRL indicates follow-the-regularized-leader. These are, arguably, the two most popular scalable stochastic algorithms for approximating Nash equilibria.  ${}^{y}{\mathrm{{QRE}}}^{\text{auto }}$  is a stochastic algorithm developed in (Gemp et al., 2022).
+
+For each of the experiments, we sweep over learning rates in log-space from  $10^{-3}$  to  $10^{2}$  in increments of 1. We also consider whether to run SGD with the projected-gradient and whether to constrain iterates to the simplex via Euclidean projection or entropic mirror descent (Beck and Teboulle, 2003). We then presented the results of the best performing hyperparameters. This was the same approach taken in (Gemp et al., 2022).
+
+Saddle Points in Blotto To confirm the existence of saddle points, we computed the Hessian of  $\mathcal{L}(\pmb{x}_{10k})$  for SGD  $(s = \infty)$ , deflated the matrix by removing from its eigenvectors all directions orthogonal to the simplex, and then computed its top-  $(n\bar{m} - n)$  eigenvalues. We do this because there always exists a  $n$ -dimensional nullspace of the Hessian at zero temperature that lies outside the tangent space of the simplex, and we only care about curvature within the tangent space. Specifically, at an equilibrium  $\pmb{x}$ , if we compute  $z^{\top}Hess(\mathcal{L})z$  where  $z$  is formed as a linear combination of the vectors  $\{[x_1, 0, \ldots, 0]^{\top}, \ldots, [0, \ldots, x_n]^{\top}\}$ , then each block  $\tilde{B}_{kl}$  is identically zero at an equilibrium:  $\tilde{B}_{kl}x_l = \sqrt{\eta_k[I - \frac{1}{m_k}\mathbf{11}^{\top}]}H_{kl}^k x_l = \sqrt{\eta_k\Pi_{T\Delta}(\nabla_{x_k}^k)} = 0$ . By Lemma 16, this implies there is zero curvature of the loss in the direction  $z$ :  $z^{\top}Hess(\mathcal{L})z = 0$ .
+
+# C.5 BLIN ON ARTIFICIAL GAME
+
+To construct the 7-player, 2-action, symmetric, artificial game in Figure 4, we used the following coefficients (discovered by trial-and-error):
+
+$$
+\left[ \begin{array}{c c c c c c c} 0. 0 9 9 0 6 8 7 3 & 0 & 0. 2 3 1 1 6 0 3 7 & 0 & 0. 6 2 7 4 3 5 2 8 & 0 & 0. 1 9 8 1 3 7 4 6 \\ 0 & 0. 3 3 0 2 2 9 0 9 & 0 & 0. 0 3 3 0 2 2 9 1 & 0 & 0. 6 2 7 4 3 5 2 8 & 0 \end{array} \right]. \tag {135}
+$$
+
+The first row indicates the payoffs received when player  $i$  plays action 0 and the background population plays any of the possible joint actions (number of combinations with replacement). For example, the first column indicates the payoff when all background players play action 0. The second column indicates all background players play action 0 except for one which plays action 1, and so on. The last column indicates all background players play action 1. These  $2n$  scalars uniquely define the payoffs of a symmetric game.
+
+Given that this game only has two actions, we represent a mixed strategy by a single scalar  $p \in [0,1]$ , i.e., the probability of the first action. Furthermore, this game is symmetric and we seek a symmetric equilibrium, so we can represent a full Nash equilibrium by this single scalar  $p$ . This reduces our search space from  $7 \times 2 = 14$  variables to 1 variable (and obviates any need for a map  $s$  from the unit hypercube to the simplex—see Lemma 22).
\ No newline at end of file
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+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
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+                                    "content": "We propose the first loss function for approximate Nash equilibria of normal-form games that is amenable to unbiased Monte Carlo estimation. This construction allows us to deploy standard non-convex stochastic optimization techniques for approximating Nash equilibria, resulting in novel algorithms with provable guarantees. We complement our theoretical analysis with experiments demonstrating that stochastic gradient descent can outperform previous state-of-the-art approaches."
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+                                    "type": "text",
+                                    "content": "1 INTRODUCTION"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
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+                                    "bbox": [
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+                                        434
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+                                    "type": "text",
+                                    "content": "Nash equilibrium (NE) famously encodes stable behavioral outcomes in multi-agent systems and is arguably the most influential solution concept in game theory. Formally speaking, if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        346,
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " players independently choose "
+                                },
+                                {
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+                                        434
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n"
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ", possibly mixed, strategies "
+                                },
+                                {
+                                    "bbox": [
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+                                        346,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(x_{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        346,
+                                        506,
+                                        434
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k \\in [n]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        346,
+                                        506,
+                                        434
+                                    ],
+                                    "type": "text",
+                                    "content": ") and their joint strategy "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        346,
+                                        506,
+                                        434
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(x = \\prod_{k} x_{k})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        346,
+                                        506,
+                                        434
+                                    ],
+                                    "type": "text",
+                                    "content": " constitutes a Nash equilibrium, then no player has any incentive to unilaterally deviate from their strategy. This concept has sparked extensive research in various fields, ranging from economics (Milgrom and Weber, 1982) to machine learning (Goodfellow et al., 2014), and has even inspired behavioral theory generalizations such as quantal response equilibria (QREs) which allow for more realistic models of boundedly rational agents (McKelvey and Palfrey, 1995)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                        518
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+                                    "type": "text",
+                                    "content": "Unfortunately, when considering Nash equilibria beyond the 2-player, zero-sum scenario, two significant challenges arise. First, it becomes unclear how "
+                                },
+                                {
+                                    "bbox": [
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+                                        439,
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+                                    "type": "inline_equation",
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " independent players would collectively identify a Nash equilibrium when multiple equilibria are possible, giving rise to the equilibrium selection problem (Harsanyi et al., 1988). Secondly, even approximating a single Nash equilibrium is known to be computationally intractable and specifically PPAD-complete (Daskalakis et al., 2009). Combining both problems together, e.g., testing for the existence of equilibria with welfare greater than some fixed threshold is NP-hard, and it is in fact even hard to approximate (Austrin et al., 2011)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
+                        104,
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+                        506,
+                        578
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "From a machine learning practitioner's perspective, such computational complexity results hardly give pause for thought as collectively we have become all too familiar with the unreasonable effectiveness of heuristics in circumventing such obstacles. Famously, non-convex optimization is NP-hard, even if the goal is to compute a local minimizer (Murty and Kabadi, 1985), however, stochastic gradient descent (SGD) and variants succeed in training billion parameter models (Brown et al., 2020)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                },
+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
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+                                    "type": "text",
+                                    "content": "Unfortunately, computational techniques for Nash equilibrium have so far not achieved anywhere near the same level of success. In contrast, most modern NE solvers for "
+                                },
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+                                    "bbox": [
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+                                    "content": "-player, "
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+                                    "type": "text",
+                                    "content": "-action, general-sum, normal-form games (NFGs) are practically restricted to a handful of players and/or actions per player except in special cases, e.g., symmetric (Wiedenbeck and Brinkman, 2023) or mean-field games (Pérolat et al., 2022). For example, when running the suite of all 7 applicable methods from the hallmark gambit library (McKelvey et al., 2016) on a 4-player Blotto game, we find only brute-force pure-NE enumeration is able to return any NE within a 1 hour time limit. Scaling solvers to large games is difficult partially due to the fact that an NFG is represented by a tensor with an exponential "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "nm^n"
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+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " entries; even reading this description into memory can be computationally prohibitive. More to the point, any computational technique that presumes exact computation of the expectation of this tensor sampled according to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
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+                                        704
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
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+                                    ],
+                                    "type": "text",
+                                    "content": " similarly does not have any hope of scaling beyond small instances."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                },
+                {
+                    "bbox": [
+                        104,
+                        709,
+                        506,
+                        733
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "This inefficiency arguably lies at the core of the differential success between machine learning (ML) optimization and equilibrium computation. For example, numerous techniques exist that reduce the"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 21
+                }
+            ],
+            "discarded_blocks": [
+                {
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+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "1"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 22
+                }
+            ],
+            "page_size": [
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+            ],
+            "page_idx": 0
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
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+                        82,
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+                        128
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        128
+                                    ],
+                                    "type": "text",
+                                    "content": "problem of Nash computation to the minimization of the expectation of a random variable (Section 3). Unfortunately, unlike the source of randomness in ML applications where batch learning suffices to produce unbiased estimators, these techniques do not extend easily to game theory which incorporates non-linear functions such as maximum and best-response. This raises our motivating goal:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        153,
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+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                133,
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+                            ],
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+                                {
+                                    "bbox": [
+                                        153,
+                                        133,
+                                        457,
+                                        146
+                                    ],
+                                    "type": "text",
+                                    "content": "Can we solve for Nash equilibria via unbiased stochastic optimization?"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        156,
+                        506,
+                        257
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        156,
+                                        506,
+                                        257
+                                    ],
+                                    "type": "text",
+                                    "content": "Our results. Following in the successful steps of the interplay between ML and stochastic optimization, we reformulate the approximation of Nash equilibria in a normal-form game as a stochastic non-convex optimization problem admitting unbiased Monte-Carlo estimation. This enables the use of powerful solvers and advances in parallel computing to efficiently enumerate Nash equilibria for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        156,
+                                        506,
+                                        257
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        156,
+                                        506,
+                                        257
+                                    ],
+                                    "type": "text",
+                                    "content": "-player, general-sum games. Furthermore, this re-casting allows practitioners to incorporate other desirable objectives into the problem such as \"find an approximate Nash equilibrium with welfare above "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        156,
+                                        506,
+                                        257
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\omega"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        156,
+                                        506,
+                                        257
+                                    ],
+                                    "type": "text",
+                                    "content": "\" or \"find an approximate Nash equilibrium nearest the current observed joint strategy\" resolving the equilibrium selection problem in an effectively ad-hoc and application tailored manner. Concretely, we make the following contributions by producing:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        105,
+                        265,
+                        504,
+                        361
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 7,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                105,
+                                265,
+                                504,
+                                287
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                        504,
+                                        287
+                                    ],
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+                                        {
+                                            "bbox": [
+                                                105,
+                                                265,
+                                                504,
+                                                287
+                                            ],
+                                            "type": "text",
+                                            "content": "- A loss "
+                                        },
+                                        {
+                                            "bbox": [
+                                                105,
+                                                265,
+                                                504,
+                                                287
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{L}^{\\tau}(\\pmb{x})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                105,
+                                                265,
+                                                504,
+                                                287
+                                            ],
+                                            "type": "text",
+                                            "content": " 1) whose global minima well approximate Nash equilibria in normal form games, 2) admits unbiased Monte-Carlo estimation, and 3) is Lipschitz and bounded."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 4
+                        },
+                        {
+                            "bbox": [
+                                105,
+                                291,
+                                504,
+                                324
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                105,
+                                                291,
+                                                504,
+                                                324
+                                            ],
+                                            "type": "text",
+                                            "content": "- Efficient randomized algorithms for approximating Nash equilibria in a novel class of games. The algorithms emerge by employing the family of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                105,
+                                                291,
+                                                504,
+                                                324
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{X}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                105,
+                                                291,
+                                                504,
+                                                324
+                                            ],
+                                            "type": "text",
+                                            "content": "-armed bandit approaches to "
+                                        },
+                                        {
+                                            "bbox": [
+                                                105,
+                                                291,
+                                                504,
+                                                324
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{L}^{\\tau}(\\pmb{x})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                105,
+                                                291,
+                                                504,
+                                                324
+                                            ],
+                                            "type": "text",
+                                            "content": " and connecting their global stochastic optimization guarantees to global approximate Nash guarantees."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 5
+                        },
+                        {
+                            "bbox": [
+                                105,
+                                328,
+                                504,
+                                361
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                        328,
+                                        504,
+                                        361
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                105,
+                                                328,
+                                                504,
+                                                361
+                                            ],
+                                            "type": "text",
+                                            "content": "- An empirical comparison of SGD against state-of-the-art baselines for approximating NEs in large games. In some games, vanilla SGD actually improves upon previous state-of-the-art; in others, SGD is slowed by saddle points, a familiar challenge in deep learning (Dauphin et al., 2014)."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 6
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
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+                        504,
+                        395
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Overall, this perspective showcases a promising new route to approximating equilibria at scale in practice. We conclude the paper with discussion for future work."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
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+                        422
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
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+                            ],
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "2 PRELIMINARIES"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
+                        434,
+                        506,
+                        559
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                434,
+                                506,
+                                559
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": "In an "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": "-player, normal-form game, each player "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k \\in \\{1, \\dots, n\\} = [n]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": " has a strategy set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{A}_k = \\{a_{k1}, \\dots, a_{km_k}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": " consisting of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m_k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": " pure strategies. These strategies can be naturally indexed, so we redefine "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{A}_k = [m_k]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": " as an abuse of notation. Each player "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": " also has a utility function, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u_k: \\mathcal{A} = \\prod_k \\mathcal{A}_k \\to [0,1]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": ", (equiv. \"payoff tensor\") that maps joint actions to payoffs in the unit-interval. We denote the average cardinality of the players' action sets by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\bar{m} = \\frac{1}{n} \\sum_k m_k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": " and maximum by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m^* = \\max_k m_k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": ". Player "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": " may play a mixed strategy by sampling from a distribution over their pure strategies. Let player "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": "'s mixed strategy be represented by a vector "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_k \\in \\Delta^{m_k-1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta^{m_k-1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": " is the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(m_k-1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": "-dimensional probability simplex embedded in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{R}^{m_k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": ". Each function "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u_k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": " is then extended to this domain so that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u_k(\\boldsymbol{x}) = \\sum_{\\boldsymbol{a} \\in \\mathcal{A}} u_k(\\boldsymbol{a}) \\prod_j x_{ja_j}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\boldsymbol{x} = (x_1, \\dots, x_n)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "a_j \\in \\mathcal{A}_j"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": " denotes player "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "j"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": "'s component of the joint action "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\boldsymbol{a} \\in \\mathcal{A}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": ". For convenience, let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_{-k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": " denote all components of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\boldsymbol{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": " belonging to players other than player "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        434,
+                                        506,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        563,
+                        504,
+                        599
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                563,
+                                504,
+                                599
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": "The joint strategy "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x} \\in \\prod_{k} \\Delta^{m_k - 1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": " is a Nash equilibrium if and only if, for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k \\in [n]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u_k(z_k, x_{-k}) \\leq u_k(\\pmb{x})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z_k \\in \\Delta^{m_k - 1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": ", i.e., no player has any incentive to unilaterally deviate from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": ". Nash is typically relaxed with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\epsilon"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": "-Nash, our focus: "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u_k(z_k, x_{-k}) \\leq u_k(\\pmb{x}) + \\epsilon"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z_k \\in \\Delta^{m_k - 1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        104,
+                        603,
+                        507,
+                        733
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                603,
+                                507,
+                                733
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "As an abuse of notation, let the atomic action "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "a_{k} = e_{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " also denote the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m_{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "-dimensional \"one-hot\" vector with all zeros aside from a 1 at index "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "a_{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "; its use should be clear from the context. We also introduce "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\nabla_{x_k}^k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " as player "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "'s utility gradient. And for convenience, denote by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H_{kl}^{k} = \\mathbb{E}_{x_{-kl}}[u_{k}(a_{k},a_{l},x_{-kl})]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " the bimatrix game approximation (Janovskaja, 1968) between players "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "l"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " with all other players marginalized out; "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_{-kl}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " denotes all strategies belonging to players other than "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "l"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u_{k}(a_{k},a_{l},x_{-kl})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " separates out "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "l"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "'s strategy "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_{l}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " from the rest of the players "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_{-k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ". Similarly, denote by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T_{klq}^{k} = \\mathbb{E}_{x_{-klq}}[u_{k}(a_{k},a_{l},a_{q},x_{-klq})]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " the 3-player tensor approximation to the game. Note player "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "'s utility can now be written succinctly as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u_{k}(x_{k},x_{-k}) = x_{k}^{\\top}\\nabla_{x_{k}}^{k} = x_{k}^{\\top}H_{kl}^{k}x_{l} = T_{klq}^{k}x_{k}x_{l}x_{q}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "l,q"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " where we use Einstein notation for tensor arithmetic. For convenience, define "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{diag}(z)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " as the function that places a vector "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " on the diagonal of a square matrix, and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{diag}3: z \\in \\mathbb{R}^{d} \\to \\mathbb{R}^{d \\times d \\times d}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " as a 3-tensor of shape "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(d,d,d)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
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+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{diag}3(z)_{lll} = z_l"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        603,
+                                        507,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
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+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
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+                                    "type": "text",
+                                    "content": "2"
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+                }
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+                                            "type": "table",
+                                            "html": "<table><tr><td>Loss</td><td>Function</td><td>Obstacle</td></tr><tr><td>Exploitability (ε)</td><td>maxkεk(x)</td><td>max of r.v.</td></tr><tr><td>Nikaido-Isoda (NI)</td><td>∑kεk(x)</td><td>max of r.v.</td></tr><tr><td>Fully-Diff. Exp</td><td>∑k∑ak∈A[k][max(0,uk(ak,x-k)-uk(x))]2</td><td>max of r.v.</td></tr><tr><td>Gradient-based NI</td><td>NI w/ BRk← aBRk = ΠΔ(xk+η∇xuk(x))</td><td>ΠΔ of r.v.</td></tr><tr><td>Unconstrained</td><td>Loss + Simplex Deviation Penalty</td><td>sampling from xk∈Rmk</td></tr></table>",
+                                            "image_path": "399ce5aa934ea3dd6ccee6516dd105581301c8e63ec58799b4562f9bcc185322.jpg"
+                                        }
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+                        189
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+                            "bbox": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Table 1: Previous loss functions for NFGs and their obstacles to unbiased estimation. Note that "
+                                },
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\epsilon_{k}(\\pmb{x}) = \\max_{z}u_{k}(z,x_{-k}) - u_{k}(\\pmb{x})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        164,
+                                        506,
+                                        189
+                                    ],
+                                    "type": "text",
+                                    "content": " contains a max operator (see equivalent definition in equation (1))."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2,
+                    "angle": 0,
+                    "type": "text"
+                },
+                {
+                    "bbox": [
+                        104,
+                        206,
+                        506,
+                        275
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                206,
+                                506,
+                                275
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        506,
+                                        275
+                                    ],
+                                    "type": "text",
+                                    "content": "Following convention from differential geometry, let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        506,
+                                        275
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T_v\\mathcal{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        506,
+                                        275
+                                    ],
+                                    "type": "text",
+                                    "content": " be the tangent space of a manifold "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        506,
+                                        275
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        506,
+                                        275
+                                    ],
+                                    "type": "text",
+                                    "content": " at "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        506,
+                                        275
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        506,
+                                        275
+                                    ],
+                                    "type": "text",
+                                    "content": ". For the interior of the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        506,
+                                        275
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        506,
+                                        275
+                                    ],
+                                    "type": "text",
+                                    "content": "-action simplex "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        506,
+                                        275
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta^{d-1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        506,
+                                        275
+                                    ],
+                                    "type": "text",
+                                    "content": ", the tangent space is the same at every point, so we drop the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        506,
+                                        275
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        506,
+                                        275
+                                    ],
+                                    "type": "text",
+                                    "content": " subscript, i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        506,
+                                        275
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T\\Delta^{d-1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        506,
+                                        275
+                                    ],
+                                    "type": "text",
+                                    "content": ". We denote the projection of a vector "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        506,
+                                        275
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z \\in \\mathbb{R}^d"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        506,
+                                        275
+                                    ],
+                                    "type": "text",
+                                    "content": " onto this tangent space as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        506,
+                                        275
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Pi_{T\\Delta^{d-1}}(z) = [I - \\frac{1}{d}\\mathbf{11}^\\top]z"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        506,
+                                        275
+                                    ],
+                                    "type": "text",
+                                    "content": " and call "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        506,
+                                        275
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Pi_{T\\Delta^{d-1}}(\\nabla_{x_k}^k)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        506,
+                                        275
+                                    ],
+                                    "type": "text",
+                                    "content": " a projected-gradient. We drop "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        506,
+                                        275
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d-1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        506,
+                                        275
+                                    ],
+                                    "type": "text",
+                                    "content": " when the dimensionality is clear from the context. Finally, let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        506,
+                                        275
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{U}(S)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        506,
+                                        275
+                                    ],
+                                    "type": "text",
+                                    "content": " denote a discrete uniform distribution over elements from set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        506,
+                                        275
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        206,
+                                        506,
+                                        275
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        105,
+                        290,
+                        212,
+                        303
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                290,
+                                212,
+                                303
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                        303
+                                    ],
+                                    "type": "text",
+                                    "content": "3 RELATED WORK"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        315,
+                        504,
+                        350
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                {
+                                    "bbox": [
+                                        104,
+                                        315,
+                                        504,
+                                        350
+                                    ],
+                                    "type": "text",
+                                    "content": "Representing the problem of computing a Nash equilibrium as an optimization problem is not new. A variety of loss functions and pseudo-distance functions have been proposed. Most of them measure some function of how much each player can exploit the joint strategy by unilaterally deviating:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        167,
+                        354,
+                        504,
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                {
+                                    "bbox": [
+                                        167,
+                                        354,
+                                        504,
+                                        376
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\epsilon_ {k} (\\boldsymbol {x}) \\stackrel {\\text {d e f}} {=} u _ {k} \\left(\\operatorname {B R} _ {k}, x _ {- k}\\right) - u _ {k} (\\boldsymbol {x}) \\text {w h e r e} \\operatorname {B R} _ {k} \\in \\underset {z} {\\arg \\max } u _ {k} (z, x _ {- k}). \\tag {1}",
+                                    "image_path": "436ab24d34406924b3878b73e638fbc60d180cd9c3a5f0bfd72ae4a00c69cb74.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        385,
+                        506,
+                        552
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                385,
+                                506,
+                                552
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        385,
+                                        506,
+                                        552
+                                    ],
+                                    "type": "text",
+                                    "content": "As argued in the introduction, we believe it is important to be able to subsample payoff tensors of normal-form games in order to scale to large instances. As Nash equilibria can consist of mixed strategies, it is advantageous to be able to sample from an equilibrium to estimate its exploitability "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        385,
+                                        506,
+                                        552
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\epsilon"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        385,
+                                        506,
+                                        552
+                                    ],
+                                    "type": "text",
+                                    "content": ". However none of these losses is amenable to unbiased estimation under sampled play. Each of the functions currently explored in the literature is biased under sampled play either because 1) a random variable appears as the argument of a complex, nonlinear (non-polynomial) function or because 2) how to sample play is unclear. Exploitability, Nikaido-Isoda (NI) (Nikaidô and Isoda, 1955) (also known by NashConv (Lanctot et al., 2017) and ADI (Gemp et al., 2022)), as well as fully-differentiable options (Shoham and Leyton-Brown, 2008, p. 106, Eqn 4.31) introduce bias when a max over payoffs is estimated using samples from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        385,
+                                        506,
+                                        552
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        385,
+                                        506,
+                                        552
+                                    ],
+                                    "type": "text",
+                                    "content": ". Gradient-based NI (Raghunathan et al., 2019) requires projecting the result of a gradient-ascent step onto the simplex; for the same reason as the max, this projection is prohibitive because it is a nonlinear operation which introduces bias. Lastly, unconstrained optimization approaches (Shoham and Leyton-Brown, 2008, p. 106) that instead penalize deviation from the simplex lose the ability to sample from strategies when each iterate "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        385,
+                                        506,
+                                        552
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        385,
+                                        506,
+                                        552
+                                    ],
+                                    "type": "text",
+                                    "content": " is no longer a distribution (i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        385,
+                                        506,
+                                        552
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_{k} \\notin \\Delta^{m_{k}-1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        385,
+                                        506,
+                                        552
+                                    ],
+                                    "type": "text",
+                                    "content": "). Table 1 summarizes these complications."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        105,
+                        567,
+                        395,
+                        578
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                567,
+                                395,
+                                578
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        567,
+                                        395,
+                                        578
+                                    ],
+                                    "type": "text",
+                                    "content": "4 NASH EQUILIBRIUM AS STOCHASTIC OPTIMIZATION"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        590,
+                        506,
+                        657
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        590,
+                                        506,
+                                        657
+                                    ],
+                                    "type": "text",
+                                    "content": "We will now develop our proposed loss function which is amenable to unbiased estimation. Sub-sections 4.1-4.4 provide a warm-up in which we assume an interior (fully-mixed) Nash equilibrium exists. Subsection 4.5 then shows how to relax that assumption allowing us to approximate partially mixed equilibria as well (including pure equilibria). Our key technical insight is to pay special attention to the geometry of the simplex. To our knowledge, prior works have failed to recognize the role of the tangent space "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        590,
+                                        506,
+                                        657
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T\\Delta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        590,
+                                        506,
+                                        657
+                                    ],
+                                    "type": "text",
+                                    "content": ". Proofs are in the appendix."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        105,
+                        670,
+                        316,
+                        681
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                670,
+                                316,
+                                681
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "4.1 STATIONARITY ON THE SIMPLEX INTERIOR"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
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+                        506,
+                        715
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                {
+                                    "bbox": [
+                                        104,
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+                                        715
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 1. Assuming player "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        691,
+                                        506,
+                                        715
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        691,
+                                        506,
+                                        715
+                                    ],
+                                    "type": "text",
+                                    "content": "'s utility, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        691,
+                                        506,
+                                        715
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u_{k}(x_{k}, x_{-k})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        691,
+                                        506,
+                                        715
+                                    ],
+                                    "type": "text",
+                                    "content": ", is concave in its own strategy "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        691,
+                                        506,
+                                        715
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        691,
+                                        506,
+                                        715
+                                    ],
+                                    "type": "text",
+                                    "content": ", a strategy in the interior of the simplex is a best response "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        691,
+                                        506,
+                                        715
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "BR_{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        691,
+                                        506,
+                                        715
+                                    ],
+                                    "type": "text",
+                                    "content": " if and only if it has zero projected-gradient"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        691,
+                                        506,
+                                        715
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "^{1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        691,
+                                        506,
+                                        715
+                                    ],
+                                    "type": "text",
+                                    "content": " norm."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
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+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
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+                    "type": "page_footnote",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "<sup>1</sup>Not to be confused with the nonlinear (biased) projected gradient operator in (Hazan et al., 2017)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "3"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 2
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "In NFGs, each player's utility is linear in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
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+                                        95
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
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+                                        95
+                                    ],
+                                    "type": "text",
+                                    "content": ", thereby satisfying the concavity condition of Lemma 1."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
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+                    ],
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+                                        309,
+                                        118
+                                    ],
+                                    "type": "text",
+                                    "content": "4.2 PROJECTED-GRADIENT NORM AS A LOSS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        127,
+                        504,
+                        161
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                127,
+                                504,
+                                161
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        127,
+                                        504,
+                                        161
+                                    ],
+                                    "type": "text",
+                                    "content": "An equivalent description of a Nash equilibrium is a joint strategy "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        127,
+                                        504,
+                                        161
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        127,
+                                        504,
+                                        161
+                                    ],
+                                    "type": "text",
+                                    "content": " where every player's strategy is a best response to the equilibrium (i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        127,
+                                        504,
+                                        161
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_{k} = \\mathbb{R}_{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        127,
+                                        504,
+                                        161
+                                    ],
+                                    "type": "text",
+                                    "content": " so that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        127,
+                                        504,
+                                        161
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\epsilon_k(\\pmb {x}) = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        127,
+                                        504,
+                                        161
+                                    ],
+                                    "type": "text",
+                                    "content": "). Lemma 1 states that any interior best response has zero projected-gradient norm, which inspires the following loss function"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        242,
+                        163,
+                        504,
+                        187
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                242,
+                                163,
+                                504,
+                                187
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        242,
+                                        163,
+                                        504,
+                                        187
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathcal {L} (\\boldsymbol {x}) = \\sum_ {k} \\eta_ {k} \\left\\| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) \\right\\| ^ {2} \\tag {2}",
+                                    "image_path": "7894eac1bafc1a529693ef211c102125c469c07761ed3cccd8aaffdeae5d5c33.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        190,
+                        485,
+                        201
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                190,
+                                485,
+                                201
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        485,
+                                        201
+                                    ],
+                                    "type": "text",
+                                    "content": "where each "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        485,
+                                        201
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\eta_{k} > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        485,
+                                        201
+                                    ],
+                                    "type": "text",
+                                    "content": " represents a scalar weight, or equivalently, a step size to be explained next."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        202,
+                        506,
+                        227
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                202,
+                                506,
+                                227
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        202,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": "Proposition 1. The loss "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        202,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        202,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": " is equivalent to NashConv, but where player "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        202,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        202,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": "'s best response is approximated by a single step of projected-gradient ascent with step size "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        202,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\eta_k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        202,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": ": "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        202,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "aBR_k = x_k + \\eta_k\\Pi_{T\\Delta}(\\nabla_{x_k}^k)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        202,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        232,
+                        506,
+                        256
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                232,
+                                506,
+                                256
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        506,
+                                        256
+                                    ],
+                                    "type": "text",
+                                    "content": "This connection was already pointed out in prior work for unconstrained problems (Gemp et al., 2022; Raghunathan et al., 2019), but this result is the first for strategies constrained to the simplex."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        105,
+                        268,
+                        306,
+                        278
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                268,
+                                306,
+                                278
+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
+                                        268,
+                                        306,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": "4.3 CONNECTION TO TRUE EXPLOITABILITY"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        289,
+                        504,
+                        312
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                                504,
+                                312
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        289,
+                                        504,
+                                        312
+                                    ],
+                                    "type": "text",
+                                    "content": "In general, we can bound exploitability in terms of the projected-gradient norm as long as each player's utility is concave (this result extends to subgradients of non-smooth functions)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
+                        313,
+                        504,
+                        335
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                                335
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        313,
+                                        504,
+                                        335
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 2. The amount a player can gain by exploiting a joint strategy "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        313,
+                                        504,
+                                        335
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        313,
+                                        504,
+                                        335
+                                    ],
+                                    "type": "text",
+                                    "content": " is upper bounded by a quantity proportional to the norm of the projected-gradient:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        248,
+                        337,
+                        504,
+                        353
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        248,
+                                        337,
+                                        504,
+                                        353
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\epsilon_ {k} (\\boldsymbol {x}) \\leq \\sqrt {2} | | \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) | |. \\tag {3}",
+                                    "image_path": "91a710249bf1bd667678ba8d486a53c4b44602c384dc1184639c5d71caa2741b.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        104,
+                        358,
+                        504,
+                        415
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                415
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        358,
+                                        504,
+                                        415
+                                    ],
+                                    "type": "text",
+                                    "content": "This bound is not tight on the boundary of the simplex, which can be seen clearly by considering "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        358,
+                                        504,
+                                        415
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        358,
+                                        504,
+                                        415
+                                    ],
+                                    "type": "text",
+                                    "content": " to be part of a pure strategy equilibrium. In that case, this analysis assumes "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        358,
+                                        504,
+                                        415
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        358,
+                                        504,
+                                        415
+                                    ],
+                                    "type": "text",
+                                    "content": " can be improved upon by a projected-gradient ascent step (via the equivalence pointed out in Proposition 1). However, that is false because the probability of a pure strategy cannot be increased beyond 1. We mention this to provide further intuition for why our \"warm-up\" loss "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        358,
+                                        504,
+                                        415
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}(\\boldsymbol{x})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        358,
+                                        504,
+                                        415
+                                    ],
+                                    "type": "text",
+                                    "content": " is only valid for interior equilibria."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        104,
+                        419,
+                        504,
+                        445
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                419,
+                                504,
+                                445
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        419,
+                                        504,
+                                        445
+                                    ],
+                                    "type": "text",
+                                    "content": "Note that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        419,
+                                        504,
+                                        445
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "||\\Pi_{T\\Delta}(\\nabla_{x_k}^k)|| \\leq ||\\nabla_{x_k}^k||"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        419,
+                                        504,
+                                        445
+                                    ],
+                                    "type": "text",
+                                    "content": " because "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        419,
+                                        504,
+                                        445
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Pi_{T\\Delta}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        419,
+                                        504,
+                                        445
+                                    ],
+                                    "type": "text",
+                                    "content": " is a projection. Therefore, this improves the naive bounds on exploitability and distance to best responses given using the \"raw\" gradient "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        419,
+                                        504,
+                                        445
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\nabla_{x_k}^k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        419,
+                                        504,
+                                        445
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        104,
+                        447,
+                        473,
+                        459
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                                473,
+                                459
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        447,
+                                        473,
+                                        459
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 3. The exploitability of a joint strategy "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        447,
+                                        473,
+                                        459
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        447,
+                                        473,
+                                        459
+                                    ],
+                                    "type": "text",
+                                    "content": ", is upper bounded by a function of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        447,
+                                        473,
+                                        459
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}(\\pmb{x})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        447,
+                                        473,
+                                        459
+                                    ],
+                                    "type": "text",
+                                    "content": ":"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        237,
+                        461,
+                        504,
+                        488
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                237,
+                                461,
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+                                488
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        237,
+                                        461,
+                                        504,
+                                        488
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\epsilon \\leq \\sqrt {\\frac {2 n}{\\min  _ {k} \\eta_ {k}}} \\sqrt {\\mathcal {L} (\\boldsymbol {x})} \\stackrel {\\text {d e f}} {=} f (\\mathcal {L}). \\tag {4}",
+                                    "image_path": "5f61609a27c35e410de95c3fc8f7004208fa49901be1a5fda6bbcc7ef095b61c.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
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+                        509
+                    ],
+                    "type": "title",
+                    "angle": 0,
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "4.4 UNBIASED ESTIMATION"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        104,
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+                        564
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        519,
+                                        506,
+                                        564
+                                    ],
+                                    "type": "text",
+                                    "content": "As discussed in Section 3, a primary obstacle to unbiased estimation of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        519,
+                                        506,
+                                        564
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}(\\pmb{x})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        519,
+                                        506,
+                                        564
+                                    ],
+                                    "type": "text",
+                                    "content": " is the presence of complex, nonlinear functions of random variables, with the projection of a point onto the simplex being one such example (see "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        519,
+                                        506,
+                                        564
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Pi_{\\Delta}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        519,
+                                        506,
+                                        564
+                                    ],
+                                    "type": "text",
+                                    "content": " in Table 1). However, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        519,
+                                        506,
+                                        564
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Pi_{T\\Delta}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        519,
+                                        506,
+                                        564
+                                    ],
+                                    "type": "text",
+                                    "content": ", the projection onto the tangent space of the simplex, is linear! This is the insight that allows us to design an unbiased estimator (Lemma 5)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        104,
+                        569,
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+                        689
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                        689
+                                    ],
+                                    "type": "text",
+                                    "content": "Our proposed loss requires computing the squared norm of the expected value of the projected-gradient under the players' mixed strategies, i.e., the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "l"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
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+                                    ],
+                                    "type": "text",
+                                    "content": "-th entry of player "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
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+                                        689
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
+                                        506,
+                                        689
+                                    ],
+                                    "type": "text",
+                                    "content": "'s gradient equals "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
+                                        506,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\nabla_{x_{kl}}^{k} = \\mathbb{E}_{a_{-k} \\sim x_{-k}} u_{k}(a_{kl}, a_{-k})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
+                                        506,
+                                        689
+                                    ],
+                                    "type": "text",
+                                    "content": ". By analogy, consider a random variable "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
+                                        506,
+                                        689
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Y"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
+                                        506,
+                                        689
+                                    ],
+                                    "type": "text",
+                                    "content": ". In general, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
+                                        506,
+                                        689
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{E}[Y]^2 \\neq \\mathbb{E}[Y^2]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
+                                        506,
+                                        689
+                                    ],
+                                    "type": "text",
+                                    "content": ". This means that we cannot just sample projected-gradients and then compute their average norm to estimate our loss. However, consider taking two independent samples from two corresponding identically distributed, independent random variables "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
+                                        506,
+                                        689
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Y^{(1)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
+                                        506,
+                                        689
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
+                                        506,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Y^{(2)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
+                                        506,
+                                        689
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
+                                        506,
+                                        689
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{E}[Y^{(1)}]^2 = \\mathbb{E}[Y^{(1)}] \\mathbb{E}[Y^{(2)}] = \\mathbb{E}[Y^{(1)}Y^{(2)}]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
+                                        506,
+                                        689
+                                    ],
+                                    "type": "text",
+                                    "content": " by properties of expected value over products of independent random variables. This is a common technique to construct unbiased estimates of expectations over polynomial functions of random variables. Proceeding in this way, define "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
+                                        506,
+                                        689
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\nabla_{x_k}^{k(1)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
+                                        506,
+                                        689
+                                    ],
+                                    "type": "text",
+                                    "content": " as a random, unbiased gradient estimate (see Table 2). Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
+                                        506,
+                                        689
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\nabla_{x_k}^{k(2)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
+                                        506,
+                                        689
+                                    ],
+                                    "type": "text",
+                                    "content": " be independent and distributed identically to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
+                                        506,
+                                        689
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\nabla_{x_k}^{k(1)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        569,
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+                                    ],
+                                    "type": "text",
+                                    "content": ". Then Lemma 5 shows"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\mathcal {L} (\\boldsymbol {x}) = \\mathbb {E} [ \\sum_ {k} \\eta_ {k} (\\underbrace {\\hat {\\nabla} _ {x _ {k}} ^ {k (1)} - \\frac {1}{m _ {k}} \\left(\\mathbf {1} ^ {\\top} \\hat {\\nabla} _ {x _ {k}} ^ {k (1)}\\right) \\mathbf {1}} _ {\\text {p r o j e c t e d - g r a d i e n t 1}}) ^ {\\top} (\\underbrace {\\hat {\\nabla} _ {x _ {k}} ^ {k (2)} - \\frac {1}{m _ {k}} \\left(\\mathbf {1} ^ {\\top} \\hat {\\nabla} _ {x _ {k}} ^ {k (2)}\\right) \\mathbf {1}} _ {\\text {p r o j e c t e d - g r a d i e n t 2}}) ] \\tag {5}",
+                                    "image_path": "6916e6c736184a761a084821196a3bc2bb739527822a5307e187327ae91086a3.jpg"
+                                }
+                            ]
+                        }
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+                    "index": 19
+                }
+            ],
+            "discarded_blocks": [
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+                    "type": "header",
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "type": "text",
+                                    "content": "4"
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+                                            "type": "table",
+                                            "html": "<table><tr><td></td><td>Exact</td><td>Sample Others</td><td>Sample All</td></tr><tr><td>Estimator of ∇k(xk)</td><td>[uk(akl, x-k)]l</td><td>[uk(akl, a-k ~ x-k)]l</td><td>mkuk(akl ~ U(Ak), a-k ~ x-k)el</td></tr><tr><td>ˆ∇k(xk) Bounds</td><td>[0, 1]</td><td>[0, 1]</td><td>[0, mk]</td></tr><tr><td>ˆ∇k(xk) Query Cost</td><td>Πn{k=1}mk</td><td>mk</td><td>1</td></tr><tr><td>ˆL Bounds</td><td>±1/4 Σkηkmk</td><td>±1/4 Σkηkmk</td><td>±1/4 Σkηkm3</td></tr><tr><td>ˆL Query Cost</td><td>n Πn{k=1}mk</td><td>2nm</td><td>2n</td></tr></table>",
+                                            "image_path": "576cad80c74d2c237fdf399b3c2a2b2781eb7705b85c3e786323caf01c63c39e.jpg"
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+                                    "type": "text",
+                                    "content": "Table 2: Examples and Properties of Unbiased Estimators of Loss and Player Gradients "
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+                                    "type": "inline_equation",
+                                    "content": "(\\hat{\\nabla}_{x_k}^{k(p)})"
+                                }
+                            ]
+                        }
+                    ],
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+                    "angle": 0,
+                    "type": "text"
+                },
+                {
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "where "
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+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\nabla}_{x_k}^{k(p)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        205,
+                                        506,
+                                        318
+                                    ],
+                                    "type": "text",
+                                    "content": " is an unbiased estimator of player "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        205,
+                                        506,
+                                        318
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        205,
+                                        506,
+                                        318
+                                    ],
+                                    "type": "text",
+                                    "content": "'s gradient. This estimator can be constructed in several ways. The most expensive, an exact estimator, is constructed by marginalizing player "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        205,
+                                        506,
+                                        318
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        205,
+                                        506,
+                                        318
+                                    ],
+                                    "type": "text",
+                                    "content": "'s payoff tensor over all other players' strategies. However, a cheaper estimate can be obtained at the expense of higher variance by approximating this marginalization with a Monte Carlo (MC) estimate of the expectation. Specifically, if we sample a single action for each of the remaining players, we can construct an unbiased estimate of player "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        205,
+                                        506,
+                                        318
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        205,
+                                        506,
+                                        318
+                                    ],
+                                    "type": "text",
+                                    "content": "'s gradient by considering the payoff of each of its actions against the sampled background strategy. Lastly, we can consider constructing an estimate of player "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        205,
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+                                        318
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        205,
+                                        506,
+                                        318
+                                    ],
+                                    "type": "text",
+                                    "content": "'s gradient by sampling only a single action from player "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        205,
+                                        506,
+                                        318
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        205,
+                                        506,
+                                        318
+                                    ],
+                                    "type": "text",
+                                    "content": " to represent their entire gradient. Each of these approaches is outlined in Table 2 along with the query complexity (Babichenko, 2016) of computing the estimator and bounds on the values it can take (Lemma 9)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        322,
+                        504,
+                        353
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        322,
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+                                        353
+                                    ],
+                                    "type": "text",
+                                    "content": "We can extend Lemma 3 to one that holds under "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        322,
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+                                        353
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        322,
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+                                        353
+                                    ],
+                                    "type": "text",
+                                    "content": " samples with probability "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        322,
+                                        504,
+                                        353
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "1 - \\delta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        322,
+                                        504,
+                                        353
+                                    ],
+                                    "type": "text",
+                                    "content": " by applying, for example, a Hoeffding bound: "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        322,
+                                        504,
+                                        353
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\epsilon \\leq f\\bigl (\\hat{\\mathcal{L}} (\\pmb {x}) + \\mathcal{O}\\bigl (\\sqrt{\\frac{1}{T}\\ln(1 / \\delta)}\\bigr)\\bigr)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        322,
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+                                        353
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        322,
+                                        504,
+                                        353
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\mathcal{L}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        322,
+                                        504,
+                                        353
+                                    ],
+                                    "type": "text",
+                                    "content": " is an MC estimate of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        322,
+                                        504,
+                                        353
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        322,
+                                        504,
+                                        353
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
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+                                    "type": "text",
+                                    "content": "4.5 INTERIOR EQUILIBRIA"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "We discussed earlier that "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}(\\pmb{x})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        390,
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+                                    ],
+                                    "type": "text",
+                                    "content": " captures interior equilibria. But some games may only have partially mixed equilibria, i.e., equilibria that lie on the boundary of the simplex. We show how to circumvent this shortcoming by considering quantal response equilibria (QREs), specifically, logit equilibria. By adding an entropy bonus to each player's utility, we can"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        132,
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+                        504,
+                        504
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 10,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                132,
+                                445,
+                                285,
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+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                            "bbox": [
+                                                132,
+                                                445,
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+                                            ],
+                                            "type": "text",
+                                            "content": "- guarantee all equilibria are interior,"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 7
+                        },
+                        {
+                            "bbox": [
+                                132,
+                                464,
+                                311,
+                                475
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                            "bbox": [
+                                                132,
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+                                            ],
+                                            "type": "text",
+                                            "content": "- still obtain unbiased estimates of our loss,"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 8
+                        },
+                        {
+                            "bbox": [
+                                132,
+                                481,
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+                                504
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                132,
+                                                481,
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+                                                504
+                                            ],
+                                            "type": "text",
+                                            "content": "- maintain an upper bound on the exploitability "
+                                        },
+                                        {
+                                            "bbox": [
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+                                                504
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\epsilon"
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                481,
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+                                                504
+                                            ],
+                                            "type": "text",
+                                            "content": " of any approximate Nash equilibrium in the original game (i.e., the game without an entropy bonus)."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 9
+                        }
+                    ],
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+                },
+                {
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Define "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        514,
+                                        504,
+                                        571
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u_{k}^{\\tau}(\\boldsymbol{x}) = u_{k}(\\boldsymbol{x}) + \\tau S(x_{k})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        514,
+                                        504,
+                                        571
+                                    ],
+                                    "type": "text",
+                                    "content": " where Shannon entropy "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        514,
+                                        504,
+                                        571
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S(x_{k}) = -\\sum_{l}x_{kl}\\ln (x_{kl})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        514,
+                                        504,
+                                        571
+                                    ],
+                                    "type": "text",
+                                    "content": " is 1-strongly concave with respect to the 1-norm (Beck and Teboulle, 2003). It is known that Nash equilibria of entropy-regularized games satisfy the conditions for logit equilibria (Leonardos et al., 2021), which are solutions to the fixed point equation "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        514,
+                                        504,
+                                        571
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_{k} = \\mathrm{softmax}\\left(\\frac{1}{\\tau}\\nabla_{x_{k}}^{k}\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        514,
+                                        504,
+                                        571
+                                    ],
+                                    "type": "text",
+                                    "content": ". The softmax should make it clear to the reader that all probabilities have positive mass at positive temperature."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        104,
+                        575,
+                        506,
+                        639
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "Recall that in order to construct an unbiased estimate of our loss, we simply needed to construct unbiased estimates of player gradients. The introduction of the entropy term to player "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        506,
+                                        639
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        506,
+                                        639
+                                    ],
+                                    "type": "text",
+                                    "content": "'s utility is special in that it depends entirely on known quantities, i.e., the player's own mixed strategy. We can directly and deterministically compute "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        506,
+                                        639
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau \\frac{dS}{dx_k} = -\\tau (\\ln (x_k) + 1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        506,
+                                        639
+                                    ],
+                                    "type": "text",
+                                    "content": " and add this to our estimator of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        506,
+                                        639
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\nabla_{x_k}^{k(p)}\\colon \\hat{\\nabla}_{x_k}^{k\\tau (p)} = \\hat{\\nabla}_{x_k}^{k(p)} + \\tau \\frac{dS}{dx_k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        506,
+                                        639
+                                    ],
+                                    "type": "text",
+                                    "content": ". Consider our loss function refined from (2) with changes in blue:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        238,
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+                        504,
+                        684
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
+                                        238,
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+                                        504,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) = \\sum_ {k} \\eta_ {k} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right| \\right| ^ {2}. \\tag {6}",
+                                    "image_path": "4a75318a417389c4f0a10693c61770d6977e8337e076ec3fb922e86507bfc125.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
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+                                    "type": "text",
+                                    "content": "As mentioned above, the utilities with entropy bonuses are still concave, therefore, a similar bound to Lemma 2 applies. We use this to prove the QRE counterpart to Lemma 3 where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        698,
+                                        504,
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+                                    "type": "inline_equation",
+                                    "content": "\\epsilon_{QRE}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        698,
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+                                    ],
+                                    "type": "text",
+                                    "content": " is the exploitability of an approximate equilibrium in a game with entropy bonuses."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                }
+            ],
+            "discarded_blocks": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
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+                                    "type": "text",
+                                    "content": "5"
+                                }
+                            ]
+                        }
+                    ],
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+                }
+            ],
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+                                            "bbox": [
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+                                                291
+                                            ],
+                                            "type": "text",
+                                            "content": "Figure 1: Effect of Sampled Play on a Biased Loss. The first row displays the expectation of the upper bound guaranteed by our proposed loss "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                222,
+                                                506,
+                                                291
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{L}^{\\tau}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                222,
+                                                506,
+                                                291
+                                            ],
+                                            "type": "text",
+                                            "content": " with "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                222,
+                                                506,
+                                                291
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\eta_{k} = 1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                222,
+                                                506,
+                                                291
+                                            ],
+                                            "type": "text",
+                                            "content": " for all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                222,
+                                                506,
+                                                291
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "k"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                222,
+                                                506,
+                                                291
+                                            ],
+                                            "type": "text",
+                                            "content": ". The second row displays the expectation of NashConv under sampled play, i.e., "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                222,
+                                                506,
+                                                291
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\sum_{k}\\epsilon_{k}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                222,
+                                                506,
+                                                291
+                                            ],
+                                            "type": "text",
+                                            "content": " where "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                222,
+                                                506,
+                                                291
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\epsilon_{k} = \\mathbb{E}_{a_{-k}\\sim x_{-k}}[\\max_{a_{k}}u_{k}^{\\tau}(\\pmb{a})] - \\mathbb{E}_{\\pmb{a}\\sim \\pmb{x}}[u_{k}^{\\tau}(\\pmb{a})]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                222,
+                                                506,
+                                                291
+                                            ],
+                                            "type": "text",
+                                            "content": ". To be consistent, we subtract the offset "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                222,
+                                                506,
+                                                291
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tau \\log (m^{2})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                222,
+                                                506,
+                                                291
+                                            ],
+                                            "type": "text",
+                                            "content": " from "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                222,
+                                                506,
+                                                291
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "f_{\\tau}(\\mathcal{L}^{\\tau})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                222,
+                                                506,
+                                                291
+                                            ],
+                                            "type": "text",
+                                            "content": " per Lemma 13, which relates the exploitability at positive temperature to that at zero temperature. The resulting loss surface clearly shows NashConv fails to recognize any interior Nash equilibrium due to its inherent bias."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 2,
+                            "angle": 0,
+                            "type": "image_caption"
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
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+                        325
+                    ],
+                    "type": "text",
+                    "angle": 0,
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+                            ],
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+                                    "bbox": [
+                                        104,
+                                        312,
+                                        506,
+                                        325
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 4. The entropy regularized exploitability, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        312,
+                                        506,
+                                        325
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\epsilon_{QRE}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        312,
+                                        506,
+                                        325
+                                    ],
+                                    "type": "text",
+                                    "content": ", of a joint strategy "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        312,
+                                        506,
+                                        325
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        312,
+                                        506,
+                                        325
+                                    ],
+                                    "type": "text",
+                                    "content": ", is upper bounded as:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        222,
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+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        222,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\epsilon_ {Q R E} \\leq \\sqrt {\\frac {2 n}{\\min  _ {k} \\eta_ {k}}} \\sqrt {\\mathcal {L} ^ {\\tau} (\\boldsymbol {x})} \\stackrel {\\mathrm {d e f}} {=} f \\left(\\mathcal {L} ^ {\\tau}\\right). \\tag {7}",
+                                    "image_path": "91de8f08a0132c11c7a144e829b6767a3073396349fdbc96874a5c002fbb3570.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
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+                    ],
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+                    "angle": 0,
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Lastly, we establish a connection between quantal response equilibria and Nash equilibria that allows us to approximate Nash equilibria in the original game via minimizing our modified loss "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}^{\\tau}(\\pmb{x})"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Lemma 13 ("
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}^{\\tau}"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        396,
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+                                    ],
+                                    "type": "text",
+                                    "content": " Scores Nash Equilibria). Let "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        396,
+                                        504,
+                                        421
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}^{\\tau}(\\pmb{x})"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        396,
+                                        504,
+                                        421
+                                    ],
+                                    "type": "text",
+                                    "content": " be our proposed entropy regularized loss function and "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        396,
+                                        504,
+                                        421
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        396,
+                                        504,
+                                        421
+                                    ],
+                                    "type": "text",
+                                    "content": " be an approximate QRE. Then it holds that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
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+                    ],
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+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\epsilon \\leq \\tau \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {\\frac {2 n}{\\min  _ {k} \\eta_ {k}}} \\sqrt {\\mathcal {L} ^ {\\tau} (\\boldsymbol {x})} \\stackrel {\\text {d e f}} {=} f _ {\\tau} \\left(\\mathcal {L} ^ {\\tau}\\right). \\tag {8}",
+                                    "image_path": "d02b24f57ddead94b47dd797baee9067bbbe09ecc374be60d64921c5fa8a307d.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "This upper bound is plotted as a heatmap for a familiar Chicken game in the top row of Figure 1. First, notice how pure equilibria are not visible as minima for zero temperature, but appear for slightly warmer temperatures. Secondly, notice that NashConv in the bottom row is unable to capture the interior Nash equilibrium because of its high bias under sampled play. In contrast, our proposed loss "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        506,
+                                        526
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}^{\\tau}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        506,
+                                        526
+                                    ],
+                                    "type": "text",
+                                    "content": " is guaranteed to capture all equilibria at low temperature "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        506,
+                                        526
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
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+                                        526
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
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+                    "type": "title",
+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "5 ANALYSIS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
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+                                    "type": "text",
+                                    "content": "In the preceding section we established a loss function that upper bounds the exploitability of an approximate equilibrium. In addition, the zeros of this loss function have a one-to-one correspondence with quantal response equilibria (which approximate Nash equilibria at low temperature)."
+                                }
+                            ]
+                        }
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+                                    "type": "text",
+                                    "content": "Here, we derive properties that suggest it is \"easy\" to optimize. While this function is generally non-convex and may suffer from a proliferation of saddle points (Figure 2), it is Lipschitz continuous (over the relevant subset of the interior) and bounded. These are two commonly made assumptions in the literature on non-convex optimization, which we leverage in Section 6. In addition, we can derive its gradient, its Hessian, and characterize its behavior around global minima."
+                                }
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+                                    "type": "text",
+                                    "content": "Lemma 14. The gradient of "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}^{\\tau}(\\pmb{x})"
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+                                    "type": "text",
+                                    "content": " with respect to player "
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+                                    "type": "text",
+                                    "content": "'s strategy "
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+                                    "type": "inline_equation",
+                                    "content": "x_{l}"
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+                                    "type": "text",
+                                    "content": " is"
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+                                    "type": "interline_equation",
+                                    "content": "\\nabla_ {x _ {l}} \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) = 2 \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\tag {9}",
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+                                    "type": "text",
+                                    "content": "where "
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+                                    "type": "inline_equation",
+                                    "content": "B_{ll} = -\\tau \\left[I - \\frac{1}{m_l}\\mathbf{11}^\\top \\right]diag\\left(\\frac{1}{x_l}\\right)"
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+                                    "type": "inline_equation",
+                                    "content": "B_{kl} = \\left[I - \\frac{1}{m_k}\\mathbf{11}^\\top \\right]H_{kl}^k"
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": " for "
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+                                    "type": "inline_equation",
+                                    "content": "k\\neq l"
+                                }
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                            "type": "text",
+                                            "content": "Figure 2: Analysis of Loss Landscape. We reapply the analysis of (Dauphin et al., 2014), originally designed to understand the success of SGD in deep learning, to \"slices\" of several popular extensive form games. To construct a slice (or meta-game), we randomly sample 6 deterministic policies and then consider the corresponding "
+                                        },
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+                                            "type": "text",
+                                            "content": "-player, 6-action normal-form game at "
+                                        },
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tau = 0.1"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": " (with payoffs normalized to [0, 1]). The index of a critical point "
+                                        },
+                                        {
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "x_{c}"
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+                                            ],
+                                            "type": "text",
+                                            "content": " ("
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+                                            "bbox": [
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\nabla_{\\pmb{x}}\\mathcal{L}^{\\tau}(\\pmb{x}_{c}) = \\mathbf{0}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
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+                                            ],
+                                            "type": "text",
+                                            "content": ") indicates the fraction of negative eigenvalues in the Hessian of "
+                                        },
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{L}^{\\tau}"
+                                        },
+                                        {
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+                                            ],
+                                            "type": "text",
+                                            "content": " at "
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "x_{c}"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": "; "
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\alpha = 0"
+                                        },
+                                        {
+                                            "bbox": [
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+                                                148,
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+                                            ],
+                                            "type": "text",
+                                            "content": " indicates a local minimum, 1 a maximum, else a saddle point. We see a positive correlation between exploitability ("
+                                        },
+                                        {
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "y"
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+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": "-axis), projected-gradient norm ("
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "x"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": "-axis), and "
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\alpha"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": " (color) indicating a lower prevalence of local minima at high exploitability."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 2,
+                            "angle": 0,
+                            "type": "image_caption"
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
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+                    ],
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Lemma 16. The Hessian of "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        254,
+                                        309,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}^{\\tau}(\\pmb{x})"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " can be written"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
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+                    ],
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+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\operatorname {H e s s} \\left(\\mathcal {L} ^ {\\tau}\\right) = 2 \\left[ \\tilde {B} ^ {\\top} \\tilde {B} + T \\Pi_ {T \\Delta} \\left(\\tilde {\\nabla} ^ {\\tau}\\right) \\right] \\tag {10}",
+                                    "image_path": "d5d14474db549c0831924327ad9b37aa530c24ecda0887acf56b91c9c41bdff1.jpg"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{B}_{kl} = \\sqrt{\\eta_k} B_{kl}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Pi_{T\\Delta}(\\tilde{\\nabla}^{\\tau}) = [\\eta_1\\Pi_{T\\Delta}(\\nabla_{x_1}^{1\\tau}),\\dots ,\\eta_n\\Pi_{T\\Delta}(\\nabla_{x_n}^{n\\tau})]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                        317
+                                    ],
+                                    "type": "text",
+                                    "content": ", and we augment "
+                                },
+                                {
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " (the 3-player approximation to the game, "
+                                },
+                                {
+                                    "bbox": [
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+                                        317
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T_{lqk}^{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                        317
+                                    ],
+                                    "type": "text",
+                                    "content": ") so that "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T_{lll}^{l} = \\tau \\text{diag} 3\\left(\\frac{1}{x_{l}^{2}}\\right)"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
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+                    "angle": 0,
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "At an NE, the latter term disappears because "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Pi_{T\\Delta}(\\nabla_{x_k}^{k\\tau}) = 0"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
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+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " (Lemma 1). If "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " was "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{R}^{n\\bar{m}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
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+                                        387
+                                    ],
+                                    "type": "text",
+                                    "content": ", then we could simply check if "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{B}"
+                                },
+                                {
+                                    "bbox": [
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+                                        325,
+                                        506,
+                                        387
+                                    ],
+                                    "type": "text",
+                                    "content": " is full-rank to determine if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        506,
+                                        387
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Hess\\succ 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        506,
+                                        387
+                                    ],
+                                    "type": "text",
+                                    "content": ", i.e., if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        506,
+                                        387
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}^{\\tau}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        506,
+                                        387
+                                    ],
+                                    "type": "text",
+                                    "content": " is locally strongly-convex. However, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        506,
+                                        387
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        506,
+                                        387
+                                    ],
+                                    "type": "text",
+                                    "content": " is a simplex product, and we only care about curvature in directions toward which we can update our strategy profile "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        506,
+                                        387
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        506,
+                                        387
+                                    ],
+                                    "type": "text",
+                                    "content": ". Toward that end, define "
+                                },
+                                {
+                                    "bbox": [
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+                                        325,
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+                                        387
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " to be the "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n(\\bar{m} +1)\\times n\\bar{m}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " matrix that stacks "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{B}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " on top of a repeated identity matrix that encodes orthogonality to the simplex:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "M (\\boldsymbol {x}) = \\left[ \\begin{array}{c c c c} - \\tau \\sqrt {\\eta_ {1}} \\Pi_ {T \\Delta} \\left(\\frac {1}{x _ {1}}\\right) & \\sqrt {\\eta_ {1}} \\Pi_ {T \\Delta} \\left(H _ {1 2} ^ {1}\\right) & \\dots & \\sqrt {\\eta_ {1}} \\Pi_ {T \\Delta} \\left(H _ {1 n} ^ {1}\\right) \\\\ \\vdots & \\vdots & \\vdots & \\vdots \\\\ \\sqrt {\\eta_ {n}} \\Pi_ {T \\Delta} \\left(H _ {n 1} ^ {n}\\right) & \\dots & \\sqrt {\\eta_ {n}} \\Pi_ {T \\Delta} \\left(H _ {n, n - 1} ^ {n}\\right) & - \\tau \\sqrt {\\eta_ {n}} \\Pi_ {T \\Delta} \\left(\\frac {1}{x _ {n}}\\right) \\\\ \\mathbf {1} _ {1} ^ {\\top} & 0 & \\dots & 0 \\\\ \\vdots & \\vdots & \\vdots & \\vdots \\\\ 0 & \\dots & 0 & \\mathbf {1} _ {n} ^ {\\top} \\end{array} \\right] \\tag {11}",
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+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Pi_{T\\Delta}(z\\in \\mathbb{R}^{a\\times b}) = [I_a - \\frac{1}{a}\\mathbf{1}_a\\mathbf{1}_a^\\top ]z"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " subtracts the mean from each column of "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z"
+                                },
+                                {
+                                    "bbox": [
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+                                        486,
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+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\frac{1}{x_k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "text",
+                                    "content": " is shorthand for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{diag}\\left(\\frac{1}{x_k}\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "text",
+                                    "content": ". If "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M(\\pmb {x})z = \\mathbf{0}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "text",
+                                    "content": " for a nonzero vector "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z\\in \\mathbb{R}^{n\\bar{m}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "text",
+                                    "content": ", this implies there exists a "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "text",
+                                    "content": " that 1) is orthogonal to the ones vectors of each simplex (i.e., is a valid equilibrium update direction) and 2) achieves zero curvature in the direction "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "text",
+                                    "content": ", i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z^{\\top}(\\tilde{B}^{\\top}\\tilde{B})z = z^{\\top}(Hess)z = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "text",
+                                    "content": ", and so "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Hess"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "text",
+                                    "content": " is not positive definite. Conversely, if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M(\\pmb {x})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "text",
+                                    "content": " is of rank "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "nm"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "text",
+                                    "content": " for a quantal response equilibrium "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "text",
+                                    "content": ", then the Hessian of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}^{\\tau}"
+                                },
+                                {
+                                    "bbox": [
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+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "text",
+                                    "content": " at "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "text",
+                                    "content": " in the tangent space of the simplex product "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\mathcal{X} = \\prod_{k}\\mathcal{X}_{k})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "text",
+                                    "content": " is positive definite. In this case, we call "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "text",
+                                    "content": " polymatrix-isolated: polymatrix because we only require information of the local polymatrix approximation of the game (i.e., the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H_{kl}^{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "text",
+                                    "content": " matrices) to construct "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "text",
+                                    "content": " and isolated because it implies "
+                                },
+                                {
+                                    "bbox": [
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+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        486,
+                                        506,
+                                        592
+                                    ],
+                                    "type": "text",
+                                    "content": " is not connected to any other equilibria."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Definition 1 (Polymatrix-Isolated Equilibrium). A Nash equilibrium "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x}^*"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " is polymatrix-isolated iff "
+                                },
+                                {
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x}^*"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        594,
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+                                        617
+                                    ],
+                                    "type": "text",
+                                    "content": " is isolated according to its local polymatrix game approximation."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
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+                        670
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
+                                        104,
+                                        624,
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+                                        670
+                                    ],
+                                    "type": "text",
+                                    "content": "By analyzing the rank of "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        624,
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+                                    ],
+                                    "type": "text",
+                                    "content": ", we can confirm that many classical matrix games including Rock-Paper-Scissors, Chicken, Matching Pennies, and Shapley's game all induce strongly convex "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        624,
+                                        506,
+                                        670
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}^{\\tau}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        624,
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+                                    ],
+                                    "type": "text",
+                                    "content": "'s at zero temperature (i.e., they have unique mixed Nash equilibria). In contrast, a game like Prisoner's Dilemma has a unique pure strategy that will not be captured by our loss at zero temperature."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
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+                    "type": "title",
+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "6 ALGORITHMS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "We have formally transformed the approximation of Nash equilibria in NFGs into a stochastic optimization problem. To our knowledge, this is the first such formulation that allows one-shot"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                }
+            ],
+            "discarded_blocks": [
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
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+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": "Figure 3: Comparison of SGD on "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                157,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{L}^{\\tau = 0}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
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+                                            ],
+                                            "type": "text",
+                                            "content": " against baselines on four games evaluated in (Gemp et al., 2022). The number of samples used to estimate each update iteration (i.e., minibatch size) is indicated by "
+                                        },
+                                        {
+                                            "bbox": [
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+                                            "type": "inline_equation",
+                                            "content": "s"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": ". From left to right: 2-player, 3-action, nonsymmetric; 6-player, 5-action, nonsymmetric; 4-player, 66-action, symmetric; 3-player, 286-action, symmetric. SGD struggles at saddle points in Blotto."
+                                        }
+                                    ]
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+                            "angle": 0,
+                            "type": "image_caption"
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+                    ],
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+                },
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+                    ],
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+                    "angle": 0,
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "unbiased Monte-Carlo estimation which is critical to introduce the use of powerful algorithms capable of solving high dimensional optimization problems. We explore two off-the-shelf approaches."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
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+                    ],
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+                                    "type": "text",
+                                    "content": "6.1 STOCHASTIC GRADIENT DESCENT"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Stochastic gradient descent is the workhorse of high-dimensional stochastic optimization. It is guaranteed to converge to stationary points (Cutkosky et al., 2023), however, it may converge to local, rather than global minima. It also enjoys implicit gradient regularization (Barrett and Dherin, 2020), seeking \"flat\" minima and performs approximate Bayesian inference (Mandt et al., 2017). Despite the lack of global convergence guarantee, we find it performs well empirically in games previously examined by the literature: modified Shapley's (Ostrovski and van Strien, 2014), GAMUT D7 (Nudelman et al., 2004), Blotto (Arad and Rubinstein, 2012). Figure 3 shows SGD is competitive with scalable techniques to approximating NEs: FTRL (Shalev-Shwartz and Singer, 2006; Shalev-Shwartz et al., 2012), Regret Matching (Hart and Mas-Colell, 2000), ADIDAS/ "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        276,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "^yQRRe^{auto}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        276,
+                                        506,
+                                        398
+                                    ],
+                                    "type": "text",
+                                    "content": " (Gemp et al., 2022). Shapley's game induces a strongly convex "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        276,
+                                        506,
+                                        398
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        276,
+                                        506,
+                                        398
+                                    ],
+                                    "type": "text",
+                                    "content": " (see Section 5) leading to SGD's strong performance. Blotto reaches low, but nonzero "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        276,
+                                        506,
+                                        398
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\epsilon"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        276,
+                                        506,
+                                        398
+                                    ],
+                                    "type": "text",
+                                    "content": ", demonstrating the challenges of saddle points."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
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+                        421
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
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+                                    "type": "text",
+                                    "content": "6.2 HIGH PROBABILITY, GLOBAL POLYNOMIAL CONVERGENCE RATES VIA BANDITS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "We explore one other algorithmic approach to non-convex optimization based on minimizing regret, which enjoys finite time global convergence rates. "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        430,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        430,
+                                        506,
+                                        509
+                                    ],
+                                    "type": "text",
+                                    "content": "-armed bandits (Bubeck et al., 2011) systematically explore the space of solutions by refining a mesh over the joint strategy space, trading off exploration versus exploitation of promising regions. Several approaches exist (Bartlett et al., 2019; Valko et al., 2013) with open source implementations, e.g., (Li et al., 2023). Applying "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        430,
+                                        506,
+                                        509
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        430,
+                                        506,
+                                        509
+                                    ],
+                                    "type": "text",
+                                    "content": "-armed bandits to our "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        430,
+                                        506,
+                                        509
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}^{\\tau}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        430,
+                                        506,
+                                        509
+                                    ],
+                                    "type": "text",
+                                    "content": " can be thought of as a stochastic generalization of the exclusion method and other bandit approaches for Nash equilibria (Berg and Sandholm, 2017; Zhou et al., 2017)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        513,
+                                        505,
+                                        587
+                                    ],
+                                    "type": "text",
+                                    "content": "Equipped with these techniques, we establish a high probability polynomial-time global convergence rate to Nash equilibria in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        513,
+                                        505,
+                                        587
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        513,
+                                        505,
+                                        587
+                                    ],
+                                    "type": "text",
+                                    "content": "-player, general-sum games given all QREs("
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        513,
+                                        505,
+                                        587
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        513,
+                                        505,
+                                        587
+                                    ],
+                                    "type": "text",
+                                    "content": ") are polymatrix-isolated. The quality of this approximation improves as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        513,
+                                        505,
+                                        587
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau \\to 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        513,
+                                        505,
+                                        587
+                                    ],
+                                    "type": "text",
+                                    "content": ", at the same time increasing the constant on the convergence rate via the Lipschitz constant "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        513,
+                                        505,
+                                        587
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sqrt{\\hat{L}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        513,
+                                        505,
+                                        587
+                                    ],
+                                    "type": "text",
+                                    "content": " defined below. For clarity, we assume users provide a temperature in the form "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        513,
+                                        505,
+                                        587
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau = \\frac{1}{\\ln(1/p)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        513,
+                                        505,
+                                        587
+                                    ],
+                                    "type": "text",
+                                    "content": " with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        513,
+                                        505,
+                                        587
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p \\in (0,1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        513,
+                                        505,
+                                        587
+                                    ],
+                                    "type": "text",
+                                    "content": " which ensures all equilibria have probability mass greater than "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        513,
+                                        505,
+                                        587
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\frac{p}{m^*}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        513,
+                                        505,
+                                        587
+                                    ],
+                                    "type": "text",
+                                    "content": " for all actions (Lemma 11). Lower "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        513,
+                                        505,
+                                        587
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        513,
+                                        505,
+                                        587
+                                    ],
+                                    "type": "text",
+                                    "content": " corresponds with lower temperature."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        104,
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+                        616
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        590,
+                                        504,
+                                        616
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem 4 (BLiN PAC Rate). Assume "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        590,
+                                        504,
+                                        616
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\eta_{k} = \\eta = 2 / \\hat{L}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        590,
+                                        504,
+                                        616
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        590,
+                                        504,
+                                        616
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau = \\frac{1}{\\ln(1 / p)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        590,
+                                        504,
+                                        616
+                                    ],
+                                    "type": "text",
+                                    "content": ", and a previously pulled arm is returned uniformly at random (i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        590,
+                                        504,
+                                        616
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t \\sim U([T])"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        590,
+                                        504,
+                                        616
+                                    ],
+                                    "type": "text",
+                                    "content": "). Then for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        590,
+                                        504,
+                                        616
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w > 0"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        151,
+                        618,
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        151,
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+                                        504,
+                                        648
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\epsilon_ {t} \\leq w \\left[ \\frac {1}{\\ln (1 / p)} \\log \\left(\\prod_ {k} m _ {k}\\right) + 2 \\left(1 + \\left(4 c ^ {2} C _ {z}\\right) ^ {1 / 3}\\right) \\sqrt {2 n \\hat {L}} \\left(\\frac {\\ln T}{T}\\right) ^ {\\frac {1}{2 (d z + 2)}} \\right] \\tag {12}",
+                                    "image_path": "fe97d35b9007fd4d7d0088485ffc30575d352135b4d89c3cc64c5aa645cf6afe.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        104,
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+                        504,
+                        703
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                        703
+                                    ],
+                                    "type": "text",
+                                    "content": "with probability "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        652,
+                                        504,
+                                        703
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(1 - w^{-1})(1 - 2T^{-2})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        652,
+                                        504,
+                                        703
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        652,
+                                        504,
+                                        703
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m^{*} = \\max_{k} m_{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        652,
+                                        504,
+                                        703
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "2|\\mathcal{L}^{\\tau}| \\leq c \\leq \\frac{1}{4}\\left(\\frac{\\ln(m^{*})}{\\ln(1 / p)} + 2\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        652,
+                                        504,
+                                        703
+                                    ],
+                                    "type": "text",
+                                    "content": " (Corollary 1), "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        652,
+                                        504,
+                                        703
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{L} = \\left(\\frac{\\ln(m^{*})}{\\ln(1 / p)} + 2\\right)\\left(\\frac{m^{*2}}{p\\ln(1 / p)} + n\\bar{m}\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        652,
+                                        504,
+                                        703
+                                    ],
+                                    "type": "text",
+                                    "content": " (Corollary 2), the zooming dimension "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        652,
+                                        504,
+                                        703
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d_{z} = \\frac{1}{2} n\\bar{m}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        652,
+                                        504,
+                                        703
+                                    ],
+                                    "type": "text",
+                                    "content": ", and the zooming constant "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        652,
+                                        504,
+                                        703
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_{z} = |\\mathcal{X}^{*}|^{-1}\\left(\\frac{1}{4} r_{\\eta}^{2}\\sigma_{-\\infty}\\right)^{-n\\bar{m}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        652,
+                                        504,
+                                        703
+                                    ],
+                                    "type": "text",
+                                    "content": " (Corollary 30)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        104,
+                        709,
+                        504,
+                        733
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
+                                        104,
+                                        709,
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+                                    ],
+                                    "type": "text",
+                                    "content": "The convergence rate for BLiN (Feng et al., 2022) depends on bounds on the exploitability in terms of the loss (Lemma 13), bounds on estimates of the loss (Corollary 1), Lipschitz bounds on the"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                }
+            ],
+            "discarded_blocks": [
+                {
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+                    ],
+                    "type": "header",
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+                                    "bbox": [
+                                        105,
+                                        26,
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+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
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+                    ],
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "8"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 7
+        },
+        {
+            "para_blocks": [
+                {
+                    "type": "image",
+                    "bbox": [
+                        121,
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+                    ],
+                    "blocks": [
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+                            "bbox": [
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+                                83,
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+                                205
+                            ],
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                        205
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                121,
+                                                83,
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+                                                205
+                                            ],
+                                            "type": "image",
+                                            "image_path": "6737fba7ce2c95955cadad5307a45761a3fc43900fed5e9e197dd9926c74b122.jpg"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 1,
+                            "angle": 0,
+                            "type": "image_body"
+                        },
+                        {
+                            "bbox": [
+                                104,
+                                219,
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+                                331
+                            ],
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                        331
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                104,
+                                                219,
+                                                506,
+                                                331
+                                            ],
+                                            "type": "text",
+                                            "content": "Figure 4: Bandit-based (BLiN) Nash solver applied to an artificial 7-player, symmetric, 2-action game. We search for a symmetric equilibrium, which is represented succinctly as the probability of selecting action 1. The plot shows the true exploitability "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                219,
+                                                506,
+                                                331
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\epsilon"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                219,
+                                                506,
+                                                331
+                                            ],
+                                            "type": "text",
+                                            "content": " of all symmetric strategies in black and indicates there exist potentially 5 NEs (the dips in the curve). Upper bounds on our unregularized loss "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                219,
+                                                506,
+                                                331
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{L}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                219,
+                                                506,
+                                                331
+                                            ],
+                                            "type": "text",
+                                            "content": " capture 4 of these equilibria, missing only the pure NE on the right. By considering our regularized loss, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                219,
+                                                506,
+                                                331
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{L}^{\\tau}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                219,
+                                                506,
+                                                331
+                                            ],
+                                            "type": "text",
+                                            "content": ", we are able to capture this pure NE (see zoomed inset). The bandit algorithm selects strategies to evaluate, using 10 Monte-Carlo samples for each evaluation (arm pull) of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                219,
+                                                506,
+                                                331
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{L}^{\\tau}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                219,
+                                                506,
+                                                331
+                                            ],
+                                            "type": "text",
+                                            "content": ". These samples are displayed as vertical bars above with the height of the vertical bar representing additional arm pulls. The best arms throughout search are denoted by green circles (darker indicates later in the search). The boxed numbers near equilibria display the welfare of the strategy profile."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 2,
+                            "angle": 0,
+                            "type": "image_caption"
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        346,
+                        504,
+                        426
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                346,
+                                504,
+                                426
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        346,
+                                        504,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": "infinity norm of the gradient (Corollary 2), and the number of distinct strategies "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        346,
+                                        504,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(n\\bar{m} = \\sum_{k}m_{k})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        346,
+                                        504,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": ". This result further depends on the near-optimality or zooming-dimension "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        346,
+                                        504,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d_z"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        346,
+                                        504,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": " and zooming constant "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        346,
+                                        504,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_z"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        346,
+                                        504,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": " which quantify the number of near optimal states. In particular, we assume "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        346,
+                                        504,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}(s(z))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        346,
+                                        504,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": " is locally "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        346,
+                                        504,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\sigma_{-\\infty})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        346,
+                                        504,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": "-strongly convex with respect to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        346,
+                                        504,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\| \\cdot \\|_{\\infty}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        346,
+                                        504,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": " about each global optimum within a ball of radius "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        346,
+                                        504,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r_\\eta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        346,
+                                        504,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": ". Here, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        346,
+                                        504,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s:[0,1]^{n(\\bar{m} -1)}\\to \\prod_i\\Delta^{m_i - 1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        346,
+                                        504,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": " is any function that maps from the unit hypercube to a product of simplices; we analyze two such maps in Appendix B.1. Next, we present an additional convergence rate result using an alternative "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        346,
+                                        504,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        346,
+                                        504,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": "-bandit approach, StoSOO (Valko et al., 2013)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        426,
+                        504,
+                        450
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                426,
+                                504,
+                                450
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem 5 (StoSOO Rate). Corollary 1 of Valko et al. (2013) implies that with probability "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(1 - w^{-1})(1 - \\delta)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "text",
+                                    "content": " for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "text",
+                                    "content": ", a uniformly randomly drawn arm (i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t \\sim U([T]))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        426,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "text",
+                                    "content": " achieves"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        149,
+                        453,
+                        504,
+                        491
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                149,
+                                453,
+                                504,
+                                491
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        149,
+                                        453,
+                                        504,
+                                        491
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\epsilon_ {t} \\leq w \\left[ \\frac {1}{\\ln (1 / p)} \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {n \\hat {L}} \\sqrt {\\xi_ {1} \\sqrt {\\frac {\\log_ {b} (T k / \\delta)}{2 \\log_ {b} (e) k}} + \\xi_ {2} b ^ {- \\frac {1}{d C} \\sqrt {T / k}}} \\right] \\tag {13}",
+                                    "image_path": "96dcfa2922b2fde49c76fef65f23363bda2b2e84b3f6c16660cd818c17948009.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        103,
+                        492,
+                        504,
+                        529
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                103,
+                                492,
+                                504,
+                                529
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        103,
+                                        492,
+                                        504,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        492,
+                                        504,
+                                        529
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d = n(\\bar{m} -1)"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        492,
+                                        504,
+                                        529
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\xi_{1} = (2 + 2^{2 / d})"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        492,
+                                        504,
+                                        529
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\xi_{2} = \\frac{1}{4} db^{2(1 + 2 / d)}"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        492,
+                                        504,
+                                        529
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k = T\\log_b(T)^{-3}"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        492,
+                                        504,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": " , "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        492,
+                                        504,
+                                        529
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "b"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        492,
+                                        504,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": " is the branching factor for partitioning cells, and the near-optimality constant "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        492,
+                                        504,
+                                        529
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C = |\\mathcal{X}^* |^{-1}\\sqrt{2\\pi d}\\Big(\\frac{b^2d^2}{5r_\\eta^2\\sigma_{-2}}\\Big)^{d / 2}"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        492,
+                                        504,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": " (Lemma 36)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        536,
+                        504,
+                        572
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                536,
+                                504,
+                                572
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        536,
+                                        504,
+                                        572
+                                    ],
+                                    "type": "text",
+                                    "content": "Here we assume "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        536,
+                                        504,
+                                        572
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}(s(z))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        536,
+                                        504,
+                                        572
+                                    ],
+                                    "type": "text",
+                                    "content": " is locally "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        536,
+                                        504,
+                                        572
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\sigma_{-2})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        536,
+                                        504,
+                                        572
+                                    ],
+                                    "type": "text",
+                                    "content": "-strongly convex with respect to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        536,
+                                        504,
+                                        572
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "||\\cdot ||_2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        536,
+                                        504,
+                                        572
+                                    ],
+                                    "type": "text",
+                                    "content": " about each global optimum within a ball of radius "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        536,
+                                        504,
+                                        572
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r_\\eta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        536,
+                                        504,
+                                        572
+                                    ],
+                                    "type": "text",
+                                    "content": ". Theorem 5 implies a "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        536,
+                                        504,
+                                        572
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\mathcal{O}}(T^{-1/4})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        536,
+                                        504,
+                                        572
+                                    ],
+                                    "type": "text",
+                                    "content": " global convergence rate (Proposition 2), however this is achieved only after an exponential number of burn-in iterations."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
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+                                    "content": "7 CONCLUSION"
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+                                    "content": "In this work, we proposed a stochastic loss for approximate Nash equilibria in normal-form games. An unbiased loss estimator of Nash equilibria is the \"key\" to the stochastic optimization \"door\" which holds a wealth of research innovations uncovered over several decades. Thus, it allows the development of new algorithmic techniques for computing equilibria. We consider bandit and vanilla SGD methods in this work, but these are only two of the many options now at our disposal (e.g., adaptive methods (Antonakopoulos et al., 2022), Gaussian processes (Calandriello et al., 2022), evolutionary algorithms (Hansen et al., 2003), etc.). Such approaches as well as generalizations of these techniques to extensive-form, imperfect-information games are promising directions for future work. Similarly to how deep learning research first balked at and then marched on to train neural networks via NP-hard non-convex optimization, we hope computational game theory can march ahead to make useful equilibrium predictions of large multiplayer systems."
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+                                    "content": "REFERENCES"
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+                                            "type": "text",
+                                            "content": "C.5 BLiN on Artificial Game 47"
+                                        }
+                                    ]
+                                }
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "content": "13"
+                                }
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+                                    "type": "text",
+                                    "content": "A LOSS WITH ITS PROPERTIES AND DERIVATIVES"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
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+                                    "type": "text",
+                                    "content": "In this section, we construct our loss function and derive many of its properties and derivatives (e.g., gradient and hessian) useful for analyzing and executing optimization algorithms."
+                                }
+                            ]
+                        }
+                    ],
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+                                    "type": "text",
+                                    "content": "A.1 LOSS: CONNECTION TO EXPLOITABILITY, UNBIASED ESTIMATION, AND UPPER BOUNDS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
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+                                    "type": "text",
+                                    "content": "A.1.1 KKT CONDITIONS IMPLY FIXED POINT SUFFICIENCY"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
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+                                    "type": "text",
+                                    "content": "Consider the following constrained optimization problem:"
+                                }
+                            ]
+                        }
+                    ],
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\max  _ {\\boldsymbol {x} \\in \\mathbb {R} ^ {d}} f (\\boldsymbol {x}) \\tag {14a}",
+                                    "image_path": "28a990549bce6ffc312d0db3eeb4c75ace7f6c10f08a27777d866a3417f6602d.jpg"
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+                                    "type": "interline_equation",
+                                    "content": "s. t. g _ {i} (\\boldsymbol {x}) \\leq 0 \\forall i \\tag {14b}",
+                                    "image_path": "73fc74ce4956caeb1696779598053ad00aea83c0e7007e614796bf497e8d681c.jpg"
+                                }
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+                                    "type": "interline_equation",
+                                    "content": "h _ {j} (\\boldsymbol {x}) = 0 \\forall j \\tag {14c}",
+                                    "image_path": "2524c9f42553603be0c54922fb84e73eac813289b88370af2f9c6f75db717c83.jpg"
+                                }
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+                                    "type": "text",
+                                    "content": "where "
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": " is concave and "
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+                                    "type": "inline_equation",
+                                    "content": "g_{i}"
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+                                    ],
+                                    "type": "text",
+                                    "content": " and "
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+                                    "type": "inline_equation",
+                                    "content": "h_j"
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+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " represent inequality and equality constraints respectively. If "
+                                },
+                                {
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g_{i}"
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+                                {
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+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h_i"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " are affine functions, then any maximizer "
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+                                {
+                                    "bbox": [
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+                                        253,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x^{*}"
+                                },
+                                {
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+                                    ],
+                                    "type": "text",
+                                    "content": " of "
+                                },
+                                {
+                                    "bbox": [
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+                                        253,
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+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " must satisfy the following necessary and sufficient KKT conditions (Ghofogh et al., 2021; Boyd and Vandenberghe, 2004):"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
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+                                            "type": "text",
+                                            "content": "- Stationarity: "
+                                        },
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+                                            "type": "inline_equation",
+                                            "content": "\\mathbf{0} \\in \\partial f(\\mathbf{x}^{*}) - \\sum_{j} \\lambda_{j} \\partial h_{j}(\\mathbf{x}^{*}) - \\sum_{i} \\mu_{i} \\partial g_{i}(\\mathbf{x}^{*})"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 10
+                        },
+                        {
+                            "bbox": [
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+                                            "type": "text",
+                                            "content": "- Primal feasibility: "
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "h_j(\\pmb{x}^*) = 0"
+                                        },
+                                        {
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+                                            ],
+                                            "type": "text",
+                                            "content": " for all "
+                                        },
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+                                                325
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "j"
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+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
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+                                            "type": "inline_equation",
+                                            "content": "g_i(\\pmb{x}^*) \\leq 0"
+                                        },
+                                        {
+                                            "bbox": [
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+                                                312,
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+                                                325
+                                            ],
+                                            "type": "text",
+                                            "content": " for all "
+                                        },
+                                        {
+                                            "bbox": [
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+                                                312,
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+                                                325
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "i"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 11
+                        },
+                        {
+                            "bbox": [
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+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
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+                                            "type": "text",
+                                            "content": "- Dual feasibility: "
+                                        },
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mu_{i} \\geq 0"
+                                        },
+                                        {
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+                                            ],
+                                            "type": "text",
+                                            "content": " for all "
+                                        },
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+                                                328,
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+                                                340
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "i"
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+                            "index": 12
+                        },
+                        {
+                            "bbox": [
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+                                356
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+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
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+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": "- Complementary slackness: "
+                                        },
+                                        {
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mu_{i}g_{i}(\\pmb{x}^{*}) = 0"
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                343,
+                                                343,
+                                                356
+                                            ],
+                                            "type": "text",
+                                            "content": " for all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                343,
+                                                343,
+                                                356
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "i"
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                343,
+                                                343,
+                                                356
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 13
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 1. Assuming player "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "'s utility, "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u_{k}(x_{k}, x_{-k})"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ", is concave in its own strategy "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": ", a strategy in the interior of the simplex is a best response "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "BR_{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " if and only if it has zero projected-gradient² norm:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
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+                                    "type": "interline_equation",
+                                    "content": "B R _ {k} \\in \\left(i n t \\Delta \\cap \\underset {z} {\\arg \\max } u _ {k} (z, x _ {- k}) - u _ {k} (x _ {k}, x _ {- k})\\right) \\iff (B R _ {k} \\in i n t \\Delta) \\wedge (\\| \\Pi_ {T \\Delta} [ \\nabla_ {B R _ {k}} ^ {k} ] \\| = 0).",
+                                    "image_path": "7ab92d2629ee372b73afd20573f3bdef05cc1bb333820e6e9d5fb2ee3ab2f9df.jpg"
+                                }
+                            ]
+                        }
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                        436,
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+                                        449
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Consider the problem of formally computing "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        436,
+                                        506,
+                                        449
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\epsilon_{k}(\\pmb{x}) = \\max_{z\\in int\\Delta}u_{k}(z,x_{-k}) - u_{k}(x_{k},x_{-k})"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
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+                                    "bbox": [
+                                        250,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\max  _ {z \\in \\mathbb {R} ^ {d}} u _ {k} (z, x _ {- k}) - u _ {k} \\left(x _ {k}, x _ {- k}\\right) \\tag {15a}",
+                                    "image_path": "1b252b1f3e2cee098c353f323b2b6423a792a3a959841cf0d5521b15b319acda.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
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+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "s. t. - z _ {k} \\leq 0 \\forall k \\tag {15b}",
+                                    "image_path": "2f5ab4ae95f688751db590f3a252d871eaf553125d2790a38f027ebb4bb53faa.jpg"
+                                }
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+                        }
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+                    "index": 19
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "1 - \\sum_ {k} z _ {k} = 0. \\tag {15c}",
+                                    "image_path": "5492d9c0085956926972a343b0f6539b99cc88748e351eab18270838b89921ff.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                        588
+                                    ],
+                                    "type": "text",
+                                    "content": "Note that the objective is linear (concave) in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        531,
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+                                        588
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        531,
+                                        504,
+                                        588
+                                    ],
+                                    "type": "text",
+                                    "content": " and the constraints are affine, therefore the KKT conditions are necessary and sufficient for optimality. Recall that we assume that the solution "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        531,
+                                        504,
+                                        588
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z^{*}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        531,
+                                        504,
+                                        588
+                                    ],
+                                    "type": "text",
+                                    "content": " lies in the interior of the simplex, i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        531,
+                                        504,
+                                        588
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z_{k}^{*} > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        531,
+                                        504,
+                                        588
+                                    ],
+                                    "type": "text",
+                                    "content": " for each "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        531,
+                                        504,
+                                        588
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        531,
+                                        504,
+                                        588
+                                    ],
+                                    "type": "text",
+                                    "content": ". Also, let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        531,
+                                        504,
+                                        588
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "e_{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        531,
+                                        504,
+                                        588
+                                    ],
+                                    "type": "text",
+                                    "content": " be a onehot vector, i.e., a zeros vector except with a 1 at index "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        531,
+                                        504,
+                                        588
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        531,
+                                        504,
+                                        588
+                                    ],
+                                    "type": "text",
+                                    "content": ". Mapping the KKT conditions onto this problem yields the following:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 21
+                },
+                {
+                    "bbox": [
+                        132,
+                        598,
+                        343,
+                        669
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 26,
+                    "blocks": [
+                        {
+                            "bbox": [
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+                                598,
+                                343,
+                                612
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                        612
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                132,
+                                                598,
+                                                343,
+                                                612
+                                            ],
+                                            "type": "text",
+                                            "content": "- Stationarity: "
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                598,
+                                                343,
+                                                612
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathbf{0} \\in \\partial u_{k}(z^{*}, x_{-k}) + \\lambda \\mathbf{1} + \\sum_{k} \\mu_{k} e_{k}"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 22
+                        },
+                        {
+                            "bbox": [
+                                132,
+                                617,
+                                299,
+                                631
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        132,
+                                        617,
+                                        299,
+                                        631
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                132,
+                                                617,
+                                                299,
+                                                631
+                                            ],
+                                            "type": "text",
+                                            "content": "- Primal feasibility: "
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                617,
+                                                299,
+                                                631
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\sum_{k}z_{k}^{*} = 1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                617,
+                                                299,
+                                                631
+                                            ],
+                                            "type": "text",
+                                            "content": " for all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                617,
+                                                299,
+                                                631
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "k"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 23
+                        },
+                        {
+                            "bbox": [
+                                132,
+                                637,
+                                275,
+                                649
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                        637,
+                                        275,
+                                        649
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                132,
+                                                637,
+                                                275,
+                                                649
+                                            ],
+                                            "type": "text",
+                                            "content": "- Dual feasibility: "
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                637,
+                                                275,
+                                                649
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mu_{i} \\geq 0"
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                637,
+                                                275,
+                                                649
+                                            ],
+                                            "type": "text",
+                                            "content": " for all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                637,
+                                                275,
+                                                649
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "k"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 24
+                        },
+                        {
+                            "bbox": [
+                                132,
+                                656,
+                                337,
+                                669
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        132,
+                                        656,
+                                        337,
+                                        669
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                132,
+                                                656,
+                                                337,
+                                                669
+                                            ],
+                                            "type": "text",
+                                            "content": "- Complementary slackness: "
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                656,
+                                                337,
+                                                669
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "-\\mu_k z_k^* = 0"
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                656,
+                                                337,
+                                                669
+                                            ],
+                                            "type": "text",
+                                            "content": " for all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                656,
+                                                337,
+                                                669
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "k"
+                                        },
+                                        {
+                                            "bbox": [
+                                                132,
+                                                656,
+                                                337,
+                                                669
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 25
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        104,
+                        680,
+                        506,
+                        703
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                680,
+                                506,
+                                703
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        680,
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+                                        703
+                                    ],
+                                    "type": "text",
+                                    "content": "For any point "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        680,
+                                        506,
+                                        703
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z^{*} \\in int\\Delta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        680,
+                                        506,
+                                        703
+                                    ],
+                                    "type": "text",
+                                    "content": ", primal feasibility will be satisfied. Given our assumption that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        680,
+                                        506,
+                                        703
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z_{k}^{*} > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        680,
+                                        506,
+                                        703
+                                    ],
+                                    "type": "text",
+                                    "content": " by complementary slackness and dual feasibility, each "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        680,
+                                        506,
+                                        703
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mu_{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        680,
+                                        506,
+                                        703
+                                    ],
+                                    "type": "text",
+                                    "content": " must be identically zero. This implies"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 27
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
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+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "page_footnote",
+                    "angle": 0,
+                    "lines": [
+                        {
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "^{2}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        711,
+                                        504,
+                                        731
+                                    ],
+                                    "type": "text",
+                                    "content": "Not to be confused with the nonlinear (i.e., introduces bias) projected gradient operator introduced in (Hazan et al., 2017)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 28
+                },
+                {
+                    "bbox": [
+                        299,
+                        750,
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+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "14"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 29
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 13
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        82,
+                        504,
+                        105
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                82,
+                                504,
+                                105
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "text",
+                                    "content": "the stationarity condition can be simplified to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{0} \\in \\partial u_k(z^*, x_{-k}) + \\lambda \\mathbf{1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "text",
+                                    "content": ". Rearranging terms (and repurposing "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\lambda"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "text",
+                                    "content": ") we find that for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z^*"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "text",
+                                    "content": ", there exists a "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\lambda"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "text",
+                                    "content": " such that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        263,
+                        111,
+                        504,
+                        124
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                263,
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+                                124
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        263,
+                                        111,
+                                        504,
+                                        124
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\lambda \\mathbf {1} \\in \\partial u _ {k} \\left(z ^ {*}, x _ {- k}\\right). \\tag {16}",
+                                    "image_path": "30abb3e2a0e43ac867aa37b2bf073cedd904a03ce79685e63b8dd007b5be9c46.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
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+                        157
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                                505,
+                                157
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        130,
+                                        505,
+                                        157
+                                    ],
+                                    "type": "text",
+                                    "content": "Equivalently, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        130,
+                                        505,
+                                        157
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\partial u_{k}(z^{*},x_{-k})\\propto \\mathbf{1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        130,
+                                        505,
+                                        157
+                                    ],
+                                    "type": "text",
+                                    "content": " at "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        130,
+                                        505,
+                                        157
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z^{*}\\in int\\Delta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        130,
+                                        505,
+                                        157
+                                    ],
+                                    "type": "text",
+                                    "content": " . Any vector proportional to the ones vector has zero projected-gradient norm, completing the claim: "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        130,
+                                        505,
+                                        157
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "[I - \\frac{1}{m_k}\\mathbf{11}^\\top ](\\lambda \\mathbf{1}) = \\lambda (\\mathbf{1} - \\frac{m_k}{m_k}\\mathbf{1}) = \\mathbf{0}."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        105,
+                        167,
+                        504,
+                        190
+                    ],
+                    "type": "title",
+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "A.1.2 NORM OF PROJECTED-GRADIENT AND EQUIVALENCE TO NFG EXPLOITABILITY WITH APPROXIMATE BEST RESPONSES"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
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+                                    "bbox": [
+                                        104,
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+                                        222
+                                    ],
+                                    "type": "text",
+                                    "content": "Proposition 1. The loss "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        506,
+                                        222
+                                    ],
+                                    "type": "text",
+                                    "content": " is equivalent to NashConv, but where player "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        506,
+                                        222
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        506,
+                                        222
+                                    ],
+                                    "type": "text",
+                                    "content": "'s best response is approximated by a single step of projected-gradient ascent with step size "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        506,
+                                        222
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\eta_k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        506,
+                                        222
+                                    ],
+                                    "type": "text",
+                                    "content": ": "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        506,
+                                        222
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "aBR_k = x_k + \\eta_k\\Pi_{T\\Delta}(\\nabla_{x_k}^k)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        506,
+                                        222
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        235,
+                        504,
+                        258
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                235,
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+                                258
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Define an approximate best response as the result of a player adjusting their strategy via a projected-gradient ascent step, i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        235,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{aBR}_k = x_k + \\eta_k\\Pi_{T\\Delta}(\\nabla_{x_k}^k)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        235,
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+                                        258
+                                    ],
+                                    "type": "text",
+                                    "content": " for player "
+                                },
+                                {
+                                    "bbox": [
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+                                        235,
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+                                        258
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        235,
+                                        504,
+                                        258
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        105,
+                        262,
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+                        275
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "In a normal form game, player "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "'s utility at this new strategy is"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "u _ {k} \\left(\\mathrm {a B R} _ {k}, x _ {- k}\\right) = \\left(\\nabla_ {x _ {k}} ^ {k}\\right) ^ {\\top} \\left(x _ {k} + \\eta_ {k} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right)\\right) = u _ {k} (\\boldsymbol {x}) + \\eta_ {k} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right). \\tag {17}",
+                                    "image_path": "964ba6c6523afa98873645f29d756b4aba2f86b9e916a7c8e11910f92a8d3d63.jpg"
+                                }
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+                        }
+                    ],
+                    "index": 8
+                },
+                {
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+                    "type": "text",
+                    "angle": 0,
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Therefore, the amount player "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " gains by playing aBR is"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
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+                    "lines": [
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+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\hat {\\epsilon} _ {k} (\\boldsymbol {x}) = u _ {k} \\left(\\mathrm {a B R} _ {k}, x _ {- k}\\right) - u _ {k} (\\boldsymbol {x}) (18a) \\\\ = \\eta_ {k} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) (18b) \\\\ = \\eta_ {k} \\left(\\nabla_ {x _ {k}} ^ {k} - \\frac {1}{m _ {k}} \\left(\\mathbf {1} ^ {\\top} \\nabla_ {x _ {k}} ^ {k}\\right) \\mathbf {1}\\right) ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) (18c) \\\\ = \\eta_ {k} \\| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) \\| ^ {2} (18d) \\\\ \\end{array}",
+                                    "image_path": "063007bf9c0d46195b60a0b0144c03bb587f59111189017453f701dd6e5960ac.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
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+                    "type": "text",
+                    "angle": 0,
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "where the third equality follows from the fact that the projected-gradient, "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\Pi_{T\\Delta}(\\nabla_{x_k}^k)"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ", is orthogonal to the ones vector."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
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+                {
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+                    "type": "title",
+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "A.1.3 CONNECTION TO TRUE EXPLOITABILITY"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Lemma 2. The amount a player can gain by deviating is upper bounded by a quantity proportional to the norm of the projected-gradient:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
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+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\epsilon_ {k} (\\boldsymbol {x}) \\leq \\sqrt {2} \\left\\| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) \\right\\|. \\tag {19}",
+                                    "image_path": "488359889359bdea4665230a502809c79bc48bb1da99ebcc0ba8b11de4f43d30.jpg"
+                                }
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Proof. Let "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " be any point on the simplex. Then by concavity of "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "u_{k}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " with respect to "
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+                                    "type": "inline_equation",
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+                                },
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ","
+                                }
+                            ]
+                        }
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+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} u _ {k} (z, x _ {- k}) - u _ {k} (\\boldsymbol {x}) \\leq \\left(\\nabla_ {x _ {k}} ^ {k}\\right) ^ {\\top} (z - x _ {k}) (20a) \\\\ = \\left(\\nabla_ {x _ {k}} ^ {k}\\right) ^ {\\top} \\left(z - x _ {k}\\right) - \\frac {1}{m _ {k}} \\left(\\mathbf {1} ^ {\\top} \\nabla_ {x _ {k}} ^ {k}\\right) \\overbrace {\\mathbf {1} ^ {\\top} \\left(z - x _ {k}\\right)} ^ {1 - 1 = 0} (20b) \\\\ = \\left(\\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right)\\right) ^ {\\top} \\underbrace {(z - x _ {k})} _ {\\text {D i a m} (\\Delta) \\leq \\sqrt {2}} (20c) \\\\ \\leq \\sqrt {2} \\| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) \\|. \\quad (\\text {C a u c h y - S c h w a r z}) (20d) \\\\ \\end{array}",
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+                                    "type": "text",
+                                    "content": "Continuing, we can prove a bound on "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "\\epsilon"
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+                                    "type": "text",
+                                    "content": " in terms of the projected-gradient loss:"
+                                }
+                            ]
+                        }
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+                    "index": 18
+                },
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+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Lemma 3. The exploitability, "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "\\epsilon"
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+                                    "type": "text",
+                                    "content": ", of a joint strategy "
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+                                    "type": "inline_equation",
+                                    "content": "x"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ", is upper bounded as a function of our proposed loss:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
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+                                    "type": "interline_equation",
+                                    "content": "\\epsilon \\leq \\sqrt {\\frac {2 n}{\\min  _ {k} \\eta_ {k}}} \\sqrt {\\mathcal {L} (\\boldsymbol {x})}. \\tag {21}",
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+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "type": "text",
+                                    "content": "15"
+                                }
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+                }
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+                                    "type": "text",
+                                    "content": "Proof."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
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+                                    "content": "\\begin{array}{l} \\epsilon = \\max  _ {k} \\max  _ {z} u _ {k} (z, x _ {- k}) - u _ {k} (\\boldsymbol {x}) (22a) \\\\ \\leq \\sum_ {k} \\max  _ {z} u _ {k} (z, x _ {- k}) - u _ {k} (\\boldsymbol {x}) (22b) \\\\ \\leq \\sum_ {k} \\sqrt {2} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) \\right| \\right| _ {2} \\quad (\\text {L e m m a 2}) (22c) \\\\ = \\sqrt {2} \\left|\\left|\\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {1}} ^ {1}\\right)\\right|\\right| _ {2}, \\dots , \\left|\\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {n}} ^ {n}\\right)\\right|\\right| _ {2} \\right|\\left. \\right| _ {1} (22d) \\\\ \\leq \\sqrt {2 n} \\left|\\left|\\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {1}} ^ {1}\\right)\\right|\\right| _ {2}, \\dots , \\left|\\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {n}} ^ {n}\\right)\\right|\\right| _ {2} \\right|\\left. \\right| _ {2} (22e) \\\\ = \\sqrt {2 n} \\sqrt {\\sum_ {k} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) \\right| \\right| _ {2} ^ {2}} (22f) \\\\ = \\sqrt {2 n} \\sqrt {\\sum_ {k} \\left(\\frac {1}{\\eta_ {k}}\\right) \\eta_ {k} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) \\right| \\right| _ {2} ^ {2}} (22g) \\\\ \\leq \\sqrt {\\frac {2 n}{\\min _ {k} \\eta_ {k}}} \\sqrt {\\sum_ {k} \\eta_ {k} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) \\right| \\right| _ {2} ^ {2}} (22h) \\\\ = \\sqrt {\\frac {2 n}{\\min _ {k} \\eta_ {k}}} \\sqrt {\\mathcal {L} (\\boldsymbol {x})}. (22i) \\\\ \\end{array}",
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+                                }
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+                    "index": 2
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+                    "type": "image",
+                    "bbox": [
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+                    ],
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+                                    "bbox": [
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+                            "index": 3,
+                            "angle": 0,
+                            "type": "image_body"
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+                    ],
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+                },
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 4. The entropy regularized exploitability, "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        376,
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+                                        399
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\epsilon_{QRE}"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        376,
+                                        504,
+                                        399
+                                    ],
+                                    "type": "text",
+                                    "content": ", of a joint strategy "
+                                },
+                                {
+                                    "bbox": [
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+                                        376,
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+                                        399
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        376,
+                                        504,
+                                        399
+                                    ],
+                                    "type": "text",
+                                    "content": ", is upper bounded as a function of our proposed loss:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
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+                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\epsilon_ {Q R E} \\leq \\sqrt {\\frac {2 n}{\\min _ {k} \\eta_ {k}}} \\sqrt {\\mathcal {L} ^ {\\tau} (\\boldsymbol {x})}. \\tag {23}",
+                                    "image_path": "29f589b0956c7a82e248645b0dde4b33f90c4b47c131a8faf4a06ad76f4217a8.jpg"
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+                    "index": 5
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Proof. Recall that "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "u_{k}^{\\tau}(x_{k}, x_{-k})"
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": " is also concave with respect to "
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+                                    "type": "inline_equation",
+                                    "content": "x_{k}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ". Then"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\epsilon_ {Q R E} = \\max  _ {k} \\max  _ {z} u _ {k} ^ {\\tau} (z, x _ {- k}) - u _ {k} ^ {\\tau} (\\boldsymbol {x}) (24a) \\\\ \\leq \\sum_ {k} \\max  _ {z} u _ {k} ^ {\\tau} (z, x _ {- k}) - u _ {k} ^ {\\tau} (\\boldsymbol {x}) (24b) \\\\ \\leq \\sum_ {k} \\sqrt {2} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right| \\right| _ {2} \\quad (\\text {L e m m a 2}) (24c) \\\\ = \\sqrt {2} \\left| \\left| | \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {1}} ^ {1 \\tau}\\right) | | _ {2}, \\dots , | | \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {n}} ^ {n \\tau}\\right) | | _ {2} \\right| \\right| _ {1} (24d) \\\\ \\leq \\sqrt {2 n} \\left| \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {1}} ^ {1 \\tau}\\right) \\right| \\right| _ {2}, \\dots , \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {n}} ^ {n \\tau}\\right) \\right| \\right| _ {2} \\right| _ {2} (24e) \\\\ = \\sqrt {2 n} \\sqrt {\\sum_ {k} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right| \\right| _ {2} ^ {2}} (24f) \\\\ \\leq \\sqrt {2 n} \\sqrt {\\sum_ {k} \\left(\\frac {1}{\\eta_ {k}}\\right) \\eta_ {k} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right| \\right| _ {2} ^ {2}} (24g) \\\\ \\leq \\sqrt {\\frac {2 n}{\\min _ {k} \\eta_ {k}}} \\sqrt {\\sum_ {k} \\eta_ {k} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right| \\right| _ {2} ^ {2}} (24h) \\\\ = \\sqrt {\\frac {2 n}{\\min _ {k} \\eta_ {k}}} \\sqrt {\\mathcal {L} ^ {\\tau} (\\boldsymbol {x})}. (24i) \\\\ \\end{array}",
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+                                }
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
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+                                    "type": "text",
+                                    "content": "16"
+                                }
+                            ]
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+                }
+            ],
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+                                    "type": "text",
+                                    "content": "A.1.4 UNBIASED ESTIMATION"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
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+                                126
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+                                    "bbox": [
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+                                        126
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 5. An unbiased estimate of "
+                                },
+                                {
+                                    "bbox": [
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+                                        102,
+                                        506,
+                                        126
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}(\\pmb{x})"
+                                },
+                                {
+                                    "bbox": [
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+                                        102,
+                                        506,
+                                        126
+                                    ],
+                                    "type": "text",
+                                    "content": " can be obtained by drawing two samples (pure strategies) from each players' mixed strategy and observing payoffs."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
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+                        165
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                165
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                        136,
+                                        505,
+                                        165
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Define "
+                                },
+                                {
+                                    "bbox": [
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+                                        136,
+                                        505,
+                                        165
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\nabla}_{x_k}^{k(1)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        136,
+                                        505,
+                                        165
+                                    ],
+                                    "type": "text",
+                                    "content": " as a random, unbiased gradient estimate (see Table 2). Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        136,
+                                        505,
+                                        165
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\nabla}_{x_k}^{k(2)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        136,
+                                        505,
+                                        165
+                                    ],
+                                    "type": "text",
+                                    "content": " be independent and distributed identically to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        136,
+                                        505,
+                                        165
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\nabla}_{x_k}^{k(1)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        136,
+                                        505,
+                                        165
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        123,
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+                                        340
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\mathbb {E} _ {\\boldsymbol {a} ^ {(1)} \\sim \\boldsymbol {x}, \\boldsymbol {a} ^ {(2)} \\sim \\boldsymbol {x}} [ \\sum_ {k} \\eta_ {k} (\\underbrace {\\hat {\\nabla} _ {x _ {k}} ^ {k (1)} - \\frac {1}{m _ {k}} \\left(\\boldsymbol {1} ^ {\\top} \\hat {\\nabla} _ {x _ {k}} ^ {k (1)}\\right) \\boldsymbol {1}} _ {\\text {p r o j e c t e d - g r a d i e n t 1}}) ^ {\\top} (\\underbrace {\\hat {\\nabla} _ {x _ {k}} ^ {k (2)} - \\frac {1}{m _ {k}} \\left(\\boldsymbol {1} ^ {\\top} \\hat {\\nabla} _ {x _ {k}} ^ {k (2)}\\right) \\boldsymbol {1}} _ {\\text {p r o j e c t e d - g r a d i e n t 2}}) ] (25a) \\\\ = \\sum_ {k} \\eta_ {k} \\mathbb {E} _ {\\boldsymbol {a} ^ {(1)} \\sim \\boldsymbol {x}, \\boldsymbol {a} ^ {(2)} \\sim \\boldsymbol {x}} [ (\\hat {\\nabla} _ {x _ {k}} ^ {k (1)} - \\frac {1}{m _ {k}} (\\boldsymbol {1} ^ {\\top} \\hat {\\nabla} _ {x _ {k}} ^ {k (1)}) \\boldsymbol {1}) ^ {\\top} (\\hat {\\nabla} _ {x _ {k}} ^ {k (2)} - \\frac {1}{m _ {k}} (\\boldsymbol {1} ^ {\\top} \\hat {\\nabla} _ {x _ {k}} ^ {k (2)}) \\boldsymbol {1}) ] (25b) \\\\ = \\sum_ {k} \\eta_ {k} \\mathbb {E} _ {\\boldsymbol {a} ^ {(1)} \\sim \\boldsymbol {x}} \\left[ \\left(\\hat {\\nabla} _ {x _ {k}} ^ {k (1)} - \\frac {1}{m _ {k}} \\left(\\boldsymbol {1} ^ {\\top} \\hat {\\nabla} _ {x _ {k}} ^ {k (1)}\\right) \\boldsymbol {1}\\right) \\right] ^ {\\top} \\mathbb {E} _ {\\boldsymbol {a} ^ {(2)} \\sim \\boldsymbol {x}} \\left[ \\left(\\hat {\\nabla} _ {x _ {k}} ^ {k (2)} - \\frac {1}{m _ {k}} \\left(\\boldsymbol {1} ^ {\\top} \\hat {\\nabla} _ {x _ {k}} ^ {k (2)}\\right) \\boldsymbol {1}\\right) \\right] (25c) \\\\ = \\sum_ {k} \\eta_ {k} \\left(\\nabla_ {x _ {k}} ^ {k} - \\frac {1}{m _ {k}} \\left(\\mathbf {1} ^ {\\top} \\nabla_ {x _ {k}} ^ {k}\\right) \\mathbf {1}\\right) ^ {\\top} \\left(\\nabla_ {x _ {k}} ^ {k} - \\frac {1}{m _ {k}} \\left(\\mathbf {1} ^ {\\top} \\nabla_ {x _ {k}} ^ {k}\\right) \\mathbf {1}\\right) (25d) \\\\ = \\sum_ {k} \\eta_ {k} \\left\\| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k}\\right) \\right\\| ^ {2} (25e) \\\\ = \\mathcal {L} (\\boldsymbol {x}) (25f) \\\\ \\end{array}",
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+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "where the first equality follows from linearity of expectation, the second from independence of random variables, and the third from "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{E}_{\\pmb{a}^{(p)}\\sim \\pmb{x}}[\\hat{\\nabla}_{x_k}^{k(p)}] = \\nabla_{x_k}^k"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ", i.e., "
+                                },
+                                {
+                                    "bbox": [
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+                                        343,
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+                                        392
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\nabla}_{x_k}^{k(p)}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " is an unbiased estimator of player "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "'s gradient. Therefore, equation (25a) comprises an unbiased estimate of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        343,
+                                        506,
+                                        392
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}(\\pmb{x})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        343,
+                                        506,
+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": " proving the claim."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "type": "image",
+                    "bbox": [
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+                    ],
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+                                        }
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+                                }
+                            ],
+                            "index": 6,
+                            "angle": 0,
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+                                {
+                                    "bbox": [
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+                                        504,
+                                        437
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 6. The loss formed as the sum of the squared norms of the projected-gradients, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        414,
+                                        504,
+                                        437
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}^{\\tau}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                        437
+                                    ],
+                                    "type": "text",
+                                    "content": ", can be decomposed into three terms as follows:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
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+                    ],
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+                                    "bbox": [
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+                                        440,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) = \\underbrace {\\sum_ {k} \\eta_ {k} x _ {q} ^ {\\top} B _ {k q} ^ {\\top} B _ {k q} x _ {q}} _ {(A)} + \\underbrace {2 \\sum_ {k} \\eta_ {k} E _ {k} ^ {\\top} B _ {k q} x _ {q}} _ {(B)} + \\underbrace {\\sum_ {k} \\eta_ {k} E _ {k} ^ {\\top} E _ {k}} _ {(C)} \\tag {26}",
+                                    "image_path": "45ac74511178f6e0d7225bd9ea6edddd41f004a73e9539f92ed2e58cd47023d5.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
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+                        498
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "q"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        487,
+                                        247,
+                                        498
+                                    ],
+                                    "type": "text",
+                                    "content": " is any player other than "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
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+                                        498
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                        537
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        509,
+                                        504,
+                                        537
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S^{\\tau} = -\\tau \\sum_{l} x_{kl} \\ln(x_{kl})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        509,
+                                        504,
+                                        537
+                                    ],
+                                    "type": "text",
+                                    "content": " so that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        509,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\frac{\\partial S^{\\tau}}{\\partial x_k} = -\\tau (\\ln(x_k) + 1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        509,
+                                        504,
+                                        537
+                                    ],
+                                    "type": "text",
+                                    "content": ". Note that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        509,
+                                        504,
+                                        537
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Pi_{T\\Delta}[\\frac{\\partial S^{\\tau}}{\\partial x_k}] = -\\tau \\Pi_{T\\Delta}[\\ln(x_k)]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        509,
+                                        504,
+                                        537
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
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+                        540,
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) = \\sum_ {k} \\eta_ {k} \\left(\\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right)\\right) ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) (27a) \\\\ = \\sum_ {k} \\eta_ {k} \\left[ H _ {k q} ^ {k} x _ {q} + \\frac {\\partial S ^ {\\tau}}{\\partial x _ {k}} \\right] ^ {\\top} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] \\left[ H _ {k q} ^ {k} x _ {q} + \\frac {\\partial S ^ {\\tau}}{\\partial x _ {k}} \\right] (27b) \\\\ = \\sum_ {k} \\eta_ {k} \\left(x _ {q} ^ {\\top} \\left[ H _ {k q} ^ {k} \\right] ^ {\\top} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] ^ {2} \\left[ H _ {k q} ^ {k} \\right] x _ {q} + 2 \\left[ \\frac {\\partial S ^ {\\tau}}{\\partial x _ {k}} \\right] ^ {\\top} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] ^ {2} \\left[ H _ {k q} ^ {k} x _ {q} \\right] \\quad (2 7 c) \\right. \\\\ \\left. + \\left[ \\frac {\\partial S ^ {\\tau}}{\\partial x _ {k}} \\right] ^ {\\top} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] ^ {2} \\left[ \\frac {\\partial S ^ {\\tau}}{\\partial x _ {k}} \\right]\\right) (27d) \\\\ = \\underbrace {\\sum_ {k} \\eta_ {k} x _ {q} ^ {\\top} B _ {k q} ^ {\\top} B _ {k q} x _ {q}} _ {(A)} + \\underbrace {2 \\sum_ {k} \\eta_ {k} E _ {k} ^ {\\top} B _ {k q} x _ {q}} _ {(B)} + \\underbrace {\\sum_ {k} \\eta_ {k} E _ {k} ^ {\\top} E _ {k}} _ {(C)} (27e) \\\\ \\end{array}",
+                                    "image_path": "ab42a8b18461938fafc9d489a3af1843ad653e9c5ba6f15745ffea52fc36ecb1.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                {
+                                    "bbox": [
+                                        104,
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+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        700,
+                                        471,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "B_{kq} = [I - \\frac{1}{m_k}\\mathbf{11}^\\top ]H_{kq}^k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        700,
+                                        471,
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+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        700,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "E_{k} = [I - \\frac{1}{m_{k}}\\mathbf{11}^{\\top}][\\frac{\\partial S^{\\tau}}{\\partial x_{k}} ] = -\\tau [I - \\frac{1}{m_{k}}\\mathbf{11}^{\\top}]\\ln (x_{k})"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                }
+            ],
+            "discarded_blocks": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
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+                                    "type": "text",
+                                    "content": "17"
+                                }
+                            ]
+                        }
+                    ],
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+                }
+            ],
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+            ],
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+        {
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+                                    "type": "text",
+                                    "content": "A.1.5 BOUND ON LOSS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
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+                    "angle": 0,
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+                                {
+                                    "bbox": [
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+                                        506,
+                                        137
+                                    ],
+                                    "type": "text",
+                                    "content": "By Proposition 1, Equation (18d), we can also rewrite this loss as a weighted sum of 2-norms, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        506,
+                                        137
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}(\\pmb{x}) = \\sum_{k}\\eta_{k}||\\nabla_{x_{k}}^{k} - \\mu_{k}||_{2}^{2}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
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+                                        137
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        506,
+                                        137
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mu_{k} = \\frac{1}{m_{k}} (\\mathbf{1}^{\\top}\\nabla_{x_{k}}^{k})\\in [0,1]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        506,
+                                        137
+                                    ],
+                                    "type": "text",
+                                    "content": " for brevity. This will allow us to more easily analyze our loss."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                        504,
+                                        166
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 7. Assume payoffs are bounded in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        138,
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+                                        166
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "[0,1]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        138,
+                                        504,
+                                        166
+                                    ],
+                                    "type": "text",
+                                    "content": ", then setting "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        138,
+                                        504,
+                                        166
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\eta_k \\leq \\frac{4}{nm_k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        138,
+                                        504,
+                                        166
+                                    ],
+                                    "type": "text",
+                                    "content": " or "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        138,
+                                        504,
+                                        166
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\eta_k \\leq \\frac{4}{nm}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        138,
+                                        504,
+                                        166
+                                    ],
+                                    "type": "text",
+                                    "content": " or "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        138,
+                                        504,
+                                        166
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sum_k \\eta_k \\leq \\frac{4}{m}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        138,
+                                        504,
+                                        166
+                                    ],
+                                    "type": "text",
+                                    "content": " ensures "
+                                },
+                                {
+                                    "bbox": [
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+                                        138,
+                                        504,
+                                        166
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "0 \\leq \\mathcal{L}(x) \\leq 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        138,
+                                        504,
+                                        166
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        138,
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+                                        166
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x \\in \\mathcal{X}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        138,
+                                        504,
+                                        166
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        105,
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+                        190
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Proof."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} 0 \\leq \\mathcal {L} (\\boldsymbol {x}) = \\sum_ {k} \\eta_ {k} \\| \\nabla_ {x _ {k}} ^ {k} - \\mu_ {k} \\| _ {2} ^ {2} (28a) \\\\ = \\sum_ {k} \\eta_ {k} m _ {k} \\left[ \\frac {1}{m _ {k}} \\sum_ {l} \\left(\\nabla_ {x _ {k l}} ^ {k} - \\mu_ {k}\\right) ^ {2} \\right] (28b) \\\\ = \\sum_ {k} \\eta_ {k} m _ {k} \\operatorname {V a r} \\left[ \\nabla_ {x _ {k}} ^ {k} \\right] (28c) \\\\ \\leq \\frac {1}{4} \\sum_ {k} \\eta_ {k} m _ {k} (28d) \\\\ \\leq \\frac {1}{4} \\left(\\max  _ {k} \\eta_ {k}\\right) \\left(\\sum_ {k} m _ {k}\\right) (28e) \\\\ = \\frac {1}{4} \\left(\\max  _ {k} \\eta_ {k}\\right) n \\bar {m} \\leq 1 (28f) \\\\ \\Rightarrow \\left(\\max  _ {k} \\eta_ {k}\\right) \\leq \\frac {4}{n \\bar {m}} (28g) \\\\ \\end{array}",
+                                    "image_path": "6b77fd59f5abdaf1d4625355eabb040036db6e717fa52308f1c386ba63ad0cef.jpg"
+                                }
+                            ]
+                        }
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+                                        393,
+                                        506,
+                                        418
+                                    ],
+                                    "type": "text",
+                                    "content": "where the first inequality follows from Popoviciu's inequality (Popoviciu, 1935): the variance of a bounded random variable "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        393,
+                                        506,
+                                        418
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "X"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        393,
+                                        506,
+                                        418
+                                    ],
+                                    "type": "text",
+                                    "content": " is upper bounded by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        393,
+                                        506,
+                                        418
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Var[X] \\leq \\frac{1}{4} (\\max_{X} - \\min_{X})^{2}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        393,
+                                        506,
+                                        418
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        105,
+                        429,
+                        299,
+                        441
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                429,
+                                299,
+                                441
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        429,
+                                        299,
+                                        441
+                                    ],
+                                    "type": "text",
+                                    "content": "Next, we establish the following useful lemmas."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        104,
+                        443,
+                        504,
+                        468
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                443,
+                                504,
+                                468
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        443,
+                                        504,
+                                        468
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 8. The matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        443,
+                                        504,
+                                        468
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "I - \\frac{1}{m_k} \\mathbf{11}^\\top"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        443,
+                                        504,
+                                        468
+                                    ],
+                                    "type": "text",
+                                    "content": " is a projection matrix and therefore idempotent. It is also symmetric, which implies it is its own square root."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        105,
+                        481,
+                        133,
+                        493
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                481,
+                                133,
+                                493
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        481,
+                                        133,
+                                        493
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        174,
+                        498,
+                        504,
+                        575
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                174,
+                                498,
+                                504,
+                                575
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        174,
+                                        498,
+                                        504,
+                                        575
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] ^ {\\top} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] = I - \\frac {2}{m _ {k}} \\mathbf {1 1} ^ {\\top} + \\frac {1}{m _ {k} ^ {2}} \\mathbf {1} \\left(\\mathbf {1} ^ {\\top} \\mathbf {1}\\right) \\mathbf {1} ^ {\\top} (29a) \\\\ = I - \\frac {2}{m _ {k}} \\mathbf {1 1} ^ {\\top} + \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} (29b) \\\\ = \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right]. (29c) \\\\ \\end{array}",
+                                    "image_path": "62123598b7f422d270c882f921936b20483914b619d1d6ca866ee20e825c0c2b.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        598,
+                        506,
+                        634
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                598,
+                                506,
+                                634
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        506,
+                                        634
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 9. The product "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        506,
+                                        634
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A[I_m - \\frac{1}{m}\\mathbf{1}_m\\mathbf{1}_m^\\top ]^p B"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        506,
+                                        634
+                                    ],
+                                    "type": "text",
+                                    "content": " for any integer "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        506,
+                                        634
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        506,
+                                        634
+                                    ],
+                                    "type": "text",
+                                    "content": " has entries whose absolute value is bounded by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        506,
+                                        634
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\frac{m}{4} (A_{\\mathrm{max}} - A_{\\mathrm{min}})(B_{\\mathrm{max}} - B_{\\mathrm{min}})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        506,
+                                        634
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        506,
+                                        634
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A_{\\mathrm{min}}, A_{\\mathrm{max}}, B_{\\mathrm{min}}, B_{\\mathrm{max}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        506,
+                                        634
+                                    ],
+                                    "type": "text",
+                                    "content": " represent the minima and maxima of the matrices respectively."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        104,
+                        644,
+                        504,
+                        659
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                644,
+                                504,
+                                659
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        644,
+                                        504,
+                                        659
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. The matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        644,
+                                        504,
+                                        659
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "[I - \\frac{1}{m}\\mathbf{11}^{\\top}]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        644,
+                                        504,
+                                        659
+                                    ],
+                                    "type": "text",
+                                    "content": " is idempotent (Lemma 8) so we can rewrite the product for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        644,
+                                        504,
+                                        659
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        644,
+                                        504,
+                                        659
+                                    ],
+                                    "type": "text",
+                                    "content": " as"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        242,
+                        664,
+                        504,
+                        687
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                242,
+                                664,
+                                504,
+                                687
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        242,
+                                        664,
+                                        504,
+                                        687
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "A \\left[ I - \\frac {1}{m} \\mathbf {1 1} ^ {\\top} \\right] \\left[ I - \\frac {1}{m} \\mathbf {1 1} ^ {\\top} \\right] B. \\tag {30}",
+                                    "image_path": "c710474f46bfc10d6f30ac3539fff13e51b5698c8582384628c3eb6ca1d95c15.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        104,
+                        693,
+                        505,
+                        733
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                693,
+                                505,
+                                733
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        693,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "The matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        693,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "[I - \\frac{1}{m}\\mathbf{11}^{\\top}]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        693,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " has the property that it removes the mean from every row of a matrix when right multiplied against it, i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        693,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A[I - \\frac{1}{m}\\mathbf{11}^{\\top}]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        693,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " removes the means from the rows of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        693,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        693,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ". Similarly, left multiplying it removes the means from the columns. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        693,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{A}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        693,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        693,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{B}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        693,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " represent these mean-centered"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
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+                                38
+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
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+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                {
+                                    "bbox": [
+                                        299,
+                                        750,
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+                                    ],
+                                    "type": "text",
+                                    "content": "18"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 17
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        82,
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": "results respectively. The absolute value of the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "ij"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": "th entry in the resulting product can then be recognized as"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        170,
+                        110,
+                        505,
+                        169
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                170,
+                                110,
+                                505,
+                                169
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        170,
+                                        110,
+                                        505,
+                                        169
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\left| \\sum_ {k} \\tilde {A} _ {i k} \\tilde {B} _ {k j} \\right| = \\left| \\sum_ {k} \\left(A _ {i k} - \\frac {1}{m} \\sum_ {k ^ {\\prime}} A _ {i k ^ {\\prime}}\\right) \\left(B _ {k j} - \\frac {1}{m} \\sum_ {k ^ {\\prime}} B _ {k ^ {\\prime} j}\\right) \\right| (31a) \\\\ = \\left| m \\cdot \\operatorname {C o r r} \\left(A _ {i,}, B _ {. j}\\right) \\cdot \\sigma_ {A _ {i,}} \\sigma_ {B _ {. j}} \\right| (31b) \\\\ \\leq m \\sigma_ {A _ {i,}} \\sigma_ {B _ {\\cdot , j}}. (31c) \\\\ \\end{array}",
+                                    "image_path": "9ae61ee02700317bc43a1e808e1e3ae3155a6e4a167d1342ae896bc71a5e7d26.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        105,
+                        172,
+                        506,
+                        218
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                172,
+                                506,
+                                218
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        172,
+                                        506,
+                                        218
+                                    ],
+                                    "type": "text",
+                                    "content": "By Popoviciu's inequality (Popoviciu, 1935), we know the variance of a bounded random variable "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        172,
+                                        506,
+                                        218
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "X"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        172,
+                                        506,
+                                        218
+                                    ],
+                                    "type": "text",
+                                    "content": " is upper bounded by "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        172,
+                                        506,
+                                        218
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Var[X] \\leq \\frac{1}{4} (\\max_{X} - \\min_{X})^{2}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        172,
+                                        506,
+                                        218
+                                    ],
+                                    "type": "text",
+                                    "content": ". Hence its standard deviation is bounded by "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        172,
+                                        506,
+                                        218
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Std[X] \\leq \\frac{1}{2} (\\max_{X} - \\min_{X})"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        172,
+                                        506,
+                                        218
+                                    ],
+                                    "type": "text",
+                                    "content": ". Plugging these bounds for "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        172,
+                                        506,
+                                        218
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        172,
+                                        506,
+                                        218
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        172,
+                                        506,
+                                        218
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "B"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        172,
+                                        506,
+                                        218
+                                    ],
+                                    "type": "text",
+                                    "content": " into equation (31c) completes the claim."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        225,
+                        331,
+                        237
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                225,
+                                331,
+                                237
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        225,
+                                        331,
+                                        237
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 10. Assume payoffs are bounded in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        225,
+                                        331,
+                                        237
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "[0,1]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        225,
+                                        331,
+                                        237
+                                    ],
+                                    "type": "text",
+                                    "content": ", then"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        208,
+                        241,
+                        505,
+                        266
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                208,
+                                241,
+                                505,
+                                266
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        208,
+                                        241,
+                                        505,
+                                        266
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left| \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) \\right| \\leq \\frac {1}{4} \\left(\\max  _ {k} \\eta_ {k}\\right) n \\bar {m} \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) ^ {2} \\tag {32}",
+                                    "image_path": "f53eb4deb20816347679a33142685c48ff88908f9813a4ced9cc31ea147b3c3c.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
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+                            "bbox": [
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+                                    "bbox": [
+                                        103,
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+                                        282
+                                    ],
+                                    "type": "text",
+                                    "content": "for any "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        270,
+                                        257,
+                                        282
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        270,
+                                        257,
+                                        282
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        270,
+                                        257,
+                                        282
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_{kl} \\geq x_{\\min} \\forall k, l"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        270,
+                                        257,
+                                        282
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        294,
+                        330,
+                        306
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                294,
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+                                306
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        294,
+                                        330,
+                                        306
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. First note that for payoffs in [0, 1], the entries in"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) = \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k} - \\tau (\\ln (x _ {k}) + 1)\\right) = \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k} - \\tau \\ln (x _ {k})\\right) \\tag {33}",
+                                    "image_path": "4f03ecff6c3a744b08bd7a66102e1858e1a2718b98f80a5e5ece84ae585d8bbb.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "are bounded within "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        356
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "[0, \\tau \\ln(\\frac{1}{x_{\\min}}) + 1]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        356
+                                    ],
+                                    "type": "text",
+                                    "content": " with a range of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        356
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau \\ln(\\frac{1}{x_{\\min}}) + 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        356
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then starting from the definition of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        356
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}^{\\tau}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        356
+                                    ],
+                                    "type": "text",
+                                    "content": " and applying Lemma 9, we find"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\left| \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) \\right| = \\left| \\sum_ {k} \\eta_ {k} \\left\\| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right\\| ^ {2} \\right| (34a) \\\\ \\leq \\frac {1}{4} \\sum_ {k} \\eta_ {k} m _ {k} \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) ^ {2} (34b) \\\\ \\leq \\frac {1}{4} \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) ^ {2} \\left(\\max  _ {k} \\eta_ {k}\\right) \\sum_ {k} m _ {k} (34c) \\\\ = \\frac {1}{4} \\left(\\max  _ {k} \\eta_ {k}\\right) n \\bar {m} \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\min }}\\right) + 1\\right) ^ {2}. (34d) \\\\ \\end{array}",
+                                    "image_path": "77ac6f49346db27f016f5df07cdd4521b35691d9b3cd41bfa01655183a1e2d52.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "type": "image",
+                    "bbox": [
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+                        487
+                    ],
+                    "blocks": [
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+                            "bbox": [
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+                            "lines": [
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+                                    "bbox": [
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+                                    "spans": [
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+                                            "type": "image",
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+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 11,
+                            "angle": 0,
+                            "type": "image_body"
+                        }
+                    ],
+                    "index": 11
+                },
+                {
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+                    ],
+                    "type": "title",
+                    "angle": 0,
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+                                    ],
+                                    "type": "text",
+                                    "content": "A.2 QRES APPROXIMATE NES AT LOW TEMPERATURE"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                        548
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 11. Setting "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        521,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau = \\ln (1 / p)^{-1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        521,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "text",
+                                    "content": " with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        521,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p\\in (0,1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        521,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "text",
+                                    "content": " and payoffs in [0,1] ensures that any QRE contains probabilities greater than "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        521,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\frac{p}{\\max_k m_k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        521,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        104,
+                        559,
+                        505,
+                        601
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                559,
+                                505,
+                                601
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        505,
+                                        601
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        505,
+                                        601
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m^{*} = \\max_{k} m_{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        505,
+                                        601
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        505,
+                                        601
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\nabla_{x_{k}}^{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        505,
+                                        601
+                                    ],
+                                    "type": "text",
+                                    "content": " be player "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        505,
+                                        601
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        505,
+                                        601
+                                    ],
+                                    "type": "text",
+                                    "content": "'s gradient. Recall that a QRE("
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        505,
+                                        601
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        505,
+                                        601
+                                    ],
+                                    "type": "text",
+                                    "content": "), specifically a logit equilibrium, satisfies the fixed point equation "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        505,
+                                        601
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_{k} = \\text{softmax}\\left(\\frac{\\nabla_{x_{k}}^{k}}{\\tau}\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        505,
+                                        601
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        505,
+                                        601
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        505,
+                                        601
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then the probability for any action in a QRE strategy profile, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        505,
+                                        601
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        505,
+                                        601
+                                    ],
+                                    "type": "text",
+                                    "content": ", is lower bounded as"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
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+                        704
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                704
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        181,
+                                        605,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} x _ {k l} \\geq \\min  _ {k} \\min  _ {l} \\left\\{x _ {k l} \\right\\} \\geq \\min  _ {\\boldsymbol {x} ^ {\\prime} \\mid \\boldsymbol {x} ^ {\\prime} \\text {i s Q R E}} \\min  _ {k} \\min  _ {l} \\left\\{x _ {k l} ^ {\\prime} \\right\\} (35a) \\\\ = \\min  _ {\\nabla_ {x _ {k}} ^ {k}} \\min  _ {k} \\min  _ {l} \\left[ \\operatorname {s o f t m a x} \\left(\\frac {\\nabla_ {x _ {k}} ^ {k}}{\\tau}\\right) \\right] _ {l} (35b) \\\\ = \\frac {e ^ {0}}{\\left(m ^ {*} - 1\\right) e ^ {\\frac {1}{\\tau}} + e ^ {0}} (35c) \\\\ \\stackrel {\\text {d e f}} {=} x _ {\\min } (35d) \\\\ \\end{array}",
+                                    "image_path": "272f74babe5f1cf9251923434e76b9cb92ed0f4fcbe1e4ac95c1f5354d08f79b.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                        732
+                                    ],
+                                    "type": "text",
+                                    "content": "where equation (35c) follows from minimizing the numerator and maximizing the denominator of the softmax formula subject to the payoff constraints."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
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+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
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+                                    "bbox": [
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+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
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+                                    "type": "text",
+                                    "content": "19"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                }
+            ],
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+            ],
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+        },
+        {
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+                    ],
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+                    "lines": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Rearranging terms, we find"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "e ^ {\\frac {1}{\\tau}} = \\frac {1}{m ^ {*} - 1} \\left(\\frac {1}{x _ {\\min }} - 1\\right) \\Rightarrow \\tau = \\frac {1}{\\ln \\left(\\frac {1}{m ^ {*} - 1} \\left(\\frac {1}{x _ {\\min }} - 1\\right)\\right)}. \\tag {36}",
+                                    "image_path": "e6213b15967904dfa5b8fe9231e8e2ab4f9817a5f01cd921ccb2500cbbcbf901.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
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+                        152
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                                321,
+                                152
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        138,
+                                        321,
+                                        152
+                                    ],
+                                    "type": "text",
+                                    "content": "Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        138,
+                                        321,
+                                        152
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p\\in (0,1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        138,
+                                        321,
+                                        152
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        138,
+                                        321,
+                                        152
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_{\\mathrm{min}} = \\frac{p}{\\max_km_k} = \\frac{p}{m^*}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        138,
+                                        321,
+                                        152
+                                    ],
+                                    "type": "text",
+                                    "content": ", then"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
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+                        285
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                {
+                                    "bbox": [
+                                        247,
+                                        156,
+                                        504,
+                                        285
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\tau = \\frac {1}{\\ln \\left(\\frac {1}{m ^ {*} - 1} \\left(\\frac {1}{x _ {\\operatorname* {m i n}}} - 1\\right)\\right)} (37a) \\\\ = \\frac {1}{\\ln \\left(\\frac {1}{m ^ {*} - 1} \\left(\\frac {m ^ {*}}{p} - 1\\right)\\right)} (37b) \\\\ = \\frac {1}{\\ln \\left(\\frac {m ^ {*} - p}{m ^ {*} - 1} \\frac {1}{p}\\right)} (37c) \\\\ \\leq \\frac {1}{\\ln \\left(\\frac {1}{p}\\right)}. (37d) \\\\ \\end{array}",
+                                    "image_path": "f7ac94edecb073f7713e51e06433f152f4c2378271a013675f52351ab72f91de.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
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+                        316
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                316
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        289,
+                                        504,
+                                        316
+                                    ],
+                                    "type": "text",
+                                    "content": "This implies if we set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        289,
+                                        504,
+                                        316
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau = \\ln (1 / p)^{-1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        289,
+                                        504,
+                                        316
+                                    ],
+                                    "type": "text",
+                                    "content": ", then we are guaranteed that all QREs contain probabilities greater than "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        289,
+                                        504,
+                                        316
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_{\\min} = \\frac{p}{\\max_k m_k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        289,
+                                        504,
+                                        316
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        320,
+                        414,
+                        333
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                320,
+                                414,
+                                333
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        320,
+                                        414,
+                                        333
+                                    ],
+                                    "type": "text",
+                                    "content": "Corollary 1. Assume payoffs are bounded in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        320,
+                                        414,
+                                        333
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "[0,1]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        320,
+                                        414,
+                                        333
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        320,
+                                        414,
+                                        333
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau = \\ln (1 / p)^{-1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        320,
+                                        414,
+                                        333
+                                    ],
+                                    "type": "text",
+                                    "content": ", then"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        216,
+                        335,
+                        504,
+                        362
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                216,
+                                335,
+                                504,
+                                362
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        216,
+                                        335,
+                                        504,
+                                        362
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left| \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) \\right| \\leq \\frac {1}{4} \\left(\\max  _ {k} \\eta_ {k}\\right) n \\bar {m} \\left(\\frac {\\ln \\left(m ^ {*}\\right)}{\\ln (1 / p)} + 2\\right) ^ {2}. \\tag {38}",
+                                    "image_path": "e075ab65875458c83b087ac38a8d2a2e20ed3d6a03de93e4f07939ac4f71ca2b.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
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+                        372,
+                        372,
+                        384
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                372,
+                                372,
+                                384
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        372,
+                                        372,
+                                        384
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Starting with Lemma 10 and applying Lemma 11, we find"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        176,
+                        387,
+                        504,
+                        493
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                176,
+                                387,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        176,
+                                        387,
+                                        504,
+                                        493
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\left| \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) \\right| \\leq \\frac {1}{4} \\left(\\max  _ {k} \\eta_ {k}\\right) n \\bar {m} \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\min }}\\right) + 1\\right) ^ {2} (39a) \\\\ = \\frac {1}{4} \\left(\\max  _ {k} \\eta_ {k}\\right) n \\bar {m} \\left(\\frac {1}{\\ln (1 / p)} \\ln \\left(\\frac {m ^ {*}}{p}\\right) + 1\\right) ^ {2} (39b) \\\\ = \\frac {1}{4} \\left(\\max  _ {k} \\eta_ {k}\\right) n \\bar {m} \\left(\\frac {1}{\\ln (1 / p)} \\left(\\ln \\left(m ^ {*}\\right) + \\ln (1 / p)\\right) + 1\\right) ^ {2} (39c) \\\\ = \\frac {1}{4} \\left(\\max  _ {k} \\eta_ {k}\\right) n \\bar {m} \\left(\\frac {\\ln \\left(m ^ {*}\\right)}{\\ln (1 / p)} + 2\\right) ^ {2}. (39d) \\\\ \\end{array}",
+                                    "image_path": "b405bc6f9c8d8cc83927eb1c931b459e6882ff95c062801106ae7bbdcf26fd4b.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
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+                        513,
+                        504,
+                        536
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                                504,
+                                536
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        513,
+                                        504,
+                                        536
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 12 (Low Temperature Approximate QREs are Approximate Nash Equilibria). Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        513,
+                                        504,
+                                        536
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\nabla_{x_k}^{k\\tau}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        513,
+                                        504,
+                                        536
+                                    ],
+                                    "type": "text",
+                                    "content": " be player "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        513,
+                                        504,
+                                        536
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        513,
+                                        504,
+                                        536
+                                    ],
+                                    "type": "text",
+                                    "content": " 's entropy regularized gradient and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        513,
+                                        504,
+                                        536
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        513,
+                                        504,
+                                        536
+                                    ],
+                                    "type": "text",
+                                    "content": " be an approximate QRE. Then it holds that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        186,
+                        539,
+                        504,
+                        555
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                186,
+                                539,
+                                504,
+                                555
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        186,
+                                        539,
+                                        504,
+                                        555
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "u _ {k} \\left(B R _ {k}, x _ {- k}\\right) - u _ {k} (\\boldsymbol {x}) \\leq \\tau \\log \\left(m _ {k}\\right) + \\sqrt {2} \\left\\| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right\\|. \\tag {40}",
+                                    "image_path": "13939b41bb07f89b58893263759286078d7444b3193997adba4aa59269684c8d.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
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+                        576
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                356,
+                                576
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        564,
+                                        356,
+                                        576
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Beginning with the definition of exploitability, we find"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
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+                        504,
+                        685
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        148,
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+                                        685
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} u _ {k} \\left(\\mathbb {B R} _ {k}, x _ {- k}\\right) - u _ {k} (\\boldsymbol {x}) = \\left(u _ {k} \\left(\\mathbb {B R} _ {k}, x _ {- k}\\right) + \\tau S \\left(\\mathbb {B R} _ {k}\\right) - \\tau S \\left(\\mathbb {B R} _ {k}\\right)\\right) (41a) \\\\ - \\left(u _ {k} (\\boldsymbol {x}) + \\tau S (x _ {k}) - \\tau S (x _ {k})\\right) \\\\ = u _ {k} ^ {\\tau} \\left(\\mathbb {B R} _ {k}, x _ {- k}\\right) - u _ {k} ^ {\\tau} (\\boldsymbol {x}) + \\tau \\left(S \\left(x _ {k}\\right) - S \\left(\\mathbb {B R} _ {k}\\right)\\right) (41b) \\\\ \\leq \\max  _ {z \\in \\Delta^ {m _ {k}} ^ {- 1}} u _ {k} ^ {\\tau} (z, x _ {- k}) - u _ {k} ^ {\\tau} (\\boldsymbol {x}) + \\tau \\max  _ {z ^ {\\prime} \\in \\Delta^ {m _ {k}} ^ {- 1}} S \\left(z ^ {\\prime}\\right) (41c) \\\\ \\leq \\sqrt {2} \\| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\| + \\tau \\max  _ {z ^ {\\prime} \\in \\Delta^ {m _ {k} - 1}} S \\left(z ^ {\\prime}\\right) (41d) \\\\ \\leq \\sqrt {2} | | \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) | | + \\tau \\log (m _ {k}) (41e) \\\\ \\end{array}",
+                                    "image_path": "70d35228cd44d7d547e7a5f3cdcef57cca986c6d3a9b914422f2f75ef9b59a37.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
+                                        104,
+                                        688,
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+                                    ],
+                                    "type": "text",
+                                    "content": "where the second equality follows from the definition of player "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "'s entropy regularized utility "
+                                },
+                                {
+                                    "bbox": [
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+                                        505,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u_{k}^{\\tau}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        688,
+                                        505,
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+                                    ],
+                                    "type": "text",
+                                    "content": ", the first inequality from nonnegativity of entropy "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        688,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                        505,
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+                                    ],
+                                    "type": "text",
+                                    "content": ", the second inequality from concavity of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        688,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u_{k}^{\\tau}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        688,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " with respect to its first argument (Lemma 2), and the last from the maximum possible value of Shannon entropy over distributions on "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        688,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m_{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        688,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " actions."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                }
+            ],
+            "discarded_blocks": [
+                {
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+                    ],
+                    "type": "header",
+                    "angle": 0,
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
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+                    "index": 0
+                },
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+                                    "type": "text",
+                                    "content": "20"
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+                            ]
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+                    ],
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+                }
+            ],
+            "page_size": [
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+            "page_idx": 19
+        },
+        {
+            "para_blocks": [
+                {
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        103,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 13 ("
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}^{\\tau}"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": " Scores Nash Equilibria). Let "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}^{\\tau}(\\pmb{x})"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": " be our proposed entropy regularized loss function and "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": " be an approximate QRE. Then it holds that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        216,
+                        114,
+                        503,
+                        143
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                216,
+                                114,
+                                503,
+                                143
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        216,
+                                        114,
+                                        503,
+                                        143
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\epsilon \\leq \\tau \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {\\frac {2 n}{\\min _ {k} \\eta_ {k}}} \\sqrt {\\mathcal {L} ^ {\\tau} (\\boldsymbol {x})}. \\tag {42}",
+                                    "image_path": "a53a272e3e28d003158e330e02d76257dae3590d16b8929742122c711e1b9fe7.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        162,
+                        456,
+                        175
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                162,
+                                456,
+                                175
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        162,
+                                        456,
+                                        175
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Beginning with the definition of exploitability and applying Lemma 12, we find"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
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+                        314
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\epsilon = \\max  _ {k} u _ {k} (\\mathbb {B R} _ {k}, x _ {- k}) - u _ {k} (\\boldsymbol {x}) \\quad \\text {(r e c a l l e a c h} \\epsilon_ {k} \\geq 0) (43a) \\\\ \\leq \\sum_ {k} u _ {k} (\\mathbb {R} _ {k}, x _ {- k}) - u _ {k} (\\boldsymbol {x}) (43b) \\\\ \\leq \\sum_ {k} \\left[ \\tau \\log \\left(m _ {k}\\right) + \\sqrt {2} \\left| \\left| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right| \\right| \\right] (43c) \\\\ = \\tau \\sum_ {k} \\log \\left(m _ {k}\\right) + \\sum_ {k} \\sqrt {2} \\left\\| \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right\\| (43d) \\\\ \\leq \\tau \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {\\frac {2 n}{\\min  _ {k} \\eta_ {k}}} \\sqrt {\\mathcal {L} ^ {\\tau} (\\boldsymbol {x})}. (43e) \\\\ \\end{array}",
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "where the last inequality follows from the same steps (24c)-(24i) outlined in Lemma 4, which established the relationship between "
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+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}(\\pmb{x})"
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+                                    ],
+                                    "type": "text",
+                                    "content": " and "
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+                                    "type": "inline_equation",
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "."
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+                        }
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+                                    "type": "text",
+                                    "content": "A.3 GRADIENT OF LOSS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Lemma 14. The gradient of "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}^{\\tau}(\\pmb{x})"
+                                },
+                                {
+                                    "bbox": [
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+                                        382,
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+                                    ],
+                                    "type": "text",
+                                    "content": " with respect to player "
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+                                {
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+                                    "type": "inline_equation",
+                                    "content": "l"
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "'s strategy "
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+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "x_{l}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " is"
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+                            ]
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\nabla_ {x _ {l}} \\mathcal {L} (\\boldsymbol {x}) = 2 \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\tag {44}",
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+                                    "type": "text",
+                                    "content": "where "
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+                                {
+                                    "bbox": [
+                                        104,
+                                        436,
+                                        431,
+                                        452
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "B_{ll} = -\\tau \\left[I - \\frac{1}{m_l}\\mathbf{11}^\\top \\right]diag\\left(\\frac{1}{x_l}\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        436,
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+                                        452
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        436,
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+                                        452
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "B_{kl} = \\left[I - \\frac{1}{m_k}\\mathbf{11}^\\top \\right]H_{kl}^k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        436,
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+                                        452
+                                    ],
+                                    "type": "text",
+                                    "content": " for "
+                                },
+                                {
+                                    "bbox": [
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+                                        436,
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+                                        452
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k\\neq l"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
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+                        483
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Proof. Recall from Lemma 6 that the loss can be decomposed as "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}^{\\tau}(\\pmb{x}) = (A) + (B) + (C)"
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": "."
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+                                    "type": "text",
+                                    "content": "Then"
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+                            ]
+                        }
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+                    "index": 11
+                },
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+                                    "type": "interline_equation",
+                                    "content": "D _ {x _ {l}} [ (A) ] = D _ {x _ {l}} \\left[ \\sum_ {k} \\eta_ {k} x _ {q} ^ {\\top} B _ {k q} ^ {\\top} B _ {k q} x _ {q} \\right] = 2 \\sum_ {k \\neq l} \\eta_ {k} B _ {k l} ^ {\\top} B _ {k l} x _ {l} \\tag {45}",
+                                    "image_path": "31b50d9cbd70d88888befc4ae0bc8e1141a3991965e8fc645becc98cfe5b97e9.jpg"
+                                }
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "where "
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+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "q \\neq k"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " and "
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+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "B_{kq} = [I - \\frac{1}{m_k}\\mathbf{11}^\\top][H_{kq}^k]"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " does not depend on "
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+                                    "type": "inline_equation",
+                                    "content": "x_{k}"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "."
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+                        }
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+                    "index": 13
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+                {
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Also, letting "
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+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "B_{ll} = -\\tau [I - \\frac{1}{m_l}\\mathbf{11}^\\top ]\\mathrm{diag}\\left(\\frac{1}{x_l}\\right)"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
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+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} D _ {x _ {l}} [ (B) ] = D _ {x _ {l}} \\left[ - 2 \\tau \\sum_ {k} \\eta_ {k} \\ln \\left(x _ {k}\\right) ^ {\\top} B _ {k q} x _ {q} \\right] (46a) \\\\ = - 2 \\tau \\left[ \\eta_ {l} D _ {x _ {l}} \\left[ \\ln \\left(x _ {l}\\right) ^ {\\top} B _ {l q} x _ {q} \\right] + \\sum_ {k \\neq l} \\eta_ {k} D _ {x _ {l}} \\left[ \\ln \\left(x _ {k}\\right) ^ {\\top} B _ {k l} x _ {l} \\right] \\right] (46b) \\\\ = - 2 \\tau \\left[ \\eta_ {l} \\operatorname {d i a g} \\left(\\frac {1}{x _ {l}}\\right) B _ {l q} x _ {q} + \\sum_ {k \\neq l} \\eta_ {k} B _ {k l} ^ {\\top} \\ln \\left(x _ {k}\\right) \\right] (46c) \\\\ = - 2 \\tau \\left[ \\eta_ {l} \\left(\\left[ I - \\frac {1}{m _ {l}} \\mathbf {1 1} ^ {\\top} \\right] \\operatorname {d i a g} \\left(\\frac {1}{x _ {l}}\\right)\\right) ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla^ {l}\\right) + \\sum_ {k \\neq l} \\eta_ {k} B _ {k l} ^ {\\top} \\ln \\left(x _ {k}\\right) \\right] (46d) \\\\ = 2 \\left[ \\eta_ {l} B _ {l l} ^ {\\top} \\Pi_ {T \\Delta} (\\nabla^ {l}) - \\tau \\sum_ {k \\neq l} \\eta_ {k} B _ {k l} ^ {\\top} \\ln (x _ {k}) \\right]. (46e) \\\\ \\end{array}",
+                                    "image_path": "d20b605c737b93999370da351388462ef3b75f6b9c5b083db7474a74586c463f.jpg"
+                                }
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "type": "text",
+                                    "content": "And"
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} D _ {x _ {l}} [ (C) ] = D _ {x _ {l}} \\left[ \\sum_ {k} \\eta_ {k} \\tau^ {2} \\ln \\left(x _ {k}\\right) ^ {\\top} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] \\ln \\left(x _ {k}\\right) \\right] (47a) \\\\ = 2 \\tau^ {2} \\left[ \\eta_ {l} \\operatorname {d i a g} \\left(\\frac {1}{x _ {l}}\\right) \\left[ I - \\frac {1}{m _ {l}} \\mathbf {1 1} ^ {\\top} \\right] \\ln \\left(x _ {l}\\right) \\right] (47b) \\\\ = - 2 \\tau \\eta_ {l} \\left(\\left[ I - \\frac {1}{m _ {l}} \\mathbf {1 1} ^ {\\top} \\right] \\operatorname {d i a g} \\left(\\frac {1}{x _ {l}}\\right)\\right) ^ {\\top} \\Pi_ {T \\Delta} (- \\tau \\ln (x _ {l})) (47c) \\\\ = 2 \\eta_ {l} B _ {l l} ^ {\\top} \\Pi_ {T \\Delta} (- \\tau \\ln (x _ {l})). (47d) \\\\ \\end{array}",
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+                                    "type": "text",
+                                    "content": "Putting these together, we find"
+                                }
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+                                    "content": "\\begin{array}{l} \\nabla_ {x _ {l}} \\mathcal {L} (\\boldsymbol {x}) = 2 \\sum_ {k \\neq l} \\eta_ {k} B _ {k l} ^ {\\top} \\left(B _ {k l} x _ {l} - \\tau \\ln \\left(x _ {k}\\right)\\right) + 2 \\eta_ {l} B _ {l l} ^ {\\top} \\left[ \\Pi_ {T \\Delta} \\left(\\nabla^ {l}\\right) + \\Pi_ {T \\Delta} (- \\tau \\ln \\left(x _ {l}\\right)) \\right] (48a) \\\\ = 2 \\eta_ {l} B _ {l l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) + 2 \\sum_ {k \\neq l} \\eta_ {k} B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) (48b) \\\\ = 2 \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right). (48c) \\\\ \\end{array}",
+                                    "image_path": "aa629c24d4963605d21c998366a365968cb2baa6b3d69b482b6e60139ccfface.jpg"
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+                                    "type": "text",
+                                    "content": "A.3.1 UNBIASED ESTIMATION"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
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+                                    "type": "text",
+                                    "content": "In order to construct an unbiased estimate for each "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "B_{kl}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ", we will need to form an independent unbiased estimate of "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "H_{kl}^{k}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ". Recall that "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H_{kl}^{k}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " is simply the expected bimatrix game between players "
+                                },
+                                {
+                                    "bbox": [
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+                                        462
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "l"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " when all other players sample their actions according to their current strategies."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
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+                    "type": "title",
+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "A.4 BOUND ON GRADIENT / LIPSCHITZ PROPERTY"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
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+                    "angle": 0,
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 15. Assume payoffs are upper bounded by 1, then the infinity norm of the gradient is bounded as"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\left| \\left| \\nabla_ {\\boldsymbol {x}} \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) \\right| \\right| _ {\\infty} \\leq \\frac {1}{2} \\left(\\max  _ {k} \\eta_ {k}\\right) \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) \\left[ \\tau m ^ {*} \\left(\\frac {1}{x _ {\\operatorname* {m i n}}} - 1\\right) + n \\bar {m} \\right]. \\tag {49}",
+                                    "image_path": "8b9841645956c6e195ba8e24a093245636e00f1b4c9a6687cc8ffa8d43789283.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
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+                    "angle": 0,
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Recall from Lemma 14 that the gradient of "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}(\\pmb{x})"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " with respect to player "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "l"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "'s strategy "
+                                },
+                                {
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_{l}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " is"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
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+                    "angle": 0,
+                    "lines": [
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+                                    "type": "interline_equation",
+                                    "content": "\\nabla_ {x _ {l}} \\mathcal {L} (\\boldsymbol {x}) = 2 \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\tag {50}",
+                                    "image_path": "9c6741e7f5df93ed5642c36e38975e31c2c15e5f3b20f9336d7ddb5afb51e9c3.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "B_{ll} = -\\tau [I - \\frac{1}{m_l}\\mathbf{11}^\\top ]\\mathrm{diag}\\left(\\frac{1}{x_l}\\right)"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "B_{kl} = [I - \\frac{1}{m_k}\\mathbf{11}^\\top ]H_{kl}^k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        672,
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+                                    "type": "text",
+                                    "content": " for "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k\\neq l"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "As noted before in the derivation of Lemma 10, for payoffs in "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "[0,1]"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ", the entries in "
+                                },
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\nabla_{x_k}^{k\\tau} = \\nabla_{x_k}^k - \\tau \\ln(x_k)"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " are bounded within "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "[0, \\tau \\ln(\\frac{1}{x_{\\min}}) + 1]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    "type": "text",
+                                    "content": " with a range "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau \\ln(\\frac{1}{x_{\\min}}) + 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    "type": "text",
+                                    "content": ". Similarly, the entries in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        693,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "-\\tau \\mathrm{diag}\\left(\\frac{1}{x_l}\\right)"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " are bounded within "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "[- \\tau \\frac{1}{x_{\\min}}, -\\tau]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " with a range of "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau\\left(\\frac{1}{x_{\\min}} - 1\\right)"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
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+                }
+            ],
+            "discarded_blocks": [
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
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+                                    "type": "text",
+                                    "content": "The infinity norm of the gradient can then be bounded as"
+                                }
+                            ]
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+                    ],
+                    "index": 1
+                },
+                {
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\left| \\left| \\nabla_ {\\boldsymbol {x}} \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) \\right| \\right| _ {\\infty} = \\max  _ {l} \\left| \\left| \\nabla_ {x _ {l}} \\mathcal {L} (\\boldsymbol {x}) \\right| \\right| _ {\\infty} (51a) \\\\ = \\max  _ {l} \\left\\| 2 \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right\\| _ {\\infty} (51b) \\\\ \\leq 2 \\sum_ {k} \\eta_ {k} \\max  _ {l} | | B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) | | _ {\\infty} (51c) \\\\ \\leq \\frac {1}{2} \\sum_ {k \\neq l ^ {*}} \\eta_ {k} m _ {k} \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) + \\frac {1}{2} \\eta_ {l ^ {*}} m _ {l ^ {*}} \\tau \\left(\\frac {1}{x _ {\\operatorname* {m i n}}} - 1\\right) \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) (51d) \\\\ = \\frac {1}{2} \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) \\left[ \\eta_ {l ^ {*}} m _ {l ^ {*}} \\tau \\left(\\frac {1}{x _ {\\operatorname* {m i n}}} - 1\\right) + \\sum_ {k \\neq l ^ {*}} \\eta_ {k} m _ {k} \\right] (51e) \\\\ \\leq \\frac {1}{2} \\left(\\max  _ {k} \\eta_ {k}\\right) \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) \\left[ \\tau m _ {l ^ {*}} \\left(\\frac {1}{x _ {\\operatorname* {m i n}}} - 1\\right) + \\sum_ {k \\neq l ^ {*}} m _ {k} \\right] (51f) \\\\ \\leq \\frac {1}{2} \\left(\\max  _ {k} \\eta_ {k}\\right) \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) \\left[ \\tau m ^ {*} \\left(\\frac {1}{x _ {\\operatorname* {m i n}}} - 1\\right) + n \\bar {m} \\right] (51g) \\\\ \\end{array}",
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+                                }
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+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
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+                        316
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
+                                        105,
+                                        304,
+                                        317,
+                                        316
+                                    ],
+                                    "type": "text",
+                                    "content": "where the second inequality follows from Lemma 9."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "type": "image",
+                    "bbox": [
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+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 4,
+                            "angle": 0,
+                            "type": "image_body"
+                        }
+                    ],
+                    "index": 4
+                },
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+                    ],
+                    "type": "text",
+                    "angle": 0,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        338,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "text",
+                                    "content": "Corollary 2. If "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        338,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        338,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "text",
+                                    "content": " is set according to Lemma 11 as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        338,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau = \\ln (1 / p)^{-1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        338,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "text",
+                                    "content": " and payoffs are in [0, 1], then the infinity norm of the gradient is bounded as"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        143,
+                        364,
+                        504,
+                        392
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                143,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        143,
+                                        364,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left| \\left| \\nabla_ {\\boldsymbol {x}} \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) \\right| \\right| _ {\\infty} \\leq \\frac {1}{2} \\left(\\max  _ {k} \\eta_ {k}\\right) \\left[ \\frac {\\ln \\left(m ^ {*}\\right)}{\\ln (1 / p)} + 2 \\right] \\left[ \\frac {m ^ {* 2}}{p \\ln (1 / p)} + n \\bar {m} \\right] = \\frac {1}{2} \\left(\\max  _ {k} \\eta_ {k}\\right) \\hat {L} \\tag {52}",
+                                    "image_path": "1f24644bd4356cd18e44a6e4c2a7dd44d80a70c7cb8984e130eb05b89b940a2f.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
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+                        456,
+                        407
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                456,
+                                407
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        396,
+                                        456,
+                                        407
+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        396,
+                                        456,
+                                        407
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m^{*} = \\max_{k} m_{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        396,
+                                        456,
+                                        407
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        396,
+                                        456,
+                                        407
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{L}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        396,
+                                        456,
+                                        407
+                                    ],
+                                    "type": "text",
+                                    "content": " is defined implicitly for convenience in other derivations."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        104,
+                        418,
+                        504,
+                        442
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                418,
+                                504,
+                                442
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        418,
+                                        504,
+                                        442
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Starting with Lemma 15 and applying Lemma 11 (i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        418,
+                                        504,
+                                        442
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau = \\ln(1/p)^{-1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        418,
+                                        504,
+                                        442
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        418,
+                                        504,
+                                        442
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_{\\min} = \\frac{p}{m^*}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        418,
+                                        504,
+                                        442
+                                    ],
+                                    "type": "text",
+                                    "content": " with payoffs in [0, 1] and where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        418,
+                                        504,
+                                        442
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m^* = \\max_k m_k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        418,
+                                        504,
+                                        442
+                                    ],
+                                    "type": "text",
+                                    "content": "), we find"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        153,
+                        445,
+                        504,
+                        525
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                153,
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+                                504,
+                                525
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        153,
+                                        445,
+                                        504,
+                                        525
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\left| \\left| \\nabla_ {\\boldsymbol {x}} \\mathcal {L} ^ {\\tau} (\\boldsymbol {x}) \\right| \\right| _ {\\infty} \\leq \\frac {1}{2} \\left(\\max  _ {k} \\eta_ {k}\\right) \\left(\\tau \\ln \\left(\\frac {1}{x _ {\\operatorname* {m i n}}}\\right) + 1\\right) \\left[ \\tau m ^ {*} \\left(\\frac {1}{x _ {\\operatorname* {m i n}}} - 1\\right) + n \\bar {m} \\right] (53a) \\\\ = \\frac {1}{2} \\left(\\max  _ {k} \\eta_ {k}\\right) \\left[ \\frac {\\ln \\left(m ^ {*} / p\\right)}{\\ln (1 / p)} + 1 \\right] \\left[ \\frac {m ^ {*}}{\\ln (1 / p)} \\left(\\frac {m ^ {*}}{p} - 1\\right) + n \\bar {m} \\right] (53b) \\\\ \\leq \\frac {1}{2} \\left(\\max  _ {k} \\eta_ {k}\\right) \\left[ \\frac {\\ln \\left(m ^ {*}\\right)}{\\ln (1 / p)} + 2 \\right] \\left[ \\frac {m ^ {* 2}}{p \\ln (1 / p)} + n \\bar {m} \\right]. (53c) \\\\ \\end{array}",
+                                    "image_path": "12f72b92111c1bdf43b88ab7b80d080eaaf442cb80208f40b20381227a0038ec.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
+                        534,
+                        506,
+                        581
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                534,
+                                506,
+                                581
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        534,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": "As "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        534,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p \\to 0^+"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        534,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": ", the norm of the gradient blows up because the gradient of Shannon entropy blows up for small probabilities. As "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        534,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p \\to 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        534,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": ", the norm of the gradient blows up because we require infinite temperature "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        534,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        534,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": " to guarantee all QREs are nearly uniform; recall "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        534,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        534,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": " is the regularization coefficient on the entropy bonus terms which means our modified utilities blow up for large "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        534,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        534,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        105,
+                        592,
+                        212,
+                        602
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        592,
+                                        212,
+                                        602
+                                    ],
+                                    "type": "text",
+                                    "content": "A.5 HESSIAN OF LOSS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        104,
+                        613,
+                        506,
+                        670
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                613,
+                                506,
+                                670
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        613,
+                                        506,
+                                        670
+                                    ],
+                                    "type": "text",
+                                    "content": "We will now derive the Hessian of our loss. This will be useful in establishing properties about global minima that enable the application of tailored minimization algorithms. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        613,
+                                        506,
+                                        670
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "D_z[f(z)]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        613,
+                                        506,
+                                        670
+                                    ],
+                                    "type": "text",
+                                    "content": " denote the differential operator applied to (possibly multivalued) function "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        613,
+                                        506,
+                                        670
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        613,
+                                        506,
+                                        670
+                                    ],
+                                    "type": "text",
+                                    "content": " with respect to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        613,
+                                        506,
+                                        670
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        613,
+                                        506,
+                                        670
+                                    ],
+                                    "type": "text",
+                                    "content": ". For example, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        613,
+                                        506,
+                                        670
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "D_{x_q}[H_{lk}^k] = D_{x_q}[x_qT_{qlk}^k] = T_{qlk}^k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        613,
+                                        506,
+                                        670
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        613,
+                                        506,
+                                        670
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T_{qlk}^k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        613,
+                                        506,
+                                        670
+                                    ],
+                                    "type": "text",
+                                    "content": " is player "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        613,
+                                        506,
+                                        670
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        613,
+                                        506,
+                                        670
+                                    ],
+                                    "type": "text",
+                                    "content": "'s payoff tensor according to the three-way approximation between players "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        613,
+                                        506,
+                                        670
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k, l"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        613,
+                                        506,
+                                        670
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        613,
+                                        506,
+                                        670
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "q"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        613,
+                                        506,
+                                        670
+                                    ],
+                                    "type": "text",
+                                    "content": " to the game at "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        613,
+                                        506,
+                                        670
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        613,
+                                        506,
+                                        670
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        105,
+                        671,
+                        310,
+                        684
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                671,
+                                310,
+                                684
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        671,
+                                        310,
+                                        684
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 16. The Hessian of "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        671,
+                                        310,
+                                        684
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}^{\\tau}(\\pmb{x})"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        671,
+                                        310,
+                                        684
+                                    ],
+                                    "type": "text",
+                                    "content": " can be written"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        230,
+                        686,
+                        504,
+                        701
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                230,
+                                686,
+                                504,
+                                701
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        230,
+                                        686,
+                                        504,
+                                        701
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "H e s s \\left(\\mathcal {L} ^ {\\tau}\\right) = 2 \\tilde {B} ^ {\\top} \\tilde {B} + T \\Pi_ {T \\Delta} \\left(\\tilde {\\nabla} ^ {\\tau}\\right) \\tag {54}",
+                                    "image_path": "f447bad14a96897942db73dae870e182d7087e3935c62a5c03f3a9bead59f40b.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        104,
+                        705,
+                        504,
+                        735
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                705,
+                                504,
+                                735
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        705,
+                                        504,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        705,
+                                        504,
+                                        735
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{B}_{kl} = \\sqrt{\\eta_k} B_{kl}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        705,
+                                        504,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        705,
+                                        504,
+                                        735
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Pi_{T\\Delta}(\\tilde{\\nabla}^{\\tau}) = [\\eta_1\\Pi_{T\\Delta}(\\nabla_{x_1}^{1\\tau}),\\dots ,\\eta_n\\Pi_{T\\Delta}(\\nabla_{x_n}^{n\\tau})]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        705,
+                                        504,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": ", and we augment "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        705,
+                                        504,
+                                        735
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    "type": "text",
+                                    "content": " (the 3-player tensor approximation to the game, "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "T_{lqk}^{k}"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ") so that "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "T_{ll}^{l} = \\tau \\text{diag} 3\\left(\\frac{1}{x_{l}^{2}}\\right)"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " and otherwise 0."
+                                }
+                            ]
+                        }
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+                    "index": 15
+                }
+            ],
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "content": "23"
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+                                    "type": "text",
+                                    "content": "Proof. Recall the gradient of our proposed loss:"
+                                }
+                            ]
+                        }
+                    ],
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+                                    "type": "interline_equation",
+                                    "content": "\\nabla_ {x _ {l}} \\mathcal {L} (\\boldsymbol {x}) = 2 \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\tag {55}",
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+                                    "type": "text",
+                                    "content": "where "
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+                                    "type": "inline_equation",
+                                    "content": "B_{ll} = -\\tau [I - \\frac{1}{m_l}\\mathbf{11}^\\top ]\\mathrm{diag}\\left(\\frac{1}{x_l}\\right)"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " and "
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+                                    "type": "inline_equation",
+                                    "content": "B_{kl} = [I - \\frac{1}{m_k}\\mathbf{11}^\\top ]H_{kl}^k"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " for "
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+                                {
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+                                    "type": "inline_equation",
+                                    "content": "k\\neq l"
+                                }
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+                        }
+                    ],
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+                {
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+                                    "type": "text",
+                                    "content": "Consider the following Jacobians, which will play an auxiliary role in our derivation of the Hessian:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
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+                                    "content": "D _ {l} \\left[ B _ {l l} \\right] = \\tau \\left[ I - \\frac {1}{m _ {l}} \\mathbf {1 1} ^ {\\top} \\right] \\operatorname {d i a g} 3 \\left(\\frac {1}{x _ {l} ^ {2}}\\right) \\tag {56a}",
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+                                    "type": "interline_equation",
+                                    "content": "D _ {q} \\left[ B _ {l l} \\right] = 0 \\tag {56b}",
+                                    "image_path": "bba8afd40d958333ce574d56f8606e8d7efc9a83e30d0d94cd5c89ea848aa694.jpg"
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+                                    "content": "D _ {l} \\left[ B _ {k l} \\right] = 0 \\tag {56c}",
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+                                    "content": "D _ {q} \\left[ B _ {k l} \\right] = \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] T _ {k l q} ^ {k} \\tag {56d}",
+                                    "image_path": "6c757cc157c39ff974f8b40f758c32b2d3b34f35c875453fbc3ca925c408b866.jpg"
+                                }
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+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} D _ {k} \\left[ \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right] = \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] D _ {k} \\left[ \\nabla_ {x _ {k}} ^ {k \\tau} \\right] (56e) \\\\ = \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] D _ {k} \\left[ \\nabla_ {x _ {k}} ^ {k} - \\tau \\ln \\left(x _ {k}\\right) \\right] (56f) \\\\ = \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] \\left[ - \\tau \\operatorname {d i a g} \\left(\\frac {1}{x _ {k}}\\right) \\right] (56g) \\\\ = B _ {k k} (56h) \\\\ \\end{array}",
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+                                }
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+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} D _ {l} \\left[ \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right] = \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] D _ {l} \\left[ \\nabla_ {x _ {k}} ^ {k \\tau} \\right] (56i) \\\\ = \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] D _ {l} \\left[ \\nabla_ {x _ {k}} ^ {k} - \\tau \\ln (x _ {k}) \\right] (56j) \\\\ = \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] \\left[ H _ {k l} ^ {k} \\right] (56k) \\\\ = B _ {k l}. (561) \\\\ \\end{array}",
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+                                    "content": "We can derive the diagonal blocks of the Hessian as"
+                                }
+                            ]
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+                                    "content": "\\begin{array}{l} D _ {l l} [ \\mathcal {L} (\\boldsymbol {x}) ] = D _ {l} \\left[ \\nabla_ {x _ {l}} \\mathcal {L} (\\boldsymbol {x}) \\right] (57a) \\\\ = 2 D _ {l} \\left[ \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right] (57b) \\\\ = 2 \\left[ \\eta_ {l} D _ {l} \\left[ B _ {l l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {l}} ^ {l \\tau}\\right) \\right] + \\sum_ {k \\neq l} \\eta_ {k} D _ {l} \\left[ B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right] \\right] (57c) \\\\ = 2 \\left[ \\eta_ {l} \\left[ D _ {l} \\left[ B _ {l l} \\right] ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {l}} ^ {l \\tau}\\right) + B _ {l l} ^ {\\top} D _ {l} \\left[ \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {l}} ^ {l \\tau}\\right) \\right] \\right] \\right. (57d) \\\\ \\left. + \\sum_ {k \\neq l} \\eta_ {k} \\left[ D _ {t} \\left[ B _ {k l} \\right] ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) + B _ {k l} ^ {\\top} D _ {l} \\left[ \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right] \\right] \\right] (57e) \\\\ = 2 \\left[ \\eta_ {l} \\left[ \\tau \\operatorname {d i a g} 3 \\left(\\frac {1}{x _ {l} ^ {2}}\\right) [ I - \\frac {1}{m _ {l}} \\mathbf {1 1} ^ {\\top} ] \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {l}} ^ {l \\tau}\\right) + B _ {l l} ^ {\\top} B _ {l l} \\right] + \\sum_ {k \\neq l} \\eta_ {k} B _ {k l} ^ {\\top} B _ {k l} \\right] (57f) \\\\ = 2 \\left[ \\tau \\eta_ {l} \\operatorname {d i a g} \\left(\\left[ \\frac {1}{x _ {l} ^ {2}} \\right] \\odot \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {l}} ^ {l \\tau}\\right)\\right) + \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} B _ {k l} \\right] (57g) \\\\ \\end{array}",
+                                    "image_path": "bea7a7cc42abd0c9a4baf5c1391c107e1213db04ba8dd8b92e26813f387430c1.jpg"
+                                }
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+                                    "type": "text",
+                                    "content": "and the off-diagonal blocks as"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
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+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} D _ {l q} [ \\mathcal {L} (\\boldsymbol {x}) ] = D _ {q} \\left[ \\nabla_ {x _ {l}} \\mathcal {L} (\\boldsymbol {x}) \\right] (58a) \\\\ = 2 D _ {q} \\left[ \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right] (58b) \\\\ = 2 \\left[ \\eta_ {l} D _ {q} \\left[ B _ {l l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {l}} ^ {l \\tau}\\right) \\right] + \\sum_ {k \\neq l} \\eta_ {k} D _ {q} \\left[ B _ {k l} ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right] \\right] (58c) \\\\ = 2 \\left[ \\eta_ {l} \\left[ D _ {q} \\left[ B _ {\\bar {l} l} \\right] ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {l}} ^ {l \\tau}\\right) + B _ {l l} ^ {\\top} D _ {q} \\left[ \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {l}} ^ {l \\tau}\\right) \\right] \\right] \\right. (58d) \\\\ \\left. + \\sum_ {k \\neq l} \\eta_ {k} \\left[ D _ {q} \\left[ B _ {k l} \\right] ^ {\\top} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) + B _ {k l} ^ {\\top} D _ {q} \\left[ \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right] \\right] \\right] (58e) \\\\ = 2 \\left[ \\eta_ {l} B _ {l l} ^ {\\top} B _ {l q} + \\sum_ {k \\neq l} \\eta_ {k} \\left[ T _ {l q k} ^ {k} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1 1} ^ {\\top} \\right] \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) + B _ {k l} ^ {\\top} B _ {k q} \\right] \\right] (58f) \\\\ = 2 \\left[ \\sum_ {k} \\eta_ {k} B _ {k l} ^ {\\top} B _ {k q} + \\sum_ {k \\neq l} \\eta_ {k} T _ {l q k} ^ {k} \\Pi_ {T \\Delta} \\left(\\nabla_ {x _ {k}} ^ {k \\tau}\\right) \\right]. (58g) \\\\ \\end{array}",
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+                                    "content": "Therefore, the Hessian can be written concisely as"
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+                                    "content": "2 \\left[ \\tilde {B} ^ {\\top} \\tilde {B} + T \\Pi_ {T \\Delta} \\left(\\tilde {\\nabla} ^ {\\tau}\\right) \\right] \\tag {59}",
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+                                    "content": "where "
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+                                    "content": "\\tilde{B}_{kl} = \\sqrt{\\eta_k} B_{kl}"
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+                                    "type": "text",
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+                                    "type": "inline_equation",
+                                    "content": "\\Pi_{T\\Delta}(\\tilde{\\nabla}^{\\tau}) = [\\eta_1\\Pi_{T\\Delta}(\\nabla_{x_1}^{1\\tau}),\\dots ,\\eta_n\\Pi_{T\\Delta}(\\nabla_{x_n}^{n\\tau})]"
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+                                    "type": "text",
+                                    "content": ", and we augment "
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+                                    "type": "text",
+                                    "content": " (the 3-player tensor approximation to the game, "
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+                                    "type": "inline_equation",
+                                    "content": "T_{lqk}^{k}"
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+                                    "type": "text",
+                                    "content": ") so that "
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+                                    "type": "inline_equation",
+                                    "content": "T_{lll}^{l} = \\tau \\mathrm{diag}3\\left(\\frac{1}{x_{l}^{2}}\\right)"
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+                                    "type": "text",
+                                    "content": " and otherwise 0."
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "type": "text",
+                                    "content": "B GLOBAL CONVERGENCE GUARANTEES"
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+                                    "content": "In this section, we analyze the application of optimization techniques such as bandit algorithms and gradient descent to our loss function."
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+                                    "type": "text",
+                                    "content": "B.1 MAPS FROM HYPERCUBE TO SIMPLEX PRODUCT"
+                                }
+                            ]
+                        }
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+                                    "type": "text",
+                                    "content": "In this subsection, we derive properties of a map "
+                                },
+                                {
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " from the unit-hypercube to the simplex product. This map is necessary to adapt our proposed loss "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}^{\\tau}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " to the commonly assumed setting in the "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "-armed bandit literature (Bubeck et al., 2011) where the feasible set is a hypercube. We derive relevant properties of two such maps: the softmax and a mapping that interprets dimensions of the hypercube as angles on a unit-sphere that are then "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\ell_1"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "-normalized."
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+                                    "type": "text",
+                                    "content": "Lemma 17. Let "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "f(x) = -\\mathcal{L}(s(x))"
+                                },
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ". Then "
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+                                    "type": "inline_equation",
+                                    "content": "||\\nabla f(x)||_{\\infty} \\leq ||J(s(x))^{\\top}||_{\\infty}||\\nabla \\mathcal{L}(s(x))||_{\\infty}"
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": "."
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+                                    "type": "text",
+                                    "content": "Proof."
+                                }
+                            ]
+                        }
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+                                    "type": "interline_equation",
+                                    "content": "\\left. \\right.\\left|\\left| \\nabla f (x) \\right|\\right| _ {\\infty} = \\left|\\left| J (s (x)) ^ {\\top} \\nabla \\mathcal {L} (s (x)) \\right|\\right| _ {\\infty} \\leq \\left|\\left| J (s (x)) ^ {\\top} \\right|\\right| _ {\\infty} \\left|\\left| \\nabla \\mathcal {L} (s (x)) \\right|\\right| _ {\\infty}. \\tag {60}",
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+                                    "type": "text",
+                                    "content": "Lemma 18. The "
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+                                    "type": "text",
+                                    "content": "-norm of the Jacobian-transpose of a transformation "
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+                                    "type": "inline_equation",
+                                    "content": "s(x)"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " applied elementwise to a product space is bounded by the "
+                                },
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+                                    "type": "text",
+                                    "content": "-norm of the Jacobian-transpose of a single transformation from that product space, i.e., "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "\\|J(s(\\pmb{x}))^\\top\\|_\\infty \\leq \\max_{x_i \\in \\mathcal{X}_i} \\|J(s(x_i))^\\top\\|_\\infty"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " for any "
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+                                    "type": "inline_equation",
+                                    "content": "i"
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+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
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+                                    "type": "text",
+                                    "content": "Proof. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        506,
+                                        389
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\boldsymbol{x} \\in \\mathcal{X} = \\prod_{i=1}^{n} \\mathcal{X}_i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        506,
+                                        389
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        506,
+                                        389
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{Z} = \\prod_{i=1}^{n} \\mathcal{Z}_i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        506,
+                                        389
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        506,
+                                        389
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S: \\mathcal{X} \\to \\mathcal{Z} = [s(x_1); \\dots; s(x_n)]^\\top"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        506,
+                                        389
+                                    ],
+                                    "type": "text",
+                                    "content": " where; denotes column-wise stacking, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        506,
+                                        389
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_i \\in \\mathcal{X}_i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        506,
+                                        389
+                                    ],
+                                    "type": "text",
+                                    "content": ". Also, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        506,
+                                        389
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}_i = \\mathcal{X}_j"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        506,
+                                        389
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        506,
+                                        389
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{Z}_i = \\mathcal{Z}_j"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        506,
+                                        389
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        506,
+                                        389
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        506,
+                                        389
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        506,
+                                        389
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "j"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        506,
+                                        389
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then the Jacobian of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        506,
+                                        389
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S(\\boldsymbol{x})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        506,
+                                        389
+                                    ],
+                                    "type": "text",
+                                    "content": " is"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        184,
+                        394,
+                        504,
+                        453
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                184,
+                                394,
+                                504,
+                                453
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        184,
+                                        394,
+                                        504,
+                                        453
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "J (S (\\boldsymbol {x})) ^ {\\top} = \\left[ \\begin{array}{c c c} J (s (x _ {1})) ^ {\\top} & 0 \\dots & 0 \\\\ 0 & J (s (x _ {2})) ^ {\\top} \\dots & 0 \\\\ 0 & 0 \\ddots & 0 \\\\ 0 & 0 \\dots & J (s (x _ {n})) ^ {\\top} \\end{array} \\right]. \\tag {61}",
+                                    "image_path": "fc8890e81b57bd38ad2e4e29a1f35f8ff7c274789032e4c2fa2d8ea52ce67df3.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        104,
+                        457,
+                        506,
+                        513
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                457,
+                                506,
+                                513
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        457,
+                                        506,
+                                        513
+                                    ],
+                                    "type": "text",
+                                    "content": "The "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        457,
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+                                        513
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\infty"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        457,
+                                        506,
+                                        513
+                                    ],
+                                    "type": "text",
+                                    "content": "-norm of this matrix is the max 1-norm of any row. This matrix is diagonal, therefore, the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        457,
+                                        506,
+                                        513
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\infty"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        457,
+                                        506,
+                                        513
+                                    ],
+                                    "type": "text",
+                                    "content": "-norm of each elementwise Jacobian-transpose represents the max 1-norm of the rows spanned by its block. Given that the domains, ranges, and transformations "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        457,
+                                        506,
+                                        513
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        457,
+                                        506,
+                                        513
+                                    ],
+                                    "type": "text",
+                                    "content": " for all blocks are the same, their "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        457,
+                                        506,
+                                        513
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\infty"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        457,
+                                        506,
+                                        513
+                                    ],
+                                    "type": "text",
+                                    "content": "-norms are also the same. The max "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        457,
+                                        506,
+                                        513
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\infty"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        457,
+                                        506,
+                                        513
+                                    ],
+                                    "type": "text",
+                                    "content": " over the blocks is then equal to the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        457,
+                                        506,
+                                        513
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\infty"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        457,
+                                        506,
+                                        513
+                                    ],
+                                    "type": "text",
+                                    "content": "-norm of any individual "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        457,
+                                        506,
+                                        513
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "J(s(x_i))^\\top"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        457,
+                                        506,
+                                        513
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        105,
+                        526,
+                        318,
+                        538
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                526,
+                                318,
+                                538
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        526,
+                                        318,
+                                        538
+                                    ],
+                                    "type": "text",
+                                    "content": "B.1.1 HESSIAN OF BANDIT REWARD FUNCTION"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        104,
+                        546,
+                        506,
+                        599
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                546,
+                                506,
+                                599
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        546,
+                                        506,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 19. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        546,
+                                        506,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s(x)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        546,
+                                        506,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": " be a function that maps the unit hypercube to the simplex product (mixed strategy space). Then the objective function "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        546,
+                                        506,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f(x) = -\\mathcal{L}(s(x))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        546,
+                                        506,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": ". The Hessian of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        546,
+                                        506,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "-f(x)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        546,
+                                        506,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": " at an optimum "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        546,
+                                        506,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x^{*}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        546,
+                                        506,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": " in direction "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        546,
+                                        506,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        546,
+                                        506,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": " is "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        546,
+                                        506,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta x^{\\top}[Ds(x)^{\\top}H_{\\mathcal{L}}(x)Ds(x)]\\Big|_{x^{*}}\\Delta x"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        546,
+                                        506,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        546,
+                                        506,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H_{\\mathcal{L}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        546,
+                                        506,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": " is the Hessian of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        546,
+                                        506,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        546,
+                                        506,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        546,
+                                        506,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Ds(x)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        546,
+                                        506,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": " is the Jacobian of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        546,
+                                        506,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s(x)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        546,
+                                        506,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        105,
+                        614,
+                        133,
+                        625
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                614,
+                                133,
+                                625
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        614,
+                                        133,
+                                        625
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        164,
+                        632,
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+                    ],
+                    "type": "interline_equation",
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+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\left. \\left(D ^ {2} (\\mathcal {L} \\circ s) \\left(x ^ {*}\\right)\\right) (\\Delta x, \\Delta x) = \\Delta x ^ {\\top} \\left[ \\sum_ {i} \\overbrace {\\partial_ {i} \\mathcal {L} (s (x))} ^ {= 0 \\text {a t} x = x ^ {*}} D ^ {2} h _ {i} (x) \\right] \\right| _ {x ^ {*}} \\Delta x (62a) \\\\ + \\Delta x ^ {\\top} [ D s (x) ^ {\\top} H _ {\\mathcal {L}} (x) D s (x) ] \\bigg | _ {x ^ {*}} \\Delta x \\\\ = \\Delta x ^ {\\top} \\left[ D s (x) ^ {\\top} H _ {\\mathcal {L}} (x) D s (x) \\right] \\Big | _ {x ^ {*}} \\Delta x. (62b) \\\\ \\end{array}",
+                                    "image_path": "e4919e925b82bcd07e1739e61202be3148b34789510fe4d6af22a04f639b36af.jpg"
+                                }
+                            ]
+                        }
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+                    "type": "image",
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+                                            "type": "image",
+                                            "image_path": "25a01722fe07d536f17dc9ae0bfaee5e73934a66ff0b933fce7eae71fe419569.jpg"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 17,
+                            "angle": 0,
+                            "type": "image_body"
+                        }
+                    ],
+                    "index": 17
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                38
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+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
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+                    "angle": 0,
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+                                {
+                                    "bbox": [
+                                        299,
+                                        750,
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+                                    "type": "text",
+                                    "content": "26"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                }
+            ],
+            "page_size": [
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+            "page_idx": 25
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
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+                        82,
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+                        115
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        103,
+                                        82,
+                                        506,
+                                        115
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 20. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        82,
+                                        506,
+                                        115
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s(x): \\mathcal{X} \\to \\prod_k \\Delta^{m_k - 1}"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        82,
+                                        506,
+                                        115
+                                    ],
+                                    "type": "text",
+                                    "content": " be an injective function, i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        82,
+                                        506,
+                                        115
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x \\neq y \\Rightarrow s(x) \\neq s(y)"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        82,
+                                        506,
+                                        115
+                                    ],
+                                    "type": "text",
+                                    "content": ". Also let "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        82,
+                                        506,
+                                        115
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "J = J(s(x))"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        82,
+                                        506,
+                                        115
+                                    ],
+                                    "type": "text",
+                                    "content": " be the Jacobian of "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        82,
+                                        506,
+                                        115
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        82,
+                                        506,
+                                        115
+                                    ],
+                                    "type": "text",
+                                    "content": " with respect to "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        82,
+                                        506,
+                                        115
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        82,
+                                        506,
+                                        115
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        82,
+                                        506,
+                                        115
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta x"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        82,
+                                        506,
+                                        115
+                                    ],
+                                    "type": "text",
+                                    "content": " be a nonzero vector in the tangent space of "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        82,
+                                        506,
+                                        115
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        82,
+                                        506,
+                                        115
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        282,
+                        118,
+                        504,
+                        129
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                282,
+                                118,
+                                504,
+                                129
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        282,
+                                        118,
+                                        504,
+                                        129
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "J \\Delta x \\neq 0. \\tag {63}",
+                                    "image_path": "d256b364b1efaa2d638498623cd6d4a6b16ae072c175b237994d54e6d4f8ceda.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        142,
+                        485,
+                        159
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                142,
+                                485,
+                                159
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        142,
+                                        485,
+                                        159
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Recall that the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        142,
+                                        485,
+                                        159
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "ij"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        142,
+                                        485,
+                                        159
+                                    ],
+                                    "type": "text",
+                                    "content": "th entry of the Jacobian represents "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        142,
+                                        485,
+                                        159
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\frac{\\partial s_i}{\\partial x_j}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        142,
+                                        485,
+                                        159
+                                    ],
+                                    "type": "text",
+                                    "content": " so that the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        142,
+                                        485,
+                                        159
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        142,
+                                        485,
+                                        159
+                                    ],
+                                    "type": "text",
+                                    "content": "th entry of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        142,
+                                        485,
+                                        159
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "J\\Delta x"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        142,
+                                        485,
+                                        159
+                                    ],
+                                    "type": "text",
+                                    "content": " is"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        241,
+                        160,
+                        504,
+                        188
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                241,
+                                160,
+                                504,
+                                188
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        241,
+                                        160,
+                                        504,
+                                        188
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "[ J \\Delta x ] _ {i} = \\sum_ {j} \\frac {\\partial s _ {i}}{\\partial x _ {j}} \\Delta x _ {j} = d s _ {i}. \\tag {64}",
+                                    "image_path": "02797e5fd031fcc1e41bde6cf117e772f3ffb6f76929318718352d509e1e8c49.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        190,
+                        506,
+                        213
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                190,
+                                506,
+                                213
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": "Assume "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "J\\Delta x = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": ". This would imply a change in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x \\in \\mathcal{X}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": " results in no change in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": " ("
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "ds = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": "), contradicting the fact that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": " is injective. Therefore, we must conclude the claim that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "J\\Delta x \\neq 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        218,
+                        504,
+                        243
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                218,
+                                504,
+                                243
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        218,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 21. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        218,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "J = J(s(x))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        218,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "text",
+                                    "content": " be the Jacobian of any composition of transformations "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        218,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s = s_t \\circ \\ldots \\circ s_1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        218,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        218,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s_t(z) = [z_i / \\sum_j z_j]_i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        218,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        218,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "J\\Delta x"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        218,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "text",
+                                    "content": " lies in the tangent space of the simplex."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        255,
+                        504,
+                        282
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                255,
+                                504,
+                                282
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        282
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. We aim to show "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        282
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{1}^{\\top}J\\Delta x = \\mathbf{0}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        282
+                                    ],
+                                    "type": "text",
+                                    "content": " for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        282
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta x"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        282
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        282
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        282
+                                    ],
+                                    "type": "text",
+                                    "content": ". By chain rule, the Jacobian of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        282
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        282
+                                    ],
+                                    "type": "text",
+                                    "content": " is "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        282
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "J = J(s) = \\prod_{t'=t}^{t'=1} J(s_t')"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        282
+                                    ],
+                                    "type": "text",
+                                    "content": ". Therefore, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        282
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{1}^{\\top}J\\Delta x = \\mathbf{1}^{\\top}(\\prod_{t'=t}^{t'=1} J(s_t'))\\Delta x"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        282
+                                    ],
+                                    "type": "text",
+                                    "content": ". Consider the first product:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        276,
+                        284,
+                        504,
+                        297
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                276,
+                                284,
+                                504,
+                                297
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        276,
+                                        284,
+                                        504,
+                                        297
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {1} ^ {\\top} J \\left(s _ {t}\\right) = \\mathbf {0} \\tag {65}",
+                                    "image_path": "97a5c715c42ba2031295aa97de6ac05d9e1ea618e932930e3d81d50332e5c884.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        300,
+                        506,
+                        336
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                300,
+                                506,
+                                336
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "text",
+                                    "content": "by Lemma 24. Therefore "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{1}^{\\top}J\\Delta x = \\mathbf{1}^{\\top}J(s_{t})(\\prod_{t^{\\prime} = t - 1}^{t^{\\prime} = 1}J(s_{t}^{\\prime}))\\Delta x = \\mathbf{0}^{\\top}(\\prod_{t^{\\prime} = t - 1}^{t^{\\prime} = 1}J(s_{t}^{\\prime}))\\Delta x = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "text",
+                                    "content": ". This implies "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "J\\Delta x"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "text",
+                                    "content": " is orthogonal to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "text",
+                                    "content": " for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x\\in \\mathcal{X}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta x"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "text",
+                                    "content": ", therefore "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "J\\Delta x"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "text",
+                                    "content": " lies in the tangent space of the simplex for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x\\in \\mathcal{X}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta x"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        105,
+                        348,
+                        209,
+                        358
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                348,
+                                209,
+                                358
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        348,
+                                        209,
+                                        358
+                                    ],
+                                    "type": "text",
+                                    "content": "B.1.2 SOFTMAX MAP"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        366,
+                        506,
+                        433
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                366,
+                                506,
+                                433
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        366,
+                                        506,
+                                        433
+                                    ],
+                                    "type": "text",
+                                    "content": "Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        366,
+                                        506,
+                                        433
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s:[0,1]^{d - 1}\\to \\Delta^{d - 1}\\in \\mathbb{R}^d"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        366,
+                                        506,
+                                        433
+                                    ],
+                                    "type": "text",
+                                    "content": " be the softmax function. See (Gao and Pavel, 2017) for an analysis of many of its properties and in the context of game theory. Note that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        366,
+                                        506,
+                                        433
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        366,
+                                        506,
+                                        433
+                                    ],
+                                    "type": "text",
+                                    "content": " maps a "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        366,
+                                        506,
+                                        433
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(d - 1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        366,
+                                        506,
+                                        433
+                                    ],
+                                    "type": "text",
+                                    "content": " dimensional variable to a "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        366,
+                                        506,
+                                        433
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        366,
+                                        506,
+                                        433
+                                    ],
+                                    "type": "text",
+                                    "content": " dimensional distribution. This can be practically handled by always appending a 0 to the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        366,
+                                        506,
+                                        433
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(d - 1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        366,
+                                        506,
+                                        433
+                                    ],
+                                    "type": "text",
+                                    "content": "-dimensional input prior to applying the standard softmax. We perform our analysis below in terms of the standard softmax, but note the norms we derive apply to our modified (invertible) softmax."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        105,
+                        439,
+                        146,
+                        449
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                439,
+                                146,
+                                449
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        439,
+                                        146,
+                                        449
+                                    ],
+                                    "type": "text",
+                                    "content": "Standard:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        237,
+                        449,
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        237,
+                                        449,
+                                        504,
+                                        478
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "s (x) = \\frac {1}{\\sum_ {j = 1} ^ {d} e ^ {x _ {j}}} \\left[ e ^ {x _ {1}}, \\dots e ^ {x _ {d}} \\right]. \\tag {66}",
+                                    "image_path": "79c5a6620351e53671dc76e4c338ff616ea110450c5bfec4206117e8d34a6272.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        105,
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+                        495
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "type": "text",
+                                    "content": "Modified:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        219,
+                        495,
+                        504,
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                524
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        219,
+                                        495,
+                                        504,
+                                        524
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "s (x) = \\frac {1}{1 + \\sum_ {j = 1} ^ {d - 1} e ^ {x _ {j}}} \\left[ e ^ {x _ {1}}, \\dots e ^ {x _ {d - 1}}, 1 \\right]. \\tag {67}",
+                                    "image_path": "58385a77e67c305d8764edc93be7f949281edaf8e9b736da99adb4c9f581d590.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        104,
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+                        539
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
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+                                        539
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 22. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        488,
+                                        539
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "J"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        488,
+                                        539
+                                    ],
+                                    "type": "text",
+                                    "content": " be the Jacobian of the softmax operator. Then "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        488,
+                                        539
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\| J\\|_{\\infty}\\leq 2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        488,
+                                        539
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        525,
+                                        488,
+                                        539
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\| J^{\\top}\\|_{\\infty}\\leq 2"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        104,
+                        549,
+                        504,
+                        572
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                549,
+                                504,
+                                572
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                        504,
+                                        572
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        572
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S_{i}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        572
+                                    ],
+                                    "type": "text",
+                                    "content": " represent the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        572
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        572
+                                    ],
+                                    "type": "text",
+                                    "content": "th entry of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        572
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S = \\mathrm{softmax}(z)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        572
+                                    ],
+                                    "type": "text",
+                                    "content": " for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        572
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z \\in \\mathbb{R}^{m}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        572
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then the 1-norm of row "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        572
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        572
+                                    ],
+                                    "type": "text",
+                                    "content": " is upper bounded as"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        226,
+                        590,
+                        504,
+                        614
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                226,
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+                                504,
+                                614
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        226,
+                                        590,
+                                        504,
+                                        614
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\Rightarrow \\sum_ {j} \\left| D _ {j} S _ {i} \\right| = \\sum_ {j} \\left| S _ {i} \\left(\\delta_ {i j} - S _ {j}\\right) \\right| \\tag {68b}",
+                                    "image_path": "c931e1eeca3e9061819d886a17969ea5b8d45dbbedbb16e34b6ab6bf6de1e7b9.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
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+                        729
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} D _ {j} S _ {i} = S _ {i} \\left(\\delta_ {i j} - S _ {j}\\right) (68a) \\\\ \\leq \\sum_ {j} \\left| \\delta_ {i j} S _ {i} \\right| + \\left| S _ {i} S _ {j} \\right| (68c) \\\\ = S _ {i} + \\sum_ {j} S _ {i} S _ {j} (68d) \\\\ = S _ {i} + S _ {i} \\sum_ {j} S _ {j} (68e) \\\\ = 2 S _ {i} (68f) \\\\ \\leq 2 \\forall i. (68g) \\\\ \\end{array}",
+                                    "image_path": "b28f461927501492ffc92f558c64e5550fe913a95b7289c09bd0318e95c532f6.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                }
+            ],
+            "discarded_blocks": [
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+                    ],
+                    "type": "header",
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
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+                                    "type": "text",
+                                    "content": "27"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                }
+            ],
+            "page_size": [
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+            ],
+            "page_idx": 26
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
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+                                    "bbox": [
+                                        104,
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+                                        95
+                                    ],
+                                    "type": "text",
+                                    "content": "Also, the 1-norm of column "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        347,
+                                        95
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "j"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        347,
+                                        95
+                                    ],
+                                    "type": "text",
+                                    "content": " is upper bounded similarly as"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        235,
+                        102,
+                        505,
+                        254
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                235,
+                                102,
+                                505,
+                                254
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        235,
+                                        102,
+                                        505,
+                                        254
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} (6 9 a) \\\\ \\sum_ {i} \\left| D _ {j} S _ {i} \\right| = \\sum_ {i} \\left| S _ {i} \\left(\\delta_ {i j} - S _ {j}\\right) \\right| (69b) \\\\ \\leq \\sum_ {i} \\left| \\delta_ {i j} S _ {i} \\right| + \\left| S _ {i} S _ {j} \\right| (69c) \\\\ = S _ {j} + \\sum_ {i} S _ {i} S _ {j} (69d) \\\\ = S _ {i} + S _ {j} \\sum_ {i} S _ {i} (69e) \\\\ = 2 S _ {j} (69f) \\\\ \\leq 2 \\forall j. (69g) \\\\ \\end{array}",
+                                    "image_path": "9f285f83fe3ddde12e798990c09eccb16d95df4ea835fc21d3cf782906972ed6.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        262,
+                        506,
+                        287
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                262,
+                                506,
+                                287
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        262,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": "The "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        262,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\infty"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        262,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": "-norm of a matrix is the maximum 1-norm of any row. Therefore, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        262,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "||J||_{\\infty}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        262,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        262,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "||J^{\\top}||_{\\infty}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        262,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": " are both upper bounded by 2."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        300,
+                        484,
+                        314
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                300,
+                                484,
+                                314
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        484,
+                                        314
+                                    ],
+                                    "type": "text",
+                                    "content": "Corollary 3. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        484,
+                                        314
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "J"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        484,
+                                        314
+                                    ],
+                                    "type": "text",
+                                    "content": " be the Jacobian of the softmax operator. Then "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        484,
+                                        314
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\| J\\| _2\\leq 2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        484,
+                                        314
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        300,
+                                        484,
+                                        314
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\| J^{\\top}\\|_{2}\\leq 2"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        335,
+                        506,
+                        402
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                335,
+                                506,
+                                402
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        506,
+                                        402
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. The Gershgorin circle theorem states that every eigenvalue of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        506,
+                                        402
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "J"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        506,
+                                        402
+                                    ],
+                                    "type": "text",
+                                    "content": " lies within one of the discs centered at the diagonal of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        506,
+                                        402
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "J"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        506,
+                                        402
+                                    ],
+                                    "type": "text",
+                                    "content": " with radius equal to the 1-norm of the row (excluding the diagonal term). The Jacobian of the softmax operator contains only nonnegative diagonal terms, therefore, the 1-norm of any entire row also represents the maximum magnitude of any eigenvalue allowed by any Gershgorin disc. In addition, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        506,
+                                        402
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "J"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        506,
+                                        402
+                                    ],
+                                    "type": "text",
+                                    "content": " is symmetric and therefore, the magnitude of its eigenvalues are equivalent to its singular values. Therefore by Lemma 22, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        506,
+                                        402
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "||J||_2 \\leq 2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        506,
+                                        402
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        105,
+                        423,
+                        214,
+                        434
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
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+                                434
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "B.1.3 SPHERICAL MAP"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
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+                                    "bbox": [
+                                        104,
+                                        443,
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+                                    ],
+                                    "type": "text",
+                                    "content": "For spherical coordinates, let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        443,
+                                        504,
+                                        468
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s(x) = n(l(c(x)))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        443,
+                                        504,
+                                        468
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        443,
+                                        504,
+                                        468
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "c(x) = \\pi / 2x"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        443,
+                                        504,
+                                        468
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        443,
+                                        504,
+                                        468
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "l(\\psi)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        443,
+                                        504,
+                                        468
+                                    ],
+                                    "type": "text",
+                                    "content": " maps angles to the unit sphere, and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        443,
+                                        504,
+                                        468
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n(z) = [z_i / \\sum_j z_j]_i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        443,
+                                        504,
+                                        468
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        104,
+                        473,
+                        488,
+                        485
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "Definition 2. Define "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        473,
+                                        488,
+                                        485
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "l(\\psi)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        473,
+                                        488,
+                                        485
+                                    ],
+                                    "type": "text",
+                                    "content": " as the transformation to the unit-sphere using spherical coordinates:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "l _ {1} (\\psi) = \\cos (\\psi_ {1}) \\tag {70a}",
+                                    "image_path": "ba9a7c780b8ce931f0ae794faa33eb8ebc763ccbbfff4d846f9d756a6168e622.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
+                                        230,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "l _ {2} (\\psi) = \\sin \\left(\\psi_ {1}\\right) \\cos \\left(\\psi_ {2}\\right) \\tag {70b}",
+                                    "image_path": "8de4304c4a7c14f5e1c824de60230c61d8509a82e2fcb610844a42ddea7fce2c.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        230,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "l _ {3} (\\psi) = \\sin (\\psi_ {1}) \\sin (\\psi_ {2}) \\cos (\\psi_ {3}) \\tag {70c}",
+                                    "image_path": "cc0fabc2927a004460df82162cc08948fe1dc81d1d60e8d40f300c1caf6cbd0a.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
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+                    ],
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+                            "bbox": [
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\vdots = \\vdots \\tag {70d}",
+                                    "image_path": "65ed8a6f8cefc8dd22ef1911a981d9ab1c046e21f2751712e760294ec3f9d1c0.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
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+                                {
+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "l _ {m - 1} (\\psi) = \\sin \\left(\\psi_ {1}\\right) \\sin \\left(\\psi_ {2}\\right) \\dots \\cos \\left(\\psi_ {m - 1}\\right) \\tag {70e}",
+                                    "image_path": "e2703153fa3dfdf1a809da7c1d956c7401f33417b12364cd7cbc56fe7f0afdac.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
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+                {
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+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "l _ {m} (\\psi) = \\sin \\left(\\psi_ {1}\\right) \\sin \\left(\\psi_ {2}\\right) \\dots \\sin \\left(\\psi_ {m - 1}\\right). \\tag {70f}",
+                                    "image_path": "7e2c93e7ae5e2af58a2f5138642e9ef76b20276664ad19dcaa2a63390603f33c.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
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+                                        617
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 23. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        617
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "J"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        617
+                                    ],
+                                    "type": "text",
+                                    "content": " be the Jacobian of the transformation to the unit-sphere using spherical coordinates, i.e. "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        617
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z = l(\\psi)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        617
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        617
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\|l\\|^2 = 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        617
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        617
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\psi_i \\in [0, \\frac{\\pi}{2}]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        617
+                                    ],
+                                    "type": "text",
+                                    "content": " represents an angle for each "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        617
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        617
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        617
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\|J\\|_F \\leq \\sqrt{m}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        617
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
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+                        648
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. The Jacobian of the transformation is"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
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+                    ],
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+                                    "type": "interline_equation",
+                                    "content": "J (l) = \\left[ \\begin{array}{c c c c} - \\sin (\\psi_ {1}) & 0 & \\dots & 0 \\\\ \\cos (\\psi_ {1}) \\cos (\\psi_ {2}) & - \\sin (\\psi_ {1}) \\sin (\\psi_ {2}) & \\dots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\cos (\\psi_ {1}) \\sin (\\psi_ {2}) \\dots \\cos (\\psi_ {m - 1}) & \\dots & \\dots & - \\sin (\\psi_ {1}) \\dots \\sin (\\psi_ {m - 2}) \\sin (\\psi_ {m - 1}) \\\\ \\cos (\\psi_ {1}) \\sin (\\psi_ {2}) \\dots \\sin (\\psi_ {m - 1}) & \\dots & \\dots & \\sin (\\psi_ {1}) \\dots \\sin (\\psi_ {m - 2}) \\cos (\\psi_ {m - 1}) \\end{array} \\right] \\tag {71}",
+                                    "image_path": "fbc00a1ad8a7f35c34c93db48b85933333c76fe7cf4e194eb88633bf83b3d209.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                }
+            ],
+            "discarded_blocks": [
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
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+                },
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+                                    "content": "28"
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+                                    "type": "text",
+                                    "content": "and it square is"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "J (l) = \\left[ \\begin{array}{c c c c} t _ {1} & 0 & \\dots & 0 \\\\ \\cos (\\psi_ {1}) ^ {2} \\cos (\\psi_ {2}) ^ {2} & \\sin (\\psi_ {1}) ^ {2} t _ {2} & \\dots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\cos (\\psi_ {1}) ^ {2} \\sin (\\psi_ {2}) ^ {2} \\dots \\cos (\\psi_ {m - 1}) ^ {2} & \\dots & \\dots & \\sin (\\psi_ {1}) ^ {2} \\dots \\sin (\\psi_ {m - 2}) ^ {2} t _ {m - 1} \\\\ \\cos (\\psi_ {1}) ^ {2} \\sin (\\psi_ {2}) ^ {2} \\dots \\sin (\\psi_ {m - 1}) ^ {2} & \\dots & \\dots & \\sin (\\psi_ {1}) ^ {2} \\dots \\sin (\\psi_ {m - 2}) ^ {2} t _ {m} \\end{array} \\right] \\tag {72}",
+                                    "image_path": "7f2d4c2e9fa1efb8e5e56d4f61ca7d3e04839c64b4cb8b1d888878d01f67c544.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
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+                    ],
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+                    "angle": 0,
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "where"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\delta_ {i m} = 1 \\text {i f} i = m, 0 \\text {e l s e} \\tag {73a}",
+                                    "image_path": "38d6df8e787f65f14596337f1baefb749b6fc53ec5867999c8da3423fe5cae52.jpg"
+                                }
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "t _ {i} = \\delta_ {i m} \\cos^ {2} \\left(\\psi_ {i - 1}\\right) + \\left(1 - \\delta_ {i m}\\right) \\sin^ {2} \\left(\\psi_ {i}\\right) \\leq 1. \\tag {73b}",
+                                    "image_path": "9cad63e9a1ec16b260cb432dc3da67cdb7b2af9ff3d978cce9f31878778419b3.jpg"
+                                }
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+                    "lines": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "To compute the Frobenius norm, we will need the sum of the squares of all entries. We will consider the sum of each row individually using the following auxiliary variable "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "R_{i,k\\leq i}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        239,
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+                                        275
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
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+                                        504,
+                                        275
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sum_{j}J_{ij}^{2} = R_{i,1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        239,
+                                        504,
+                                        275
+                                    ],
+                                    "type": "text",
+                                    "content": " and apply a recursive inequality."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        481,
+                        281,
+                        504,
+                        293
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                481,
+                                281,
+                                504,
+                                293
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        481,
+                                        281,
+                                        504,
+                                        293
+                                    ],
+                                    "type": "text",
+                                    "content": "(74a)"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        111,
+                        296,
+                        505,
+                        523
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                111,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        111,
+                                        296,
+                                        505,
+                                        523
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} R _ {i, k \\leq i} = \\sum_ {k ^ {\\prime} = k} ^ {i - 1} \\cos^ {2} \\left(\\psi_ {k ^ {\\prime}}\\right) \\left[ \\prod_ {l = k, l \\neq k ^ {\\prime}} ^ {i - 1} \\sin^ {2} \\left(\\psi_ {l}\\right) \\right] \\cos^ {2} \\left(\\psi_ {i}\\right) + t _ {i} \\prod_ {l = k} ^ {i - 1} \\sin^ {2} \\left(\\psi_ {l}\\right) (74b) \\\\ = \\cos^ {2} \\left(\\psi_ {k}\\right) \\underbrace {\\left[ \\prod_ {l = k + 1} ^ {i - 1} \\sin^ {2} \\left(\\psi_ {l}\\right) \\right] \\cos^ {2} \\left(\\psi_ {i}\\right)} _ {\\leq 1} (74c) \\\\ + \\sin^ {2} \\left(\\psi_ {k}\\right) \\sum_ {k ^ {\\prime} = k + 1} ^ {i - 1} \\cos^ {2} \\left(\\psi_ {k ^ {\\prime}}\\right) \\left[ \\prod_ {l = k + 1, l \\neq k ^ {\\prime}} ^ {i - 1} \\sin^ {2} \\left(\\psi_ {l}\\right) \\right] \\cos^ {2} \\left(\\psi_ {i}\\right) (74d) \\\\ + \\sin^ {2} \\left(\\psi_ {k}\\right) t _ {i} \\prod_ {l = k + 1} ^ {i - 1} \\sin^ {2} \\left(\\psi_ {l}\\right) (74e) \\\\ \\leq \\cos^ {2} \\left(\\psi_ {k}\\right) (74f) \\\\ + \\sin^ {2} \\left(\\psi_ {k}\\right) \\left(\\sum_ {k ^ {\\prime} = k + 1} ^ {i - 1} \\cos^ {2} \\left(\\psi_ {k ^ {\\prime}}\\right) \\left[ \\prod_ {l = k + 1, l \\neq k ^ {\\prime}} ^ {i - 1} \\sin^ {2} \\left(\\psi_ {l}\\right) \\right] \\cos^ {2} \\left(\\psi_ {i}\\right) + t _ {i} \\prod_ {l = k + 1} ^ {i - 1} \\sin^ {2} \\left(\\psi_ {l}\\right)\\right) (7 4 g) \\\\ = \\cos^ {2} (\\psi_ {k}) + \\sin^ {2} (\\psi_ {k}) R _ {i, k + 1}. (74h) \\\\ \\end{array}",
+                                    "image_path": "bc7f837b7f2e6dc53d1dfd76c5f81dac11ea4de829110b09b9d8d298edc115a3.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
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+                        553
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                        506,
+                                        553
+                                    ],
+                                    "type": "text",
+                                    "content": "Note then that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        527,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "R_{i,k+1} \\leq 1 \\Rightarrow R_{i,k} \\leq 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        527,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "text",
+                                    "content": ". We know "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        527,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "R_{i,i} = t_i \\leq 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        527,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "text",
+                                    "content": ", therefore, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        527,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "R_{i,1} \\leq 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        527,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "text",
+                                    "content": " by applying the inequality recursively. Finally, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        527,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sum_j J_{ij}^2 = R_{i,1} \\leq 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        527,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "text",
+                                    "content": " implies the claim "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        527,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "||J||_F^2 = \\sum_i R_{i,1} \\leq m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        527,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        105,
+                        559,
+                        461,
+                        574
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                559,
+                                461,
+                                574
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        559,
+                                        461,
+                                        574
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 24. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        559,
+                                        461,
+                                        574
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "J"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        559,
+                                        461,
+                                        574
+                                    ],
+                                    "type": "text",
+                                    "content": " be the Jacobian of "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        559,
+                                        461,
+                                        574
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n(z) = z / Z"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        559,
+                                        461,
+                                        574
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        559,
+                                        461,
+                                        574
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Z = \\sum_{k}z_{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        559,
+                                        461,
+                                        574
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        559,
+                                        461,
+                                        574
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{1}^{\\top}J = \\mathbf{0}^{\\top}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        559,
+                                        461,
+                                        574
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        105,
+                        584,
+                        299,
+                        596
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                584,
+                                299,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        584,
+                                        299,
+                                        596
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. The "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        584,
+                                        299,
+                                        596
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        584,
+                                        299,
+                                        596
+                                    ],
+                                    "type": "text",
+                                    "content": "th entry of the Jacobian of "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        584,
+                                        299,
+                                        596
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n(z)"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        584,
+                                        299,
+                                        596
+                                    ],
+                                    "type": "text",
+                                    "content": " is"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        247,
+                        601,
+                        505,
+                        625
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                247,
+                                601,
+                                505,
+                                625
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        247,
+                                        601,
+                                        505,
+                                        625
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "J (n) _ {i j} = \\frac {1}{Z ^ {2}} \\left(- z _ {i} + \\delta_ {i j} Z\\right). \\tag {75}",
+                                    "image_path": "3814cb7bcddcc69a47c1dd14f4254f980a14a8fa5fc89655ed84aec12bc0883d.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        104,
+                        630,
+                        505,
+                        654
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                                654
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        630,
+                                        505,
+                                        654
+                                    ],
+                                    "type": "text",
+                                    "content": "Therefore "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        630,
+                                        505,
+                                        654
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "[ \\mathbf{1}^{\\top} J ]_{j} = \\sum_{i} J(n)_{ij} = \\frac{1}{Z^{2}} (-Z + Z) = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        630,
+                                        505,
+                                        654
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        630,
+                                        505,
+                                        654
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        630,
+                                        505,
+                                        654
+                                    ],
+                                    "type": "text",
+                                    "content": " is a point on the unit-sphere in the positive orthant."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        105,
+                        666,
+                        325,
+                        678
+                    ],
+                    "type": "title",
+                    "angle": 0,
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+                            ],
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "B.2 NEAR OPTIMALITY & ZOOMING DIMENSION"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
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+                            ],
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "In this subsection, we derive bounds on the near-optimality dimension and zooming dimension of the associated bandit optimization problems we use to derive global convergence guarantees to Nash equilibria. This is written in terms of maximizing a function "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " rather than minimizing a loss to better match the bandit literature (Bubeck et al., 2011; Valko et al., 2013; Feng et al., 2022)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                }
+            ],
+            "discarded_blocks": [
+                {
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+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
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+                                    "bbox": [
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+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
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+                                    "type": "text",
+                                    "content": "29"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                }
+            ],
+            "page_size": [
+                612,
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+            ],
+            "page_idx": 28
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Assumption 1. Locally around each interior "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x^{*} \\in \\mathcal{X}^{*}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
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+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
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+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "-f(x)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": " is lower bounded by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "-f(x^{*}) + \\sigma_{-}||x - x^{*}||^{\\alpha_{hi}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": ". In other words, for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f(x) \\geq f(x^{*}) - \\eta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": ":"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
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+                        504,
+                        127
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                236,
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+                                127
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        236,
+                                        113,
+                                        504,
+                                        127
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "f \\left(x ^ {*}\\right) - f (x) \\geq \\sigma_ {-} \\| x - x ^ {*} \\| ^ {\\alpha_ {h i}} \\tag {76}",
+                                    "image_path": "0dea4593f594cd81fb976e344c9425283b68a9b86d6597c490331637f5e6c52b.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
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+                        486,
+                        148
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                134,
+                                486,
+                                148
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        134,
+                                        486,
+                                        148
+                                    ],
+                                    "type": "text",
+                                    "content": "where we have left the precise norm unspecified for generality. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        134,
+                                        486,
+                                        148
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell(x, x^{*}) = \\sigma_{+}||x - x^{*}||^{\\alpha_{lo}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        134,
+                                        486,
+                                        148
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
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+                        167
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Definition 3. "
+                                },
+                                {
+                                    "bbox": [
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+                                        167
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}_{\\epsilon} \\stackrel{\\mathrm{def}}{=} \\{x \\in \\mathcal{X} \\mid \\exists x^{*} \\in \\mathcal{X}^{*}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        152,
+                                        370,
+                                        167
+                                    ],
+                                    "type": "text",
+                                    "content": " s.t. "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        152,
+                                        370,
+                                        167
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f(x) \\geq f(x^{*}) - \\epsilon\\}"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        172,
+                        470,
+                        188
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                172,
+                                470,
+                                188
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        172,
+                                        470,
+                                        188
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition 4. "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        172,
+                                        470,
+                                        188
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}_{\\epsilon}^{lower} \\stackrel{\\mathrm{def}}{=} \\{x \\in \\mathcal{X} \\mid \\exists x^* \\in \\mathcal{X}^* \\text{ s.t. } f(x^*) - \\sigma_{-} \\|x - x^*\\|^{\\alpha_{hi}} \\geq f(x^*) - \\epsilon\\}"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        190,
+                        220,
+                        204
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                190,
+                                220,
+                                204
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        220,
+                                        204
+                                    ],
+                                    "type": "text",
+                                    "content": "Corollary 4. "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        220,
+                                        204
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}_{\\varepsilon} \\subseteq \\mathcal{X}_{\\varepsilon}^{lower}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        220,
+                                        204
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        224,
+                        506,
+                        258
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                224,
+                                506,
+                                258
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        506,
+                                        258
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. By Assumption 1, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        506,
+                                        258
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f(x^{*}) - \\sigma_{-}||x - x^{*}||^{\\alpha_{hi}}\\geq f(x)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        506,
+                                        258
+                                    ],
+                                    "type": "text",
+                                    "content": ". Therefore, any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        506,
+                                        258
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x\\in \\mathcal{X}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        506,
+                                        258
+                                    ],
+                                    "type": "text",
+                                    "content": " that satisfies the requirement for an element of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        506,
+                                        258
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}_{\\epsilon}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        506,
+                                        258
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        506,
+                                        258
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f(x)\\geq f(x^{*}) - \\epsilon"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        506,
+                                        258
+                                    ],
+                                    "type": "text",
+                                    "content": ", will also satisfy the requirement for an element of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        506,
+                                        258
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}_{\\epsilon}^{lower}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        224,
+                                        506,
+                                        258
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        104,
+                        273,
+                        506,
+                        308
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                273,
+                                506,
+                                308
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        273,
+                                        506,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition 5 ("
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        273,
+                                        506,
+                                        308
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\psi"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        273,
+                                        506,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": "-near Optimality Dimension). The "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        273,
+                                        506,
+                                        308
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\psi"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        273,
+                                        506,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": "-near optimality dimension is the smallest "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        273,
+                                        506,
+                                        308
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d' > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        273,
+                                        506,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": " such that there exists "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        273,
+                                        506,
+                                        308
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        273,
+                                        506,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": " such that for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        273,
+                                        506,
+                                        308
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\epsilon > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        273,
+                                        506,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": ", the maximum number of disjoint "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        273,
+                                        506,
+                                        308
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        273,
+                                        506,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": "-balls of radius "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        273,
+                                        506,
+                                        308
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\psi \\epsilon"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        273,
+                                        506,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": " and center in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        273,
+                                        506,
+                                        308
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}_{\\epsilon}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        273,
+                                        506,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": " is less than "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        273,
+                                        506,
+                                        308
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C \\epsilon^{-d'}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        273,
+                                        506,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        312,
+                        505,
+                        349
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                312,
+                                505,
+                                349
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        312,
+                                        505,
+                                        349
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition 6 (Zooming Dimension). The zooming dimension is the smallest "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        312,
+                                        505,
+                                        349
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d_z > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        312,
+                                        505,
+                                        349
+                                    ],
+                                    "type": "text",
+                                    "content": " such that there exists "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        312,
+                                        505,
+                                        349
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_z > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        312,
+                                        505,
+                                        349
+                                    ],
+                                    "type": "text",
+                                    "content": " such that for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        312,
+                                        505,
+                                        349
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        312,
+                                        505,
+                                        349
+                                    ],
+                                    "type": "text",
+                                    "content": ", the maximum number of disjoint "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        312,
+                                        505,
+                                        349
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        312,
+                                        505,
+                                        349
+                                    ],
+                                    "type": "text",
+                                    "content": "-balls of radius "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        312,
+                                        505,
+                                        349
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\frac{r}{2}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        312,
+                                        505,
+                                        349
+                                    ],
+                                    "type": "text",
+                                    "content": " and center in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        312,
+                                        505,
+                                        349
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}_{16r}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        312,
+                                        505,
+                                        349
+                                    ],
+                                    "type": "text",
+                                    "content": " is less than "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        312,
+                                        505,
+                                        349
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_z r^{-d_z}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        312,
+                                        505,
+                                        349
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
+                        353,
+                        505,
+                        392
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                353,
+                                505,
+                                392
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        353,
+                                        505,
+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": "Corollary 5 (Zooming Dimension from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        353,
+                                        505,
+                                        392
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\psi"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        353,
+                                        505,
+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": "-near Optimality Dimension). The zooming dimension of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        353,
+                                        505,
+                                        392
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f: x \\in [0,1]^d \\to [-1,1]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        353,
+                                        505,
+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": " under "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        353,
+                                        505,
+                                        392
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell(x,y) = \\sigma_+||x - y||^{\\alpha_{lo}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        353,
+                                        505,
+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": " is "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        353,
+                                        505,
+                                        392
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d_z = d\\left(\\frac{\\alpha_{hi} - \\alpha_{lo}}{\\alpha_{lo}\\alpha_{hi}}\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        353,
+                                        505,
+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": " with constant "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        353,
+                                        505,
+                                        392
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_z = 16^{-d'}C"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        353,
+                                        505,
+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": " with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        353,
+                                        505,
+                                        392
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\psi = \\frac{1}{32}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        353,
+                                        505,
+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": " ("
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        353,
+                                        505,
+                                        392
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\psi"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        353,
+                                        505,
+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": " is needed to compute "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        353,
+                                        505,
+                                        392
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        353,
+                                        505,
+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": ")."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        412,
+                        504,
+                        446
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                412,
+                                504,
+                                446
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        412,
+                                        504,
+                                        446
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Mapping the definition of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        412,
+                                        504,
+                                        446
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\psi"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        412,
+                                        504,
+                                        446
+                                    ],
+                                    "type": "text",
+                                    "content": "-near optimality onto zooming dimension, we find "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        412,
+                                        504,
+                                        446
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\psi \\epsilon = r / 2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        412,
+                                        504,
+                                        446
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        412,
+                                        504,
+                                        446
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\epsilon = 16r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        412,
+                                        504,
+                                        446
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then we can infer "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        412,
+                                        504,
+                                        446
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\psi = 1 / 32"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        412,
+                                        504,
+                                        446
+                                    ],
+                                    "type": "text",
+                                    "content": "; this is used to compute "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        412,
+                                        504,
+                                        446
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        412,
+                                        504,
+                                        446
+                                    ],
+                                    "type": "text",
+                                    "content": " (see Theorem 1). Rewriting the bound from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        412,
+                                        504,
+                                        446
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\psi"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        412,
+                                        504,
+                                        446
+                                    ],
+                                    "type": "text",
+                                    "content": "-near optimality in terms of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        412,
+                                        504,
+                                        446
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        412,
+                                        504,
+                                        446
+                                    ],
+                                    "type": "text",
+                                    "content": ", we find"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        197,
+                        454,
+                        504,
+                        469
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                469
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        197,
+                                        454,
+                                        504,
+                                        469
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "N _ {\\epsilon} = N _ {1 6 r} \\leq C (1 6 r) ^ {- d ^ {\\prime}} = C 1 6 ^ {- d ^ {\\prime}} r ^ {- d ^ {\\prime}} = C _ {z} r ^ {- d _ {z}} \\tag {77}",
+                                    "image_path": "d0103f047fde3edd71eb77c668386b6a53feebef86627c2b03b0f670e0157ce2.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                513
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        477,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        477,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "N_{16r}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        477,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "text",
+                                    "content": " denotes the number of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        477,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        477,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "text",
+                                    "content": "-balls of radius "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        477,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r/2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        477,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "text",
+                                    "content": " with center in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        477,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}_{16r}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        477,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        477,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d_z = d'"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        477,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        477,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_z = C16^{-d'} = C16^{-d_z}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        477,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "text",
+                                    "content": ". Therefore, this translation only effects the constant "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        477,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_z"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        477,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "text",
+                                    "content": ", not the zooming dimension."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        104,
+                        528,
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                581
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 25 ("
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "N_{\\epsilon \\leq \\eta} \\leq C_{\\epsilon \\leq \\eta}\\epsilon^{-d'}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": "). The maximum number of disjoint "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "l"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": "-balls with radius "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\psi \\in"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": " and center in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}_{\\epsilon \\leq \\eta}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "N_{\\epsilon \\leq \\eta}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": ", is upper bounded by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_{\\epsilon \\leq \\eta}\\epsilon^{-d'}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_{\\epsilon \\leq \\eta} = |\\mathcal{X}^*| \\left( \\frac{\\sigma_+}{\\psi \\sigma_-^{\\alpha_{lo} / \\alpha_{hi}}} \\right)^{d / \\alpha_{lo}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d' = d \\left( \\frac{\\alpha_{hi} - \\alpha_{lo}}{\\alpha_{lo} \\alpha_{hi}} \\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        104,
+                        600,
+                        504,
+                        622
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        600,
+                                        504,
+                                        622
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. The number of disjoint "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        600,
+                                        504,
+                                        622
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        600,
+                                        504,
+                                        622
+                                    ],
+                                    "type": "text",
+                                    "content": "-balls of radius "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        600,
+                                        504,
+                                        622
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\psi \\epsilon"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        600,
+                                        504,
+                                        622
+                                    ],
+                                    "type": "text",
+                                    "content": " and center in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        600,
+                                        504,
+                                        622
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}_{\\epsilon \\leq \\eta}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        600,
+                                        504,
+                                        622
+                                    ],
+                                    "type": "text",
+                                    "content": " can be upper bounded as follows."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        105,
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+                        268,
+                        640
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        627,
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+                                    ],
+                                    "type": "text",
+                                    "content": "Rewrite "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        627,
+                                        268,
+                                        640
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}_{\\epsilon}^{lower}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        627,
+                                        268,
+                                        640
+                                    ],
+                                    "type": "text",
+                                    "content": " by rearranging terms as"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        170,
+                                        649,
+                                        504,
+                                        673
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathcal {X} _ {\\epsilon} ^ {l o w e r} = \\left\\{x \\in \\mathcal {X} | \\exists x ^ {*} \\in \\mathcal {X} ^ {*} s. t. \\left| \\left| x - x ^ {*} \\right| \\right| \\leq \\left(\\frac {\\epsilon}{\\sigma_ {-}}\\right) ^ {1 / \\alpha_ {h i}} \\stackrel {\\text {d e f}} {=} r _ {\\epsilon} \\right\\} \\tag {78}",
+                                    "image_path": "7d7bd1a8d29f25972643b8d2a2344fdf90a9f57eb49ee63bbb6286718448ec09.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        104,
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        683,
+                                        486,
+                                        696
+                                    ],
+                                    "type": "text",
+                                    "content": "and recall that from Corollary 4 that "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}_{\\epsilon} \\subseteq \\mathcal{X}_{\\epsilon}^{lower}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        683,
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+                                    ],
+                                    "type": "text",
+                                    "content": ". Furthermore, an "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        683,
+                                        486,
+                                        696
+                                    ],
+                                    "type": "text",
+                                    "content": "-ball of radius "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        683,
+                                        486,
+                                        696
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\psi \\epsilon"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        683,
+                                        486,
+                                        696
+                                    ],
+                                    "type": "text",
+                                    "content": " implies"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
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+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\ell (x, y) = \\sigma_ {+} | | x - y | | ^ {\\alpha_ {l o}} \\leq \\psi \\epsilon \\Longrightarrow | | x - y | | \\leq \\left(\\frac {\\psi \\epsilon}{\\sigma_ {+}}\\right) ^ {1 / \\alpha_ {l o}} \\stackrel {\\text {d e f}} {=} r _ {\\ell}. \\tag {79}",
+                                    "image_path": "919838a92bd35d6204fa53cc7c8251ae0fe02cb3bd929a2abd226b40e23ef099.jpg"
+                                }
+                            ]
+                        }
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+            ],
+            "discarded_blocks": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
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+                                    "type": "text",
+                                    "content": "30"
+                                }
+                            ]
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+                }
+            ],
+            "page_size": [
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+            "para_blocks": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "The number of disjoint "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "text",
+                                    "content": "-balls that can pack into a set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}_{\\epsilon}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                        105
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "N_{\\epsilon \\leq \\eta}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
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+                                        105
+                                    ],
+                                    "type": "text",
+                                    "content": ", is upper bounded by the ratio of the volumes of the two sets:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} N _ {\\epsilon \\leq \\eta} \\leq \\frac {V o l \\left(\\mathcal {X} _ {\\epsilon}\\right)}{V o l \\left(\\mathcal {B} _ {\\ell}\\right)} (80a) \\\\ \\leq \\frac {\\operatorname {V o l} \\left(\\mathcal {X} _ {\\epsilon} ^ {\\text {l o w e r}}\\right)}{\\operatorname {V o l} \\left(\\mathcal {B} _ {\\ell}\\right)} (80b) \\\\ = \\frac {\\left| \\mathcal {X} ^ {*} \\right| S _ {d} r _ {\\epsilon} ^ {d}}{S _ {d} r _ {\\ell} ^ {d}} (80c) \\\\ \\leq \\frac {\\left| \\mathcal {X} ^ {*} \\right| \\left(\\frac {\\epsilon}{\\sigma_ {-}}\\right) ^ {d / \\alpha_ {h i}}}{\\left(\\frac {\\psi \\epsilon}{\\sigma_ {+}}\\right) ^ {d / \\alpha_ {l o}}} (80d) \\\\ = \\left| \\mathcal {X} ^ {*} \\right| \\left(\\frac {\\sigma_ {+} ^ {1 / \\alpha_ {l o}} \\psi^ {- 1 / \\alpha_ {l o}}}{\\sigma_ {-} ^ {1 / \\alpha_ {h i}}}\\right) ^ {d} \\epsilon^ {d \\left(1 / \\alpha_ {h i} - 1 / \\alpha_ {l o}\\right)} (80e) \\\\ = \\left| \\mathcal {X} ^ {*} \\right| \\left(\\frac {\\sigma_ {+}}{\\psi \\sigma_ {-} ^ {\\alpha_ {l o} / \\alpha_ {h i}}}\\right) ^ {d / \\alpha_ {l o}} \\epsilon^ {- d \\left(\\frac {\\alpha_ {h i} - \\alpha_ {l o}}{\\alpha_ {l o} \\alpha_ {h i}}\\right)} (80f) \\\\ = C _ {\\epsilon \\leq \\eta} \\epsilon^ {- d ^ {\\prime}} (80g) \\\\ \\end{array}",
+                                    "image_path": "8ece6f4683db4e6d6de868166d86b342b99918186c4409bc7c0705959bd0295c.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
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+                        366
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                366
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        330,
+                                        504,
+                                        366
+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        330,
+                                        504,
+                                        366
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_{\\epsilon \\leq \\eta} = |\\mathcal{X}^{*}|\\left(\\frac{\\sigma_{+}}{\\psi\\sigma_{-}^{\\alpha_{lo} / \\alpha_{hi}}}\\right)^{d / \\alpha_{lo}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        330,
+                                        504,
+                                        366
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        330,
+                                        504,
+                                        366
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d' = d\\left(\\frac{\\alpha_{hi} - \\alpha_{lo}}{\\alpha_{lo}\\alpha_{hi}}\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        330,
+                                        504,
+                                        366
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        330,
+                                        504,
+                                        366
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "|\\mathcal{X}^{*}|"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        330,
+                                        504,
+                                        366
+                                    ],
+                                    "type": "text",
+                                    "content": " is the number of distinct global optima, and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        330,
+                                        504,
+                                        366
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S_{d}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        330,
+                                        504,
+                                        366
+                                    ],
+                                    "type": "text",
+                                    "content": " is the volume constant for a "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        330,
+                                        504,
+                                        366
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        330,
+                                        504,
+                                        366
+                                    ],
+                                    "type": "text",
+                                    "content": "-sphere under the given norm "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        330,
+                                        504,
+                                        366
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "||\\cdot||"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        330,
+                                        504,
+                                        366
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        376,
+                        504,
+                        411
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                376,
+                                504,
+                                411
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        376,
+                                        504,
+                                        411
+                                    ],
+                                    "type": "text",
+                                    "content": "Recall, these results apply when "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        376,
+                                        504,
+                                        411
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f(x) \\geq f(x^{*}) - \\eta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        376,
+                                        504,
+                                        411
+                                    ],
+                                    "type": "text",
+                                    "content": ", i.e., when "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        376,
+                                        504,
+                                        411
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\epsilon \\leq \\eta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        376,
+                                        504,
+                                        411
+                                    ],
+                                    "type": "text",
+                                    "content": ". Otherwise, we can upper bound the number of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        376,
+                                        504,
+                                        411
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        376,
+                                        504,
+                                        411
+                                    ],
+                                    "type": "text",
+                                    "content": "-balls by considering the entire set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        376,
+                                        504,
+                                        411
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        376,
+                                        504,
+                                        411
+                                    ],
+                                    "type": "text",
+                                    "content": " which has volume 1. First, we will bound the constant associated with the volume of a "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        376,
+                                        504,
+                                        411
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        376,
+                                        504,
+                                        411
+                                    ],
+                                    "type": "text",
+                                    "content": "-sphere."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        414,
+                        505,
+                        459
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                414,
+                                505,
+                                459
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        414,
+                                        505,
+                                        459
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 26 ("
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        414,
+                                        505,
+                                        459
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "N_{\\epsilon \\geq \\eta} \\leq C_{\\epsilon \\geq \\eta}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        414,
+                                        505,
+                                        459
+                                    ],
+                                    "type": "text",
+                                    "content": "). The maximum number of disjoint "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        414,
+                                        505,
+                                        459
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        414,
+                                        505,
+                                        459
+                                    ],
+                                    "type": "text",
+                                    "content": "-balls with radius "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        414,
+                                        505,
+                                        459
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\psi \\epsilon"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        414,
+                                        505,
+                                        459
+                                    ],
+                                    "type": "text",
+                                    "content": " and center in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        414,
+                                        505,
+                                        459
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}_{\\epsilon \\geq \\eta}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        414,
+                                        505,
+                                        459
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        414,
+                                        505,
+                                        459
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "N_{\\epsilon \\geq \\eta}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        414,
+                                        505,
+                                        459
+                                    ],
+                                    "type": "text",
+                                    "content": ", is upper bounded by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        414,
+                                        505,
+                                        459
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_{\\epsilon \\geq \\eta}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        414,
+                                        505,
+                                        459
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        414,
+                                        505,
+                                        459
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_{\\epsilon \\geq \\eta} = S_d^{-1}\\left(\\frac{\\sigma_+}{\\psi\\eta}\\right)^{d / \\alpha_{lo}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        414,
+                                        505,
+                                        459
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        414,
+                                        505,
+                                        459
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S_d"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        414,
+                                        505,
+                                        459
+                                    ],
+                                    "type": "text",
+                                    "content": " is the volume constant for a "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        414,
+                                        505,
+                                        459
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        414,
+                                        505,
+                                        459
+                                    ],
+                                    "type": "text",
+                                    "content": "-sphere under a given norm."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        470,
+                        504,
+                        495
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                470,
+                                504,
+                                495
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        504,
+                                        495
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. We can upper bound the number of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        504,
+                                        495
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        504,
+                                        495
+                                    ],
+                                    "type": "text",
+                                    "content": "-balls needed to pack the entire space by considering the smallest possible radius "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        504,
+                                        495
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\psi \\eta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        470,
+                                        504,
+                                        495
+                                    ],
+                                    "type": "text",
+                                    "content": ":"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        255,
+                        499,
+                        504,
+                        630
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                255,
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+                                630
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        255,
+                                        499,
+                                        504,
+                                        630
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} N _ {\\epsilon \\geq \\eta} \\leq \\frac {V o l (\\mathcal {X})}{V o l (\\mathcal {B} _ {\\ell})} (81a) \\\\ = \\frac {1}{S _ {d} r _ {\\ell} ^ {d}} (81b) \\\\ \\leq \\frac {1}{S _ {d} \\left(\\frac {\\psi \\eta}{\\sigma_ {+}}\\right) ^ {d / \\alpha_ {l o}}} (81c) \\\\ = S _ {d} ^ {- 1} \\left(\\frac {\\sigma_ {+}}{\\psi \\eta}\\right) ^ {d / \\alpha_ {l o}} (81d) \\\\ = C _ {\\epsilon \\geq \\eta} (81e) \\\\ \\end{array}",
+                                    "image_path": "188e7997a86c7a06891a1bf5e1a4eeb8dc8359115e6a8e27c73aea34af90187a.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        105,
+                        635,
+                        504,
+                        647
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                635,
+                                504,
+                                647
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        635,
+                                        504,
+                                        647
+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        635,
+                                        504,
+                                        647
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r_l"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        635,
+                                        504,
+                                        647
+                                    ],
+                                    "type": "text",
+                                    "content": " was defined in equation (79)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        653,
+                        505,
+                        680
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                653,
+                                505,
+                                680
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        653,
+                                        505,
+                                        680
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem 1. The "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        653,
+                                        505,
+                                        680
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\psi"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        653,
+                                        505,
+                                        680
+                                    ],
+                                    "type": "text",
+                                    "content": "-near optimality dimension of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        653,
+                                        505,
+                                        680
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f: x \\in [0,1]^d \\to [-1,1]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        653,
+                                        505,
+                                        680
+                                    ],
+                                    "type": "text",
+                                    "content": " under "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        653,
+                                        505,
+                                        680
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        653,
+                                        505,
+                                        680
+                                    ],
+                                    "type": "text",
+                                    "content": " is "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        653,
+                                        505,
+                                        680
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d' = d\\left(\\frac{\\alpha_{hi} - \\alpha_{lo}}{\\alpha_{lo}\\alpha_{hi}}\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        653,
+                                        505,
+                                        680
+                                    ],
+                                    "type": "text",
+                                    "content": " with constant"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        181,
+                        685,
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+                        714
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
+                                        181,
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+                                        714
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "C = \\max  \\left\\{1, S _ {d} ^ {- 1} \\left(r _ {\\eta} ^ {\\frac {\\alpha_ {h i}}{\\alpha_ {l o}}} \\sigma_ {-} ^ {\\left(\\frac {\\alpha_ {h i} - \\alpha_ {l o}}{\\alpha_ {l o} \\alpha_ {h i}}\\right)}\\right) ^ {- d} \\right\\} \\left(\\frac {\\sigma_ {+}}{\\psi \\sigma_ {-} ^ {\\alpha_ {l o} / \\alpha_ {h i}}}\\right) ^ {d / \\alpha_ {l o}} \\tag {82}",
+                                    "image_path": "7026819b857b616cbe7516c0703f5962eaa66cd01e2a7062025791fae313aa9d.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
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+                    "bbox": [
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+                            "bbox": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
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+                                    "type": "inline_equation",
+                                    "content": "S_{d}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        720,
+                                        403,
+                                        732
+                                    ],
+                                    "type": "text",
+                                    "content": " is the volume constant for a "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        720,
+                                        403,
+                                        732
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        720,
+                                        403,
+                                        732
+                                    ],
+                                    "type": "text",
+                                    "content": "-sphere under the same norm as "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        720,
+                                        403,
+                                        732
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        720,
+                                        403,
+                                        732
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                }
+            ],
+            "discarded_blocks": [
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+                    ],
+                    "type": "header",
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        127
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. First, let us define "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        127
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r_{\\eta} = \\left(\\frac{\\eta}{\\sigma_{-}}\\right)^{1 / \\alpha_{hi}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        127
+                                    ],
+                                    "type": "text",
+                                    "content": " as in equation (78) which implies "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        127
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\eta = \\sigma_{-}r_{\\eta}^{\\alpha_{hi}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        127
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then apply Lemmas 25 ("
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        127
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "N_{\\epsilon \\leq \\eta} \\leq C_{\\epsilon \\leq \\eta}\\epsilon^{-d'}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        127
+                                    ],
+                                    "type": "text",
+                                    "content": ") and 26 ("
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        127
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "N_{\\epsilon \\geq \\eta} \\leq C_{\\epsilon \\geq \\eta}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        127
+                                    ],
+                                    "type": "text",
+                                    "content": ") which bound the number of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        127
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        127
+                                    ],
+                                    "type": "text",
+                                    "content": "-balls required to pack "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        127
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}_{\\epsilon}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        127
+                                    ],
+                                    "type": "text",
+                                    "content": " when "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        127
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\epsilon"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        127
+                                    ],
+                                    "type": "text",
+                                    "content": " is less than and greater than "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        127
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\eta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        127
+                                    ],
+                                    "type": "text",
+                                    "content": " respectively:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        238,
+                        145,
+                        504,
+                        173
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                238,
+                                145,
+                                504,
+                                173
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        238,
+                                        145,
+                                        504,
+                                        173
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "C _ {\\epsilon \\leq \\eta} = \\left| \\mathcal {X} ^ {*} \\right| \\left(\\frac {\\sigma_ {+}}{\\psi \\sigma_ {-} ^ {\\alpha_ {l o} / \\alpha_ {h i}}}\\right) ^ {d / \\alpha_ {l o}} \\tag {83}",
+                                    "image_path": "93f5ae651515464829ccf28f24fd7aea13b370ccc9d062c28cc7692915fe0cab.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        252,
+                        177,
+                        504,
+                        200
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                252,
+                                177,
+                                504,
+                                200
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        252,
+                                        177,
+                                        504,
+                                        200
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "d ^ {\\prime} = d \\left(\\frac {\\alpha_ {h i} - \\alpha_ {l o}}{\\alpha_ {l o} \\alpha_ {h i}}\\right) \\tag {84}",
+                                    "image_path": "9ce662c354e6160fe416a87a3614ce86dc91ba6c15f959b15fd3a116297f4c73.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        105,
+                        203,
+                        123,
+                        214
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                203,
+                                123,
+                                214
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        203,
+                                        123,
+                                        214
+                                    ],
+                                    "type": "text",
+                                    "content": "and"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        185,
+                        217,
+                        504,
+                        243
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                185,
+                                217,
+                                504,
+                                243
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        185,
+                                        217,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "C _ {\\epsilon \\geq \\eta} = S _ {d} ^ {- 1} \\left(\\frac {\\sigma_ {+}}{\\psi \\eta}\\right) ^ {d / \\alpha_ {l o}} \\tag {85a}",
+                                    "image_path": "56ec96a5f364b18654623da373453c9179b526a492e254d35223b7b0a6630e96.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        210,
+                        244,
+                        504,
+                        272
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                210,
+                                244,
+                                504,
+                                272
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        210,
+                                        244,
+                                        504,
+                                        272
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "= \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} S _ {d} ^ {- 1} \\eta^ {- d / \\alpha_ {l o}} \\sigma_ {-} ^ {d / \\alpha_ {h i}} \\left| \\mathcal {X} ^ {*} \\right| \\left(\\frac {\\sigma_ {+}}{\\psi \\sigma_ {-} ^ {\\alpha_ {l o} / \\alpha_ {h i}}}\\right) ^ {d / \\alpha_ {l o}} \\tag {85b}",
+                                    "image_path": "86fbf2813b874b8ad7fe499c0ddb77cf489ccefc4b9bb620644ef074cd62dc3a.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        210,
+                        275,
+                        504,
+                        291
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                210,
+                                275,
+                                504,
+                                291
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        210,
+                                        275,
+                                        504,
+                                        291
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "= \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} S _ {d} ^ {- 1} \\eta^ {- d / \\alpha_ {l o}} \\sigma_ {-} ^ {d / \\alpha_ {h i}} C _ {\\epsilon \\leq \\eta} \\tag {85c}",
+                                    "image_path": "dac07ea68b4e86ff0a14af88ca5aeeeeeef1f6de6ae0ff3a428666c25727528b.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        210,
+                        293,
+                        504,
+                        310
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                210,
+                                293,
+                                504,
+                                310
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        210,
+                                        293,
+                                        504,
+                                        310
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "= \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} S _ {d} ^ {- 1} r _ {\\eta} ^ {- d \\alpha_ {h i} / \\alpha_ {l o}} \\sigma_ {-} ^ {- d / \\alpha_ {l o}} \\sigma_ {-} ^ {d / \\alpha_ {h i}} C _ {\\epsilon \\leq \\eta} \\tag {85d}",
+                                    "image_path": "a0bc8114913259345eb73363453a7a24212064a0716e07d09211888714817ac1.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        210,
+                        312,
+                        504,
+                        333
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                210,
+                                312,
+                                504,
+                                333
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        210,
+                                        312,
+                                        504,
+                                        333
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "= \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} S _ {d} ^ {- 1} r _ {\\eta} ^ {- d \\frac {\\alpha_ {h i}}{\\alpha_ {l o}}} \\sigma_ {-} ^ {- d \\left(\\frac {\\alpha_ {h i} - \\alpha_ {l o}}{\\alpha_ {l o} \\alpha_ {h i}}\\right)} C _ {\\epsilon \\leq \\eta} \\tag {85e}",
+                                    "image_path": "3fcbc35e1add49a4f5a4bd1fcc3c327399aaa19db5b5c4969ef1fee0e15c1005.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        210,
+                        335,
+                        504,
+                        360
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                210,
+                                335,
+                                504,
+                                360
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        210,
+                                        335,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "= \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} S _ {d} ^ {- 1} \\left(r _ {\\eta} ^ {\\frac {\\alpha_ {h i}}{\\alpha_ {l o}}} \\sigma_ {-} ^ {\\left(\\frac {\\alpha_ {h i} - \\alpha_ {l o}}{\\alpha_ {l o} \\alpha_ {h i}}\\right)}\\right) ^ {- d} C _ {\\epsilon \\leq \\eta} \\tag {85f}",
+                                    "image_path": "642af73ebc19f0dddfba9f825ec5c150a7ba28377c9de8a24bcfc3565af64fdb.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        364,
+                        504,
+                        401
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                364,
+                                504,
+                                401
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        504,
+                                        401
+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        504,
+                                        401
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S_{d}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        504,
+                                        401
+                                    ],
+                                    "type": "text",
+                                    "content": " is the volume constant for a "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        504,
+                                        401
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        504,
+                                        401
+                                    ],
+                                    "type": "text",
+                                    "content": "-sphere under the given norm. "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        504,
+                                        401
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S_{d}^{-1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        504,
+                                        401
+                                    ],
+                                    "type": "text",
+                                    "content": " has been upper bounded for the 2-norm in Lemma 35. For the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        504,
+                                        401
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\infty"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        504,
+                                        401
+                                    ],
+                                    "type": "text",
+                                    "content": "-norm, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        504,
+                                        401
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S_{d}^{-1} = 2^{-d}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        504,
+                                        401
+                                    ],
+                                    "type": "text",
+                                    "content": ". We have written "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        504,
+                                        401
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_{\\epsilon \\geq \\eta}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        504,
+                                        401
+                                    ],
+                                    "type": "text",
+                                    "content": " in terms of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        504,
+                                        401
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_{\\epsilon \\leq \\eta}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        504,
+                                        401
+                                    ],
+                                    "type": "text",
+                                    "content": " to clarify which is larger."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        105,
+                        405,
+                        149,
+                        416
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                405,
+                                149,
+                                416
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        405,
+                                        149,
+                                        416
+                                    ],
+                                    "type": "text",
+                                    "content": "Therefore,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        164,
+                        422,
+                        504,
+                        477
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                164,
+                                422,
+                                504,
+                                477
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        164,
+                                        422,
+                                        504,
+                                        477
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} C = \\max  \\left\\{1, | \\mathcal {X} ^ {*} | ^ {- 1} S _ {d} ^ {- 1} \\left(r \\eta^ {\\frac {\\alpha_ {h i}}{\\alpha_ {l o}}} \\sigma_ {-} ^ {\\left(\\frac {\\alpha_ {h i} - \\alpha_ {l o}}{\\alpha_ {l o} \\alpha_ {h i}}\\right)}\\right) ^ {- d} \\right\\} C _ {\\epsilon \\leq \\eta} (86a) \\\\ = \\max  \\left\\{1, \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} S _ {d} ^ {- 1} \\left(r _ {\\eta} ^ {\\frac {\\alpha_ {h i}}{\\alpha_ {l o}}} \\sigma_ {-} ^ {\\left(\\frac {\\alpha_ {h i} - \\alpha_ {l o}}{\\alpha_ {l o} \\alpha_ {h i}}\\right)}\\right) ^ {- d} \\right\\} \\left(\\frac {\\sigma_ {+}}{\\psi \\sigma_ {-} ^ {\\alpha_ {l o} / \\alpha_ {h i}}}\\right) ^ {d / \\alpha_ {l o}}. (86b) \\\\ \\end{array}",
+                                    "image_path": "190f6677e3426e77e7b50e0cb2c6c6fe1db382c6de075848f0c7193e226282ef.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        104,
+                        488,
+                        506,
+                        533
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                488,
+                                506,
+                                533
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        506,
+                                        533
+                                    ],
+                                    "type": "text",
+                                    "content": "Intuitively, if the radius for which the polynomial bounds hold "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        506,
+                                        533
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(r_{\\eta})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        506,
+                                        533
+                                    ],
+                                    "type": "text",
+                                    "content": " is large and the minimum curvature constant "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        506,
+                                        533
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma_{-}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        506,
+                                        533
+                                    ],
+                                    "type": "text",
+                                    "content": " is also large, then the bound "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        506,
+                                        533
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_{\\epsilon \\leq \\eta}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        506,
+                                        533
+                                    ],
+                                    "type": "text",
+                                    "content": " holds for large deviations from optimality "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        506,
+                                        533
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\eta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        506,
+                                        533
+                                    ],
+                                    "type": "text",
+                                    "content": ". The number of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        506,
+                                        533
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\eta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        506,
+                                        533
+                                    ],
+                                    "type": "text",
+                                    "content": "-radius "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        506,
+                                        533
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        506,
+                                        533
+                                    ],
+                                    "type": "text",
+                                    "content": "-balls required to cover the remaining space, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        506,
+                                        533
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_{\\epsilon \\geq \\eta}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
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+                                        533
+                                    ],
+                                    "type": "text",
+                                    "content": ", will be comparatively small."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        105,
+                        545,
+                        170,
+                        557
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
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+                            ],
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "B.3 D-BLIN"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        104,
+                        566,
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        566,
+                                        504,
+                                        612
+                                    ],
+                                    "type": "text",
+                                    "content": "The regret bound for Doubling BLiN (Feng et al., 2022) was originally proved assuming a standard normal distribution, however, the authors state their proof can be easily adapted to any sub-Gaussian distribution, which includes bounded random variables. This matches our setting with bounded payoffs, so we repeat their analysis here for that setting."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        104,
+                        616,
+                        506,
+                        694
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                616,
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+                                694
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        694
+                                    ],
+                                    "type": "text",
+                                    "content": "In what follows, in an effort to match notation in (Feng et al., 2022), let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        694
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mu(x) = -\\mathcal{L}^{\\tau}(s(x))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        694
+                                    ],
+                                    "type": "text",
+                                    "content": " denote the expected reward for evaluating "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        694
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x \\in [0,1]^d"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        694
+                                    ],
+                                    "type": "text",
+                                    "content": "; "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        694
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\mu}(x)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        694
+                                    ],
+                                    "type": "text",
+                                    "content": " is an unbiased estimate of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        694
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mu(x)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        694
+                                    ],
+                                    "type": "text",
+                                    "content": ". Here, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        694
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s: [0,1]^{n(\\tilde{m} - 1)} \\to \\prod_i \\Delta^{m_i - 1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        694
+                                    ],
+                                    "type": "text",
+                                    "content": " is any function that maps from the unit hypercube to a product of simplices; we analyze two such maps in Appendix B.1. Also let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        694
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{A}_m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        694
+                                    ],
+                                    "type": "text",
+                                    "content": " denote the collection of hypercubes to be investigated during the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        694
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        694
+                                    ],
+                                    "type": "text",
+                                    "content": "th batch of arm pulls, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        694
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C \\in \\mathcal{A}_m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        694
+                                    ],
+                                    "type": "text",
+                                    "content": " denote a hypercube partition in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        694
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        694
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        694
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n_m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        694
+                                    ],
+                                    "type": "text",
+                                    "content": " indicates the number of times each cube in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        694
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{A}_m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        694
+                                    ],
+                                    "type": "text",
+                                    "content": " is played in batch "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        694
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        694
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        694
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "B_{stop}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        694
+                                    ],
+                                    "type": "text",
+                                    "content": " denote the last batch."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        104,
+                        696,
+                        504,
+                        735
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        696,
+                                        504,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition 7 (Global Arm Accuracy). "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        696,
+                                        504,
+                                        735
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{E} \\stackrel{\\mathrm{def}}{=} \\left\\{ |\\mu(x) - \\hat{\\mu}_m(C)| \\leq r_m + \\sqrt{c_1 \\frac{\\ln T}{n_m}}, \\forall 1 \\leq m \\leq B_{stop} - 1, \\forall C \\in \\mathcal{A}_m, \\forall x \\in C \\right\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        696,
+                                        504,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
+                        750,
+                        311,
+                        761
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                299,
+                                750,
+                                311,
+                                761
+                            ],
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+                                {
+                                    "bbox": [
+                                        299,
+                                        750,
+                                        311,
+                                        761
+                                    ],
+                                    "type": "text",
+                                    "content": "32"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 31
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        80,
+                        282,
+                        101
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                80,
+                                282,
+                                101
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        282,
+                                        101
+                                    ],
+                                    "type": "text",
+                                    "content": "Define: "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        282,
+                                        101
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n_m = c_2 \\frac{\\ln T}{r_m^2} \\Rightarrow r_m = \\sqrt{c_2 \\frac{\\ln T}{n_m}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        282,
+                                        101
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        103,
+                        504,
+                        137
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                103,
+                                504,
+                                137
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        103,
+                                        504,
+                                        137
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition 8 (Elimination Rule). Eliminate "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        103,
+                                        504,
+                                        137
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C \\in \\mathcal{A}_m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        103,
+                                        504,
+                                        137
+                                    ],
+                                    "type": "text",
+                                    "content": " if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        103,
+                                        504,
+                                        137
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\mu}_m^{\\max} - \\hat{\\mu}_m(C) \\geq 2(1 + \\sqrt{c_1 / c_2})r_m = 2(\\sqrt{c_2} + \\sqrt{c_1})\\sqrt{\\frac{\\ln T}{n_m}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        103,
+                                        504,
+                                        137
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        103,
+                                        504,
+                                        137
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\mu}_m^{\\max} \\stackrel{\\mathrm{def}}{=} \\max_{C \\in \\mathcal{A}_m} \\hat{\\mu}_m(C)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        103,
+                                        504,
+                                        137
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        105,
+                        140,
+                        277,
+                        155
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                140,
+                                277,
+                                155
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        140,
+                                        277,
+                                        155
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 27. "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        140,
+                                        277,
+                                        155
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Pr[\\mathcal{E}] \\geq 1 - 2T^{-2(c_1 / c^2 - 1)}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        140,
+                                        277,
+                                        155
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        167,
+                        506,
+                        205
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                167,
+                                506,
+                                205
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        167,
+                                        506,
+                                        205
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        167,
+                                        506,
+                                        205
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\mu}_m(C) = \\frac{1}{n_m}\\sum_{i=1}^{n_m}y_{C,i}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        167,
+                                        506,
+                                        205
+                                    ],
+                                    "type": "text",
+                                    "content": " be the unbiased estimate of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        167,
+                                        506,
+                                        205
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mu(x\\in C)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        167,
+                                        506,
+                                        205
+                                    ],
+                                    "type": "text",
+                                    "content": " using "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        167,
+                                        506,
+                                        205
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n_m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        167,
+                                        506,
+                                        205
+                                    ],
+                                    "type": "text",
+                                    "content": " samples. Assume each "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        167,
+                                        506,
+                                        205
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "y_{C,i}\\in [a,b]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        167,
+                                        506,
+                                        205
+                                    ],
+                                    "type": "text",
+                                    "content": " with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        167,
+                                        506,
+                                        205
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "c = b - a"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        167,
+                                        506,
+                                        205
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        167,
+                                        506,
+                                        205
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\mu}(C) = \\frac{1}{n_m}\\sum_{i=1}^{n_m}y_{C,i}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        167,
+                                        506,
+                                        205
+                                    ],
+                                    "type": "text",
+                                    "content": ". Applying a Hoeffding inequality gives"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        193,
+                        211,
+                        505,
+                        272
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                193,
+                                211,
+                                505,
+                                272
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        193,
+                                        211,
+                                        505,
+                                        272
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} P r \\left[ | \\hat {\\mu} (C) - \\mathbb {E} [ \\hat {\\mu} (C) ] | \\geq \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m}}} \\right] \\leq 2 e ^ {- 2 c _ {1} \\ln T / c ^ {2}} (87a) \\\\ = 2 \\left(e ^ {\\ln T}\\right) ^ {- 2 c _ {1} / c ^ {2}} (87b) \\\\ = 2 T ^ {- 2 c _ {1} / c ^ {2}} \\forall C. (87c) \\\\ \\end{array}",
+                                    "image_path": "51aa6868bc60bb3bd1009ab61f0a82ea523f8a7a91fca52949c851694927a076.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        105,
+                        278,
+                        250,
+                        290
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                278,
+                                250,
+                                290
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        278,
+                                        250,
+                                        290
+                                    ],
+                                    "type": "text",
+                                    "content": "By Lipschitzness of "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        278,
+                                        250,
+                                        290
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mu"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        278,
+                                        250,
+                                        290
+                                    ],
+                                    "type": "text",
+                                    "content": " we also have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left| \\mathbb {E} \\left[ \\hat {\\mu} (C) \\right] - \\mu (x) \\right| \\leq r _ {m}, \\forall x \\in C. \\tag {88}",
+                                    "image_path": "3ae36a485938adc61c1f352826032438413c9aff4832834d6179f653599baffd.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
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+                        332
+                    ],
+                    "type": "text",
+                    "angle": 0,
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+                                    "bbox": [
+                                        105,
+                                        321,
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+                                        332
+                                    ],
+                                    "type": "text",
+                                    "content": "Then consider"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
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+                    ],
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+                                    "bbox": [
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+                                        434
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\sup  _ {x \\in C} | \\mu (x) - \\hat {\\mu} (C) | = \\sup  _ {x \\in C} | \\mu (x) - \\mathbb {E} [ \\hat {\\mu} (C) ] + \\mathbb {E} [ \\hat {\\mu} (C) ] - \\hat {\\mu} (C) | (89a) \\\\ \\leq \\sup  _ {x \\in C} \\left(| \\mu (x) - \\mathbb {E} [ \\hat {\\mu} (C) ] | + | \\mathbb {E} [ \\hat {\\mu} (C) ] - \\hat {\\mu} (C) |\\right) (89b) \\\\ = \\sup  _ {x \\in C} | \\mu (x) - \\mathbb {E} [ \\hat {\\mu} (C) ] | + | \\mathbb {E} [ \\hat {\\mu} (C) ] - \\hat {\\mu} (C) | (89c) \\\\ \\leq \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m}}} + r _ {m} (89d) \\\\ \\end{array}",
+                                    "image_path": "12cec17ce56b65a7c4e3c86132ba7241fbcd186104d7474a97bba0fea12e698d.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
+                        441,
+                        504,
+                        465
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                441,
+                                504,
+                                465
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        441,
+                                        504,
+                                        465
+                                    ],
+                                    "type": "text",
+                                    "content": "with probability "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        441,
+                                        504,
+                                        465
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "1 - 2T^{-2c_1 / c^2}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        441,
+                                        504,
+                                        465
+                                    ],
+                                    "type": "text",
+                                    "content": ". The first inequality follows by triangle inequality and the second follows from equation (88) and considering the complement of equation (87c)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        105,
+                        471,
+                        323,
+                        483
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        471,
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+                                        483
+                                    ],
+                                    "type": "text",
+                                    "content": "The complement of this result occurs with probability"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        186,
+                        489,
+                        504,
+                        515
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                    "bbox": [
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+                                        515
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "P r \\left[ \\sup  _ {x \\in C} | \\mu (x) - \\hat {\\mu} (C) | \\geq r _ {m} + \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m}}} \\right] \\leq 2 T ^ {- 2 c _ {1} / c ^ {2}}. \\tag {90}",
+                                    "image_path": "87824ff2ddad7855341c9a861d93e3190f24befb2c3b7180d99fab27a18f1f8f.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
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+                        563
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
+                                        104,
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+                                        563
+                                    ],
+                                    "type": "text",
+                                    "content": "At least 1 arm is played in each cube "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C \\in \\mathcal{A}_m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "text",
+                                    "content": " for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "1 \\leq m \\leq B_{stop} - 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "text",
+                                    "content": ", therefore, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "|\\mathcal{A}_m| \\leq T"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "text",
+                                    "content": " must be true given the exit condition of the algorithm. In addition, assume "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "B_{stop} \\leq T"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "text",
+                                    "content": " ("
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "B_{stop}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "text",
+                                    "content": " will be defined such that this is true). Then a union bound over all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        528,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "text",
+                                    "content": " events gives"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        133,
+                        570,
+                        505,
+                        689
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                133,
+                                570,
+                                505,
+                                689
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        133,
+                                        570,
+                                        505,
+                                        689
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} P r \\left[ \\exists m \\in [ 1, B _ {s t o p} - 1 ], C \\in \\mathcal {A} _ {m} s. t. \\sup  _ {x \\in C} | \\mu (x) - \\hat {\\mu} (C) | \\geq r _ {m} + \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m}}} \\right] (91a) \\\\ \\leq \\sum_ {m = 1} ^ {B _ {s t o p} - 1} \\sum_ {C \\in \\mathcal {A} _ {m}} P r \\left[ \\sup  _ {x \\in C} | \\mu (x) - \\hat {\\mu} (C) | \\geq r _ {m} + \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m}}} \\right] (91b) \\\\ \\leq \\sum_ {m = 1} ^ {B _ {s t o p} - 1} \\sum_ {C \\in \\mathcal {A} _ {m}} 2 T ^ {- 2 c _ {1} / c ^ {2}} (91c) \\\\ \\leq 2 T ^ {- 2 c _ {1} / c ^ {2}} T ^ {2}. (91d) \\\\ \\end{array}",
+                                    "image_path": "0bac28001a620bd7c9ac3b0fa7e70b4de8c03a9a3de8ac313d0206e7e2c1a5f4.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        104,
+                        696,
+                        504,
+                        735
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                696,
+                                504,
+                                735
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        696,
+                                        504,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": "Taking the complement of this event and noting that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        696,
+                                        504,
+                                        735
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sup_{x\\in C}|\\mu (x) - \\hat{\\mu} (C)|\\leq r_m + \\sqrt{c_1\\frac{\\ln T}{n_m}}\\Rightarrow |\\mu (x) - \\hat{\\mu} (C)|\\leq r_m + \\sqrt{c_1\\frac{\\ln T}{n_m}}\\forall x\\in C"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        696,
+                                        504,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": " gives the desired result."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
+                        750,
+                        311,
+                        760
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                299,
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+                                {
+                                    "bbox": [
+                                        299,
+                                        750,
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+                                        760
+                                    ],
+                                    "type": "text",
+                                    "content": "33"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 32
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        82,
+                        506,
+                        106
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                82,
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+                                106
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 28 (Optimal Arm Survives). Under event "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{E}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": ", the optimal arm "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x^{*} = \\arg \\max \\mu(x)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": " is not eliminated after the first "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "B_{stop} - 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": " batches."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        116,
+                        505,
+                        140
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                116,
+                                505,
+                                140
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        116,
+                                        505,
+                                        140
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        116,
+                                        505,
+                                        140
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_m^*"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        116,
+                                        505,
+                                        140
+                                    ],
+                                    "type": "text",
+                                    "content": " denote the cube containing "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        116,
+                                        505,
+                                        140
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x^*"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        116,
+                                        505,
+                                        140
+                                    ],
+                                    "type": "text",
+                                    "content": " in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        116,
+                                        505,
+                                        140
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{A}_m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        116,
+                                        505,
+                                        140
+                                    ],
+                                    "type": "text",
+                                    "content": ". Under event "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        116,
+                                        505,
+                                        140
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{E}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        116,
+                                        505,
+                                        140
+                                    ],
+                                    "type": "text",
+                                    "content": ", for any cube "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        116,
+                                        505,
+                                        140
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C \\in \\mathcal{A}_m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        116,
+                                        505,
+                                        140
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        116,
+                                        505,
+                                        140
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x \\in C"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        116,
+                                        505,
+                                        140
+                                    ],
+                                    "type": "text",
+                                    "content": ", the following relation shows that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        116,
+                                        505,
+                                        140
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_m^*"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        116,
+                                        505,
+                                        140
+                                    ],
+                                    "type": "text",
+                                    "content": " avoids the elimination rule in round "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        116,
+                                        505,
+                                        140
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        116,
+                                        505,
+                                        140
+                                    ],
+                                    "type": "text",
+                                    "content": ":"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        136,
+                        143,
+                        504,
+                        264
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                136,
+                                143,
+                                504,
+                                264
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        136,
+                                        143,
+                                        504,
+                                        264
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\hat {\\mu} (C) - \\hat {\\mu} \\left(C _ {m} ^ {*}\\right) \\leq \\left(\\mu (x) + r _ {m} + \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m}}}\\right) + \\left(- \\mu \\left(x ^ {*}\\right) + r _ {m} + \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m}}}\\right) (92a) \\\\ = \\underbrace {(\\mu (x) - \\mu \\left(x ^ {*}\\right))} _ {\\leq 0} + 2 r _ {m} + 2 \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m}}} (92b) \\\\ \\leq 2 \\sqrt {c _ {2} \\frac {\\ln T}{n _ {m}}} + 2 \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m}}} (92c) \\\\ = 2 \\left(\\sqrt {c _ {1}} + \\sqrt {c _ {2}}\\right) \\sqrt {\\frac {\\ln T}{n _ {m}}} (92d) \\\\ \\end{array}",
+                                    "image_path": "a6d8e174b5cf82b89d75762abffe80917b3f7b4d64ddd8fd270b33bac395b284.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        265,
+                        506,
+                        298
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                265,
+                                506,
+                                298
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        265,
+                                        506,
+                                        298
+                                    ],
+                                    "type": "text",
+                                    "content": "where the first inequality follows from applying Lemma 27 to upper bound "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        265,
+                                        506,
+                                        298
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\mu} (C)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        265,
+                                        506,
+                                        298
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        265,
+                                        506,
+                                        298
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\mu} (C_m^*)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        265,
+                                        506,
+                                        298
+                                    ],
+                                    "type": "text",
+                                    "content": " individually. The remaining steps use the optimality of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        265,
+                                        506,
+                                        298
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x^{*}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        265,
+                                        506,
+                                        298
+                                    ],
+                                    "type": "text",
+                                    "content": ", the definition of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        265,
+                                        506,
+                                        298
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r_m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        265,
+                                        506,
+                                        298
+                                    ],
+                                    "type": "text",
+                                    "content": ", and the elimination rule."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        304,
+                        491,
+                        317
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                304,
+                                491,
+                                317
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        491,
+                                        317
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 29. Under event "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        491,
+                                        317
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{E}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        491,
+                                        317
+                                    ],
+                                    "type": "text",
+                                    "content": ", for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        491,
+                                        317
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "1 \\leq m \\leq B_{stop}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        491,
+                                        317
+                                    ],
+                                    "type": "text",
+                                    "content": ", any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        491,
+                                        317
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C \\in A_m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        491,
+                                        317
+                                    ],
+                                    "type": "text",
+                                    "content": " and any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        491,
+                                        317
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x \\in C"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        491,
+                                        317
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        491,
+                                        317
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta_x"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        491,
+                                        317
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfies"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        249,
+                        319,
+                        504,
+                        334
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                249,
+                                319,
+                                504,
+                                334
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        249,
+                                        319,
+                                        504,
+                                        334
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\Delta_ {x} \\leq 4 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {m - 1} \\tag {93}",
+                                    "image_path": "41beb415e5554aa753249af253069ce3553cfaa8293df839ed4a548286289f88.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        344,
+                        504,
+                        370
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                344,
+                                504,
+                                370
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        344,
+                                        504,
+                                        370
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. For "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        344,
+                                        504,
+                                        370
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m = 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        344,
+                                        504,
+                                        370
+                                    ],
+                                    "type": "text",
+                                    "content": ", recall that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        344,
+                                        504,
+                                        370
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r_m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        344,
+                                        504,
+                                        370
+                                    ],
+                                    "type": "text",
+                                    "content": " is the side length of a cube "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        344,
+                                        504,
+                                        370
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C \\in \\mathcal{A}_m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        344,
+                                        504,
+                                        370
+                                    ],
+                                    "type": "text",
+                                    "content": ", therefore, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        344,
+                                        504,
+                                        370
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta_x \\leq r_{m-1} \\leq 4(1 + \\sqrt{c_1 / c_2})r_{m-1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        344,
+                                        504,
+                                        370
+                                    ],
+                                    "type": "text",
+                                    "content": " holds directly from the Lipschitzness of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        344,
+                                        504,
+                                        370
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mu"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        344,
+                                        504,
+                                        370
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        104,
+                        374,
+                        506,
+                        409
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                374,
+                                506,
+                                409
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        506,
+                                        409
+                                    ],
+                                    "type": "text",
+                                    "content": "For "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        506,
+                                        409
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m > 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        506,
+                                        409
+                                    ],
+                                    "type": "text",
+                                    "content": ", let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        506,
+                                        409
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_{m - 1}^{*}\\in \\mathcal{A}_{m - 1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        506,
+                                        409
+                                    ],
+                                    "type": "text",
+                                    "content": " be the cube containing "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        506,
+                                        409
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x^{*}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        506,
+                                        409
+                                    ],
+                                    "type": "text",
+                                    "content": ". From Lemma 28, this cube has not been eliminated under event "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        506,
+                                        409
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{E}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        506,
+                                        409
+                                    ],
+                                    "type": "text",
+                                    "content": ". For any cube "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        506,
+                                        409
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C\\in \\mathcal{A}_m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        506,
+                                        409
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        506,
+                                        409
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x\\in C"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        506,
+                                        409
+                                    ],
+                                    "type": "text",
+                                    "content": ", it is clear that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        506,
+                                        409
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        506,
+                                        409
+                                    ],
+                                    "type": "text",
+                                    "content": " is also in the parent of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        506,
+                                        409
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        506,
+                                        409
+                                    ],
+                                    "type": "text",
+                                    "content": ", denoted "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        506,
+                                        409
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_{par}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        506,
+                                        409
+                                    ],
+                                    "type": "text",
+                                    "content": " ("
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        506,
+                                        409
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x\\in C\\subset C_{par}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        506,
+                                        409
+                                    ],
+                                    "type": "text",
+                                    "content": "). Then for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        506,
+                                        409
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x\\in C"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        374,
+                                        506,
+                                        409
+                                    ],
+                                    "type": "text",
+                                    "content": ", it holds that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        129,
+                        411,
+                        504,
+                        632
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                411,
+                                504,
+                                632
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        411,
+                                        504,
+                                        632
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\Delta_ {x} = \\mu \\left(x ^ {*}\\right) - \\mu (x) \\leq \\left(\\hat {\\mu} _ {m - 1} \\left(C _ {m - 1} ^ {*}\\right) + r _ {m - 1} + \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m - 1}}}\\right) (94a) \\\\ + \\left(- \\hat {\\mu} _ {m - 1} (C _ {p a r}) + r _ {m - 1} + \\sqrt {c _ {1} \\frac {\\ln T}{n _ {m - 1}}}\\right) \\\\ = \\left(\\hat {\\mu} _ {m - 1} \\left(C _ {m - 1} ^ {*}\\right) - \\hat {\\mu} _ {m - 1} \\left(C _ {p a r}\\right)\\right) + 2 \\left(\\sqrt {c _ {1}} + \\sqrt {c _ {2}}\\right) \\sqrt {\\frac {\\ln T}{n _ {m - 1}}} (94b) \\\\ \\leq \\left(\\hat {\\mu} _ {m - 1} ^ {\\max } - \\hat {\\mu} _ {m - 1} \\left(C _ {p a r}\\right)\\right) + 2 \\left(\\sqrt {c _ {1}} + \\sqrt {c _ {2}}\\right) \\sqrt {\\frac {\\ln T}{n _ {m - 1}}} (94c) \\\\ \\leq 2 \\left(\\sqrt {c _ {1}} + \\sqrt {c _ {2}}\\right) \\sqrt {\\frac {\\ln T}{n _ {m - 1}}} + 2 \\left(\\sqrt {c _ {1}} + \\sqrt {c _ {2}}\\right) \\sqrt {\\frac {\\ln T}{n _ {m - 1}}} (94d) \\\\ = 4 \\left(\\sqrt {c _ {1}} + \\sqrt {c _ {2}}\\right) \\sqrt {\\frac {\\ln T}{n _ {m - 1}}} (94e) \\\\ = 4 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {m - 1} (94f) \\\\ \\end{array}",
+                                    "image_path": "5998b8f2a3f178c8ddfb3b652b598e17271b4a39eb8a00e45662b1d9a4cc27fd.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
+                        633,
+                        506,
+                        657
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                633,
+                                506,
+                                657
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        633,
+                                        506,
+                                        657
+                                    ],
+                                    "type": "text",
+                                    "content": "where we have applied Lemma 27 similarly as in Lemma 28 and also used the definition of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        633,
+                                        506,
+                                        657
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r_{m - 1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        633,
+                                        506,
+                                        657
+                                    ],
+                                    "type": "text",
+                                    "content": ". The last two inequalities use the fact that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        633,
+                                        506,
+                                        657
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\mu}_{m - 1}(C_{m - 1}^{*})\\leq \\hat{\\mu}_{m - 1}^{\\max}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        633,
+                                        506,
+                                        657
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        633,
+                                        506,
+                                        657
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_{par}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        633,
+                                        506,
+                                        657
+                                    ],
+                                    "type": "text",
+                                    "content": " was not eliminated."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        663,
+                        505,
+                        688
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                663,
+                                505,
+                                688
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        505,
+                                        688
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem 2 (BLiN Regret Rate). With probability exceeding "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        505,
+                                        688
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "1 - 2T^{-2(c_1/c^2 - 1)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        505,
+                                        688
+                                    ],
+                                    "type": "text",
+                                    "content": ", the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        505,
+                                        688
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        505,
+                                        688
+                                    ],
+                                    "type": "text",
+                                    "content": "-step total regret "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        505,
+                                        688
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "R(T)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        505,
+                                        688
+                                    ],
+                                    "type": "text",
+                                    "content": " of BLiN with Doubling Edge-length Sequence ("
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        505,
+                                        688
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "D"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        505,
+                                        688
+                                    ],
+                                    "type": "text",
+                                    "content": "-BLiN) Feng et al. (2022) satisfies"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        201,
+                        690,
+                        504,
+                        706
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                201,
+                                690,
+                                504,
+                                706
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        201,
+                                        690,
+                                        504,
+                                        706
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "R (T) \\leq 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left(2 c _ {2} + 1\\right) \\ln (T) ^ {\\frac {1}{d _ {z} + 2}} T ^ {\\frac {d _ {z} + 1}{d _ {z} + 2}} \\tag {95}",
+                                    "image_path": "07931c1a9012748fe305609cd8e52740b9ea03710abfc267e6fbd096b912264a.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        104,
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+                        735
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        708,
+                                        504,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        708,
+                                        504,
+                                        735
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d_{z}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        708,
+                                        504,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": " is the zooming dimension of the problem instance. In addition, D-BLiN only needs no more than "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        708,
+                                        504,
+                                        735
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "B^{*} = \\frac{\\log 2(T) - \\log 2(\\ln(T))}{d_{z} + 2} + 2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        708,
+                                        504,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": " rounds of communications to achieve this regret rate."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
+                        750,
+                        312,
+                        760
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                760
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        299,
+                                        750,
+                                        312,
+                                        760
+                                    ],
+                                    "type": "text",
+                                    "content": "34"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 33
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        82,
+                        506,
+                        120
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                82,
+                                506,
+                                120
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        120
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Since "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        120
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r_m = \\frac{r_{m-1}}{2} \\Rightarrow r_{m-1} = 2r_m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        120
+                                    ],
+                                    "type": "text",
+                                    "content": " for the Doubling Edge-length Sequence, Lemma 29 implies that every cube "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        120
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C \\in A_m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        120
+                                    ],
+                                    "type": "text",
+                                    "content": " is a subset of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        120
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S(8(1 + \\sqrt{c_1 / c_2})r_m)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        120
+                                    ],
+                                    "type": "text",
+                                    "content": ". Thus from the definition of zooming dimension (Corollary 5 with appropriate condition), we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        253,
+                        147,
+                        504,
+                        160
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                160
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        253,
+                                        147,
+                                        504,
+                                        160
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left| \\mathcal {A} _ {m} \\right| \\leq N _ {r _ {m}} \\leq C _ {z} r _ {m} ^ {- d _ {z}}. \\tag {96}",
+                                    "image_path": "cbb65f1ce2d67f99247d7eaca5f7912c4245274b827c3cbb1831377f6cae0b0e.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        194,
+                        504,
+                        243
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                194,
+                                504,
+                                243
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        194,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "text",
+                                    "content": "Fix any positive number "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        194,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "B"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        194,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "text",
+                                    "content": ". Also by Lemma 29, we know that any arm played after batch "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        194,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "B"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        194,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "text",
+                                    "content": " incurs a regret bounded by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        194,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "8(1 + \\sqrt{c_1 / c_2})r_B"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        194,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "text",
+                                    "content": ", since the cubes played after batch "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        194,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "B"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        194,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "text",
+                                    "content": " have edge length no larger than "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        194,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r_B"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        194,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then the total regret that occurs after batch "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        194,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "B"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        194,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "text",
+                                    "content": " is bounded by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        194,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "8(1 + \\sqrt{c_1 / c_2})r_BT"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        194,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "text",
+                                    "content": " (where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        194,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        194,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "text",
+                                    "content": " is an upper bound on the number of arms)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        105,
+                        247,
+                        244,
+                        258
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                247,
+                                244,
+                                258
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        247,
+                                        244,
+                                        258
+                                    ],
+                                    "type": "text",
+                                    "content": "Thus the regret can be bounded as"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        196,
+                        286,
+                        504,
+                        321
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                196,
+                                286,
+                                504,
+                                321
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        196,
+                                        286,
+                                        504,
+                                        321
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "R (T) \\leq \\sum_ {m = 1} ^ {B} \\sum_ {C \\in \\mathcal {A} _ {m}} \\sum_ {i = 1} ^ {n _ {m}} \\Delta_ {x _ {C, i}} + 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {B} T \\tag {97}",
+                                    "image_path": "ddd1c99e1d462d31018942f4c18772ee3eec3f632809fa3d7c5e0beefe13f4b6.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        347,
+                        504,
+                        384
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                347,
+                                504,
+                                384
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        347,
+                                        504,
+                                        384
+                                    ],
+                                    "type": "text",
+                                    "content": "where the first term bounds the regret in the first "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        347,
+                                        504,
+                                        384
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "B"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        347,
+                                        504,
+                                        384
+                                    ],
+                                    "type": "text",
+                                    "content": " batches of D-BLiN, and the second term bounds the regret after the first "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        347,
+                                        504,
+                                        384
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "B"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        347,
+                                        504,
+                                        384
+                                    ],
+                                    "type": "text",
+                                    "content": " batches. If the algorithm stops at batch "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        347,
+                                        504,
+                                        384
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{B} < B"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        347,
+                                        504,
+                                        384
+                                    ],
+                                    "type": "text",
+                                    "content": ", we define "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        347,
+                                        504,
+                                        384
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{A}_m ="
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        347,
+                                        504,
+                                        384
+                                    ],
+                                    "type": "text",
+                                    "content": " for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        347,
+                                        504,
+                                        384
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{B} < m \\leq B"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        347,
+                                        504,
+                                        384
+                                    ],
+                                    "type": "text",
+                                    "content": " and inequality equation (97) still holds."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        390,
+                        506,
+                        415
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                390,
+                                506,
+                                415
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        390,
+                                        506,
+                                        415
+                                    ],
+                                    "type": "text",
+                                    "content": "By Lemma 29, we have "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        390,
+                                        506,
+                                        415
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta_{x_{C,i}}\\leq 8(1 + \\sqrt{c_1 / c_2})r_m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        390,
+                                        506,
+                                        415
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        390,
+                                        506,
+                                        415
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C\\in \\mathcal{A}_m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        390,
+                                        506,
+                                        415
+                                    ],
+                                    "type": "text",
+                                    "content": ". We can thus bound equation (97) by"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        161,
+                        441,
+                        504,
+                        578
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                161,
+                                441,
+                                504,
+                                578
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        161,
+                                        441,
+                                        504,
+                                        578
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} R (T) \\leq \\sum_ {m = 1} ^ {B} | \\mathcal {A} _ {m} | \\cdot n _ {m} \\cdot 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {m} + 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {B} T (98a) \\\\ \\leq \\sum_ {m = 1} ^ {B} N _ {r _ {m}} \\cdot n _ {m} \\cdot 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {m} + 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {B} T (98b) \\\\ = \\sum_ {m = 1} ^ {B} N _ {r _ {m}} \\cdot c _ {2} \\frac {\\ln T}{r _ {m} ^ {2}} \\cdot 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {m} + 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {B} T (98c) \\\\ = \\sum_ {m = 1} ^ {B} N _ {r _ {m}} \\cdot \\frac {\\ln T}{r _ {m}} \\cdot 8 c _ {2} \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) + 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) r _ {B} T (98d) \\\\ \\end{array}",
+                                    "image_path": "4cc80ac1e352d58415efbe4537c15d4e07f3a72732536c4079c78ab84bcd603c.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        606,
+                        504,
+                        635
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                606,
+                                504,
+                                635
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        606,
+                                        504,
+                                        635
+                                    ],
+                                    "type": "text",
+                                    "content": "where equation (98b) uses equation (96), and equation (98c) uses equality "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        606,
+                                        504,
+                                        635
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n_m = c_2 \\frac{\\ln T}{r_m^2}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        606,
+                                        504,
+                                        635
+                                    ],
+                                    "type": "text",
+                                    "content": ". Since "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        606,
+                                        504,
+                                        635
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r_m = 2^{-m + 1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        606,
+                                        504,
+                                        635
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        606,
+                                        504,
+                                        635
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "N_{r_m} \\leq C_z r_m^{-d_z} \\leq C_z 2^{(m - 1)d_z}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        606,
+                                        504,
+                                        635
+                                    ],
+                                    "type": "text",
+                                    "content": ", we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        127,
+                        662,
+                        505,
+                        730
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                127,
+                                662,
+                                505,
+                                730
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        127,
+                                        662,
+                                        505,
+                                        730
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} R (T) \\leq \\sum_ {m = 1} ^ {B} C _ {z} 2 ^ {(m - 1) d _ {z}} \\cdot \\frac {\\ln T}{2 ^ {- m + 1}} \\cdot 8 c _ {2} \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) + 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) 2 ^ {- B + 1} T (99a) \\\\ = 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ c _ {2} C _ {z} \\ln T \\sum_ {m = 1} ^ {B} 2 ^ {(m - 1) (d _ {z} + 1)} + 2 ^ {- B + 1} T \\right]. (99b) \\\\ \\end{array}",
+                                    "image_path": "2ca5d633fff74c8b23b00fdae3953caeba6f104163f6ad93ef05fae6b505cc2d.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                38
+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "page_number",
+                    "angle": 0,
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+                            "spans": [
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+                                    "type": "text",
+                                    "content": "35"
+                                }
+                            ]
+                        }
+                    ],
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+                }
+            ],
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+                                    "type": "text",
+                                    "content": "Continuing we find"
+                                }
+                            ]
+                        }
+                    ],
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+                },
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+                                    "type": "interline_equation",
+                                    "content": "R (T) \\leq 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ c _ {2} C _ {z} \\ln T \\sum_ {m = 1} ^ {B} 2 ^ {(m - 1) (d _ {z} + 1)} + 2 ^ {- B + 1} T \\right] \\tag {100a}",
+                                    "image_path": "da066972cc3a3d301bc64581ee2fd1b7c92e4316a88e3591eb6261ba3a703e7d.jpg"
+                                }
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+                                    "type": "interline_equation",
+                                    "content": "= 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ c _ {2} C _ {z} \\ln T \\sum_ {m = 1} ^ {B} \\left(2 ^ {d _ {z} + 1}\\right) ^ {m - 1} + 2 ^ {- B + 1} T \\right] \\tag {100b}",
+                                    "image_path": "6c48d3e6e98a3c2264730db5375c36c84ea6e1d86d0c2003ba1c0f790faa65de.jpg"
+                                }
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+                                    "bbox": [
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+                                        170,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "= 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ c _ {2} C _ {z} \\ln T \\sum_ {m = 0} ^ {B - 1} \\left(2 ^ {d _ {z} + 1}\\right) ^ {m} + 2 ^ {- B + 1} T \\right] \\tag {100c}",
+                                    "image_path": "d17d79e1052968cedd8dcd2f2a5b1faf1769bbac85c5178006d21cc0029e31e1.jpg"
+                                }
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+                    "angle": 0,
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "= 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ c _ {2} C _ {z} \\ln T \\left(\\frac {2 ^ {B (d _ {z} + 1)} - 1}{2 ^ {d _ {z} + 1} - 1}\\right) + 2 ^ {- B + 1} T \\right] \\text {v i a g e o m e t r i c s e r i e s} \\tag {100d}",
+                                    "image_path": "cd280ad91cf7ed063940152ab33e0a9b821682ed4849e3b5b1bde4aed63b4ef3.jpg"
+                                }
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+                {
+                    "bbox": [
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+                    ],
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+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        143,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\leq 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ c _ {2} C _ {z} \\ln T \\left(\\frac {2 ^ {B (d _ {z} + 1)}}{2 ^ {d _ {z} + 1} - 1}\\right) + 2 ^ {- B + 1} T \\right] \\tag {100e}",
+                                    "image_path": "11122ccf29da156f3d3fc5f9df1e268f0b004081d6ce801c286c23f9546c72d0.jpg"
+                                }
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+                    "angle": 0,
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\leq 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ c _ {2} C _ {z} \\ln T \\left(2 \\cdot \\frac {2 ^ {B (d _ {z} + 1)}}{2 ^ {d _ {z} + 1}}\\right) + 2 ^ {- B + 1} T \\right] \\tag {100f}",
+                                    "image_path": "d3a76eb0bc4364fcc0a3559bfd5da0680e211b39bbfbd37cb798642f226def01.jpg"
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+                                    "type": "interline_equation",
+                                    "content": "= 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ 2 c _ {2} C _ {z} 2 ^ {(B - 1) (d _ {z} + 1)} \\ln T + 2 ^ {- (B - 1)} T \\right]. \\tag {100g}",
+                                    "image_path": "381361986a294ada53265946fd1f0c3900ced5e5ba43f4657eff70094768d59e.jpg"
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+                                    "type": "text",
+                                    "content": "This inequality holds for any positive "
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+                                    "type": "inline_equation",
+                                    "content": "B"
+                                },
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ". By choosing "
+                                },
+                                {
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "B^{*} = 1 + \\frac{\\log_{2}\\left(\\frac{T}{\\ln T}\\right)}{d_{z} + 2}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
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+                                    "type": "text",
+                                    "content": ", we have"
+                                }
+                            ]
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+                    ],
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+                },
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+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} R (T) \\leq 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ 2 c _ {2} C _ {z} \\left(\\frac {T}{\\ln T}\\right) ^ {\\frac {(d _ {z} + 1)}{(d _ {z} + 2)}} \\ln T + \\left(\\frac {\\ln T}{T}\\right) ^ {\\frac {1}{(d _ {z} + 2)}} T \\right] (101a) \\\\ = 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ 2 c _ {2} C _ {z} T ^ {\\frac {(d _ {z} + 1)}{(d _ {z} + 2)}} \\ln T ^ {1 - \\frac {(d _ {z} + 1)}{(d _ {z} + 2)}} + T ^ {1 - \\frac {1}{(d _ {z} + 2)}} \\ln T ^ {\\frac {1}{(d _ {z} + 2)}} \\right] (101b) \\\\ = 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left[ 2 c _ {2} C _ {z} T ^ {\\frac {(d _ {z} + 1)}{(d _ {z} + 2)}} \\ln T ^ {\\frac {1}{(d _ {z} + 2)}} + T ^ {\\frac {(d _ {z} + 1)}{(d _ {z} + 2)}} \\ln T ^ {\\frac {1}{(d _ {z} + 2)}} \\right] (101c) \\\\ = 8 \\left(1 + \\sqrt {c _ {1} / c _ {2}}\\right) \\left(2 c _ {2} C _ {z} + 1\\right) T ^ {\\frac {(d _ {z} + 1)}{(d _ {z} + 2)}} \\ln T ^ {\\frac {1}{(d _ {z} + 2)}}. (101d) \\\\ \\end{array}",
+                                    "image_path": "4707d575a8ba3a4ba582753ea7b0cd0c21034473b85fdd13734bb2b613f0ab52.jpg"
+                                }
+                            ]
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Corollary 6 (BLiN Regret Rate Refined). Setting "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "c_{1} = 2c^{2}"
+                                },
+                                {
+                                    "bbox": [
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+                                        441,
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+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "c_{2} = 2\\left(\\frac{c}{4C_{z}}\\right)^{2 / 3}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " simplifies Theorem 2 such that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
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+                                    "type": "interline_equation",
+                                    "content": "R (T) \\leq 8 \\left(1 + \\left(4 c ^ {2} C _ {z}\\right) ^ {1 / 3}\\right) ^ {2} T ^ {\\frac {(d _ {z} + 1)}{(d _ {z} + 2)}} \\ln T ^ {\\frac {1}{(d _ {z} + 2)}}. \\tag {102}",
+                                    "image_path": "ed517588e39fd85a5a1180613f06cff4f47b83216dffb59a2b7215fb1aff5f10.jpg"
+                                }
+                            ]
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+                },
+                {
+                    "bbox": [
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+                    "angle": 0,
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "with probability "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "1 - 2T^{-2}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
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+                    "index": 13
+                },
+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. If we set "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "c_1 = 2c^2"
+                                },
+                                {
+                                    "bbox": [
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+                                        521,
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+                                        555
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
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+                                        555
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "c_2 = 2\\left(\\frac{c}{4C_z}\\right)^{2/3}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        521,
+                                        504,
+                                        555
+                                    ],
+                                    "type": "text",
+                                    "content": ", then "
+                                },
+                                {
+                                    "bbox": [
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+                                        504,
+                                        555
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sqrt{c_1 / c_2} = c\\left(\\frac{4C_z}{c}\\right)^{1/3} = \\left(4c^2 C_z\\right)^{1/3} = 2c_2C_z"
+                                },
+                                {
+                                    "bbox": [
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+                                        555
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 30. The zooming dimension and zooming constant under the "
+                                },
+                                {
+                                    "bbox": [
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+                                        572
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell (x,y) = ||x - y||_{\\infty}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        560,
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+                                        572
+                                    ],
+                                    "type": "text",
+                                    "content": " norm are"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
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+                        {
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "d _ {z} = \\frac {1}{2} n \\bar {m} \\tag {103}",
+                                    "image_path": "7814a20221427de367b43d35f196a745b3ed69ea7548d97d3c040f5b5582e94b.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "C _ {z} = \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\left(\\frac {4}{r _ {\\eta} ^ {2} \\sigma_ {- \\infty}}\\right) ^ {n \\bar {m}} \\tag {104}",
+                                    "image_path": "40512c6fe4f3401a70b8be2a74ca38a1f3ce6b36629bc35a2b8f93d0a3dfcbca.jpg"
+                                }
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+                {
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+                    "angle": 0,
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma_{-\\infty} = ||\\text{Hess}(-f(x))^{-1}||_{\\infty}"
+                                },
+                                {
+                                    "bbox": [
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+                                        632,
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+                                    ],
+                                    "type": "text",
+                                    "content": " is an upper bound on the infinity norm of the inverse Hessian matrix of the function at every equilibrium."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                667,
+                                504,
+                                704
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        667,
+                                        504,
+                                        704
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Recall from Theorem 1 that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        667,
+                                        504,
+                                        704
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d_{z} = d\\left(\\frac{\\alpha_{hi} - \\alpha_{lo}}{\\alpha_{lo}\\alpha_{hi}}\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        667,
+                                        504,
+                                        704
+                                    ],
+                                    "type": "text",
+                                    "content": " with constant "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        667,
+                                        504,
+                                        704
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_z = 16^{-d'}C"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        667,
+                                        504,
+                                        704
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        667,
+                                        504,
+                                        704
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        667,
+                                        504,
+                                        704
+                                    ],
+                                    "type": "text",
+                                    "content": " is defined below. In addition, BLiN assumes "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        667,
+                                        504,
+                                        704
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell (x,y) = ||x - y||_{\\infty}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        667,
+                                        504,
+                                        704
+                                    ],
+                                    "type": "text",
+                                    "content": ". Matching to Assumption 1, we see that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        667,
+                                        504,
+                                        704
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma_{+} = \\alpha_{lo} = 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        667,
+                                        504,
+                                        704
+                                    ],
+                                    "type": "text",
+                                    "content": ". Under the infinity norm, the volume constant "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        667,
+                                        504,
+                                        704
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S_{d} = 2^{d}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        667,
+                                        504,
+                                        704
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                },
+                {
+                    "bbox": [
+                        104,
+                        709,
+                        504,
+                        732
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
+                                        104,
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+                                        732
+                                    ],
+                                    "type": "text",
+                                    "content": "We will define the other constants with respect to properties of the Hessian of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        709,
+                                        504,
+                                        732
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f(x)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        709,
+                                        504,
+                                        732
+                                    ],
+                                    "type": "text",
+                                    "content": " about each equilibrium, specifically the infinity norm of the inverse Hessian so that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        709,
+                                        504,
+                                        732
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma_{-} = \\sigma_{-\\infty}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        709,
+                                        504,
+                                        732
+                                    ],
+                                    "type": "text",
+                                    "content": ". This means"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                38
+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
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+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "36"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 21
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 35
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        82,
+                        504,
+                        117
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                82,
+                                504,
+                                117
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        117
+                                    ],
+                                    "type": "text",
+                                    "content": "we will bound the function locally with a quadratic, i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        117
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha_{hi} = 2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        117
+                                    ],
+                                    "type": "text",
+                                    "content": ". Lastly, recall from Corollary 5 that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        117
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\psi = \\frac{1}{32}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        117
+                                    ],
+                                    "type": "text",
+                                    "content": " and the dimension of our search space (the product space of player mixed strategies) is "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        117
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d = n(\\bar{m} - 1) \\leq nm"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        117
+                                    ],
+                                    "type": "text",
+                                    "content": " for simplicity."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        105,
+                        121,
+                        310,
+                        133
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        121,
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+                                        133
+                                    ],
+                                    "type": "text",
+                                    "content": "Plugging this information into Theorem 1, we find"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        270,
+                        140,
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "d _ {z} = d ^ {\\prime} = \\frac {1}{2} n \\bar {m} \\tag {105}",
+                                    "image_path": "cd1d1bc59943defc9b258bbeefc62930e447176b9f71e41a949fbcce0493ce7f.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                {
+                                    "bbox": [
+                                        105,
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+                                        182
+                                    ],
+                                    "type": "text",
+                                    "content": "and"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        165,
+                        189,
+                        505,
+                        219
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                219
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        165,
+                                        189,
+                                        505,
+                                        219
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "C = \\max  \\left\\{1, | \\mathcal {X} ^ {*} | ^ {- 1} S _ {d} ^ {- 1} \\left(r _ {\\eta} ^ {\\frac {\\alpha_ {h i}}{\\alpha_ {l o}}} \\sigma_ {-} ^ {\\left(\\frac {\\alpha_ {h i} - \\alpha_ {l o}}{\\alpha_ {l o} \\alpha_ {h i}}\\right)}\\right) ^ {- d} \\right\\} \\left(\\frac {\\sigma_ {+}}{\\psi \\sigma_ {-} ^ {\\alpha_ {l o} / \\alpha_ {h i}}}\\right) ^ {d / \\alpha_ {l o}} \\tag {106a}",
+                                    "image_path": "5e90b0f890005c3dbb50a36b537884b51be5661cc904fc2ecead352ddf070ab0.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        177,
+                                        220,
+                                        504,
+                                        250
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "= \\max  \\left\\{1, \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} 2 ^ {- d} \\left(r _ {\\eta} ^ {2} \\sigma_ {- \\infty} ^ {\\frac {1}{2}}\\right) ^ {- d} \\right\\} \\left(\\frac {3 2}{\\sigma_ {- \\infty} ^ {1 / 2}}\\right) ^ {d} \\tag {106b}",
+                                    "image_path": "bd72d7a2f5937ef253e10c8e3c5dbe43757a928e8afc4d8c4f74b8899ee6870d.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        177,
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+                        504,
+                        281
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                177,
+                                251,
+                                504,
+                                281
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        177,
+                                        251,
+                                        504,
+                                        281
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "= \\max  \\left\\{1, \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\left(2 r _ {\\eta} ^ {2} \\sigma_ {- \\infty} ^ {\\frac {1}{2}}\\right) ^ {- d} \\right\\} \\left(\\frac {3 2}{\\sigma_ {- \\infty} ^ {1 / 2}}\\right) ^ {d} \\tag {106c}",
+                                    "image_path": "9d6be0b7e39507834e2d73d531be5d3e98bfb99096a1c7d260bab33b51299e65.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        176,
+                        282,
+                        504,
+                        311
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                176,
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+                                504,
+                                311
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        176,
+                                        282,
+                                        504,
+                                        311
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\stackrel {h a r d} {=} \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\left(2 r _ {\\eta} ^ {2} \\sigma_ {- \\infty} ^ {\\frac {1}{2}}\\right) ^ {- d} \\left(\\frac {3 2}{\\sigma_ {- \\infty} ^ {1 / 2}}\\right) ^ {d} \\tag {106d}",
+                                    "image_path": "e9f01dac7f0445d8ebf31a43455a6f2737a12042f55e930a75c947e75b1b5025.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                340
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        176,
+                                        312,
+                                        504,
+                                        340
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\stackrel {h a r d} {=} \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\left(\\frac {1 6}{r _ {\\eta} ^ {2} \\sigma_ {- \\infty}}\\right) ^ {n \\bar {m}} \\tag {106e}",
+                                    "image_path": "840cf9a8672549bf08979a50020e31dc9fcb52451089e9d6f288950a4b38acfc.jpg"
+                                }
+                            ]
+                        }
+                    ],
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+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        347,
+                                        504,
+                                        369
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\text{hard}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        347,
+                                        504,
+                                        369
+                                    ],
+                                    "type": "text",
+                                    "content": " indicates we are assuming "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        347,
+                                        504,
+                                        369
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r_{\\eta}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        347,
+                                        504,
+                                        369
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        347,
+                                        504,
+                                        369
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma_{-\\infty}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        347,
+                                        504,
+                                        369
+                                    ],
+                                    "type": "text",
+                                    "content": " are small enough to dominate the other operand of the max."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        105,
+                        375,
+                        422,
+                        387
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        375,
+                                        422,
+                                        387
+                                    ],
+                                    "type": "text",
+                                    "content": "Finally, converting the near optimality constant to a zooming constant, we find"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
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+                    ],
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+                    "angle": 0,
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+                                    "bbox": [
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+                                        449
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} C _ {z} = 1 6 ^ {- d _ {z}} \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\left(\\frac {1 6}{r _ {\\eta} ^ {2} \\sigma_ {- \\infty}}\\right) ^ {n \\bar {m}} (107a) \\\\ = \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\left(\\frac {4}{r _ {\\eta} ^ {2} \\sigma_ {- \\infty}}\\right) ^ {n \\bar {m}}. (107b) \\\\ \\end{array}",
+                                    "image_path": "90e2e13fb1be4dfd63ffd09ace5f5898ec1b8a2dfa13de7614fcd90a4813a748.jpg"
+                                }
+                            ]
+                        }
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+                {
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+                                    "bbox": [
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+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 13,
+                            "angle": 0,
+                            "type": "image_body"
+                        }
+                    ],
+                    "index": 13
+                },
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+                    ],
+                    "type": "title",
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+                                    "type": "text",
+                                    "content": "B.4 BOUNDED DIAMETERS AND WELL-SHAPED CELLS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
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+                                    "type": "text",
+                                    "content": "We assume the feasible set is a unit-hypercube of dimensionality "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "d"
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+                                    "type": "text",
+                                    "content": " where cells are evenly split along the longest edge to give "
+                                },
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+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "b"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " new partitions and "
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+                                {
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_{h,i}"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " represents the center of each cell."
+                                }
+                            ]
+                        }
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+                                    "bbox": [
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+                                    "content": "There exists a decreasing sequence "
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+                                    "content": ", such that for any depth "
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+                                    "content": "\\mathcal{X}_{h,i}"
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+                                    "type": "text",
+                                    "content": " of depth "
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+                                    "type": "text",
+                                    "content": ", we have "
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+                                    "type": "inline_equation",
+                                    "content": "\\sup_{x \\in \\mathcal{X}_{h,i}} \\ell(x_{h,i}, x) \\leq w(h)"
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": ". Moreover, there exists "
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+                                    "type": "inline_equation",
+                                    "content": "\\nu > 0"
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+                                    "type": "text",
+                                    "content": " such that for any depth "
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+                                    "type": "inline_equation",
+                                    "content": "h \\geq 0"
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+                                    "type": "text",
+                                    "content": ", any cell "
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+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}_{h,i}"
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+                                    "type": "text",
+                                    "content": " contains an "
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+                                    "type": "inline_equation",
+                                    "content": "\\ell"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "-ball of radius "
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+                                    "type": "inline_equation",
+                                    "content": "\\nu w(h)"
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+                                    "type": "text",
+                                    "content": " centered at "
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+                                    "type": "inline_equation",
+                                    "content": "x_{h,i}"
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+                                    "type": "text",
+                                    "content": "."
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+                                            "html": "<table><tr><td>l(x,y)</td><td>c</td><td>γ</td><td>ν</td></tr><tr><td>l(x,y)=||x-y||2/2</td><td>dα/2(b/2)α</td><td>b-α/d</td><td>d-α/2b-2α</td></tr><tr><td>l(x,y)=||x-y||∞/∞</td><td>(b/2)α</td><td>b-α/d</td><td>b-2α</td></tr></table>",
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+                                    "content": "Table 3: Bounding Constants: "
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+                                    "content": "\\sup_{x \\in \\mathcal{X}_{h,i}} \\ell(x_{h,i}, x) \\leq w(h) = c\\gamma^h"
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+                                    "type": "text",
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+                                    "content": "B.4.1 "
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+                                    "type": "inline_equation",
+                                    "content": "L_{2}"
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+                                    "type": "text",
+                                    "content": "-NORM"
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+                                    "type": "text",
+                                    "content": "Lemma 31 (L2-Norm Bounding Ball). Let "
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+                                    "type": "inline_equation",
+                                    "content": "\\ell(x, y) = ||x - y||_2^\\alpha"
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": ". Then "
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+                                    "type": "inline_equation",
+                                    "content": "\\sup_{x \\in \\mathcal{X}_{h,i}} \\ell(x_{h,i}, x) \\leq w_2(h) = c\\gamma^h"
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": " where "
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+                                    "content": "c = \\left(\\frac{db^2}{4}\\right)^{\\alpha/2}"
+                                },
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+                                    "type": "text",
+                                    "content": " and "
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+                                    "type": "inline_equation",
+                                    "content": "\\gamma = b^{-\\alpha/d}"
+                                },
+                                {
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+                                    ],
+                                    "type": "text",
+                                    "content": "."
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
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+                                    "type": "interline_equation",
+                                    "content": "w (0) = \\left[ \\sum_ {i = 1} ^ {d} (1 / 2) ^ {2} \\right] ^ {\\alpha / 2} = \\left(\\frac {d}{4}\\right) ^ {\\alpha / 2} \\tag {108a}",
+                                    "image_path": "c57e1283b7699dc44628ded72d225946876f89c3a50d50837072a549c339e2d0.jpg"
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+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} w (1) = \\left[ (1 / b \\cdot 1 / 2) ^ {2} + \\sum_ {i = 2} ^ {d} (1 / 2) ^ {2} \\right] ^ {\\alpha / 2} = \\left[ (1 / b ^ {2}) (1 / 4) + (d - 1) (1 / 4) \\right] ^ {\\alpha / 2} (108b) \\\\ = \\left(\\frac {d - 1 + 1 / b ^ {2}}{4}\\right) ^ {\\alpha / 2} (108c) \\\\ \\end{array}",
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} w (d) = \\left[ \\sum_ {i = 1} ^ {d} (1 / b \\cdot 1 / 2) ^ {2} \\right] ^ {\\alpha / 2} = \\left(\\frac {d}{4 \\cdot b ^ {2}}\\right) ^ {\\alpha / 2} (108d) \\\\ w (h) = \\left[ r (1 / b) ^ {2 (q + 1)} (1 / 2) ^ {2} + \\sum_ {i = r} ^ {d} (1 / b) ^ {2 q} (1 / 2) ^ {2} \\right] ^ {\\alpha / 2} (108e) \\\\ = \\left[ (1 / b) ^ {2 q} (1 / 2) ^ {2} (r (1 / b) ^ {2} + (d - r) \\right] ^ {\\alpha / 2} (108f) \\\\ = \\left[ (1 / b ^ {2}) ^ {q} (1 / 4) \\left(d - r \\left(1 - \\frac {1}{b ^ {2}}\\right)\\right) \\right] ^ {\\alpha / 2} (108g) \\\\ \\leq \\left[ (1 / b ^ {2}) ^ {q} (1 / 4) d \\right] ^ {\\alpha / 2} (108h) \\\\ \\leq \\left[ \\left(1 / b ^ {2}\\right) ^ {h / d - 1} (1 / 4) d \\right] ^ {\\alpha / 2} (108i) \\\\ = \\left[ (1 / b ^ {2}) ^ {h / d} (b ^ {2} / 4) d \\right] ^ {\\alpha / 2} (108j) \\\\ = \\left(\\frac {d b ^ {2}}{4}\\right) ^ {\\alpha / 2} (1 / b) ^ {\\frac {\\alpha}{d} h} (108k) \\\\ = c \\gamma^ {h} (1081) \\\\ \\end{array}",
+                                    "image_path": "adca2b0b3198ce708a4f481aff901785d2adc92065762f0da2d1b98965b07f00.jpg"
+                                }
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+                                    "bbox": [
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+                                    "content": "where "
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+                                    "type": "inline_equation",
+                                    "content": "q, r = \\text{divmod}(h, d) \\Rightarrow q \\geq h / d - 1, c = \\left(\\frac{db^2}{4}\\right)^{\\alpha / 2}"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ", and "
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+                                    "bbox": [
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+                                        419
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\gamma = (1 / b)^{\\alpha / d} = b^{-\\alpha / d}"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "."
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+                    "index": 5
+                },
+                {
+                    "bbox": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Lemma 32 ("
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "L_2"
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+                                    "type": "text",
+                                    "content": "-Norm Inner Ball). Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        423,
+                                        506,
+                                        449
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell(x, y) = ||x - y||_2^\\alpha"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        423,
+                                        506,
+                                        449
+                                    ],
+                                    "type": "text",
+                                    "content": ". Any cell "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        423,
+                                        506,
+                                        449
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}_{h,i}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        423,
+                                        506,
+                                        449
+                                    ],
+                                    "type": "text",
+                                    "content": " contains an "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        423,
+                                        506,
+                                        449
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        423,
+                                        506,
+                                        449
+                                    ],
+                                    "type": "text",
+                                    "content": "-ball of radius "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        423,
+                                        506,
+                                        449
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\nu w_2(h)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        423,
+                                        506,
+                                        449
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        423,
+                                        506,
+                                        449
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\nu = (db^4)^{-\\alpha/2}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        423,
+                                        506,
+                                        449
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        460,
+                        403,
+                        472
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                460,
+                                403,
+                                472
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        460,
+                                        403,
+                                        472
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Any cell "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        460,
+                                        403,
+                                        472
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}_{h,i}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        460,
+                                        403,
+                                        472
+                                    ],
+                                    "type": "text",
+                                    "content": " contains an "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        460,
+                                        403,
+                                        472
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        460,
+                                        403,
+                                        472
+                                    ],
+                                    "type": "text",
+                                    "content": "-ball of radius equal to its shortest axis:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        239,
+                        474,
+                        505,
+                        558
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                239,
+                                474,
+                                505,
+                                558
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        239,
+                                        474,
+                                        505,
+                                        558
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} r _ {\\min } = \\left[ (1 / 4) \\left(1 / b ^ {2}\\right) ^ {\\lceil h / d \\rceil} \\right] ^ {\\alpha / 2} (109a) \\\\ \\geq \\left[ (1 / 4) \\left(1 / b ^ {2}\\right) ^ {h / d + 1} \\right] ^ {\\alpha / 2} (109b) \\\\ = \\left(\\frac {1}{b ^ {2} \\cdot 4}\\right) ^ {\\alpha / 2} (1 / b) ^ {\\frac {\\alpha}{d} h} (109c) \\\\ = w (h) \\cdot \\left(\\frac {1}{d b ^ {4}}\\right) ^ {\\alpha / 2}. (109d) \\\\ \\end{array}",
+                                    "image_path": "80d414bf2fac3dc572ab104a79dbe7aa7e66f1e617bdb8c06e1f019849a67eb2.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
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+                        594
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                582,
+                                188,
+                                594
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        582,
+                                        188,
+                                        594
+                                    ],
+                                    "type": "text",
+                                    "content": "B.4.2 "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        582,
+                                        188,
+                                        594
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "L_{\\infty}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        582,
+                                        188,
+                                        594
+                                    ],
+                                    "type": "text",
+                                    "content": "-NORM"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
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+                        600,
+                        504,
+                        628
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                600,
+                                504,
+                                628
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        600,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 33 ("
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        600,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "L_{\\infty}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        600,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": "-Norm Bounding Ball). Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        600,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell(x, y) = ||x - y||_{\\infty}^{\\alpha}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        600,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        600,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sup_{x \\in \\mathcal{X}_{h,i}} \\ell(x_{h,i}, x) \\leq w_{\\infty}(h) = c\\gamma^{h}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        600,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        600,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "c = \\left(\\frac{b}{2}\\right)^{\\alpha}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        600,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        600,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\gamma = b^{-\\alpha / d}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        600,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        639,
+                        428,
+                        651
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                639,
+                                428,
+                                651
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        639,
+                                        428,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Any cell "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        639,
+                                        428,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}_{h,i}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        639,
+                                        428,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": " is contained by an "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        639,
+                                        428,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        639,
+                                        428,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": "-ball of radius equal to its longest axis:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        239,
+                        653,
+                        505,
+                        730
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                239,
+                                653,
+                                505,
+                                730
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        239,
+                                        653,
+                                        505,
+                                        730
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} r _ {\\max } = \\left[ (1 / 4) \\left(1 / b ^ {2}\\right) ^ {\\lfloor h / d \\rfloor} \\right] ^ {\\alpha / 2} (110a) \\\\ \\leq \\left[ (1 / 4) \\left(1 / b ^ {2}\\right) ^ {h / d - 1} \\right] ^ {\\alpha / 2} (110b) \\\\ = \\left(\\frac {b ^ {2}}{4}\\right) ^ {\\alpha / 2} (1 / b) ^ {\\frac {\\alpha}{d} h} (110c) \\\\ = c \\gamma^ {h} (110d) \\\\ \\end{array}",
+                                    "image_path": "36ca72410863d16ea10eb563795b63c70d2ea2ea76b1826a9114652631945f3f.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
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+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
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+                                    "type": "text",
+                                    "content": "38"
+                                }
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+                }
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        302,
+                                        97
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "c = \\left(\\frac{b^2}{4}\\right)^{\\alpha / 2}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        302,
+                                        97
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        302,
+                                        97
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\gamma = (1 / b)^{\\alpha / d} = b^{-\\alpha / d}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        302,
+                                        97
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "type": "image",
+                    "bbox": [
+                        494,
+                        84,
+                        505,
+                        94
+                    ],
+                    "blocks": [
+                        {
+                            "bbox": [
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+                                84,
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+                                94
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+                            "lines": [
+                                {
+                                    "bbox": [
+                                        494,
+                                        84,
+                                        505,
+                                        94
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+                                    "spans": [
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+                                            "bbox": [
+                                                494,
+                                                84,
+                                                505,
+                                                94
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+                                            "type": "image",
+                                            "image_path": "bceee6c50ff66f24cad0648f2ad82e3b0b2853e2fd8d46f2a2a53b40be08fb15.jpg"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 2,
+                            "angle": 0,
+                            "type": "image_body"
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        103,
+                        506,
+                        128
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                103,
+                                506,
+                                128
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        103,
+                                        506,
+                                        128
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 34 ("
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        103,
+                                        506,
+                                        128
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "L_{\\infty}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        103,
+                                        506,
+                                        128
+                                    ],
+                                    "type": "text",
+                                    "content": "-Norm Inner Ball). Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        103,
+                                        506,
+                                        128
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell(x, y) = ||x - y||_{\\infty}^{\\alpha}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        103,
+                                        506,
+                                        128
+                                    ],
+                                    "type": "text",
+                                    "content": ". Any cell "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        103,
+                                        506,
+                                        128
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}_{h,i}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        103,
+                                        506,
+                                        128
+                                    ],
+                                    "type": "text",
+                                    "content": " contains an "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        103,
+                                        506,
+                                        128
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        103,
+                                        506,
+                                        128
+                                    ],
+                                    "type": "text",
+                                    "content": "-ball of radius "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        103,
+                                        506,
+                                        128
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\nu w_{\\infty}(h)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        103,
+                                        506,
+                                        128
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        103,
+                                        506,
+                                        128
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\nu = b^{-2\\alpha}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        103,
+                                        506,
+                                        128
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        141,
+                        403,
+                        154
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                141,
+                                403,
+                                154
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        141,
+                                        403,
+                                        154
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Any cell "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        141,
+                                        403,
+                                        154
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X}_{h,i}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        141,
+                                        403,
+                                        154
+                                    ],
+                                    "type": "text",
+                                    "content": " contains an "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        141,
+                                        403,
+                                        154
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        141,
+                                        403,
+                                        154
+                                    ],
+                                    "type": "text",
+                                    "content": "-ball of radius equal to its shortest axis:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        242,
+                        161,
+                        504,
+                        243
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                242,
+                                161,
+                                504,
+                                243
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        242,
+                                        161,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} r _ {\\min } = \\left[ (1 / 4) \\left(1 / b ^ {2}\\right) ^ {\\lceil h / d \\rceil} \\right] ^ {\\alpha / 2} (111a) \\\\ \\geq \\left[ (1 / 4) \\left(1 / b ^ {2}\\right) ^ {h / d + 1} \\right] ^ {\\alpha / 2} (111b) \\\\ = \\left(\\frac {1}{b ^ {2} \\cdot 4}\\right) ^ {\\alpha / 2} (1 / b) ^ {\\frac {\\alpha}{d} h} (111c) \\\\ = w (h) \\cdot \\left(\\frac {1}{b ^ {4}}\\right) ^ {\\alpha / 2}. (111d) \\\\ \\end{array}",
+                                    "image_path": "90e2152d3b30a8b2ed8cd85aa960181cb7807f96570f168737f8c105349ff05d.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
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+                {
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+                                    "type": "text",
+                                    "content": "B.5 STOCHASTIC SIMULTANEOUS OPTIMISTIC OPTIMIZATION"
+                                }
+                            ]
+                        }
+                    ],
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+                },
+                {
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+                                    "type": "text",
+                                    "content": "StoSOO is flexible in its choice of metric and partitioning structure. In StoSOO, we may choose "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "\\ell(x,y) = ||x - y||_2^2"
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Lemma 35. The volume of a "
+                                },
+                                {
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+                                    ],
+                                    "type": "inline_equation",
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "-sphere with radius "
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r"
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
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+                                        322,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " even is given by "
+                                },
+                                {
+                                    "bbox": [
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+                                        358
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S_{d}r^{d}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        322,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
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+                                        322,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S_{d}^{-1} \\leq \\sqrt{2\\pi d}\\left(\\frac{d}{2\\pi e}\\right)^{d/2}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        322,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
+                        371,
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+                        397
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                397
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Proof. First, we recall Stirling's bounds on the factorial: "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        371,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sqrt{2\\pi n} \\left( \\frac{n}{e} \\right)^n e^{\\frac{1}{12n + 1}} < n! < \\sqrt{2\\pi n} \\left( \\frac{n}{e} \\right)^n e^{\\frac{1}{12n}}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ". This will be useful for bounding the Gamma function: "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        371,
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+                                        397
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Gamma(d) = (d - 1)!"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        371,
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+                                        397
+                                    ],
+                                    "type": "text",
+                                    "content": " for even "
+                                },
+                                {
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+                                        397
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                        397
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
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+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "Given "
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+                                    "type": "inline_equation",
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " is even, we start with the exact formula for "
+                                },
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S_{d}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ":"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
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+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} S _ {d} ^ {- 1} = \\frac {\\Gamma (d / 2 + 1)}{\\pi^ {d / 2}} (112a) \\\\ = \\frac {(d / 2) !}{\\pi^ {d / 2}} (112b) \\\\ <   \\frac {\\sqrt {2 \\pi (d / 2)} \\left(\\frac {d / 2}{e}\\right) ^ {d / 2} e ^ {\\frac {1}{1 2 (d / 2)}}}{\\pi^ {d / 2}} (112c) \\\\ = \\frac {\\pi^ {1 / 2} d ^ {1 / 2} \\left(\\frac {d}{2 e}\\right) ^ {d / 2} e ^ {\\frac {1}{6 d}}}{\\pi^ {d / 2}} (112d) \\\\ = \\frac {\\pi^ {1 / 2} d ^ {(d + 1) / 2} e ^ {\\frac {1}{6 d}}}{(2 \\pi e) ^ {d / 2}} (112e) \\\\ \\leq \\sqrt {2 \\pi d} \\left(\\frac {d}{2 \\pi e}\\right) ^ {d / 2}. (112f) \\\\ \\end{array}",
+                                    "image_path": "320e69cfb2a20a288d8e4f7bf3a92a2a0ff2c9ad77dbed1726700906f788a822.jpg"
+                                }
+                            ]
+                        }
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+                            "index": 13,
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 36. The near optimality dimension and constant under the "
+                                },
+                                {
+                                    "bbox": [
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+                                        499,
+                                        617
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell(x, y) = ||x - y||_2^2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                        499,
+                                        617
+                                    ],
+                                    "type": "text",
+                                    "content": " norm are"
+                                }
+                            ]
+                        }
+                    ],
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+                },
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+                                    "type": "interline_equation",
+                                    "content": "d ^ {\\prime} = 0 \\tag {113}",
+                                    "image_path": "07be73d1481b57035949be774dcb5b6c354915972f43d614aec325dbf301e33b.jpg"
+                                }
+                            ]
+                        }
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+                                    "type": "interline_equation",
+                                    "content": "C = \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\sqrt {2 \\pi n \\bar {m}} \\left(\\frac {n \\bar {m}}{5 \\nu r _ {\\eta} ^ {2} \\sigma_ {- 2}}\\right) ^ {n \\bar {m} / 2} \\tag {114}",
+                                    "image_path": "171fdaf253305b894fd77b90ef6a896828202c163e13d3bc7f1ac7942e03da79.jpg"
+                                }
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+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                        693
+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        670,
+                                        504,
+                                        693
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma_{-2}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        670,
+                                        504,
+                                        693
+                                    ],
+                                    "type": "text",
+                                    "content": " is a lower bound on the singular value of the Hessian matrix of the function at every equilibrium and "
+                                },
+                                {
+                                    "bbox": [
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+                                        670,
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+                                        693
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\nu"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        670,
+                                        504,
+                                        693
+                                    ],
+                                    "type": "text",
+                                    "content": " is defined in Table 3 for the corresponding "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        670,
+                                        504,
+                                        693
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell(x, y)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                        504,
+                                        693
+                                    ],
+                                    "type": "text",
+                                    "content": "-ball."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
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+                        733
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Recall from Theorem 1 that "
+                                },
+                                {
+                                    "bbox": [
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+                                        706,
+                                        506,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d' = d\\left(\\frac{\\alpha_{hi} - \\alpha_{lo}}{\\alpha_{lo}\\alpha_{hi}}\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        706,
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+                                    ],
+                                    "type": "text",
+                                    "content": " with constant "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        706,
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+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " defined below. In addition, we will analyze StoSOO with the choice "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        706,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell(x,y) = ||x - y||_2^2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        706,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ". Matching to Assumption 1, we see"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                }
+            ],
+            "discarded_blocks": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
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+                },
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+                                    "type": "text",
+                                    "content": "39"
+                                }
+                            ]
+                        }
+                    ],
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+                }
+            ],
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+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "that "
+                                },
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+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\sigma_{+} = 1"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " and "
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+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\alpha_{lo} = 2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
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+                                        112
+                                    ],
+                                    "type": "text",
+                                    "content": ". Under the 2-norm, the volume constant "
+                                },
+                                {
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S_{d}^{-1} \\leq \\sqrt{2\\pi d}\\left(\\frac{d}{2\\pi e}\\right)^{d/2}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        80,
+                                        504,
+                                        112
+                                    ],
+                                    "type": "text",
+                                    "content": " (see Lemma 35)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        117,
+                        505,
+                        152
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                117,
+                                505,
+                                152
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        117,
+                                        505,
+                                        152
+                                    ],
+                                    "type": "text",
+                                    "content": "We will bound the function locally with a quadratic, i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        117,
+                                        505,
+                                        152
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha_{hi} = 2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        117,
+                                        505,
+                                        152
+                                    ],
+                                    "type": "text",
+                                    "content": ". Lastly, from (Valko et al., 2013, Corollary 1), "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        117,
+                                        505,
+                                        152
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\psi = \\frac{\\nu}{3}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        117,
+                                        505,
+                                        152
+                                    ],
+                                    "type": "text",
+                                    "content": " with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        117,
+                                        505,
+                                        152
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\nu = \\frac{1}{db^2}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        117,
+                                        505,
+                                        152
+                                    ],
+                                    "type": "text",
+                                    "content": " defined in Table 3 and the dimension of our search space (the product space of player mixed strategies) is "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        117,
+                                        505,
+                                        152
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d = n(\\bar{m} - 1) \\leq n\\bar{m}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        117,
+                                        505,
+                                        152
+                                    ],
+                                    "type": "text",
+                                    "content": " for simplicity."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        105,
+                        156,
+                        356,
+                        168
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                156,
+                                356,
+                                168
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        156,
+                                        356,
+                                        168
+                                    ],
+                                    "type": "text",
+                                    "content": "Plugging this information into Theorem 1, we find "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        156,
+                                        356,
+                                        168
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d' = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        156,
+                                        356,
+                                        168
+                                    ],
+                                    "type": "text",
+                                    "content": " and"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        165,
+                        175,
+                        505,
+                        205
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                165,
+                                175,
+                                505,
+                                205
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        165,
+                                        175,
+                                        505,
+                                        205
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "C = \\max  \\left\\{1, \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} S _ {d} ^ {- 1} \\left(r _ {\\eta} ^ {\\frac {\\alpha_ {h i}}{\\alpha_ {l o}}} \\sigma_ {-} ^ {\\left(\\frac {\\alpha_ {h i} - \\alpha_ {l o}}{\\alpha_ {l o} \\alpha_ {h i}}\\right)}\\right) ^ {- d} \\right\\} \\left(\\frac {\\sigma_ {+}}{\\psi \\sigma_ {-} ^ {\\alpha_ {l o} / \\alpha_ {h i}}}\\right) ^ {d / \\alpha_ {l o}} \\tag {115a}",
+                                    "image_path": "c807c3054ed41dc85bc074505ce35550ddbebc6375a67eed8c316c83f8b8c623.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        177,
+                        207,
+                        505,
+                        233
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                177,
+                                207,
+                                505,
+                                233
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        177,
+                                        207,
+                                        505,
+                                        233
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "= \\max  \\left\\{1, \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} S _ {d} ^ {- 1} r _ {\\eta} ^ {- d} \\right\\} \\left(\\frac {1}{\\psi \\sigma_ {- 2}}\\right) ^ {d / 2} \\tag {115b}",
+                                    "image_path": "059093aaf3bbbf9f1c376dcdc9bb65b26aaa23024abdc7ce42a0cac001d4f616.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        176,
+                        235,
+                        505,
+                        259
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                176,
+                                235,
+                                505,
+                                259
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        176,
+                                        235,
+                                        505,
+                                        259
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\stackrel {h a r d} {=} \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} S _ {d} ^ {- 1} r _ {\\eta} ^ {- d} \\left(\\frac {3}{\\nu \\sigma_ {- 2}}\\right) ^ {d / 2} \\tag {115c}",
+                                    "image_path": "5be90318f09b80bef1a313ad09572ce52263389b57b38ee927d7c269d4cf29b1.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        176,
+                        262,
+                        505,
+                        289
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                176,
+                                262,
+                                505,
+                                289
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        176,
+                                        262,
+                                        505,
+                                        289
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\leq^ {h a r d} \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\sqrt {2 \\pi d} \\left(\\frac {d}{2 \\pi e}\\right) ^ {d / 2} \\left(\\frac {3}{\\nu r _ {\\eta} ^ {2} \\sigma_ {- 2}}\\right) ^ {d / 2} \\tag {115d}",
+                                    "image_path": "20b86c6617829b0398684ba5ce5d55bc6850aa7f66c4022ba09d5e50097c2121.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        176,
+                        291,
+                        505,
+                        319
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                176,
+                                291,
+                                505,
+                                319
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        176,
+                                        291,
+                                        505,
+                                        319
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\leq^ {h a r d} \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\sqrt {2 \\pi d} \\left(\\frac {b ^ {2} d ^ {2}}{5 r _ {\\eta} ^ {2} \\sigma_ {- 2}}\\right) ^ {d / 2} \\tag {115e}",
+                                    "image_path": "3513c22827e7901cb10a3ef46b8a5c411f9692bb756454fccdee68d216fdf65f.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        176,
+                        320,
+                        505,
+                        349
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                176,
+                                320,
+                                505,
+                                349
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        176,
+                                        320,
+                                        505,
+                                        349
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\stackrel {h a r d} {=} \\left| \\mathcal {X} ^ {*} \\right| ^ {- 1} \\sqrt {2 \\pi n \\bar {m}} \\left(\\frac {b ^ {2} n ^ {2} \\bar {m} ^ {2}}{5 r _ {\\eta} ^ {2} \\sigma_ {- 2}}\\right) ^ {n \\bar {m} / 2} \\tag {115f}",
+                                    "image_path": "4b3a3b58b325c5f1ed740c329447edcfbc02ed6a98d311b8273d6f6b2c0c5565.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
+                        354,
+                        504,
+                        378
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                354,
+                                504,
+                                378
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        504,
+                                        378
+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        504,
+                                        378
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\text{hard}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        504,
+                                        378
+                                    ],
+                                    "type": "text",
+                                    "content": " indicates we are assuming "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        504,
+                                        378
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r_{\\eta}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        504,
+                                        378
+                                    ],
+                                    "type": "text",
+                                    "content": ", the radius of the ball under which the local polynomial bounds are accurate, is small enough to dominate the other operand of the max."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        386,
+                        504,
+                        410
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                386,
+                                504,
+                                410
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        386,
+                                        504,
+                                        410
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem 3 (StoSOO Regret Rate). Corollary 1 of Valko et al. (2013) implies that with probability "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        386,
+                                        504,
+                                        410
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "1 - \\delta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        386,
+                                        504,
+                                        410
+                                    ],
+                                    "type": "text",
+                                    "content": ", the regret, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        386,
+                                        504,
+                                        410
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "R_{t}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        386,
+                                        504,
+                                        410
+                                    ],
+                                    "type": "text",
+                                    "content": ", of StoSOO after "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        386,
+                                        504,
+                                        410
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        386,
+                                        504,
+                                        410
+                                    ],
+                                    "type": "text",
+                                    "content": " pulls is upper bounded as"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        201,
+                        416,
+                        505,
+                        449
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                201,
+                                416,
+                                505,
+                                449
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        201,
+                                        416,
+                                        505,
+                                        449
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "(2 + b ^ {2 / d}) \\sqrt {\\frac {\\log_ {b} (t k / \\delta)}{2 \\log_ {b} (e) k}} + \\frac {1}{4} d b ^ {2 (1 + 2 / d)} b ^ {- \\frac {1}{d C} \\sqrt {t / k}} \\tag {116}",
+                                    "image_path": "e14cc354569fbec7662ab2bd7c3d917d2ecc2236e0126a340d61a484c6241f37.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        104,
+                        455,
+                        506,
+                        478
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                455,
+                                506,
+                                478
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        455,
+                                        506,
+                                        478
+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        455,
+                                        506,
+                                        478
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d = n(\\bar{m} - 1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        455,
+                                        506,
+                                        478
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        455,
+                                        506,
+                                        478
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "b"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        455,
+                                        506,
+                                        478
+                                    ],
+                                    "type": "text",
+                                    "content": " is the branching factor for partitioning cells, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        455,
+                                        506,
+                                        478
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        455,
+                                        506,
+                                        478
+                                    ],
+                                    "type": "text",
+                                    "content": " is the near-optimality constant, and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        455,
+                                        506,
+                                        478
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        455,
+                                        506,
+                                        478
+                                    ],
+                                    "type": "text",
+                                    "content": " is the maximum number of evaluations per node."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        104,
+                        493,
+                        504,
+                        516
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                493,
+                                504,
+                                516
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        493,
+                                        504,
+                                        516
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Plugging the constants "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        493,
+                                        504,
+                                        516
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "c, \\gamma"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        493,
+                                        504,
+                                        516
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        493,
+                                        504,
+                                        516
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\nu"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        493,
+                                        504,
+                                        516
+                                    ],
+                                    "type": "text",
+                                    "content": " defined in Table 3 for the 2-norm into Corollary 1 of Valko et al. (2013), we find with probability "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        493,
+                                        504,
+                                        516
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "1 - \\delta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        493,
+                                        504,
+                                        516
+                                    ],
+                                    "type": "text",
+                                    "content": ":"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        180,
+                        522,
+                        504,
+                        665
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                180,
+                                522,
+                                504,
+                                665
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        180,
+                                        522,
+                                        504,
+                                        665
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} R _ {t} \\leq (2 + 1 / \\gamma) \\epsilon + c \\gamma^ {\\frac {1}{2 C} \\sqrt {t / k} - 2} (117a) \\\\ = (2 + 1 / \\gamma) \\sqrt {\\frac {\\log (t k / \\delta)}{2 k}} + c \\gamma^ {\\frac {1}{2 C} \\sqrt {t / k} - 2} (117b) \\\\ = \\left(2 + b ^ {2 / d}\\right) \\sqrt {\\frac {\\log (t k / \\delta)}{2 k}} + \\frac {1}{4} d b ^ {2} \\left(b ^ {- 2 / d}\\right) ^ {\\frac {1}{2 C} \\sqrt {t / k} - 2} (117c) \\\\ = (2 + b ^ {2 / d}) \\sqrt {\\frac {\\log_ {b} (t k / \\delta)}{2 \\log_ {b} (e) k}} + \\frac {1}{4} d b ^ {2 (1 + 2 / d)} \\left(b ^ {- 2 / d}\\right) ^ {\\frac {1}{2 C} \\sqrt {t / k}} (117d) \\\\ = \\left(2 + b ^ {2 / d}\\right) \\sqrt {\\frac {\\log_ {b} (t k / \\delta)}{2 \\log_ {b} (e) k}} + \\frac {1}{4} d b ^ {2 (1 + 2 / d)} b ^ {- \\frac {1}{d C} \\sqrt {t / k}}. (117e) \\\\ \\end{array}",
+                                    "image_path": "f1f845bff7db7c7f3bb59ab1c13fe91da8632aae009866cd2e0c333e14556014.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "type": "image",
+                    "bbox": [
+                        494,
+                        670,
+                        504,
+                        681
+                    ],
+                    "blocks": [
+                        {
+                            "bbox": [
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+                                670,
+                                504,
+                                681
+                            ],
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
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+                                                670,
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+                                                681
+                                            ],
+                                            "type": "image",
+                                            "image_path": "9ace8b6d2efdd07243cdf25690d81420b8365d871deb581108c33f389410ec51.jpg"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 16,
+                            "angle": 0,
+                            "type": "image_body"
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        104,
+                        691,
+                        506,
+                        733
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                691,
+                                506,
+                                733
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        691,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "Proposition 2. Assume the same conditions as Theorem 3 and let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        691,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k = t\\log_b(t)^{-\\rho}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        691,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        691,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho \\geq 3"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        691,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then the StoSOO (Algorithm 1 of Valko et al. (2013)) bound requires "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        691,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t = b^{\\left(\\frac{dC}{2}\\right)^{\\frac{2}{\\rho - 2}}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        691,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " pulls before exhibiting a "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        691,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\mathcal{O}}(T^{-1/2})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        691,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " decay in regret."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
+                        751,
+                        311,
+                        760
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                299,
+                                751,
+                                311,
+                                760
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        299,
+                                        751,
+                                        311,
+                                        760
+                                    ],
+                                    "type": "text",
+                                    "content": "40"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 39
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        81,
+                        506,
+                        159
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                81,
+                                506,
+                                159
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        159
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. We know from the given definition of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        159
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        159
+                                    ],
+                                    "type": "text",
+                                    "content": " that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        159
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sqrt{t / k} = \\log_b(t)^{\\rho /2}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        159
+                                    ],
+                                    "type": "text",
+                                    "content": ". We will analyze the second term in the regret formula of Corollary 3. Note that the second term, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        159
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "b^{-\\frac{1}{dC}\\sqrt{t / k}} = (b\\sqrt{t / k})^{-\\frac{1}{dC}} = (b^{\\log_b(t)^{\\rho /2}})^{-\\frac{1}{dC}} = (b^{\\log_b(t)\\cdot \\log_b(t)^{(\\rho -2) / 2}})^{-\\frac{1}{dC}} = (t^{\\log_b(t)^{(\\rho -2) / 2}})^{-\\frac{1}{dC}} = t^{-\\frac{\\log_b(t)^{(\\rho - 2) / 2}}{dC}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        159
+                                    ],
+                                    "type": "text",
+                                    "content": ". To achieve a "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        159
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t^{-1 / 2}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        159
+                                    ],
+                                    "type": "text",
+                                    "content": " convergence rate (or better), we need "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        159
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t\\geq b^{(\\frac{dC}{2})^{\\frac{2}{\\rho - 2}}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        159
+                                    ],
+                                    "type": "text",
+                                    "content": ". For convenience, we report the entire simplification of the bound below:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        132,
+                        168,
+                        505,
+                        365
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                132,
+                                168,
+                                505,
+                                365
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        132,
+                                        168,
+                                        505,
+                                        365
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} R _ {t} \\leq (2 + b ^ {2 / d}) \\sqrt {\\frac {\\log_ {b} (t k / \\delta)}{2 \\log_ {b} (e) k}} + \\frac {1}{4} d b ^ {2 (1 + 2 / d)} b ^ {- \\frac {1}{d C}} \\sqrt {t / k} (118a) \\\\ = (2 + b ^ {2 / d}) \\sqrt {\\frac {\\log_ {b} \\left(t ^ {2} / \\left(\\delta \\log_ {b} (t) ^ {\\rho}\\right)\\right) \\log_ {b} (t) ^ {\\rho}}{2 \\log_ {b} (e) t}} + \\frac {1}{4} d b ^ {2 (1 + 2 / d)} t ^ {- \\frac {\\log_ {b} (t) ^ {(\\rho - 2) / 2}}{d C}} (118b) \\\\ \\leq \\left(2 + b ^ {2 / d}\\right) \\sqrt {\\frac {\\log_ {b} \\left(t ^ {2} / \\delta\\right) \\log_ {b} \\left(t / \\delta\\right) ^ {\\rho}}{2 \\log_ {b} (e) t}} + \\frac {1}{4} d b ^ {2 (1 + 2 / d)} t ^ {- \\frac {\\log_ {b} (t) ^ {(\\rho - 2) / 2}}{d C}} (118c) \\\\ = (2 + b ^ {2 / d}) \\sqrt {\\frac {2 \\log_ {b} (t / \\delta) ^ {\\rho + 1}}{2 \\log_ {b} (e) t}} + \\frac {1}{4} d b ^ {2 (1 + 2 / d)} t ^ {- \\frac {\\log_ {b} (t) ^ {(\\rho - 2) / 2}}{d C}} (118d) \\\\ = \\frac {\\left(2 + b ^ {2 / d}\\right)}{\\sqrt {\\log_ {b} (e)}} \\frac {\\log_ {b} (t / \\delta) ^ {(\\rho + 1) / 2}}{\\sqrt {t}} + \\frac {1}{4} d b ^ {2 (1 + 2 / d)} t ^ {- \\frac {\\log_ {b} (t) ^ {(\\rho - 2) / 2}}{d C}} (118e) \\\\ = \\frac {\\left(2 + b ^ {2 / d}\\right)}{\\sqrt {\\log_ {b} (e)}} \\frac {\\left[ \\log_ {b} (t) - \\log_ {b} (\\delta) \\right] ^ {(\\rho + 1) / 2}}{\\sqrt {t}} + \\frac {1}{4} d b ^ {2 (1 + 2 / d)} t ^ {- \\frac {\\log_ {b} (t) ^ {(\\rho - 2) / 2}}{d C}}. (118f) \\\\ \\end{array}",
+                                    "image_path": "a11e2fb9b8de2605000d70b11b07c15ec29b8a15061290d746d8ee64d90cbc95.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "type": "image",
+                    "bbox": [
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+                        381
+                    ],
+                    "blocks": [
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+                            "bbox": [
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+                                    "bbox": [
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+                                            ],
+                                            "type": "image",
+                                            "image_path": "2d3ff3d62dd76cd6834c1121350eb79246f1e3f2f88373c316c4cc70635be961.jpg"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 3,
+                            "angle": 0,
+                            "type": "image_body"
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
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+                        409
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "B.6 REGRET TO PAC BOUNDS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
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+                        464
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                464
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        419,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 37. [Loss Regret to Exploitability Regret] Assume exploitability of a joint strategy "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        419,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        419,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": " is upper bounded by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        419,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f(\\mathcal{L}^{\\tau}(\\pmb{x}))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        419,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        419,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        419,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": " is a concave function and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        419,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}^{\\tau}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        419,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": " is a loss function. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        419,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x}_t"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        419,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": " be a joint strategy randomly drawn from the set of predictions made by an online learning algorithm "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        419,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{A}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        419,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": " over "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        419,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        419,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": " steps. Then the expected exploitability of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        419,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x}_t"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        419,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": " is bounded by the average regret of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        419,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{A}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        419,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": ":"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        260,
+                        471,
+                        504,
+                        495
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                260,
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+                                504,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        260,
+                                        471,
+                                        504,
+                                        495
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbb {E} _ {t} [ \\epsilon_ {t} ] \\leq f \\left(\\frac {1}{T} R (T)\\right). \\tag {119}",
+                                    "image_path": "1c0afe4078fb0512c557fd109c8abb315a61f2c8d924718ce3d0320271475418.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
+                                        511,
+                                        133,
+                                        523
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        154,
+                        530,
+                        504,
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbb {E} _ {t} \\left[ \\epsilon_ {t} \\right] = \\mathbb {E} _ {t} \\left[ f \\left(\\mathcal {L} \\left(\\boldsymbol {x} _ {t}\\right)\\right) \\right] \\leq f \\left(\\mathbb {E} _ {t} \\left[ \\mathcal {L} \\left(\\boldsymbol {x} _ {t}\\right) \\right]\\right) = f \\left(\\frac {1}{T} \\sum_ {t} \\mathcal {L} \\left(\\boldsymbol {x} _ {t}\\right)\\right) = f \\left(\\frac {1}{T} R (T)\\right) \\tag {120}",
+                                    "image_path": "f2e826cd63790f2aa80f26a5fcea861394543d8f54181c2046c65a043352d51c.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                {
+                                    "bbox": [
+                                        105,
+                                        565,
+                                        323,
+                                        578
+                                    ],
+                                    "type": "text",
+                                    "content": "where the inequality follows from Jensen's inequality."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "type": "image",
+                    "bbox": [
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+                    ],
+                    "blocks": [
+                        {
+                            "bbox": [
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+                                            "image_path": "8cb08643aa045aa04bdc3c88894e7dc948749456006babe3f63950ec4680e0d7.jpg"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 10,
+                            "angle": 0,
+                            "type": "image_body"
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        590,
+                        506,
+                        628
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                590,
+                                506,
+                                628
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        590,
+                                        506,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem 4 (BLiN PAC Rate). Assume "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        590,
+                                        506,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\eta_{k} = \\eta = 2 / \\hat{L}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        590,
+                                        506,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": " as defined in Lemma 2, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        590,
+                                        506,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau = \\frac{1}{\\ln(1 / p)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        590,
+                                        506,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": " so that all equilibria place at least "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        590,
+                                        506,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\frac{p}{m^*}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        590,
+                                        506,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": " mass on each strategy, and a previously pulled arm is returned uniformly at random (i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        590,
+                                        506,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t \\sim U(T)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        590,
+                                        506,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": "). Then for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        590,
+                                        506,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        590,
+                                        506,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": ","
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        165,
+                        635,
+                        504,
+                        666
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                165,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        165,
+                                        635,
+                                        504,
+                                        666
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\epsilon_ {t} \\leq w \\left[ \\tau \\log \\left(\\prod_ {k} m _ {k}\\right) + 2 \\left(1 + \\left(4 c ^ {2} C _ {z}\\right) ^ {1 / 3}\\right) \\sqrt {2 n \\hat {L}} \\left(\\frac {\\ln T}{T}\\right) ^ {\\frac {1}{2 \\left(d _ {z} + 2\\right)}} \\right] \\tag {121}",
+                                    "image_path": "27ba56afb8c35a478f70a1de34aa5562b181d25e6d02509ef537299f68648aed.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        104,
+                        673,
+                        506,
+                        735
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                673,
+                                506,
+                                735
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        673,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": "with probability "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        673,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(1 - w^{-1})(1 - 2T^{-2})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        673,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        673,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m^{*} = \\max_{k} m_{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        673,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        673,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "c \\leq \\frac{n\\bar{m}}{\\hat{L}}\\left(\\frac{\\ln(m^{*})}{\\ln(1/p)} + 2\\right)^{2}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        673,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": " is an upper bound on the range of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        673,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\mathcal{L}}^{\\tau}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        673,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": " (Corollary 1), "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        673,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{L} = \\left(\\frac{\\ln(m^{*})}{\\ln(1/p)} + 2\\right)\\left(\\frac{m^{*2}}{p\\ln(1/p)} + n\\bar{m}\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        673,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": " (Corollary 2), the zooming dimension "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        673,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d_{z} = \\frac{1}{2} n\\bar{m}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        673,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": ", and the zooming constant "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        673,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_{z} = |\\mathcal{X}^{*}|^{-1}\\left(\\frac{4}{r_{\\eta}^{2}\\sigma_{-\\infty}}\\right)^{n\\bar{m}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        673,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": " (Corollary 30)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
+                        751,
+                        310,
+                        760
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                310,
+                                760
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        299,
+                                        751,
+                                        310,
+                                        760
+                                    ],
+                                    "type": "text",
+                                    "content": "41"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 40
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        81,
+                        506,
+                        118
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                81,
+                                506,
+                                118
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        118
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Assume "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        118
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\eta_{k} = \\eta = \\frac{2}{L}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        118
+                                    ],
+                                    "type": "text",
+                                    "content": " as defined in Lemma 2 so that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        118
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}^{\\tau}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        118
+                                    ],
+                                    "type": "text",
+                                    "content": " is 1-Lipschitz with respect to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        118
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "||\\cdot ||_{\\infty}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        118
+                                    ],
+                                    "type": "text",
+                                    "content": ". Also assume a previously pulled arm is returned uniformly at random. Starting with Lemma 13 and applying Corollary 6, we find"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        133,
+                        125,
+                        505,
+                        220
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                133,
+                                125,
+                                505,
+                                220
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        133,
+                                        125,
+                                        505,
+                                        220
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\mathbb {E} \\left[ \\epsilon_ {t} \\right] \\leq \\tau \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {\\frac {2 n}{\\min  _ {k} \\eta_ {k}}} \\sqrt {\\frac {1}{T} \\sum_ {t} \\mathcal {L} ^ {\\tau} \\left(\\boldsymbol {x} _ {t}\\right)} (122a) \\\\ \\leq \\frac {1}{\\ln (1 / p)} \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {n \\hat {L}} \\sqrt {8 \\left(1 + \\left(4 c ^ {2}\\right) ^ {1 / 3}\\right) ^ {2} T ^ {\\frac {- 1}{\\left(d z + 2\\right)}} \\ln T ^ {\\frac {1}{\\left(d z + 2\\right)}}} (122b) \\\\ = \\frac {1}{\\ln (1 / p)} \\log \\left(\\prod_ {k} m _ {k}\\right) + 2 \\left(1 + \\left(4 c ^ {2} C _ {z}\\right) ^ {1 / 3}\\right) \\sqrt {2 n \\hat {L}} \\left(\\frac {\\ln T}{T}\\right) ^ {\\frac {1}{2 \\left(d _ {z} + 2\\right)}} (122c) \\\\ \\end{array}",
+                                    "image_path": "b69f1cd7327537d6b780b984fedff50895e3bab5b87d53a7c7e6dac5b2f004f9.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        228,
+                        504,
+                        262
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                228,
+                                504,
+                                262
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        228,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "text",
+                                    "content": "with probability "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        228,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "1 - 2T^{-2}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        228,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        228,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m^{*} = \\max_{k} m_{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        228,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        228,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "c \\leq \\frac{n\\bar{m}}{\\hat{L}} \\left( \\frac{\\ln(m^{*})}{\\ln(1/p)} + 2 \\right)^{2}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        228,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "text",
+                                    "content": " is an upper bound on the range of sampled values from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        228,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{\\mathcal{L}}^{\\tau}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        228,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "text",
+                                    "content": " (see Corollary 1)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        266,
+                        505,
+                        289
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                266,
+                                505,
+                                289
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        266,
+                                        505,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": "Markov's inequality then allows us to bound the pointwise exploitability of any arm returned by the algorithm as"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        150,
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+                        326
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                150,
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+                                326
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        150,
+                                        297,
+                                        505,
+                                        326
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\epsilon_ {t} \\leq w \\left[ \\frac {1}{\\ln (1 / p)} \\log \\left(\\prod_ {k} m _ {k}\\right) + 2 \\left(1 + \\left(4 c ^ {2} C _ {z}\\right) ^ {1 / 3}\\right) \\sqrt {2 n \\hat {L}} \\left(\\frac {\\ln T}{T}\\right) ^ {\\frac {1}{2 \\left(d _ {z} + 2\\right)}} \\right] \\tag {123}",
+                                    "image_path": "3951ba69e9c16d42495257fc92acfcda09d80d756343eaedafcce31c30759844.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        333,
+                        327,
+                        346
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                333,
+                                327,
+                                346
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        333,
+                                        327,
+                                        346
+                                    ],
+                                    "type": "text",
+                                    "content": "with probability "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        333,
+                                        327,
+                                        346
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(1 - w^{-1})(1 - 2T^{-2})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        333,
+                                        327,
+                                        346
+                                    ],
+                                    "type": "text",
+                                    "content": " for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        333,
+                                        327,
+                                        346
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        333,
+                                        327,
+                                        346
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        354,
+                        504,
+                        380
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                354,
+                                504,
+                                380
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        504,
+                                        380
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem 5 (StoSOO PAC Rate). Corollary 1 of Valko et al. (2013) implies that with probability "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        504,
+                                        380
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(1 - w^{-1})(1 - \\delta)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        504,
+                                        380
+                                    ],
+                                    "type": "text",
+                                    "content": " for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        504,
+                                        380
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        504,
+                                        380
+                                    ],
+                                    "type": "text",
+                                    "content": ", a uniformly randomly drawn arm (i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        504,
+                                        380
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t \\sim U([T])"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        504,
+                                        380
+                                    ],
+                                    "type": "text",
+                                    "content": ") achieves"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        148,
+                        385,
+                        504,
+                        424
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                148,
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+                                424
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        148,
+                                        385,
+                                        504,
+                                        424
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\epsilon_ {t} \\leq w \\left[ \\frac {1}{\\ln (1 / p)} \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {n \\hat {L}} \\sqrt {\\xi_ {1} \\sqrt {\\frac {\\log_ {b} (T k / \\delta)}{2 \\log_ {b} (e) k}} + \\xi_ {2} b ^ {- \\frac {1}{d C} \\sqrt {T / k}}} \\right] \\tag {124}",
+                                    "image_path": "e711a776a2ee1305710192c4f9af1dcb3e991bf4b317b94d7dc3365b9cd317d2.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        103,
+                        430,
+                        506,
+                        467
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                103,
+                                430,
+                                506,
+                                467
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        103,
+                                        430,
+                                        506,
+                                        467
+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        430,
+                                        506,
+                                        467
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d = n(\\bar{m} - 1)"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        430,
+                                        506,
+                                        467
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        430,
+                                        506,
+                                        467
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\xi_1 = (2 + 2^{2/d})"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        430,
+                                        506,
+                                        467
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        430,
+                                        506,
+                                        467
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\xi_2 = \\frac{1}{4} db^{2(1 + 2/d)}"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        430,
+                                        506,
+                                        467
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        430,
+                                        506,
+                                        467
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k = T \\log_b(T)^{-3}"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        430,
+                                        506,
+                                        467
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        430,
+                                        506,
+                                        467
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "b"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        430,
+                                        506,
+                                        467
+                                    ],
+                                    "type": "text",
+                                    "content": " is the branching factor for partitioning cells, and the near-optimality constant "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        430,
+                                        506,
+                                        467
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C = |\\mathcal{X}^*|^{-1}\\sqrt{2\\pi d}\\left(\\frac{b^2d^2}{5r_\\eta^2\\sigma_{-2}}\\right)^{d/2}"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        430,
+                                        506,
+                                        467
+                                    ],
+                                    "type": "text",
+                                    "content": " (Lemma 36)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
+                        482,
+                        504,
+                        507
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                482,
+                                504,
+                                507
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        482,
+                                        504,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Assume "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        482,
+                                        504,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\eta_{k} = \\eta = \\frac{2}{L}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        482,
+                                        504,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": " as defined in Lemma 2. Also assume a previously pulled arm is returned uniformly at random. Starting with Lemma 13 and applying Theorem 3 and Lemma 37, we find"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        137,
+                        514,
+                        505,
+                        586
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                137,
+                                514,
+                                505,
+                                586
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        137,
+                                        514,
+                                        505,
+                                        586
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\mathbb {E} \\left[ \\epsilon_ {t} \\right] \\leq \\tau \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {\\frac {2 n}{\\min  _ {k} \\eta_ {k}}} \\sqrt {\\frac {1}{T} \\sum_ {t} \\mathcal {L} ^ {\\tau} \\left(\\boldsymbol {x} _ {t}\\right)} (125a) \\\\ \\leq \\frac {1}{\\ln (1 / p)} \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {n \\hat {L}} \\sqrt {\\xi_ {1} \\sqrt {\\frac {\\log_ {b} (T k / \\delta)}{2 \\log_ {b} (e) k}} + \\xi_ {2} b ^ {- \\frac {1}{d C} \\sqrt {T / k}}} (125b) \\\\ \\end{array}",
+                                    "image_path": "9b63eaca726a17206d07875ec1f56daafd3069ffd500a76bfe81ac709e6c48b9.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        104,
+                        593,
+                        506,
+                        641
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                593,
+                                506,
+                                641
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        641
+                                    ],
+                                    "type": "text",
+                                    "content": "with probability "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        641
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "1 - \\delta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        641
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        641
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m^{*} = \\max_{k} m_{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        641
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        641
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d = n(\\bar{m} - 1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        641
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        641
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\xi_{1} = (2 + 2^{2 / d})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        641
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        641
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\xi_{2} = \\frac{1}{4} db^{2(1 + 2 / d)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        641
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        641
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k = T\\log_b(T)^{-3}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        641
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        641
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "b"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        641
+                                    ],
+                                    "type": "text",
+                                    "content": " is the branching factor for partitioning cells, and the near-optimality constant "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        641
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C = |\\mathcal{X}^{*}|^{-1}\\sqrt{2\\pi d}\\left(\\frac{b^2d^2}{5r_\\eta^2\\sigma_{-2}}\\right)^{d / 2}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        506,
+                                        641
+                                    ],
+                                    "type": "text",
+                                    "content": " (Lemma 36)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        104,
+                        644,
+                        504,
+                        668
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                644,
+                                504,
+                                668
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        644,
+                                        504,
+                                        668
+                                    ],
+                                    "type": "text",
+                                    "content": "Markov's inequality then allows us to bound the pointwise exploitability of any arm returned by the algorithm as"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        148,
+                        674,
+                        505,
+                        713
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                148,
+                                674,
+                                505,
+                                713
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        148,
+                                        674,
+                                        505,
+                                        713
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\epsilon_ {t} \\leq w \\left[ \\frac {1}{\\ln (1 / p)} \\log \\left(\\prod_ {k} m _ {k}\\right) + \\sqrt {n \\hat {L}} \\sqrt {\\xi_ {1} \\sqrt {\\frac {\\log_ {b} (T k / \\delta)}{2 \\log_ {b} (e) k}} + \\xi_ {2} b ^ {- \\frac {1}{d C} \\sqrt {T / k}}} \\right] \\tag {126}",
+                                    "image_path": "1e0d48c03868597679ddb4b562efe93f937b9d6b71fb5c1a52928053da8009c2.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        104,
+                        719,
+                        309,
+                        733
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                719,
+                                309,
+                                733
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        719,
+                                        309,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "with probability "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        719,
+                                        309,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(1 - w^{-1})(1 - \\delta)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        719,
+                                        309,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        719,
+                                        309,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        719,
+                                        309,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
+                        751,
+                        311,
+                        760
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                299,
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+                                311,
+                                760
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        299,
+                                        751,
+                                        311,
+                                        760
+                                    ],
+                                    "type": "text",
+                                    "content": "42"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 41
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        82,
+                        299,
+                        94
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                82,
+                                299,
+                                94
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        82,
+                                        299,
+                                        94
+                                    ],
+                                    "type": "text",
+                                    "content": "B.7 COMPLEXITY OF POLYMATRIX GAMES"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        102,
+                        506,
+                        138
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                102,
+                                506,
+                                138
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        506,
+                                        138
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma 38. For a polymatrix game defined by the set of bimatrix games with payoff matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        506,
+                                        138
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{kl}^{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        506,
+                                        138
+                                    ],
+                                    "type": "text",
+                                    "content": " for every player "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        506,
+                                        138
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        506,
+                                        138
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        506,
+                                        138
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "l \\neq k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        506,
+                                        138
+                                    ],
+                                    "type": "text",
+                                    "content": ", the rank of the matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        506,
+                                        138
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M(x)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        506,
+                                        138
+                                    ],
+                                    "type": "text",
+                                    "content": " defined in equation (11) can be equivalently studied by replacing all instances of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        506,
+                                        138
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H_{kl}^{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        506,
+                                        138
+                                    ],
+                                    "type": "text",
+                                    "content": " with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        506,
+                                        138
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{kl}^{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        506,
+                                        138
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        148,
+                        506,
+                        205
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                148,
+                                506,
+                                205
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        148,
+                                        506,
+                                        205
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Consider the polymatrix game given by the set of matrices "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        148,
+                                        506,
+                                        205
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{kl}^{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        148,
+                                        506,
+                                        205
+                                    ],
+                                    "type": "text",
+                                    "content": " for every "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        148,
+                                        506,
+                                        205
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        148,
+                                        506,
+                                        205
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        148,
+                                        506,
+                                        205
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "l \\neq k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        148,
+                                        506,
+                                        205
+                                    ],
+                                    "type": "text",
+                                    "content": ". The polymatrix game can be equivalently written in normal form, albeit, less concisely. Note that the polymatrix approximation, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        148,
+                                        506,
+                                        205
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H_{kl}^{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        148,
+                                        506,
+                                        205
+                                    ],
+                                    "type": "text",
+                                    "content": ", as we have defined it (see Section 2 Preliminaries) of this normal-form representation between players "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        148,
+                                        506,
+                                        205
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        148,
+                                        506,
+                                        205
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        148,
+                                        506,
+                                        205
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "l"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        148,
+                                        506,
+                                        205
+                                    ],
+                                    "type": "text",
+                                    "content": " with all other players' strategies marginalized out is related to the true underlying bimatrix game between them as follows:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        197,
+                        209,
+                        504,
+                        300
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                300
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        197,
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+                                        300
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} H _ {k l} ^ {k} [ a _ {k}, a _ {l} ] = \\mathbb {E} _ {x _ {- k l}} [ u _ {k} (a _ {k}, a _ {l}, x _ {- k l}) ] \\forall a _ {k}, a _ {l} (127a) \\\\ = a _ {k} ^ {\\top} \\left(P _ {k l} ^ {k} a _ {l} + \\sum_ {j \\notin \\{k, l \\}} P _ {k j} ^ {k} x _ {j}\\right) \\forall a _ {k}, a _ {l} (127b) \\\\ = a _ {k} ^ {\\top} P _ {k l} ^ {k} a _ {l} + \\underbrace {a _ {k} ^ {\\top} \\left(\\sum_ {j \\notin \\{k , l \\}} P _ {k j} ^ {k} x _ {j}\\right)} _ {p _ {k}} \\forall a _ {k}, a _ {l} (127c) \\\\ \\end{array}",
+                                    "image_path": "212e1610aa24afa7dba14f33a6c2c7d979fda27514c7822f871be364af240a78.jpg"
+                                }
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+                    ],
+                    "index": 4
+                },
+                {
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+                    "angle": 0,
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
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+                                        304,
+                                        504,
+                                        327
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{kl}^{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    "type": "text",
+                                    "content": " is player "
+                                },
+                                {
+                                    "bbox": [
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+                                        304,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        327
+                                    ],
+                                    "type": "text",
+                                    "content": "'s payoff matrix for the bimatrix game between players "
+                                },
+                                {
+                                    "bbox": [
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+                                        327
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        327
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "l"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
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+                                        327
+                                    ],
+                                    "type": "text",
+                                    "content": " in a polymatrix game and "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p_k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        327
+                                    ],
+                                    "type": "text",
+                                    "content": " does not depend on player "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        327
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "l"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        327
+                                    ],
+                                    "type": "text",
+                                    "content": "'s strategy."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
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+                        345
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                332,
+                                411,
+                                345
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
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+                                        345
+                                    ],
+                                    "type": "text",
+                                    "content": "This implies that "
+                                },
+                                {
+                                    "bbox": [
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+                                        345
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H_{kl}^{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        411,
+                                        345
+                                    ],
+                                    "type": "text",
+                                    "content": " is equal to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        411,
+                                        345
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{kl}^{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        411,
+                                        345
+                                    ],
+                                    "type": "text",
+                                    "content": " up to a constant offset of the rows, i.e.,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        270,
+                        349,
+                        504,
+                        364
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                270,
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+                                364
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        270,
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+                                        504,
+                                        364
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "H _ {k l} ^ {k} = P _ {k l} ^ {k} + C _ {k} \\tag {128}",
+                                    "image_path": "8bfed2336b9149a482ae79674faa174dafcdf7d94f964e8794b329b57cb81454.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                378
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        367,
+                                        271,
+                                        378
+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        367,
+                                        271,
+                                        378
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        367,
+                                        271,
+                                        378
+                                    ],
+                                    "type": "text",
+                                    "content": " is a matrix with constant rows."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        383,
+                        504,
+                        437
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                383,
+                                504,
+                                437
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        383,
+                                        504,
+                                        437
+                                    ],
+                                    "type": "text",
+                                    "content": "Consider the matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        383,
+                                        504,
+                                        437
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M(x)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        383,
+                                        504,
+                                        437
+                                    ],
+                                    "type": "text",
+                                    "content": " which contains "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        383,
+                                        504,
+                                        437
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H_{kl}^{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        383,
+                                        504,
+                                        437
+                                    ],
+                                    "type": "text",
+                                    "content": " blocks. Recall that the bottom rows of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        383,
+                                        504,
+                                        437
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M(x)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        383,
+                                        504,
+                                        437
+                                    ],
+                                    "type": "text",
+                                    "content": " contain rows of 1's matching each column of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        383,
+                                        504,
+                                        437
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H_{kl}^{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        383,
+                                        504,
+                                        437
+                                    ],
+                                    "type": "text",
+                                    "content": " blocks. Consider multiplying the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        383,
+                                        504,
+                                        437
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "l"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        383,
+                                        504,
+                                        437
+                                    ],
+                                    "type": "text",
+                                    "content": "th row of 1's (which contains 0's on all columns not in the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        383,
+                                        504,
+                                        437
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "l"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        383,
+                                        504,
+                                        437
+                                    ],
+                                    "type": "text",
+                                    "content": "th block) by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        383,
+                                        504,
+                                        437
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sqrt{\\eta_k} [I - \\frac{1}{m_k}\\mathbf{1}_k\\mathbf{1}_k^\\top ]C_k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        383,
+                                        504,
+                                        437
+                                    ],
+                                    "type": "text",
+                                    "content": " and subtracting it from the block containing "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        383,
+                                        504,
+                                        437
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sqrt{\\eta_k} [I - \\frac{1}{m_k}\\mathbf{1}_k\\mathbf{1}_k^\\top ]H_{kl}^k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        383,
+                                        504,
+                                        437
+                                    ],
+                                    "type": "text",
+                                    "content": ","
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        185,
+                        440,
+                        504,
+                        464
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                185,
+                                440,
+                                504,
+                                464
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        185,
+                                        440,
+                                        504,
+                                        464
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\sqrt {\\eta_ {k}} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1} _ {k} \\mathbf {1} _ {k} ^ {\\top} \\right] \\left[ H _ {k l} ^ {k} - C _ {k} \\right] = \\sqrt {\\eta_ {k}} \\left[ I - \\frac {1}{m _ {k}} \\mathbf {1} _ {k} \\mathbf {1} _ {k} ^ {\\top} \\right] P _ {k l} ^ {k}. \\tag {129}",
+                                    "image_path": "66a154a1f8ba296a6a90d0c4c83238076b639fe2ba6bc4693414dc50c799eb82.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        468,
+                        506,
+                        526
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                468,
+                                506,
+                                526
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        468,
+                                        506,
+                                        526
+                                    ],
+                                    "type": "text",
+                                    "content": "Note that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        468,
+                                        506,
+                                        526
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sqrt{\\eta_k} [I - \\frac{1}{m_k}\\mathbf{1}_k\\mathbf{1}_k^\\top ]C_k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        468,
+                                        506,
+                                        526
+                                    ],
+                                    "type": "text",
+                                    "content": " still remains a matrix with constant rows (the preconditioner effectively subtracts a constant matrix from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        468,
+                                        506,
+                                        526
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        468,
+                                        506,
+                                        526
+                                    ],
+                                    "type": "text",
+                                    "content": "). This multiplying and subtracting a row from another is an elementary operation on the matrix, meaning it does not change its row rank. Therefore, for a polymatrix game, we can reason about the positive definiteness of the Hessian at equilibria by examining the matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        468,
+                                        506,
+                                        526
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M(x)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        468,
+                                        506,
+                                        526
+                                    ],
+                                    "type": "text",
+                                    "content": " with all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        468,
+                                        506,
+                                        526
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H_{kl}^{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        468,
+                                        506,
+                                        526
+                                    ],
+                                    "type": "text",
+                                    "content": "'s swapped for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        468,
+                                        506,
+                                        526
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{kl}^{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        468,
+                                        506,
+                                        526
+                                    ],
+                                    "type": "text",
+                                    "content": "'s."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        104,
+                        537,
+                        506,
+                        616
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                537,
+                                506,
+                                616
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        537,
+                                        506,
+                                        616
+                                    ],
+                                    "type": "text",
+                                    "content": "Interestingly, at zero temperature (where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        537,
+                                        506,
+                                        616
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{QRE} = \\mathrm{Nash}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        537,
+                                        506,
+                                        616
+                                    ],
+                                    "type": "text",
+                                    "content": "), "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        537,
+                                        506,
+                                        616
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        537,
+                                        506,
+                                        616
+                                    ],
+                                    "type": "text",
+                                    "content": " is constant for a polymatrix game, so the rank of this matrix can be computed just once to extract information about all possible interior equilibria in the game. Furthermore, the Hessian is positive semi-definite over the entire joint strategy space, implying the loss function is convex (see Figure 5 (left) for empirical support). This indicates, by convex optimization theory, 1) all mixed Nash equilibria in polymatrix games form a convex set (i.e., they are connected) and 2) assuming mixed equilibria exist, they can be computed simply by stochastic gradient descent on "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        537,
+                                        506,
+                                        616
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        537,
+                                        506,
+                                        616
+                                    ],
+                                    "type": "text",
+                                    "content": ". If "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        537,
+                                        506,
+                                        616
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        537,
+                                        506,
+                                        616
+                                    ],
+                                    "type": "text",
+                                    "content": " is rank-"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        537,
+                                        506,
+                                        616
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n\\bar{m}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        537,
+                                        506,
+                                        616
+                                    ],
+                                    "type": "text",
+                                    "content": ", then this interior equilibrium is unique."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        104,
+                        626,
+                        507,
+                        704
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                626,
+                                507,
+                                704
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        507,
+                                        704
+                                    ],
+                                    "type": "text",
+                                    "content": "**Complexity** Approximation of Nash equilibria in polymatrix games is known to be PPAD-hard (Deligkas et al., 2022). In contrast, if we restrict our class of polymatrix games to those with at least one interior Nash equilibrium, our analysis proves we can find an approximate Nash equilibrium in deterministic, polynomial time (Corollary 7). This follows directly from the fact that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        507,
+                                        704
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        507,
+                                        704
+                                    ],
+                                    "type": "text",
+                                    "content": " is convex, our decision set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        507,
+                                        704
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{X} = \\prod_{k} \\mathcal{X}_{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        507,
+                                        704
+                                    ],
+                                    "type": "text",
+                                    "content": " is convex, and convex optimization theory admits polynomial time approximation algorithms (e.g., gradient descent). We consider the assumption of the existence of an interior Nash equilibrium to be relatively mild"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        507,
+                                        704
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "^3"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        507,
+                                        704
+                                    ],
+                                    "type": "text",
+                                    "content": ", so this positive complexity result is surprising."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                }
+            ],
+            "discarded_blocks": [
+                {
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+                    "angle": 0,
+                    "lines": [
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+                            ],
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "3Marris et al. (2023) shows 2-player, 2-action polymatrix games with interior Nash equilibria make up a non-trivial "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        711,
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+                                    "type": "inline_equation",
+                                    "content": "1/4"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        711,
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+                                    ],
+                                    "type": "text",
+                                    "content": " of the space of possible "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        711,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "2 \\times 2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        711,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " games."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        299,
+                        751,
+                        310,
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+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "43"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 42
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        82,
+                        504,
+                        139
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                82,
+                                504,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        139
+                                    ],
+                                    "type": "text",
+                                    "content": "Also, note that the Hessian of the loss at Nash equilibria is encoded entirely by the polymatrix approximation at the equilibrium. Therefore, approximating the Hessian of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        139
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        139
+                                    ],
+                                    "type": "text",
+                                    "content": " about the equilibrium (which amounts to observing near-equilibrium behavior (Ling et al., 2018)) allows one to recover this polymatrix approximation (up to constant offsets of the columns which equilibria are invariant to (Marris et al., 2023))."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        140,
+                        507,
+                        175
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                140,
+                                507,
+                                175
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        140,
+                                        507,
+                                        175
+                                    ],
+                                    "type": "text",
+                                    "content": "Corollary 7 (Approximating Nash Equilibria of Polymatrix Games with Interior Equilibria). Consider the class of polymatrix games with interior Nash equilibria. This class of games admits a fully polynomial time deterministic approximation scheme (FPTAS)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        186,
+                        506,
+                        364
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                186,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        364
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Lemma 3 relates the approximation of Nash equilibria to the minimization of the loss function "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        364
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}(\\pmb{x})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        364
+                                    ],
+                                    "type": "text",
+                                    "content": ". By Lemma 1, this loss function attains its minimum value of zero if and only if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        364
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        364
+                                    ],
+                                    "type": "text",
+                                    "content": " is a Nash equilibrium. For polymatrix games, the Hessian of this loss function is everywhere finite and positive definite (Lemma 16), therefore, this loss function is convex. The decision set for this minimization problem is the product space of simplices, therefore it is also convex. Given that we only consider polymatrix games with interior Nash equilibria, we know that our loss function attains a global minimum within this set. By convex optimization theory, this function can be approximately minimized in a polynomial number of steps by, for example, (projected) gradient descent (Boyd and Vandenberghe, 2004). Gradient descent requires computing the gradient of the loss function at each step. From Lemma 14, we see that computing the gradient (at zero temperature) simply requires reading the polymatrix description of the game (i.e., each bi-matrix game "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        364
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H_{kl}^{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        364
+                                    ],
+                                    "type": "text",
+                                    "content": " between players), which is clearly polynomial in the size of the input (the polymatrix description). The remaining computational steps of gradient descent (e.g., projection onto simplices) are polynomial as well. In conclusion, gradient descent approximates a Nash equilibrium in polynomial number of steps (logarithmic if strongly-convex (Mairal, 2013)), each of which costs polynomial time, therefore the entire scheme is polynomial."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                }
+            ],
+            "discarded_blocks": [
+                {
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+                    ],
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
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+                    ],
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+                    "angle": 0,
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "44"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 43
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        81,
+                        320,
+                        94
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
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+                            ],
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+                                {
+                                    "bbox": [
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+                                        81,
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+                                    ],
+                                    "type": "text",
+                                    "content": "C EXPERIMENTAL SETUP AND DETAILS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        105,
+                        106,
+                        313,
+                        118
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                313,
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+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
+                                        106,
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+                                    ],
+                                    "type": "text",
+                                    "content": "Here we provide further details on the experiments."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        105,
+                        133,
+                        176,
+                        144
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                133,
+                                176,
+                                144
+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
+                                        133,
+                                        176,
+                                        144
+                                    ],
+                                    "type": "text",
+                                    "content": "C.1 GAMBIT"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        154,
+                        506,
+                        200
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                154,
+                                506,
+                                200
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        154,
+                                        506,
+                                        200
+                                    ],
+                                    "type": "text",
+                                    "content": "The seven methods from the gambit (McKelvey et al., 2016) library that we tested on the 3-player and 4-player Blotto games are listed below (with runtimes). Only gambit-enumpoly and gambit-enumpure are able to return any NE for 3-player Blotto within a 1 hour time limit (and only pure equilibria). And only gambit-enumpure returns any NE for the 4-player game."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        132,
+                        208,
+                        375,
+                        316
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 12,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                132,
+                                208,
+                                375,
+                                220
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        132,
+                                        208,
+                                        375,
+                                        220
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                132,
+                                                208,
+                                                375,
+                                                220
+                                            ],
+                                            "type": "text",
+                                            "content": "- gambit-enumpoly [73 sec 3-player, timeout 4-player]"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 5
+                        },
+                        {
+                            "bbox": [
+                                132,
+                                224,
+                                369,
+                                236
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        132,
+                                        224,
+                                        369,
+                                        236
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                132,
+                                                224,
+                                                369,
+                                                236
+                                            ],
+                                            "type": "text",
+                                            "content": "- gambit-enumpure [72 sec 3-player, 45 sec 4-player]"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 6
+                        },
+                        {
+                            "bbox": [
+                                132,
+                                241,
+                                204,
+                                251
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        132,
+                                        241,
+                                        204,
+                                        251
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                132,
+                                                241,
+                                                204,
+                                                251
+                                            ],
+                                            "type": "text",
+                                            "content": "- gambit-gnm"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 7
+                        },
+                        {
+                            "bbox": [
+                                132,
+                                257,
+                                203,
+                                268
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        132,
+                                        257,
+                                        203,
+                                        268
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                132,
+                                                257,
+                                                203,
+                                                268
+                                            ],
+                                            "type": "text",
+                                            "content": "- gambit-ipa"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 8
+                        },
+                        {
+                            "bbox": [
+                                132,
+                                274,
+                                209,
+                                285
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        132,
+                                        274,
+                                        209,
+                                        285
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                132,
+                                                274,
+                                                209,
+                                                285
+                                            ],
+                                            "type": "text",
+                                            "content": "- gambit-liap"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 9
+                        },
+                        {
+                            "bbox": [
+                                132,
+                                289,
+                                215,
+                                300
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        132,
+                                        289,
+                                        215,
+                                        300
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                132,
+                                                289,
+                                                215,
+                                                300
+                                            ],
+                                            "type": "text",
+                                            "content": "- gambit-logit"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 10
+                        },
+                        {
+                            "bbox": [
+                                132,
+                                305,
+                                227,
+                                316
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        132,
+                                        305,
+                                        227,
+                                        316
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                132,
+                                                305,
+                                                227,
+                                                316
+                                            ],
+                                            "type": "text",
+                                            "content": "- gambit-simpdiv"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 11
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        105,
+                        331,
+                        301,
+                        342
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
+                                        331,
+                                        301,
+                                        342
+                                    ],
+                                    "type": "text",
+                                    "content": "C.2 LOSS VISUALIZATION AND RANK TEST"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        104,
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                        387
+                                    ],
+                                    "type": "text",
+                                    "content": "Figure 5 and claims made in Section 5 analyze several classical matrix games. We report the payoff matrices in standard row-player / column-player payoff form below. All games are then shifted and scaled so payoffs lie in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        352,
+                                        506,
+                                        387
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "[0,1]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        352,
+                                        506,
+                                        387
+                                    ],
+                                    "type": "text",
+                                    "content": " (i.e., first by subtracting the minimum and then scaling by the max)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "RPS:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
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+                    "type": "interline_equation",
+                    "angle": 0,
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+                                    "bbox": [
+                                        245,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left[ \\begin{array}{c c c} 0 / 0 & - 1 / 1 & 1 / - 1 \\\\ 1 / - 1 & 0 / 0 & - 1 / 1 \\\\ - 1 / 1 & 1 / - 1 & 0 / 0 \\end{array} \\right]. \\tag {130}",
+                                    "image_path": "547c933de72b1d8b2d20eec474cb31439854438140d82223d1d7d30a0d79d5d1.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                {
+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "type": "text",
+                                    "content": "Chicken:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
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+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                                {
+                                    "bbox": [
+                                        260,
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+                                        522
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left[ \\begin{array}{c c} 0 / 0 & - 1 / 1 \\\\ 1 / - 1 & - 3 / - 3 \\end{array} \\right]. \\tag {131}",
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+                                    "content": "C.2.1 LOSS ON FAMILIAR GAMES"
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+                                    "type": "text",
+                                    "content": "We visualize our proposed loss "
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+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}^{\\tau}"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " on two classic 2-player, general-sum games: Chicken, payoff matrix (131), and the Prisoner's Dilemma, payoff matrix (134). Each plot in Figure 5 visualizes the loss at various strategy profiles in probability-space; each strategy profile is represented by each player's probability of playing action 1 of 2 (top row / first column of payoff matrix). Temperature "
+                                },
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": " is varied across the plots. Figure 5 repeats this same visualization but in logit-space to better show the equilibria closest to the boundaries."
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+                                            "content": "Figure 5: Upper Bound "
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+                                            "type": "inline_equation",
+                                            "content": "(\\epsilon \\leq f_{\\tau}(\\mathcal{L}^{\\tau}))"
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+                                            "type": "text",
+                                            "content": " Heatmap Visualization. The first row examines the loss landscape for the classic anti-coordination game of Chicken (Nash equilibria: "
+                                        },
+                                        {
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+                                                330,
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+                                            "type": "inline_equation",
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+                                            "type": "text",
+                                            "content": ", "
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+                                            "type": "text",
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+                                            "type": "inline_equation",
+                                            "content": "(2/3,1/3))"
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+                                        {
+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": " while the second row examines the Prisoner's dilemma (Unique Nash equilibrium: "
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+                                            "type": "inline_equation",
+                                            "content": "(0,0))"
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+                                            ],
+                                            "type": "text",
+                                            "content": ". For improved visibility, we subtract the offset "
+                                        },
+                                        {
+                                            "bbox": [
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+                                                330,
+                                                506,
+                                                430
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tau \\log(m^{2})"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": " from "
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+                                                330,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "f_{\\tau}(\\mathcal{L}^{\\tau})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                330,
+                                                506,
+                                                430
+                                            ],
+                                            "type": "text",
+                                            "content": " per Lemma 13, which relates the exploitability at positive temperature to that at zero temperature. Temperature increases for each plot moving to the right. For high temperatures, interior (fully-mixed) strategies are incentivized while for lower temperatures, nearly pure strategies can achieve minimum exploitability. For zero temperature, pure strategy equilibria (e.g., defect-defect) are not captured by the loss as illustrated by the bottom-left Prisoner's Dilemma plot with a constant loss surface."
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+                                            "type": "text",
+                                            "content": "Figure 6: [Figure 5 Repeated in Logit-Space "
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+                                            "type": "inline_equation",
+                                            "content": "(\\ln \\left(\\frac{p}{1 - p}\\right))"
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+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": " Rather than Probability-Space "
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+                                            "type": "inline_equation",
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+                                            "type": "text",
+                                            "content": " Upper Bound "
+                                        },
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+                                            "type": "inline_equation",
+                                            "content": "(\\epsilon \\leq f_{\\tau}(\\mathcal{L}^{\\tau}))"
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+                                        {
+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": " Heatmap Visualization. The first row examines the loss landscape for the classic anti-coordination game of Chicken (Nash equilibria: "
+                                        },
+                                        {
+                                            "bbox": [
+                                                104,
+                                                602,
+                                                506,
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+                                            "type": "inline_equation",
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+                                            "type": "inline_equation",
+                                            "content": "(2/3,1/3))"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": " while the second row examines the Prisoner's dilemma (Unique Nash equilibrium: "
+                                        },
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+                                            "type": "inline_equation",
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+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": "). Temperature increases for each plot moving to the right. For improved visibility, we subtract the offset "
+                                        },
+                                        {
+                                            "bbox": [
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+                                                506,
+                                                716
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tau \\log(m^{2})"
+                                        },
+                                        {
+                                            "bbox": [
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+                                                602,
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+                                            ],
+                                            "type": "text",
+                                            "content": " from "
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+                                                506,
+                                                716
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "f_{\\tau}(\\mathcal{L}^{\\tau})"
+                                        },
+                                        {
+                                            "bbox": [
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+                                                602,
+                                                506,
+                                                716
+                                            ],
+                                            "type": "text",
+                                            "content": " per Lemma 13, which relates the exploitability at positive temperature to that at zero temperature. For high temperatures, interior (fully-mixed) strategies are incentivized while for lower temperatures, nearly pure strategies can achieve minimum exploitability. For zero temperature, pure strategy equilibria (e.g., defect-defect) are not captured by the loss as illustrated by the bottom-left Prisoner's Dilemma plot with a constant loss surface."
+                                        }
+                                    ]
+                                }
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+                                246,
+                                94
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        82,
+                                        246,
+                                        94
+                                    ],
+                                    "type": "text",
+                                    "content": "C.3 SADDLE POINT ANALYSIS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        103,
+                        506,
+                        182
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                103,
+                                506,
+                                182
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        103,
+                                        506,
+                                        182
+                                    ],
+                                    "type": "text",
+                                    "content": "To generate Figure 2, we follow a procedure similar to the study of MNIST in (Dauphin et al., 2014) (Section 3 of Supp.). Their recommended procedure searches for critical points in two ways. The first repeats a randomized, iterative optimization process 20 times. They then sample one these 20 trials at random, select a random point along the descent trajectory, and search for a critical point (using Newton's method) nearby. They repeat this sampling process 100 times. The second approach randomly selects a feasible point in the decision set and searches for a critical point nearby (again using Newton's method). They also perform this 100 times."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        186,
+                        506,
+                        255
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                186,
+                                506,
+                                255
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        255
+                                    ],
+                                    "type": "text",
+                                    "content": "Our protocol differs from theirs slightly in a few respects. One, we use SGD, rather than the saddle-free Newton algorithm to trace out an initial descent trajectory. Two, we do not add noise to strategies along the descent trajectory prior to looking for critical points. Thirdly, we minimize gradient norm rather than use Newton's method to look for critical points. Lastly, we use different experimental hyperparameters. We run SGD for 1000 iterations rather than 20 epochs and rerun SGD 100 times rather than 20. We sample 1000 points for each of the two approaches for finding critical points."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        105,
+                        269,
+                        257,
+                        280
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                269,
+                                257,
+                                280
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        269,
+                                        257,
+                                        280
+                                    ],
+                                    "type": "text",
+                                    "content": "C.4 SGD ON CLASSICAL GAMES"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        290,
+                        504,
+                        314
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        290,
+                                        504,
+                                        314
+                                    ],
+                                    "type": "text",
+                                    "content": "The games examined in Figure 3 were all taken from (Gemp et al., 2022). Each is available via open source implementations in OpenSpiel (Lanctot et al., 2019) or GAMUT (Nudelman et al., 2004)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        318,
+                        505,
+                        363
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                318,
+                                505,
+                                363
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        318,
+                                        505,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": "We compare against several other baselines, replicating the experiments in (Gemp et al., 2022). RM indicates regret-matching and FTRL indicates follow-the-regularized-leader. These are, arguably, the two most popular scalable stochastic algorithms for approximating Nash equilibria. "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        318,
+                                        505,
+                                        363
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "{}^{y}{\\mathrm{{QRE}}}^{\\text{auto }}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        318,
+                                        505,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": " is a stochastic algorithm developed in (Gemp et al., 2022)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        367,
+                        506,
+                        425
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                367,
+                                506,
+                                425
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        367,
+                                        506,
+                                        425
+                                    ],
+                                    "type": "text",
+                                    "content": "For each of the experiments, we sweep over learning rates in log-space from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        367,
+                                        506,
+                                        425
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "10^{-3}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        367,
+                                        506,
+                                        425
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        367,
+                                        506,
+                                        425
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "10^{2}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        367,
+                                        506,
+                                        425
+                                    ],
+                                    "type": "text",
+                                    "content": " in increments of 1. We also consider whether to run SGD with the projected-gradient and whether to constrain iterates to the simplex via Euclidean projection or entropic mirror descent (Beck and Teboulle, 2003). We then presented the results of the best performing hyperparameters. This was the same approach taken in (Gemp et al., 2022)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        104,
+                        437,
+                        506,
+                        545
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                                545
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        437,
+                                        506,
+                                        545
+                                    ],
+                                    "type": "text",
+                                    "content": "Saddle Points in Blotto To confirm the existence of saddle points, we computed the Hessian of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        437,
+                                        506,
+                                        545
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}(\\pmb{x}_{10k})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        437,
+                                        506,
+                                        545
+                                    ],
+                                    "type": "text",
+                                    "content": " for SGD "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        437,
+                                        506,
+                                        545
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(s = \\infty)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        437,
+                                        506,
+                                        545
+                                    ],
+                                    "type": "text",
+                                    "content": ", deflated the matrix by removing from its eigenvectors all directions orthogonal to the simplex, and then computed its top- "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        437,
+                                        506,
+                                        545
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(n\\bar{m} - n)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        437,
+                                        506,
+                                        545
+                                    ],
+                                    "type": "text",
+                                    "content": " eigenvalues. We do this because there always exists a "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        437,
+                                        506,
+                                        545
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        437,
+                                        506,
+                                        545
+                                    ],
+                                    "type": "text",
+                                    "content": "-dimensional nullspace of the Hessian at zero temperature that lies outside the tangent space of the simplex, and we only care about curvature within the tangent space. Specifically, at an equilibrium "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        437,
+                                        506,
+                                        545
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{x}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        437,
+                                        506,
+                                        545
+                                    ],
+                                    "type": "text",
+                                    "content": ", if we compute "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        437,
+                                        506,
+                                        545
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z^{\\top}Hess(\\mathcal{L})z"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        437,
+                                        506,
+                                        545
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        437,
+                                        506,
+                                        545
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        437,
+                                        506,
+                                        545
+                                    ],
+                                    "type": "text",
+                                    "content": " is formed as a linear combination of the vectors "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        437,
+                                        506,
+                                        545
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{[x_1, 0, \\ldots, 0]^{\\top}, \\ldots, [0, \\ldots, x_n]^{\\top}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        437,
+                                        506,
+                                        545
+                                    ],
+                                    "type": "text",
+                                    "content": ", then each block "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        437,
+                                        506,
+                                        545
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{B}_{kl}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        437,
+                                        506,
+                                        545
+                                    ],
+                                    "type": "text",
+                                    "content": " is identically zero at an equilibrium: "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        437,
+                                        506,
+                                        545
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{B}_{kl}x_l = \\sqrt{\\eta_k[I - \\frac{1}{m_k}\\mathbf{11}^{\\top}]}H_{kl}^k x_l = \\sqrt{\\eta_k\\Pi_{T\\Delta}(\\nabla_{x_k}^k)} = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        437,
+                                        506,
+                                        545
+                                    ],
+                                    "type": "text",
+                                    "content": ". By Lemma 16, this implies there is zero curvature of the loss in the direction "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        437,
+                                        506,
+                                        545
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        437,
+                                        506,
+                                        545
+                                    ],
+                                    "type": "text",
+                                    "content": ": "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        437,
+                                        506,
+                                        545
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z^{\\top}Hess(\\mathcal{L})z = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        437,
+                                        506,
+                                        545
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        105,
+                        559,
+                        255,
+                        571
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                571
+                            ],
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+                                    "bbox": [
+                                        105,
+                                        559,
+                                        255,
+                                        571
+                                    ],
+                                    "type": "text",
+                                    "content": "C.5 BLIN ON ARTIFICIAL GAME"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
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+                        604
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        580,
+                                        504,
+                                        604
+                                    ],
+                                    "type": "text",
+                                    "content": "To construct the 7-player, 2-action, symmetric, artificial game in Figure 4, we used the following coefficients (discovered by trial-and-error):"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        106,
+                        612,
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        106,
+                                        612,
+                                        517,
+                                        658
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left[ \\begin{array}{c c c c c c c} 0. 0 9 9 0 6 8 7 3 & 0 & 0. 2 3 1 1 6 0 3 7 & 0 & 0. 6 2 7 4 3 5 2 8 & 0 & 0. 1 9 8 1 3 7 4 6 \\\\ 0 & 0. 3 3 0 2 2 9 0 9 & 0 & 0. 0 3 3 0 2 2 9 1 & 0 & 0. 6 2 7 4 3 5 2 8 & 0 \\end{array} \\right]. \\tag {135}",
+                                    "image_path": "d4eba5f76e8bba30e1003d8d0114d13ee0229f431a4269d8f264d24d6c1002b2.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        104,
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        665,
+                                        507,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": "The first row indicates the payoffs received when player "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        665,
+                                        507,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        665,
+                                        507,
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+                                    ],
+                                    "type": "text",
+                                    "content": " plays action 0 and the background population plays any of the possible joint actions (number of combinations with replacement). For example, the first column indicates the payoff when all background players play action 0. The second column indicates all background players play action 0 except for one which plays action 1, and so on. The last column indicates all background players play action 1. These "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        665,
+                                        507,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "2n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        665,
+                                        507,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": " scalars uniquely define the payoffs of a symmetric game."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
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+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
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+                    ],
+                    "type": "page_number",
+                    "angle": 0,
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+                                {
+                                    "bbox": [
+                                        299,
+                                        750,
+                                        311,
+                                        760
+                                    ],
+                                    "type": "text",
+                                    "content": "47"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 46
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        82,
+                        506,
+                        140
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                82,
+                                506,
+                                140
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        140
+                                    ],
+                                    "type": "text",
+                                    "content": "Given that this game only has two actions, we represent a mixed strategy by a single scalar "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        140
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p \\in [0,1]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        140
+                                    ],
+                                    "type": "text",
+                                    "content": ", i.e., the probability of the first action. Furthermore, this game is symmetric and we seek a symmetric equilibrium, so we can represent a full Nash equilibrium by this single scalar "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        140
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        140
+                                    ],
+                                    "type": "text",
+                                    "content": ". This reduces our search space from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        140
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "7 \\times 2 = 14"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        140
+                                    ],
+                                    "type": "text",
+                                    "content": " variables to 1 variable (and obviates any need for a map "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        140
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        140
+                                    ],
+                                    "type": "text",
+                                    "content": " from the unit hypercube to the simplex—see Lemma 22)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
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+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
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diff --git a/2024/Batched Low-Rank Adaptation of Foundation Models/698c37fd-a34d-43c4-aeed-9c12baba206a_content_list.json b/2024/Batched Low-Rank Adaptation of Foundation Models/698c37fd-a34d-43c4-aeed-9c12baba206a_content_list.json
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+++ b/2024/Batched Low-Rank Adaptation of Foundation Models/698c37fd-a34d-43c4-aeed-9c12baba206a_content_list.json	
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+[
+    {
+        "type": "text",
+        "text": "BATCHED LOW-RANK ADAPTATION OF FOUNDATION MODELS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            99,
+            643,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yeming Wen & Swarat Chaudhuri*",
+        "bbox": [
+            181,
+            170,
+            433,
+            184
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Department of Computer Science",
+        "bbox": [
+            181,
+            185,
+            405,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The University of Texas at Austin",
+        "bbox": [
+            181,
+            198,
+            405,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT",
+        "text_level": 1,
+        "bbox": [
+            450,
+            250,
+            547,
+            263
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Low-Rank Adaptation (LoRA) has recently gained attention for fine-tuning foundation models by incorporating trainable low-rank matrices, thereby reducing the number of trainable parameters. While LoRA offers numerous advantages, its applicability for real-time serving to a diverse and global user base is constrained by its incapability to handle multiple task-specific adapters efficiently. This imposes a performance bottleneck in scenarios requiring personalized, task-specific adaptations for each incoming request.",
+        "bbox": [
+            228,
+            280,
+            766,
+            378
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To mitigate this constraint, we introduce Fast LoRA (FLORA), a framework in which each input example in a minibatch can be associated with its unique low-rank adaptation weights, allowing for efficient batching of heterogeneous requests. We empirically demonstrate that FLORA retains the performance merits of LoRA, showcasing competitive results on the MultiPL-E code generation benchmark spanning over 8 languages and a multilingual speech recognition task across 6 languages.",
+        "bbox": [
+            228,
+            380,
+            766,
+            477
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            503,
+            336,
+            518
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Transformer-based foundation models have showcased remarkable performance across various natural language processing tasks, as evidenced by the successes of ChatGPT (OpenAI, 2023), GitHub Copilot (Chen et al., 2021) and Speech Recognition (Radford et al., 2022) among others. The practice of fine-tuning these models for specific domains or specialized needs, such as instruction-tuning, has become increasingly prevalent (Wang et al., 2022c; Honovich et al., 2022; Taori et al., 2023; Chiang et al., 2023). This is driven by the requirements of real-world applications, which often demand models tailored to specific domains, tasks, or even individual user preferences (Ouyang et al., 2022). However, the extensive number of parameters in foundation models poses computational and memory challenges for task-specific fine-tuning.",
+        "bbox": [
+            169,
+            535,
+            826,
+            662
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Low-Rank Adaptation (LORA) emerged as a solution to this challenge by incorporating trainable low-rank matrices (Hu et al., 2021) which significantly reduces the number of trainable parameters during fine-tuning. LoRA's success stems from its ability to achieve domain adaptation without retraining the entire model (Taori et al., 2023; Dettmers et al., 2023; Lee et al., 2023). However, a practical challenge arises in real-time serving scenarios. Batching is the practice of aggregating multiple data points into a single computation. It is a common technique to leverage parallel processing capabilities in GPUs, ensuring higher throughput and lower serving cost. It becomes especially crucial when serving world-wide users where many requests could flood in every second. The intrinsic design of LoRA dictates that every example within a batch shares the same adapter, which is suboptimal for real-world serving scenarios where each request may require a unique adapter.",
+        "bbox": [
+            169,
+            667,
+            826,
+            808
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Consider a scenario where users from various locations and professions demand different language and occupation adapters as illustrated in Fig. 1. With LoRA, the batch processing would either force all these diverse requests to share the same adapter or process them sequentially, both of which are impractical. These limitations emphasize the need for a solution that can not only utilize the advantages of LoRA but also serve multiple adapters in parallel, catering to the diverse and simultaneous requests encountered in reality.",
+        "bbox": [
+            169,
+            813,
+            828,
+            898
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "page_footnote",
+        "text": "$^{*}$ ywen@utexas.edu, swarat@cs.utexas.edu",
+        "bbox": [
+            189,
+            910,
+            517,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "page_number",
+        "text": "1",
+        "bbox": [
+            493,
+            948,
+            504,
+            959
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/320d41ac7fa023cfe4d8924c9dbfc31f82c620dfe92ae5454aab3fa9f52061c6.jpg",
+        "image_caption": [
+            "Figure 1: This shows a pragmatic scenario where a foundation model in production receives four incoming requests, each requiring distinct adapters. Omitting two adapters in step 2 & 3 for presentation simplicity. FLORA facilitates batching in such serving circumstances, provided the adapters are of low rank, thereby sustaining high throughput and low latency. Detailed discussion on vectorization is provided in §3.2."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            192,
+            104,
+            821,
+            410
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We posit that it is critical to develop a more flexible adaptation mechanism that is compatible with diverse real-world user queries. We introduce fast LORA (FLORA), a modification of LORA, enabling individual examples in a minibatch to be associated with distinct low-rank weights without compromising the expressive power. This modification promises the benefits of domain adaptation, as heralded by LORA, but without the batching limitation.",
+        "bbox": [
+            169,
+            512,
+            823,
+            583
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our contributions can be summarized as follows:",
+        "bbox": [
+            171,
+            589,
+            495,
+            604
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1. We propose FLORA, a framework that augments LoRA by allowing each example in a minibatch to have its unique low-rank adapters, facilitating efficient batching.",
+            "2. We provided an analytical analysis describing the scenarios where FLORA would be preferred over LoRA in practical applications. This analysis is further substantiated by the empirical evidence where FLORA achieves a 2X throughput improvement on the state-of-the-art code LLM StarCoder 15B in the low-rank setting when diverse adapters are required for incoming examples. Additionally, FLORA reduces the latency by half under the same low-rank setting.",
+            "3. We demonstrate that FLORA does not sacrifice accuracy compared to LORA on a multilingual code generation task across 8 programming languages, and maintains accuracy in speech recognition tasks over five languages."
+        ],
+        "bbox": [
+            210,
+            617,
+            823,
+            789
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 PROBLEM FORMULATION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            820,
+            418,
+            834
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this section, we outline the problem tackled in this work, illustrating the constraints and objectives that drive the development of the proposed FLORA methodology. Let  $\\mathcal{M}$  denote a foundation model parameterized by  $\\theta$ , with a total number of parameters  $N$ . The common practice is to fine-tune this foundational model for various specific tasks, ranging from multilingual code generation to speech recognition as demonstrated in §4.2 and §4.3.",
+        "bbox": [
+            169,
+            854,
+            823,
+            925
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "page_number",
+        "text": "2",
+        "bbox": [
+            493,
+            948,
+            503,
+            959
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 LORA ADAPTERS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            103,
+            341,
+            118
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Fine-tuning the entire model  $\\mathcal{M}$  for a specific task is usually computationally expensive due to the massive parameter count. LoRA (Low-rank Adaptation, (Hu et al., 2021)) was introduced to facilitate domain-specific adaptations with a significantly reduced parameter footprint, with the hypothesis that low-rank adaptation is sufficient for fine-tuning domain specific foundation models.",
+        "bbox": [
+            169,
+            130,
+            823,
+            186
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Given the pre-trained weight matrix  $W_0 \\in \\mathbb{R}^{d \\times k}$ , LoRA posits that the weight matrix of the adapted foundation model can be expressed as  $W_0 + \\Delta W = W_0 + BA$ , where  $\\Delta W$  has a low-rank structure. This matrix  $\\Delta W$  is factorized into two smaller, trainable matrices:  $B \\in \\mathbb{R}^{d \\times r}$  and  $A \\in \\mathbb{R}^{r \\times k}$ , such that  $\\Delta W = BA$  where  $r$  stands for the rank. For a given input  $\\mathbf{x}_i$ , the output  $\\mathbf{y}_i$  is given by:",
+        "bbox": [
+            169,
+            191,
+            825,
+            250
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {y} _ {i} = \\mathcal {M} \\left(\\mathbf {x} _ {i} \\mid \\Delta W, W _ {0}, \\theta\\right) \\tag {1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            411,
+            268,
+            825,
+            287
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 Batching & THROUGHPUT",
+        "text_level": 1,
+        "bbox": [
+            171,
+            303,
+            408,
+            316
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Batching is a common practice where multiple data points  $(\\mathbf{x}_1, \\mathbf{x}_2, \\ldots, \\mathbf{x}_m)$  are aggregated into a single batch  $\\mathcal{B}$ . Consequently, the forward passes of these data points are processed concurrently rather than individually. This practice leverages the parallel processing capability of a modern GPU, thereby significantly improving the throughput  $T$ , i.e., the number of data points processed per unit of time. In the context of foundation models, throughput of a batch  $\\mathcal{B}$  can be defined as  $T = \\sum_{i=1}^{m} |\\mathbf{y}_i| / \\Delta t$ , where  $|\\mathbf{y}_i|$  is the number of tokens generated for each example  $\\mathbf{x}_i$  in the batch,  $\\Delta t$  is the total time taken to process the batch, and  $m$  is the number of examples in the batch. Note that batching incurs minimal latency penalties. However, given its substantial increase in throughput, batching and its variants are widely used in the state-of-the-art foundation models serving framework such as vLLM (Kwon et al., 2023) to achieve the best balance between throughput and latency.",
+        "bbox": [
+            169,
+            329,
+            826,
+            470
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 OBJECTIVE",
+        "text_level": 1,
+        "bbox": [
+            171,
+            486,
+            294,
+            500
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Batching typically assumes the same model parameters are utilized for every input example within a minibatch. Hence, a straightforward application of batching in LoRA requires that the adapter matrix  $\\Delta W$  be shared across all inputs in the batch  $\\mathcal{B}$ . The challenge arises when considering a scenario where each input example in the batch might originate from a different task. Sharing the same  $\\Delta W$  for all  $\\mathbf{x}_i$  in  $\\mathcal{B}$  becomes suboptimal where each input potentially demands a unique adapter. The limitation is particularly acute when the model is expected to serve a world-wide user base with diverse incoming requests.",
+        "bbox": [
+            169,
+            512,
+            823,
+            611
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Given the limitations of LORA in batching, our objective is to maximize the throughput  $T$  in global user serving scenarios by maintaining the batching mechanism. Formally, for each  $\\mathbf{x}_i \\in \\mathcal{B}$ , we aim to compute  $\\mathbf{y}_i = \\mathcal{M}(\\mathbf{x}_i | \\Delta W_i, W_0, \\theta)$ , where  $\\Delta W_i$  is the adapter matrix corresponding to the input example  $x_i$ . Therefore,  $\\Delta W_i$  can be unique across  $\\mathcal{B}$  and specific to a domain or user preference.",
+        "bbox": [
+            169,
+            616,
+            825,
+            674
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 FLORA: FAST LOW RANK ADAPTATION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            695,
+            522,
+            709
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As shown in §2.3, adapter sharing is often impractical in real-world serving scenarios. The innovation of FLORA is the introduction of example-specific adapter  $\\Delta W_{i}$  for each  $\\mathbf{x}_i$  in a minibatch. In FLORA, the weight matrix  $W_{i}$  for each example  $\\mathbf{x}_i$  in the minibatch is calculated as  $W_{i} = \\Delta W_{i} \\circ W_{0}$ , where  $\\circ$  denotes element-wise multiplication,  $W_{0}$  is the pre-trained weight matrix and  $\\Delta W_{i}$  is a low-rank adaptation specifically designed for  $\\mathbf{x}_i$ . Similar to Hu et al. (2021),  $\\Delta W_{i}$  is decomposed into two trainable matrices:  $B_{i} \\in \\mathbb{R}^{d \\times r}$  and  $A_{i} \\in \\mathbb{R}^{r \\times k}$ , such that  $\\Delta W_{i} = B_{i} A_{i}$ , as shown in Fig. 1. Note that FLORA has the same expressive power as LORA by its construction.",
+        "bbox": [
+            169,
+            726,
+            823,
+            825
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 FORWARD PASS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            840,
+            326,
+            854
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The advantage of FLORA is that computations on a minibatch can be written in terms of matrix multiplications. This enables efficient batched implementations on modern accelerators such as GPUs. Let  $x_{i}$  denote the activations in one layer of a neural net, which is a vertical vector of length  $d$ . The next layer's activations are given by",
+        "bbox": [
+            169,
+            867,
+            823,
+            925
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "page_number",
+        "text": "3",
+        "bbox": [
+            493,
+            948,
+            504,
+            959
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} y _ {i} = \\phi \\left(W _ {i} ^ {T} x _ {i}\\right) (2) \\\\ = \\phi \\left(\\left(W _ {0} ^ {T} \\circ \\Delta W _ {i} ^ {T}\\right) x _ {i}\\right) (3) \\\\ = \\phi \\left(\\left(W _ {0} ^ {T} \\circ \\left(B _ {i} A _ {i}\\right) ^ {T}\\right) x _ {i}\\right) (4) \\\\ = \\phi \\left(A _ {i} \\circ \\left(W _ {0} ^ {T} \\left(B _ {i} \\circ x _ {i}\\right)\\right)\\right) (5) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            395,
+            125,
+            823,
+            213
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "When the rank is greater than one, we extend the use of the symbol “o” to denote potential broadcasting. Additionally, a dimension reduction operation such as torch.mean is required prior to applying the activation function  $\\phi$ .",
+        "bbox": [
+            169,
+            227,
+            823,
+            268
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The key to FLORA's flexibility lies in the low rank decomposition enables the incorporation of example-specific adapters directly into the forward pass, as demonstrated in the equations above. Crucially, each of these operations—the element-wise multiplication between  $A_{i}$  and  $x_{i}$ , and between  $B_{i}$  and  $y_{i}$  — is inherently batch-friendly. Consequently, FLORA allows for simultaneous processing of multiple requests, each requiring its own adapter, within a single minibatch. To vectorize all adapters in the minibatch, we define matrices  $\\mathbf{A}$  and  $\\mathbf{B}$  whose rows correspond to the adapters  $A_{i}$  and  $B_{i}$  for all examples in the minibatch. The above equation is vectorized as:",
+        "bbox": [
+            169,
+            275,
+            826,
+            376
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {Y} = \\phi \\left(\\mathbf {A} \\circ \\left((\\mathbf {B} \\circ \\mathbf {X}) W _ {0}\\right)\\right) \\tag {6}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            401,
+            392,
+            823,
+            417
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 COMPUTATIONAL EFFICIENCY",
+        "text_level": 1,
+        "bbox": [
+            171,
+            433,
+            426,
+            448
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The computational analysis primarily concentrates on the examination of fully connected layers within a transformer architecture, given that LoRA is specifically applied to these layers, such as query and key projections. To begin, we analyze a baseline that leverages batch matrix multiplication to facilitate the serving of LoRA with multiple adapters. This operation is possible under the assumption that every adapter required by the input examples in the minibatch shares the same shape, specifically, the same rank. The batch matrix multiplication (BMM) can be implemented using the torch.bmm operator in deep learning frameworks such as PyTorch (Paszke et al., 2019). Note that the BMM operator is typically unfavorable in practical settings due to the significant overhead it introduces (Abdelfattah et al., 2016). This overhead diminishes the throughput and increases latency, which is detrimental in serving scenarios where response times are crucial for maintaining a good user experience.",
+        "bbox": [
+            169,
+            459,
+            826,
+            616
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Let  $b$  and  $l$  denote the batch size and the maximum sequence length in the input batch  $\\mathcal{B}$ . Revisiting the notation introduced in §3, where  $W_0 \\in \\mathbb{R}^{d \\times k}$ ,  $B_i \\in \\mathbb{R}^{d \\times r}$  and  $A_i \\in \\mathbb{R}^{r \\times k}$ , the operations required to compute the pre-activation for an input batch  $\\mathcal{B}$  with dimensions  $[b, l, d]$  consist of one matrix multiplication and two BMMs. The matrix multiplication occurs between the input batch  $\\mathbf{X}$  and the pre-trained weight  $W_0$ . The two BMM operations are conducted firstly between the input batch  $\\mathbf{X}$  and  $\\mathbf{B}$ , and secondly between the result of the prior computation and  $\\mathbf{A}$ , where  $\\mathbf{A}$  and  $\\mathbf{B}$  are matrices whose rows correspond to the adapters  $A_i$  and  $B_i$  for all examples in the minibatch respectively. Assuming for simplicity that the layer neither upscales nor downscales the hidden dimension (i.e.  $d = k$ ), the upper bound complexity of this layer is discerned as  $2c_1(dblr) + c_2(bld^2)$ , with  $c_1$  and  $c_2$  representing the computational coefficients of BMM and matrix multiplication respectively. Note that  $c_1 >> c_2$  because the BMM operator is more expensive than matrix multiplication.",
+        "bbox": [
+            169,
+            619,
+            826,
+            773
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For FLORA, the cost is one matrix multiplication which is  $c_{2}(rbld^{2})$  where  $r$  denotes the rank of the adapters. We omit the cost of element-wise multiplication in this analysis because it is negligible to the matrix multiplication cost. Comparing the computational cost of FLORA and LORA boils down to the following inequality",
+        "bbox": [
+            169,
+            779,
+            826,
+            837
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\frac {2 c _ {1}}{d c _ {2}} + \\frac {1}{r} \\geq 1 \\tag {7}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            450,
+            853,
+            823,
+            885
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "FLORA exhibits a lower computational cost than bmm LoRA whenever the above inequality holds true. The benefit of FLORA over LoRA is notably pronounced when  $r = 1$ . As the rank increases,",
+        "bbox": [
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+        "text": "4",
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+    {
+        "type": "text",
+        "text": "LORA gradually becomes less costly. From the established inequality, a variety of scenarios can be inferred where FLORA has an advantage over LORA. Firstly, the advantage of FLORA is significantly apparent when the rank of adapters is small. Secondly, in configurations where the model has fewer hidden units but an increased number of layers, FLORA tends to outperform LORA due to the smaller value of  $d$  in the denominator of Eq. (7).",
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+        "type": "text",
+        "text": "Another advantage of FLORA is the cost remains invariant to the number of adapters required by the input batch. While the preceding analysis assumes that every token in an example  $x_{i}$  shares the same adapter, it is possible to apply multiple adapters to a single example by dividing the example into chunks, and then applying different adapters to each chunk. This approach is commonly observed in the Mixture of Experts framework (Fedus et al., 2021; Lepikhin et al., 2020; Puigcerver et al., 2023). Incorporating several adapters in an input example notably amplifies the ratio  $c_{1} / c_{2}$  in Eq. (7), thereby significantly increasing LORA's cost.",
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+        "type": "text",
+        "text": "The ratio  $c_{1} / c_{2}$  might not be the same across different transformer architectures. §4.1 is designed to provide a deeper insight into how comparative serving efficiency of FLORA and LORA changes under various architectures. Additionally, it's worth noting that LORA does not apply to the self-attention layers, which constitute a non-trivial portion of the computational cost, thereby overshadowing the advantage of FLORA. However, as efficient self-attention mechanisms such as flash attention (Dao et al., 2022) get adopted, the advantage of FLORA over LORA is likely to get larger.",
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+    {
+        "type": "text",
+        "text": "Connection to IA3. IA3 was proposed in Liu et al. (2022a) featuring fast adaptation of LLM. It introduces a learned vector  $l$  which re-scales the activation by  $y_{i} = l\\circ \\phi (W_{0}^{T}x_{i}) = \\phi \\bigl (l\\circ (W_{0}^{T}x_{i})\\bigr)$ . This can be viewed as a special case of FLORA - a rank 0.5 variant - which only re-scales the columns instead of the entire pre-trained weights. It has a limited expressive power compared to FLORA and LoRA.",
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+        "type": "text",
+        "text": "4 EXPERIMENTS",
+        "text_level": 1,
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+        "text": "In this section, we compare FLORA to LoRA and other notable baselines across various metrics and tasks. To begin with, we delve into a computational analysis to substantiate the enhanced throughput and the reduced latency achieved by FLORA in the case of low rank. Subsequently, we pivot towards analyzing the accuracy of FLORA in multilingual code generation tasks spanning across different languages. The goal is to discern the proficiency of FLORA in maintaining, if not enhancing, the model's accuracy as compared to LoRA. Progressing further, we replicate a similar analysis but in the domain of multilingual speech recognition.",
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+        "text": "4.1 SERVING ANALYSIS",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The primary objective of this serving analysis is to measure the maximum throughput both FLORA and LoRA can attain under varied rank configurations. We carried out this exploration on the state-of-the-art code Large Language Model (LLM) StarCoder (Li et al., 2023), evaluating models of different number of parameters namely 1B, 3B, and 15B. The dataset facilitating this analysis has been sourced from the vLLM throughput benchmark. Noteworthy, this dataset was previously used to fine-tune the English Vicuna model, a state-of-the-art chat LLMs (Chiang et al., 2023). To expedite the benchmarking process, we extracted a subset of 1,000 samples from the original dataset, ensuring a diverse range of sample lengths varying from 50 to 2,000 tokens.",
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+        "text": "In setting up the computational analysis, our primary intent is to compare FLORA and LoRA in the real-world serving scenario. See Appendix B.1 on how bmm LoRA is implemented. The vLLM framework (Kwon et al., 2023) $^{2}$ , with its implementation of continuous batching, presents an ideal setup for this analysis. The continuous batching mechanism in vLLM, as inspired by the principles delineated in Orca (Yu & Jeong, 2022), facilitates a more efficient utilization of GPU resource by allowing new sequence to be inserted immediately once any sequence in the current batch is completed. This continuous flow significantly enhances GPU utilization as compared to static batching, where the GPU awaits the completion of all sequences in a batch before initiating a new batch processing. The comparison of FLORA and LoRA within this setup offers a compelling evidence of",
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+        "text": "Published as a conference paper at ICLR 2024",
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+        "type": "page_footnote",
+        "text": "<sup>1</sup>The batch size in BMM operator increases from  $b$  to  $b \\times m$  where  $m$  is the number of adapters per example.",
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+        "text": "2The version is 0.1.3.",
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+        "text": "5",
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+    {
+        "type": "image",
+        "img_path": "images/ed93ba9b562aa6fa23d4c29067f423da49c97808341cacf67371ac664858c667.jpg",
+        "image_caption": [
+            "Figure 2: Left: Generation throughput vs. rank for FLORA and torch.bmm implementation of LORA, measured in tokens per second (token/s). The experiments were conducted on three star-coder models: StarCoder 15B, StarCoderbase 3B and StarCoderbase 1B. FLORA has great throughput advantage over LoRA when the rank is small. As the rank increases, the torch.bmm implementation of LoRA finally has a better throughput. Right: Latency vs. rank on StarCoder-15B. Requests are coming at the speed of 8 requests per second."
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+    {
+        "type": "text",
+        "text": "their respective throughput and latency in the real-world serving scenario. It is worth noting that the experiments were conducted without Flash-attention (Dao et al., 2022).",
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+        "text": "Throughput experiment. In Fig. 2, the throughput results for both FLORA and bmm LORA are illustrated across different rank configurations on three StarCoder models with different number of parameters, namely 1B, 3B and 15B. All experiments were conducted on an NVIDIA H100 GPU with a float16 precision<sup>3</sup>. The maximum number of batched tokens is 8,192, which is the same as the model context length. Evidently, FLORA shows a superior throughput over LORA in the lower rank configurations. At rank 1, the throughput of FLORA is more than threefold higher than that of LORA, thereby highlighting the considerable serving performance boost FLORA provides. The advantage of FLORA continues as the rank increases, albeit with a diminishing rate. For instance, at rank 2, FLORA's throughput is around 2.5 times higher, and this multiplicative advantage decreases as the rank increases further. This performance advantage continues up until the rank of 8 in the StarCoder 15B model, where LoRA starts to outperform FLORA. This inflection point suggests that the advantages of FLORA in terms of throughput are more pronounced in lower rank.",
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+        "type": "text",
+        "text": "Notice that the inflection points occurred at a higher rank when serving a smaller LLM as illustrated in (Fig. 2, left). This demonstrates a significant potential of FLORA, especially when considering future applications of quantization techniques to serving LLMs. By applying quantization, such as 8-bit or 4-bit inference, the effective size of the model is reduced, akin to serving a smaller LLM thus potentially extending the rank at which FLORA maintains a throughput advantage over LORA.",
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+        "type": "text",
+        "text": "Latency experiment. We assessed the latency-versus-rank performance on the Starcoder 15B model, using the same dataset as the throughput experiment. This evaluation was conducted under the conditions where requests arrive at a rate of 8 per second. The default maximum number of batched tokens in vLLM serving launcher is 2,560. The results, as shown in (Fig. 2, right), measure latency in terms of seconds per output token. Remarkably, in the lower rank regime (ranging from rank 1 to rank 4), FLORA exhibited a 2-5X reduction in latency compared to LoRA. Notably, the latency for LoRA stood at approximately 2.3 seconds per output token, is impractical for serving due to the poor user experience it would cause. This experiment further highlights the superior capabilities of FLORA in efficiently catering to diverse incoming user requests.",
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+        "type": "text",
+        "text": "These findings validate the theoretical analysis in §4.1, confirming that FLORA provides significant throughput advantages, particularly in settings with lower to moderate ranks. This positions FLORA",
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+        "text": "Published as a conference paper at ICLR 2024",
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+        "text": "3The quantization in vLLM is still under development.",
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+        "type": "page_number",
+        "text": "6",
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+    {
+        "type": "table",
+        "img_path": "images/63fb72defddfc43c9382e9d77680467431d34217ddf999542abbee004b482e82.jpg",
+        "table_caption": [],
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+        "table_body": "<table><tr><td rowspan=\"2\">Language</td><td rowspan=\"2\">Base Model</td><td colspan=\"3\">pass@1 (Relative Improvement)</td></tr><tr><td>FLORA</td><td>IA3</td><td>LoRA</td></tr><tr><td></td><td colspan=\"4\">StarCoder 15B</td></tr><tr><td>Dlang</td><td>14.13</td><td>17.26 (22.14%)</td><td>15.26 (7.99%)</td><td>17.15 (21.37%)</td></tr><tr><td>Perl</td><td>17.05</td><td>21.44 (25.76%)</td><td>17.71 (3.90%)</td><td>21.46 (25.76%)</td></tr><tr><td>Ruby</td><td>1.39</td><td>24.94 (1692.86%)</td><td>20.80 (1394.64%)</td><td>23.76 (1608.04%)</td></tr><tr><td>Rust</td><td>22.40</td><td>26.24 (17.14%)</td><td>23.53 (5.04%)</td><td>26.87 (19.95%)</td></tr><tr><td>Racket</td><td>10.20</td><td>12.41 (21.61%)</td><td>11.53 (12.96%)</td><td>12.51 (22.58%)</td></tr><tr><td>Swift</td><td>16.91</td><td>20.38 (20.51%)</td><td>18.13 (7.19%)</td><td>20.35 (20.36%)</td></tr><tr><td></td><td colspan=\"4\">StarCoderBase 3B</td></tr><tr><td>Dlang</td><td>5.65</td><td>5.72 (1.20%)</td><td>5.72 (1.20%)</td><td>6.97 (23.34%)</td></tr><tr><td>Perl</td><td>10.73</td><td>13.01 (21.25%)</td><td>11.46 (6.83%)</td><td>13.31 (27.51%)</td></tr><tr><td>Ruby</td><td>5.33</td><td>14.48 (171.68%)</td><td>7.88 (47.90%)</td><td>13.89 (160.68%)</td></tr><tr><td>Rust</td><td>17.18</td><td>21.00 (22.24%)</td><td>17.28 (0.60%)</td><td>20.67 (20.31%)</td></tr><tr><td>Racket</td><td>8.04</td><td>9.16 (13.99%)</td><td>8.40 (4.48%)</td><td>8.80 (9.48%)</td></tr><tr><td>Swift</td><td>10.04</td><td>15.69 (56.21%)</td><td>12.54 (24.83%)</td><td>15.04 (49.76%)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 1: Comparison of three fine-tuning methods FLORA, IA3, and LORA on StarCoder 15B and StarCoderBase 3B across various low-resource programming languages in the MultiPL-E benchmark. The table presents Pass@1 accuracy of each method alongside the relative improvement over the baseline. The standard errors are less than  $0.3\\%$  in all cells in the table, therefore we exclude that for clear presentation.",
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+        "type": "text",
+        "text": "as a compelling alternative for efficiently serving adapted foundation models, especially in scenarios where lower ranks suffice the desired model accuracy, as further demonstrated in the subsequent accuracy analysis sections. Moreover, if an enterprise chooses to serve foundation models with a substantial number of diverse adapters, for instance, a personalized LLM, then a low rank or even a rank one is imperative to avoid excessive storage costs.",
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+        "text": "4.2 MULTILINGUAL CODE GENERATION",
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+        "text": "A key aspect to examine before applying FLORA in real-world LLM serving is to scrutinize any potential degradation in performance. In this section, we consider multilingual code generation as the testbed for comparing FLORA and LoRA due to its alignment with real-world applications, where the necessity to cater to diverse programming languages is paramount. Low-resource languages, as referred to in this context, are languages that appear much less frequently than other languages in the pre-training data. Orlanski et al. (2023) showed that the performance of code LLMs can be notably enhanced on low-resource programming languages such as PERL by recalibrating the language distribution in the pre-training data. This suggests that fine-tuning a trained LLM on such low-resource languages could potentially boost its performance on the same language. Hence, by employing multilingual code generation as a benchmark, we can make an informed evaluation of adaptability and performance enhancements that FLORA and LoRA can achieve.",
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+        "type": "text",
+        "text": "Additionally, a comparison is made against a third baseline, IA3, which can be considered as a special case of FLORA. Essentially, IA3 can be seen as a rank 0.5 variant of FLORA, thereby facing a constrained expressive power in comparison to both FLORA and LoRA. For FLORA and LoRA, we conducted fine-tuning across a range of rank choices, spanning from 1 to 8. It emerges that within the scope of this multilingual code generation task, a rank of one typically suffices to achieve optimal results, with the exception of Racket and Lua. Consequently, the results shown in §4.2 are predominantly at rank 1, barring Racket and Lua, which are presented at rank 4.",
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+        "text": "Fine-tuning. In our effort to evaluate the performance of FLORA, LORA, and IA3 on the multilingual code generation task, we fine-tuned these models on state-of-the-art multilingual code LLMs, StarCoder 15B and StarCoderBase 3B as introduced in (Li et al., 2023). A pivotal aspect of our fine-tuning process was the utilization of existing data, negating the need for creating new data for",
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+        "text": "Published as a conference paper at ICLR 2024",
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+        "text": "7",
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+    {
+        "type": "table",
+        "img_path": "images/e5e12cbc36ea7108dc7b0c49f35badf93c2d636539ec51664adf097b4e0e4e16.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Arabic</td><td>Czech</td><td>Lithuanian</td><td>Marathi</td><td>Mongolian</td><td>Hindi</td></tr><tr><td>Whisper</td><td>46.03</td><td>23.19</td><td>46.09</td><td>84.84</td><td>115.20</td><td>43.41</td></tr><tr><td>FLORA</td><td>30.21 ± 0.25</td><td>10.76 ± 0.22</td><td>20.39 ± 0.32</td><td>33.17 ± 0.17</td><td>42.12 ± 0.63</td><td>23.58 ± 0.65</td></tr><tr><td>IA3</td><td>30.48 ± 0.22</td><td>11.61 ± 0.33</td><td>21.41 ± 0.20</td><td>35.03 ± 0.61</td><td>46.47 ± 0.53</td><td>25.64 ± 0.78</td></tr><tr><td>LORA</td><td>30.18 ± 0.21</td><td>10.77 ± 0.38</td><td>20.50 ± 0.21</td><td>31.94 ± 0.33</td><td>41.57 ± 0.69</td><td>23.82 ± 0.74</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 2: Mean WER with standard deviation for different models and languages. The WER is calculated with respect to the unnormalized tokenizer. The base model here is Whisper-1.5B.",
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+    {
+        "type": "text",
+        "text": "low-resource programming languages. We leveraged the same pre-training data that was used for pre-training StarCoder, specifically, the Stack dataset, which contains over 6TB of permissively-licensed source code files covering 358 programming languages. For each low-resource language in our experiment, we fine-tuned on its corresponding split from the Stack dataset for a total of 1,500 steps, along with batch size 8. More fine-tuning details are given in Appendix A.",
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+    {
+        "type": "text",
+        "text": "Evaluation. The evaluation of FLORA, LORA, and IA3 was conducted on the MultiPL-E benchmark (Cassano et al., 2023), which contains the translation of two unit-test driven Python benchmarks (HumanEval and MBPP) to 18 other programming languages. We used the HumanEval split of the benchmark to evaluate the fine-tuned models. As for the metrics, We adopted the  $pass@k$  metrics from Chen et al. (2021); Austin et al. (2021), which is calculated as the fraction of problems with at least one correct sample given  $k$  samples. Similar to Chen et al. (2021), we drew 100 samples for computing  $pass@1$ , with a sampling temperature set at 0.1.",
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+    {
+        "type": "text",
+        "text": "Main results. The result in §4.2 exhibits the performance of three methods across various programming languages on both StarCoder 15B and StarCoderBase 3B models. The average relative improvement achieved by FLORA and LoRA is roughly  $20\\%$  in the selected low-resource programming languages. FLORA consistently outperforms IA3 on all languages, especially on StarCoder 15B, denoting its efficiency in leveraging the model expressive power to improve multilingual code generation. It is notable that StarCoder 15B has an unforeseen issue regarding Ruby generation, where it yields lower accuracy compared to the 3B model. However, all methods are able to fix its abnormal performance.",
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+    {
+        "type": "text",
+        "text": "On StarCoderBase 3B, a smaller model, it is evident that the baseline performance drops, yet FLoRA and LoRA still manage to exhibit substantial relative improvement over the baseline, especially in languages like Swift and Ruby. This suggests that both methods benefit from continuous training on the low-resource language split of the pre-training data, although the advantages may diminish with a reduction in model size. While the absolute performance (pass@1 accuracy) varies among languages, the relative improvements highlight the effectiveness of the tested methods in enhancing multilingual code generation.",
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+    {
+        "type": "text",
+        "text": "4.3 MULTILINGUAL SPEECH RECOGNITION",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Following our analysis in multilingual code generation, we shift our focus to another substantial application of large language models — multilingual speech recognition (MSR). This domain has seen increasing demand due to the necessity of serving various linguistic interfaces. The capabilities of FLORA and LORA to adapt to multiple languages in code generation presents a tempting premise to investigate their effectiveness in the MSR domain.",
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+        "type": "text",
+        "text": "Fine-tuning. A crucial aspect of our analysis involves fine-tuning the foundation model on a multilingual speech recognition dataset. We selected the Whisper large 1.5B (Radford et al., 2022) as our base model to conduct the subsequent analysis. Whisper is an encoder-decoder transformer model trained on 680k hours of labeled speech data through large-scale weak supervision. Despite being pre-trained on a multilingual speech corpus, the model's predictive capabilities can be improved further for certain languages and tasks through fine-tuning. We use the Common Voice benchmark (Ardila et al., 2020) — containing a total of 38 languages and 2,500 hours of collected audio — for the fine-tuning process, with a particular focus on low-resource languages. For each low-resource language enumerated in §4.2, we fine-tuned on its training split within the Common",
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+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+        "type": "page_footnote",
+        "text": "Dataset statistics can be found here https://commonvoice.—.org/en/datasets.",
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+        "text": "8",
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+        "type": "text",
+        "text": "Voice dataset for approximately 5,000 steps, with a batch size of 6. More fine-tuning details are given in Appendix A. Similar to §4.2, we found that rank one is sufficient to achieve the best result in most cases, with the exception of Marathi, which requires a rank of 4.",
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+        "text": "Main results. We present the main results on the MSR task in §4.2. Notice that the evaluation is conducted with the unnormalized tokenizer.<sup>5</sup> The Whisper model exhibited a relatively higher Word Error Rate (WER) across all low-resource languages when compared to FLORA, LORA and IA3 models, reflecting a significant room for enhancement in its multilingual speech recognition capabilities. Particularly, its performance was found to be subpar for the Marathi and Mongolian languages, with WERs of 84.84 and 115.20 respectively. All examined models significantly outperform the base Whisper model by a wide margin across all languages, highlighting the effective fine-tuning of FLORA, LORA and IA3 models. For instance, in Arabic, FLORA reduces the WER to 30.21 from Whisper's 46.03, showcasing a significant enhancement in speech recognition accuracy. Particularly, FLORA and LORA consistently outperform IA3, indicating a better expressive power for multilingual speech recognition tasks. For example, in Czech, FLORA has a mean WER of 10.76, which is slightly better compared to IA3's 11.61 and LORA's 11.47. A similar trend is observed in Lithuanian and Hindi. In summary, this table shows the superior performance of FLORA and LORA over IA3 in the speech recognition task across diverse languages. The consistent low WERs attained by FLORA demonstrates its potential as a viable model for MSR tasks.",
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+        "type": "text",
+        "text": "5 RELATED WORK",
+        "text_level": 1,
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+        "text": "Parameter-Efficient Fine-Tuning (PEFT) methods are broadly partitioned into two categories: weight-based and non-weight-based approaches. MC-dropout (Lakshminarayanan et al., 2016) stands as an early example for the non-weight-based approach, where distinct dropout masks are allocated to various tasks. Recently, prompt tuning techniques have emerged as a prevalent stream within this category (Li & Liang, 2021; Lester et al., 2021), facilitating efficient adaptation with minimal modifications to models. Successive endeavors aimed to enhance this class of methods, delving into aspects such as optimization (Mao et al., 2021; Diao et al., 2022), transferability (Wang et al., 2021; Vu et al., 2021; He et al., 2022b), and the usage of discrete prompts (Schick & Schütze, 2020a;b; Gao et al., 2021; Malkin et al., 2021), among others (Liu et al., 2022b; 2021).",
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+        "text": "We focus on weight-based approaches in this work, which has a weight interpretation as exemplified by LoRA (Hu et al., 2021). This line of research can be traced back to Progressive Network (Rusu et al., 2016), which inserts a sub-network when a new task arrives. This principle was later adapted widely in foundation models as represented by adapter based methods (Houlsby et al., 2019; Mahabadi et al., 2021; Davison, 2021; Ding et al., 2022; Wang et al., 2022b). In particular, BitFit (BenZaken et al., 2021) was introduced to solely update the bias parameters, while IA3 (Liu et al., 2022a) was proposed to rescale the activations. Additionally, approaches such Fish (Sung et al., 2021) and Diff-pruning (Guo et al., 2020) leverage sparsity to facilitate efficient adaptation of foundation models. A separate vein of research aims to improve LoRA by reducing its computational and memory costs (Zhang et al., 2023b;a; Malladi et al., 2023). He et al. (2022a) explored how to unify different PEFT methods. Dettmers et al. (2023) quantized LoRA to reduce memory footprint. Chavan et al. (2023) generalized LoRA by learning individual adapters in each layer. Several other works focus on building mixture of adapters (Wang et al., 2022a; Diao et al., 2023).",
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+        "type": "text",
+        "text": "6 CONCLUSION",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We introduced FLORA, an extension of LORA, facilitating efficient batching. Empirical evaluations demonstrated that FLORA enhances throughput and latency in practical serving scenarios, all while preserving the accuracy of LORA. Through FLORA, we aim to facilitate a more efficient adaptation of large language models to diverse and real-world user requests.",
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+    {
+        "type": "text",
+        "text": "Limitations. Despite its parameter efficiency, FLORA still requires fine-tuning. A promising future work could be to derive FLORA weights from a trained LoRA model, given that LORA remains",
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+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+    {
+        "type": "page_footnote",
+        "text": "This is because the normalized tokenizer has a bug in the official leaderboard implementation at the time of submission.",
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+        "type": "page_number",
+        "text": "9",
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+    {
+        "type": "ref_text",
+        "text": "the most prevalent type of adapter as per (Huang et al., 2023). This adaptation could potentially obviate the requirement for fine-tuning, thereby accelerating the process of model adaptation.",
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+    {
+        "type": "text",
+        "text": "REFERENCES",
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+            "Rohan Taori, Ishaan Gulrajani, Tianyi Zhang, Yann Dubois, Xuechen Li, Carlos Guestrin, Percy Liang, and Tatsunori B. Hashimoto. Stanford alpaca: An instruction-following llama model. https://github.com/tatsu-lab/stanford_alpaca, 2023.",
+            "Hugo Touvron, Louis Martin, Kevin R. Stone, Peter Albert, Amjad Almahairi, Yasmine Babaei, Nikolay Bashlykov, Soumya Batra, Prajwal Bhargava, Shruti Bhosale, Daniel M. Bikel, Lukas Blecher, Cristian Cantón Ferrer, Moya Chen, Guillem Cucurull, David Esiobu, Jude Fernandes, Jeremy Fu, Wenyin Fu, Brian Fuller, Cynthia Gao, Vedanuj Goswami, Naman Goyal, Anthony S. Hartshorn, Saghar Hosseini, Rui Hou, Hakan Inan, Marcin Kardas, Viktor Kerkez, Madian Khabsa, Isabel M. Kloumann, A. V. Korenev, Punit Singh Koura, Marie-Anne Lachaux, Thibaut Lavril, Jenya Lee, Diana Liskovich, Yinghai Lu, Yuning Mao, Xavier Martinet, Todor Mihaylov, Pushkar Mishra, Igor Molybog, Yixin Nie, Andrew Poulton, Jeremy Reizenstein, Rashi Rungta, Kalyan Saladi, Alan Schelten, Ruan Silva, Eric Michael Smith, R. Subramanian, Xia"
+        ],
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+    },
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+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
+            "Tan, Binh Tang, Ross Taylor, Adina Williams, Jian Xiang Kuan, Puxin Xu, Zhengxu Yan, Iliyan Zarov, Yuchen Zhang, Angela Fan, Melanie Kambadur, Sharan Narang, Aurelien Rodriguez, Robert Stojnic, Sergey Edunov, and Thomas Scialom. Llama 2: Open foundation and fine-tuned chat models. ArXiv, abs/2307.09288, 2023. URL https://api_semanticscholar.org/CorpusID:259950998.",
+            "Tu Vu, Brian Lester, Noah Constant, Rami Al-Rfou, and Daniel Matthew Cer. Spot: Better frozen model adaptation through soft prompt transfer. ArXiv, abs/2110.07904, 2021. URL https://api-semanticscholar.org/CorpusID:239009558.",
+            "Chengyu Wang, J. Wang, Minghui Qiu, Jun Huang, and Ming Gao. Transprompt: Towards an automatic transferable prompting framework for few-shot text classification. In Conference on Empirical Methods in Natural Language Processing, 2021. URL https://apisemantic scholar.org/CorpusID:243865402.",
+            "Yaqing Wang, Subhabrata Mukherjee, Xiaodong Liu, Jing Gao, Ahmed Hassan Awadallah, and Jianfeng Gao. Adamix: Mixture-of-adapter for parameter-efficient tuning of large language models. ArXiv, abs/2205.12410, 2022a. URL https://api_semanticscholar.org/ CorpusID:249063002.",
+            "Yaqing Wang, Subhabrata Mukherjee, Xiaodong Liu, Jing Gao, Ahmed Hassan Awadallah, and Jianfeng Gao. Parameter-efficient tuning of large language models. 2022b. URL https:// api.sementicscholar.org/CorpusID:249536106.",
+            "Yizhong Wang, Yeganeh Kordi, Swaroop Mishra, Alisa Liu, Noah A. Smith, Daniel Khashabi, and Hannaneh Hajishirzi. Self-instruct: Aligning language model with self generated instructions. ArXiv, abs/2212.10560, 2022c.",
+            "Gyeong-In Yu and Joo Seong Jeong. Orca: A distributed serving system for transformer-based generative models. In USENIX Symposium on Operating Systems Design and Implementation, 2022. URL https://api-semanticscholar.org/CorpusID:251734964.",
+            "Longteng Zhang, Lin Zhang, Shaohuai Shi, Xiaowen Chu, and Bo Li. Lora-fa: Memory-efficient low-rank adaptation for large language models fine-tuning. ArXiv, abs/2308.03303, 2023a. URL https://api-semanticscholar.org/CorpusID:260683267.",
+            "Qingru Zhang, Minshuo Chen, Alexander W. Bukharin, Pengcheng He, Yu Cheng, Weizhu Chen, and Tuo Zhao. Adaptive budget allocation for parameter-efficient fine-tuning. ArXiv, abs/2303.10512, 2023b. URL https://api_semanticscholar.org/CorpusID: 257631760."
+        ],
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A ADDITIONAL TRAINING DETAILS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            652,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We delve into additional details regarding the models used in §4.2. We conducted fine-tuning on two StarCoder models (Li et al., 2023), specifically the 15B and 3B models. For both models, we employed LoRA (Hu et al., 2021) on the default layers within the Parameter Efficient Fine-Tuning (PEFT) library $^6$ , targeting only the key and query projections in the self-attention layers. This incurs roughly  $0.03\\%$  number of training parameters in the rank 1 setting. For a fair comparison, we applied FLORA to the same layers where LoRA was applied. Regarding the other baseline IA3 (Liu et al., 2022a), we executed the fine-tuning in two distinct settings: one with the default configuration in PEFT, applying IA3 to the key projection layers and the first fully-connected layer in the feedforward layers; and another aligning with the LoRA setting, restricting application to the key and query projection layers. The latter setup proved to be less effective than the default setting, thus the results presented in §4.2 adhere to the default configuration.",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Despite StarCoder's pre-training on the Stack dataset, which contains over 6TB of source code files covering 358 programming languages, we still fine-tuned all methods on this dataset for the low-resource programming languages. This eliminates the time-consuming process to create new data. Our observations revealed that both IA3 and FLORA require a higher learning rate compared to",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
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+            47
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "page_footnote",
+        "text": "$^{6}$ https://github.com/huggingface/peft/tree/main",
+        "bbox": [
+            192,
+            909,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "page_number",
+        "text": "14",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/4b7844a51e05ee1d97b5c69628a9146f68ff4ce5a083447a1174d49c029a81da.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Language</td><td rowspan=\"2\">Base Model</td><td colspan=\"3\">pass@1 (Relative Improvement)</td></tr><tr><td>FLORA</td><td>IA3</td><td>LORA</td></tr><tr><td></td><td colspan=\"4\">StarCoder 15B</td></tr><tr><td>Julia</td><td>23.31</td><td>26.56 (13.92%)</td><td>23.90 (2.51%)</td><td>25.62 (10.45%)</td></tr><tr><td>Lua</td><td>26.60</td><td>30.20 (13.57%)</td><td>28.40 (6.77%)</td><td>29.04 (9.18%)</td></tr><tr><td>R</td><td>16.07</td><td>17.55 (9.24%)</td><td>16.65 (3.59%)</td><td>18.60 (15.8%)</td></tr><tr><td></td><td colspan=\"4\">StarCoderBase 3B</td></tr><tr><td>Julia</td><td>15.62</td><td>19.11 (22.39%)</td><td>16.72 (7.05%)</td><td>19.77 (26.59%)</td></tr><tr><td>Lua</td><td>17.63</td><td>19.46 (10.39%)</td><td>18.04 (2.33%)</td><td>20.23 (14.75%)</td></tr><tr><td>R</td><td>10.00</td><td>10.70 (7.02%)</td><td>9.98 (-0.25%)</td><td>10.06 (0.62%)</td></tr></table>",
+        "bbox": [
+            243,
+            99,
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+            291
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Table 3: Comparison of three fine-tuning methods FLORA, IA3, and LoRA on StarCoder 15B and StarCoderBase 3B across additional low-resource programming languages in the MultiPL-E benchmark.",
+        "bbox": [
+            169,
+            299,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "LORA. For LORA, a learning rate of  $1\\mathrm{e} - 4$  sufficed, whereas IA3 required  $5\\mathrm{e} - 3$ , and FLORA demanded an even higher rate of  $8\\mathrm{e} - 3$ . We conjecture that this stems from FLORA and IA3 incorporating multiplicative weight perturbation, which typically requires a larger learning rate. All models were fine-tuned using 8 bit quantization featuring lower training memory cost.",
+        "bbox": [
+            169,
+            369,
+            823,
+            426
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We present results for additional low-resource languages in Appendix A. The improvements provided by PEFT methods are less pronounced in these languages compared to those documented in §4.2. Nevertheless, it can be observed that LORA and FLORA still outperform IA3, despite the limited room for enhancement in these languages.",
+        "bbox": [
+            169,
+            431,
+            823,
+            491
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Next, we discuss further details concerning the models used in §4.3. We performed fine-tuning on the Whisper large model (Radford et al., 2022), which consists of 1.5 billion parameters. Unlike the code generation task, we discovered that using the default modules in the PEFT library does not yield the best performance. Instead, we applied FLORA, LoRA, and IA3 to every fully-connected layer in the Whisper transformer. We fine-tuned all methods on the Common Voice benchmark (Ardila et al., 2020) for each low-resource language showcased in §4.2. We continued to employ larger learning rates for FLORA and IA3. Specifically, we maintained a learning rate of  $2\\mathrm{e} - 4$  for LORA, while using  $4\\mathrm{e} - 3$  and  $6\\mathrm{e} - 3$  for IA3 and FLORA respectively. Additionally, we utilized 8-bit quantization during training to minimize the GPU memory requirement.",
+        "bbox": [
+            169,
+            494,
+            826,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "B ADDITIONAL SERVING RESULTS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            642,
+            475,
+            657
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "B.1 BMM IMPLEMENTATION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            672,
+            390,
+            686
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We provide further details concerning the experiments in §4.1. To begin, we illustrate the implementation of \"torch.bmm\" as follows. Upon computing the adaptersout, it can be added to the output of the standard LLM layer. This method facilitates the computation of diverse adapters' outputs within a batch.",
+        "bbox": [
+            169,
+            699,
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+            756
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "code",
+        "sub_type": "code",
+        "code_caption": [],
+        "code_body": "inputs = torch Randn(batch, seq_length, d_model)  \nadapter_b # shape of (batch, d_model, rank)  \nadapter_a # shape of (batch, rank, d_model)  \nhidden = torch.bmm(inputs, adapter_b)  \nadaptersout = torch.bmm(hidden, adapter_a)",
+        "guess_lang": "julia",
+        "bbox": [
+            169,
+            768,
+            640,
+            839
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "B.2 ADDITIONAL RESULTS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            854,
+            375,
+            869
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We further conducted a throughput experiment, employing a setup similar to the one described in §4.1, on the newly proposed Llama-2 foundation models (Touvron et al., 2023). The results for both the 13B and 7B models are illustrated in Fig. 3 (left). When compared against the plot in",
+        "bbox": [
+            169,
+            881,
+            823,
+            925
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "page_number",
+        "text": "15",
+        "bbox": [
+            490,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Fig. 2 (left), it is evident that the inflection rank at which LoRA begins to outperform FLORA is lower for the Llama-2 models than for the StarCoder 15B. This shows that the ratio  $c_{1} / c_{2}$  from Eq. (7) varies across different architectures. One plausible explanation could be that, being a recent architecture, the Llama-2 model has not yet been fully optimized within the vLLM framework, potentially impacting throughput performance.",
+        "bbox": [
+            169,
+            103,
+            823,
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "The right panel of Fig. 3 presents the serving results concerning StarCoder-3B. Notably, the inflection rank at which LORA begins to outperform FLORA is lower here as compared to the one observed in StarCoder 15B, as seen in Fig. 2 (right). This is attributed to the relatively low latency on StarCoder-3B, which implies that a request rate of 8 requests per second fails to saturate the LLM server, thereby diminishing the advantage of FLORA. Upon escalating the request rate to 15 requests per second, the inflection rank correspondingly increases to around 12. This illustrates that the latency improvement exhibited in Fig. 2 (left) represents a theoretical enhancement. The actual improvement in serving depends on model architectures, request rates, and other practical considerations.",
+        "bbox": [
+            169,
+            180,
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/1aeccb6df31c6f2249796b5ddbff36f5f1fddab42a39da232b9d477586f581d6.jpg",
+        "image_caption": [
+            "Figure 3: Left: Generation throughput vs. rank for FLORA and torch.bmm implementation of LORA, measured in tokens per second (token/s). The experiments were conducted on two Llama-2 models: 13B and 7B (Touvron et al., 2023). FLORA has great throughput advantage over LORA when the rank is small. As the rank increases, the torch.bmm implementation of LORA finally has a better throughput. Right: Latency vs. rank on StarCoder-3B. Requests are coming at the speed of 8 requests per second and 15 requests per second."
+        ],
+        "image_footnote": [],
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+            47
+        ],
+        "page_idx": 15
+    },
+    {
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+    }
+]
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diff --git a/2024/Batched Low-Rank Adaptation of Foundation Models/698c37fd-a34d-43c4-aeed-9c12baba206a_model.json b/2024/Batched Low-Rank Adaptation of Foundation Models/698c37fd-a34d-43c4-aeed-9c12baba206a_model.json
new file mode 100644
index 0000000000000000000000000000000000000000..a0c07394ad28d672362d31aa6e783f427f563c22
--- /dev/null
+++ b/2024/Batched Low-Rank Adaptation of Foundation Models/698c37fd-a34d-43c4-aeed-9c12baba206a_model.json	
@@ -0,0 +1,2333 @@
+[
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.1,
+                0.645,
+                0.147
+            ],
+            "angle": 0,
+            "content": "BATCHED LOW-RANK ADAPTATION OF FOUNDATION MODELS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.182,
+                0.171,
+                0.434,
+                0.185
+            ],
+            "angle": 0,
+            "content": "Yeming Wen & Swarat Chaudhuri*"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.182,
+                0.186,
+                0.406,
+                0.199
+            ],
+            "angle": 0,
+            "content": "Department of Computer Science"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.182,
+                0.199,
+                0.406,
+                0.213
+            ],
+            "angle": 0,
+            "content": "The University of Texas at Austin"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.451,
+                0.25,
+                0.548,
+                0.264
+            ],
+            "angle": 0,
+            "content": "ABSTRACT"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.23,
+                0.281,
+                0.767,
+                0.379
+            ],
+            "angle": 0,
+            "content": "Low-Rank Adaptation (LoRA) has recently gained attention for fine-tuning foundation models by incorporating trainable low-rank matrices, thereby reducing the number of trainable parameters. While LoRA offers numerous advantages, its applicability for real-time serving to a diverse and global user base is constrained by its incapability to handle multiple task-specific adapters efficiently. This imposes a performance bottleneck in scenarios requiring personalized, task-specific adaptations for each incoming request."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.23,
+                0.381,
+                0.767,
+                0.478
+            ],
+            "angle": 0,
+            "content": "To mitigate this constraint, we introduce Fast LoRA (FLORA), a framework in which each input example in a minibatch can be associated with its unique low-rank adaptation weights, allowing for efficient batching of heterogeneous requests. We empirically demonstrate that FLORA retains the performance merits of LoRA, showcasing competitive results on the MultiPL-E code generation benchmark spanning over 8 languages and a multilingual speech recognition task across 6 languages."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.505,
+                0.338,
+                0.52
+            ],
+            "angle": 0,
+            "content": "1 INTRODUCTION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.536,
+                0.827,
+                0.663
+            ],
+            "angle": 0,
+            "content": "Transformer-based foundation models have showcased remarkable performance across various natural language processing tasks, as evidenced by the successes of ChatGPT (OpenAI, 2023), GitHub Copilot (Chen et al., 2021) and Speech Recognition (Radford et al., 2022) among others. The practice of fine-tuning these models for specific domains or specialized needs, such as instruction-tuning, has become increasingly prevalent (Wang et al., 2022c; Honovich et al., 2022; Taori et al., 2023; Chiang et al., 2023). This is driven by the requirements of real-world applications, which often demand models tailored to specific domains, tasks, or even individual user preferences (Ouyang et al., 2022). However, the extensive number of parameters in foundation models poses computational and memory challenges for task-specific fine-tuning."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.668,
+                0.827,
+                0.809
+            ],
+            "angle": 0,
+            "content": "Low-Rank Adaptation (LORA) emerged as a solution to this challenge by incorporating trainable low-rank matrices (Hu et al., 2021) which significantly reduces the number of trainable parameters during fine-tuning. LoRA's success stems from its ability to achieve domain adaptation without retraining the entire model (Taori et al., 2023; Dettmers et al., 2023; Lee et al., 2023). However, a practical challenge arises in real-time serving scenarios. Batching is the practice of aggregating multiple data points into a single computation. It is a common technique to leverage parallel processing capabilities in GPUs, ensuring higher throughput and lower serving cost. It becomes especially crucial when serving world-wide users where many requests could flood in every second. The intrinsic design of LoRA dictates that every example within a batch shares the same adapter, which is suboptimal for real-world serving scenarios where each request may require a unique adapter."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.814,
+                0.829,
+                0.9
+            ],
+            "angle": 0,
+            "content": "Consider a scenario where users from various locations and professions demand different language and occupation adapters as illustrated in Fig. 1. With LoRA, the batch processing would either force all these diverse requests to share the same adapter or process them sequentially, both of which are impractical. These limitations emphasize the need for a solution that can not only utilize the advantages of LoRA but also serve multiple adapters in parallel, catering to the diverse and simultaneous requests encountered in reality."
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
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+                0.519,
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+            ],
+            "angle": 0,
+            "content": "\\(^{*}\\)ywen@utexas.edu, swarat@cs.utexas.edu"
+        },
+        {
+            "type": "page_number",
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+            "bbox": [
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+            "angle": 0,
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+        {
+            "type": "image",
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+            ],
+            "angle": 0,
+            "content": "Figure 1: This shows a pragmatic scenario where a foundation model in production receives four incoming requests, each requiring distinct adapters. Omitting two adapters in step 2 & 3 for presentation simplicity. FLORA facilitates batching in such serving circumstances, provided the adapters are of low rank, thereby sustaining high throughput and low latency. Detailed discussion on vectorization is provided in §3.2."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.513,
+                0.825,
+                0.584
+            ],
+            "angle": 0,
+            "content": "We posit that it is critical to develop a more flexible adaptation mechanism that is compatible with diverse real-world user queries. We introduce fast LORA (FLORA), a modification of LORA, enabling individual examples in a minibatch to be associated with distinct low-rank weights without compromising the expressive power. This modification promises the benefits of domain adaptation, as heralded by LORA, but without the batching limitation."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.59,
+                0.496,
+                0.605
+            ],
+            "angle": 0,
+            "content": "Our contributions can be summarized as follows:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.618,
+                0.825,
+                0.647
+            ],
+            "angle": 0,
+            "content": "1. We propose FLORA, a framework that augments LoRA by allowing each example in a minibatch to have its unique low-rank adapters, facilitating efficient batching."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.655,
+                0.825,
+                0.739
+            ],
+            "angle": 0,
+            "content": "2. We provided an analytical analysis describing the scenarios where FLORA would be preferred over LoRA in practical applications. This analysis is further substantiated by the empirical evidence where FLORA achieves a 2X throughput improvement on the state-of-the-art code LLM StarCoder 15B in the low-rank setting when diverse adapters are required for incoming examples. Additionally, FLORA reduces the latency by half under the same low-rank setting."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.747,
+                0.825,
+                0.79
+            ],
+            "angle": 0,
+            "content": "3. We demonstrate that FLORA does not sacrifice accuracy compared to LORA on a multilingual code generation task across 8 programming languages, and maintains accuracy in speech recognition tasks over five languages."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.211,
+                0.618,
+                0.825,
+                0.79
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.821,
+                0.419,
+                0.835
+            ],
+            "angle": 0,
+            "content": "2 PROBLEM FORMULATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.855,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "In this section, we outline the problem tackled in this work, illustrating the constraints and objectives that drive the development of the proposed FLORA methodology. Let \\(\\mathcal{M}\\) denote a foundation model parameterized by \\(\\theta\\), with a total number of parameters \\(N\\). The common practice is to fine-tune this foundational model for various specific tasks, ranging from multilingual code generation to speech recognition as demonstrated in §4.2 and §4.3."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.504,
+                0.96
+            ],
+            "angle": 0,
+            "content": "2"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.104,
+                0.342,
+                0.119
+            ],
+            "angle": 0,
+            "content": "2.1 LORA ADAPTERS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.131,
+                0.825,
+                0.188
+            ],
+            "angle": 0,
+            "content": "Fine-tuning the entire model \\(\\mathcal{M}\\) for a specific task is usually computationally expensive due to the massive parameter count. LoRA (Low-rank Adaptation, (Hu et al., 2021)) was introduced to facilitate domain-specific adaptations with a significantly reduced parameter footprint, with the hypothesis that low-rank adaptation is sufficient for fine-tuning domain specific foundation models."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.193,
+                0.826,
+                0.251
+            ],
+            "angle": 0,
+            "content": "Given the pre-trained weight matrix \\( W_0 \\in \\mathbb{R}^{d \\times k} \\), LoRA posits that the weight matrix of the adapted foundation model can be expressed as \\( W_0 + \\Delta W = W_0 + BA \\), where \\( \\Delta W \\) has a low-rank structure. This matrix \\( \\Delta W \\) is factorized into two smaller, trainable matrices: \\( B \\in \\mathbb{R}^{d \\times r} \\) and \\( A \\in \\mathbb{R}^{r \\times k} \\), such that \\( \\Delta W = BA \\) where \\( r \\) stands for the rank. For a given input \\( \\mathbf{x}_i \\), the output \\( \\mathbf{y}_i \\) is given by:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.413,
+                0.27,
+                0.826,
+                0.288
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {y} _ {i} = \\mathcal {M} \\left(\\mathbf {x} _ {i} \\mid \\Delta W, W _ {0}, \\theta\\right) \\tag {1}\n\\]"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.304,
+                0.41,
+                0.318
+            ],
+            "angle": 0,
+            "content": "2.2 Batching & THROUGHPUT"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.33,
+                0.827,
+                0.471
+            ],
+            "angle": 0,
+            "content": "Batching is a common practice where multiple data points \\((\\mathbf{x}_1, \\mathbf{x}_2, \\ldots, \\mathbf{x}_m)\\) are aggregated into a single batch \\(\\mathcal{B}\\). Consequently, the forward passes of these data points are processed concurrently rather than individually. This practice leverages the parallel processing capability of a modern GPU, thereby significantly improving the throughput \\(T\\), i.e., the number of data points processed per unit of time. In the context of foundation models, throughput of a batch \\(\\mathcal{B}\\) can be defined as \\(T = \\sum_{i=1}^{m} |\\mathbf{y}_i| / \\Delta t\\), where \\(|\\mathbf{y}_i|\\) is the number of tokens generated for each example \\(\\mathbf{x}_i\\) in the batch, \\(\\Delta t\\) is the total time taken to process the batch, and \\(m\\) is the number of examples in the batch. Note that batching incurs minimal latency penalties. However, given its substantial increase in throughput, batching and its variants are widely used in the state-of-the-art foundation models serving framework such as vLLM (Kwon et al., 2023) to achieve the best balance between throughput and latency."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.487,
+                0.295,
+                0.501
+            ],
+            "angle": 0,
+            "content": "2.3 OBJECTIVE"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.513,
+                0.825,
+                0.612
+            ],
+            "angle": 0,
+            "content": "Batching typically assumes the same model parameters are utilized for every input example within a minibatch. Hence, a straightforward application of batching in LoRA requires that the adapter matrix \\(\\Delta W\\) be shared across all inputs in the batch \\(\\mathcal{B}\\). The challenge arises when considering a scenario where each input example in the batch might originate from a different task. Sharing the same \\(\\Delta W\\) for all \\(\\mathbf{x}_i\\) in \\(\\mathcal{B}\\) becomes suboptimal where each input potentially demands a unique adapter. The limitation is particularly acute when the model is expected to serve a world-wide user base with diverse incoming requests."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.617,
+                0.826,
+                0.675
+            ],
+            "angle": 0,
+            "content": "Given the limitations of LORA in batching, our objective is to maximize the throughput \\( T \\) in global user serving scenarios by maintaining the batching mechanism. Formally, for each \\( \\mathbf{x}_i \\in \\mathcal{B} \\), we aim to compute \\( \\mathbf{y}_i = \\mathcal{M}(\\mathbf{x}_i | \\Delta W_i, W_0, \\theta) \\), where \\( \\Delta W_i \\) is the adapter matrix corresponding to the input example \\( x_i \\). Therefore, \\( \\Delta W_i \\) can be unique across \\( \\mathcal{B} \\) and specific to a domain or user preference."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.696,
+                0.524,
+                0.71
+            ],
+            "angle": 0,
+            "content": "3 FLORA: FAST LOW RANK ADAPTATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.727,
+                0.825,
+                0.826
+            ],
+            "angle": 0,
+            "content": "As shown in §2.3, adapter sharing is often impractical in real-world serving scenarios. The innovation of FLORA is the introduction of example-specific adapter \\(\\Delta W_{i}\\) for each \\(\\mathbf{x}_i\\) in a minibatch. In FLORA, the weight matrix \\(W_{i}\\) for each example \\(\\mathbf{x}_i\\) in the minibatch is calculated as \\(W_{i} = \\Delta W_{i} \\circ W_{0}\\), where \\(\\circ\\) denotes element-wise multiplication, \\(W_{0}\\) is the pre-trained weight matrix and \\(\\Delta W_{i}\\) is a low-rank adaptation specifically designed for \\(\\mathbf{x}_i\\). Similar to Hu et al. (2021), \\(\\Delta W_{i}\\) is decomposed into two trainable matrices: \\(B_{i} \\in \\mathbb{R}^{d \\times r}\\) and \\(A_{i} \\in \\mathbb{R}^{r \\times k}\\), such that \\(\\Delta W_{i} = B_{i} A_{i}\\), as shown in Fig. 1. Note that FLORA has the same expressive power as LORA by its construction."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.842,
+                0.327,
+                0.856
+            ],
+            "angle": 0,
+            "content": "3.1 FORWARD PASS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.868,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "The advantage of FLORA is that computations on a minibatch can be written in terms of matrix multiplications. This enables efficient batched implementations on modern accelerators such as GPUs. Let \\( x_{i} \\) denote the activations in one layer of a neural net, which is a vertical vector of length \\( d \\). The next layer's activations are given by"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.505,
+                0.96
+            ],
+            "angle": 0,
+            "content": "3"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.397,
+                0.126,
+                0.825,
+                0.214
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} y _ {i} = \\phi \\left(W _ {i} ^ {T} x _ {i}\\right) (2) \\\\ = \\phi \\left(\\left(W _ {0} ^ {T} \\circ \\Delta W _ {i} ^ {T}\\right) x _ {i}\\right) (3) \\\\ = \\phi \\left(\\left(W _ {0} ^ {T} \\circ \\left(B _ {i} A _ {i}\\right) ^ {T}\\right) x _ {i}\\right) (4) \\\\ = \\phi \\left(A _ {i} \\circ \\left(W _ {0} ^ {T} \\left(B _ {i} \\circ x _ {i}\\right)\\right)\\right) (5) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.228,
+                0.825,
+                0.27
+            ],
+            "angle": 0,
+            "content": "When the rank is greater than one, we extend the use of the symbol “o” to denote potential broadcasting. Additionally, a dimension reduction operation such as torch.mean is required prior to applying the activation function \\(\\phi\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.276,
+                0.827,
+                0.377
+            ],
+            "angle": 0,
+            "content": "The key to FLORA's flexibility lies in the low rank decomposition enables the incorporation of example-specific adapters directly into the forward pass, as demonstrated in the equations above. Crucially, each of these operations—the element-wise multiplication between \\( A_{i} \\) and \\( x_{i} \\), and between \\( B_{i} \\) and \\( y_{i} \\) — is inherently batch-friendly. Consequently, FLORA allows for simultaneous processing of multiple requests, each requiring its own adapter, within a single minibatch. To vectorize all adapters in the minibatch, we define matrices \\( \\mathbf{A} \\) and \\( \\mathbf{B} \\) whose rows correspond to the adapters \\( A_{i} \\) and \\( B_{i} \\) for all examples in the minibatch. The above equation is vectorized as:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.402,
+                0.393,
+                0.825,
+                0.418
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbf {Y} = \\phi \\left(\\mathbf {A} \\circ \\left((\\mathbf {B} \\circ \\mathbf {X}) W _ {0}\\right)\\right) \\tag {6}\n\\]"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.434,
+                0.427,
+                0.449
+            ],
+            "angle": 0,
+            "content": "3.2 COMPUTATIONAL EFFICIENCY"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.46,
+                0.827,
+                0.617
+            ],
+            "angle": 0,
+            "content": "The computational analysis primarily concentrates on the examination of fully connected layers within a transformer architecture, given that LoRA is specifically applied to these layers, such as query and key projections. To begin, we analyze a baseline that leverages batch matrix multiplication to facilitate the serving of LoRA with multiple adapters. This operation is possible under the assumption that every adapter required by the input examples in the minibatch shares the same shape, specifically, the same rank. The batch matrix multiplication (BMM) can be implemented using the torch.bmm operator in deep learning frameworks such as PyTorch (Paszke et al., 2019). Note that the BMM operator is typically unfavorable in practical settings due to the significant overhead it introduces (Abdelfattah et al., 2016). This overhead diminishes the throughput and increases latency, which is detrimental in serving scenarios where response times are crucial for maintaining a good user experience."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.62,
+                0.827,
+                0.775
+            ],
+            "angle": 0,
+            "content": "Let \\( b \\) and \\( l \\) denote the batch size and the maximum sequence length in the input batch \\( \\mathcal{B} \\). Revisiting the notation introduced in §3, where \\( W_0 \\in \\mathbb{R}^{d \\times k} \\), \\( B_i \\in \\mathbb{R}^{d \\times r} \\) and \\( A_i \\in \\mathbb{R}^{r \\times k} \\), the operations required to compute the pre-activation for an input batch \\( \\mathcal{B} \\) with dimensions \\( [b, l, d] \\) consist of one matrix multiplication and two BMMs. The matrix multiplication occurs between the input batch \\( \\mathbf{X} \\) and the pre-trained weight \\( W_0 \\). The two BMM operations are conducted firstly between the input batch \\( \\mathbf{X} \\) and \\( \\mathbf{B} \\), and secondly between the result of the prior computation and \\( \\mathbf{A} \\), where \\( \\mathbf{A} \\) and \\( \\mathbf{B} \\) are matrices whose rows correspond to the adapters \\( A_i \\) and \\( B_i \\) for all examples in the minibatch respectively. Assuming for simplicity that the layer neither upscales nor downscales the hidden dimension (i.e. \\( d = k \\)), the upper bound complexity of this layer is discerned as \\( 2c_1(dblr) + c_2(bld^2) \\), with \\( c_1 \\) and \\( c_2 \\) representing the computational coefficients of BMM and matrix multiplication respectively. Note that \\( c_1 >> c_2 \\) because the BMM operator is more expensive than matrix multiplication."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.78,
+                0.827,
+                0.838
+            ],
+            "angle": 0,
+            "content": "For FLORA, the cost is one matrix multiplication which is \\( c_{2}(rbld^{2}) \\) where \\( r \\) denotes the rank of the adapters. We omit the cost of element-wise multiplication in this analysis because it is negligible to the matrix multiplication cost. Comparing the computational cost of FLORA and LORA boils down to the following inequality"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.452,
+                0.854,
+                0.825,
+                0.886
+            ],
+            "angle": 0,
+            "content": "\\[\n\\frac {2 c _ {1}}{d c _ {2}} + \\frac {1}{r} \\geq 1 \\tag {7}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.896,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "FLORA exhibits a lower computational cost than bmm LoRA whenever the above inequality holds true. The benefit of FLORA over LoRA is notably pronounced when \\( r = 1 \\). As the rank increases,"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.505,
+                0.96
+            ],
+            "angle": 0,
+            "content": "4"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.104,
+                0.825,
+                0.176
+            ],
+            "angle": 0,
+            "content": "LORA gradually becomes less costly. From the established inequality, a variety of scenarios can be inferred where FLORA has an advantage over LORA. Firstly, the advantage of FLORA is significantly apparent when the rank of adapters is small. Secondly, in configurations where the model has fewer hidden units but an increased number of layers, FLORA tends to outperform LORA due to the smaller value of \\(d\\) in the denominator of Eq. (7)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.181,
+                0.827,
+                0.281
+            ],
+            "angle": 0,
+            "content": "Another advantage of FLORA is the cost remains invariant to the number of adapters required by the input batch. While the preceding analysis assumes that every token in an example \\( x_{i} \\) shares the same adapter, it is possible to apply multiple adapters to a single example by dividing the example into chunks, and then applying different adapters to each chunk. This approach is commonly observed in the Mixture of Experts framework (Fedus et al., 2021; Lepikhin et al., 2020; Puigcerver et al., 2023). Incorporating several adapters in an input example notably amplifies the ratio \\( c_{1} / c_{2} \\) in Eq. (7), thereby significantly increasing LORA's cost."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.286,
+                0.825,
+                0.372
+            ],
+            "angle": 0,
+            "content": "The ratio \\( c_{1} / c_{2} \\) might not be the same across different transformer architectures. §4.1 is designed to provide a deeper insight into how comparative serving efficiency of FLORA and LORA changes under various architectures. Additionally, it's worth noting that LORA does not apply to the self-attention layers, which constitute a non-trivial portion of the computational cost, thereby overshadowing the advantage of FLORA. However, as efficient self-attention mechanisms such as flash attention (Dao et al., 2022) get adopted, the advantage of FLORA over LORA is likely to get larger."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.376,
+                0.827,
+                0.448
+            ],
+            "angle": 0,
+            "content": "Connection to IA3. IA3 was proposed in Liu et al. (2022a) featuring fast adaptation of LLM. It introduces a learned vector \\( l \\) which re-scales the activation by \\( y_{i} = l\\circ \\phi (W_{0}^{T}x_{i}) = \\phi \\bigl (l\\circ (W_{0}^{T}x_{i})\\bigr) \\). This can be viewed as a special case of FLORA - a rank 0.5 variant - which only re-scales the columns instead of the entire pre-trained weights. It has a limited expressive power compared to FLORA and LoRA."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.466,
+                0.33,
+                0.483
+            ],
+            "angle": 0,
+            "content": "4 EXPERIMENTS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.498,
+                0.825,
+                0.598
+            ],
+            "angle": 0,
+            "content": "In this section, we compare FLORA to LoRA and other notable baselines across various metrics and tasks. To begin with, we delve into a computational analysis to substantiate the enhanced throughput and the reduced latency achieved by FLORA in the case of low rank. Subsequently, we pivot towards analyzing the accuracy of FLORA in multilingual code generation tasks spanning across different languages. The goal is to discern the proficiency of FLORA in maintaining, if not enhancing, the model's accuracy as compared to LoRA. Progressing further, we replicate a similar analysis but in the domain of multilingual speech recognition."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.613,
+                0.356,
+                0.627
+            ],
+            "angle": 0,
+            "content": "4.1 SERVING ANALYSIS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.638,
+                0.826,
+                0.751
+            ],
+            "angle": 0,
+            "content": "The primary objective of this serving analysis is to measure the maximum throughput both FLORA and LoRA can attain under varied rank configurations. We carried out this exploration on the state-of-the-art code Large Language Model (LLM) StarCoder (Li et al., 2023), evaluating models of different number of parameters namely 1B, 3B, and 15B. The dataset facilitating this analysis has been sourced from the vLLM throughput benchmark. Noteworthy, this dataset was previously used to fine-tune the English Vicuna model, a state-of-the-art chat LLMs (Chiang et al., 2023). To expedite the benchmarking process, we extracted a subset of 1,000 samples from the original dataset, ensuring a diverse range of sample lengths varying from 50 to 2,000 tokens."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.757,
+                0.827,
+                0.884
+            ],
+            "angle": 0,
+            "content": "In setting up the computational analysis, our primary intent is to compare FLORA and LoRA in the real-world serving scenario. See Appendix B.1 on how bmm LoRA is implemented. The vLLM framework (Kwon et al., 2023)\\(^{2}\\), with its implementation of continuous batching, presents an ideal setup for this analysis. The continuous batching mechanism in vLLM, as inspired by the principles delineated in Orca (Yu & Jeong, 2022), facilitates a more efficient utilization of GPU resource by allowing new sequence to be inserted immediately once any sequence in the current batch is completed. This continuous flow significantly enhances GPU utilization as compared to static batching, where the GPU awaits the completion of all sequences in a batch before initiating a new batch processing. The comparison of FLORA and LoRA within this setup offers a compelling evidence of"
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.195,
+                0.897,
+                0.825,
+                0.911
+            ],
+            "angle": 0,
+            "content": "<sup>1</sup>The batch size in BMM operator increases from \\( b \\) to \\( b \\times m \\) where \\( m \\) is the number of adapters per example."
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.197,
+                0.911,
+                0.327,
+                0.923
+            ],
+            "angle": 0,
+            "content": "2The version is 0.1.3."
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+        {
+            "type": "header",
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+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "image",
+            "bbox": [
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+        },
+        {
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+        },
+        {
+            "type": "image_caption",
+            "bbox": [
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+            "angle": 0,
+            "content": "Figure 2: Left: Generation throughput vs. rank for FLORA and torch.bmm implementation of LORA, measured in tokens per second (token/s). The experiments were conducted on three star-coder models: StarCoder 15B, StarCoderbase 3B and StarCoderbase 1B. FLORA has great throughput advantage over LoRA when the rank is small. As the rank increases, the torch.bmm implementation of LoRA finally has a better throughput. Right: Latency vs. rank on StarCoder-15B. Requests are coming at the speed of 8 requests per second."
+        },
+        {
+            "type": "text",
+            "bbox": [
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+            "angle": 0,
+            "content": "their respective throughput and latency in the real-world serving scenario. It is worth noting that the experiments were conducted without Flash-attention (Dao et al., 2022)."
+        },
+        {
+            "type": "text",
+            "bbox": [
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+            "content": "Throughput experiment. In Fig. 2, the throughput results for both FLORA and bmm LORA are illustrated across different rank configurations on three StarCoder models with different number of parameters, namely 1B, 3B and 15B. All experiments were conducted on an NVIDIA H100 GPU with a float16 precision<sup>3</sup>. The maximum number of batched tokens is 8,192, which is the same as the model context length. Evidently, FLORA shows a superior throughput over LORA in the lower rank configurations. At rank 1, the throughput of FLORA is more than threefold higher than that of LORA, thereby highlighting the considerable serving performance boost FLORA provides. The advantage of FLORA continues as the rank increases, albeit with a diminishing rate. For instance, at rank 2, FLORA's throughput is around 2.5 times higher, and this multiplicative advantage decreases as the rank increases further. This performance advantage continues up until the rank of 8 in the StarCoder 15B model, where LoRA starts to outperform FLORA. This inflection point suggests that the advantages of FLORA in terms of throughput are more pronounced in lower rank."
+        },
+        {
+            "type": "text",
+            "bbox": [
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+            "angle": 0,
+            "content": "Notice that the inflection points occurred at a higher rank when serving a smaller LLM as illustrated in (Fig. 2, left). This demonstrates a significant potential of FLORA, especially when considering future applications of quantization techniques to serving LLMs. By applying quantization, such as 8-bit or 4-bit inference, the effective size of the model is reduced, akin to serving a smaller LLM thus potentially extending the rank at which FLORA maintains a throughput advantage over LORA."
+        },
+        {
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+            "angle": 0,
+            "content": "Latency experiment. We assessed the latency-versus-rank performance on the Starcoder 15B model, using the same dataset as the throughput experiment. This evaluation was conducted under the conditions where requests arrive at a rate of 8 per second. The default maximum number of batched tokens in vLLM serving launcher is 2,560. The results, as shown in (Fig. 2, right), measure latency in terms of seconds per output token. Remarkably, in the lower rank regime (ranging from rank 1 to rank 4), FLORA exhibited a 2-5X reduction in latency compared to LoRA. Notably, the latency for LoRA stood at approximately 2.3 seconds per output token, is impractical for serving due to the poor user experience it would cause. This experiment further highlights the superior capabilities of FLORA in efficiently catering to diverse incoming user requests."
+        },
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+            "content": "These findings validate the theoretical analysis in §4.1, confirming that FLORA provides significant throughput advantages, particularly in settings with lower to moderate ranks. This positions FLORA"
+        },
+        {
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+            "bbox": [
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+            "angle": 0,
+            "content": "3The quantization in vLLM is still under development."
+        },
+        {
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+        }
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+    [
+        {
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+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table",
+            "bbox": [
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+            "content": "<table><tr><td rowspan=\"2\">Language</td><td rowspan=\"2\">Base Model</td><td colspan=\"3\">pass@1 (Relative Improvement)</td></tr><tr><td>FLORA</td><td>IA3</td><td>LoRA</td></tr><tr><td></td><td colspan=\"4\">StarCoder 15B</td></tr><tr><td>Dlang</td><td>14.13</td><td>17.26 (22.14%)</td><td>15.26 (7.99%)</td><td>17.15 (21.37%)</td></tr><tr><td>Perl</td><td>17.05</td><td>21.44 (25.76%)</td><td>17.71 (3.90%)</td><td>21.46 (25.76%)</td></tr><tr><td>Ruby</td><td>1.39</td><td>24.94 (1692.86%)</td><td>20.80 (1394.64%)</td><td>23.76 (1608.04%)</td></tr><tr><td>Rust</td><td>22.40</td><td>26.24 (17.14%)</td><td>23.53 (5.04%)</td><td>26.87 (19.95%)</td></tr><tr><td>Racket</td><td>10.20</td><td>12.41 (21.61%)</td><td>11.53 (12.96%)</td><td>12.51 (22.58%)</td></tr><tr><td>Swift</td><td>16.91</td><td>20.38 (20.51%)</td><td>18.13 (7.19%)</td><td>20.35 (20.36%)</td></tr><tr><td></td><td colspan=\"4\">StarCoderBase 3B</td></tr><tr><td>Dlang</td><td>5.65</td><td>5.72 (1.20%)</td><td>5.72 (1.20%)</td><td>6.97 (23.34%)</td></tr><tr><td>Perl</td><td>10.73</td><td>13.01 (21.25%)</td><td>11.46 (6.83%)</td><td>13.31 (27.51%)</td></tr><tr><td>Ruby</td><td>5.33</td><td>14.48 (171.68%)</td><td>7.88 (47.90%)</td><td>13.89 (160.68%)</td></tr><tr><td>Rust</td><td>17.18</td><td>21.00 (22.24%)</td><td>17.28 (0.60%)</td><td>20.67 (20.31%)</td></tr><tr><td>Racket</td><td>8.04</td><td>9.16 (13.99%)</td><td>8.40 (4.48%)</td><td>8.80 (9.48%)</td></tr><tr><td>Swift</td><td>10.04</td><td>15.69 (56.21%)</td><td>12.54 (24.83%)</td><td>15.04 (49.76%)</td></tr></table>"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
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+            "angle": 0,
+            "content": "Table 1: Comparison of three fine-tuning methods FLORA, IA3, and LORA on StarCoder 15B and StarCoderBase 3B across various low-resource programming languages in the MultiPL-E benchmark. The table presents Pass@1 accuracy of each method alongside the relative improvement over the baseline. The standard errors are less than \\(0.3\\%\\) in all cells in the table, therefore we exclude that for clear presentation."
+        },
+        {
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+            "bbox": [
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+            "angle": 0,
+            "content": "as a compelling alternative for efficiently serving adapted foundation models, especially in scenarios where lower ranks suffice the desired model accuracy, as further demonstrated in the subsequent accuracy analysis sections. Moreover, if an enterprise chooses to serve foundation models with a substantial number of diverse adapters, for instance, a personalized LLM, then a low rank or even a rank one is imperative to avoid excessive storage costs."
+        },
+        {
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+            "content": "4.2 MULTILINGUAL CODE GENERATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
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+            "content": "A key aspect to examine before applying FLORA in real-world LLM serving is to scrutinize any potential degradation in performance. In this section, we consider multilingual code generation as the testbed for comparing FLORA and LoRA due to its alignment with real-world applications, where the necessity to cater to diverse programming languages is paramount. Low-resource languages, as referred to in this context, are languages that appear much less frequently than other languages in the pre-training data. Orlanski et al. (2023) showed that the performance of code LLMs can be notably enhanced on low-resource programming languages such as PERL by recalibrating the language distribution in the pre-training data. This suggests that fine-tuning a trained LLM on such low-resource languages could potentially boost its performance on the same language. Hence, by employing multilingual code generation as a benchmark, we can make an informed evaluation of adaptability and performance enhancements that FLORA and LoRA can achieve."
+        },
+        {
+            "type": "text",
+            "bbox": [
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+            "angle": 0,
+            "content": "Additionally, a comparison is made against a third baseline, IA3, which can be considered as a special case of FLORA. Essentially, IA3 can be seen as a rank 0.5 variant of FLORA, thereby facing a constrained expressive power in comparison to both FLORA and LoRA. For FLORA and LoRA, we conducted fine-tuning across a range of rank choices, spanning from 1 to 8. It emerges that within the scope of this multilingual code generation task, a rank of one typically suffices to achieve optimal results, with the exception of Racket and Lua. Consequently, the results shown in §4.2 are predominantly at rank 1, barring Racket and Lua, which are presented at rank 4."
+        },
+        {
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+            "content": "Fine-tuning. In our effort to evaluate the performance of FLORA, LORA, and IA3 on the multilingual code generation task, we fine-tuned these models on state-of-the-art multilingual code LLMs, StarCoder 15B and StarCoderBase 3B as introduced in (Li et al., 2023). A pivotal aspect of our fine-tuning process was the utilization of existing data, negating the need for creating new data for"
+        },
+        {
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+            "bbox": [
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+            "angle": 0,
+            "content": "7"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
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+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table",
+            "bbox": [
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+            "angle": 0,
+            "content": "<table><tr><td>Model</td><td>Arabic</td><td>Czech</td><td>Lithuanian</td><td>Marathi</td><td>Mongolian</td><td>Hindi</td></tr><tr><td>Whisper</td><td>46.03</td><td>23.19</td><td>46.09</td><td>84.84</td><td>115.20</td><td>43.41</td></tr><tr><td>FLORA</td><td>30.21 ± 0.25</td><td>10.76 ± 0.22</td><td>20.39 ± 0.32</td><td>33.17 ± 0.17</td><td>42.12 ± 0.63</td><td>23.58 ± 0.65</td></tr><tr><td>IA3</td><td>30.48 ± 0.22</td><td>11.61 ± 0.33</td><td>21.41 ± 0.20</td><td>35.03 ± 0.61</td><td>46.47 ± 0.53</td><td>25.64 ± 0.78</td></tr><tr><td>LORA</td><td>30.18 ± 0.21</td><td>10.77 ± 0.38</td><td>20.50 ± 0.21</td><td>31.94 ± 0.33</td><td>41.57 ± 0.69</td><td>23.82 ± 0.74</td></tr></table>"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
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+            "angle": 0,
+            "content": "Table 2: Mean WER with standard deviation for different models and languages. The WER is calculated with respect to the unnormalized tokenizer. The base model here is Whisper-1.5B."
+        },
+        {
+            "type": "text",
+            "bbox": [
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+            "angle": 0,
+            "content": "low-resource programming languages. We leveraged the same pre-training data that was used for pre-training StarCoder, specifically, the Stack dataset, which contains over 6TB of permissively-licensed source code files covering 358 programming languages. For each low-resource language in our experiment, we fine-tuned on its corresponding split from the Stack dataset for a total of 1,500 steps, along with batch size 8. More fine-tuning details are given in Appendix A."
+        },
+        {
+            "type": "text",
+            "bbox": [
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+            ],
+            "angle": 0,
+            "content": "Evaluation. The evaluation of FLORA, LORA, and IA3 was conducted on the MultiPL-E benchmark (Cassano et al., 2023), which contains the translation of two unit-test driven Python benchmarks (HumanEval and MBPP) to 18 other programming languages. We used the HumanEval split of the benchmark to evaluate the fine-tuned models. As for the metrics, We adopted the \\(pass@k\\) metrics from Chen et al. (2021); Austin et al. (2021), which is calculated as the fraction of problems with at least one correct sample given \\(k\\) samples. Similar to Chen et al. (2021), we drew 100 samples for computing \\(pass@1\\), with a sampling temperature set at 0.1."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
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+            ],
+            "angle": 0,
+            "content": "Main results. The result in §4.2 exhibits the performance of three methods across various programming languages on both StarCoder 15B and StarCoderBase 3B models. The average relative improvement achieved by FLORA and LoRA is roughly \\(20\\%\\) in the selected low-resource programming languages. FLORA consistently outperforms IA3 on all languages, especially on StarCoder 15B, denoting its efficiency in leveraging the model expressive power to improve multilingual code generation. It is notable that StarCoder 15B has an unforeseen issue regarding Ruby generation, where it yields lower accuracy compared to the 3B model. However, all methods are able to fix its abnormal performance."
+        },
+        {
+            "type": "text",
+            "bbox": [
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+            "angle": 0,
+            "content": "On StarCoderBase 3B, a smaller model, it is evident that the baseline performance drops, yet FLoRA and LoRA still manage to exhibit substantial relative improvement over the baseline, especially in languages like Swift and Ruby. This suggests that both methods benefit from continuous training on the low-resource language split of the pre-training data, although the advantages may diminish with a reduction in model size. While the absolute performance (pass@1 accuracy) varies among languages, the relative improvements highlight the effectiveness of the tested methods in enhancing multilingual code generation."
+        },
+        {
+            "type": "title",
+            "bbox": [
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+            "angle": 0,
+            "content": "4.3 MULTILINGUAL SPEECH RECOGNITION"
+        },
+        {
+            "type": "text",
+            "bbox": [
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+            "content": "Following our analysis in multilingual code generation, we shift our focus to another substantial application of large language models — multilingual speech recognition (MSR). This domain has seen increasing demand due to the necessity of serving various linguistic interfaces. The capabilities of FLORA and LORA to adapt to multiple languages in code generation presents a tempting premise to investigate their effectiveness in the MSR domain."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
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+            "angle": 0,
+            "content": "Fine-tuning. A crucial aspect of our analysis involves fine-tuning the foundation model on a multilingual speech recognition dataset. We selected the Whisper large 1.5B (Radford et al., 2022) as our base model to conduct the subsequent analysis. Whisper is an encoder-decoder transformer model trained on 680k hours of labeled speech data through large-scale weak supervision. Despite being pre-trained on a multilingual speech corpus, the model's predictive capabilities can be improved further for certain languages and tasks through fine-tuning. We use the Common Voice benchmark (Ardila et al., 2020) — containing a total of 38 languages and 2,500 hours of collected audio — for the fine-tuning process, with a particular focus on low-resource languages. For each low-resource language enumerated in §4.2, we fine-tuned on its training split within the Common"
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.195,
+                0.91,
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+            ],
+            "angle": 0,
+            "content": "Dataset statistics can be found here https://commonvoice.—.org/en/datasets."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
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+            "angle": 0,
+            "content": "8"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
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+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.825,
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+            ],
+            "angle": 0,
+            "content": "Voice dataset for approximately 5,000 steps, with a batch size of 6. More fine-tuning details are given in Appendix A. Similar to §4.2, we found that rank one is sufficient to achieve the best result in most cases, with the exception of Marathi, which requires a rank of 4."
+        },
+        {
+            "type": "text",
+            "bbox": [
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+            ],
+            "angle": 0,
+            "content": "Main results. We present the main results on the MSR task in §4.2. Notice that the evaluation is conducted with the unnormalized tokenizer.<sup>5</sup> The Whisper model exhibited a relatively higher Word Error Rate (WER) across all low-resource languages when compared to FLORA, LORA and IA3 models, reflecting a significant room for enhancement in its multilingual speech recognition capabilities. Particularly, its performance was found to be subpar for the Marathi and Mongolian languages, with WERs of 84.84 and 115.20 respectively. All examined models significantly outperform the base Whisper model by a wide margin across all languages, highlighting the effective fine-tuning of FLORA, LORA and IA3 models. For instance, in Arabic, FLORA reduces the WER to 30.21 from Whisper's 46.03, showcasing a significant enhancement in speech recognition accuracy. Particularly, FLORA and LORA consistently outperform IA3, indicating a better expressive power for multilingual speech recognition tasks. For example, in Czech, FLORA has a mean WER of 10.76, which is slightly better compared to IA3's 11.61 and LORA's 11.47. A similar trend is observed in Lithuanian and Hindi. In summary, this table shows the superior performance of FLORA and LORA over IA3 in the speech recognition task across diverse languages. The consistent low WERs attained by FLORA demonstrates its potential as a viable model for MSR tasks."
+        },
+        {
+            "type": "title",
+            "bbox": [
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+            "angle": 0,
+            "content": "5 RELATED WORK"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.413,
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+                0.54
+            ],
+            "angle": 0,
+            "content": "Parameter-Efficient Fine-Tuning (PEFT) methods are broadly partitioned into two categories: weight-based and non-weight-based approaches. MC-dropout (Lakshminarayanan et al., 2016) stands as an early example for the non-weight-based approach, where distinct dropout masks are allocated to various tasks. Recently, prompt tuning techniques have emerged as a prevalent stream within this category (Li & Liang, 2021; Lester et al., 2021), facilitating efficient adaptation with minimal modifications to models. Successive endeavors aimed to enhance this class of methods, delving into aspects such as optimization (Mao et al., 2021; Diao et al., 2022), transferability (Wang et al., 2021; Vu et al., 2021; He et al., 2022b), and the usage of discrete prompts (Schick & Schütze, 2020a;b; Gao et al., 2021; Malkin et al., 2021), among others (Liu et al., 2022b; 2021)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.545,
+                0.827,
+                0.728
+            ],
+            "angle": 0,
+            "content": "We focus on weight-based approaches in this work, which has a weight interpretation as exemplified by LoRA (Hu et al., 2021). This line of research can be traced back to Progressive Network (Rusu et al., 2016), which inserts a sub-network when a new task arrives. This principle was later adapted widely in foundation models as represented by adapter based methods (Houlsby et al., 2019; Mahabadi et al., 2021; Davison, 2021; Ding et al., 2022; Wang et al., 2022b). In particular, BitFit (BenZaken et al., 2021) was introduced to solely update the bias parameters, while IA3 (Liu et al., 2022a) was proposed to rescale the activations. Additionally, approaches such Fish (Sung et al., 2021) and Diff-pruning (Guo et al., 2020) leverage sparsity to facilitate efficient adaptation of foundation models. A separate vein of research aims to improve LoRA by reducing its computational and memory costs (Zhang et al., 2023b;a; Malladi et al., 2023). He et al. (2022a) explored how to unify different PEFT methods. Dettmers et al. (2023) quantized LoRA to reduce memory footprint. Chavan et al. (2023) generalized LoRA by learning individual adapters in each layer. Several other works focus on building mixture of adapters (Wang et al., 2022a; Diao et al., 2023)."
+        },
+        {
+            "type": "title",
+            "bbox": [
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+            "angle": 0,
+            "content": "6 CONCLUSION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.778,
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+                0.835
+            ],
+            "angle": 0,
+            "content": "We introduced FLORA, an extension of LORA, facilitating efficient batching. Empirical evaluations demonstrated that FLORA enhances throughput and latency in practical serving scenarios, all while preserving the accuracy of LORA. Through FLORA, we aim to facilitate a more efficient adaptation of large language models to diverse and real-world user requests."
+        },
+        {
+            "type": "text",
+            "bbox": [
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+                0.84,
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+            "angle": 0,
+            "content": "Limitations. Despite its parameter efficiency, FLORA still requires fine-tuning. A promising future work could be to derive FLORA weights from a trained LoRA model, given that LORA remains"
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+            "content": "This is because the normalized tokenizer has a bug in the official leaderboard implementation at the time of submission."
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+            "angle": 0,
+            "content": "the most prevalent type of adapter as per (Huang et al., 2023). This adaptation could potentially obviate the requirement for fine-tuning, thereby accelerating the process of model adaptation."
+        },
+        {
+            "type": "title",
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+            "content": "REFERENCES"
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+            "content": "Hugo Touvron, Louis Martin, Kevin R. Stone, Peter Albert, Amjad Almahairi, Yasmine Babaei, Nikolay Bashlykov, Soumya Batra, Prajwal Bhargava, Shruti Bhosale, Daniel M. Bikel, Lukas Blecher, Cristian Cantón Ferrer, Moya Chen, Guillem Cucurull, David Esiobu, Jude Fernandes, Jeremy Fu, Wenyin Fu, Brian Fuller, Cynthia Gao, Vedanuj Goswami, Naman Goyal, Anthony S. Hartshorn, Saghar Hosseini, Rui Hou, Hakan Inan, Marcin Kardas, Viktor Kerkez, Madian Khabsa, Isabel M. Kloumann, A. V. Korenev, Punit Singh Koura, Marie-Anne Lachaux, Thibaut Lavril, Jenya Lee, Diana Liskovich, Yinghai Lu, Yuning Mao, Xavier Martinet, Todor Mihaylov, Pushkar Mishra, Igor Molybog, Yixin Nie, Andrew Poulton, Jeremy Reizenstein, Rashi Rungta, Kalyan Saladi, Alan Schelten, Ruan Silva, Eric Michael Smith, R. Subramanian, Xia"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.173,
+                0.103,
+                0.901,
+                0.925
+            ],
+            "angle": 0,
+            "content": null
+        },
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+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
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+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "ref_text",
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+                0.103,
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+            "content": "Tan, Binh Tang, Ross Taylor, Adina Williams, Jian Xiang Kuan, Puxin Xu, Zhengxu Yan, Iliyan Zarov, Yuchen Zhang, Angela Fan, Melanie Kambadur, Sharan Narang, Aurelien Rodriguez, Robert Stojnic, Sergey Edunov, and Thomas Scialom. Llama 2: Open foundation and fine-tuned chat models. ArXiv, abs/2307.09288, 2023. URL https://api_semanticscholar.org/CorpusID:259950998."
+        },
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+            "bbox": [
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+                0.183,
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+            "angle": 0,
+            "content": "Tu Vu, Brian Lester, Noah Constant, Rami Al-Rfou, and Daniel Matthew Cer. Spot: Better frozen model adaptation through soft prompt transfer. ArXiv, abs/2110.07904, 2021. URL https://api-semanticscholar.org/CorpusID:239009558."
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
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+            "content": "Chengyu Wang, J. Wang, Minghui Qiu, Jun Huang, and Ming Gao. Transprompt: Towards an automatic transferable prompting framework for few-shot text classification. In Conference on Empirical Methods in Natural Language Processing, 2021. URL https://apisemantic scholar.org/CorpusID:243865402."
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+            "content": "Yizhong Wang, Yeganeh Kordi, Swaroop Mishra, Alisa Liu, Noah A. Smith, Daniel Khashabi, and Hannaneh Hajishirzi. Self-instruct: Aligning language model with self generated instructions. ArXiv, abs/2212.10560, 2022c."
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+            "type": "ref_text",
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+            "angle": 0,
+            "content": "Gyeong-In Yu and Joo Seong Jeong. Orca: A distributed serving system for transformer-based generative models. In USENIX Symposium on Operating Systems Design and Implementation, 2022. URL https://api-semanticscholar.org/CorpusID:251734964."
+        },
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+                0.174,
+                0.519,
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+            ],
+            "angle": 0,
+            "content": "Longteng Zhang, Lin Zhang, Shaohuai Shi, Xiaowen Chu, and Bo Li. Lora-fa: Memory-efficient low-rank adaptation for large language models fine-tuning. ArXiv, abs/2308.03303, 2023a. URL https://api-semanticscholar.org/CorpusID:260683267."
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
+                0.174,
+                0.571,
+                0.826,
+                0.627
+            ],
+            "angle": 0,
+            "content": "Qingru Zhang, Minshuo Chen, Alexander W. Bukharin, Pengcheng He, Yu Cheng, Weizhu Chen, and Tuo Zhao. Adaptive budget allocation for parameter-efficient fine-tuning. ArXiv, abs/2303.10512, 2023b. URL https://api_semanticscholar.org/CorpusID: 257631760."
+        },
+        {
+            "type": "list",
+            "bbox": [
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+                0.103,
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+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
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+            ],
+            "angle": 0,
+            "content": "A ADDITIONAL TRAINING DETAILS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.684,
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+            "angle": 0,
+            "content": "We delve into additional details regarding the models used in §4.2. We conducted fine-tuning on two StarCoder models (Li et al., 2023), specifically the 15B and 3B models. For both models, we employed LoRA (Hu et al., 2021) on the default layers within the Parameter Efficient Fine-Tuning (PEFT) library\\(^6\\), targeting only the key and query projections in the self-attention layers. This incurs roughly \\(0.03\\%\\) number of training parameters in the rank 1 setting. For a fair comparison, we applied FLORA to the same layers where LoRA was applied. Regarding the other baseline IA3 (Liu et al., 2022a), we executed the fine-tuning in two distinct settings: one with the default configuration in PEFT, applying IA3 to the key projection layers and the first fully-connected layer in the feedforward layers; and another aligning with the LoRA setting, restricting application to the key and query projection layers. The latter setup proved to be less effective than the default setting, thus the results presented in §4.2 adhere to the default configuration."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.845,
+                0.826,
+                0.903
+            ],
+            "angle": 0,
+            "content": "Despite StarCoder's pre-training on the Stack dataset, which contains over 6TB of source code files covering 358 programming languages, we still fine-tuned all methods on this dataset for the low-resource programming languages. This eliminates the time-consuming process to create new data. Our observations revealed that both IA3 and FLORA require a higher learning rate compared to"
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.194,
+                0.91,
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+            ],
+            "angle": 0,
+            "content": "\\(^{6}\\)https://github.com/huggingface/peft/tree/main"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
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+                0.948,
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+            ],
+            "angle": 0,
+            "content": "14"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.244,
+                0.101,
+                0.756,
+                0.292
+            ],
+            "angle": 0,
+            "content": "<table><tr><td rowspan=\"2\">Language</td><td rowspan=\"2\">Base Model</td><td colspan=\"3\">pass@1 (Relative Improvement)</td></tr><tr><td>FLORA</td><td>IA3</td><td>LORA</td></tr><tr><td></td><td colspan=\"4\">StarCoder 15B</td></tr><tr><td>Julia</td><td>23.31</td><td>26.56 (13.92%)</td><td>23.90 (2.51%)</td><td>25.62 (10.45%)</td></tr><tr><td>Lua</td><td>26.60</td><td>30.20 (13.57%)</td><td>28.40 (6.77%)</td><td>29.04 (9.18%)</td></tr><tr><td>R</td><td>16.07</td><td>17.55 (9.24%)</td><td>16.65 (3.59%)</td><td>18.60 (15.8%)</td></tr><tr><td></td><td colspan=\"4\">StarCoderBase 3B</td></tr><tr><td>Julia</td><td>15.62</td><td>19.11 (22.39%)</td><td>16.72 (7.05%)</td><td>19.77 (26.59%)</td></tr><tr><td>Lua</td><td>17.63</td><td>19.46 (10.39%)</td><td>18.04 (2.33%)</td><td>20.23 (14.75%)</td></tr><tr><td>R</td><td>10.00</td><td>10.70 (7.02%)</td><td>9.98 (-0.25%)</td><td>10.06 (0.62%)</td></tr></table>"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.17,
+                0.3,
+                0.825,
+                0.342
+            ],
+            "angle": 0,
+            "content": "Table 3: Comparison of three fine-tuning methods FLORA, IA3, and LoRA on StarCoder 15B and StarCoderBase 3B across additional low-resource programming languages in the MultiPL-E benchmark."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.37,
+                0.825,
+                0.428
+            ],
+            "angle": 0,
+            "content": "LORA. For LORA, a learning rate of \\(1\\mathrm{e} - 4\\) sufficed, whereas IA3 required \\(5\\mathrm{e} - 3\\), and FLORA demanded an even higher rate of \\(8\\mathrm{e} - 3\\). We conjecture that this stems from FLORA and IA3 incorporating multiplicative weight perturbation, which typically requires a larger learning rate. All models were fine-tuned using 8 bit quantization featuring lower training memory cost."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.433,
+                0.825,
+                0.492
+            ],
+            "angle": 0,
+            "content": "We present results for additional low-resource languages in Appendix A. The improvements provided by PEFT methods are less pronounced in these languages compared to those documented in §4.2. Nevertheless, it can be observed that LORA and FLORA still outperform IA3, despite the limited room for enhancement in these languages."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.496,
+                0.827,
+                0.624
+            ],
+            "angle": 0,
+            "content": "Next, we discuss further details concerning the models used in §4.3. We performed fine-tuning on the Whisper large model (Radford et al., 2022), which consists of 1.5 billion parameters. Unlike the code generation task, we discovered that using the default modules in the PEFT library does not yield the best performance. Instead, we applied FLORA, LoRA, and IA3 to every fully-connected layer in the Whisper transformer. We fine-tuned all methods on the Common Voice benchmark (Ardila et al., 2020) for each low-resource language showcased in §4.2. We continued to employ larger learning rates for FLORA and IA3. Specifically, we maintained a learning rate of \\(2\\mathrm{e} - 4\\) for LORA, while using \\(4\\mathrm{e} - 3\\) and \\(6\\mathrm{e} - 3\\) for IA3 and FLORA respectively. Additionally, we utilized 8-bit quantization during training to minimize the GPU memory requirement."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.643,
+                0.476,
+                0.658
+            ],
+            "angle": 0,
+            "content": "B ADDITIONAL SERVING RESULTS"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.674,
+                0.391,
+                0.688
+            ],
+            "angle": 0,
+            "content": "B.1 BMM IMPLEMENTATION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.7,
+                0.825,
+                0.757
+            ],
+            "angle": 0,
+            "content": "We provide further details concerning the experiments in §4.1. To begin, we illustrate the implementation of \"torch.bmm\" as follows. Upon computing the adaptersout, it can be added to the output of the standard LLM layer. This method facilitates the computation of diverse adapters' outputs within a batch."
+        },
+        {
+            "type": "code",
+            "bbox": [
+                0.171,
+                0.769,
+                0.642,
+                0.84
+            ],
+            "angle": 0,
+            "content": "inputs = torch Randn(batch, seq_length, d_model)  \nadapter_b # shape of (batch, d_model, rank)  \nadapter_a # shape of (batch, rank, d_model)  \nhidden = torch.bmm(inputs, adapter_b)  \nadaptersout = torch.bmm(hidden, adapter_a)"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.856,
+                0.377,
+                0.87
+            ],
+            "angle": 0,
+            "content": "B.2 ADDITIONAL RESULTS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.882,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "We further conducted a throughput experiment, employing a setup similar to the one described in §4.1, on the newly proposed Llama-2 foundation models (Touvron et al., 2023). The results for both the 13B and 7B models are illustrated in Fig. 3 (left). When compared against the plot in"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.491,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "15"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.104,
+                0.825,
+                0.176
+            ],
+            "angle": 0,
+            "content": "Fig. 2 (left), it is evident that the inflection rank at which LoRA begins to outperform FLORA is lower for the Llama-2 models than for the StarCoder 15B. This shows that the ratio \\( c_{1} / c_{2} \\) from Eq. (7) varies across different architectures. One plausible explanation could be that, being a recent architecture, the Llama-2 model has not yet been fully optimized within the vLLM framework, potentially impacting throughput performance."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.181,
+                0.827,
+                0.306
+            ],
+            "angle": 0,
+            "content": "The right panel of Fig. 3 presents the serving results concerning StarCoder-3B. Notably, the inflection rank at which LORA begins to outperform FLORA is lower here as compared to the one observed in StarCoder 15B, as seen in Fig. 2 (right). This is attributed to the relatively low latency on StarCoder-3B, which implies that a request rate of 8 requests per second fails to saturate the LLM server, thereby diminishing the advantage of FLORA. Upon escalating the request rate to 15 requests per second, the inflection rank correspondingly increases to around 12. This illustrates that the latency improvement exhibited in Fig. 2 (left) represents a theoretical enhancement. The actual improvement in serving depends on model architectures, request rates, and other practical considerations."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.182,
+                0.322,
+                0.49,
+                0.556
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.504,
+                0.322,
+                0.816,
+                0.556
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.17,
+                0.562,
+                0.828,
+                0.646
+            ],
+            "angle": 0,
+            "content": "Figure 3: Left: Generation throughput vs. rank for FLORA and torch.bmm implementation of LORA, measured in tokens per second (token/s). The experiments were conducted on two Llama-2 models: 13B and 7B (Touvron et al., 2023). FLORA has great throughput advantage over LORA when the rank is small. As the rank increases, the torch.bmm implementation of LORA finally has a better throughput. Right: Latency vs. rank on StarCoder-3B. Requests are coming at the speed of 8 requests per second and 15 requests per second."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.491,
+                0.948,
+                0.51,
+                0.96
+            ],
+            "angle": 0,
+            "content": "16"
+        }
+    ]
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+# BATCHED LOW-RANK ADAPTATION OF FOUNDATION MODELS
+
+Yeming Wen & Swarat Chaudhuri*
+
+Department of Computer Science
+
+The University of Texas at Austin
+
+# ABSTRACT
+
+Low-Rank Adaptation (LoRA) has recently gained attention for fine-tuning foundation models by incorporating trainable low-rank matrices, thereby reducing the number of trainable parameters. While LoRA offers numerous advantages, its applicability for real-time serving to a diverse and global user base is constrained by its incapability to handle multiple task-specific adapters efficiently. This imposes a performance bottleneck in scenarios requiring personalized, task-specific adaptations for each incoming request.
+
+To mitigate this constraint, we introduce Fast LoRA (FLORA), a framework in which each input example in a minibatch can be associated with its unique low-rank adaptation weights, allowing for efficient batching of heterogeneous requests. We empirically demonstrate that FLORA retains the performance merits of LoRA, showcasing competitive results on the MultiPL-E code generation benchmark spanning over 8 languages and a multilingual speech recognition task across 6 languages.
+
+# 1 INTRODUCTION
+
+Transformer-based foundation models have showcased remarkable performance across various natural language processing tasks, as evidenced by the successes of ChatGPT (OpenAI, 2023), GitHub Copilot (Chen et al., 2021) and Speech Recognition (Radford et al., 2022) among others. The practice of fine-tuning these models for specific domains or specialized needs, such as instruction-tuning, has become increasingly prevalent (Wang et al., 2022c; Honovich et al., 2022; Taori et al., 2023; Chiang et al., 2023). This is driven by the requirements of real-world applications, which often demand models tailored to specific domains, tasks, or even individual user preferences (Ouyang et al., 2022). However, the extensive number of parameters in foundation models poses computational and memory challenges for task-specific fine-tuning.
+
+Low-Rank Adaptation (LORA) emerged as a solution to this challenge by incorporating trainable low-rank matrices (Hu et al., 2021) which significantly reduces the number of trainable parameters during fine-tuning. LoRA's success stems from its ability to achieve domain adaptation without retraining the entire model (Taori et al., 2023; Dettmers et al., 2023; Lee et al., 2023). However, a practical challenge arises in real-time serving scenarios. Batching is the practice of aggregating multiple data points into a single computation. It is a common technique to leverage parallel processing capabilities in GPUs, ensuring higher throughput and lower serving cost. It becomes especially crucial when serving world-wide users where many requests could flood in every second. The intrinsic design of LoRA dictates that every example within a batch shares the same adapter, which is suboptimal for real-world serving scenarios where each request may require a unique adapter.
+
+Consider a scenario where users from various locations and professions demand different language and occupation adapters as illustrated in Fig. 1. With LoRA, the batch processing would either force all these diverse requests to share the same adapter or process them sequentially, both of which are impractical. These limitations emphasize the need for a solution that can not only utilize the advantages of LoRA but also serve multiple adapters in parallel, catering to the diverse and simultaneous requests encountered in reality.
+
+![](images/320d41ac7fa023cfe4d8924c9dbfc31f82c620dfe92ae5454aab3fa9f52061c6.jpg)  
+Figure 1: This shows a pragmatic scenario where a foundation model in production receives four incoming requests, each requiring distinct adapters. Omitting two adapters in step 2 & 3 for presentation simplicity. FLORA facilitates batching in such serving circumstances, provided the adapters are of low rank, thereby sustaining high throughput and low latency. Detailed discussion on vectorization is provided in §3.2.
+
+We posit that it is critical to develop a more flexible adaptation mechanism that is compatible with diverse real-world user queries. We introduce fast LORA (FLORA), a modification of LORA, enabling individual examples in a minibatch to be associated with distinct low-rank weights without compromising the expressive power. This modification promises the benefits of domain adaptation, as heralded by LORA, but without the batching limitation.
+
+Our contributions can be summarized as follows:
+
+1. We propose FLORA, a framework that augments LoRA by allowing each example in a minibatch to have its unique low-rank adapters, facilitating efficient batching.  
+2. We provided an analytical analysis describing the scenarios where FLORA would be preferred over LoRA in practical applications. This analysis is further substantiated by the empirical evidence where FLORA achieves a 2X throughput improvement on the state-of-the-art code LLM StarCoder 15B in the low-rank setting when diverse adapters are required for incoming examples. Additionally, FLORA reduces the latency by half under the same low-rank setting.  
+3. We demonstrate that FLORA does not sacrifice accuracy compared to LORA on a multilingual code generation task across 8 programming languages, and maintains accuracy in speech recognition tasks over five languages.
+
+# 2 PROBLEM FORMULATION
+
+In this section, we outline the problem tackled in this work, illustrating the constraints and objectives that drive the development of the proposed FLORA methodology. Let  $\mathcal{M}$  denote a foundation model parameterized by  $\theta$ , with a total number of parameters  $N$ . The common practice is to fine-tune this foundational model for various specific tasks, ranging from multilingual code generation to speech recognition as demonstrated in §4.2 and §4.3.
+
+# 2.1 LORA ADAPTERS
+
+Fine-tuning the entire model  $\mathcal{M}$  for a specific task is usually computationally expensive due to the massive parameter count. LoRA (Low-rank Adaptation, (Hu et al., 2021)) was introduced to facilitate domain-specific adaptations with a significantly reduced parameter footprint, with the hypothesis that low-rank adaptation is sufficient for fine-tuning domain specific foundation models.
+
+Given the pre-trained weight matrix  $W_0 \in \mathbb{R}^{d \times k}$ , LoRA posits that the weight matrix of the adapted foundation model can be expressed as  $W_0 + \Delta W = W_0 + BA$ , where  $\Delta W$  has a low-rank structure. This matrix  $\Delta W$  is factorized into two smaller, trainable matrices:  $B \in \mathbb{R}^{d \times r}$  and  $A \in \mathbb{R}^{r \times k}$ , such that  $\Delta W = BA$  where  $r$  stands for the rank. For a given input  $\mathbf{x}_i$ , the output  $\mathbf{y}_i$  is given by:
+
+$$
+\mathbf {y} _ {i} = \mathcal {M} \left(\mathbf {x} _ {i} \mid \Delta W, W _ {0}, \theta\right) \tag {1}
+$$
+
+# 2.2 Batching & THROUGHPUT
+
+Batching is a common practice where multiple data points  $(\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_m)$  are aggregated into a single batch  $\mathcal{B}$ . Consequently, the forward passes of these data points are processed concurrently rather than individually. This practice leverages the parallel processing capability of a modern GPU, thereby significantly improving the throughput  $T$ , i.e., the number of data points processed per unit of time. In the context of foundation models, throughput of a batch  $\mathcal{B}$  can be defined as  $T = \sum_{i=1}^{m} |\mathbf{y}_i| / \Delta t$ , where  $|\mathbf{y}_i|$  is the number of tokens generated for each example  $\mathbf{x}_i$  in the batch,  $\Delta t$  is the total time taken to process the batch, and  $m$  is the number of examples in the batch. Note that batching incurs minimal latency penalties. However, given its substantial increase in throughput, batching and its variants are widely used in the state-of-the-art foundation models serving framework such as vLLM (Kwon et al., 2023) to achieve the best balance between throughput and latency.
+
+# 2.3 OBJECTIVE
+
+Batching typically assumes the same model parameters are utilized for every input example within a minibatch. Hence, a straightforward application of batching in LoRA requires that the adapter matrix  $\Delta W$  be shared across all inputs in the batch  $\mathcal{B}$ . The challenge arises when considering a scenario where each input example in the batch might originate from a different task. Sharing the same  $\Delta W$  for all  $\mathbf{x}_i$  in  $\mathcal{B}$  becomes suboptimal where each input potentially demands a unique adapter. The limitation is particularly acute when the model is expected to serve a world-wide user base with diverse incoming requests.
+
+Given the limitations of LORA in batching, our objective is to maximize the throughput  $T$  in global user serving scenarios by maintaining the batching mechanism. Formally, for each  $\mathbf{x}_i \in \mathcal{B}$ , we aim to compute  $\mathbf{y}_i = \mathcal{M}(\mathbf{x}_i | \Delta W_i, W_0, \theta)$ , where  $\Delta W_i$  is the adapter matrix corresponding to the input example  $x_i$ . Therefore,  $\Delta W_i$  can be unique across  $\mathcal{B}$  and specific to a domain or user preference.
+
+# 3 FLORA: FAST LOW RANK ADAPTATION
+
+As shown in §2.3, adapter sharing is often impractical in real-world serving scenarios. The innovation of FLORA is the introduction of example-specific adapter  $\Delta W_{i}$  for each  $\mathbf{x}_i$  in a minibatch. In FLORA, the weight matrix  $W_{i}$  for each example  $\mathbf{x}_i$  in the minibatch is calculated as  $W_{i} = \Delta W_{i} \circ W_{0}$ , where  $\circ$  denotes element-wise multiplication,  $W_{0}$  is the pre-trained weight matrix and  $\Delta W_{i}$  is a low-rank adaptation specifically designed for  $\mathbf{x}_i$ . Similar to Hu et al. (2021),  $\Delta W_{i}$  is decomposed into two trainable matrices:  $B_{i} \in \mathbb{R}^{d \times r}$  and  $A_{i} \in \mathbb{R}^{r \times k}$ , such that  $\Delta W_{i} = B_{i} A_{i}$ , as shown in Fig. 1. Note that FLORA has the same expressive power as LORA by its construction.
+
+# 3.1 FORWARD PASS
+
+The advantage of FLORA is that computations on a minibatch can be written in terms of matrix multiplications. This enables efficient batched implementations on modern accelerators such as GPUs. Let  $x_{i}$  denote the activations in one layer of a neural net, which is a vertical vector of length  $d$ . The next layer's activations are given by
+
+$$
+\begin{array}{l} y _ {i} = \phi \left(W _ {i} ^ {T} x _ {i}\right) (2) \\ = \phi \left(\left(W _ {0} ^ {T} \circ \Delta W _ {i} ^ {T}\right) x _ {i}\right) (3) \\ = \phi \left(\left(W _ {0} ^ {T} \circ \left(B _ {i} A _ {i}\right) ^ {T}\right) x _ {i}\right) (4) \\ = \phi \left(A _ {i} \circ \left(W _ {0} ^ {T} \left(B _ {i} \circ x _ {i}\right)\right)\right) (5) \\ \end{array}
+$$
+
+When the rank is greater than one, we extend the use of the symbol “o” to denote potential broadcasting. Additionally, a dimension reduction operation such as torch.mean is required prior to applying the activation function  $\phi$ .
+
+The key to FLORA's flexibility lies in the low rank decomposition enables the incorporation of example-specific adapters directly into the forward pass, as demonstrated in the equations above. Crucially, each of these operations—the element-wise multiplication between  $A_{i}$  and  $x_{i}$ , and between  $B_{i}$  and  $y_{i}$  — is inherently batch-friendly. Consequently, FLORA allows for simultaneous processing of multiple requests, each requiring its own adapter, within a single minibatch. To vectorize all adapters in the minibatch, we define matrices  $\mathbf{A}$  and  $\mathbf{B}$  whose rows correspond to the adapters  $A_{i}$  and  $B_{i}$  for all examples in the minibatch. The above equation is vectorized as:
+
+$$
+\mathbf {Y} = \phi \left(\mathbf {A} \circ \left((\mathbf {B} \circ \mathbf {X}) W _ {0}\right)\right) \tag {6}
+$$
+
+# 3.2 COMPUTATIONAL EFFICIENCY
+
+The computational analysis primarily concentrates on the examination of fully connected layers within a transformer architecture, given that LoRA is specifically applied to these layers, such as query and key projections. To begin, we analyze a baseline that leverages batch matrix multiplication to facilitate the serving of LoRA with multiple adapters. This operation is possible under the assumption that every adapter required by the input examples in the minibatch shares the same shape, specifically, the same rank. The batch matrix multiplication (BMM) can be implemented using the torch.bmm operator in deep learning frameworks such as PyTorch (Paszke et al., 2019). Note that the BMM operator is typically unfavorable in practical settings due to the significant overhead it introduces (Abdelfattah et al., 2016). This overhead diminishes the throughput and increases latency, which is detrimental in serving scenarios where response times are crucial for maintaining a good user experience.
+
+Let  $b$  and  $l$  denote the batch size and the maximum sequence length in the input batch  $\mathcal{B}$ . Revisiting the notation introduced in §3, where  $W_0 \in \mathbb{R}^{d \times k}$ ,  $B_i \in \mathbb{R}^{d \times r}$  and  $A_i \in \mathbb{R}^{r \times k}$ , the operations required to compute the pre-activation for an input batch  $\mathcal{B}$  with dimensions  $[b, l, d]$  consist of one matrix multiplication and two BMMs. The matrix multiplication occurs between the input batch  $\mathbf{X}$  and the pre-trained weight  $W_0$ . The two BMM operations are conducted firstly between the input batch  $\mathbf{X}$  and  $\mathbf{B}$ , and secondly between the result of the prior computation and  $\mathbf{A}$ , where  $\mathbf{A}$  and  $\mathbf{B}$  are matrices whose rows correspond to the adapters  $A_i$  and  $B_i$  for all examples in the minibatch respectively. Assuming for simplicity that the layer neither upscales nor downscales the hidden dimension (i.e.  $d = k$ ), the upper bound complexity of this layer is discerned as  $2c_1(dblr) + c_2(bld^2)$ , with  $c_1$  and  $c_2$  representing the computational coefficients of BMM and matrix multiplication respectively. Note that  $c_1 >> c_2$  because the BMM operator is more expensive than matrix multiplication.
+
+For FLORA, the cost is one matrix multiplication which is  $c_{2}(rbld^{2})$  where  $r$  denotes the rank of the adapters. We omit the cost of element-wise multiplication in this analysis because it is negligible to the matrix multiplication cost. Comparing the computational cost of FLORA and LORA boils down to the following inequality
+
+$$
+\frac {2 c _ {1}}{d c _ {2}} + \frac {1}{r} \geq 1 \tag {7}
+$$
+
+FLORA exhibits a lower computational cost than bmm LoRA whenever the above inequality holds true. The benefit of FLORA over LoRA is notably pronounced when  $r = 1$ . As the rank increases,
+
+LORA gradually becomes less costly. From the established inequality, a variety of scenarios can be inferred where FLORA has an advantage over LORA. Firstly, the advantage of FLORA is significantly apparent when the rank of adapters is small. Secondly, in configurations where the model has fewer hidden units but an increased number of layers, FLORA tends to outperform LORA due to the smaller value of  $d$  in the denominator of Eq. (7).
+
+Another advantage of FLORA is the cost remains invariant to the number of adapters required by the input batch. While the preceding analysis assumes that every token in an example  $x_{i}$  shares the same adapter, it is possible to apply multiple adapters to a single example by dividing the example into chunks, and then applying different adapters to each chunk. This approach is commonly observed in the Mixture of Experts framework (Fedus et al., 2021; Lepikhin et al., 2020; Puigcerver et al., 2023). Incorporating several adapters in an input example notably amplifies the ratio  $c_{1} / c_{2}$  in Eq. (7), thereby significantly increasing LORA's cost.
+
+The ratio  $c_{1} / c_{2}$  might not be the same across different transformer architectures. §4.1 is designed to provide a deeper insight into how comparative serving efficiency of FLORA and LORA changes under various architectures. Additionally, it's worth noting that LORA does not apply to the self-attention layers, which constitute a non-trivial portion of the computational cost, thereby overshadowing the advantage of FLORA. However, as efficient self-attention mechanisms such as flash attention (Dao et al., 2022) get adopted, the advantage of FLORA over LORA is likely to get larger.
+
+Connection to IA3. IA3 was proposed in Liu et al. (2022a) featuring fast adaptation of LLM. It introduces a learned vector  $l$  which re-scales the activation by  $y_{i} = l\circ \phi (W_{0}^{T}x_{i}) = \phi \bigl (l\circ (W_{0}^{T}x_{i})\bigr)$ . This can be viewed as a special case of FLORA - a rank 0.5 variant - which only re-scales the columns instead of the entire pre-trained weights. It has a limited expressive power compared to FLORA and LoRA.
+
+# 4 EXPERIMENTS
+
+In this section, we compare FLORA to LoRA and other notable baselines across various metrics and tasks. To begin with, we delve into a computational analysis to substantiate the enhanced throughput and the reduced latency achieved by FLORA in the case of low rank. Subsequently, we pivot towards analyzing the accuracy of FLORA in multilingual code generation tasks spanning across different languages. The goal is to discern the proficiency of FLORA in maintaining, if not enhancing, the model's accuracy as compared to LoRA. Progressing further, we replicate a similar analysis but in the domain of multilingual speech recognition.
+
+# 4.1 SERVING ANALYSIS
+
+The primary objective of this serving analysis is to measure the maximum throughput both FLORA and LoRA can attain under varied rank configurations. We carried out this exploration on the state-of-the-art code Large Language Model (LLM) StarCoder (Li et al., 2023), evaluating models of different number of parameters namely 1B, 3B, and 15B. The dataset facilitating this analysis has been sourced from the vLLM throughput benchmark. Noteworthy, this dataset was previously used to fine-tune the English Vicuna model, a state-of-the-art chat LLMs (Chiang et al., 2023). To expedite the benchmarking process, we extracted a subset of 1,000 samples from the original dataset, ensuring a diverse range of sample lengths varying from 50 to 2,000 tokens.
+
+In setting up the computational analysis, our primary intent is to compare FLORA and LoRA in the real-world serving scenario. See Appendix B.1 on how bmm LoRA is implemented. The vLLM framework (Kwon et al., 2023) $^{2}$ , with its implementation of continuous batching, presents an ideal setup for this analysis. The continuous batching mechanism in vLLM, as inspired by the principles delineated in Orca (Yu & Jeong, 2022), facilitates a more efficient utilization of GPU resource by allowing new sequence to be inserted immediately once any sequence in the current batch is completed. This continuous flow significantly enhances GPU utilization as compared to static batching, where the GPU awaits the completion of all sequences in a batch before initiating a new batch processing. The comparison of FLORA and LoRA within this setup offers a compelling evidence of
+
+![](images/ed93ba9b562aa6fa23d4c29067f423da49c97808341cacf67371ac664858c667.jpg)  
+Figure 2: Left: Generation throughput vs. rank for FLORA and torch.bmm implementation of LORA, measured in tokens per second (token/s). The experiments were conducted on three star-coder models: StarCoder 15B, StarCoderbase 3B and StarCoderbase 1B. FLORA has great throughput advantage over LoRA when the rank is small. As the rank increases, the torch.bmm implementation of LoRA finally has a better throughput. Right: Latency vs. rank on StarCoder-15B. Requests are coming at the speed of 8 requests per second.
+
+![](images/3c18fccff2618301be1b089888cfaea229f6dbe29568ccba3a1feab2c51d877a.jpg)
+
+their respective throughput and latency in the real-world serving scenario. It is worth noting that the experiments were conducted without Flash-attention (Dao et al., 2022).
+
+Throughput experiment. In Fig. 2, the throughput results for both FLORA and bmm LORA are illustrated across different rank configurations on three StarCoder models with different number of parameters, namely 1B, 3B and 15B. All experiments were conducted on an NVIDIA H100 GPU with a float16 precision<sup>3</sup>. The maximum number of batched tokens is 8,192, which is the same as the model context length. Evidently, FLORA shows a superior throughput over LORA in the lower rank configurations. At rank 1, the throughput of FLORA is more than threefold higher than that of LORA, thereby highlighting the considerable serving performance boost FLORA provides. The advantage of FLORA continues as the rank increases, albeit with a diminishing rate. For instance, at rank 2, FLORA's throughput is around 2.5 times higher, and this multiplicative advantage decreases as the rank increases further. This performance advantage continues up until the rank of 8 in the StarCoder 15B model, where LoRA starts to outperform FLORA. This inflection point suggests that the advantages of FLORA in terms of throughput are more pronounced in lower rank.
+
+Notice that the inflection points occurred at a higher rank when serving a smaller LLM as illustrated in (Fig. 2, left). This demonstrates a significant potential of FLORA, especially when considering future applications of quantization techniques to serving LLMs. By applying quantization, such as 8-bit or 4-bit inference, the effective size of the model is reduced, akin to serving a smaller LLM thus potentially extending the rank at which FLORA maintains a throughput advantage over LORA.
+
+Latency experiment. We assessed the latency-versus-rank performance on the Starcoder 15B model, using the same dataset as the throughput experiment. This evaluation was conducted under the conditions where requests arrive at a rate of 8 per second. The default maximum number of batched tokens in vLLM serving launcher is 2,560. The results, as shown in (Fig. 2, right), measure latency in terms of seconds per output token. Remarkably, in the lower rank regime (ranging from rank 1 to rank 4), FLORA exhibited a 2-5X reduction in latency compared to LoRA. Notably, the latency for LoRA stood at approximately 2.3 seconds per output token, is impractical for serving due to the poor user experience it would cause. This experiment further highlights the superior capabilities of FLORA in efficiently catering to diverse incoming user requests.
+
+These findings validate the theoretical analysis in §4.1, confirming that FLORA provides significant throughput advantages, particularly in settings with lower to moderate ranks. This positions FLORA
+
+<table><tr><td rowspan="2">Language</td><td rowspan="2">Base Model</td><td colspan="3">pass@1 (Relative Improvement)</td></tr><tr><td>FLORA</td><td>IA3</td><td>LoRA</td></tr><tr><td></td><td colspan="4">StarCoder 15B</td></tr><tr><td>Dlang</td><td>14.13</td><td>17.26 (22.14%)</td><td>15.26 (7.99%)</td><td>17.15 (21.37%)</td></tr><tr><td>Perl</td><td>17.05</td><td>21.44 (25.76%)</td><td>17.71 (3.90%)</td><td>21.46 (25.76%)</td></tr><tr><td>Ruby</td><td>1.39</td><td>24.94 (1692.86%)</td><td>20.80 (1394.64%)</td><td>23.76 (1608.04%)</td></tr><tr><td>Rust</td><td>22.40</td><td>26.24 (17.14%)</td><td>23.53 (5.04%)</td><td>26.87 (19.95%)</td></tr><tr><td>Racket</td><td>10.20</td><td>12.41 (21.61%)</td><td>11.53 (12.96%)</td><td>12.51 (22.58%)</td></tr><tr><td>Swift</td><td>16.91</td><td>20.38 (20.51%)</td><td>18.13 (7.19%)</td><td>20.35 (20.36%)</td></tr><tr><td></td><td colspan="4">StarCoderBase 3B</td></tr><tr><td>Dlang</td><td>5.65</td><td>5.72 (1.20%)</td><td>5.72 (1.20%)</td><td>6.97 (23.34%)</td></tr><tr><td>Perl</td><td>10.73</td><td>13.01 (21.25%)</td><td>11.46 (6.83%)</td><td>13.31 (27.51%)</td></tr><tr><td>Ruby</td><td>5.33</td><td>14.48 (171.68%)</td><td>7.88 (47.90%)</td><td>13.89 (160.68%)</td></tr><tr><td>Rust</td><td>17.18</td><td>21.00 (22.24%)</td><td>17.28 (0.60%)</td><td>20.67 (20.31%)</td></tr><tr><td>Racket</td><td>8.04</td><td>9.16 (13.99%)</td><td>8.40 (4.48%)</td><td>8.80 (9.48%)</td></tr><tr><td>Swift</td><td>10.04</td><td>15.69 (56.21%)</td><td>12.54 (24.83%)</td><td>15.04 (49.76%)</td></tr></table>
+
+Table 1: Comparison of three fine-tuning methods FLORA, IA3, and LORA on StarCoder 15B and StarCoderBase 3B across various low-resource programming languages in the MultiPL-E benchmark. The table presents Pass@1 accuracy of each method alongside the relative improvement over the baseline. The standard errors are less than  $0.3\%$  in all cells in the table, therefore we exclude that for clear presentation.
+
+as a compelling alternative for efficiently serving adapted foundation models, especially in scenarios where lower ranks suffice the desired model accuracy, as further demonstrated in the subsequent accuracy analysis sections. Moreover, if an enterprise chooses to serve foundation models with a substantial number of diverse adapters, for instance, a personalized LLM, then a low rank or even a rank one is imperative to avoid excessive storage costs.
+
+# 4.2 MULTILINGUAL CODE GENERATION
+
+A key aspect to examine before applying FLORA in real-world LLM serving is to scrutinize any potential degradation in performance. In this section, we consider multilingual code generation as the testbed for comparing FLORA and LoRA due to its alignment with real-world applications, where the necessity to cater to diverse programming languages is paramount. Low-resource languages, as referred to in this context, are languages that appear much less frequently than other languages in the pre-training data. Orlanski et al. (2023) showed that the performance of code LLMs can be notably enhanced on low-resource programming languages such as PERL by recalibrating the language distribution in the pre-training data. This suggests that fine-tuning a trained LLM on such low-resource languages could potentially boost its performance on the same language. Hence, by employing multilingual code generation as a benchmark, we can make an informed evaluation of adaptability and performance enhancements that FLORA and LoRA can achieve.
+
+Additionally, a comparison is made against a third baseline, IA3, which can be considered as a special case of FLORA. Essentially, IA3 can be seen as a rank 0.5 variant of FLORA, thereby facing a constrained expressive power in comparison to both FLORA and LoRA. For FLORA and LoRA, we conducted fine-tuning across a range of rank choices, spanning from 1 to 8. It emerges that within the scope of this multilingual code generation task, a rank of one typically suffices to achieve optimal results, with the exception of Racket and Lua. Consequently, the results shown in §4.2 are predominantly at rank 1, barring Racket and Lua, which are presented at rank 4.
+
+Fine-tuning. In our effort to evaluate the performance of FLORA, LORA, and IA3 on the multilingual code generation task, we fine-tuned these models on state-of-the-art multilingual code LLMs, StarCoder 15B and StarCoderBase 3B as introduced in (Li et al., 2023). A pivotal aspect of our fine-tuning process was the utilization of existing data, negating the need for creating new data for
+
+<table><tr><td>Model</td><td>Arabic</td><td>Czech</td><td>Lithuanian</td><td>Marathi</td><td>Mongolian</td><td>Hindi</td></tr><tr><td>Whisper</td><td>46.03</td><td>23.19</td><td>46.09</td><td>84.84</td><td>115.20</td><td>43.41</td></tr><tr><td>FLORA</td><td>30.21 ± 0.25</td><td>10.76 ± 0.22</td><td>20.39 ± 0.32</td><td>33.17 ± 0.17</td><td>42.12 ± 0.63</td><td>23.58 ± 0.65</td></tr><tr><td>IA3</td><td>30.48 ± 0.22</td><td>11.61 ± 0.33</td><td>21.41 ± 0.20</td><td>35.03 ± 0.61</td><td>46.47 ± 0.53</td><td>25.64 ± 0.78</td></tr><tr><td>LORA</td><td>30.18 ± 0.21</td><td>10.77 ± 0.38</td><td>20.50 ± 0.21</td><td>31.94 ± 0.33</td><td>41.57 ± 0.69</td><td>23.82 ± 0.74</td></tr></table>
+
+Table 2: Mean WER with standard deviation for different models and languages. The WER is calculated with respect to the unnormalized tokenizer. The base model here is Whisper-1.5B.
+
+low-resource programming languages. We leveraged the same pre-training data that was used for pre-training StarCoder, specifically, the Stack dataset, which contains over 6TB of permissively-licensed source code files covering 358 programming languages. For each low-resource language in our experiment, we fine-tuned on its corresponding split from the Stack dataset for a total of 1,500 steps, along with batch size 8. More fine-tuning details are given in Appendix A.
+
+Evaluation. The evaluation of FLORA, LORA, and IA3 was conducted on the MultiPL-E benchmark (Cassano et al., 2023), which contains the translation of two unit-test driven Python benchmarks (HumanEval and MBPP) to 18 other programming languages. We used the HumanEval split of the benchmark to evaluate the fine-tuned models. As for the metrics, We adopted the  $pass@k$  metrics from Chen et al. (2021); Austin et al. (2021), which is calculated as the fraction of problems with at least one correct sample given  $k$  samples. Similar to Chen et al. (2021), we drew 100 samples for computing  $pass@1$ , with a sampling temperature set at 0.1.
+
+Main results. The result in §4.2 exhibits the performance of three methods across various programming languages on both StarCoder 15B and StarCoderBase 3B models. The average relative improvement achieved by FLORA and LoRA is roughly  $20\%$  in the selected low-resource programming languages. FLORA consistently outperforms IA3 on all languages, especially on StarCoder 15B, denoting its efficiency in leveraging the model expressive power to improve multilingual code generation. It is notable that StarCoder 15B has an unforeseen issue regarding Ruby generation, where it yields lower accuracy compared to the 3B model. However, all methods are able to fix its abnormal performance.
+
+On StarCoderBase 3B, a smaller model, it is evident that the baseline performance drops, yet FLoRA and LoRA still manage to exhibit substantial relative improvement over the baseline, especially in languages like Swift and Ruby. This suggests that both methods benefit from continuous training on the low-resource language split of the pre-training data, although the advantages may diminish with a reduction in model size. While the absolute performance (pass@1 accuracy) varies among languages, the relative improvements highlight the effectiveness of the tested methods in enhancing multilingual code generation.
+
+# 4.3 MULTILINGUAL SPEECH RECOGNITION
+
+Following our analysis in multilingual code generation, we shift our focus to another substantial application of large language models — multilingual speech recognition (MSR). This domain has seen increasing demand due to the necessity of serving various linguistic interfaces. The capabilities of FLORA and LORA to adapt to multiple languages in code generation presents a tempting premise to investigate their effectiveness in the MSR domain.
+
+Fine-tuning. A crucial aspect of our analysis involves fine-tuning the foundation model on a multilingual speech recognition dataset. We selected the Whisper large 1.5B (Radford et al., 2022) as our base model to conduct the subsequent analysis. Whisper is an encoder-decoder transformer model trained on 680k hours of labeled speech data through large-scale weak supervision. Despite being pre-trained on a multilingual speech corpus, the model's predictive capabilities can be improved further for certain languages and tasks through fine-tuning. We use the Common Voice benchmark (Ardila et al., 2020) — containing a total of 38 languages and 2,500 hours of collected audio — for the fine-tuning process, with a particular focus on low-resource languages. For each low-resource language enumerated in §4.2, we fine-tuned on its training split within the Common
+
+Voice dataset for approximately 5,000 steps, with a batch size of 6. More fine-tuning details are given in Appendix A. Similar to §4.2, we found that rank one is sufficient to achieve the best result in most cases, with the exception of Marathi, which requires a rank of 4.
+
+Main results. We present the main results on the MSR task in §4.2. Notice that the evaluation is conducted with the unnormalized tokenizer.<sup>5</sup> The Whisper model exhibited a relatively higher Word Error Rate (WER) across all low-resource languages when compared to FLORA, LORA and IA3 models, reflecting a significant room for enhancement in its multilingual speech recognition capabilities. Particularly, its performance was found to be subpar for the Marathi and Mongolian languages, with WERs of 84.84 and 115.20 respectively. All examined models significantly outperform the base Whisper model by a wide margin across all languages, highlighting the effective fine-tuning of FLORA, LORA and IA3 models. For instance, in Arabic, FLORA reduces the WER to 30.21 from Whisper's 46.03, showcasing a significant enhancement in speech recognition accuracy. Particularly, FLORA and LORA consistently outperform IA3, indicating a better expressive power for multilingual speech recognition tasks. For example, in Czech, FLORA has a mean WER of 10.76, which is slightly better compared to IA3's 11.61 and LORA's 11.47. A similar trend is observed in Lithuanian and Hindi. In summary, this table shows the superior performance of FLORA and LORA over IA3 in the speech recognition task across diverse languages. The consistent low WERs attained by FLORA demonstrates its potential as a viable model for MSR tasks.
+
+# 5 RELATED WORK
+
+Parameter-Efficient Fine-Tuning (PEFT) methods are broadly partitioned into two categories: weight-based and non-weight-based approaches. MC-dropout (Lakshminarayanan et al., 2016) stands as an early example for the non-weight-based approach, where distinct dropout masks are allocated to various tasks. Recently, prompt tuning techniques have emerged as a prevalent stream within this category (Li & Liang, 2021; Lester et al., 2021), facilitating efficient adaptation with minimal modifications to models. Successive endeavors aimed to enhance this class of methods, delving into aspects such as optimization (Mao et al., 2021; Diao et al., 2022), transferability (Wang et al., 2021; Vu et al., 2021; He et al., 2022b), and the usage of discrete prompts (Schick & Schütze, 2020a;b; Gao et al., 2021; Malkin et al., 2021), among others (Liu et al., 2022b; 2021).
+
+We focus on weight-based approaches in this work, which has a weight interpretation as exemplified by LoRA (Hu et al., 2021). This line of research can be traced back to Progressive Network (Rusu et al., 2016), which inserts a sub-network when a new task arrives. This principle was later adapted widely in foundation models as represented by adapter based methods (Houlsby et al., 2019; Mahabadi et al., 2021; Davison, 2021; Ding et al., 2022; Wang et al., 2022b). In particular, BitFit (BenZaken et al., 2021) was introduced to solely update the bias parameters, while IA3 (Liu et al., 2022a) was proposed to rescale the activations. Additionally, approaches such Fish (Sung et al., 2021) and Diff-pruning (Guo et al., 2020) leverage sparsity to facilitate efficient adaptation of foundation models. A separate vein of research aims to improve LoRA by reducing its computational and memory costs (Zhang et al., 2023b;a; Malladi et al., 2023). He et al. (2022a) explored how to unify different PEFT methods. Dettmers et al. (2023) quantized LoRA to reduce memory footprint. Chavan et al. (2023) generalized LoRA by learning individual adapters in each layer. Several other works focus on building mixture of adapters (Wang et al., 2022a; Diao et al., 2023).
+
+# 6 CONCLUSION
+
+We introduced FLORA, an extension of LORA, facilitating efficient batching. Empirical evaluations demonstrated that FLORA enhances throughput and latency in practical serving scenarios, all while preserving the accuracy of LORA. Through FLORA, we aim to facilitate a more efficient adaptation of large language models to diverse and real-world user requests.
+
+Limitations. Despite its parameter efficiency, FLORA still requires fine-tuning. A promising future work could be to derive FLORA weights from a trained LoRA model, given that LORA remains
+
+the most prevalent type of adapter as per (Huang et al., 2023). This adaptation could potentially obviate the requirement for fine-tuning, thereby accelerating the process of model adaptation.
+
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+Yizhong Wang, Yeganeh Kordi, Swaroop Mishra, Alisa Liu, Noah A. Smith, Daniel Khashabi, and Hannaneh Hajishirzi. Self-instruct: Aligning language model with self generated instructions. ArXiv, abs/2212.10560, 2022c.  
+Gyeong-In Yu and Joo Seong Jeong. Orca: A distributed serving system for transformer-based generative models. In USENIX Symposium on Operating Systems Design and Implementation, 2022. URL https://api-semanticscholar.org/CorpusID:251734964.  
+Longteng Zhang, Lin Zhang, Shaohuai Shi, Xiaowen Chu, and Bo Li. Lora-fa: Memory-efficient low-rank adaptation for large language models fine-tuning. ArXiv, abs/2308.03303, 2023a. URL https://api-semanticscholar.org/CorpusID:260683267.  
+Qingru Zhang, Minshuo Chen, Alexander W. Bukharin, Pengcheng He, Yu Cheng, Weizhu Chen, and Tuo Zhao. Adaptive budget allocation for parameter-efficient fine-tuning. ArXiv, abs/2303.10512, 2023b. URL https://api_semanticscholar.org/CorpusID: 257631760.
+
+# A ADDITIONAL TRAINING DETAILS
+
+We delve into additional details regarding the models used in §4.2. We conducted fine-tuning on two StarCoder models (Li et al., 2023), specifically the 15B and 3B models. For both models, we employed LoRA (Hu et al., 2021) on the default layers within the Parameter Efficient Fine-Tuning (PEFT) library $^6$ , targeting only the key and query projections in the self-attention layers. This incurs roughly  $0.03\%$  number of training parameters in the rank 1 setting. For a fair comparison, we applied FLORA to the same layers where LoRA was applied. Regarding the other baseline IA3 (Liu et al., 2022a), we executed the fine-tuning in two distinct settings: one with the default configuration in PEFT, applying IA3 to the key projection layers and the first fully-connected layer in the feedforward layers; and another aligning with the LoRA setting, restricting application to the key and query projection layers. The latter setup proved to be less effective than the default setting, thus the results presented in §4.2 adhere to the default configuration.
+
+Despite StarCoder's pre-training on the Stack dataset, which contains over 6TB of source code files covering 358 programming languages, we still fine-tuned all methods on this dataset for the low-resource programming languages. This eliminates the time-consuming process to create new data. Our observations revealed that both IA3 and FLORA require a higher learning rate compared to
+
+<table><tr><td rowspan="2">Language</td><td rowspan="2">Base Model</td><td colspan="3">pass@1 (Relative Improvement)</td></tr><tr><td>FLORA</td><td>IA3</td><td>LORA</td></tr><tr><td></td><td colspan="4">StarCoder 15B</td></tr><tr><td>Julia</td><td>23.31</td><td>26.56 (13.92%)</td><td>23.90 (2.51%)</td><td>25.62 (10.45%)</td></tr><tr><td>Lua</td><td>26.60</td><td>30.20 (13.57%)</td><td>28.40 (6.77%)</td><td>29.04 (9.18%)</td></tr><tr><td>R</td><td>16.07</td><td>17.55 (9.24%)</td><td>16.65 (3.59%)</td><td>18.60 (15.8%)</td></tr><tr><td></td><td colspan="4">StarCoderBase 3B</td></tr><tr><td>Julia</td><td>15.62</td><td>19.11 (22.39%)</td><td>16.72 (7.05%)</td><td>19.77 (26.59%)</td></tr><tr><td>Lua</td><td>17.63</td><td>19.46 (10.39%)</td><td>18.04 (2.33%)</td><td>20.23 (14.75%)</td></tr><tr><td>R</td><td>10.00</td><td>10.70 (7.02%)</td><td>9.98 (-0.25%)</td><td>10.06 (0.62%)</td></tr></table>
+
+Table 3: Comparison of three fine-tuning methods FLORA, IA3, and LoRA on StarCoder 15B and StarCoderBase 3B across additional low-resource programming languages in the MultiPL-E benchmark.
+
+LORA. For LORA, a learning rate of  $1\mathrm{e} - 4$  sufficed, whereas IA3 required  $5\mathrm{e} - 3$ , and FLORA demanded an even higher rate of  $8\mathrm{e} - 3$ . We conjecture that this stems from FLORA and IA3 incorporating multiplicative weight perturbation, which typically requires a larger learning rate. All models were fine-tuned using 8 bit quantization featuring lower training memory cost.
+
+We present results for additional low-resource languages in Appendix A. The improvements provided by PEFT methods are less pronounced in these languages compared to those documented in §4.2. Nevertheless, it can be observed that LORA and FLORA still outperform IA3, despite the limited room for enhancement in these languages.
+
+Next, we discuss further details concerning the models used in §4.3. We performed fine-tuning on the Whisper large model (Radford et al., 2022), which consists of 1.5 billion parameters. Unlike the code generation task, we discovered that using the default modules in the PEFT library does not yield the best performance. Instead, we applied FLORA, LoRA, and IA3 to every fully-connected layer in the Whisper transformer. We fine-tuned all methods on the Common Voice benchmark (Ardila et al., 2020) for each low-resource language showcased in §4.2. We continued to employ larger learning rates for FLORA and IA3. Specifically, we maintained a learning rate of  $2\mathrm{e} - 4$  for LORA, while using  $4\mathrm{e} - 3$  and  $6\mathrm{e} - 3$  for IA3 and FLORA respectively. Additionally, we utilized 8-bit quantization during training to minimize the GPU memory requirement.
+
+# B ADDITIONAL SERVING RESULTS
+
+# B.1 BMM IMPLEMENTATION
+
+We provide further details concerning the experiments in §4.1. To begin, we illustrate the implementation of "torch.bmm" as follows. Upon computing the adaptersout, it can be added to the output of the standard LLM layer. This method facilitates the computation of diverse adapters' outputs within a batch.
+
+```julia
+inputs = torch Randn(batch, seq_length, d_model)  
+adapter_b # shape of (batch, d_model, rank)  
+adapter_a # shape of (batch, rank, d_model)  
+hidden = torch.bmm(inputs, adapter_b)  
+adaptersout = torch.bmm(hidden, adapter_a)
+```
+
+# B.2 ADDITIONAL RESULTS
+
+We further conducted a throughput experiment, employing a setup similar to the one described in §4.1, on the newly proposed Llama-2 foundation models (Touvron et al., 2023). The results for both the 13B and 7B models are illustrated in Fig. 3 (left). When compared against the plot in
+
+Fig. 2 (left), it is evident that the inflection rank at which LoRA begins to outperform FLORA is lower for the Llama-2 models than for the StarCoder 15B. This shows that the ratio  $c_{1} / c_{2}$  from Eq. (7) varies across different architectures. One plausible explanation could be that, being a recent architecture, the Llama-2 model has not yet been fully optimized within the vLLM framework, potentially impacting throughput performance.
+
+The right panel of Fig. 3 presents the serving results concerning StarCoder-3B. Notably, the inflection rank at which LORA begins to outperform FLORA is lower here as compared to the one observed in StarCoder 15B, as seen in Fig. 2 (right). This is attributed to the relatively low latency on StarCoder-3B, which implies that a request rate of 8 requests per second fails to saturate the LLM server, thereby diminishing the advantage of FLORA. Upon escalating the request rate to 15 requests per second, the inflection rank correspondingly increases to around 12. This illustrates that the latency improvement exhibited in Fig. 2 (left) represents a theoretical enhancement. The actual improvement in serving depends on model architectures, request rates, and other practical considerations.
+
+![](images/1aeccb6df31c6f2249796b5ddbff36f5f1fddab42a39da232b9d477586f581d6.jpg)  
+Figure 3: Left: Generation throughput vs. rank for FLORA and torch.bmm implementation of LORA, measured in tokens per second (token/s). The experiments were conducted on two Llama-2 models: 13B and 7B (Touvron et al., 2023). FLORA has great throughput advantage over LORA when the rank is small. As the rank increases, the torch.bmm implementation of LORA finally has a better throughput. Right: Latency vs. rank on StarCoder-3B. Requests are coming at the speed of 8 requests per second and 15 requests per second.
+
+![](images/8fbba7a8dc39dc3f748d891d8ece27df4f5a63831bf343d85289781bbe3847d7.jpg)
\ No newline at end of file
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+                                    "content": "BATCHED LOW-RANK ADAPTATION OF FOUNDATION MODELS"
+                                }
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+                        }
+                    ],
+                    "index": 1
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+                                    "type": "text",
+                                    "content": "Yeming Wen & Swarat Chaudhuri*"
+                                }
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+                                    "type": "text",
+                                    "content": "ABSTRACT"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Low-Rank Adaptation (LoRA) has recently gained attention for fine-tuning foundation models by incorporating trainable low-rank matrices, thereby reducing the number of trainable parameters. While LoRA offers numerous advantages, its applicability for real-time serving to a diverse and global user base is constrained by its incapability to handle multiple task-specific adapters efficiently. This imposes a performance bottleneck in scenarios requiring personalized, task-specific adaptations for each incoming request."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
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+                                    "type": "text",
+                                    "content": "To mitigate this constraint, we introduce Fast LoRA (FLORA), a framework in which each input example in a minibatch can be associated with its unique low-rank adaptation weights, allowing for efficient batching of heterogeneous requests. We empirically demonstrate that FLORA retains the performance merits of LoRA, showcasing competitive results on the MultiPL-E code generation benchmark spanning over 8 languages and a multilingual speech recognition task across 6 languages."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
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+                                    "type": "text",
+                                    "content": "1 INTRODUCTION"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
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+                                    "content": "Transformer-based foundation models have showcased remarkable performance across various natural language processing tasks, as evidenced by the successes of ChatGPT (OpenAI, 2023), GitHub Copilot (Chen et al., 2021) and Speech Recognition (Radford et al., 2022) among others. The practice of fine-tuning these models for specific domains or specialized needs, such as instruction-tuning, has become increasingly prevalent (Wang et al., 2022c; Honovich et al., 2022; Taori et al., 2023; Chiang et al., 2023). This is driven by the requirements of real-world applications, which often demand models tailored to specific domains, tasks, or even individual user preferences (Ouyang et al., 2022). However, the extensive number of parameters in foundation models poses computational and memory challenges for task-specific fine-tuning."
+                                }
+                            ]
+                        }
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+                                    "content": "Low-Rank Adaptation (LORA) emerged as a solution to this challenge by incorporating trainable low-rank matrices (Hu et al., 2021) which significantly reduces the number of trainable parameters during fine-tuning. LoRA's success stems from its ability to achieve domain adaptation without retraining the entire model (Taori et al., 2023; Dettmers et al., 2023; Lee et al., 2023). However, a practical challenge arises in real-time serving scenarios. Batching is the practice of aggregating multiple data points into a single computation. It is a common technique to leverage parallel processing capabilities in GPUs, ensuring higher throughput and lower serving cost. It becomes especially crucial when serving world-wide users where many requests could flood in every second. The intrinsic design of LoRA dictates that every example within a batch shares the same adapter, which is suboptimal for real-world serving scenarios where each request may require a unique adapter."
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+                                    "content": "Consider a scenario where users from various locations and professions demand different language and occupation adapters as illustrated in Fig. 1. With LoRA, the batch processing would either force all these diverse requests to share the same adapter or process them sequentially, both of which are impractical. These limitations emphasize the need for a solution that can not only utilize the advantages of LoRA but also serve multiple adapters in parallel, catering to the diverse and simultaneous requests encountered in reality."
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+                                    "content": "We posit that it is critical to develop a more flexible adaptation mechanism that is compatible with diverse real-world user queries. We introduce fast LORA (FLORA), a modification of LORA, enabling individual examples in a minibatch to be associated with distinct low-rank weights without compromising the expressive power. This modification promises the benefits of domain adaptation, as heralded by LORA, but without the batching limitation."
+                                }
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+                                    "content": "Our contributions can be summarized as follows:"
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+                    "index": 8,
+                    "blocks": [
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+                            "bbox": [
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+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": "1. We propose FLORA, a framework that augments LoRA by allowing each example in a minibatch to have its unique low-rank adapters, facilitating efficient batching."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 5
+                        },
+                        {
+                            "bbox": [
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+                            "type": "text",
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+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": "2. We provided an analytical analysis describing the scenarios where FLORA would be preferred over LoRA in practical applications. This analysis is further substantiated by the empirical evidence where FLORA achieves a 2X throughput improvement on the state-of-the-art code LLM StarCoder 15B in the low-rank setting when diverse adapters are required for incoming examples. Additionally, FLORA reduces the latency by half under the same low-rank setting."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 6
+                        },
+                        {
+                            "bbox": [
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+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": "3. We demonstrate that FLORA does not sacrifice accuracy compared to LORA on a multilingual code generation task across 8 programming languages, and maintains accuracy in speech recognition tasks over five languages."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 7
+                        }
+                    ],
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+                                    "type": "text",
+                                    "content": "2 PROBLEM FORMULATION"
+                                }
+                            ]
+                        }
+                    ],
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+                },
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+                                    "type": "text",
+                                    "content": "In this section, we outline the problem tackled in this work, illustrating the constraints and objectives that drive the development of the proposed FLORA methodology. Let "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{M}"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " denote a foundation model parameterized by "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "\\theta"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ", with a total number of parameters "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "N"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ". The common practice is to fine-tune this foundational model for various specific tasks, ranging from multilingual code generation to speech recognition as demonstrated in §4.2 and §4.3."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                }
+            ],
+            "discarded_blocks": [
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+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
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+                                    "type": "text",
+                                    "content": "2"
+                                }
+                            ]
+                        }
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+                }
+            ],
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+                                    "type": "text",
+                                    "content": "2.1 LORA ADAPTERS"
+                                }
+                            ]
+                        }
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+                                    "type": "text",
+                                    "content": "Fine-tuning the entire model "
+                                },
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{M}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " for a specific task is usually computationally expensive due to the massive parameter count. LoRA (Low-rank Adaptation, (Hu et al., 2021)) was introduced to facilitate domain-specific adaptations with a significantly reduced parameter footprint, with the hypothesis that low-rank adaptation is sufficient for fine-tuning domain specific foundation models."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
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+                    ],
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+                    "angle": 0,
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Given the pre-trained weight matrix "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "W_0 \\in \\mathbb{R}^{d \\times k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        152,
+                                        505,
+                                        198
+                                    ],
+                                    "type": "text",
+                                    "content": ", LoRA posits that the weight matrix of the adapted foundation model can be expressed as "
+                                },
+                                {
+                                    "bbox": [
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+                                        198
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+                                    "type": "inline_equation",
+                                    "content": "W_0 + \\Delta W = W_0 + BA"
+                                },
+                                {
+                                    "bbox": [
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+                                        198
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+                                    "type": "text",
+                                    "content": ", where "
+                                },
+                                {
+                                    "bbox": [
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+                                        198
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+                                    "type": "inline_equation",
+                                    "content": "\\Delta W"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " has a low-rank structure. This matrix "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta W"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " is factorized into two smaller, trainable matrices: "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "B \\in \\mathbb{R}^{d \\times r}"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " and "
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+                                {
+                                    "bbox": [
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+                                        198
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A \\in \\mathbb{R}^{r \\times k}"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ", such that "
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+                                {
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+                                        198
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+                                    "type": "inline_equation",
+                                    "content": "\\Delta W = BA"
+                                },
+                                {
+                                    "bbox": [
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+                                        505,
+                                        198
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
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+                                        152,
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+                                        198
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r"
+                                },
+                                {
+                                    "bbox": [
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+                                        198
+                                    ],
+                                    "type": "text",
+                                    "content": " stands for the rank. For a given input "
+                                },
+                                {
+                                    "bbox": [
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+                                        198
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_i"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ", the output "
+                                },
+                                {
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{y}_i"
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": " is given by:"
+                                }
+                            ]
+                        }
+                    ],
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+                },
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {y} _ {i} = \\mathcal {M} \\left(\\mathbf {x} _ {i} \\mid \\Delta W, W _ {0}, \\theta\\right) \\tag {1}",
+                                    "image_path": "720db5d4e3638dee3f9508d636eb2780b0c893fba1c571d705798143a3c5585c.jpg"
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+                                    "type": "text",
+                                    "content": "2.2 Batching & THROUGHPUT"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
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+                                    "type": "text",
+                                    "content": "Batching is a common practice where multiple data points "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\mathbf{x}_1, \\mathbf{x}_2, \\ldots, \\mathbf{x}_m)"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " are aggregated into a single batch "
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+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{B}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ". Consequently, the forward passes of these data points are processed concurrently rather than individually. This practice leverages the parallel processing capability of a modern GPU, thereby significantly improving the throughput "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "T"
+                                },
+                                {
+                                    "bbox": [
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+                                        373
+                                    ],
+                                    "type": "text",
+                                    "content": ", i.e., the number of data points processed per unit of time. In the context of foundation models, throughput of a batch "
+                                },
+                                {
+                                    "bbox": [
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+                                        261,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{B}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " can be defined as "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T = \\sum_{i=1}^{m} |\\mathbf{y}_i| / \\Delta t"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        261,
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+                                    "type": "text",
+                                    "content": ", where "
+                                },
+                                {
+                                    "bbox": [
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+                                        261,
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+                                        373
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "|\\mathbf{y}_i|"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        261,
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+                                        373
+                                    ],
+                                    "type": "text",
+                                    "content": " is the number of tokens generated for each example "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        261,
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+                                        373
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_i"
+                                },
+                                {
+                                    "bbox": [
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+                                        373
+                                    ],
+                                    "type": "text",
+                                    "content": " in the batch, "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta t"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        261,
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+                                        373
+                                    ],
+                                    "type": "text",
+                                    "content": " is the total time taken to process the batch, and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        261,
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+                                        373
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " is the number of examples in the batch. Note that batching incurs minimal latency penalties. However, given its substantial increase in throughput, batching and its variants are widely used in the state-of-the-art foundation models serving framework such as vLLM (Kwon et al., 2023) to achieve the best balance between throughput and latency."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
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+                                    "type": "text",
+                                    "content": "2.3 OBJECTIVE"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Batching typically assumes the same model parameters are utilized for every input example within a minibatch. Hence, a straightforward application of batching in LoRA requires that the adapter matrix "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta W"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        406,
+                                        504,
+                                        484
+                                    ],
+                                    "type": "text",
+                                    "content": " be shared across all inputs in the batch "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        406,
+                                        504,
+                                        484
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{B}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        406,
+                                        504,
+                                        484
+                                    ],
+                                    "type": "text",
+                                    "content": ". The challenge arises when considering a scenario where each input example in the batch might originate from a different task. Sharing the same "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        406,
+                                        504,
+                                        484
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta W"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        406,
+                                        504,
+                                        484
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        406,
+                                        504,
+                                        484
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        406,
+                                        504,
+                                        484
+                                    ],
+                                    "type": "text",
+                                    "content": " in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        406,
+                                        504,
+                                        484
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{B}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        406,
+                                        504,
+                                        484
+                                    ],
+                                    "type": "text",
+                                    "content": " becomes suboptimal where each input potentially demands a unique adapter. The limitation is particularly acute when the model is expected to serve a world-wide user base with diverse incoming requests."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        488,
+                        505,
+                        534
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                                534
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        505,
+                                        534
+                                    ],
+                                    "type": "text",
+                                    "content": "Given the limitations of LORA in batching, our objective is to maximize the throughput "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        505,
+                                        534
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        505,
+                                        534
+                                    ],
+                                    "type": "text",
+                                    "content": " in global user serving scenarios by maintaining the batching mechanism. Formally, for each "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        505,
+                                        534
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_i \\in \\mathcal{B}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        505,
+                                        534
+                                    ],
+                                    "type": "text",
+                                    "content": ", we aim to compute "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        505,
+                                        534
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{y}_i = \\mathcal{M}(\\mathbf{x}_i | \\Delta W_i, W_0, \\theta)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        505,
+                                        534
+                                    ],
+                                    "type": "text",
+                                    "content": ", where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        505,
+                                        534
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta W_i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        505,
+                                        534
+                                    ],
+                                    "type": "text",
+                                    "content": " is the adapter matrix corresponding to the input example "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        505,
+                                        534
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        505,
+                                        534
+                                    ],
+                                    "type": "text",
+                                    "content": ". Therefore, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        505,
+                                        534
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta W_i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        505,
+                                        534
+                                    ],
+                                    "type": "text",
+                                    "content": " can be unique across "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        505,
+                                        534
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{B}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        488,
+                                        505,
+                                        534
+                                    ],
+                                    "type": "text",
+                                    "content": " and specific to a domain or user preference."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        105,
+                        551,
+                        320,
+                        562
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "3 FLORA: FAST LOW RANK ADAPTATION"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "As shown in §2.3, adapter sharing is often impractical in real-world serving scenarios. The innovation of FLORA is the introduction of example-specific adapter "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    "type": "inline_equation",
+                                    "content": "\\Delta W_{i}"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " for each "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        504,
+                                        654
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        504,
+                                        654
+                                    ],
+                                    "type": "text",
+                                    "content": " in a minibatch. In FLORA, the weight matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
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+                                    "type": "inline_equation",
+                                    "content": "W_{i}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " for each example "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " in the minibatch is calculated as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
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+                                        654
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "W_{i} = \\Delta W_{i} \\circ W_{0}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        504,
+                                        654
+                                    ],
+                                    "type": "text",
+                                    "content": ", where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
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+                                        654
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\circ"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " denotes element-wise multiplication, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        504,
+                                        654
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "W_{0}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        504,
+                                        654
+                                    ],
+                                    "type": "text",
+                                    "content": " is the pre-trained weight matrix and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        504,
+                                        654
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta W_{i}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        504,
+                                        654
+                                    ],
+                                    "type": "text",
+                                    "content": " is a low-rank adaptation specifically designed for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        504,
+                                        654
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{x}_i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        504,
+                                        654
+                                    ],
+                                    "type": "text",
+                                    "content": ". Similar to Hu et al. (2021), "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        504,
+                                        654
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta W_{i}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        504,
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+                                    ],
+                                    "type": "text",
+                                    "content": " is decomposed into two trainable matrices: "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                        504,
+                                        654
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "B_{i} \\in \\mathbb{R}^{d \\times r}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        504,
+                                        654
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        504,
+                                        654
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A_{i} \\in \\mathbb{R}^{r \\times k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
+                                        504,
+                                        654
+                                    ],
+                                    "type": "text",
+                                    "content": ", such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Delta W_{i} = B_{i} A_{i}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        575,
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+                                        654
+                                    ],
+                                    "type": "text",
+                                    "content": ", as shown in Fig. 1. Note that FLORA has the same expressive power as LORA by its construction."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        105,
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+                        677
+                    ],
+                    "type": "title",
+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "3.1 FORWARD PASS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
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+                                    "type": "text",
+                                    "content": "The advantage of FLORA is that computations on a minibatch can be written in terms of matrix multiplications. This enables efficient batched implementations on modern accelerators such as GPUs. Let "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "x_{i}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    "type": "text",
+                                    "content": " denote the activations in one layer of a neural net, which is a vertical vector of length "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "d"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ". The next layer's activations are given by"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                }
+            ],
+            "discarded_blocks": [
+                {
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+                    ],
+                    "type": "header",
+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
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+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} y _ {i} = \\phi \\left(W _ {i} ^ {T} x _ {i}\\right) (2) \\\\ = \\phi \\left(\\left(W _ {0} ^ {T} \\circ \\Delta W _ {i} ^ {T}\\right) x _ {i}\\right) (3) \\\\ = \\phi \\left(\\left(W _ {0} ^ {T} \\circ \\left(B _ {i} A _ {i}\\right) ^ {T}\\right) x _ {i}\\right) (4) \\\\ = \\phi \\left(A _ {i} \\circ \\left(W _ {0} ^ {T} \\left(B _ {i} \\circ x _ {i}\\right)\\right)\\right) (5) \\\\ \\end{array}",
+                                    "image_path": "80e22d4303c4e8655343c3789c848cce2f527e9e4456b32dd88de4f97442247e.jpg"
+                                }
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+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "When the rank is greater than one, we extend the use of the symbol “o” to denote potential broadcasting. Additionally, a dimension reduction operation such as torch.mean is required prior to applying the activation function "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
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+                    "angle": 0,
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "The key to FLORA's flexibility lies in the low rank decomposition enables the incorporation of example-specific adapters directly into the forward pass, as demonstrated in the equations above. Crucially, each of these operations—the element-wise multiplication between "
+                                },
+                                {
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": " and "
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+                                    "type": "text",
+                                    "content": ", and between "
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": " and "
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+                                    "type": "inline_equation",
+                                    "content": "y_{i}"
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+                                    ],
+                                    "type": "text",
+                                    "content": " — is inherently batch-friendly. Consequently, FLORA allows for simultaneous processing of multiple requests, each requiring its own adapter, within a single minibatch. To vectorize all adapters in the minibatch, we define matrices "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        218,
+                                        506,
+                                        298
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{A}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        218,
+                                        506,
+                                        298
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        218,
+                                        506,
+                                        298
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{B}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        218,
+                                        506,
+                                        298
+                                    ],
+                                    "type": "text",
+                                    "content": " whose rows correspond to the adapters "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        218,
+                                        506,
+                                        298
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A_{i}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        218,
+                                        506,
+                                        298
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        218,
+                                        506,
+                                        298
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "B_{i}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        218,
+                                        506,
+                                        298
+                                    ],
+                                    "type": "text",
+                                    "content": " for all examples in the minibatch. The above equation is vectorized as:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        246,
+                        311,
+                        504,
+                        331
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                246,
+                                311,
+                                504,
+                                331
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        246,
+                                        311,
+                                        504,
+                                        331
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathbf {Y} = \\phi \\left(\\mathbf {A} \\circ \\left((\\mathbf {B} \\circ \\mathbf {X}) W _ {0}\\right)\\right) \\tag {6}",
+                                    "image_path": "09343e2e578eda97ac5fa339de2dda8d3f8d03d87c1898fcb9b452b465cb00e0.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
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+                        343,
+                        261,
+                        355
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
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+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
+                                        343,
+                                        261,
+                                        355
+                                    ],
+                                    "type": "text",
+                                    "content": "3.2 COMPUTATIONAL EFFICIENCY"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        364,
+                        506,
+                        488
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "The computational analysis primarily concentrates on the examination of fully connected layers within a transformer architecture, given that LoRA is specifically applied to these layers, such as query and key projections. To begin, we analyze a baseline that leverages batch matrix multiplication to facilitate the serving of LoRA with multiple adapters. This operation is possible under the assumption that every adapter required by the input examples in the minibatch shares the same shape, specifically, the same rank. The batch matrix multiplication (BMM) can be implemented using the torch.bmm operator in deep learning frameworks such as PyTorch (Paszke et al., 2019). Note that the BMM operator is typically unfavorable in practical settings due to the significant overhead it introduces (Abdelfattah et al., 2016). This overhead diminishes the throughput and increases latency, which is detrimental in serving scenarios where response times are crucial for maintaining a good user experience."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        491,
+                        506,
+                        613
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                491,
+                                506,
+                                613
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": "Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "b"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "l"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": " denote the batch size and the maximum sequence length in the input batch "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{B}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": ". Revisiting the notation introduced in §3, where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "W_0 \\in \\mathbb{R}^{d \\times k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "B_i \\in \\mathbb{R}^{d \\times r}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A_i \\in \\mathbb{R}^{r \\times k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": ", the operations required to compute the pre-activation for an input batch "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{B}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": " with dimensions "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "[b, l, d]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": " consist of one matrix multiplication and two BMMs. The matrix multiplication occurs between the input batch "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{X}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": " and the pre-trained weight "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "W_0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": ". The two BMM operations are conducted firstly between the input batch "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{X}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{B}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": ", and secondly between the result of the prior computation and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{A}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": ", where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{A}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{B}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": " are matrices whose rows correspond to the adapters "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A_i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "B_i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": " for all examples in the minibatch respectively. Assuming for simplicity that the layer neither upscales nor downscales the hidden dimension (i.e. "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d = k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": "), the upper bound complexity of this layer is discerned as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "2c_1(dblr) + c_2(bld^2)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": ", with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "c_1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "c_2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": " representing the computational coefficients of BMM and matrix multiplication respectively. Note that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "c_1 >> c_2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        491,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": " because the BMM operator is more expensive than matrix multiplication."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        104,
+                        617,
+                        506,
+                        663
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                617,
+                                506,
+                                663
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        617,
+                                        506,
+                                        663
+                                    ],
+                                    "type": "text",
+                                    "content": "For FLORA, the cost is one matrix multiplication which is "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        617,
+                                        506,
+                                        663
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "c_{2}(rbld^{2})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        617,
+                                        506,
+                                        663
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        617,
+                                        506,
+                                        663
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        617,
+                                        506,
+                                        663
+                                    ],
+                                    "type": "text",
+                                    "content": " denotes the rank of the adapters. We omit the cost of element-wise multiplication in this analysis because it is negligible to the matrix multiplication cost. Comparing the computational cost of FLORA and LORA boils down to the following inequality"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        276,
+                        676,
+                        504,
+                        701
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                276,
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+                                701
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        276,
+                                        676,
+                                        504,
+                                        701
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\frac {2 c _ {1}}{d c _ {2}} + \\frac {1}{r} \\geq 1 \\tag {7}",
+                                    "image_path": "403016e926042be35485fb6affc9642210f2f389dda49f267a245cd21c3c9bc6.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
+                        709,
+                        504,
+                        733
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                            ],
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+                                {
+                                    "bbox": [
+                                        104,
+                                        709,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "FLORA exhibits a lower computational cost than bmm LoRA whenever the above inequality holds true. The benefit of FLORA over LoRA is notably pronounced when "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        709,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r = 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        709,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ". As the rank increases,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
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+                                38
+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        302,
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+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                {
+                                    "bbox": [
+                                        302,
+                                        751,
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+                                        760
+                                    ],
+                                    "type": "text",
+                                    "content": "4"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 3
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        82,
+                        504,
+                        139
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                82,
+                                504,
+                                139
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        139
+                                    ],
+                                    "type": "text",
+                                    "content": "LORA gradually becomes less costly. From the established inequality, a variety of scenarios can be inferred where FLORA has an advantage over LORA. Firstly, the advantage of FLORA is significantly apparent when the rank of adapters is small. Secondly, in configurations where the model has fewer hidden units but an increased number of layers, FLORA tends to outperform LORA due to the smaller value of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        139
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        139
+                                    ],
+                                    "type": "text",
+                                    "content": " in the denominator of Eq. (7)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        143,
+                        506,
+                        222
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                143,
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+                                    "type": "text",
+                                    "content": "Another advantage of FLORA is the cost remains invariant to the number of adapters required by the input batch. While the preceding analysis assumes that every token in an example "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "x_{i}"
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+                                    "type": "text",
+                                    "content": " shares the same adapter, it is possible to apply multiple adapters to a single example by dividing the example into chunks, and then applying different adapters to each chunk. This approach is commonly observed in the Mixture of Experts framework (Fedus et al., 2021; Lepikhin et al., 2020; Puigcerver et al., 2023). Incorporating several adapters in an input example notably amplifies the ratio "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "c_{1} / c_{2}"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " in Eq. (7), thereby significantly increasing LORA's cost."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
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+                                    "type": "text",
+                                    "content": "The ratio "
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+                                    "type": "inline_equation",
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+                                },
+                                {
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+                                    "type": "text",
+                                    "content": " might not be the same across different transformer architectures. §4.1 is designed to provide a deeper insight into how comparative serving efficiency of FLORA and LORA changes under various architectures. Additionally, it's worth noting that LORA does not apply to the self-attention layers, which constitute a non-trivial portion of the computational cost, thereby overshadowing the advantage of FLORA. However, as efficient self-attention mechanisms such as flash attention (Dao et al., 2022) get adopted, the advantage of FLORA over LORA is likely to get larger."
+                                }
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+                        }
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+                    "index": 3
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+                {
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+                                    "type": "text",
+                                    "content": "Connection to IA3. IA3 was proposed in Liu et al. (2022a) featuring fast adaptation of LLM. It introduces a learned vector "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "l"
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+                                    "type": "text",
+                                    "content": " which re-scales the activation by "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "y_{i} = l\\circ \\phi (W_{0}^{T}x_{i}) = \\phi \\bigl (l\\circ (W_{0}^{T}x_{i})\\bigr)"
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": ". This can be viewed as a special case of FLORA - a rank 0.5 variant - which only re-scales the columns instead of the entire pre-trained weights. It has a limited expressive power compared to FLORA and LoRA."
+                                }
+                            ]
+                        }
+                    ],
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+                                    "type": "text",
+                                    "content": "4 EXPERIMENTS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
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+                                    "content": "In this section, we compare FLORA to LoRA and other notable baselines across various metrics and tasks. To begin with, we delve into a computational analysis to substantiate the enhanced throughput and the reduced latency achieved by FLORA in the case of low rank. Subsequently, we pivot towards analyzing the accuracy of FLORA in multilingual code generation tasks spanning across different languages. The goal is to discern the proficiency of FLORA in maintaining, if not enhancing, the model's accuracy as compared to LoRA. Progressing further, we replicate a similar analysis but in the domain of multilingual speech recognition."
+                                }
+                            ]
+                        }
+                    ],
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+                },
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+                                    "type": "text",
+                                    "content": "4.1 SERVING ANALYSIS"
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+                                    "content": "The primary objective of this serving analysis is to measure the maximum throughput both FLORA and LoRA can attain under varied rank configurations. We carried out this exploration on the state-of-the-art code Large Language Model (LLM) StarCoder (Li et al., 2023), evaluating models of different number of parameters namely 1B, 3B, and 15B. The dataset facilitating this analysis has been sourced from the vLLM throughput benchmark. Noteworthy, this dataset was previously used to fine-tune the English Vicuna model, a state-of-the-art chat LLMs (Chiang et al., 2023). To expedite the benchmarking process, we extracted a subset of 1,000 samples from the original dataset, ensuring a diverse range of sample lengths varying from 50 to 2,000 tokens."
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+                                    "type": "text",
+                                    "content": "In setting up the computational analysis, our primary intent is to compare FLORA and LoRA in the real-world serving scenario. See Appendix B.1 on how bmm LoRA is implemented. The vLLM framework (Kwon et al., 2023)"
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "^{2}"
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+                                    "type": "text",
+                                    "content": ", with its implementation of continuous batching, presents an ideal setup for this analysis. The continuous batching mechanism in vLLM, as inspired by the principles delineated in Orca (Yu & Jeong, 2022), facilitates a more efficient utilization of GPU resource by allowing new sequence to be inserted immediately once any sequence in the current batch is completed. This continuous flow significantly enhances GPU utilization as compared to static batching, where the GPU awaits the completion of all sequences in a batch before initiating a new batch processing. The comparison of FLORA and LoRA within this setup offers a compelling evidence of"
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+            ],
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+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "content": "<sup>1</sup>The batch size in BMM operator increases from "
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+                                            "content": "Figure 2: Left: Generation throughput vs. rank for FLORA and torch.bmm implementation of LORA, measured in tokens per second (token/s). The experiments were conducted on three star-coder models: StarCoder 15B, StarCoderbase 3B and StarCoderbase 1B. FLORA has great throughput advantage over LoRA when the rank is small. As the rank increases, the torch.bmm implementation of LoRA finally has a better throughput. Right: Latency vs. rank on StarCoder-15B. Requests are coming at the speed of 8 requests per second."
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+                                    "type": "text",
+                                    "content": "their respective throughput and latency in the real-world serving scenario. It is worth noting that the experiments were conducted without Flash-attention (Dao et al., 2022)."
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+                                    "type": "text",
+                                    "content": "Throughput experiment. In Fig. 2, the throughput results for both FLORA and bmm LORA are illustrated across different rank configurations on three StarCoder models with different number of parameters, namely 1B, 3B and 15B. All experiments were conducted on an NVIDIA H100 GPU with a float16 precision<sup>3</sup>. The maximum number of batched tokens is 8,192, which is the same as the model context length. Evidently, FLORA shows a superior throughput over LORA in the lower rank configurations. At rank 1, the throughput of FLORA is more than threefold higher than that of LORA, thereby highlighting the considerable serving performance boost FLORA provides. The advantage of FLORA continues as the rank increases, albeit with a diminishing rate. For instance, at rank 2, FLORA's throughput is around 2.5 times higher, and this multiplicative advantage decreases as the rank increases further. This performance advantage continues up until the rank of 8 in the StarCoder 15B model, where LoRA starts to outperform FLORA. This inflection point suggests that the advantages of FLORA in terms of throughput are more pronounced in lower rank."
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+                                    "content": "Notice that the inflection points occurred at a higher rank when serving a smaller LLM as illustrated in (Fig. 2, left). This demonstrates a significant potential of FLORA, especially when considering future applications of quantization techniques to serving LLMs. By applying quantization, such as 8-bit or 4-bit inference, the effective size of the model is reduced, akin to serving a smaller LLM thus potentially extending the rank at which FLORA maintains a throughput advantage over LORA."
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+                                    "content": "Latency experiment. We assessed the latency-versus-rank performance on the Starcoder 15B model, using the same dataset as the throughput experiment. This evaluation was conducted under the conditions where requests arrive at a rate of 8 per second. The default maximum number of batched tokens in vLLM serving launcher is 2,560. The results, as shown in (Fig. 2, right), measure latency in terms of seconds per output token. Remarkably, in the lower rank regime (ranging from rank 1 to rank 4), FLORA exhibited a 2-5X reduction in latency compared to LoRA. Notably, the latency for LoRA stood at approximately 2.3 seconds per output token, is impractical for serving due to the poor user experience it would cause. This experiment further highlights the superior capabilities of FLORA in efficiently catering to diverse incoming user requests."
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+                                    "content": "3The quantization in vLLM is still under development."
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+                                    "content": "6"
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+                                            "html": "<table><tr><td rowspan=\"2\">Language</td><td rowspan=\"2\">Base Model</td><td colspan=\"3\">pass@1 (Relative Improvement)</td></tr><tr><td>FLORA</td><td>IA3</td><td>LoRA</td></tr><tr><td></td><td colspan=\"4\">StarCoder 15B</td></tr><tr><td>Dlang</td><td>14.13</td><td>17.26 (22.14%)</td><td>15.26 (7.99%)</td><td>17.15 (21.37%)</td></tr><tr><td>Perl</td><td>17.05</td><td>21.44 (25.76%)</td><td>17.71 (3.90%)</td><td>21.46 (25.76%)</td></tr><tr><td>Ruby</td><td>1.39</td><td>24.94 (1692.86%)</td><td>20.80 (1394.64%)</td><td>23.76 (1608.04%)</td></tr><tr><td>Rust</td><td>22.40</td><td>26.24 (17.14%)</td><td>23.53 (5.04%)</td><td>26.87 (19.95%)</td></tr><tr><td>Racket</td><td>10.20</td><td>12.41 (21.61%)</td><td>11.53 (12.96%)</td><td>12.51 (22.58%)</td></tr><tr><td>Swift</td><td>16.91</td><td>20.38 (20.51%)</td><td>18.13 (7.19%)</td><td>20.35 (20.36%)</td></tr><tr><td></td><td colspan=\"4\">StarCoderBase 3B</td></tr><tr><td>Dlang</td><td>5.65</td><td>5.72 (1.20%)</td><td>5.72 (1.20%)</td><td>6.97 (23.34%)</td></tr><tr><td>Perl</td><td>10.73</td><td>13.01 (21.25%)</td><td>11.46 (6.83%)</td><td>13.31 (27.51%)</td></tr><tr><td>Ruby</td><td>5.33</td><td>14.48 (171.68%)</td><td>7.88 (47.90%)</td><td>13.89 (160.68%)</td></tr><tr><td>Rust</td><td>17.18</td><td>21.00 (22.24%)</td><td>17.28 (0.60%)</td><td>20.67 (20.31%)</td></tr><tr><td>Racket</td><td>8.04</td><td>9.16 (13.99%)</td><td>8.40 (4.48%)</td><td>8.80 (9.48%)</td></tr><tr><td>Swift</td><td>10.04</td><td>15.69 (56.21%)</td><td>12.54 (24.83%)</td><td>15.04 (49.76%)</td></tr></table>",
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+                                    "type": "text",
+                                    "content": "Table 1: Comparison of three fine-tuning methods FLORA, IA3, and LORA on StarCoder 15B and StarCoderBase 3B across various low-resource programming languages in the MultiPL-E benchmark. The table presents Pass@1 accuracy of each method alongside the relative improvement over the baseline. The standard errors are less than "
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+                                    "content": " in all cells in the table, therefore we exclude that for clear presentation."
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+                                    "type": "text",
+                                    "content": "as a compelling alternative for efficiently serving adapted foundation models, especially in scenarios where lower ranks suffice the desired model accuracy, as further demonstrated in the subsequent accuracy analysis sections. Moreover, if an enterprise chooses to serve foundation models with a substantial number of diverse adapters, for instance, a personalized LLM, then a low rank or even a rank one is imperative to avoid excessive storage costs."
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+                                    "content": "4.2 MULTILINGUAL CODE GENERATION"
+                                }
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+                                    "content": "A key aspect to examine before applying FLORA in real-world LLM serving is to scrutinize any potential degradation in performance. In this section, we consider multilingual code generation as the testbed for comparing FLORA and LoRA due to its alignment with real-world applications, where the necessity to cater to diverse programming languages is paramount. Low-resource languages, as referred to in this context, are languages that appear much less frequently than other languages in the pre-training data. Orlanski et al. (2023) showed that the performance of code LLMs can be notably enhanced on low-resource programming languages such as PERL by recalibrating the language distribution in the pre-training data. This suggests that fine-tuning a trained LLM on such low-resource languages could potentially boost its performance on the same language. Hence, by employing multilingual code generation as a benchmark, we can make an informed evaluation of adaptability and performance enhancements that FLORA and LoRA can achieve."
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+                                    "type": "text",
+                                    "content": "Additionally, a comparison is made against a third baseline, IA3, which can be considered as a special case of FLORA. Essentially, IA3 can be seen as a rank 0.5 variant of FLORA, thereby facing a constrained expressive power in comparison to both FLORA and LoRA. For FLORA and LoRA, we conducted fine-tuning across a range of rank choices, spanning from 1 to 8. It emerges that within the scope of this multilingual code generation task, a rank of one typically suffices to achieve optimal results, with the exception of Racket and Lua. Consequently, the results shown in §4.2 are predominantly at rank 1, barring Racket and Lua, which are presented at rank 4."
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+                                    "type": "text",
+                                    "content": "Fine-tuning. In our effort to evaluate the performance of FLORA, LORA, and IA3 on the multilingual code generation task, we fine-tuned these models on state-of-the-art multilingual code LLMs, StarCoder 15B and StarCoderBase 3B as introduced in (Li et al., 2023). A pivotal aspect of our fine-tuning process was the utilization of existing data, negating the need for creating new data for"
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+                                            "html": "<table><tr><td>Model</td><td>Arabic</td><td>Czech</td><td>Lithuanian</td><td>Marathi</td><td>Mongolian</td><td>Hindi</td></tr><tr><td>Whisper</td><td>46.03</td><td>23.19</td><td>46.09</td><td>84.84</td><td>115.20</td><td>43.41</td></tr><tr><td>FLORA</td><td>30.21 ± 0.25</td><td>10.76 ± 0.22</td><td>20.39 ± 0.32</td><td>33.17 ± 0.17</td><td>42.12 ± 0.63</td><td>23.58 ± 0.65</td></tr><tr><td>IA3</td><td>30.48 ± 0.22</td><td>11.61 ± 0.33</td><td>21.41 ± 0.20</td><td>35.03 ± 0.61</td><td>46.47 ± 0.53</td><td>25.64 ± 0.78</td></tr><tr><td>LORA</td><td>30.18 ± 0.21</td><td>10.77 ± 0.38</td><td>20.50 ± 0.21</td><td>31.94 ± 0.33</td><td>41.57 ± 0.69</td><td>23.82 ± 0.74</td></tr></table>",
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+                                    "content": "Table 2: Mean WER with standard deviation for different models and languages. The WER is calculated with respect to the unnormalized tokenizer. The base model here is Whisper-1.5B."
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+                    ],
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+                                    "type": "text",
+                                    "content": "low-resource programming languages. We leveraged the same pre-training data that was used for pre-training StarCoder, specifically, the Stack dataset, which contains over 6TB of permissively-licensed source code files covering 358 programming languages. For each low-resource language in our experiment, we fine-tuned on its corresponding split from the Stack dataset for a total of 1,500 steps, along with batch size 8. More fine-tuning details are given in Appendix A."
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+                                    "type": "text",
+                                    "content": "Evaluation. The evaluation of FLORA, LORA, and IA3 was conducted on the MultiPL-E benchmark (Cassano et al., 2023), which contains the translation of two unit-test driven Python benchmarks (HumanEval and MBPP) to 18 other programming languages. We used the HumanEval split of the benchmark to evaluate the fine-tuned models. As for the metrics, We adopted the "
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+                                    "type": "text",
+                                    "content": " metrics from Chen et al. (2021); Austin et al. (2021), which is calculated as the fraction of problems with at least one correct sample given "
+                                },
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+                                    "type": "text",
+                                    "content": " samples. Similar to Chen et al. (2021), we drew 100 samples for computing "
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+                                    "type": "text",
+                                    "content": ", with a sampling temperature set at 0.1."
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+                                    "type": "text",
+                                    "content": "Main results. The result in §4.2 exhibits the performance of three methods across various programming languages on both StarCoder 15B and StarCoderBase 3B models. The average relative improvement achieved by FLORA and LoRA is roughly "
+                                },
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+                                    "type": "text",
+                                    "content": " in the selected low-resource programming languages. FLORA consistently outperforms IA3 on all languages, especially on StarCoder 15B, denoting its efficiency in leveraging the model expressive power to improve multilingual code generation. It is notable that StarCoder 15B has an unforeseen issue regarding Ruby generation, where it yields lower accuracy compared to the 3B model. However, all methods are able to fix its abnormal performance."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
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+                {
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+                                    "type": "text",
+                                    "content": "On StarCoderBase 3B, a smaller model, it is evident that the baseline performance drops, yet FLoRA and LoRA still manage to exhibit substantial relative improvement over the baseline, especially in languages like Swift and Ruby. This suggests that both methods benefit from continuous training on the low-resource language split of the pre-training data, although the advantages may diminish with a reduction in model size. While the absolute performance (pass@1 accuracy) varies among languages, the relative improvements highlight the effectiveness of the tested methods in enhancing multilingual code generation."
+                                }
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+                                    "content": "4.3 MULTILINGUAL SPEECH RECOGNITION"
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+                                    "content": "Following our analysis in multilingual code generation, we shift our focus to another substantial application of large language models — multilingual speech recognition (MSR). This domain has seen increasing demand due to the necessity of serving various linguistic interfaces. The capabilities of FLORA and LORA to adapt to multiple languages in code generation presents a tempting premise to investigate their effectiveness in the MSR domain."
+                                }
+                            ]
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+                                    "content": "Fine-tuning. A crucial aspect of our analysis involves fine-tuning the foundation model on a multilingual speech recognition dataset. We selected the Whisper large 1.5B (Radford et al., 2022) as our base model to conduct the subsequent analysis. Whisper is an encoder-decoder transformer model trained on 680k hours of labeled speech data through large-scale weak supervision. Despite being pre-trained on a multilingual speech corpus, the model's predictive capabilities can be improved further for certain languages and tasks through fine-tuning. We use the Common Voice benchmark (Ardila et al., 2020) — containing a total of 38 languages and 2,500 hours of collected audio — for the fine-tuning process, with a particular focus on low-resource languages. For each low-resource language enumerated in §4.2, we fine-tuned on its training split within the Common"
+                                }
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+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "type": "text",
+                                    "content": "Dataset statistics can be found here https://commonvoice.—.org/en/datasets."
+                                }
+                            ]
+                        }
+                    ],
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+                },
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+                                    "content": "8"
+                                }
+                            ]
+                        }
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+                }
+            ],
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+                                    "content": "Voice dataset for approximately 5,000 steps, with a batch size of 6. More fine-tuning details are given in Appendix A. Similar to §4.2, we found that rank one is sufficient to achieve the best result in most cases, with the exception of Marathi, which requires a rank of 4."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
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+                                    "type": "text",
+                                    "content": "Main results. We present the main results on the MSR task in §4.2. Notice that the evaluation is conducted with the unnormalized tokenizer.<sup>5</sup> The Whisper model exhibited a relatively higher Word Error Rate (WER) across all low-resource languages when compared to FLORA, LORA and IA3 models, reflecting a significant room for enhancement in its multilingual speech recognition capabilities. Particularly, its performance was found to be subpar for the Marathi and Mongolian languages, with WERs of 84.84 and 115.20 respectively. All examined models significantly outperform the base Whisper model by a wide margin across all languages, highlighting the effective fine-tuning of FLORA, LORA and IA3 models. For instance, in Arabic, FLORA reduces the WER to 30.21 from Whisper's 46.03, showcasing a significant enhancement in speech recognition accuracy. Particularly, FLORA and LORA consistently outperform IA3, indicating a better expressive power for multilingual speech recognition tasks. For example, in Czech, FLORA has a mean WER of 10.76, which is slightly better compared to IA3's 11.61 and LORA's 11.47. A similar trend is observed in Lithuanian and Hindi. In summary, this table shows the superior performance of FLORA and LORA over IA3 in the speech recognition task across diverse languages. The consistent low WERs attained by FLORA demonstrates its potential as a viable model for MSR tasks."
+                                }
+                            ]
+                        }
+                    ],
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+                                    "type": "text",
+                                    "content": "5 RELATED WORK"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
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+                                    "type": "text",
+                                    "content": "Parameter-Efficient Fine-Tuning (PEFT) methods are broadly partitioned into two categories: weight-based and non-weight-based approaches. MC-dropout (Lakshminarayanan et al., 2016) stands as an early example for the non-weight-based approach, where distinct dropout masks are allocated to various tasks. Recently, prompt tuning techniques have emerged as a prevalent stream within this category (Li & Liang, 2021; Lester et al., 2021), facilitating efficient adaptation with minimal modifications to models. Successive endeavors aimed to enhance this class of methods, delving into aspects such as optimization (Mao et al., 2021; Diao et al., 2022), transferability (Wang et al., 2021; Vu et al., 2021; He et al., 2022b), and the usage of discrete prompts (Schick & Schütze, 2020a;b; Gao et al., 2021; Malkin et al., 2021), among others (Liu et al., 2022b; 2021)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
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+                    "angle": 0,
+                    "lines": [
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+                                    "type": "text",
+                                    "content": "We focus on weight-based approaches in this work, which has a weight interpretation as exemplified by LoRA (Hu et al., 2021). This line of research can be traced back to Progressive Network (Rusu et al., 2016), which inserts a sub-network when a new task arrives. This principle was later adapted widely in foundation models as represented by adapter based methods (Houlsby et al., 2019; Mahabadi et al., 2021; Davison, 2021; Ding et al., 2022; Wang et al., 2022b). In particular, BitFit (BenZaken et al., 2021) was introduced to solely update the bias parameters, while IA3 (Liu et al., 2022a) was proposed to rescale the activations. Additionally, approaches such Fish (Sung et al., 2021) and Diff-pruning (Guo et al., 2020) leverage sparsity to facilitate efficient adaptation of foundation models. A separate vein of research aims to improve LoRA by reducing its computational and memory costs (Zhang et al., 2023b;a; Malladi et al., 2023). He et al. (2022a) explored how to unify different PEFT methods. Dettmers et al. (2023) quantized LoRA to reduce memory footprint. Chavan et al. (2023) generalized LoRA by learning individual adapters in each layer. Several other works focus on building mixture of adapters (Wang et al., 2022a; Diao et al., 2023)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
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+                                    "type": "text",
+                                    "content": "6 CONCLUSION"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
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+                                    "type": "text",
+                                    "content": "We introduced FLORA, an extension of LORA, facilitating efficient batching. Empirical evaluations demonstrated that FLORA enhances throughput and latency in practical serving scenarios, all while preserving the accuracy of LORA. Through FLORA, we aim to facilitate a more efficient adaptation of large language models to diverse and real-world user requests."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
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+                                    "type": "text",
+                                    "content": "Limitations. Despite its parameter efficiency, FLORA still requires fine-tuning. A promising future work could be to derive FLORA weights from a trained LoRA model, given that LORA remains"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
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+            ],
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+                                    "content": "Published as a conference paper at ICLR 2024"
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+                        }
+                    ],
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+                },
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+                                    "type": "text",
+                                    "content": "This is because the normalized tokenizer has a bug in the official leaderboard implementation at the time of submission."
+                                }
+                            ]
+                        }
+                    ],
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+                                    "content": "9"
+                                }
+                            ]
+                        }
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+            ],
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+                                    "content": "the most prevalent type of adapter as per (Huang et al., 2023). This adaptation could potentially obviate the requirement for fine-tuning, thereby accelerating the process of model adaptation."
+                                }
+                            ]
+                        }
+                    ],
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+                                    "content": "REFERENCES"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
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+                    "angle": 0,
+                    "index": 15,
+                    "blocks": [
+                        {
+                            "bbox": [
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+                            "type": "ref_text",
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+                                        }
+                                    ]
+                                }
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+                                        }
+                                    ]
+                                }
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+                                        {
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+                                            "type": "text",
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+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 5
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+                        {
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+                            ],
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+                            "angle": 0,
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+                                        {
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+                                            "type": "text",
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+                                        }
+                                    ]
+                                }
+                            ],
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+                            ],
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+                            "angle": 0,
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+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
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+                                        }
+                                    ]
+                                }
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+                                            ],
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+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 9
+                        },
+                        {
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+                                            "content": "Longteng Zhang, Lin Zhang, Shaohuai Shi, Xiaowen Chu, and Bo Li. Lora-fa: Memory-efficient low-rank adaptation for large language models fine-tuning. ArXiv, abs/2308.03303, 2023a. URL https://api-semanticscholar.org/CorpusID:260683267."
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+                                    "content": "A ADDITIONAL TRAINING DETAILS"
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+                                    "content": "We delve into additional details regarding the models used in §4.2. We conducted fine-tuning on two StarCoder models (Li et al., 2023), specifically the 15B and 3B models. For both models, we employed LoRA (Hu et al., 2021) on the default layers within the Parameter Efficient Fine-Tuning (PEFT) library"
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+                                    "content": ", targeting only the key and query projections in the self-attention layers. This incurs roughly "
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+                                    "content": "0.03\\%"
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+                                    "type": "text",
+                                    "content": " number of training parameters in the rank 1 setting. For a fair comparison, we applied FLORA to the same layers where LoRA was applied. Regarding the other baseline IA3 (Liu et al., 2022a), we executed the fine-tuning in two distinct settings: one with the default configuration in PEFT, applying IA3 to the key projection layers and the first fully-connected layer in the feedforward layers; and another aligning with the LoRA setting, restricting application to the key and query projection layers. The latter setup proved to be less effective than the default setting, thus the results presented in §4.2 adhere to the default configuration."
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+                                    "content": "Despite StarCoder's pre-training on the Stack dataset, which contains over 6TB of source code files covering 358 programming languages, we still fine-tuned all methods on this dataset for the low-resource programming languages. This eliminates the time-consuming process to create new data. Our observations revealed that both IA3 and FLORA require a higher learning rate compared to"
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+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                            "html": "<table><tr><td rowspan=\"2\">Language</td><td rowspan=\"2\">Base Model</td><td colspan=\"3\">pass@1 (Relative Improvement)</td></tr><tr><td>FLORA</td><td>IA3</td><td>LORA</td></tr><tr><td></td><td colspan=\"4\">StarCoder 15B</td></tr><tr><td>Julia</td><td>23.31</td><td>26.56 (13.92%)</td><td>23.90 (2.51%)</td><td>25.62 (10.45%)</td></tr><tr><td>Lua</td><td>26.60</td><td>30.20 (13.57%)</td><td>28.40 (6.77%)</td><td>29.04 (9.18%)</td></tr><tr><td>R</td><td>16.07</td><td>17.55 (9.24%)</td><td>16.65 (3.59%)</td><td>18.60 (15.8%)</td></tr><tr><td></td><td colspan=\"4\">StarCoderBase 3B</td></tr><tr><td>Julia</td><td>15.62</td><td>19.11 (22.39%)</td><td>16.72 (7.05%)</td><td>19.77 (26.59%)</td></tr><tr><td>Lua</td><td>17.63</td><td>19.46 (10.39%)</td><td>18.04 (2.33%)</td><td>20.23 (14.75%)</td></tr><tr><td>R</td><td>10.00</td><td>10.70 (7.02%)</td><td>9.98 (-0.25%)</td><td>10.06 (0.62%)</td></tr></table>",
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+                                    "type": "text",
+                                    "content": "Table 3: Comparison of three fine-tuning methods FLORA, IA3, and LoRA on StarCoder 15B and StarCoderBase 3B across additional low-resource programming languages in the MultiPL-E benchmark."
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+                                    "content": "LORA. For LORA, a learning rate of "
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+                                    "type": "text",
+                                    "content": " sufficed, whereas IA3 required "
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+                                    "content": ", and FLORA demanded an even higher rate of "
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+                                    "content": ". We conjecture that this stems from FLORA and IA3 incorporating multiplicative weight perturbation, which typically requires a larger learning rate. All models were fine-tuned using 8 bit quantization featuring lower training memory cost."
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+                                    "content": "We present results for additional low-resource languages in Appendix A. The improvements provided by PEFT methods are less pronounced in these languages compared to those documented in §4.2. Nevertheless, it can be observed that LORA and FLORA still outperform IA3, despite the limited room for enhancement in these languages."
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+                                    "type": "text",
+                                    "content": "Next, we discuss further details concerning the models used in §4.3. We performed fine-tuning on the Whisper large model (Radford et al., 2022), which consists of 1.5 billion parameters. Unlike the code generation task, we discovered that using the default modules in the PEFT library does not yield the best performance. Instead, we applied FLORA, LoRA, and IA3 to every fully-connected layer in the Whisper transformer. We fine-tuned all methods on the Common Voice benchmark (Ardila et al., 2020) for each low-resource language showcased in §4.2. We continued to employ larger learning rates for FLORA and IA3. Specifically, we maintained a learning rate of "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "2\\mathrm{e} - 4"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " for LORA, while using "
+                                },
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+                                    "type": "inline_equation",
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " and "
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "6\\mathrm{e} - 3"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        392,
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+                                    ],
+                                    "type": "text",
+                                    "content": " for IA3 and FLORA respectively. Additionally, we utilized 8-bit quantization during training to minimize the GPU memory requirement."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
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+                                    "content": "B ADDITIONAL SERVING RESULTS"
+                                }
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+                                }
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+                    "index": 7
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+                                    "content": "We provide further details concerning the experiments in §4.1. To begin, we illustrate the implementation of \"torch.bmm\" as follows. Upon computing the adaptersout, it can be added to the output of the standard LLM layer. This method facilitates the computation of diverse adapters' outputs within a batch."
+                                }
+                            ]
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+                                            "content": "inputs = torch Randn(batch, seq_length, d_model)  \nadapter_b # shape of (batch, d_model, rank)  \nadapter_a # shape of (batch, rank, d_model)  \nhidden = torch.bmm(inputs, adapter_b)  \nadaptersout = torch.bmm(hidden, adapter_a)"
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+                                    "content": "B.2 ADDITIONAL RESULTS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
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+                                    "content": "We further conducted a throughput experiment, employing a setup similar to the one described in §4.1, on the newly proposed Llama-2 foundation models (Touvron et al., 2023). The results for both the 13B and 7B models are illustrated in Fig. 3 (left). When compared against the plot in"
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+                                    "type": "text",
+                                    "content": "Fig. 2 (left), it is evident that the inflection rank at which LoRA begins to outperform FLORA is lower for the Llama-2 models than for the StarCoder 15B. This shows that the ratio "
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": " from Eq. (7) varies across different architectures. One plausible explanation could be that, being a recent architecture, the Llama-2 model has not yet been fully optimized within the vLLM framework, potentially impacting throughput performance."
+                                }
+                            ]
+                        }
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+                                    "content": "The right panel of Fig. 3 presents the serving results concerning StarCoder-3B. Notably, the inflection rank at which LORA begins to outperform FLORA is lower here as compared to the one observed in StarCoder 15B, as seen in Fig. 2 (right). This is attributed to the relatively low latency on StarCoder-3B, which implies that a request rate of 8 requests per second fails to saturate the LLM server, thereby diminishing the advantage of FLORA. Upon escalating the request rate to 15 requests per second, the inflection rank correspondingly increases to around 12. This illustrates that the latency improvement exhibited in Fig. 2 (left) represents a theoretical enhancement. The actual improvement in serving depends on model architectures, request rates, and other practical considerations."
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+                                            "content": "Figure 3: Left: Generation throughput vs. rank for FLORA and torch.bmm implementation of LORA, measured in tokens per second (token/s). The experiments were conducted on two Llama-2 models: 13B and 7B (Touvron et al., 2023). FLORA has great throughput advantage over LORA when the rank is small. As the rank increases, the torch.bmm implementation of LORA finally has a better throughput. Right: Latency vs. rank on StarCoder-3B. Requests are coming at the speed of 8 requests per second and 15 requests per second."
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diff --git a/2024/Beyond Weisfeiler-Lehman_ A Quantitative Framework for GNN Expressiveness/fa29bbb1-c854-4aeb-a13c-5bf1509f8f38_content_list.json b/2024/Beyond Weisfeiler-Lehman_ A Quantitative Framework for GNN Expressiveness/fa29bbb1-c854-4aeb-a13c-5bf1509f8f38_content_list.json
new file mode 100644
index 0000000000000000000000000000000000000000..2a3de04168a5114a487c51f9fff5c44167c4d220
--- /dev/null
+++ b/2024/Beyond Weisfeiler-Lehman_ A Quantitative Framework for GNN Expressiveness/fa29bbb1-c854-4aeb-a13c-5bf1509f8f38_content_list.json	
@@ -0,0 +1,11582 @@
+[
+    {
+        "type": "text",
+        "text": "BEYOND WEISFEILER-LEHMAN: A QUANTITATIVE FRAMEWORK FOR GNN EXPRESSIVENESS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            99,
+            789,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Bohang Zhang $^{1* \\dagger}$  Jingchu Gai $^{2*}$  Yiheng Du $^{3}$  Qiwei Ye $^{4}$  Di He $^{1 \\boxtimes}$  Liwei Wang $^{1,5 \\boxtimes}$",
+        "bbox": [
+            181,
+            167,
+            820,
+            185
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "$^{1}$ National Key Laboratory of General Artificial Intelligence, SIST, Peking University",
+        "bbox": [
+            183,
+            185,
+            684,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "$^{2}$ School of Mathematical Science, Peking University  $^{3}$ Yuanpei College, Peking University",
+        "bbox": [
+            183,
+            198,
+            728,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "$^{4}$ Beijing Academy of Artificial Intelligence  $^{5}$ Center for Machine Learning Research, Peking University zhangbohang@pku.edu.cn, {gaijingchu,duyiheng}@stu.pku.edu.cn",
+        "bbox": [
+            183,
+            212,
+            803,
+            239
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "qwye@baai.ac.cn, {dihe, wanglw}@pku.edu.cn",
+        "bbox": [
+            183,
+            241,
+            544,
+            253
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT",
+        "text_level": 1,
+        "bbox": [
+            450,
+            272,
+            545,
+            286
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Designing expressive Graph Neural Networks (GNNs) is a fundamental topic in the graph learning community. So far, GNN expressiveness has been primarily assessed via the Weisfeiler-Lehman (WL) hierarchy. However, such an expressivity measure has notable limitations: it is inherently coarse, qualitative, and may not well reflect practical requirements (e.g., the ability to encode substructures). In this paper, we introduce a novel framework for quantitatively studying the expressiveness of GNN architectures, addressing all the above limitations. Specifically, we identify a fundamental expressivity measure termed homomorphism expressivity, which quantifies the ability of GNN models to count graphs under homomorphism. Homomorphism expressivity offers a complete and practical assessment tool: the completeness enables direct expressivity comparisons between GNN models, while the practicality allows for understanding concrete GNN abilities such as subgraph counting. By examining four classes of prominent GNNs as case studies, we derive simple, unified, and elegant descriptions of their homomorphism expressivity for both invariant and equivariant settings. Our results provide novel insights into a series of previous work, unify the landscape of different subareas in the community, and settle several open questions. Empirically, extensive experiments on both synthetic and real-world tasks verify our theory, showing that the practical performance of GNN models aligns well with the proposed metric.",
+        "bbox": [
+            228,
+            300,
+            767,
+            566
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION",
+        "text_level": 1,
+        "bbox": [
+            173,
+            587,
+            336,
+            602
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Owing to the ubiquity of graph-structured data in numerous applications, Graph Neural Networks (GNNs) have achieved enormous success in the field of machine learning over the past few years. However, one of the most prominent drawbacks of popular GNNs lies in the limited expressive power. In particular, Morris et al. (2019); Xu et al. (2019) showed that Message Passing GNNs (MPNNs) are intrinsically bounded by the 1-dimensional Weisfeiler-Lehman test (1-WL) in distinguishing non-isomorphic graphs (Weisfeiler & Lehman, 1968). Since then, the Weisfeiler-Lehman hierarchy has become a yardstick to measure the expressiveness and guide designing more powerful GNN architectures (see Appendix A.1 for an overview of representative approaches in this area).",
+        "bbox": [
+            169,
+            618,
+            826,
+            731
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "However, as more and more architectures have been proposed, the limitations of the WL hierarchy are becoming increasingly evident. First, the WL hierarchy is arguably too coarse to evaluate the expressive power of practical GNN models (Morris et al., 2022; Puny et al., 2023). On one hand, architectures inspired by higher-order WL tests (Maron et al., 2019b;a; Morris et al., 2019) often suffer from substantial computation/memory costs. On the other hand, most practical and efficient GNNs are only proved to be strictly more expressive than 1-WL by leveraging toy example graphs (e.g., Zhang & Li, 2021; Bevilacqua et al., 2022; Wijesinghe & Wang, 2022a). Such a qualitative characterization may provide little insight into the models' true expressiveness. Besides, the expressive power brought from the WL hierarchy often does not align well with the one required in practice (Veličković, 2022). Hence, how to study the expressiveness of GNN models in a quantitative, systematic, and practical way remains a central research direction for the GNN community.",
+        "bbox": [
+            169,
+            736,
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+            891
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "page_footnote",
+        "text": "*Equal technical contributions.",
+        "bbox": [
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+            896,
+            377,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "page_footnote",
+        "text": "$\\dagger$  Project lead.",
+        "bbox": [
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+            922
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "page_number",
+        "text": "1",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To address the above limitations, this paper takes a different approach by studying GNN expressivity from the following practical angle: What structural information can a GNN model encode? Since the ability to detect/count graph substructures is crucial in various real-world applications (Chen et al., 2020; Huang et al., 2023; Tahmasebi et al., 2023), many expressive GNNs have been proposed based on preprocessing substructure information (Bouritsas et al., 2022; Barceló et al., 2021; Bodnar et al., 2021b;a). However, instead of augmenting GNNs by manually preprocessed (task-specific) substructures, it is nowadays more desirable to design generic, domain-agnostic GNNs that can end-to-end learn different structural information suitable for diverse applications. This naturally gives rise to the fundamental question of characterizing the complete set of substructures prevalent GNN models can encode. Unfortunately, this problem is widely recognized as challenging even when examining simple structures like cycles (Fürer, 2017; Arvind et al., 2020; Huang et al., 2023).",
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+        "text": "Our contributions. Motivated by GNNs' ability to encode substructures, this paper presents a novel framework for quantitatively analyzing the expressive power of GNN models. Our approach is rooted in a critical discovery: given a GNN model  $M$ , the model's output representation for any graph  $G$  can be fully determined by the structural information of  $G$  over some pattern family  $\\mathcal{F}^M$ , where  $\\mathcal{F}^M$  corresponds to precisely all (and only) those substructures that can be \"coded\" by model  $M$ . In this way, the set  $\\mathcal{F}^M$  can be naturally viewed as an expressivity description of  $M$ : after identifying  $\\mathcal{F}^M$  for each model  $M$ , the expressivity of different models can then be qualitatively/quantitatively compared by simply looking at their set inclusion relation and set difference.",
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+        "type": "text",
+        "text": "The crux here is to define an appropriate notion of \"encodability\" so that  $\\mathcal{F}^M$  can admit a simple description. We identify that a good candidate is the homomorphism expressivity: i.e.,  $\\mathcal{F}^M$  consists of all substructures that can be counted by model  $M$  under homomorphism (see Section 2 for a formal definition). Homomorphism is a foundational concept in graph theory (Lovasz, 2012) and is linked to many important topics such as graph coloring, graph matching, and subgraph counting. With this concept, we are able to give complete, unified, and surprisingly elegant descriptions of the pattern family  $\\mathcal{F}^M$  for a wide range of mainstream GNN architectures listed below:",
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+        "type": "list",
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+        "list_items": [
+            "- MPNN (e.g., Gilmer et al., 2017; Hamilton et al., 2017; Kipf & Welling, 2017; Xu et al., 2019);",
+            "- Subgraph GNN (You et al., 2021; Zhang & Li, 2021; Bevilacqua et al., 2022; Qian et al., 2022);",
+            "- Local GNN (Morris et al., 2020; 2022; Zhang et al., 2023a; Frasca et al., 2022);",
+            "- Folklore-type GNN (Maron et al., 2019a; Zhang et al., 2023a; Feng et al., 2023)."
+        ],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Technically, the descriptions are based on a novel application and extension of the concept of nested ear decomposition (NED) in graph theory (Eppstein, 1992). We prove that: (i) (necessity) each model  $M$  above can count (under homomorphism) a specific family of patterns  $\\mathcal{F}^M$ , characterized by a specific type of NED; (ii) (sufficiency) any pattern  $F \\notin \\mathcal{F}^M$  cannot be counted under homomorphism by model  $M$ ; (iii) (completeness) for any graph, information collected from the homomorphism count in pattern family  $\\mathcal{F}^M$  determines its representation computed by model  $M$ . Therefore, homomorphism expressivity is well-defined and is a complete expressivity measure for GNN models.",
+        "bbox": [
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+        "type": "text",
+        "text": "Our theory can be generalized in various aspects. One significant extension is the node-level and edge-level expressivity for equivariant GNNs (Azizian & Lelarge, 2021; Geerts & Reutter, 2022), which can be naturally tackled by a fine-grained analysis of NED. As another non-trivial generalization, we study higher-order GNN variants for several of the above architectures and derive results by defining higher-order NED. Both aspects demonstrate the flexibility of our proposed framework, suggesting it as a general recipe for analyzing future architectures.",
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+        "text": "Implications. Homomorphism expressivity serves as a powerful toolbox for bridging different sub-areas in the GNN community, providing fresh understandings of a series of known results that were previously proved in complex ways, and answering a set of unresolved open problems. First, our results can readily establish a complete expressiveness hierarchy among all the aforementioned architectures and their higher-order extensions. This recovers and extends a number of results in Morris et al. (2020); Qian et al. (2022); Zhang et al. (2023a); Frasca et al. (2022) and answers their open problems (Section 4.1). In fact, our results go far beyond revealing the expressivity gap between models: we essentially answer how large the gap is and establish a systematic approach to constructing counterexample graphs. Second, based on the relation between homomorphism and subgraph count, we are able to characterize the subgraph counting power of GNN models for all patterns at graph, node, and edge levels, significantly advancing an open direction initiated in Fürer (2017); Arvind et al. (2020) (Section 4.2). As a special case, our results extend recent findings in",
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
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+            47
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "page_number",
+        "text": "2",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Huang et al. (2023) about the cycle counting power of GNN models, highlighting that Local 2-GNN can already subgraph-count all cycles/paths within 7 nodes (even at edge-level). Third, our results provide a new toolbox for studying the polynomial expressivity proposed recently in Puny et al. (2023), extending it to various practical architectures and answering an open question (Section 4.3). Empirically, an extensive set of experiments verifies our theory, showing that the homomorphism expressivity of different models matches well with their practical performance in diverse tasks.",
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+    },
+    {
+        "type": "text",
+        "text": "2 PRELIMINARY",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Notations. We use  $\\{\\}$  and  $\\{\\}$  to denote sets and multisets, respectively. Given a (multi)set  $S$ , its cardinality is denoted as  $|S|$ . In this paper, we consider finite, undirected, vertex-labeled graphs with no self-loops or repeated edges. Let  $G = (V_G, E_G, \\ell_G)$  be a graph with vertex set  $V_G$ , edge set  $E_G$ , and label function  $\\ell_G$ , where each edge in  $E_G$  is a set  $\\{u, v\\} \\subset V_G$  of cardinality two, and  $\\ell_G(u)$  is the label of vertex  $u$ . The rooted graph  $G^u$  is a graph obtained from  $G$  by marking the special vertex  $u \\in V_G$ ; we can similarly consider marking two special vertices  $u, v \\in V_G$  (denote by  $G^{uv}$ ). The neighbors of vertex  $u$  is denoted as  $N_G(u) := \\{v \\in V_G : \\{u, v\\} \\in E_G\\}$ . A graph  $F = (V_F, E_F, \\ell_F)$  is a subgraph of  $G$  if  $V_F \\subset V_G$ ,  $E_F \\subset E_G$ , and  $\\ell_F(u) = \\ell_G(u)$  for all  $u \\in V_F$ . A simple path  $P$  in  $G$  is an edge set of the form  $\\{\\{w_0, w_1\\}, \\dots, \\{w_{k-1}, w_k\\}\\} \\subset E_G$  where  $w_i \\neq w_j$  for all  $i \\neq j$ . Here,  $w_0$  and  $w_k$  are called endpoints of  $P$  and other vertices are called internal points.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Homomorphism, isomorphism, and subgraph count. Given two graphs  $F$  and  $G$ , a homomorphism from  $F$  to  $G$  is a mapping  $f: V_F \\to V_G$  that preserves edges and labels, i.e.,  $\\ell_F(u) = \\ell_G(f(u))$  for all  $u \\in V_F$ , and  $\\{f(u), f(v)\\} \\in E_G$  for all  $\\{u, v\\} \\in E_F$ . When the mapping  $f$  exists, we say  $F$  is homomorphic to  $G$ . We denote by  $\\operatorname{Hom}(F, G)$  the set of all homomorphisms from  $F$  to  $G$  and define  $\\operatorname{hom}(F, G) = |\\operatorname{Hom}(F, G)|$ , which counts the number of homomorphisms for pattern  $F$  in graph  $G$ . If  $f$  is further surjective on both vertices and edges, we call  $G$  a homomorphic image of  $F$ . Denote by  $\\operatorname{Spasm}(F)$  the set of all homomorphic images of  $F$ , called the spasm of  $F$ . For rooted graphs, homomorphism should additionally preserve vertex marking: i.e., if  $f$  is a homomorphism from  $F^{uv}$  to  $G^{xy}$ , then  $f(u) = x$  and  $f(v) = y$ .",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A mapping  $f: V_F \\to V_G$  is called an isomorphism if  $f$  is a bijection and both  $f$  and its inverse  $f^{-1}$  are homomorphisms. We denote by  $\\operatorname{Sub}(F, G)$  the set of all subgraphs of  $G$  isomorphic to  $F$  and define  $\\operatorname{sub}(F, G) = |\\operatorname{Sub}(F, G)|$ , which counts the number of patterns  $F$  occurred in graph  $G$  as a subgraph. We note that a similar definition holds for rooted graphs (e.g.,  $\\operatorname{sub}(F^{uv}, G^{xy})$ ).",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Graph neural networks. GNNs can be generally described as graph functions that are invariant under isomorphism. To achieve such invariance, most popular GNN models follow a color refinement (CR) paradigm: they maintain a feature representation (color) for each vertex or vertex tuples and iteratively refine these features through equivariant aggregation layers. Finally, there is a global pooling layer to merge all features and obtain the graph representation. Below, we separately define the corresponding CR algorithms for four mainstream classes of GNNs studied in this paper.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "- MPNN. Given a graph  $G$ , MPNN maintains a color  $\\chi_G^{\\mathrm{MP}}(u)$  for each vertex  $u \\in V_G$ . Initially, the color only depends on the vertex label, i.e.,  $\\chi_G^{\\mathrm{MP},(0)}(u) = \\ell_G(u)$ . Then, in each iteration, the color is refined by the following update formula (where hash is a perfect hash function):",
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\chi_ {G} ^ {\\mathsf {M P}, (t + 1)} (u) = \\operatorname {h a s h} \\left(\\chi_ {G} ^ {\\mathsf {M P}, (t)} (u), \\{\\{\\chi_ {G} ^ {\\mathsf {M P}, (t)} (v): v \\in N _ {G} (u) \\} \\}\\right). \\tag {1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "After a sufficient number of iterations, the colors become stable. We denote by  $\\chi_G^{\\mathrm{MP}}(u)$  the stable color of  $u$ , which is also the node feature of  $u$  computed by the MPNN. The graph representation is defined as the multiset of node colors, i.e.,  $\\chi_G^{\\mathrm{MP}}(G) = \\{\\{\\chi_G^{\\mathrm{MP}}(u): u \\in V_G\\}\\}$ .",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "- Subgraph GNN. It treats a graph  $G$  as a set of subgraphs  $\\{\\{G^u : u \\in V_G\\}\\}$ , each obtained from  $G$  by marking a special vertex  $u \\in V_G$ . Subgraph GNN maintains a color  $\\chi_G^{\\mathrm{Sub}}(u, v)$  for each vertex  $v$  in graph  $G^u$ . Initially,  $\\chi_G^{\\mathrm{Sub}, (0)}(u, v) = (\\ell_G(v), \\mathbb{I}[u = v])$ , where the latter term distinguishes the special mark. It then runs MPNNs independently on each graph  $G^u$ :",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\chi_ {G} ^ {\\text {S u b}, (t + 1)} (u, v) = \\operatorname {h a s h} \\left(\\chi_ {G} ^ {\\text {S u b}, (t)} (u, v), \\{\\{\\chi_ {G} ^ {\\text {S u b}, (t)} (u, w): w \\in N _ {G} (v) \\} \\}\\right). \\tag {2}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            272,
+            849,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Denote the stable color of  $(u,v)$  as  $\\chi_G^{\\mathrm{Sub}}(u,v)$ . The node feature of  $u$  computed by Subgraph GNN is defined by merging all colors in  $G^{u}$ , i.e.,  $\\chi_G^{\\mathrm{Sub}}(u) \\coloneqq \\mathsf{hash}\\left(\\{\\chi_G^{\\mathrm{Sub}}(u,v):v\\in V_G\\}\\right)$ . Finally, the graph representation is defined as  $\\chi_G^{\\mathrm{Sub}}(G) = \\{\\{\\chi_G^{\\mathrm{Sub}}(u):u\\in V_G\\}\\}$ .",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
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+            47
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "page_number",
+        "text": "3",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "- Local GNN. Inspired by the  $k$ -WL test (Grohe, 2017), Local  $k$ -GNN is defined by replacing all global aggregations in  $k$ -WL by sparse ones that only aggregate local neighbors, yielding a much more efficient CR algorithm. As an example, the iteration of Local 2-GNN has the following form and enjoys the same computational complexity as a Subgraph GNN.",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\chi_ {G} ^ {\\mathrm {L}, (t + 1)} (u, v) = \\operatorname {h a s h} \\left(\\chi_ {G} ^ {\\mathrm {L}, (t)} (u, v), \\{\\{\\chi_ {G} ^ {\\mathrm {L}, (t)} (w, v): w \\in N _ {G} (u) \\} \\}, \\{\\{\\chi_ {G} ^ {\\mathrm {L}, (t)} (u, w): w \\in N _ {G} (v) \\} \\}\\right). \\tag {3}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+            161,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Initially,  $\\chi_G^{\\mathsf{L},(0)}(u,v) = (\\ell_G(u),\\ell_G(v),\\mathbb{I}[u = v],\\mathbb{I}[\\{u,v\\} \\in E_G])$ , which is called the isomorphism type of vertex pair  $(u,v)$ . We similarly denote the stable color as  $\\chi_G^{\\mathsf{L}}(u,v)$  and define the node feature  $\\chi_G^{\\mathsf{L}}(u)$  and graph representation  $\\chi_G^{\\mathsf{L}}(G)$  as in the Subgraph GNN.",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "- Folklore-type GNN. The Folklore GNN (FGNN) is inspired by the standard  $k$ -FWL test (Cai et al., 1992). As an example, the iteration formula of 2-FGNN is written as follows:",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\chi_ {G} ^ {\\mathsf {F}, (t + 1)} (u, v) = \\operatorname {h a s h} \\left(\\chi_ {G} ^ {\\mathsf {F}, (t)} (u, v), \\{\\left(\\chi_ {G} ^ {\\mathsf {F}, (t)} (w, v), \\chi_ {G} ^ {\\mathsf {F}, (t)} (u, w)\\right): w \\in V _ {G} \\} \\}\\right). \\tag {4}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "One can similarly consider the more efficient Local 2-FGNN by only aggregating local neighbors, which has the same computational complexity as Local 2-GNN and Subgraph GNN:",
+        "bbox": [
+            210,
+            303,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\chi_ {G} ^ {\\mathrm {L F}, (t + 1)} (u, v) = \\operatorname {h a s h} \\left(\\chi_ {G} ^ {\\mathrm {L F}, (t)} (u, v), \\{\\{\\left(\\chi_ {G} ^ {\\mathrm {L F}, (t)} (w, v), \\chi_ {G} ^ {\\mathrm {L F}, (t)} (u, w)\\right): w \\in N _ {G} (u) \\cup N _ {G} (v) \\} \\}\\right). \\\\ \\text {T h e s t a b l e c o l o r , n o d e f e a t u r e , a n d g r a p h r e p r e s e n t a t i o n c a n b e s i m a r l y d e f i n e d .} \\end{array} \\tag {5}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Finally, we note that the latter three types of GNNs can be naturally generalized into higher-order variants. We give a general definition of all these architectures in Appendix E.1. For the base case of  $k = 1$ , Subgraph  $(k - 1)$ -GNN, Local  $k$ -GNN, and Local  $k$ -FGNN all reduce to the MPNN.",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 HOMOMORPHISM EXPRESSIVITY OF GRAPH NEURAL NETWORKS",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 HOMOMORPHISM EXPRESSIVITY",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Given a GNN model  $M$  and a substructure  $F$ , we say  $M$  can count graph  $F$  under homomorphism if, for any graph  $G$ , the graph representation  $\\chi_G^M (G)$  determines the homomorphism count  $\\mathrm{hom}(F,G)$ . In other words,  $\\chi_G^M (G) = \\chi_H^M (H)$  implies  $\\mathrm{hom}(F,G) = \\mathrm{hom}(F,H)$  for any graphs  $G,H$ . The central question studied in this paper is, what substructures  $F$  can a GNN model  $M$  count under homomorphism? This gives rise to the notion of homomorphism expressivity defined below:",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Definition 3.1. The homomorphism expressivity of a GNN model  $M$ , denoted by  $\\mathcal{F}^M$ , is a family of (labeled) graphs satisfying the following conditions<sup>1</sup>:",
+        "bbox": [
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+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a) For any two graphs  $G,H,\\chi_G^M (G) = \\chi_H^M (H)$  iff  $\\operatorname {hom}(F,G) = \\operatorname {hom}(F,H)$  for all  $F\\in \\mathcal{F}^{M}$",
+            "b)  $\\mathcal{F}^M$  is maximal, i.e., for any graph  $F \\notin \\mathcal{F}^M$ , there exists a pair of graphs  $G, H$  such that  $\\chi_G^M(G) = \\chi_H^M(H)$  and  $\\operatorname{hom}(F, G) \\neq \\operatorname{hom}(F, H)$ ."
+        ],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Example 3.2. As a simple example, consider a maximally expressive GNN  $M$  that can solve the graph isomorphism problem, i.e., it computes the same representation for two graphs iff they are isomorphic. Then,  $\\mathcal{F}^M$  contains all graphs. This is a classic result proved in Lovász (1967).",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The significance of homomorphism expressivity can be justified in the following aspects. First, it is a complete expressivity measure. Based on item (a), the homomorphism count within  $\\mathcal{F}^M$  essentially captures all information embedded in the graph representation computed by model  $M$ . This contrasts with previously studied metrics such as the ability to compute biconnectivity properties (Zhang et al., 2023b) or count cycles (Huang et al., 2023), which only reflects restricted aspects of expressivity. Second, homomorphism expressivity is a quantitative measure and is much finer than qualitative expressivity results obtained from the graph isomorphism test. Specifically, by item (a), a GNN model  $M_1$  is more expressive than another model  $M_2$  in distinguishing non-isomorphic graphs iff  $\\mathcal{F}^{M_2} \\subset \\mathcal{F}^{M_1}$ . Furthermore, by item (b),  $M_1$  is strictly more expressive than  $M_2$  iff  $\\mathcal{F}^{M_2} \\subsetneq \\mathcal{F}^{M_1}$ , and the expressivity gap can be quantitatively understood via the set difference  $\\mathcal{F}^{M_1} \\setminus \\mathcal{F}^{M_2}$ .",
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+    {
+        "type": "text",
+        "text": "Consequently, by deriving which graphs are encompassed in the graph family  $\\mathcal{F}^M$ , homomorphism expressivity provides a novel way to analyze and compare the expressivity of GNN models. In the next subsection, we will give exact characterizations of  $\\mathcal{F}^M$  for all models  $M$  defined in Section 2.",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+    },
+    {
+        "type": "page_footnote",
+        "text": "<sup>1</sup>While homomorphism expressivity exists for all common GNNs such as the ones in Section 2, we note that it may not be well-defined for certain pathological GNNs. See Appendix F.1 for a deep discussion on it.",
+        "bbox": [
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+    {
+        "type": "page_number",
+        "text": "4",
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+    {
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+        "img_path": "images/367ee691a84a43507936b5344ade34b24c0f77306722872c498684d810fe1472.jpg",
+        "image_caption": [
+            "(a) Illustration of NED"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "img_path": "images/1f2dea9b8014da564ef24f81d834fcb7ed1b46ee56f322fed817aaea5b3cb53d.jpg",
+        "image_caption": [
+            "(b) Examples of endpoint-shared/strong/almost-strong/general NED"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "img_path": "images/6595a83d633e94cfae26f06d8ee5547d7af51a42b08bdbfe883a45d9e8a38b81.jpg",
+        "image_caption": [
+            "Figure 1: Illustration of NED and its variants. The number  $j$  next to each edge indicates that the edge belongs to the ear  $P_{j}$ . Different colors represent different ears. See Figure 9 for more examples."
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "3.2 MAIN RESULTS",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To derive our main results, we leverage a concept in graph theory known as nested ear decomposition (NED), which is originally introduced in Eppstein (1992). Here, we adapt the definition as follows:",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Definition 3.3. Given a graph  $G$ , a NED  $\\mathcal{P}$  is a partition of the edge set  $E_G$  into a sequence of simple paths  $P_1, \\dots, P_m$  (called ears), which satisfies the following conditions:",
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+        "list_items": [
+            "- Any two ears  $P_{i}$  and  $P_{j}$  with indices  $1 \\leq i < j \\leq c$  do not intersect, where  $c$  is the number of connected components of  $G$ .",
+            "- For each ear  $P_{j}$  with index  $j > c$ , there is an ear  $P_{i}$  with index  $1 \\leq i < j$  such that one or two endpoints of  $P_{j}$  lie in ear  $P_{i}$  (we say  $P_{j}$  is nested on  $P_{i}$ ). Moreover, except for the endpoints lying in ear  $P_{i}$ , no other vertices in  $P_{j}$  are in any previous ear  $P_{k}$  for  $1 \\leq k < j$ . If both endpoints of  $P_{j}$  lie in  $P_{i}$ , the subpath in  $P_{i}$  that shares the endpoints of  $P_{j}$  is called the nested interval of  $P_{j}$  in  $P_{i}$ , denoted as  $I(P_{j}) \\subset P_{i}$ . If only one endpoint lies in  $P_{i}$ , define  $I(P_{j}) = \\emptyset$ .",
+            "- For all ears  $P_{j}, P_{k}$  with  $c < j < k \\leq m$ , either  $I(P_{j}) \\cap I(P_{k}) = \\emptyset$  or  $I(P_{j}) \\subset I(P_{k})$ ."
+        ],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Intuitively, Definition 3.3 states that the relation between different ears forms a forest, in that each ear is nested on its parent. Moreover, the nested intervals either do not intersect or have inclusion relations for different children of the same parent ear. We give illustrations of NED in Figure 1.",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we considerably extend the concept of NED to several variants defined below:",
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+        "list_items": [
+            "- Endpoint-shared NED: a NED is called endpoint-shared if all ears with non-empty nested intervals share a common endpoint (see Figure 1(b,1)).",
+            "- **Strong NED:** a NED is called strong if for any two children  $P_j$ ,  $P_k$  ( $j < k$ ) nested on the same parent ear, we have  $I(P_j) \\subset I(P_k)$  (see Figure 1(b,2)).",
+            "- Almost-strong NED: a NED is called almost-strong if for any children  $P_j$ ,  $P_k$  ( $j < k$ ) nested on the same parent ear and  $|I(P_j)| > 1$ , we have  $I(P_j) \\subset I(P_k)$  (see Figure 1(b,3))."
+        ],
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We are now ready to present our main results:",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.4. For all GNN models  $M$  defined in Section 2, the graph family  $\\mathcal{F}^M$  satisfying Definition 3.1 exists (and is unique). Moreover, each  $\\mathcal{F}^M$  can be separately described below:",
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+            "- MPNN:  $\\mathcal{F}^{\\mathrm{MP}} = \\{F:F$  is a forest\\};",
+            "- Subgraph GNN:  $\\mathcal{F}^{\\mathrm{Sub}} = \\{F:F$  has an endpoint-shared NED\\};",
+            "- Local 2-GNN:  $\\mathcal{F}^{\\mathsf{L}} = \\{F:F$  has a strong NED\\};",
+            "- Local 2-FGNN:  $\\mathcal{F}^{\\mathrm{LF}} = \\{F:F$  has an almost-strong NED\\};",
+            "- 2-FGNN:  $\\mathcal{F}^{\\mathrm{F}} = \\{F:F$  has a NED\\}."
+        ],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Theorem 3.4 gives a unified description of the homomorphism expressivity for all popular GNN models defined in Section 2. Despite the elegant conclusion, the proof process is actually involved and represents a major technical contribution, so we present a proof sketch below. Our proof is divided into three parts, presented in Appendices C.2 to C.4. First, we show the existence of  $\\mathcal{F}^M$  for each model  $M$  based on a beautiful theory developed in Dell et al. (2018). Using the technique of unfolding tree, we prove that  $\\mathcal{F}^M$  at least contains all graphs  $F$  that allow a specific type of tree decomposition (Diestel, 2017), and the homomorphism information of these graphs determines the representation of any graph  $G$  computed by  $M$  (i.e., Definition 3.1(a) holds). However, characterizing  $\\mathcal{F}^M$  in terms of tree decomposition is sophisticated and not intuitive for most models  $M$ . In the next step, we give an equivalent description of  $\\mathcal{F}^M$  based on novel extensions of NED proposed in Definition 3.3, which is simpler and more elegant. In the last step, we prove that  $\\mathcal{F}^M$  does not contain other graphs. This is achieved by building non-trivial relations between three distinct theoretical tools: tree decomposition, pebble game (Cai et al., 1992), and Fürer graph (Fürer, 2001). Through a fine-grained analysis of the Fürer graphs expanded by  $F \\notin \\mathcal{F}^M$  (see Theorems C.47 and C.53), we show they are precisely a pair of graphs satisfying Definition 3.1(b), thus concluding the proof.",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
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+            47
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "page_number",
+        "text": "5",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Discussions with Dvořák (2010); Dell et al. (2018). Our work significantly extends a beautiful theory developed in Dvořák (2010); Dell et al. (2018), which showed that a pair of graphs  $G$ ,  $H$  are indistinguishable by 1-WL iff  $\\operatorname{hom}(F, G) = \\operatorname{hom}(F, H)$  for all trees  $F$ , and more generally, they are indistinguishable by  $k$ -FWL iff  $\\operatorname{hom}(F, G) = \\operatorname{hom}(F, H)$  for all graphs  $F$  of bounded treewidth  $k$ . In this paper, we successfully generalize these results to a broad range of practical GNN models. Moreover, two distinct contributions are worth discussing. First, we highlight a key insight that homomorphism can serve as a fundamental expressivity measure, which has far-reaching consequences as will be elaborated in Section 4. To show that  $\\mathcal{F}^M$  is a valid expressivity measure, we prove an extra and non-trivial result that  $\\mathcal{F}^M$  is maximal (Definition 3.1(b)). Without this crucial property,  $\\mathcal{F}^{M_1} \\supsetneq \\mathcal{F}^{M_2}$  will not necessarily mean that model  $M_1$  is strictly more expressive than  $M_2$ , thus preventing any quantitative comparison between models. Second, Dell et al. (2018) leveraged treewidth to describe results, which, unfortunately, cannot be applied to most GNN models studied here. Instead, we resort to the novel concept of NED, by which we successfully derive unified and elegant descriptions for all models. Moreover, as will be shown later, NED is quite flexible and can be naturally generalized to node/edge-level expressivity, which is not studied in prior work.",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Finally, we remark that one can derive an equivalent (perhaps simpler) description of  $\\mathcal{F}^{\\mathrm{Sub}}$ , based on the fact that a graph  $F$  has an endpoint-shared NED iff  $F$  becomes a forest when deleting the shared endpoint. Formally, denoting by  $F \\backslash \\{u\\}$  the induced subgraph of  $F$  over  $V_F \\backslash \\{u\\}$ , we have",
+        "bbox": [
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+            318,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Corollary 3.5.  $\\mathcal{F}^{\\mathrm{Sub}} = \\{F:\\exists u\\in V_F$  s.t.  $F\\backslash \\{u\\}$  is a forest}.",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.3 EXTENDING TO NODE/EDGE-LEVEL EXPRESSIVITY",
+        "text_level": 1,
+        "bbox": [
+            171,
+            392,
+            566,
+            406
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "So far, this paper mainly focuses on the graph-level expressivity, i.e., what information is encoded in the graph representation. In this subsection, we extend all results in Theorem 3.4 to the more fine-grained node/edge-level expressivity by answering what information is encoded in the node/edge features of a GNN (i.e.,  $\\chi_G^M(u)$  or  $\\chi_G^M(u,v)$  in Section 2). This yields the following definition:",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Definition 3.6. The node-level homomorphism expressivity of a GNN model  $M$ , denoted by  $\\mathcal{F}_{\\mathrm{n}}^{M}$ , is a family of connected rooted graphs satisfying the following conditions:",
+        "bbox": [
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+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a) For any connected graphs  $G,H$  and vertices  $u\\in V_G,v\\in V_H,\\chi_G^M (u) = \\chi_H^M (v)$  iff  $\\operatorname {hom}(F^w,G^u) = \\operatorname {hom}(F^w,H^v)$  for all  $F^w\\in \\mathcal{F}_{\\mathfrak{n}}^{M}$",
+            "b) For any connected rooted graph  $F^w \\notin \\mathcal{F}_{\\mathfrak{n}}^{M}$ , there exists a pair of connected graphs  $G, H$  and vertices  $u \\in V_G, v \\in V_H$  such that  $\\chi_G^M(u) = \\chi_H^M(v)$  and  $\\operatorname{hom}(F^w, G^u) \\neq \\operatorname{hom}(F^w, H^v)$ ."
+        ],
+        "bbox": [
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+            508,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "One can similarly define the edge-level homomorphism expressivity  $\\mathcal{F}_{\\mathrm{e}}^{M}$  to be a family of connected rooted graphs, each marking two special vertices (we omit the definition for clarity). The following result exactly characterizes  $\\mathcal{F}_{\\mathrm{n}}^{M}$  and  $\\mathcal{F}_{\\mathrm{e}}^{M}$  for all models  $M$  considered in this paper:",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.7. For all model  $M$  defined in Section 2,  $\\mathcal{F}_{\\mathfrak{n}}^{M}$  and  $\\mathcal{F}_{\\mathrm{e}}^{M}$  (except MPNN) exist. Moreover,",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- MPNN:  $\\mathcal{F}_{\\mathfrak{n}}^{\\mathsf{MP}} = \\{F^{w}: F \\text{ is a tree}\\}$ ;",
+            "- Subgraph GNN:"
+        ],
+        "bbox": [
+            197,
+            641,
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+            675
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "$\\mathcal{F}_{\\mathtt{n}}^{\\mathtt{Sub}} = \\{F^{w}:F$  has a NED with shared endpoint  $w\\} = \\{F^{w}:F\\backslash \\{w\\}$  is a forest},  $\\mathcal{F}_{\\mathtt{e}}^{\\mathtt{Sub}} = \\{F^{wx}:F$  has a NED with shared endpoint  $w\\} = \\{F^{wx}:F\\backslash \\{w\\}$  is a forest};",
+        "bbox": [
+            210,
+            674,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "- 2-FGNN: \\(\\mathcal{F}_{\\mathfrak{n}}^{\\mathsf{F}} = \\{F^{w}: F\\) has a NED where \\(w\\) is an endpoint of the first ear}\\(, \\)\\mathcal{F}_{\\mathrm{e}}^{\\mathrm{F}} = \\{F^{wx}: F\\) has a NED where \\(w\\) and \\(x\\) are endpoints of the first ear\\}$.",
+        "bbox": [
+            197,
+            707,
+            725,
+            737
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The cases of Local 2-GNN and Local 2-FGNN are similar to 2-FGNN by replacing \"NED\" with \"strong NED\" and \"almost-strong NED\", respectively.",
+        "bbox": [
+            171,
+            739,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In summary, the node/edge-level homomorphism expressivity can be naturally described using NED by further specifying the endpoints of the first ear.",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.4 EXTENDING TO HIGHER-ORDER GNNS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            816,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Finally, we discuss how our results can be naturally extended to higher-order GNNs, thus providing a complete picture of the homomorphism expressivity hierarchy for infinitely many architectures. We focus on three representative examples: Subgraph  $k$ -GNN (Qian et al., 2022), Local  $k$ -GNN (Morris et al., 2020), and  $k$ -FGNN (Azizian & Lelarge, 2021). Subgraph  $k$ -GNN extracts a graph  $G^u$  for each vertex  $k$ -tuple  $u \\in V_G^k$  and runs MPNNs independently, which recovers Subgraph GNN when  $k = 1$ . As the reader may have guessed, the following result exactly parallels Corollary 3.5:",
+        "bbox": [
+            169,
+            839,
+            823,
+            925
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "page_number",
+        "text": "6",
+        "bbox": [
+            493,
+            948,
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+            959
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.8. The homomorphism expressivity of Subgraph  $k$ -GNN exists and can be described as  $\\mathcal{F}^{\\mathrm{Sub}(k)} = \\{F : \\exists U \\subset V_F \\text{ s.t. } |U| \\leq k \\text{ and } F \\backslash U \\text{ is a forest}\\}$ .",
+        "bbox": [
+            169,
+            103,
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+            133
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We next turn to Local  $k$ -GNN. To describe the result, we introduce a novel extension of Definition 3.3, called the  $k$ -order ear. Intuitively, it is formed by a graph of no more than  $k$  vertices, plus  $k$  paths each linking to a vertex in the graph (see Figure 2(a) for an illustration). Note that a 2-order ear is exactly a simple path. Then, we can naturally define the nested \"interval\" (see the solid orange lines in Figure 2(b) for an illustration) and thus define the concept of  $k$ -order strong NED. Due to space limit, a formal definition is deferred to Definition E.3. We have the following main result:",
+        "bbox": [
+            169,
+            138,
+            544,
+            292
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/963dfb2c3d1f660e84e4e3dfed0f1c537c67de224bfb1ed809069755040a7d30.jpg",
+        "image_caption": [
+            "Figure 2: Illustration of higher-order ears. Each curve indicates a path (with possibly zero length) and each straight segment indicates an edge."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            558,
+            138,
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+            219
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.9. The homomorphism expressivity of Local  $k$ -GNN exists and can be described as  $\\mathcal{F}^{\\mathsf{L}(k)} = \\{F : F \\text{ has a } k \\text{-order strong NED}\\}$ .",
+        "bbox": [
+            169,
+            296,
+            823,
+            325
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Finally, let us consider the standard  $k$ -FGNN (or equivalently, the  $k$ -FWL). Unfortunately, we cannot find a description of its homomorphism expressivity based on some form of higher-order NED; nevertheless, it is easy to describe the results using the notion of treewidth (see Definition C.2). Specifically, denoting  $\\mathrm{tw}(F)$  to be the treewidth of graph  $F$ , we have the following result:",
+        "bbox": [
+            169,
+            330,
+            823,
+            388
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.10. The homomorphism expressivity of  $k$ -FGNN exists and can be described as  $\\mathcal{F}^{\\mathsf{F}(k)} = \\{F : \\mathrm{tw}(F) \\leq k\\}$ .",
+        "bbox": [
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+            392,
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+            422
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Interestingly, one can see that  $\\mathcal{F}^{\\mathrm{Sub}(0)}$ ,  $\\mathcal{F}^{\\mathrm{L}(1)}$ , and  $\\mathcal{F}^{\\mathrm{F}(1)}$  all degenerate to the family of forests, which coincides with the fact that all these higher-order GNNs reduces to MPNN for the base case.",
+        "bbox": [
+            169,
+            434,
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+            465
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4 IMPLICATIONS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            484,
+            328,
+            500
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The previous section has provided a complete description of the homomorphism expressivity for a variety of GNN models. In this section, we highlight the significance of these results through three different contexts. We will show how homomorphism expressivity can be used to link different GNN subareas, provide new insights into various known results, and answer a number of open problems.",
+        "bbox": [
+            169,
+            516,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.1 QUALITATIVE EXPRESSIVITY COMPARISON",
+        "text_level": 1,
+        "bbox": [
+            171,
+            590,
+            511,
+            604
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "One direct corollary of Theorem 3.4 is that it readily enables expressivity comparison among all models in Section 2. This can be summarized below:",
+        "bbox": [
+            169,
+            614,
+            823,
+            643
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Corollary 4.1. Under the notation of Theorem 3.4,  $\\mathcal{F}^{\\mathrm{MP}}\\subsetneq \\mathcal{F}^{\\mathrm{Sub}}\\subsetneq \\mathcal{F}^{\\mathrm{L}}\\subsetneq \\mathcal{F}^{\\mathrm{LF}}\\subsetneq \\mathcal{F}^{\\mathrm{F}}$ . Thus, the expressive power of the following GNN models strictly increases in order (in terms of distinguishing non-isomorphic graphs): MPNN, Subgraph GNN, Local 2-GNN, Local 2-FGNN, and 2-FGNN.",
+        "bbox": [
+            169,
+            647,
+            823,
+            691
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Proof.  $\\mathcal{F}^{\\mathrm{MP}}\\subset \\mathcal{F}^{\\mathrm{Sub}}$  follows from Corollary 3.5 and the fact that deleting any vertex of a forest yields a forest.  $\\mathcal{F}^{\\mathrm{Sub}}\\subset \\mathcal{F}^{\\mathrm{L}}$  follows by the fact that any endpoint-shared NED is a strong NED.  $\\mathcal{F}^{\\mathrm{L}}\\subset \\mathcal{F}^{\\mathrm{LF}}\\subset \\mathcal{F}^{\\mathrm{F}}$  follows similarly since any strong NED is an almost-strong NED and any almost-strong NED is a NED. To prove strict separation results, one can check that the four graphs in Figure 1(b) precisely reveal the gap between each pair of graph families, thus concluding the proof.",
+        "bbox": [
+            169,
+            698,
+            823,
+            768
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Corollary 4.1 recovers a series of results recently proved in Zhang et al. (2023a); Frasca et al. (2022). Compared to their results, our approach draws a much clearer picture of the expressivity gap between different architectures and essentially answers how large the gaps are. Moreover, we provide systematic guidance for finding counterexample graphs that unveil the expressivity gap: as shown in Corollary C.54, any graph  $F \\in \\mathcal{F}^{M_2} \\backslash \\mathcal{F}^{M_1}$  immediately gives a pair of non-isomorphic graphs that reveals the gap between models  $M_1$  and  $M_2$ . We note that this readily recovers the counterexamples constructed in Zhang et al. (2023a) and greatly simplifies their sophisticated case-by-case analysis.",
+        "bbox": [
+            169,
+            777,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We next turn to three types of higher-order GNNs studied in Section 3.4, for which we can establish a complete expressiveness hierarchy, as presented in Corollary 4.2. A graphical illustration of these results is given in Figure 3.",
+        "bbox": [
+            169,
+            881,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "page_number",
+        "text": "7",
+        "bbox": [
+            493,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/fca9ed91942914b08d04507aa0dd77121cccd68d26f7701733b23158a27efb31.jpg",
+        "image_caption": [
+            "Figure 3: Expressiveness hierarchy of MPNN, Subgraph GNN, Local GNN, and FGNN."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            209,
+            89,
+            785,
+            167
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Corollary 4.2. Under the notations in Section 3.4, for any  $k > 0$ , the following hold:",
+        "bbox": [
+            171,
+            200,
+            735,
+            215
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a)  $\\mathcal{F}^{\\mathrm{Sub}(k - 1)}\\subsetneq \\mathcal{F}^{\\mathrm{Sub}(k)}$  . I.e., the expressive power of Subgraph  $k$  GNN strictly increases with  $k$",
+            "b)  $\\mathcal{F}^{\\mathsf{L}(k)}\\subsetneq \\mathcal{F}^{\\mathsf{L}(k + 1)}$  . I.e., the expressive power of Local  $k$  -GNN strictly increases with  $k$",
+            "c)  $\\mathcal{F}^{\\mathrm{Sub}(k)}\\subsetneq \\mathcal{F}^{\\mathrm{L}(k + 1)}$  . I.e., Local  $(k + 1)$  -GNN is strictly more expressive than Subgraph  $k$  GNN;",
+            "d)  $\\mathcal{F}^{\\mathsf{F}(k)}\\subsetneq \\mathcal{F}^{\\mathsf{L}(k + 1)}\\subsetneq \\mathcal{F}^{\\mathsf{F}(k + 1)}$  . I.e., the expressive power of Local  $(k + 1)$  -GNN lies strictly between  $k$  -FWL and  $(k + 1)$  -FWL;",
+            "e)  $\\mathcal{F}^{\\mathrm{Sub}(k)}\\subsetneq \\mathcal{F}^{\\mathrm{F}(k + 1)}$ , and for all  $k > 1$ ,  $\\mathcal{F}^{\\mathrm{Sub}(k)}\\backslash \\mathcal{F}^{\\mathrm{F}(k + 1)}\\neq \\emptyset$  and  $\\mathcal{F}^{\\mathrm{F}(k + 1)}\\backslash \\mathcal{F}^{\\mathrm{Sub}(k)}\\neq \\emptyset$ . In other words, the expressive power of Subgraph  $k$ -GNN lies strictly within  $(k + 1)$ -FWL, but it is incomparable to  $k$ -FWL when  $k > 1$ ."
+        ],
+        "bbox": [
+            189,
+            215,
+            823,
+            335
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Corollary 4.2 recovers results in Morris et al. (2020); Qian et al. (2022) and further answers two open problems. First, Corollary 4.2(c) is a new result that bridges Morris et al. (2020) with Qian et al. (2022) and partially answers an open question in Zhang et al. (2023a, Appendix C). Another new result is Corollary 4.2(d), which essentially answers a fundamental open problem raised in Frasca et al. (2022, Appendix E), showing that their proposed  $\\mathsf{RelGN}(k)$  model is bounded by  $k$ -FWL with an inherent expressivity gap (see Appendix E.4 for a detailed discussion). To sum up, all these challenging open problems become straightforward through the lens of homomorphism expressivity.",
+        "bbox": [
+            169,
+            342,
+            826,
+            441
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.2 SUBGRAPH COUNTING POWER",
+        "text_level": 1,
+        "bbox": [
+            171,
+            457,
+            426,
+            470
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The significance of homomorphism expressivity can go much beyond qualitative comparisons between models. As another implication, it provides a systematic way to study GNNs' ability to encode structural information such as subgraph count, which has been found crucial in numerous practical applications. Specifically, a well-known result in graph theory states that, for any graphs  $F, G$ , the subgraph count  $\\operatorname{sub}(F, G)$  can be determined by  $\\operatorname{hom}(\\tilde{F}, G)$  where  $\\tilde{F}$  ranges over all homomorphic images of  $F$  (i.e.,  $\\operatorname{Spasm}(F)$ , see Section 2) (Lovász, 2012; Curticapean et al., 2017).",
+        "bbox": [
+            169,
+            482,
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+            570
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Mathematically, given any graph  $F$ , let  $\\operatorname{Spasm}^{\\mathcal{Z}}(F)$  be any maximal set of pairwise non-isomorphic graphs chosen from  $\\operatorname{Spasm}(F)$  (see Figure 4(a) for an illustration). Then, we have the following linear relation for all graph  $G$ :",
+        "bbox": [
+            169,
+            577,
+            823,
+            619
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\operatorname {s u b} (F, G) = \\sum_ {\\tilde {F} \\in \\operatorname {S p a s m} ^ {\\neq} (F)} \\alpha (F, \\tilde {F}) \\cdot \\operatorname {h o m} (\\tilde {F}, G), \\tag {6}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            334,
+            623,
+            823,
+            659
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "where  $\\alpha (F,\\tilde{F})\\neq 0$  is a constant scalar coefficient independent of  $G$ . Based on this formula, we can easily study the subgraph counting power of GNN models as shown in Proposition 4.4.",
+        "bbox": [
+            169,
+            665,
+            823,
+            695
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Definition 4.3. Given a GNN model  $M$ , we say  $M$  can subgraph-count graph  $F$  at graph-level if  $\\chi_G^M (G) = \\chi_H^M (H)$  implies  $\\operatorname {sub}(F,G) = \\operatorname {sub}(F,H)$  for any graphs  $G,H$ . We say  $M$  can subgraphcount rooted graph  $F^{w}$  at node-level if  $\\chi_G^M (u) = \\chi_H^M (v)$  implies  $\\operatorname {sub}(F^w,G^u) = \\operatorname {sub}(F^w,H^v)$  for any graphs  $G,H$  and vertices  $u\\in V_{G},v\\in V_{H}$ . We can similarly define the edge-level subgraph counting ability for rooted graphs marking two special vertices.",
+        "bbox": [
+            169,
+            696,
+            823,
+            768
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Proposition 4.4. For any GNN model  $M$  defined in Section 2, it can subgraph-count graph  $F$  (at graph-level) if  $\\tilde{F} \\in \\mathcal{F}^M$  for all  $\\tilde{F} \\in \\operatorname{Spasm}(F)$ . It can subgraph-count  $F^w$  (at node-level) if  $\\tilde{F}^w \\in \\mathcal{F}_{\\mathfrak{n}}^M$  for all  $\\tilde{F}^w \\in \\operatorname{Spasm}(F^w)$ . A similar result holds for edge-level subgraph counting.",
+        "bbox": [
+            169,
+            771,
+            826,
+            820
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The above proposition offers a simple way to affirm the ability of a GNN model  $M$  to subgraph-count any pattern at graph/node/edge-level. On the other hand, one may wonder whether the converse direction also holds, i.e.,  $M$  cannot subgraph-count  $F$  if there exists a homomorphic image  $\\tilde{F} \\in \\mathsf{Spasm}(F)$  such that  $\\tilde{F} \\notin \\mathcal{F}^M$ . We find that it is indeed the case. Specifically, if the set  $\\mathsf{Spasm}(F) \\backslash \\mathcal{F}^M$  is not empty, then one can always find a pair of counterexample graphs  $G, H$  such that  $\\chi_G^M(G) = \\chi_H^M(H)$  but  $\\mathsf{sub}(F, G) \\neq \\mathsf{sub}(F, H)$ . We eventually arrive at the following main theorem (see Appendix G.1 for a proof):",
+        "bbox": [
+            169,
+            824,
+            825,
+            925
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "page_number",
+        "text": "8",
+        "bbox": [
+            493,
+            948,
+            503,
+            959
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4.5. For any GNN model  $M$  such that their homomorphism expressivity  $\\mathcal{F}^M$  exists,  $M$  can subgraph-count  $F$  iff  $\\operatorname{Spasm}(F) \\subset \\mathcal{F}^M$ . Similar results hold for rooted graphs  $F^u / F^{uv}$  by replacing  $\\mathcal{F}^M$  with node/edge-level homomorphism expressivity  $\\mathcal{F}_{\\mathfrak{n}}^M / \\mathcal{F}_{\\mathfrak{e}}^M$ .",
+        "bbox": [
+            169,
+            103,
+            823,
+            148
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Example 4.6. As an example, we can readily characterize the cycle/path counting power of various GNNs. Denote by  $C_n / P_n$  the simple cycle/path of  $n$  vertices. Let  $\\{u, v\\} \\in E_{C_n}$  be any edge in  $C_n$ , and  $\\{w, x\\} \\in E_{P_n}$  be any edge in  $P_n$  where  $w$  is an endpoint of  $P_n$ . The following table lists exactly all cycles/paths each model can count at graph/node/edge-level.",
+        "bbox": [
+            169,
+            148,
+            823,
+            205
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/a5c6deac3ad30fe57c2b93dc89c2ec7c1e0350d5f908173ffcce4592f6ef8727.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Structure Model</td><td colspan=\"3\">Cycle</td><td colspan=\"3\">Path</td></tr><tr><td>\\( C_n \\)</td><td>\\( C_n^u \\)</td><td>\\( C_n^{uv} \\)</td><td>\\( P_n \\)</td><td>\\( P_n^w \\)</td><td>\\( P_n^{wx} \\)</td></tr><tr><td>MPNN</td><td>None</td><td>None</td><td>None</td><td>n≤3</td><td>n≤3</td><td>n≤3</td></tr><tr><td>Subgraph GNN</td><td>n≤7</td><td>n≤4</td><td>n≤4</td><td>n≤7</td><td>n≤4</td><td>n≤4</td></tr><tr><td>Local 2-GNN</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Local 2-FGNN</td><td colspan=\"3\">n≤7</td><td colspan=\"3\">n≤7</td></tr><tr><td>2-FGNN</td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>",
+        "bbox": [
+            173,
+            210,
+            529,
+            306
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/0b8c8167c57ee57277750953d00c94ad88af1049db49016e3f2c9112fed7d752.jpg",
+        "image_caption": [
+            "(a)  $\\mathsf{Spasm}^{\\neq}(C_6)$  has 10 graphs. (b) Rooted  $C_6$  Figure 4: Illustration of homomorphic images of the 6-cycle and rooted 6-cycle."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            558,
+            208,
+            714,
+            263
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/b30b49dc435d40d1f52f70b72039407f218c2615658acf1500e994bd1a523e89.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            754,
+            208,
+            818,
+            263
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "(b) Rooted  $C_6$",
+        "bbox": [
+            743,
+            267,
+            830,
+            280
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "**Discussions with prior work.** Our results significantly extend Huang et al. (2023) in several aspects. First, we show Subgraph GNN can count 6-cycle at graph-level by simply enumerating its spasm (see Figure 4(a)). However, it cannot count rooted 5/6-cycle at node-level because the homomorphic image can contain cycles that do not pass the marked vertex (see Figure 4(b)). This provides novel insights into Huang et al. (2023) and extends their results (albeit with a simpler analysis). Second, we reveal that Local 2-GNN can already count all cycles/pathways that 2-FWL can count (even at edge-level). This identifies a new architecture with both efficiency and strong expressiveness in subgraph counting, considerably extending the finding in the concurrent work of Zhou et al. (2023b).",
+        "bbox": [
+            169,
+            318,
+            823,
+            431
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In Appendix G.2 (Tables 4 and 5), we summarize the statistics of all moderate-size patterns each model can count under homomorphisms/subgraphs, which enables quantitative expressivity comparisons of different models in a clear and exact manner. We also comprehensively list the counting ability of all moderate-size patterns in Table 6, which we believe can be helpful for future research.",
+        "bbox": [
+            169,
+            436,
+            823,
+            494
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.3 POLYNOMIAL EXPRESSIVITY",
+        "text_level": 1,
+        "bbox": [
+            171,
+            508,
+            415,
+            523
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "As the third implication, homomorphism expressivity is closely related to the polynomial expressivity recently proposed in Puny et al. (2023). Concretely, given a model  $M$ , a graph  $F$  is in  $\\mathcal{F}^M$  if  $M$  can express the invariant graph polynomial  $P_F$  (defined in Puny et al. (2023), Section 2.2), and a rooted graph  $F^{uv}$  is in  $\\mathcal{F}_{\\mathrm{e}}^{M}$  if  $M$  can express the equivariant graph polynomial  $P_{F^{uv}}$ . Based on this connection, our work introduces a novel toolbox for studying polynomial expressivity via the NED framework and offers new insights into which graph polynomials can be computed for a variety of practical GNNs. Moreover, we readily settle an open question in Puny et al. (2023), which upper bounds the polynomial expressivity for their proposed PPGN++.",
+        "bbox": [
+            169,
+            535,
+            823,
+            647
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Corollary 4.7.  $PPGN++$  is bounded by (and thus as expressive as) the Prototypical edge-based model defined in Puny et al. (2023) for computing equivariant graph polynomials.",
+        "bbox": [
+            169,
+            648,
+            823,
+            679
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Due to space limit, please refer to Appendix H for proof and more discussions.",
+        "bbox": [
+            169,
+            686,
+            689,
+            702
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            720,
+            328,
+            736
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This section aims to verify our theory through a comprehensive set of experiments. In each experiment, we implement four types of GNN models listed in Section 2, i.e., MPNN, Subgraph GNN, Local 2-GNN, and Local 2-FGNN. Note that all of these models are much more efficient than 2-FWL. Our primary objective here is not to produce SOTA results, but rather to provide a unified and equitable empirical comparison among these models. To ensure fairness, we employ the same GIN-based design (Xu et al., 2019) for all models and control their model sizes and training budgets to be roughly the same on each task. Details of model configurations are given in Appendix I. Our code is available at https://github.com/subgraph23/homomorphism-expressivity.",
+        "bbox": [
+            169,
+            750,
+            823,
+            862
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Synthetic task. We first test whether these GNN models can easily learn homomorphism information from data as our theory predicts. We use the benchmark dataset from Zhao et al. (2022a) and comprehensively test the homomorphism expressivity at graph/node/edge-level by carefully selecting 8 substructures shown in Table 1. The reported performance is measured by the normalized",
+        "bbox": [
+            169,
+            867,
+            823,
+            925
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "page_number",
+        "text": "9",
+        "bbox": [
+            493,
+            948,
+            503,
+            959
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/e62b09685c256037bc23b38112b0f58f72dd12c72c5105853273a37fd43524b7.jpg",
+        "table_caption": [
+            "Table 1: Experimental results on homomorphism counting. Red/blue nodes indicate marked vertices."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Task Model</td><td colspan=\"3\">Graph-level</td><td colspan=\"2\">Node-level</td><td colspan=\"3\">Edge-level</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>MPNN</td><td>.300</td><td>.233</td><td>.254</td><td>.505</td><td>.478</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Subgraph GNN</td><td>.011</td><td>.015</td><td>.012</td><td>.004</td><td>.058</td><td>.003</td><td>.058</td><td>.048</td></tr><tr><td>Local 2-GNN</td><td>.008</td><td>.008</td><td>.010</td><td>.003</td><td>.004</td><td>.005</td><td>.006</td><td>.008</td></tr><tr><td>Local 2-FGNN</td><td>.003</td><td>.005</td><td>.004</td><td>.005</td><td>.005</td><td>.007</td><td>.007</td><td>.008</td></tr></table>",
+        "bbox": [
+            178,
+            132,
+            506,
+            208
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/5a90209828fc05203b99208c20760d031ba5b56b6aa476ccbf3fb19cfdfeb4c0.jpg",
+        "table_caption": [
+            "Table 2: Experimental results on ZINC and Alchemy datasets. See Appendix I.4 for comparisons of more GNN models in literature."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Task \nModel</td><td colspan=\"2\">ZINC</td><td rowspan=\"2\">Alchemy</td></tr><tr><td>Subset</td><td>Full</td></tr><tr><td>MPNN</td><td>.138 ± .006</td><td>.030 ± .002</td><td>.122 ± .002</td></tr><tr><td>Subgraph GNN</td><td>.110 ± .007</td><td>.028 ± .002</td><td>.116 ± .001</td></tr><tr><td>Local 2-GNN</td><td>.069 ± .001</td><td>.024 ± .002</td><td>.114 ± .001</td></tr><tr><td>Local 2-FGNN</td><td>.064 ± .002</td><td>.023 ± .001</td><td>.111 ± .001</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/89a5cbe27b4547b0d4d3e4f3ab03559de369f305a6ff1696267918849f9cd4ba.jpg",
+        "table_caption": [
+            "Table 3: Experimental results on the (Chordal) Cycle Counting task."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Task Model</td><td colspan=\"6\">Graph-level</td><td colspan=\"6\">Node-level</td><td colspan=\"6\">Edge-level</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>MPNN</td><td>.358</td><td>.208</td><td>.188</td><td>.146</td><td>.261</td><td>.205</td><td>.600</td><td>.413</td><td>.300</td><td>.207</td><td>.318</td><td>.237</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Subgraph GNN</td><td>.010</td><td>.020</td><td>.024</td><td>.046</td><td>.007</td><td>.027</td><td>.003</td><td>.005</td><td>.092</td><td>.082</td><td>.050</td><td>.073</td><td>.001</td><td>.003</td><td>.090</td><td>.096</td><td>.038</td><td>.065</td></tr><tr><td>Local 2-GNN</td><td>.008</td><td>.011</td><td>.017</td><td>.034</td><td>.007</td><td>.016</td><td>.002</td><td>.005</td><td>.010</td><td>.023</td><td>.004</td><td>.015</td><td>.001</td><td>.005</td><td>.010</td><td>.019</td><td>.005</td><td>.014</td></tr><tr><td>Local 2-FGNN</td><td>.003</td><td>.004</td><td>.010</td><td>.020</td><td>.003</td><td>.010</td><td>.004</td><td>.006</td><td>.012</td><td>.021</td><td>.004</td><td>.014</td><td>.003</td><td>.006</td><td>.012</td><td>.022</td><td>.005</td><td>.012</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Mean Absolute Error (MAE) on the test dataset. It can be seen that the model performance indeed correlates to our theoretical predictions: (i) MPNN cannot encode any substructure under homomorphism; (ii) Subgraph GNN cannot encode the 2th, 3rd, 5th, 7th, 8th substructures; (iii) Local 2-GNN cannot encode the 3rd and 8th substructures; (iv) Local 2-FGNN can encode all substructures.",
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+    {
+        "type": "text",
+        "text": "Cycle counting power. Cycles are important structures in numerous graph learning tasks, yet encoding them is notoriously hard for GNNs. We next test the ability of different GNN models to subgraph-count (chordal) cycles at graph/node/edge-level. We follow the setting in Frasca et al. (2022); Zhang et al. (2023a); Huang et al. (2023) and present results in Table 3 (measured by the normalized test MAE). Remarkably, despite the same computational cost and model size, Local 2-(F)GNN performs significantly better than Subgraph GNN and achieves good performance for counting all  $3/4/5/6$ -cycles as well as chordal  $4/5$ -cycles (even at edge-level). These results match Example 4.6 and may suggest Local 2-(F)GNN as generic, efficient, yet powerful architectures in solving chemical and biological tasks where counting cycles is essential (e.g., benzene rings).",
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+    {
+        "type": "text",
+        "text": "Real-world tasks. We finally test these GNN models on three real-world benchmarks: ZINC-subset, ZINC-full (Dwivedi et al., 2020), and Alchemy (Chen et al., 2019a). Following the standard configuration, all models obey a 500K parameter budget. The results are shown in Table 2. It can be seen that the performance continues to improve when a more expressive model is used. In particular, Local 2-FGNN achieves the best performance on all tasks, suggesting that its theoretical expressivity guarantee can translate to practical performance in real-world settings.",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "In this paper, we present a new framework for systematically and quantitatively studying the expressive power of various GNN architectures. Through the lens of homomorphism expressivity, we give exact descriptions of the graph family each model can encode in terms of homomorphism counting. Our framework stands as a valuable toolbox to unify the landscape between different subareas in the GNN community, providing deep insights into a number of prior works and answering their open problems. In particular, one can establish a complete expressiveness hierarchy between models, determine the subgraph counting capabilities of GNNs at graph/node/edge-level, and understand their polynomial expressivity. On the theoretical side, our results establish deep connections with a series of fundamental topics in graph theory (see Appendix A.2); On the practical side, these results closely correlate with the empirical performance of GNN models, as demonstrated through extensive experiments. Finally, Appendix B outlines several open directions for further exploration, and we believe that the homomorphism expressivity framework paves a fresh way for future study of more expressive GNNs.",
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+        "type": "text",
+        "text": "Acknowledgement Bohang Zhang would like to thank Shengjie Luo for helpful discussions. We sincerely appreciate all reviewers for the valuable suggestions. This work is supported by National Science and Technology Major Project (2022ZD0114902) and National Science Foundation of China (NSFC62276005, NSFC62376007).",
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+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+    {
+        "type": "text",
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+        ],
+        "bbox": [
+            171,
+            102,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "page_number",
+        "text": "15",
+        "bbox": [
+            490,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
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+        ],
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+    },
+    {
+        "type": "page_number",
+        "text": "16",
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+    },
+    {
+        "type": "text",
+        "text": "Appendix",
+        "text_level": 1,
+        "bbox": [
+            171,
+            130,
+            316,
+            159
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Table of Contents",
+        "text_level": 1,
+        "bbox": [
+            171,
+            181,
+            352,
+            200
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "A More Related Work 18",
+        "bbox": [
+            212,
+            209,
+            785,
+            223
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "A.1 Expressive Graph Neural Networks 18",
+            "A.2 Broader impacts and additional discussions 20"
+        ],
+        "bbox": [
+            233,
+            224,
+            785,
+            253
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "B Limitations and Open Directions 21",
+            "C Proof of Theorem 3.4 21"
+        ],
+        "bbox": [
+            212,
+            266,
+            785,
+            306
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "C.1 Preliminary 22",
+            "C.2 Part 1: tree decomposition 25",
+            "C.3 Part 2: nested ear decomposition 39",
+            "C.4 Part 3: pebble game 43"
+        ],
+        "bbox": [
+            233,
+            308,
+            785,
+            367
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "D Node/edge-level Expressivity 50",
+        "bbox": [
+            212,
+            378,
+            785,
+            393
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "D.1 Related to tree decomposition and ear decomposition 51",
+            "D.2 Counterexamples 52"
+        ],
+        "bbox": [
+            233,
+            393,
+            785,
+            422
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "E Higher-order GNNs 54",
+        "bbox": [
+            212,
+            436,
+            785,
+            449
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "E.1 Definition of higher-order GNNs 54",
+            "E.2 Higher-order strong NED 55",
+            "E.3 Proofs in Section 3.4 55",
+            "E.4 Expressivity gap between higher-order GNNs 56"
+        ],
+        "bbox": [
+            233,
+            450,
+            785,
+            510
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "F Additional Discussions 57",
+        "bbox": [
+            212,
+            523,
+            785,
+            536
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "F.1 Regarding the definition of homomorphism expressivity 57",
+            "F.2 Regarding the definition of NED 57"
+        ],
+        "bbox": [
+            233,
+            537,
+            785,
+            566
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "G Homomorphism and Subgraph Counting Power 58",
+        "bbox": [
+            212,
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+            593
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "G.1 Proof of Theorem 4.5 58",
+            "G.2 Graph statistics and examples 59"
+        ],
+        "bbox": [
+            233,
+            594,
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "H Polynomial Expressivity 69",
+        "bbox": [
+            212,
+            636,
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+            650
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "I Experimental Details 70",
+        "bbox": [
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+            676
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "I.1 Datasets 70",
+            "I.2 Model details 70",
+            "I.3 Training details 72",
+            "I.4 Performance of baseline models in literature 72"
+        ],
+        "bbox": [
+            233,
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "page_number",
+        "text": "17",
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+    },
+    {
+        "type": "text",
+        "text": "A MORE RELATED WORK",
+        "text_level": 1,
+        "bbox": [
+            171,
+            102,
+            408,
+            118
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "A.1 EXPRESSIVE GRAPH NEURAL NETWORKS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            133,
+            511,
+            148
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Since Morris et al. (2019); Xu et al. (2019) discovered the limited expressive power of MPNNs in distinguishing non-isomorphic graphs, a large amount of work has been devoted to developing GNNs with better expressiveness. Below, we briefly review representative approaches in this area. For a comprehensive survey on expressive GNNs, we refer readers to Morris et al. (2023).",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Higher-order GNNs. Inspired by the relation between MPNNs and the 1-WL test, a natural approach to designing provably more expressive GNNs is to mimic the higher-order WL tests. This gives rise to two fundamental types of higher-order GNNs. One type of GNNs mimics the  $k$ -WL test (Grohe, 2017), and representative architectures include  $k$ -GNN (Morris et al., 2019) and  $k$ -IGN (Maron et al., 2019b;c); the other type of GNNs mimics the  $k$ -FWL (Folklore WL) test (Cai et al., 1992) and is referred to as the the  $k$ -FGNN (Maron et al., 2019a). Azizian & Lelarge (2021); Geerts & Reutter (2022) proved that each of these architectures is exactly as expressive as the corresponding higher-order WL/FWL test. Therefore, the expressiveness grows strictly as the order  $k$  increases; when  $k$  approaches infinity, they can universally approximate any continuous graph functions (Chen et al., 2019b; Keriven & Peyré, 2019). However, due to the inherent computation/memory complexity, these architectures are generally not practical in real-world applications.",
+        "bbox": [
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+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Local GNNs. To improve computational efficiency, a subsequent line of work seeks to develop more scalable and practical GNN architectures by taking into account the local/sparse nature of graphs. Locality/sparsity is also an important inductive bias for graphs but is not well-exploited in higher-order GNNs, since their layer aggregation is inherently global and the graph adjacency information is only encoded in initial node features. To address these shortcomings, Morris et al. (2020) proposed the Local  $k$ -GNN (and several variants) as a replacement of  $k$ -GNN, which directly incorporates graph adjacency into network layers and only aggregates neighboring information instead of the global one. The authors further proved that Local  $k$ -GNN is strictly more expressive than  $k$ -GNN. Building upon Local  $k$ -GNN, Morris et al. (2022) proposed the  $(k,s)$ -SpeqNet that further reduces the computational cost by considering a subset of  $k$ -tuples whose vertices can be grouped into no more than  $s$  connected components. A similar idea appeared in Zhao et al. (2022b), in which the authors proposed the  $(k,s)$ -SetGNN by considering  $k$ -sets instead of  $k$ -tuples. Besides Local  $k$ -GNN, recent architectures proposed in Frasca et al. (2022) and Zhang et al. (2023a) can be analogously understood as Local  $k$ -IGN and Local  $k$ -FGNN, respectively. Very recently, Feng et al. (2023); Zhou et al. (2023b) generalized the Local  $k$ -FGNN to a broad class of Folklore-type GNNs and achieved good performance on several benchmark datasets.",
+        "bbox": [
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+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Subgraph GNNs. Graphs that are indistinguishable by WL tests typically possess a high degree of symmetry. In light of this observation, Subgraph GNNs have recently emerged as a compelling approach to designing expressive GNNs. The basic idea is to break symmetry by transforming the original graph into a collection of slightly modified subgraphs and feeding these subgraphs into a GNN model. The earliest forms of Subgraph GNNs may track back to Cotta et al. (2021); Papp et al. (2021) (albeit with a different motivation), where the authors proposed to feed node-deleted subgraphs into an MPNN. Papp & Wattenhofer (2022) later argued to use node marking instead of node deletion for better expressive power, resulting in the standard Subgraph GNN studied in this paper. Zhang & Li (2021); You et al. (2021) proposed the Nested GNN and Identity-aware GNN, both of which can be treated as variants of Subgraph GNNs that use ego networks as subgraphs. In particular, the heterogeneous message passing proposed in You et al. (2021) can also be seen as a form of node marking. We note that the model proposed in Vignac et al. (2020) can also be interpreted as a Subgraph GNN. Qian et al. (2022) proposed the higher-order Subgraph GNN by marking  $k$  nodes per subgraph, resulting in  $n^k$  different subgraphs when the original graph has  $n$  vertices. We call this architecture Subgraph  $k$ -GNN in this paper. The authors proved that Subgraph  $k$ -GNN is strictly bounded by  $(k + 1)$ -FWL and is incomparable to  $k$ -FWL when  $k > 1$ . Zhou et al. (2023a) further generalized Subgraph  $k$ -GNN to  $(l, k)$ -GNN by using  $l$ -GNN instead of MPNN to process each subgraph. It was proved that  $(l, k)$ -GNN is bounded by  $(k + l)$ -GNN for  $l \\geq 2$ .",
+        "bbox": [
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+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Recently, Subgraph GNNs have been greatly extended by further allowing interactions between subgraphs. This is achieved by designing cross-subgraph aggregation layers (rather than feeding each subgraph independently into a GNN). Bevilacqua et al. (2022) developed the Equivariant Subgraph Aggregation Network that introduces a global aggregation between subgraphs. A similar design is",
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+        "page_idx": 17
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "page_number",
+        "text": "18",
+        "bbox": [
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+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "proposed in the concurrent work of Zhao et al. (2022a). These architectures were later proved to strictly improve the expressivity of the original Subgraph GNNs (Zhang et al., 2023b;a). Frasca et al. (2022) built a general design space of Subgraph GNNs that unifies prior work and showed that all models in this space are bounded by a variant of 2-IGN (dubbed the Local 2-IGN in this paper), which is then bounded by 2-FWL. Zhang et al. (2023a) later proved that Local 2-IGN is as expressive as Local 2-GNN and strictly less expressive than 2-FWL (2-FGNN). In this paper, we still use the term \"Subgraph GNN\" to refer to the original architecture in the previous paragraph, while using \"Local 2-GNN\" to refer to the general architecture in Frasca et al. (2022); Zhang et al. (2023a).",
+        "bbox": [
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+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Substructure-based GNNs. Another line of work sought to develop expressive GNNs from practical considerations. In particular, Chen et al. (2020) pointed out that the ability of GNNs to detect/count graph substructures like path, cycle, and clique is crucial in numerous applications. Yet, MPNNs cannot subgraph-count any cycles/cliques. While higher-order WL tests can be more powerful in counting cycles (Fürer, 2017; Arvind et al., 2020), they suffer from substantial computational cost. As such, several works proposed to directly incorporate substructure/homomorphism counting into the node features as a preprocessing step to boost the expressiveness of MPNNs (Bouritsas et al., 2022; Barceló et al., 2021) or serve as an expressive graph kernel (Grohe, 2020; Nguyen & Maehara, 2020; Welke et al., 2023). Going beyond node features, Bodnar et al. (2021b;a); Giusti et al. (2023) further proposed a message-passing framework that enables interaction between nodes, edges, and higher-order substructures like cycles and cliques. We note that the Autobahn, TOGL, and Cy2C-GNN proposed in Thiede et al. (2021); Horn et al. (2022); Choi et al. (2022) can also be viewed as Substructure-based GNNs. However, most of the above approaches consider a fixed, predefined set of substructures rather than designing generic architectures that can learn substructures in an end-to-end fashion. Recently, Tahmasebi et al. (2023) proposed the RNP-GNN, an architecture that can count any substructure by recursively splitting a graph into a collection of vertex-marked subgraphs. We note that this design shares interesting similarities to higher-order subgraph GNNs and also the SpeqNet (Morris et al., 2022). Huang et al. (2023) proposed a generic model called  $\\mathrm{I}^2$ -GNN based on a variant of Subgraph 2-GNN, which can count 6-cycle at node-level. In this paper, we show the Local 2-GNN can already count all cycles/paths within 7 nodes even at edge-level while being more efficient than  $\\mathrm{I}^2$ -GNN (when using a similar ego network design). Finally, we remark that the polynomial expressivity proposed in Puny et al. (2023) can also be seen as a generalization of substructure counting, which further takes into account the real-valued node/edge features.",
+        "bbox": [
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+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Distance-based GNNs. Besides structural information, distance serves as another fundamental attribute of a graph, which, again, is not captured by MPNNs and the 1-WL test. Li et al. (2020) first proposed to improve the expressive power of GNNs by augmenting node features with Distance Encoding (DE). Related to DE, another approach to injecting distance information is the  $k$ -hop MPNN, which aggregates  $k$ -hop neighbors in a message-passing layer (Feng et al., 2022; Abboud et al., 2022; Wang et al., 2023). Feng et al. (2022); Zhang et al. (2023a) proved that the expressive power of  $k$ -hop MPNN is strictly bounded by 2-FWL. Distance can also be naturally incorporated in Graph Transformers through relative positional encoding, yielding the Graphormer architecture that has achieved remarkable performance across various benchmarks (Ying et al., 2021). Recently, Zhang et al. (2023b) built an interesting connection between distance and biconnectivity properties, showing that distance-enhanced GNNs can detect cut vertices and cut edges of a graph. This provides insights into the practical superiority of these models as biconnectivity is closely linked to real applications in chemistry and social network analysis. Zhang et al. (2023a) proved that Local 2-GNN can provably encode the distance (and thus biconnectivity) of a graph.",
+        "bbox": [
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+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Spectral-based GNNs. Graph spectra are also a class of fundamental properties and have long been used to design GNN models (Bruna et al., 2014; Defferrard et al., 2016). Balcilar et al. (2021b;a) showed that designing GNNs in the spectral domain can easily break the 1-WL expressivity. For Graph Transformers, Kreuzer et al. (2021); Dwivedi & Bresson (2020); Dwivedi et al. (2022) proposed to incorporate the spectra of the graph Laplacian matrix to boost the expressive power beyond the 1-WL test. Lim et al. (2023) further designed a principled equivariant architecture that takes the Laplacian eigenvalues and eigenvectors as inputs, which generalizes prior work.",
+        "bbox": [
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+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Other approaches. Murphy et al. (2019); Chen et al. (2020) proposed Relational Pooling as a general approach to designing expressive GNN architectures, whose basic idea is to implement a permutation-invariant GNN by symmetrizing permutation-sensitive base models. Wijesinghe & Wang (2022b) proposed the GraphSNN, which improves the expressive power of MPNNs by using more distinguishing edge features. Specifically, each edge feature encodes the structure of the over-",
+        "bbox": [
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+        "page_idx": 18
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
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+            47
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "page_number",
+        "text": "19",
+        "bbox": [
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+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "lap subgraph of two 1-hop ego networks centered on the two endpoints of the edge. Very recently, Dimitrov et al. (2023) designed a GNN model that can distinguish all planar graphs, thus achieving a strong expressivity in chemical applications since molecular graphs are often planar.",
+        "bbox": [
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+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "A.2 BROADER IMPACTS AND ADDITIONAL DISCUSSIONS",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Broader impact in graph theory. Due to the fundamental nature of GNN architectures studied in this paper, our theoretical results may potentially have broader impacts on the graph theory community. Specifically, we study the color refinement (CR) algorithms corresponding to four types of (higher-order) GNNs: Subgraph  $(k - 1)$ -GNN, Local  $k$ -GNN, Local  $k$ -FGNN, and  $k$ -FGNN. All algorithms can be seen as natural extensions of the 1-WL test since they all reduce to 1-WL when  $k = 1$ . In the graph theory community, Subgraph GNN has another name known as the vertex-individualized CR algorithm, which appears widely in literature (Babai, 2016; Rattan & Seppelt, 2023; Neuen & Schweitzer, 2018) and has become part of the core algorithm for fast graph isomorphism testing software (e.g., McKay & Piperno, 2014). On the other hand, Local  $k$ -GNN and Local  $k$ -FGNN are surprisingly related to the guarded logic (Barceló et al., 2020; De Rijke, 2000; Baader, 2003), since the aggregations are purely local (guarded by the edge). From this perspective, these CR algorithms can be seen as natural extensions of guarded logic in higher-order scenarios.",
+        "bbox": [
+            169,
+            186,
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+            354
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Besides these CR algorithms, our new extensions of NED may also have implications in graph theory. In particular, the strong NED (as well as its higher-order version) is elegant and may serve as a descriptive tool to characterize certain graph families. In addition, we establish intrinsic connections between NED and tree decomposition, which may have value in understanding other graph topics related to tree decomposition.",
+        "bbox": [
+            169,
+            361,
+            823,
+            431
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Finally, to our knowledge, the node/edge-level homomorphism and the corresponding subgraph counting abilities of different CR algorithms do not seem to have been systematically investigated before. Whereas in this paper, all graph/node/edge-level expressivity is studied in a unified manner. To achieve this, we introduce additional proof techniques which we believe may facilitate future study in related areas. For example, the original technique for constructing counterexample graphs satisfying Definition 3.1(b) does not apply to node/edge-level settings, since they are no longer counterexample graphs satisfying Definition 3.6(b) (no matter which vertices  $u \\in V_G, v \\in V_H$  are marked). To address the problem, we propose clique-augmented Fürer graphs, a novel class of counterexample graphs that extend several prior works (e.g., Cai et al., 1992; Fürer, 2001), and conduct a fine-grained analysis of their automorphism property (see Appendix D.2). We believe this new technique can be used to generalize other results from graph-level to node/edge-level settings.",
+        "bbox": [
+            169,
+            436,
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+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Discussions with Barceló et al. (2021). In the GNN community, Barceló et al. (2021) first proposed to incorporate the homomorphism count of predefined substructures into node features as an approach to enhancing the expressivity of MPNNs. They systematically investigated the questions of what substructures are useful and how the homomorphism information of these substructures can boost the model expressivity to even break out  $k$ -FWL. Yet, they only gave a partial (incomplete) characterization of the substructures that can be counted by the specific  $\\mathcal{F}$ -MPNN architecture and did not answer what substructures cannot be encoded, whereas our paper fully addresses both questions for a variety of popular GNN models. Note that these aspects are crucial to ensure that homomorphism expressivity is well-defined. As such, our paper first identifies that homomorphism expressivity is a complete, quantitative expressivity measure to compare different GNN models.",
+        "bbox": [
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+            597,
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+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Discussions with the concurrent work of Neuen (2023). After the initial submission, we became aware of a concurrent work (Neuen, 2023), which proved that  $k$ -FWL cannot count any graph with treewidth larger than  $k$  under homomorphism. In our context, this result exactly shows that Definition 3.1(b) is satisfied, and thus the homomorphism expressivity of  $k$ -FWL is well-defined. Notably, their construction of counterexample graphs is also based on Fürer graphs. Nevertheless, the proof technique between the two works is quite different: the proof in Neuen (2023) is built upon a key concept called oddomorphism, while our proof is based on the relation between tree decomposition and the simplified pebble game developed in Zhang et al. (2023a). It is essential to underscore that our results and proof technique are more general and go beyond the standard  $k$ -FWL, in that (i) it applies to a broad range of color refinement algorithms related to practical GNN architectures and (ii) it further extends to node/edge-level homomorphism expressivity. Our theoretical results thus strictly incorporate the results in Neuen (2023). The approach in Neuen (2023) (based on oddomorphism) cannot be easily generalized to these settings.",
+        "bbox": [
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+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "page_number",
+        "text": "20",
+        "bbox": [
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+            946,
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+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "B LIMITATIONS AND OPEN DIRECTIONS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            102,
+            524,
+            118
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "There are still several open questions that are not fully explored in this paper. We list them below as promising directions for future study.",
+        "bbox": [
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+            135,
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+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Existence of homomorphism expressivity for refinement-based GNN architectures. In this paper, we prove that homomorphism expressivity exists for a wide range of architectures defined in Section 2. On the other hand, we also note that it may not be well-defined for certain pathological GNNs, as illustrated in Appendix F.1. Given this observation, a fundamental question is: what conditions can guarantee that the homomorphism expressivity of a GNN exists? Here, we hypothesize that a very mild condition can suffice. Specifically, we conjecture that as long as a GNN architecture is defined following a general form of color refinement procedure that outputs stable color mappings, its homomorphism expressivity always exists. We leave this conjecture as an important open problem for future study.",
+        "bbox": [
+            169,
+            170,
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+            297
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Regarding higher-order Local FGNN. This paper characterizes the homomorphism expressivity for three classes of higher-order GNNs: Subgraph  $k$ -GNN, Local  $k$ -GNN, and  $k$ -FGNN. In particular, we introduce the  $k$ -order ear and  $k$ -order strong NED as a way to describe the homomorphism expressivity of Local  $k$ -GNN. However, it remains unclear how to give a simple description of the homomorphism expressivity for Local  $k$ -FGNN that can generalize the concept of almost-strong NED for  $k = 2$ . As such, our current expressiveness hierarchy (Figure 3) does not support Local  $k$ -FGNN yet. We leave this as an open problem and make the following conjecture below. We note that a similar open question has been informally raised in Zhang et al. (2023a).",
+        "bbox": [
+            169,
+            303,
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+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Conjecture B.1. For all  $k \\geq 2$ , Local  $k$ -FGNN is strictly more expressive than Local  $k$ -GNN and strictly less expressive than  $k$ -FGNN.",
+        "bbox": [
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+            420,
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+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Expressivity gap between Local 2-(F)GNN and 2-FGNN in practical aspects. We have proved that 2-FGNN is strictly more expressive than Local 2-(F)GNN. However, from a practical perspective, we surprisingly find that the subgraph counting ability of Local 2-(F)GNN matches that of 2-FGNN for all structures within a moderate size (see Table 5), although the former is much more efficient. This leads to the intriguing question of what fundamental gaps exist between the two models in practical aspects, or is the efficiency gain free?",
+        "bbox": [
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+            462,
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+            547
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Other architectures. In this paper, we comprehensively study a variety of popular GNN architectures ranging from Subgraph GNNs and Local GNNs to higher-order GNNs, and further link these architectures to Substructure-based GNNs (see Appendix A.1). Yet, we still do not cover all popular GNN architectures, such as the GSWL-based Subgraph GNN (Zhang et al., 2023a; Bevilacqua et al., 2022), SpeqNet (Morris et al., 2022), and  $\\mathrm{I}^2$ -GNN (Huang et al., 2023). Moreover, the classes of Distance-based GNNs and Spectral-based GNNs (Appendix A.1) are also widely used in practice, which deserve future study. We would like to raise the question of characterizing the homomorphism expressivity of Distance-based GNNs and Spectral-based GNNs as an important open question. In this way, one can gain deep insights into these models' true expressivity and enable quantitative comparisons between all mainstream architectures. Moreover, it will become clear to what extent other GNN models can encode distance and spectral information about a graph.",
+        "bbox": [
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+            553,
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+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "C PROOF OF THEOREM 3.4",
+        "text_level": 1,
+        "bbox": [
+            171,
+            729,
+            416,
+            746
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "This section gives the proof of the main theorem. For ease of reading, we first restate Theorem 3.4:",
+        "bbox": [
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+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.4. For all GNN models  $M$  defined in Section 2, the graph family  $\\mathcal{F}^M$  satisfying Definition 3.1 exists (and is unique). Moreover, each  $\\mathcal{F}^M$  can be separately described below:",
+        "bbox": [
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+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- Subgraph GNN:  $\\mathcal{F}^{\\mathrm{Sub}} = \\{F:F$  has an endpoint-shared NED\\};",
+            "- Local 2-GNN:  $\\mathcal{F}^{\\mathsf{L}} = \\{F:F$  has a strong NED\\};",
+            "- Local 2-FGNN:  $\\mathcal{F}^{\\mathrm{LF}} = \\{F:F$  has an almost-strong NED\\};",
+            "- 2-FGNN:  $\\mathcal{F}^{\\mathrm{F}} = \\{F:F$  has a NED\\}."
+        ],
+        "bbox": [
+            196,
+            819,
+            640,
+            902
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "For MPNN, since it is a special case of Subgraph  $k$ -GNN, the proof can be found in Appendix E.3.",
+        "bbox": [
+            169,
+            909,
+            821,
+            925
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "page_number",
+        "text": "21",
+        "bbox": [
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+            946,
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+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "C.1 PRELIMINARY",
+        "text_level": 1,
+        "bbox": [
+            171,
+            103,
+            316,
+            118
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "Additional notations and concepts. Besides the notations defined in Section 2, we further define the following notations. We use the symbol  $\\mathcal{G}$  to denote the set of all finite, simple, undirected, labeled graphs. Let  $G = (V_G,E_G,\\ell_G)$  be a graph in  $\\mathcal{G}$ . When the label is the same for all vertices, we can omit the term  $\\ell_G$  and write  $G = (V_G,E_G)$ . Given vertex  $u\\in V_{G}$ , denote the degree of  $u$  as  $\\deg_G(u) = |N_G(u)|$ , and denote the closed neighborhood of  $u$  as  $N_G[u] = N_G(u)\\cup \\{u\\}$ . The shortest path distance between vertices  $u$  and  $v$  is denoted by  $\\mathrm{dis}_G(u,v)$ .",
+        "bbox": [
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+            128,
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+            214
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "Given a vertex set  $S \\subset V_G$ , the induced subgraph of  $G$  over  $S$ , denoted as  $G[S]$ , is the subgraph of  $G$  with vertex set  $S$  and edge set  $\\{\\{u, v\\} \\in E_G : u, v \\in S\\}$ . Similarly, without abuse of notation, given an edge set  $R \\subset E_G$ , the induced subgraph of  $G$  over  $R$ , denoted as  $G[R]$ , is the subgraph of  $G$  with vertex set  $\\bigcup_{\\{u, v\\} \\in R} \\{u, v\\}$  and edge set  $R$ . Given a vertex tuple  $\\mathbf{u} = (u_1, \\dots, u_k)$ , denote by  $G^{\\mathbf{u}} = G^{u_1, \\dots, u_k}$  the rooted graph obtained from  $G$  by marking vertices  $u_1, \\dots, u_k$ . The atomic type of  $G$  over  $\\mathbf{u}$ , denoted by  $\\mathrm{atp}_G(\\mathbf{u})$ , is a  $k \\times k$  matrix where the element at position  $(i, j)$  is the tuple  $(\\mathbb{I}[u_i = u_j], \\mathbb{I}[\\{u_i, u_j\\} \\in E_G])$ . Given two graphs  $G, H$ , the graph union  $G \\cup H$  is defined to be the graph  $(V_G \\cup V_H, E_G \\cup E_H, \\ell_{G \\cup H})$ , where  $\\ell_{G \\cup H}(u) := \\ell_G(u)$  for all  $u \\in V_G$  and  $\\ell_{G \\cup H}(u) := \\ell_H(u)$  for all  $u \\in V_H$ . It is well-defined iff  $\\ell_G(u) = \\ell_H(u)$  for all  $u \\in V_G \\cap V_H$ . Finally, we use the notation  $G \\simeq H$  to denote that  $G$  and  $H$  are isomorphic graphs.",
+        "bbox": [
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+            219,
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+            362
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "A graph  $T = (V_T, E_T, \\ell_T)$  is called a tree if it does not contain cycles. Let  $T^r$  be a rooted tree where  $r$  is the root vertex. For each vertex  $t \\in V_T$ , define its depth  $\\mathsf{dep}_{T^r}(t) := \\mathsf{dis}_T(t, r)$  to be the distance to the root, and denote by  $\\mathsf{Desc}_{T^r}(t)$  the set of descendants of  $t$ , namely,  $s \\in \\mathsf{Desc}_{T^r}(t)$  iff  $\\mathsf{dep}_{T^r}(s) = \\mathsf{dep}_{T^r}(t) + \\mathsf{dis}_T(t, s)$ . For each  $t \\in V_T \\setminus \\{r\\}$ , denote by  $\\mathsf{pa}_{T^r}(t)$  the parent vertex of  $t$ , i.e., the unique vertex  $s \\in N_T(t)$  satisfying  $\\mathsf{dep}_{T^r}(t) = \\mathsf{dep}_{T^r}(s) + 1$ . Define the subtree of  $T^r$  rooted at node  $t$  by  $T^r[t]$ , which is exactly the induced subgraph  $T[\\mathsf{Desc}_{T^r}(t)]^t$  with root  $t$ .",
+        "bbox": [
+            169,
+            368,
+            823,
+            454
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "Tree decomposition. Our proof is based on a central concept in graph theory, called tree decomposition. It can be formally defined below:",
+        "bbox": [
+            169,
+            459,
+            823,
+            487
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "Definition C.1 (Tree decomposition). Given a graph  $G = (V_G, E_G, \\ell_G)$ , its tree decomposition is a tree  $T = (V_T, E_T, \\beta_T)$ , where the label function  $\\beta_T : V_T \\to 2^{V_G}$  satisfies the following conditions:",
+        "bbox": [
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+            489,
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+            520
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a) Each tree node  $t \\in V_T$  is associated to a non-empty subset of vertices  $\\beta_T(t) \\subset V_G$  in  $G$ , called a bag. We say tree node  $t$  contains vertex  $u$  if  $u \\in \\beta_T(t)$ ;",
+            "b) For each edge  $\\{u,v\\} \\in V_G$  , there exists at least one tree node  $t\\in V_T$  that contains the edge, i.e.,  $\\{u,v\\} \\subset \\beta_T(t)$",
+            "c) For each vertex  $u \\in V_G$ , all tree nodes  $t$  containing  $u$  form a (non-empty) connected subtree. Formally, denoting  $B_T(u) = \\{t \\in V_T : u \\in \\beta_T(t)\\}$ , then  $T[B_T(u)]$  is connected."
+        ],
+        "bbox": [
+            189,
+            525,
+            821,
+            619
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "If  $T$  is a tree decomposition of  $G$ , we call the pair  $(G, T)$  a tree-decomposed graph.",
+        "bbox": [
+            169,
+            626,
+            718,
+            642
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "We remark that given a graph  $G$ , there are multiple ways to decompose it and thus its tree decomposition is not unique. Several examples of tree decomposition are given in Figure 5.",
+        "bbox": [
+            169,
+            652,
+            823,
+            681
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "Definition C.2 (Treewidth). The width of a tree decomposition is defined as one less than the maximum bag size, i.e.,  $\\max_{t\\in T}|\\beta_T(t)| - 1$ . The treewidth of a graph  $G$ , denoted as  $\\mathrm{tw}(G)$ , is the minimum positive integer  $k$  such that there exists a tree decomposition of width  $k$ .",
+        "bbox": [
+            169,
+            684,
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+            727
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "Some important facts about treewidth are listed below:",
+        "bbox": [
+            171,
+            737,
+            532,
+            751
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "Fact C.3. For any graph  $G$ , the following hold:",
+        "bbox": [
+            171,
+            755,
+            488,
+            770
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- The treewidth of  $G$  is at most  $|V_G| - 1$ , i.e., a trivial tree decomposition that only has one node  $t$  and  $\\beta_T(t) = V_G$ .",
+            "-  $\\operatorname{tw}(G) = |V_G| - 1$  iff  $G$  is a clique.",
+            "-  $\\operatorname{tw}(G) = 1$  iff  $G$  is a forest."
+        ],
+        "bbox": [
+            196,
+            776,
+            821,
+            842
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "The above definition of tree decomposition is quite flexible without constraints on the structure of the tree or the size of each bag. Below, we define several restricted variants of tree decomposition, which (we will later see) are closely related to the GNN architectures studied in this paper. To begin with, we first define a general concept that slight modifies the original definition (Definition C.1) such that the tree becomes rooted and each bag is a multiset of vertices rather than a set.",
+        "bbox": [
+            169,
+            854,
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+            925
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "page_number",
+        "text": "22",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "image",
+        "img_path": "images/f880c48c3a49ad53d640c5d9cb66668fb8cf6308777fdb2eb36d2e6debcf0923.jpg",
+        "image_caption": [
+            "Figure 5: Illustration of tree decomposition."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            85,
+            823,
+            309
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "Definition C.4 (Canonical tree decomposition). Given a graph  $G = (V_G, E_G, \\ell_G)$ , a canonical tree decomposition of width  $k$  is a rooted tree  $T^r = (V_T, E_T, \\beta_T)$  satisfying the following conditions:",
+        "bbox": [
+            169,
+            358,
+            823,
+            388
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a) The depth of  $T$  is even, i.e.  $\\max_{t\\in V_T}\\mathsf{dep}_{T^r}(t)$  is even;",
+            "b) Each tree node  $t \\in V_T$  is associated to a multiset of vertices  $\\beta_T(t) \\subset V_G$ , called a bag. Moreover,  $|\\beta_T(t)| = k$  if  $\\mathsf{dep}_{T^r}(t)$  is even and  $|\\beta_T(t)| = k + 1$  if  $\\mathsf{dep}_{T^r}(t)$  is odd;",
+            "c) For all tree edges  $\\{s,t\\} \\in E_T$  where  $\\mathrm{dep}_{T^r}(s)$  is even and  $\\mathrm{dep}_{T^r}(t)$  is odd,  $\\beta_{T}(s)\\subset \\beta_{T}(t)$  (where “ $\\subset$ ” denotes the multiset inclusion relation);",
+            "d) The conditions (b) and (c) in Definition C.1 are satisfied."
+        ],
+        "bbox": [
+            189,
+            393,
+            823,
+            492
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "As examples, one can check that the tree decomposition of all graphs in Figure 5(b) is canonical, but the tree decomposition in Figure 5(a) is not. An important observation about canonical tree decomposition is shown below:",
+        "bbox": [
+            169,
+            503,
+            823,
+            546
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "Proposition C.5. Let  $(F, T^r)$  be any tree-decomposed graph where  $T^r$  is a canonical tree decomposition of  $F$ . For any vertices  $u, v \\in V_F$ , either of the following holds:",
+        "bbox": [
+            169,
+            550,
+            823,
+            580
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "-  $u$  and  $v$  are in the same bag of  $T^r$ , i.e., there is a node  $t \\in V_T$  such that  $\\{\\{u, v\\}\\} \\subset \\beta_T(t)$ ;",
+            "- The induced subgraph  $T[B_T(u) \\cup B_T(v)]$  is disconnected."
+        ],
+        "bbox": [
+            196,
+            585,
+            800,
+            622
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "Proof. Assume that  $u$  and  $v$  are not in the same bag, i.e.,  $B_{T}(u) \\cap B_{T}(v) = \\emptyset$ . Pick  $s \\in B_{T}(u)$  and  $t \\in B_{T}(v)$  such that  $\\mathsf{dep}_{T^r}(s)$  and  $\\mathsf{dep}_{T^r}(t)$  are minimal, respectively. Without loss of generality, assume that  $\\mathsf{dep}_{T^r}(s) \\leq \\mathsf{dep}_{T^r}(t)$ . Then,  $t \\neq r$  is not the root node and thus we can pick its parent  $\\mathsf{pa}_{T^r}(t)$ . It follows that  $\\mathsf{dep}_{T^r}(t)$  is odd and  $\\beta_{T}(\\mathsf{pa}_{T^{r}}(t)) \\subset \\beta_{T}(t)$  by definition of canonical tree decomposition. Therefore,  $u \\notin \\beta_{T}(\\mathsf{pa}_{T^{r}}(t))$ . Moreover, by the assumption that  $\\mathsf{dep}_{T^r}(s) \\leq \\mathsf{dep}_{T^r}(t)$ , any node in  $B_{T}(u)$  is not a descendant of  $t$ . We thus conclude that there does not exist a tree edge such that the two endpoints contain  $u$  and  $v$ , respectively.",
+        "bbox": [
+            169,
+            636,
+            826,
+            736
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "Now we are ready to define several restricted variants of canonical tree decomposition:",
+        "bbox": [
+            169,
+            750,
+            743,
+            763
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "Definition C.6. Define four families of tree-decomposed graphs  $S^{\\mathrm{Sub}}$ ,  $S^{\\mathrm{L}}$ ,  $S^{\\mathrm{LF}}$ , and  $S^{\\mathrm{F}}$  as follows:",
+        "bbox": [
+            169,
+            767,
+            821,
+            782
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a)  $(F,T^{r})\\in S^{\\mathsf{F}}$  iff  $(F,T^{r})$  satisfies Definition C.4 with width  $k = 2$",
+            "b)  $(F,T^{r})\\in S^{\\mathrm{LF}}$  iff  $(F,T^{r})$  satisfies Definition C.4 with width  $k = 2$ , and for any tree node  $t$  of odd depth, it has only one child if  $w\\notin \\{v:v\\in N_G[u],u\\in \\beta_T(s)\\}$  where  $s$  is the parent node of  $t$  and  $w$  is the unique vertex in  $\\beta_T(t)\\backslash \\beta_T(s)$ ;",
+            "c)  $(F,T^r)\\in S^{\\mathsf{L}}$  iff  $(F,T^{r})$  satisfies Definition C.4 with width  $k = 2$ , and any tree node  $t$  of odd depth has only one child;",
+            "d)  $(F, T^r) \\in S^{\\mathrm{Sub}}$  iff  $(F, T^r)$  satisfies Definition C.4 with width  $k = 2$ , and there exists a vertex  $u \\in V_G$  such that  $u \\in \\beta_T(t)$  for all  $t \\in V_T$ ."
+        ],
+        "bbox": [
+            189,
+            787,
+            825,
+            924
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "page_number",
+        "text": "23",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "Examples of tree-decomposed graphs in the four families are illustrated in Figure 5(b).",
+        "bbox": [
+            169,
+            103,
+            743,
+            119
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "Before closing this subsection, we define several notations for tree-decomposed graphs:",
+        "bbox": [
+            169,
+            125,
+            746,
+            140
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "Definition C.7. Given canonical tree-decomposed graph  $(G,T^r)$  and node  $t\\in V_T$  , denote by  $G[T^{r}[t]]$  the subgraph of  $G$  induced by the vertex set  $\\{u:u\\in \\beta_{T^r}(s),s\\in \\mathsf{Desc}_{T^r}(t)\\}$",
+        "bbox": [
+            169,
+            142,
+            823,
+            172
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "Definition C.8. Given two canonical tree-decomposed graphs  $(G, T^r)$  and  $(\\tilde{G}, \\tilde{T}^s)$ , a pair of mappings  $(\\rho, \\tau)$  is called an isomorphism from  $(G, T^r)$  to  $(\\tilde{G}, \\tilde{T}^s)$ , denoted by  $(G, T^r) \\simeq (\\tilde{G}, \\tilde{T}^s)$ , if the following hold:",
+        "bbox": [
+            169,
+            176,
+            826,
+            222
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a)  $\\rho$  is an isomorphism from  $G$  to  $\\tilde{G}$ ;",
+            "b)  $\\tau$  is an isomorphism from  $T^r$  to  $\\tilde{T}^s$  (ignoring labels  $\\beta$ );",
+            "c) For any  $t\\in T^r$ $\\rho (\\beta_T(t)) = \\beta_{\\bar{T}}(\\tau (t))$"
+        ],
+        "bbox": [
+            189,
+            229,
+            580,
+            290
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "Equivalent formulation of GNN architectures. Below, we give equivalent definitions for several GNN architectures presented in Section 2, which will be used in subsequent analysis. Let  $G$  be a graph and  $u,v\\in V_G$ . For all models  $M$  including Subgraph GNN, Local 2-GNN, Local 2-FGNN, and 2-FGNN, we define the initial color  $\\tilde{\\chi}_G^{M,(0)}(u,v)$  to be the isomorphism type of vertex pair  $(u,v)$  (i.e.,  $\\mathsf{atp}_G(u,v)$  plus labels of each vertex). Note that this matches the original definition except for Subgraph GNN. Then in each iteration  $t$ , the color is updated according to the following formula. Here, for clarity, we denote  $\\tilde{\\chi}_G^{M,(t)}(u,S) = \\{\\{\\tilde{\\chi}_G^{M,(t)}(u,v):v\\in S\\}\\}$  and  $\\tilde{\\chi}_G^{M,(t)}(S,v) = \\{\\{\\tilde{\\chi}_G^{M,(t)}(u,v):u\\in S\\}\\}$  for any model  $M$  and set  $S$ .",
+        "bbox": [
+            169,
+            299,
+            826,
+            424
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "- Subgraph GNN:",
+        "bbox": [
+            197,
+            429,
+            323,
+            444
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\tilde {\\chi} _ {G} ^ {\\text {S u b}, (t + 1)} (u, v) = \\operatorname {h a s h} \\left(\\tilde {\\chi} _ {G} ^ {\\text {S u b}, (t)} (u, v), \\tilde {\\chi} _ {G} ^ {\\text {S u b}, (t)} (u, N _ {G} (v)), \\tilde {\\chi} _ {G} ^ {\\text {S u b}, (t)} (u, V _ {G})\\right). \\tag {7}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            259,
+            448,
+            825,
+            474
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "- Local 2-GNN:",
+        "bbox": [
+            197,
+            479,
+            313,
+            494
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\tilde {\\chi} _ {G} ^ {\\mathsf {L}, (t + 1)} (u, v) = \\operatorname {h a s h} \\left(\\tilde {\\chi} _ {G} ^ {\\mathsf {L}, (t)} (u, v), \\tilde {\\chi} _ {G} ^ {\\mathsf {L}, (t)} (u, N _ {G} (v)), \\tilde {\\chi} _ {G} ^ {\\mathsf {L}, (t)} \\left(N _ {G} (u), v\\right), \\right. \\tag {8} \\\\ \\left. \\tilde {\\chi} _ {G} ^ {\\mathsf {L}, (t)} (u, V _ {G}), \\tilde {\\chi} _ {G} ^ {\\mathsf {L}, (t)} (V _ {G}, v)\\right). \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            276,
+            498,
+            823,
+            551
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "- Local 2-FGNN:",
+        "bbox": [
+            197,
+            556,
+            320,
+            571
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\tilde {\\chi} _ {G} ^ {\\mathsf {L F}, (t + 1)} (u, v) = \\mathsf {h a s h} \\left(\\tilde {\\chi} _ {G} ^ {\\mathsf {L F}, (t)} (u, v), \\{\\{\\left(\\tilde {\\chi} _ {G} ^ {\\mathsf {L F}, (t)} (w, v), \\tilde {\\chi} _ {G} ^ {\\mathsf {L F}, (t)} (u, w)\\right): w \\in N _ {G} [ u ] \\cup N _ {G} [ v ] \\}, \\right. \\\\ \\left. \\tilde {\\chi} _ {G} ^ {\\mathrm {L F}, (t)} \\left(u, V _ {G}\\right), \\tilde {\\chi} _ {G} ^ {\\mathrm {L F}, (t)} \\left(V _ {G}, v\\right)\\right). \\tag {9} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            212,
+            574,
+            823,
+            640
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "2-FGNN:",
+        "bbox": [
+            197,
+            642,
+            279,
+            654
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\tilde {\\chi} _ {G} ^ {\\mathsf {F}, (t + 1)} (u, v) = \\operatorname {h a s h} \\left(\\tilde {\\chi} _ {G} ^ {\\mathsf {F}, (t)} (u, v), \\{\\{\\left(\\tilde {\\chi} _ {G} ^ {\\mathsf {F}, (t)} (w, v), \\tilde {\\chi} _ {G} ^ {\\mathsf {F}, (t)} (u, w)\\right): w \\in V _ {G} \\} \\}, \\right. \\tag {10} \\\\ \\left. \\tilde {\\chi} _ {G} ^ {\\mathsf {F}, (t)} (u, V _ {G}), \\tilde {\\chi} _ {G} ^ {\\mathsf {F}, (t)} (V _ {G}, v)\\right). \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            245,
+            657,
+            823,
+            712
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "It can be seen that we additionally add global aggregations for these architectures. Moreover, in Local 2-FGNN we replace the neighbors by closed neighbors  $N_G[u] \\cup N_G[v]$ . The stable color of  $(u,v)$  for different models is denoted by  $\\tilde{\\chi}_G^{\\mathrm{Sub}}(u,v)$ ,  $\\tilde{\\chi}_G^{\\mathrm{L}}(u,v)$ ,  $\\tilde{\\chi}_G^{\\mathrm{LF}}(u,v)$ ,  $\\tilde{\\chi}_G^{\\mathrm{F}}(u,v)$ , respectively. We have the following result:",
+        "bbox": [
+            169,
+            719,
+            826,
+            773
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "Proposition C.9. Let  $M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}$  be any model. For any graphs  $G, H$ ,  $\\tilde{\\chi}_G^M(G) = \\tilde{\\chi}_H^M(H)$  iff  $\\chi_G^M(G) = \\chi_H^M(H)$ . Furthermore, if  $G$  and  $H$  are connected, then for any vertices  $u, v \\in V_G$  and  $x, y \\in V_H$ ,  $\\tilde{\\chi}_G^M(u, v) = \\tilde{\\chi}_H^M(x, y)$  iff  $\\chi_G^M(u, v) = \\chi_H^M(x, y)$ . In other words, the color mapping  $\\tilde{\\chi}^M$  is as fine as the original one  $\\chi^M$ .",
+        "bbox": [
+            169,
+            777,
+            826,
+            839
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "Proof. The proof simply follows from Zhang et al. (2023a, Proposition 4.2 and Theorem 4.4), because (i) node marking in the initial color of Subgraph GNN is as expressive as using the isomorphism type, (ii) the global aggregation does not improve the expressivity when the corresponding local aggregation is presented, (iii) the single-point aggregation does not improve the expressivity in Local 2-FGNN.  $\\square$",
+        "bbox": [
+            169,
+            853,
+            826,
+            922
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "page_number",
+        "text": "24",
+        "bbox": [
+            488,
+            946,
+            509,
+            959
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "text",
+        "text": "C.2 PART 1: TREE DECOMPOSITION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            103,
+            434,
+            118
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "We first define the unfolding tree of different CR algorithms, which is a standard tool in analyzing GNN expressivity.",
+        "bbox": [
+            169,
+            128,
+            823,
+            157
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "Definition C.10 (Unfolding tree of Subgraph GNN). Given a graph  $G$ , vertices  $u, v \\in V_G$ , and a non-negative integer  $D$ , the depth-2D Subgraph GNN unfolding tree of graph  $G$  at  $(u, v)$ , denoted as  $\\left(F_G^{\\mathrm{Sub},(D)}(u, v), T_G^{\\mathrm{Sub},(D)}(u, v)\\right)$ , is a tree-decomposed graph  $(F, T^r) \\in S^{\\mathrm{Sub}}$  constructed as follows:",
+        "bbox": [
+            169,
+            159,
+            826,
+            224
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1. Initialization. At the beginning,  $F = G[\\{u,v\\}]$  (if  $u = v$ ,  $F$  only has one vertex), and  $T$  only has a root node  $r$  with  $\\beta_T(r) = \\{\\{u,v\\}\\}$ . Define a mapping  $\\pi : V_F \\to V_G$  as  $\\pi(u) = u$  and  $\\pi(v) = v$ .",
+            "2. Loop for  $D$  rounds. For each leaf node  $t$  in  $T^r$ , do the following procedure:"
+        ],
+        "bbox": [
+            189,
+            231,
+            825,
+            295
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "Let  $\\beta_T(t) = \\{\\{u,x\\}\\}$ . For each  $w \\in V_G$ , add a fresh child node  $t_w$  to  $T^r$  and designate  $t$  as its parent. Then, consider the following three cases:",
+        "bbox": [
+            210,
+            295,
+            823,
+            324
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a) If  $w \\neq \\pi(u)$  and  $w \\neq \\pi(x)$ , then add a fresh vertex  $z$  to  $F$  and extend  $\\pi$  with  $\\pi(z) = w$ . The label of  $z$  in  $F$  is set by  $\\ell_F(z) = \\ell_G(w)$ . Define  $\\beta_T(t_w) = \\beta_T(t) \\cup \\{\\{z\\}\\}$ . Then, we add edges between  $z$  and  $\\beta_T(t)$ , so that  $\\pi$  is an isomorphism from  $F[\\beta_T(t_w)]$  to  $G[\\pi(\\beta_T(t_w))]$ .",
+            "b) If  $w = \\pi(u)$ , then we simply set  $\\beta_T(t_w) = \\beta_T(t) \\cup \\{\\{u\\}\\}$  without modifying graph  $F$ .",
+            "c) If  $w = \\pi(x)$ , then we simply set  $\\beta_T(t_w) = \\beta_T(t) \\cup \\{x\\}$  without modifying graph  $F$ ."
+        ],
+        "bbox": [
+            230,
+            329,
+            825,
+            416
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "Finally, add a fresh child node  $t_w'$  to  $T^r$ , designate  $t_w$  as its parent, and set  $\\beta_T(t_w')$  based on the following three cases:",
+        "bbox": [
+            210,
+            421,
+            823,
+            450
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a) If  $w \\neq \\pi(u)$  and  $w \\neq \\pi(x)$ , then  $\\beta_T(t_w') = \\{\\{u, z\\}\\}$ .",
+            "b) If  $w = \\pi(u)$ , then  $\\beta_T(t_w') = \\{\\{u, u\\}\\}$ .",
+            "c) If  $w = \\pi(x)$ , then  $\\beta_T(t_w') = \\{\\{u, x\\}\\}$ ."
+        ],
+        "bbox": [
+            230,
+            455,
+            598,
+            501
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "It is easy to see that the depth of tree  $T^r$  increases by 2 after each round,  $T^r$  is always a canonical tree decomposition of  $F$ , and  $(F, T^r) \\in S^{\\mathsf{Sub}}$ . An illustration of the construction of unfolding tree is given in Figure 6(a).",
+        "bbox": [
+            169,
+            507,
+            825,
+            551
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "We next define the unfolding tree of Local 2-GNN, which differs in the loop part such that the bags  $\\beta_{T}(t_{w})$  and  $\\beta_{T}(t_{w}^{\\prime})$  now do not necessarily contain  $u$ .",
+        "bbox": [
+            169,
+            559,
+            823,
+            588
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "Definition C.11 (Unfolding tree of Local 2-GNN). Given a graph  $G$ , vertices  $u, v \\in V_G$ , and a non-negative integer  $D$ , the depth-2D Local 2-GNN unfolding tree of graph  $G$  at  $(u, v)$ , denoted as  $\\left(F_G^{\\mathsf{L},(D)}(u, v), T_G^{\\mathsf{L},(D)}(u, v)\\right)$ , is a tree-decomposed graph  $(F, T^r) \\in S^\\mathsf{L}$  constructed as follows:",
+        "bbox": [
+            169,
+            589,
+            825,
+            642
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1. Initialization. The procedure is exactly the same as Subgraph GNN (Definition C.10).",
+            "2. Loop for  $D$  rounds. For each leaf node  $t$  in  $T^r$ , do the following procedure:"
+        ],
+        "bbox": [
+            191,
+            647,
+            782,
+            683
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "Let  $\\beta_T(t) = \\{\\{x,y\\}\\}$ . For each  $w \\in V_G$ , add a fresh child node  $t_w$  to  $T$  and designate  $t$  as its parent. Then, consider the following three cases:",
+        "bbox": [
+            210,
+            683,
+            823,
+            710
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a) If  $w \\neq \\pi(x)$  and  $w \\neq \\pi(y)$ , then add a fresh vertex  $z$  to  $F$  and extend  $\\pi$  with  $\\pi(z) = w$ . The label of  $z$  in  $F$  is set by  $\\ell_F(z) = \\ell_G(w)$ . Define  $\\beta_T(t_w) = \\beta_T(t) \\cup \\{\\{z\\}\\}$ . Then, we add edges between  $z$  and  $\\beta_T(t)$ , so that  $\\pi$  is an isomorphism from  $F[\\beta_T(t_w)]$  to  $G[\\pi(\\beta_T(t_w))]$ .",
+            "b) If  $w = \\pi(x)$ , then we simply set  $\\beta_T(t_w) = \\beta_T(t) \\cup \\{\\{x\\}\\}$  without modifying graph  $F$ .",
+            "c) If  $w = \\pi(y)$ , then we simply set  $\\beta_T(t_w) = \\beta_T(t) \\cup \\{\\{y\\}\\}$  without modifying graph  $F$ ."
+        ],
+        "bbox": [
+            230,
+            718,
+            825,
+            804
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "Next, add a fresh child node  $t_w'$  in  $T^r$ , designate  $t_w$  as its parent, and set  $\\beta_T(t_w')$  based on the following three cases:",
+        "bbox": [
+            210,
+            809,
+            823,
+            839
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a) If  $w \\neq \\pi(x)$  and  $w \\neq \\pi(y)$ , then  $\\beta_T(t'_w) = \\{\\{x, z\\}\\}$ .",
+            "b) If  $w = \\pi(x)$ , then  $\\beta_T(t'_w) = \\{ \\{x, x\\} \\}$ .",
+            "c) If  $w = \\pi(y)$ , then  $\\beta_T(t'_w) = \\{ \\{x, y\\} \\}$ ."
+        ],
+        "bbox": [
+            230,
+            844,
+            596,
+            888
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "text",
+        "text": "Finally, we repeat the above procedure (point 2) once more, but this time the bag  $\\beta_{T}(t_{w}^{\\prime})$  is replaced by the following three cases (changing  $x$  to  $y$ ):",
+        "bbox": [
+            210,
+            895,
+            823,
+            925
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+        ],
+        "page_idx": 24
+    },
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+        "text": "25",
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+        "page_idx": 24
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+        "image_caption": [
+            "(a)"
+        ],
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+        "page_idx": 25
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+        "type": "image",
+        "img_path": "images/a29759fcfc14622ca3b6dfb31fdd215d97ea759c5e7316ee08dd80f7289f62eb.jpg",
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+        "img_path": "images/be2394636d3e86199f69686eb0f0270d9b6185180e6484739b5b7398b8625fcd.jpg",
+        "image_caption": [
+            "G"
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+        "img_path": "images/6631bd04768c7185219cc1ca7a5e3965c85649a0466c8f69e7b21241f8453157.jpg",
+        "image_caption": [
+            "$F_{G}^{\\mathrm{Sub},(1)}(1,2)$",
+            "$F_{G}^{\\mathrm{L},(1)}(1,2)$"
+        ],
+        "image_footnote": [],
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+        "img_path": "images/3516d23cb44d351fed1e422bf9fed664343c07d6e0a360cebbb829994a4633d3.jpg",
+        "image_caption": [
+            "$④^{\\prime \\prime}$"
+        ],
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+        "img_path": "images/a7ab05cc814a7595f7b27f26a603cbf6b6f76e8d7c57596895f586e833eb22cd.jpg",
+        "image_caption": [
+            "$T_{G}^{\\mathrm{Sub},(1)}(1,2)$",
+            "$T_{G}^{\\mathbf{L},(1)}(1,2)$"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "image_caption": [
+            "(c)",
+            "G",
+            "(d)",
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+        "image_caption": [
+            "$F_{G}^{\\mathrm{LF},(1)}(1,2)$",
+            "$F_{G}^{\\mathsf{F},(1)}(1,2)$"
+        ],
+        "image_footnote": [],
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+        "image_caption": [
+            "$T_{G}^{\\mathrm{LF},(1)}$  (1,2)"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/50334674602320888695697b2c7dc06d34e49c04fca5e6372ad63b686fb99431.jpg",
+        "image_caption": [
+            "$T_{G}^{\\mathsf{F},(1)}(1,2)$",
+            "Figure 6: The depth-2 unfolding tree of graph  $G$  at vertex pair (1,2) for Subgraph GNN, Local 2-GNN, Local 2-FGNN, and 2-FGNN, respectively."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            527,
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+        ],
+        "page_idx": 25
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+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a) If  $w \\neq \\pi(x)$  and  $w \\neq \\pi(y)$ , then  $\\beta_T(t'_w) = \\{\\{y, z\\}\\}$ .",
+            "b) If  $w = \\pi(y)$ , then  $\\beta_T(t'_w) = \\{\\{y, y\\}\\}$ .",
+            "c) If  $w = \\pi(x)$ , then  $\\beta_T(t_w') = \\{\\{x, y\\}\\}$ ."
+        ],
+        "bbox": [
+            230,
+            595,
+            596,
+            643
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "An illustration of the construction of unfolding tree is given in Figure 6(b).",
+        "bbox": [
+            171,
+            648,
+            663,
+            664
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "We next define the unfolding tree of Local 2-FGNN, which differs in the loop part such that the node  $t_w$  can have two children under certain conditions.",
+        "bbox": [
+            169,
+            675,
+            823,
+            703
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "Definition C.12 (Unfolding tree of Local 2-FGNN). Given a graph  $G$ , vertices  $u, v \\in V_G$ , and a non-negative integer  $D$ , the depth-2D Local 2-FGNN unfolding tree of graph  $G$  at  $(u, v)$ , denoted as  $\\left(F_G^{\\mathsf{LF},(D)}(u, v), T_G^{\\mathsf{LF},(D)}(u, v)\\right)$ , is a tree-decomposed graph  $(F, T^r) \\in S^{\\mathsf{LF}}$  constructed as follows:",
+        "bbox": [
+            171,
+            708,
+            823,
+            761
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1. Initialization. The procedure is exactly the same as Subgraph GNN (Definition C.10).",
+            "2. Loop for  $D$  rounds. For each leaf node  $t$  in  $T^r$ , do the following procedure:"
+        ],
+        "bbox": [
+            189,
+            766,
+            782,
+            801
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "Let  $\\beta_T(t) = \\{\\{x,y\\}\\}$ . For each  $w \\in N_G[\\pi(x)] \\cup N_G[\\pi(y)]$ , add a fresh child node  $t_w$  to  $T$  and designate  $t$  as its parent. Then, consider the following three cases:",
+        "bbox": [
+            210,
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+            830
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a) If  $w \\neq \\pi(x)$  and  $w \\neq \\pi(y)$ , then add a fresh vertex  $z$  to  $F$  and extend  $\\pi$  with  $\\pi(z) = w$ . The label of  $z$  in  $F$  is set by  $\\ell_F(z) = \\ell_G(w)$ . Define  $\\beta_T(t_w) = \\beta_T(t) \\cup \\{\\{z\\}\\}$ . Then, we add edges between  $z$  and  $\\beta_T(t)$ , so that  $\\pi$  is an isomorphism from  $F[\\beta_T(t_w)]$  to  $G[\\pi(\\beta_T(t_w))]$ .",
+            "b) If  $w = \\pi(x)$ , then we simply set  $\\beta_T(t_w) = \\beta_T(t) \\cup \\{\\{x\\}\\}$  without modifying graph  $F$ .",
+            "c) If  $w = \\pi(y)$ , then we simply set  $\\beta_T(t_w) = \\beta_T(t) \\cup \\{\\{y\\}\\}$  without modifying graph  $F$ ."
+        ],
+        "bbox": [
+            230,
+            837,
+            825,
+            925
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "page_number",
+        "text": "26",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "text",
+        "text": "Next, add two fresh children  $t_w'$  and  $t_w''$  in  $T^r$ , designate  $t_w$  as their parent, and set  $\\beta_T(t_w')$  and  $\\beta_T(t_w'')$  based on the following three cases:",
+        "bbox": [
+            210,
+            103,
+            823,
+            133
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a) If  $w \\neq \\pi(x)$  and  $w \\neq \\pi(y)$ , then  $\\beta_T(t_w') = \\{\\{x, z\\}\\}$  and  $\\beta_T(t_w'') = \\{\\{y, z\\}\\}$ .",
+            "b) If  $w = \\pi(x)$ , then  $\\beta_T(t_w') = \\{ \\{x, x\\} \\}$  and  $\\beta_T(t_w'') = \\{ \\{x, y\\} \\}$ .",
+            "c) If  $w = \\pi(y)$ , then  $\\beta_T(t_w') = \\{ \\{x, y\\} \\}$  and  $\\beta_T(t_w'') = \\{ \\{y, x\\} \\}$ ."
+        ],
+        "bbox": [
+            230,
+            138,
+            746,
+            184
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "For each  $w \\notin N_G[\\pi(x)] \\cup N_G[\\pi(y)]$ , follow the same procedure as Local 2-GNN (Definition C.11).",
+        "bbox": [
+            210,
+            189,
+            823,
+            218
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "An illustration of the construction of unfolding tree is given in Figure 6(c).",
+        "bbox": [
+            171,
+            224,
+            663,
+            239
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "We finally define the unfolding tree of 2-FGNN, which differs in the loop part such that all nodes  $t_w$  have two children.",
+        "bbox": [
+            169,
+            251,
+            823,
+            277
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "Definition C.13 (Unfolding tree of 2-FGNN). Given a graph  $G$ , vertices  $u, v \\in V_G$ , and a non-negative integer  $D$ , the depth-2D 2-FGNN unfolding tree of graph  $G$  at  $(u, v)$ , denoted as  $\\left(F_G^{\\mathrm{Sub},(D)}(u, v), T_G^{\\mathrm{Sub},(D)}(u, v)\\right)$ , is a tree-decomposed graph  $(F, T^r) \\in S^{\\mathsf{F}}$  constructed as follows:",
+        "bbox": [
+            171,
+            282,
+            825,
+            335
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1. Initialization. The procedure is exactly the same as Local 2-FGNN (Definition C.12).",
+            "2. Loop for  $D$  rounds. The procedure is similar to Local 2-FGNN (Definition C.12) except that the condition  $w \\in N_G[\\pi(x)] \\cup N_G[\\pi(y)]$  is relaxed to all vertices."
+        ],
+        "bbox": [
+            191,
+            340,
+            823,
+            391
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "An illustration of the construction of unfolding tree is given in Figure 6(d).",
+        "bbox": [
+            171,
+            397,
+            663,
+            412
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "We are now ready to present the first core result:",
+        "bbox": [
+            171,
+            422,
+            491,
+            436
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "Lemma C.14. Let  $M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}$  be any model. For any two graphs  $G, H$ , any vertices  $u, v \\in V_G, x, y \\in V_H$ , and any non-negative integer  $D$ ,  $\\tilde{\\chi}_G^{M,(D)}(u,v) = \\tilde{\\chi}_H^{M,(D)}(x,y)$  iff there exists an isomorphism  $(\\rho, \\tau)$  from  $\\left(F_G^{M,(D)}(u,v), T_G^{M,(D)}(u,v)\\right)$  to  $\\left(F_H^{M,(D)}(x,y), T_H^{M,(D)}(x,y)\\right)$  such that  $\\rho(u) = x, \\rho(v) = y$ .",
+        "bbox": [
+            169,
+            440,
+            823,
+            512
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "Proof. Here, we only give the proof for Local 2-GNN, and the proofs for Subgraph GNN, Local 2-FGNN and 2-FGNN are almost the same so we omit them for clarity.",
+        "bbox": [
+            169,
+            526,
+            823,
+            556
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "Proof for Local 2-GNN. The proof is based on induction over  $D$ . When  $D = 0$ , the theorem obviously holds. Now assume that the theorem holds for  $D \\leq d$ , and consider  $D = d + 1$ . Below, we omit  $\\mathsf{L}$  in the corner mark for clarity.",
+        "bbox": [
+            169,
+            561,
+            823,
+            604
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "1. We first prove that  $\\tilde{\\chi}_G^{(d + 1)}(u,v) = \\tilde{\\chi}_H^{(d + 1)}(x,y)$  implies that there exists an isomorphism  $(\\rho ,\\tau)$  from  $\\left(F_G^{(d + 1)}(u,v),T_G^{(d + 1)}(u,v)\\right)$  to  $\\left(F_H^{(d + 1)}(x,y),T_H^{(d + 1)}(x,y)\\right)$  such that  $\\rho (u) = x,\\rho (v) = y$  . If  $\\tilde{\\chi}_G^{(d + 1)}(u,v) = \\tilde{\\chi}_H^{(d + 1)}(x,y)$  , then",
+        "bbox": [
+            191,
+            611,
+            823,
+            672
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\left\\{\\left\\{\\left(\\tilde {\\chi} _ {G} ^ {(d)} (u, w), \\mathsf {a t p} _ {G} (u, v, w)\\right): w \\in V _ {G} \\right\\} \\right\\} = \\left\\{\\left\\{\\left(\\tilde {\\chi} _ {H} ^ {(d)} (x, z), \\mathsf {a t p} _ {H} (x, y, z)\\right): z \\in V _ {H} \\right\\}, \\right. (11) \\\\ \\{\\{(\\tilde {\\chi} _ {G} ^ {(d)} (w, v), \\mathsf {a t p} _ {G} (u, v, w)): w \\in V _ {G} \\} \\} = \\{\\{(\\tilde {\\chi} _ {H} ^ {(d)} (z, y), \\mathsf {a t p} _ {H} (x, y, z)): z \\in V _ {H} \\}. (12) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            225,
+            676,
+            823,
+            720
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "Let  $n = |V_G| = |V_H|$ . Thus, we can denote  $V_G = \\{w_1, \\dots, w_n\\} = \\{w_1', \\dots, w_n'\\}$  and  $V_H = \\{z_1, \\dots, z_n\\} = \\{z_1', \\dots, z_n'\\}$  such that",
+        "bbox": [
+            210,
+            723,
+            823,
+            753
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "-  $(\\tilde{\\chi}_G^{(d)}(u, w_i), \\mathsf{atp}_G(u, v, w_i)) = (\\tilde{\\chi}_H^{(d)}(x, z_i), \\mathsf{atp}_H(x, y, z_i))$  for all  $i \\in [n]$ ;",
+            "-  $(\\tilde{\\chi}_G^{(d)}(w_i', v), \\mathsf{atp}_G(u, v, w_i')) = (\\tilde{\\chi}_H^{(d)}(z_i', y), \\mathsf{atp}_H(x, y, z_i'))$  for all  $i \\in [n]$ ."
+        ],
+        "bbox": [
+            236,
+            758,
+            746,
+            797
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "text",
+        "text": "On the other hand, by definition of tree unfolding, we have",
+        "bbox": [
+            210,
+            803,
+            601,
+            816
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nF _ {G} ^ {(d + 1)} (u, v) = \\left(\\bigcup_ {w _ {i}} F _ {G} ^ {(d)} (u, w _ {i})\\right) \\cup \\left(\\bigcup_ {w _ {i} ^ {\\prime}} F _ {G} ^ {(d)} (w _ {i} ^ {\\prime}, v)\\right) \\cup F _ {G} ^ {(1)} (u, v),\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            282,
+            821,
+            751,
+            869
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nF _ {H} ^ {(d + 1)} (x, y) = \\left(\\bigcup_ {z _ {i}} F _ {H} ^ {(d)} (x, z _ {i})\\right) \\cup \\left(\\bigcup_ {z _ {i} ^ {\\prime}} F _ {H} ^ {(d)} (z _ {i} ^ {\\prime}, y)\\right) \\cup F _ {H} ^ {(1)} (x, y),\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            284,
+            872,
+            743,
+            921
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "page_number",
+        "text": "27",
+        "bbox": [
+            488,
+            946,
+            506,
+            959
+        ],
+        "page_idx": 26
+    },
+    {
+        "type": "image",
+        "img_path": "images/674f09770f4f210781d42e760384377169b82fa8963ecb20080e13cd099ef355.jpg",
+        "image_caption": [
+            "Figure 7: Illustration of the proof of Lemma C.14."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            99,
+            459,
+            238
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "image",
+        "img_path": "images/bed23a54f04d7099280690fd9344766e4f3300f57d3451cb048c4c79d9187c73.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            470,
+            99,
+            810,
+            237
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "where  $\\cup$  represents the graph union. Here, all  $w_{i}, w_{j}^{\\prime} \\notin \\{u, v\\}$  in different graphs are treated as different vertices when taking the union, while  $u, v$  in different graphs are shared. See Figure 7 for an illustration of the above equations.",
+        "bbox": [
+            210,
+            287,
+            823,
+            330
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "By induction, there exists an isomorphism  $(\\rho_i,\\tau_i)$  from  $\\left(F_G^{(d)}(u,w_i),T_G^{(d)}(u,w_i)\\right)$  to  $\\left(F_H^{(d)}(x,z_i),T_H^{(d)}(x,z_i)\\right)$  such that  $\\rho_{i}(u) = x$ ,  $\\rho_{i}(w_{i}) = z_{i}$ $(i\\in [n])$ , and there exist an isomorphism  $\\rho_{i}^{\\prime}$  from  $\\left(F_G^{(d)}(w_i',v),T_G^{(d)}(w_i',v)\\right)$  to  $\\left(F_H^{(d)}(z_i',y),T_H^{(d)}(z_i',y)\\right)$  such that  $\\rho_{i}^{\\prime}(v) = y$ ,  $\\rho_{i}^{\\prime}(w_{i}^{\\prime}) = z_{i}^{\\prime}$ $(i\\in [n])$ . Moreover, we have  $\\mathsf{atp}_G(u,v,w_i) = \\mathsf{atp}_H(x,y,z_i)$  and  $\\mathsf{atp}_G(u,v,w_i') = \\mathsf{atp}_H(x,y,z_i')$  for  $i\\in [n]$ , which implies that  $F_{G}^{(1)}(u,v)$  is isomorphic to  $F_H^{(1)}(x,y)$ . Therefore, if we construct  $\\tilde{\\rho}$  by merging all  $\\rho_{i}$  and  $\\rho_{i}'$ $(i\\in [n])$ , and construct  $\\tilde{\\tau}$  by merging all  $\\tau_{i}$  and  $\\tau_{i}'$  and further specifying an appropriate mapping between tree nodes of depth no more than 1 in  $T_G^{(d + 1)}(u,v)$  and  $T_H^{(d + 1)}(x,y)$ , then it is straightforward to see that  $(\\tilde{\\rho},\\tilde{\\tau})$  is well-defined and is an isomorphism from  $\\left(F_G^{(d + 1)}(u,v),T_G^{(d + 1)}(u,v)\\right)$  to  $\\left(F_H^{(d + 1)}(x,y),T_H^{(d + 1)}(x,y)\\right)$  such that  $\\tilde{\\rho}(u) = x$ ,  $\\tilde{\\rho}(v) = y$ .",
+        "bbox": [
+            210,
+            337,
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+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "2. We next prove that if there exists an isomorphism  $(\\rho, \\tau)$  from the tree-decomposed graph  $\\left(F_G^{(d+1)}(u, v), T_G^{(d+1)}(u, v)\\right)$  to  $\\left(F_H^{(d+1)}(x, y), T_H^{(d+1)}(x, y)\\right)$  such that  $\\rho(u) = x$ ,  $\\rho(v) = y$ , then  $\\tilde{\\chi}_G^{(d+1)}(u, v) = \\tilde{\\chi}_H^{(d+1)}(x, y)$ .",
+        "bbox": [
+            191,
+            544,
+            823,
+            602
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "Without loss of generality, assume  $u \\neq v$  and  $x \\neq y$ . Since  $\\tau$  is an isomorphism from  $T_G^{(d+1)}(u,v)$  to  $T_H^{(d+1)}(x,y)$ ,  $\\tau$  maps all tree nodes of depth 1 in  $T_G^{(d+1)}(u,v)$  to all tree nodes of depth 1 in  $T_H^{(d+1)}(x,y)$ . Let  $s_1, \\dots, s_n$  be all nodes of depth 2 in  $T_G^{(d+1)}(u,v)$  such that  $u \\in \\beta_{T_G^{(d+1)}(u,v)}(s_i)$  (it follows that  $n = |V_G|$ ), and let  $s_i'$  be the parent of  $s_i$ . Similarly, let  $t_1, \\dots, t_n$  be all nodes of depth 2 in  $T_H^{(d+1)}(x,y)$  such that  $x \\in \\beta_{T_H^{(d+1)}(x,y)}(t_i)$ , and let  $t_i'$  be the parent of  $t_i$ . Moreover, we can arrange the order so that the following are satisfied (for each  $i \\in [n]$ ):",
+        "bbox": [
+            210,
+            601,
+            825,
+            719
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a)  $\\tau$  is an isomorphism from the subtree  $T_{G}^{(d + 1)}(u,v)[s_i]$  to the subtree  $T_{H}^{(d + 1)}(x,y)[t_i]$ .",
+            "b) For all  $s \\in \\mathsf{Desc}_{T_G^{(d + 1)}(u,v)}(s_i)$ ,  $\\rho(\\beta_{T_G^{(d + 1)}(u,v)}(s)) = \\beta_{T_H^{(d + 1)}(x,y)}(\\tau(s))$ .",
+            "c) By definition of the unfolding tree,  $\\rho$  is an isomorphism from the induced subgraph  $F_{G}^{(d + 1)}(u,v)[T_{G}^{(d + 1)}(u,v)[s_{i}]]$  to the induced subgraph  $F_{H}^{(d + 1)}(x,y)[T_{H}^{(d + 1)}(x,y)[t_{i}]]$  (see Definition C.7).",
+            "d) Let  $\\beta_{T_G^{(d + 1)}(u,v)}(s_i') = \\{\\{u,v,\\tilde{w}_i\\}\\}$  and  $\\beta_{T_H^{(d + 1)}(x,y)}(t_i') = \\{\\{x,y,\\tilde{z}_i\\}\\}$ . Then,  $\\rho (\\tilde{w}_i) = \\tilde{z}_i$ , and thus  $\\{v,\\tilde{w}_i\\} \\in E_{F_G^{(d + 1)}(u,v)}$  iff  $\\{y,\\tilde{z}_i\\} \\in E_{F_H^{(d + 1)}(x,y)}$ ."
+        ],
+        "bbox": [
+            230,
+            724,
+            823,
+            849
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "By items (a) to (c),  $\\left(F_{G}^{(d + 1)}(u,v)\\left[T_{G}^{(d + 1)}(u,v)[s_{i}]\\right],T_{G}^{(d + 1)}(u,v)[s_{i}]\\right)$  is isomorphic to  $\\left(F_H^{(d + 1)}(x,y)\\left[T_H^{(d + 1)}(x,y)[t_i]\\right],T_H^{(d + 1)}(x,y)[t_i]\\right)$ . On the other hand, by definition of the unfolding tree,  $\\left(F_G^{(d + 1)}(u,v)\\left[T_G^{(d + 1)}(u,v)[s_i]\\right],T_G^{(d + 1)}(u,v)[s_i]\\right)$  is isomorphic to",
+        "bbox": [
+            210,
+            854,
+            826,
+            928
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "page_number",
+        "text": "28",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 27
+    },
+    {
+        "type": "text",
+        "text": "the depth-2d unfolding tree  $\\left(F_{G}^{(d)}(u,w_{i}),T_{G}^{(d)}(u,w_{i})\\right)$  for some  $w_{i}\\in V_{G}$  satisfying that  $\\{w_i,v\\} \\in E_G$  iff  $\\{\\tilde{w}_i,v\\} \\in E_{F_G^{(d + 1)}(u,v)}$",
+        "bbox": [
+            210,
+            101,
+            823,
+            143
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "Similarly,  $\\left(F_H^{(d + 1)}(x,y)\\left[T_H^{(d + 1)}(x,y)[t_i]\\right],T_H^{(d + 1)}(x,y)[t_i]\\right)$  is isomorphic to  $\\left(F_{H}^{(d)}(x,z_{i}),T_{H}^{(d)}(x,z_{i})\\right)$  for some  $z_{i}\\in V_{H}$  satisfying that  $\\{z_i,y\\} \\in E_H$  iff  $\\{\\tilde{z}_i,y\\} \\in E_{F_H^{(d + 1)}(x,y)}$ . Combining all the above equivalence yields that  $\\left(F_G^{(d)}(u,w_i),T_G^{(d)}(u,w_i)\\right)$  is isomorphic to  $\\left(F_H^{(d)}(x,z_i),T_H^{(d)}(x,z_i)\\right)$ , and  $\\{w_i,v\\} \\in E_G$  iff  $\\{z_i,y\\} \\in E_H$ .",
+        "bbox": [
+            210,
+            143,
+            823,
+            251
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "By induction, we have  $\\tilde{\\chi}_G^{(d)}(u,w_i) = \\tilde{\\chi}_H^{(d)}(x,z_i)$ . Moreover, we clearly have that  $\\{u,w_i\\} \\in E_G$  iff  $\\{x,z_i\\} \\in E_H$ , and  $\\{u,v\\} \\in E_G$  iff  $\\{x,y\\} \\in E_H$ . Therefore,",
+        "bbox": [
+            210,
+            255,
+            823,
+            287
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left(\\tilde {\\chi} _ {G} ^ {(d)} (u, w _ {i}), \\mathsf {a t p} _ {G} (u, v, w _ {i})\\right) = \\left(\\tilde {\\chi} _ {H} ^ {(d)} (x, z _ {i}), \\mathsf {a t p} _ {H} (x, y, z _ {i})\\right). \\tag {13}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            292,
+            823,
+            316
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "Next, note that  $\\tilde{w}_i$  are different from each other for  $i\\in [n]$  by definition of unfolding tree. Thus,  $w_{i}$  are also different from each other. It follows that",
+        "bbox": [
+            210,
+            319,
+            823,
+            349
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\{\\{(\\tilde {\\chi} _ {G} ^ {(d)} (u, w), \\mathsf {a t p} _ {G} (u, v, w)): w \\in V _ {G} \\} \\} = \\{\\{(\\tilde {\\chi} _ {H} ^ {(d)} (x, z), \\mathsf {a t p} _ {H} (x, y, z)): z \\in V _ {H} \\} \\}. \\tag {14}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            223,
+            354,
+            823,
+            377
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "Again using the same analysis as before, we obtain",
+        "bbox": [
+            212,
+            380,
+            552,
+            395
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\{\\{(\\tilde {\\chi} _ {G} ^ {(d)} (w, v), \\mathsf {a t p} _ {G} (u, v, w)): w \\in V _ {G} \\} \\} = \\{\\{(\\tilde {\\chi} _ {H} ^ {(d)} (z, y), \\mathsf {a t p} _ {H} (x, y, z)): z \\in V _ {H} \\} \\}. \\tag {15}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            225,
+            401,
+            823,
+            422
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "It remains to prove that  $\\tilde{\\chi}_G^{(d)}(u,v) = \\tilde{\\chi}_H^{(d)}(x,y)$ . To prove this, note that Equation (14) implies that",
+        "bbox": [
+            210,
+            428,
+            823,
+            459
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left\\{\\left\\{\\left(\\tilde {\\chi} _ {G} ^ {(d ^ {\\prime})} (u, w), \\mathsf {a t p} _ {G} (u, v, w)\\right): w \\in V _ {G} \\right\\} \\right\\} = \\left\\{\\left\\{\\left(\\tilde {\\chi} _ {H} ^ {(d ^ {\\prime})} (x, z), \\mathsf {a t p} _ {H} (x, y, z)\\right): z \\in V _ {H} \\right\\} \\right\\} \\tag {16}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            222,
+            464,
+            823,
+            486
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "holds for all  $0 \\leq d' \\leq d$ , and Equation (15) implies that",
+        "bbox": [
+            210,
+            489,
+            580,
+            506
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left\\{\\left\\{\\left(\\tilde {\\chi} _ {G} ^ {(d ^ {\\prime})} (w, v), \\mathsf {a t p} _ {G} (u, v, w)\\right): w \\in V _ {G} \\right\\}\\right\\} = \\left\\{ \\right.\\left\\{\\left(\\tilde {\\chi} _ {H} ^ {(d)} (z, y), \\mathsf {a t p} _ {H} (x, y, z)\\right): z \\in V _ {H} \\right\\}. \\tag {17}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            222,
+            512,
+            823,
+            534
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "holds for all  $0 \\leq d' \\leq d$ . Combined with Equations (16) and (17) and the fact that  $\\tilde{\\chi}_{G}^{(0)}(u,v) = \\tilde{\\chi}_{H}^{(0)}(x,y)$ , we can incrementally prove that  $\\tilde{\\chi}_{G}^{(d')}(u,v) = \\tilde{\\chi}_{H}^{(d')}(x,y)$  for all  $d' \\leq d + 1$ .",
+        "bbox": [
+            210,
+            537,
+            823,
+            585
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "We have thus concluded the proof.",
+        "bbox": [
+            171,
+            590,
+            401,
+            606
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "Definition C.15. Let  $M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}$  be any model. Given a graph  $G$  and a tree-decomposed graph  $(F, T^r)$ , define",
+        "bbox": [
+            169,
+            614,
+            823,
+            643
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathsf {c n t} ^ {M} \\left(\\left(F, T ^ {r}\\right), G\\right) := \\left| \\left\\{\\left(u, v\\right) \\in V _ {G} ^ {2}: \\exists D \\in \\mathbb {N} _ {+} \\text {s . t .} \\left(F _ {G} ^ {M, (D)} (u, v), T _ {G} ^ {M, (D)} (u, v)\\right) \\simeq \\left(F, T ^ {r}\\right) \\right\\} \\right|,\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            169,
+            648,
+            828,
+            676
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "where  $\\left(F_G^{M,(D)}(u,v),T_G^{M,(D)}(u,v)\\right)$  is the depth-2D unfolding tree of  $G$  at  $(u,v)$  for model  $M$ .",
+        "bbox": [
+            169,
+            683,
+            813,
+            707
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "Corollary C.16. Let  $M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}$  be any model. For any graphs  $G, H$ ,  $\\chi_G^M(G) = \\chi_H^M(H)$  iff  $\\mathsf{cnt}^M((F, T^r), G) = \\mathsf{cnt}^M((F, T^r), H)$  holds for all  $(F, T^r) \\in S^M$ .",
+        "bbox": [
+            169,
+            710,
+            823,
+            743
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "Proof. “ $\\Rightarrow$ ”. If  $\\chi_G^M (G) = \\chi_H^M (H)$ , then  $\\{\\{\\chi_G^M (u,v):u,v\\in V_G\\} \\} = \\{\\{\\chi_H^M (x,y):x,y\\in V_H\\}\\}$ . For each color  $c$  in the above multiset, pick  $u,v\\in V_{G}$  with  $\\chi_G^M (u,v) = c$ . It follows that if  $(F,T^{r})\\simeq (F_{G}^{M,(D)}(u,v),T_{G}^{M,(D)}(u,v))\\in S^{M}$  for some  $D$ , then  $\\mathsf{cnt}^M ((F,T^r),G) = |\\{(u,v)\\in V_G^2:\\chi_G^M (u,v) = c\\} | = |\\{(x,y)\\in V_H^2:\\chi_H^M (x,y) = c\\} | = \\mathsf{cnt}^M ((F,T^r),H)$  by Lemma C.14. On the other hand, if  $(F,T^{r})\\neq (F_{G}^{M,(D)}(u,v),T_{G}^{M,(D)}(u,v))$  for all  $u,v\\in V_{G}$  and all  $D$ , then clearly  $\\mathsf{cnt}^M ((F,T^r),G) = \\mathsf{cnt}^M ((F,T^r),H) = 0$ .",
+        "bbox": [
+            169,
+            755,
+            825,
+            854
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "$\\Longleftarrow$  . If  $\\mathsf{cnt}^M ((F,T^r),G) = \\mathsf{cnt}^M ((F,T^r),H)$  holds for all  $(F,T^{r})\\in S^{M}$  , it clearly holds for all  $(F_M^{(D)}(u,v),T_M^{(D)}(u,v))$  with  $u,v\\in V_G$  and a sufficiently large  $D$  . This guarantees that for all color  $c,\\left|\\{(u,v)\\in V_G^2:\\chi_G^M (u,v) = c\\} \\right| = \\left|\\{(x,y)\\in V_H^2:\\chi_H^M (x,y) = c\\} \\right|$  by Lemma C.14. Therefore,  $\\{\\{\\chi_G^M (u,v):u,v\\in V_G\\} \\} = \\{\\{\\chi_H^M (x,y):x,y\\in V_H\\} \\}$  , concluding the proof.",
+        "bbox": [
+            169,
+            859,
+            825,
+            926
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "page_number",
+        "text": "29",
+        "bbox": [
+            488,
+            946,
+            508,
+            960
+        ],
+        "page_idx": 28
+    },
+    {
+        "type": "text",
+        "text": "We next define an important concept called bag isomorphism (Dell et al., 2018).",
+        "bbox": [
+            169,
+            103,
+            699,
+            118
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Definition C.17. Given a tree-decomposed graph  $(F, T^r)$  and a graph  $G$ , a bag isomorphism from  $(F, T^r)$  to  $G$  (abbreviated as \"bIso\") is a homomorphism  $f$  from  $F$  to  $G$  such that, for all  $t \\in V_T$ ,  $f$  is an isomorphism from  $F[\\beta_T(t)]$  to  $G[f(\\beta_T(t))]$ . Denote  $\\mathsf{BIso}((F, T^r), G)$  to be the set of all bag isomorphisms from  $(F, T^r)$  to  $G$ , and denote  $\\mathsf{bIso}((F, T^r), G) = |\\mathsf{BIso}((F, T^r), G)|$ .",
+        "bbox": [
+            169,
+            122,
+            826,
+            180
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Remark C.18. To prove that a mapping  $f$  from  $(F, T^r)$  to  $G$  is a bIso, it suffices to prove the following conditions:",
+        "bbox": [
+            169,
+            184,
+            823,
+            212
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1. For any two different vertices  $u, v \\in V_G$  in the same bag,  $f(u) \\neq f(v)$ ;",
+            "2. For any two vertices  $u, v \\in V_G$  in the same bag,  $\\{u, v\\} \\in E_F$  iff  $\\{f(u), f(v)\\} \\in E_G$ ;",
+            "3. For any  $u\\in V_G,\\ell_F(u) = \\ell_G(f(u))$"
+        ],
+        "bbox": [
+            189,
+            215,
+            774,
+            270
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "The following fact is straightforward from the construction of the unfolding tree:",
+        "bbox": [
+            169,
+            281,
+            702,
+            296
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Fact C.19. Let  $M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}$  be any model considered above. For any graph  $G$ , any vertex pair  $(u, v) \\in V_G^2$ , and any non-negative integer  $D$ , there is a bIso  $\\pi$  from  $\\left(F_G^{M,(D)}(u, v), T_G^{M,(D)}(u, v)\\right)$  to  $G$ .",
+        "bbox": [
+            169,
+            300,
+            823,
+            354
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Similarly, we need the following concept to describe the relation between two tree-decomposed graphs. We note that these technical concepts also appeared in Dell et al. (2018).",
+        "bbox": [
+            169,
+            364,
+            823,
+            393
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Definition C.20. Given two tree-decomposed graphs  $(F, T^r)$  and  $(\\tilde{F}, \\tilde{T}^s)$ , a pair of mappings  $(\\rho, \\tau)$  is called homomorphism from  $(F, T^r)$  to  $(\\tilde{F}, \\tilde{T}^s)$  if it satisfies the following conditions:",
+        "bbox": [
+            169,
+            398,
+            823,
+            431
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a)  $\\tau$  is a homomorphism from  $T$  to  $\\tilde{T}$  (ignoring labels  $\\beta$ ) and is depth-preserving, i.e.,  $\\mathsf{dep}_{T^r}(t) = \\mathsf{dep}_{\\tilde{T}^s}(\\tau(t))$  for all  $t \\in V_T$ ;",
+            "b) For all  $t \\in V_T$ ,  $\\rho$  is a homomorphism from  $F[\\beta_T(t)]$  to  $\\tilde{F}[\\beta_{\\tilde{T}}(\\tau(t))]$ . Note that this implies that  $\\rho$  is a homomorphism from  $F$  to  $\\tilde{F}$ .",
+            "c) The depth of  $T^r$  is equal to the depth of  $\\hat{T}^s$ ."
+        ],
+        "bbox": [
+            189,
+            438,
+            823,
+            527
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Definition C.21. Under Definition C.20,  $(\\rho, \\tau)$  is further called a bag-isomorphism homomorphism (abbreviated as \"bIsoHom\") from  $(F, T^r)$  to  $(\\tilde{F}, \\tilde{T}^s)$  if it is a homomorphism satisfying that, for all  $t \\in V_T$ ,  $\\rho$  is an isomorphism from  $F[\\beta_T(t)]$  to  $\\tilde{F}[\\beta_{\\tilde{T}}(\\tau(t))]$ . Furthermore,  $(\\rho, \\tau)$  is called a bIsoSurj if  $\\tau$  is surjective; and  $(\\rho, \\tau)$  is called a bIsoInj if  $\\tau$  is injective. We use BIsoHom  $\\left((F, T^r), (\\tilde{F}, \\tilde{T}^s)\\right)$  to denote the set of bIsoHoms from  $(F, T^r)$  to  $(\\tilde{F}, \\tilde{T}^s)$ , and let bIsoHom  $\\left((F, T^r), (\\tilde{F}, \\tilde{T}^s)\\right) = \\left|\\text{BIsoHom}\\left((F, T^r), (\\tilde{F}, \\tilde{T}^s)\\right)\\right|$ . The notations BIsoSurj  $\\left((F, T^r), (\\tilde{F}, \\tilde{T}^s)\\right)$ , bIsoSurj  $\\left((F, T^r), (\\tilde{F}, \\tilde{T}^s)\\right)$ , BIsolnj  $\\left((F, T^r), (\\tilde{F}, \\tilde{T}^s)\\right)$ , and bIsolnj  $\\left((F, T^r), (\\tilde{F}, \\tilde{T}^s)\\right)$  are defined accordingly.",
+        "bbox": [
+            169,
+            541,
+            826,
+            698
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Remark C.22. In the above definition, the depth of a tree  $T^r$  is the maximal depth among all tree nodes in  $T^r$ . Note that we do not require that all leaf nodes have the same depth in  $T^r$ .",
+        "bbox": [
+            169,
+            702,
+            823,
+            732
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "We are now ready to present the second core result:",
+        "bbox": [
+            169,
+            743,
+            513,
+            758
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Lemma C.23. Let  $M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}$  be any model. For any graph  $G$  and tree-decomposed graph  $(F, T^r) \\in S^M$ ,",
+        "bbox": [
+            169,
+            762,
+            826,
+            792
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\operatorname {b l s o} \\left(\\left(F, T ^ {r}\\right), G\\right) = \\sum_ {\\left(\\tilde {F}, \\tilde {T} ^ {s}\\right) \\in \\mathcal {S} ^ {M}} \\operatorname {b l s o H o m} \\left(\\left(F, T ^ {r}\\right), \\left(\\tilde {F}, \\tilde {T} ^ {s}\\right)\\right) \\cdot \\operatorname {c n t} ^ {M} \\left(\\left(\\tilde {F}, \\tilde {T} ^ {s}\\right), G\\right). \\tag {18}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            196,
+            800,
+            823,
+            838
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Here, the summation ranges over all non-isomorphic (tree-decomposed) graphs in  $S^M$  and is well-defined as there are only a finite number of graphs making the value in the summation non-zero.",
+        "bbox": [
+            169,
+            847,
+            823,
+            877
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Proof. Here, we only give the proof for Local 2-GNN, and the proofs for Subgraph GNN, Local 2-FGNN and 2-FGNN are almost the same so we omit them for clarity.",
+        "bbox": [
+            169,
+            895,
+            823,
+            925
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "page_number",
+        "text": "30",
+        "bbox": [
+            488,
+            946,
+            509,
+            959
+        ],
+        "page_idx": 29
+    },
+    {
+        "type": "text",
+        "text": "Proof for Local 2-GNN. We assume that the root bag of  $(F, T^r)$  is  $\\{\\{u, v\\}\\}$ , and the depth of  $(F, T^r)$  is  $2d$ . Let  $y, z \\in V_G$  be any vertices in  $G$ , and denote  $(F_G^{(d)}(y, z), T_G^{(d)}(y, z))$  as the depth-2d Local 2-GNN unfolding tree at  $(y, z)$ . Define the following two sets:",
+        "bbox": [
+            169,
+            103,
+            823,
+            151
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} S _ {1} (y, z) = \\left\\{g: g \\in \\mathsf {B l s o} \\left((F, T ^ {r}), G\\right), g (u) = y, g (v) = z \\right\\}, \\\\ S _ {2} (y, z) = \\left\\{(\\rho , \\tau): (\\rho , \\tau) \\in \\mathsf {B l s o H o m} \\left((F, T ^ {r}), \\left(F _ {G} ^ {(d)} (y, z), T _ {G} ^ {(d)} (y, z)\\right)\\right), \\rho (u) = y, \\rho (v) = z \\right\\}. \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            171,
+            159,
+            821,
+            205
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "text",
+        "text": "Then, Lemma C.23 is equivalent to the following equation:",
+        "bbox": [
+            171,
+            213,
+            563,
+            229
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\sum_ {y, z \\in V _ {G}} | S _ {1} (y, z) | = \\sum_ {y, z \\in V _ {G}} | S _ {2} (y, z) |.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            370,
+            238,
+            622,
+            271
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "text",
+        "text": "We will prove that  $|S_{1}(y,z)| = |S_{2}(y,z)|$  for all  $y,z\\in V_G$ .",
+        "bbox": [
+            171,
+            281,
+            565,
+            297
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "text",
+        "text": "Given  $y, z \\in V_G$ , according to Fact C.19, there exists a bIso  $\\pi$  from  $(F_G^{(d)}(y,z), T_G^{(d)}(y,z))$  to graph  $G$ . Define a mapping  $\\sigma$  such that  $\\sigma(\\rho, \\tau) = \\pi \\circ \\rho$  for all  $(\\rho, \\tau) \\in S_2(y,z)$ . It suffices to prove that  $\\sigma$  is a bijection from  $S_2(y,z)$  to  $S_1(y,z)$ .",
+        "bbox": [
+            169,
+            305,
+            823,
+            351
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "text",
+        "text": "1. We first prove that  $\\sigma$  is a mapping from  $S_{2}(y,z)$  to  $S_{1}(y,z)$ , i.e.,  $\\pi \\circ \\rho \\in S_{1}(y,z)$  for all  $(\\rho, \\tau) \\in S_{2}(y,z)$ . First, we clearly have  $(\\pi \\circ \\rho)(u) = \\pi(y) = y$ ,  $(\\pi \\circ \\rho)(v) = \\pi(z) = z$ . We next prove that  $\\pi \\circ \\rho \\in \\mathsf{BlsO}((F, T^{r}), G)$ . The proof is based on Remark C.18.",
+        "bbox": [
+            191,
+            354,
+            825,
+            398
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a) Let  $w, x \\in V_F$ ,  $w \\neq x$  be any vertices in the same bag of  $T^r$ . Since  $(\\rho, \\tau)$  is a bIsoHom,  $\\rho(w) \\neq \\rho(x)$  and  $\\rho(w)$  and  $\\rho(x)$  are in the same bag of  $T_G^{(d)}(y, z)$ . Again, since  $\\pi$  is a bIso, we have  $\\pi(\\rho(w)) \\neq \\pi(\\rho(x))$ .",
+            "b) Let  $w, x \\in V_F$  be any vertices in the same bag of  $T^r$ . Since  $(\\rho, \\tau)$  is a bIso-Hom,  $\\rho(w)$  and  $\\rho(x)$  are in the same bag of  $T_G^{(d)}(y,z)$ , and  $\\{w,x\\} \\in E_F$  iff  $\\{\\rho(w),\\rho(x)\\} \\in E_{F_G^{(d)}(y,z)}$ . Again, since  $\\pi$  is a bIso,  $\\{\\rho(w),\\rho(x)\\} \\in E_{F_G^{(d)}(y,z)}$  iff  $\\{\\pi(\\rho(w)),\\pi(\\rho(x))\\} \\in E_G$ . Therefore,  $\\{w,x\\} \\in E_F$  iff  $\\{\\pi(\\rho(w)),\\pi(\\rho(x))\\} \\in E_G$ .",
+            "c) We clearly have  $\\ell_F(w) = \\ell_{F_G^{(d)}(y,z)}(\\rho (w)) = \\ell_G(\\pi (\\rho (w)))$"
+        ],
+        "bbox": [
+            230,
+            404,
+            825,
+            537
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "text",
+        "text": "We have proved that  $\\pi \\circ \\rho \\in \\mathrm{BlsO}((F,T^r),G)$",
+        "bbox": [
+            210,
+            542,
+            529,
+            559
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "text",
+        "text": "2. We then prove that  $\\sigma$  is a surjection. For all  $g\\in S_1(y,z)$ , we define a mapping  $(\\rho ,\\tau)$  from  $(F,T^{r})$  to  $(F_{G}^{(d)}(y,z),T_{G}^{(d)}(y,z))$  as follows. First define  $\\rho (u) = y$ ,  $\\rho (v) = z$ , and set  $\\tau (r)$  to be the root of  $(F_{G}^{(d)}(y,z),T_{G}^{(d)}(y,z))$ . Let  $w_{1},\\dots ,w_{m}\\in V_{F}$  and  $w_1^{\\prime},\\dots ,w_{m^{\\prime}}^{\\prime}\\in V_{F}$  be vertices such that all  $\\{u,w_i\\}$  and  $\\{w_i',v\\}$  correspond to bags of  $T^{r}$  associated to all tree nodes of depth 2. Similarly, by definition of the Local 2-GNN unfolding tree, let  $x_{1},\\dots ,x_{n}\\in V_{F_{G}^{(d)}(y,z)}$  be different vertices and  $x_1^\\prime ,\\dots ,x_n^\\prime \\in V_{F_G^{(d)}(y,z)}$  be different vertices such that all  $\\{y,x_i\\}$  and  $\\{x_i',z\\}$  correspond to bags of  $T_{G}^{(d)}(y,z)$  associated to all tree nodes of depth 2. Since  $g$  and  $\\pi$  are bIsos, we have:",
+        "bbox": [
+            191,
+            565,
+            825,
+            693
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- For every  $w_{i}$  ( $i \\in [m]$ ), there exists  $x_{j}$  ( $j \\in [n]$ ), such that  $g(w_{i}) = \\pi(x_{j}) = \\tilde{x}_{j}$  for some  $\\tilde{x}_{j} \\in V_{G}$  and  $F[\\{\\{u, v, w_{i}\\}\\}] \\simeq F_{G}^{(d)}(y, z)[\\{\\{y, z, x_{j}\\}\\}] \\simeq G[\\{\\{y, z, \\tilde{x}_{j}\\}\\}$ ;",
+            "- For every  $w_{i}^{\\prime}$  ( $i \\in [m']$ ), there exists  $x_{j}^{\\prime}$  ( $j \\in [n]$ ), such that  $g(w_{i}^{\\prime}) = \\pi(x_{j}^{\\prime}) = \\tilde{x}_{j}^{\\prime}$  for some  $\\tilde{x}_{j}^{\\prime} \\in V_{G}$  and  $F[\\{\\{u, v, w_{i}^{\\prime}\\}\\}] \\simeq F_{G}^{(d)}(y, z)[\\{\\{y, z, x_{j}^{\\prime}\\}\\}] \\simeq G[\\{\\{y, z, \\tilde{x}_{j}^{\\prime}\\}]$ ."
+        ],
+        "bbox": [
+            236,
+            699,
+            823,
+            770
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "text",
+        "text": "We then define  $\\rho(w_i) = x_j$  for each  $i \\in [m]$  and  $\\rho(w_i') = x_j'$  for each  $i \\in [m']$ . Based on the above two items, one can easily define  $\\tau$  such that each node  $s$  in  $T^r$  of depth 1 or 2 is mapped by  $\\tau$  to a node  $t$  in  $T_G^{(d)}(y,z)$  of the same depth,  $\\rho(\\beta_T(s)) = \\beta_{T_G^{(d)}}(t)$ , and  $\\rho$  is an isomorphism from  $F[\\beta_T(s)]$  to  $F_G^{(d)}(y,z)[\\beta_{T_G^{(d)}(y,z)}(t)]$ .",
+        "bbox": [
+            210,
+            775,
+            825,
+            851
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "text",
+        "text": "Next, we can recursively define  $\\rho$ 's image on  $F[T^r[s]]$  for each tree node  $s$  of depth 2 following the same construction above. This is because  $g$  is still a bIso from  $(F[T^r[s]], T^r[s])$  to  $G$ ,  $\\pi$  is still a bIso from  $\\left(F_G^{(d)}(y,z)[T_G^{(d)}(y,z)[\\tau(s)]], T_G^{(d)}(y,z)[\\tau(s)]\\right)$  to  $G$ , and  $g(\\beta_T(s)) = \\pi(\\beta_{T_G^{(d)}}(\\tau(s)))$ . Recursively applying this procedure, we can construct  $(\\rho, \\tau)$  such that it is a",
+        "bbox": [
+            210,
+            854,
+            826,
+            928
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "page_number",
+        "text": "31",
+        "bbox": [
+            488,
+            946,
+            506,
+            959
+        ],
+        "page_idx": 30
+    },
+    {
+        "type": "text",
+        "text": "bIsoHom from  $(F, T^r)$  to  $\\left(F_G^{(d)}(y, z), T_G^{(d)}(y, z)\\right)$ . That is to say, we have proved that for all  $g \\in S_1(u, v)$ , there is a preimage  $(\\rho, \\tau) \\in S_2(y, z)$  such that  $\\sigma(\\rho, \\tau) = g$ .",
+        "bbox": [
+            209,
+            102,
+            823,
+            140
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "text",
+        "text": "3. We finally prove that  $\\sigma$  is an injection. Let  $(\\rho_1, \\tau_1), (\\rho_2, \\tau_2) \\in S_2(y, z)$  such that  $\\pi \\circ \\rho_1 = \\pi \\circ \\rho_2$ . Let  $w_1, \\dots, w_m \\in V_F$ ,  $w_1', \\dots, w_{m'}' \\in V_F$ ,  $x_1, \\dots, x_n \\in V_{F_G^{(d)}(y,z)}$ , and  $x_1', \\dots, x_n' \\in V_{F_G^{(d)}(y,z)}$  be defined as in the previous item. For each  $i \\in [m]$ , let  $j_1(i)$  and  $j_2(i)$  be indices satisfying  $\\rho_1(w_i) = x_{j_1(i)}$  and  $\\rho_2(w_i) = x_{j_2(i)}$ . It follows that  $\\pi(x_{j_1(i)}) = \\pi(x_{j_2(i)})$ . By definition of the Local 2-GNN unfolding tree, we must have  $x_{j_1(i)} = x_{j_2(i)}$ , and thus  $\\rho_1(w_i) = \\rho_2(w_i)$ . Using a similar approach, we can prove that  $\\rho_1(w_i') = \\rho_2(w_i')$  for each  $i \\in [m']$ . Next, we can recursively apply the above procedure to the subtree  $T^r[s]$  for each tree node of depth 2 following the previous item, and finally prove that  $\\rho_1 = \\rho_2$ . Therefore,  $\\sigma$  is an injection.",
+        "bbox": [
+            191,
+            145,
+            826,
+            286
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "text",
+        "text": "Combining the above three items completes the proof.",
+        "bbox": [
+            169,
+            291,
+            529,
+            306
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "text",
+        "text": "Proposition C.24. Under Definition C.21, (i) if  $(\\rho, \\tau)$  is a bIsoSurj, then  $\\rho$  is a surjection from  $F$  to  $\\tilde{F}$  on both vertices and edges; (ii) if  $(\\rho, \\tau)$  is a bIsoInj, then  $\\rho$  is an injection from  $F$  to  $\\tilde{F}$  on both vertices and edges.",
+        "bbox": [
+            169,
+            313,
+            823,
+            359
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "text",
+        "text": "Proof. We first prove that  $\\rho$  is surjective if  $(\\rho, \\tau)$  is a bIsoSurj. We will only prove that  $\\rho$  is surjective on edges, as proving that  $\\rho$  is surjective on vertices is almost the same. For any  $\\{x, y\\} \\in E_{\\tilde{F}}$ , by Definition C.1(b) we can pick  $\\tilde{t} \\in V_{\\tilde{T}}$  such that  $\\{x, y\\} \\in \\beta_{\\tilde{T}}(\\tilde{t})$ . Since  $\\tau$  is surjective, there exists  $t \\in V_T$  such that  $\\tau(t) = \\tilde{t}$ . By definition of bag isomorphism, there exists  $u, v \\in \\beta_T(t)$  such that  $\\rho(u) = x$ ,  $\\rho(v) = y$ , and  $F[\\{u, v\\}] \\simeq \\tilde{F}[\\{x, y\\}]$ . Therefore,  $\\{u, v\\} \\in E_F$ .",
+        "bbox": [
+            169,
+            373,
+            823,
+            452
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "text",
+        "text": "We next prove that  $\\rho$  is injective if  $(\\rho, \\tau)$  is a bIsoInj. Pick any  $u \\in V_F$ . It suffices to prove that  $\\rho(u) = \\rho(v)$  iff  $u = v$  for any  $v \\in V_F$ . If the result does not hold, consider two cases:",
+        "bbox": [
+            169,
+            455,
+            823,
+            486
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- There exists  $v \\in V_F, v \\neq u$  such that  $\\rho(u) = \\rho(v)$  and  $\\{u, v\\}$  are in the same bag of  $F$ . This contradicts the definition of bag isomorphism.",
+            "- For all  $v \\in V_F$  such that  $v \\neq u$  and  $\\rho(u) = \\rho(v)$ ,  $\\{u, v\\}$  are not in the same bag of  $F$ . By Proposition C.5,  $T\\left[\\bigcup_{v \\in V_F: \\rho(u) = \\rho(v)} B_T(v)\\right]$  is disconnected. We can thus pick a path  $P$  in  $T$  such that the endpoints  $t_1$  and  $t_2$  are in different connected components of  $T\\left[\\bigcup_{v \\in V_F: \\rho(u) = \\rho(v)} B_T(v)\\right]$ . This implies that there is a tree node  $t_3$  in  $P$  such that  $\\beta_T(t_3) \\cap \\{v \\in V_F: \\rho(u) = \\rho(v)\\} = \\emptyset$ . Consequently,  $\\rho(u) \\in \\rho(\\beta_T(t_1)) = \\beta_{\\tilde{T}}(\\tau(t_1))$ ,  $\\rho(u) \\in \\rho(\\beta_T(t_2)) = \\beta_{\\tilde{T}}(\\tau(t_2))$ , but  $\\rho(u) \\notin \\beta_{\\tilde{T}}(\\tau(t_3))$ . On the other hand, since  $\\tau$  is injective,  $\\tilde{T}\\left[\\tau(V_P)\\right]$  is also a path and  $\\tau(t_3)$  is on the path between  $\\tau(t_1)$  and  $\\tau(t_2)$  in  $\\tilde{T}$ . This contradicts the definition of tree decomposition (Definition C.1(c))."
+        ],
+        "bbox": [
+            196,
+            491,
+            826,
+            661
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "text",
+        "text": "Combining the two cases concludes the proof.",
+        "bbox": [
+            169,
+            667,
+            477,
+            683
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "text",
+        "text": "Lemma C.25. Let  $M \\in \\{\\mathrm{Sub},\\mathrm{L},\\mathrm{LF},\\mathrm{F}\\}$  be any model. For any tree-decomposed graphs  $(F,T^r),(\\tilde{F},\\tilde{T}^s) \\in S^M$ ,",
+        "bbox": [
+            169,
+            689,
+            823,
+            722
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathsf {b l s o H o m} ((F, T ^ {r}), (\\tilde {F}, \\tilde {T} ^ {s})) = \\sum_ {(\\widehat {F}, \\widehat {T} ^ {t}) \\in \\mathcal {S} ^ {M}} \\frac {\\mathsf {b l s o S u r j} \\left((F , T ^ {r}) , (\\widehat {F} , \\widehat {T} ^ {t})\\right) \\cdot \\mathsf {b l s o l n j} \\left((\\widehat {F} , \\widehat {T} ^ {t}) , (\\tilde {F} , \\tilde {T} ^ {s})\\right)}{\\mathsf {a u t} (\\widehat {F} , \\widehat {T} ^ {t})},\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            727,
+            813,
+            779
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "text",
+        "text": "where  $\\operatorname{aut}(\\widehat{F},\\widehat{T}^t)$  denotes the number of automorphisms of  $(\\widehat{F},\\widehat{T}^t)$ . Here, the summation ranges over all non-isomorphic (tree-decomposed) graphs in  $S^M$  and is well-defined as there are only a finite number of graphs making the value in the summation non-zero.",
+        "bbox": [
+            168,
+            787,
+            823,
+            832
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "text",
+        "text": "Proof. We define the following set of three-tuples:",
+        "bbox": [
+            169,
+            845,
+            506,
+            862
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} S = \\left\\{\\left((\\widehat {F}, \\widehat {T} ^ {t}), (\\rho^ {\\mathsf {S}}, \\tau^ {\\mathsf {S}}), (\\rho^ {\\mathsf {I}}, \\tau^ {\\mathsf {I}})\\right): (\\widehat {F}, \\widehat {T} ^ {t}) \\in \\mathcal {S} ^ {M}, \\right. \\\\ \\left. \\left(\\rho^ {\\mathsf {S}}, \\tau^ {\\mathsf {S}}\\right) \\in \\mathsf {B l s o S u r j} \\left(\\left(F, T ^ {r}\\right), \\left(\\widehat {F}, \\widehat {T} ^ {t}\\right)\\right), \\left(\\rho^ {\\mathsf {I}}, \\tau^ {\\mathsf {I}}\\right) \\in \\mathsf {B l s o l n j} \\left(\\left(\\widehat {F}, \\widehat {T} ^ {t}\\right), \\left(\\tilde {F}, \\tilde {T} ^ {s}\\right)\\right) \\right\\}. \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            215,
+            867,
+            777,
+            922
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "page_number",
+        "text": "32",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 31
+    },
+    {
+        "type": "text",
+        "text": "Define a mappings  $\\sigma$  such that",
+        "bbox": [
+            171,
+            104,
+            375,
+            119
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\sigma \\left(\\left(\\widehat {F}, \\widehat {T} ^ {t}\\right), \\left(\\rho^ {\\mathsf {S}}, \\tau^ {\\mathsf {S}}\\right), \\left(\\rho^ {\\mathsf {I}}, \\tau^ {\\mathsf {I}}\\right)\\right) = \\left(\\rho^ {\\mathsf {I}} \\circ \\rho^ {\\mathsf {S}}, \\tau^ {\\mathsf {I}} \\circ \\tau^ {\\mathsf {S}}\\right)\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            331,
+            122,
+            663,
+            148
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "for all  $\\left((\\widehat{F},\\widehat{T}^t),(\\rho^\\mathsf{S},\\tau^\\mathsf{S}),(\\rho^\\mathsf{I},\\tau^\\mathsf{I})\\right)\\in S.$  It suffices to prove the following three statements:",
+        "bbox": [
+            169,
+            152,
+            767,
+            176
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1.  $\\sigma$  is a mapping from  $S$  to  $\\mathsf{B}\\mathsf{I}\\mathsf{o}\\mathsf{s}\\mathsf{H}\\mathsf{o}\\mathsf{m}((F,T^r),(\\tilde{F},\\tilde{T}^s))$",
+            "2.  $\\sigma$  is surjective;",
+            "3.  $\\sigma\\left((\\widehat{F}_1, \\widehat{T}_1^{t_1}), (\\rho_1^{\\mathsf{S}}, \\tau_1^{\\mathsf{S}}), (\\rho_1^{\\mathsf{I}}, \\tau_1^{\\mathsf{I}})\\right) = \\sigma\\left((\\widehat{F}_2, \\widehat{T}_2^{t_2}), (\\rho_2^{\\mathsf{S}}, \\tau_2^{\\mathsf{S}}), (\\rho_2^{\\mathsf{I}}, \\tau_2^{\\mathsf{I}})\\right)$  iff there exists an isomorphism  $(\\widehat{\\rho}, \\widehat{\\tau})$  from  $(\\widehat{F}_1, \\widehat{T}_1^{t_1})$  to  $(\\widehat{F}_2, \\widehat{T}_2^{t_2})$  such that  $\\widehat{\\rho} \\circ \\rho_1^{\\mathsf{S}} = \\rho_2^{\\mathsf{S}}, \\widehat{\\tau} \\circ \\tau_1^{\\mathsf{S}} = \\tau_2^{\\mathsf{S}}, \\rho_1^{\\mathsf{I}} = \\rho_2^{\\mathsf{I}} \\circ \\widehat{\\rho}, \\tau_1^{\\mathsf{I}} = \\tau_2^{\\mathsf{I}} \\circ \\widehat{\\tau}$ ."
+        ],
+        "bbox": [
+            189,
+            183,
+            826,
+            284
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "We will prove these statements one by one.",
+        "bbox": [
+            171,
+            290,
+            457,
+            306
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1. We first prove that  $\\sigma$  is a mapping from  $S$  to  $\\mathsf{BIsoHom}((F,T^r),(\\tilde{F},\\tilde{T}^s))$ . This simply follows from the fact that both bIsoSurj and bIsoInj are bIsoHom, and the composition of two bIsoHoms are still a bIsoHom.",
+            "2. We next prove that  $\\sigma$  is surjective. Given  $(\\rho^{\\mathsf{H}},\\tau^{\\mathsf{H}})\\in \\mathrm{BlsoHom}((F,T^{r}),(\\tilde{F},\\tilde{T}^{s}))$ , we define  $(\\widehat{F},\\widehat{T}^{t})$ ,  $(\\rho^{\\mathsf{S}},\\tau^{\\mathsf{S}})$ , and  $(\\rho^{\\mathsf{I}},\\tau^{\\mathsf{I}})$  as follows:"
+        ],
+        "bbox": [
+            189,
+            311,
+            825,
+            395
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a) Let  $\\widehat{F} = \\tilde{F}[\\rho^{\\mathsf{H}}(V_F)]$  and  $\\widehat{T}^t = \\tilde{T}^s[\\tau^{\\mathsf{H}}(V_T)]$ . We clearly have  $(\\widehat{F}, \\widehat{T}^t) \\in S^M$ .",
+            "b) Let  $\\rho^{\\mathsf{S}} = \\rho^{\\mathsf{H}}$  and  $\\tau^{\\mathsf{S}} = \\tau^{\\mathsf{H}}$ . Obviously,  $(\\rho^{\\mathsf{S}},\\tau^{\\mathsf{S}})$  is a bIsoSurj from  $(F,T^{r})$  to  $(\\widehat{F},\\widehat{T}^{t})$ .",
+            "c) Define identity mappings  $\\rho^{\\mathsf{I}}(u) = u$  for all  $u\\in V_{\\widehat{F}}$  and  $\\tau^1 (t) = t$  for all  $t\\in V_{\\widehat{T}}$ . Obviously,  $(\\rho^{\\mathsf{I}},\\tau^{\\mathsf{I}})$  is a bIsoInj from  $(\\widehat{F},\\widehat{T}^{t})$  to  $(\\tilde{F},\\tilde{T}^{s})$ ."
+        ],
+        "bbox": [
+            230,
+            400,
+            821,
+            469
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "We clearly have  $\\rho^{\\mathsf{H}} = \\rho^{\\mathsf{I}}\\circ \\rho^{\\mathsf{S}}$  and  $\\tau^{\\mathrm{H}} = \\tau^{\\mathrm{I}}\\circ \\tau^{\\mathrm{S}}$ . Thus,  $\\sigma$  is a surjection.",
+        "bbox": [
+            210,
+            473,
+            683,
+            489
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "3. We finally prove the aforementioned item 3. It suffices to prove only one direction, namely,  $\\sigma\\left((\\widehat{F}_1,\\widehat{T}_1^{t_1}),(p_1^{\\mathsf{S}},\\tau_1^{\\mathsf{S}}),(p_1^{\\mathsf{l}},\\tau_1^{\\mathsf{l}})\\right) = \\sigma\\left((\\widehat{F}_2,\\widehat{T}_2^{t_2}),(p_2^{\\mathsf{S}},\\tau_2^{\\mathsf{S}}),(p_2^{\\mathsf{l}},\\tau_2^{\\mathsf{l}})\\right)$  implies that there exists an isomorphism  $(\\widehat{\\rho},\\widehat{\\tau})$  from  $(\\widehat{F}_1,\\widehat{T}_1^{t_1})$  to  $(\\widehat{F}_2,\\widehat{T}_2^{t_2})$  such that  $\\widehat{\\rho}\\circ \\rho_{1}^{\\mathsf{S}} = \\rho_{2}^{\\mathsf{S}},\\widehat{\\tau}\\circ \\tau_{1}^{\\mathsf{S}} = \\tau_{2}^{\\mathsf{S}},$ $\\rho_1^{\\mathrm{l}} = \\rho_2^{\\mathrm{l}}\\circ \\widehat{\\rho},\\tau_1^{\\mathrm{l}} = \\tau_2^{\\mathrm{l}}\\circ \\widehat{\\tau}$",
+        "bbox": [
+            191,
+            496,
+            823,
+            568
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "a) We first prove that  $\\widehat{F}_1\\simeq \\widehat{F}_2$  and  $\\widehat{T}_1^{t_1}\\simeq \\widehat{T}_2^{t_2}$",
+        "bbox": [
+            230,
+            573,
+            544,
+            590
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "For any  $u, v \\in V_F$ , if  $\\rho_1^{\\mathsf{S}}(u) \\neq \\rho_1^{\\mathsf{S}}(v)$ , then  $\\rho_1^{\\mathsf{I}}(\\rho_1^{\\mathsf{S}}(u)) \\neq \\rho_1^{\\mathsf{I}}(\\rho_1^{\\mathsf{S}}(v))$  since  $\\rho_1^{\\mathsf{I}}$  is an injection (by Proposition C.24). Therefore,  $\\rho_2^{\\mathsf{I}}(\\rho_2^{\\mathsf{S}}(u)) \\neq \\rho_2^{\\mathsf{I}}(\\rho_2^{\\mathsf{S}}(v))$ , and thus  $\\rho_2^{\\mathsf{S}}(u) \\neq \\rho_2^{\\mathsf{S}}(v)$ . By symmetry, we also have that  $\\rho_2^{\\mathsf{S}}(u) \\neq \\rho_2^{\\mathsf{S}}(v)$  implies  $\\rho_1^{\\mathsf{S}}(u) \\neq \\rho_1^{\\mathsf{S}}(v)$ . This proves that  $\\rho_1^{\\mathsf{S}}(u) = \\rho_1^{\\mathsf{S}}(v)$  iff  $\\rho_2^{\\mathsf{S}}(u) = \\rho_2^{\\mathsf{S}}(v)$ .",
+        "bbox": [
+            250,
+            590,
+            823,
+            648
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "For any  $u, v \\in V_F$ , if  $\\{\\rho_1^S(u), \\rho_1^S(v)\\} \\in E_{\\widehat{F}_1}$ , then  $\\{\\rho_1^1(\\rho_1^S(u)), \\rho_1^1(\\rho_1^S(v))\\} \\in E_{\\widehat{F}}$  since  $\\rho_1^1$  is a homomorphism. Therefore,  $\\{\\rho_2^1(\\rho_2^S(u)), \\rho_2^1(\\rho_2^S(v))\\} \\in E_{\\widehat{F}}$ . This implies that  $\\rho_2^1(\\rho_2^S(u))$  and  $\\rho_2^1(\\rho_2^S(v))$  are in the same bag of  $\\tilde{T}^s$ . Therefore, there are vertices  $x, y \\in V_{\\widehat{F}_2}$  in the same bag of  $\\widehat{T}_2^{s_2}$  such that  $\\rho_2^1(x) = \\rho_2^1(\\rho_2^S(u))$  and  $\\rho_2^1(y) = \\rho_2^1(\\rho_2^S(v))$ , and by definition of bag isomorphism we have  $\\{x, y\\} \\in E_{\\widehat{F}_2}$ . Since  $\\rho_2$  is injective,  $x = \\rho_2^S(u)$  and  $y = \\rho_2^S(v)$ , namely,  $\\{\\rho_2^S(u), \\rho_2^S(v)\\} \\in E_{\\widehat{F}_2}$ . By symmetry, we can prove that  $\\{\\rho_1^S(u), \\rho_1^S(v)\\} \\in E_{\\widehat{F}_1}$  iff  $\\{\\rho_2^S(u), \\rho_2^S(v)\\} \\in E_{\\widehat{F}_2}$ .",
+        "bbox": [
+            250,
+            646,
+            823,
+            765
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "Finally, noting that  $\\rho_1^{\\mathsf{S}}$  and  $\\rho_2^{\\mathsf{S}}$  are surjective (by Proposition C.24) and  $\\ell_{\\widehat{F}_1}(\\rho_1^{\\mathsf{S}}(u)) = \\ell_F(u) = \\ell_{\\widehat{F}_2}(\\rho_2^{\\mathsf{S}}(u))$  for all  $u \\in V_F$ , we obtain that  $\\widehat{F}_1 \\simeq \\widehat{F}_2$ . Following the same procedure, we can prove that  $\\widehat{T}_1^{t_1} \\simeq \\widehat{T}_2^{t_2}$ .",
+        "bbox": [
+            251,
+            763,
+            823,
+            820
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "b) Consequently, there exist isomorphisms  $\\widehat{\\rho}$  and  $\\widehat{\\tau}$  such that  $\\widehat{\\rho} \\circ \\rho_1^{\\mathsf{S}} = \\rho_2^{\\mathsf{S}}, \\widehat{\\tau} \\circ \\tau_1^{\\mathsf{S}} = \\tau_2^{\\mathsf{S}}$ . For any node  $q \\in V_T$ ,",
+        "bbox": [
+            230,
+            818,
+            823,
+            849
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\widehat {\\rho} (\\beta_ {\\widehat {T} _ {1}} (\\tau_ {1} ^ {\\mathsf {S}} (q))) = \\widehat {\\rho} (\\rho_ {1} ^ {\\mathsf {S}} (\\beta_ {T} (q))) = \\rho_ {2} ^ {\\mathsf {S}} (\\beta_ {T} (q))) = \\beta_ {\\widehat {T} _ {2}} (\\tau_ {2} ^ {\\mathsf {S}} (q)) = \\beta_ {\\widehat {T} _ {2}} (\\widehat {\\tau} (\\tau_ {1} ^ {\\mathsf {S}} (q))).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            274,
+            851,
+            800,
+            873
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "Since  $\\tau_1^S$  is surjective,  $\\tau_1^S (q)$  ranges over all nodes in  $\\widehat{T}_1^{s_1}$  when  $q$  ranges over  $V_{T}$ . We thus conclude that  $(\\rho ,\\tau)$  is an isomorphism from  $(\\widehat{F}_1,\\widehat{T}_1^{t_1})$  to  $(\\widehat{F}_2,\\widehat{T}_2^{t_2})$  (see Definition C.8).",
+        "bbox": [
+            251,
+            876,
+            823,
+            924
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "page_number",
+        "text": "33",
+        "bbox": [
+            488,
+            946,
+            506,
+            959
+        ],
+        "page_idx": 32
+    },
+    {
+        "type": "text",
+        "text": "c) We finally prove that  $\\rho_1^{\\mathsf{I}} = \\rho_2^{\\mathsf{I}}\\circ \\widehat{\\rho}$  and  $\\tau_1^{\\mathsf{I}} = \\tau_2^{\\mathsf{I}}\\circ \\widehat{\\tau}$ . Pick any  $u\\in V_F$ , we have  $\\rho_2^{\\mathsf{I}}(\\widehat{\\rho} (\\rho_1^{\\mathsf{S}}(u))) = \\rho_2^{\\mathsf{I}}(\\rho_2^{\\mathsf{S}}(u)) = \\rho_1^{\\mathsf{I}}(\\rho_1^{\\mathsf{S}}(u))$ . Since  $\\rho_{1}^{\\mathsf{S}}$  is surjective,  $\\rho_{1}^{\\mathsf{S}}(u)$  ranges over all nodes in  $\\widehat{F}_1$  when  $u$  ranges over  $V_{F}$ . This proves that  $\\rho_1^{\\mathsf{I}} = \\rho_2^{\\mathsf{I}}\\circ \\widehat{\\rho}$ . Following the same procedure, we can prove that  $\\tau_1^{\\mathsf{I}} = \\tau_2^{\\mathsf{I}}\\circ \\widehat{\\tau}$ .",
+        "bbox": [
+            230,
+            102,
+            823,
+            164
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "Combining the above three items concludes the proof.",
+        "bbox": [
+            171,
+            169,
+            529,
+            185
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "image",
+        "img_path": "images/2c363b7f4c30baf190b496a4a88db627c65ea10677b711930d0c51fc336999c8.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            170,
+            823,
+            181
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "In the following, we further define two technical concepts that will be used to describe the next result. We note that these technical concepts also appeared in Dell et al. (2018).",
+        "bbox": [
+            169,
+            202,
+            823,
+            232
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "Definition C.26. A tree-decomposed graph  $(\\tilde{F},\\tilde{T}^{s})$  is called a bag extension of another tree-decomposed graph  $(F,T^r)$  if there is a graph  $H$  and a mapping  $(\\rho ,\\tau)$  such that  $F$  is a subgraph of  $H$  and  $(\\rho ,\\tau)$  is an isomorphism from the tree-decomposed graph  $(H,T^r)$  to  $(\\tilde{F},\\tilde{T}^{s})$ . Define  $\\mathsf{BExt}\\left((F,T^r),(\\tilde{F},\\tilde{T}^s)\\right)$  to be the set of all mappings  $(\\rho ,\\tau)$  that satisfies the above conditions, and define  $\\mathsf{bExt}\\left((F,T^r),(\\tilde{F},\\tilde{T}^s)\\right) = \\left|\\mathsf{BExt}\\left((F,T^r),(\\tilde{F},\\tilde{T}^s)\\right)\\right|$ .",
+        "bbox": [
+            169,
+            237,
+            823,
+            330
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "Remark C.27. In other words, a bag extension of a tree-decomposed graph  $(F,T^r)$  can be obtained by adding an arbitrary number of edges to  $F$  while ensuring that each added edge is contained in a tree node in  $T^r$ . A immediate fact is that  $(\\rho ,\\tau)$  is a homomorphism from  $(F,T^{r})$  to  $(\\tilde{F},\\tilde{T}^{s})$ .",
+        "bbox": [
+            169,
+            334,
+            823,
+            380
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "Definition C.28. Given a tree-decomposed graph  $(F,T^r)$  and a graph  $G$ , a bag-strong homomorphism from  $(F,T^r)$  to  $G$  (abbreviated as \"bStrHom\") is a homomorphism  $f$  from  $F$  to  $G$  such that, for all  $t\\in V_T$ ,  $f$  is a strong homomorphism from  $F[\\beta_T(t)]$  to  $G[f(\\beta_T(t))]$ , i.e.,  $\\{u,v\\} \\in E_{F[\\beta_T(t)]}$  iff  $\\{f(u),f(v)\\} \\in E_{G[f(\\beta_T(t))]}$ . Denote BStrHom  $((F,T^r),G)$  to be the set of all bStrHom from  $(F,T^r)$  to  $G$ , and denote bStrHom  $((F,T^r),G) = |\\mathsf{BStrHom}((F,T^r),G)|$ .",
+        "bbox": [
+            169,
+            383,
+            823,
+            458
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "The following equation is straightforward:",
+        "bbox": [
+            171,
+            468,
+            454,
+            484
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "Lemma C.29. Let  $M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}$  be any model. For any graph  $G$  and tree-decomposed graph  $(F, T^r) \\in S^M$ ,",
+        "bbox": [
+            169,
+            487,
+            826,
+            517
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\operatorname {h o m} (F, G) = \\sum_ {(\\tilde {F}, \\tilde {T} ^ {s}) \\in \\mathcal {S} ^ {M}} \\frac {\\mathrm {b E x t} ((F , T ^ {r}) , (\\tilde {F} , \\tilde {T} ^ {s})) \\cdot \\mathrm {b S t r H o m} ((\\tilde {F} , \\tilde {T} ^ {s}) , G)}{\\operatorname {a u t} (\\tilde {F} , \\tilde {T} ^ {s})},\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            241,
+            523,
+            753,
+            575
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "where  $\\operatorname{aut}(\\tilde{F},\\tilde{T}^{s})$  denotes the number of automorphisms of  $(\\tilde{F},\\tilde{T}^{s})$ . Here, the summation ranges over all non-isomorphic (tree-decomposed) graphs in  $S^M$  and is well-defined as there are only a finite number of graphs making the value in the summation non-zero.",
+        "bbox": [
+            169,
+            585,
+            823,
+            630
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "Proof. The proof has a similar structure to the previous lemma. We define the following set of three-tuples:",
+        "bbox": [
+            169,
+            647,
+            826,
+            676
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nS = \\left\\{\\left((\\tilde {F}, \\tilde {T} ^ {s}), (\\rho , \\tau), g\\right): (\\tilde {F}, \\tilde {T} ^ {s}) \\in \\mathcal {S} ^ {M}, (\\rho , \\tau) \\in \\mathsf {B E x t} \\Big ((F, T ^ {r}), (\\tilde {F}, \\tilde {T} ^ {s}) \\Big), g \\in \\mathsf {B S t r H o m} ((\\tilde {F}, \\tilde {T} ^ {s}), G) \\right\\}.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            171,
+            684,
+            862,
+            712
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "Define a mappings  $\\sigma$  such that  $\\sigma\\left((\\tilde{F},\\tilde{T}^s),(\\rho,\\tau),g\\right)=g\\circ\\rho$  for all  $\\left((\\tilde{F},\\tilde{T}^s),(\\rho,\\tau),g\\right)\\in S$ . It is straightforward to see that  $\\sigma$  is a mapping from  $S$  to  $\\mathsf{BlsoHom}((F,T^r),(\\tilde{F},\\tilde{T}^s))$ , namely,  $g\\circ\\rho$  is a homomorphism from  $F$  to  $G$  for all  $\\left((\\tilde{F},\\tilde{T}^s),(\\rho,\\tau),g\\right)\\in S$ .",
+        "bbox": [
+            169,
+            719,
+            823,
+            785
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "We then prove that  $\\sigma$  is surjective. Given  $h\\in \\mathrm{Hom}(F,G)$ , define  $(\\tilde{F},\\tilde{T}^{s})$ ,  $(\\rho ,\\tau)$ , and  $g$  as follows:",
+        "bbox": [
+            171,
+            790,
+            821,
+            809
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a) Define  $\\tilde{F}$  be the graph obtained from  $F$  by adding edges  $\\{\\{u,v\\} : \\exists t \\in V_T \\text{s.t.} u,v \\in \\beta_T(t), \\{h(u),h(v)\\} \\in E_G\\}$ , and let  $\\tilde{T}^s = T^r$ . Clearly,  $(\\tilde{F},\\tilde{T}^s)$  is a bag extension of  $(F,T^r)$ .",
+            "b) Define identity mappings  $\\rho(u) = u$  for all  $u \\in V_F$  and  $\\tau(t) = t$  for all  $t \\in V_T$ . Clearly,  $(\\rho, \\tau) \\in \\mathsf{BExt}((F, T^r), (\\tilde{F}, \\tilde{T}^s))$ .",
+            "c) Let  $g = h$ . It is easy to see that  $g$  is a strong homomorphism from  $\\tilde{F}[\\beta_{\\tilde{T}}(t)]$  to  $G[g(\\beta_{\\tilde{T}}(t))]$  for each  $t \\in V_{\\tilde{T}}$ . Thus,  $g \\in \\mathsf{BStrHom}((\\tilde{F}, \\tilde{T}^s), G)$ ."
+        ],
+        "bbox": [
+            189,
+            814,
+            825,
+            925
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "page_number",
+        "text": "34",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 33
+    },
+    {
+        "type": "text",
+        "text": "Noting that  $h = g = g \\circ \\rho$ , we have proved that  $\\sigma$  is a surjection.",
+        "bbox": [
+            169,
+            103,
+            599,
+            119
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "text",
+        "text": "We finally prove that  $\\sigma\\left((\\tilde{F}_1, \\tilde{T}_1^{s_1}), (\\rho_1, \\tau_1), g_1\\right) = \\sigma\\left((\\tilde{F}_2, \\tilde{T}_2^{s_2}), (\\rho_2, \\tau_2), g_2\\right)$  implies that there exists an isomorphism  $(\\tilde{\\rho}, \\tilde{\\tau})$  from  $(\\tilde{F}_1, \\tilde{T}_1^{s_1})$  to  $(\\tilde{F}_2, \\tilde{T}_2^{s_2})$  such that  $\\tilde{\\rho} \\circ \\rho_1 = \\rho_2$ ,  $\\tilde{\\tau} \\circ \\tau_1 = \\tau_2$ ,  $g_1 = g_2 \\circ \\tilde{\\rho}$ . We first prove that  $\\tilde{F}_1 \\simeq \\tilde{F}_2$  and  $\\tilde{T}_1^{s_1} \\simeq \\tilde{T}_2^{s_2}$ . (i) For any  $u, v \\in V_F$ , we obviously have  $\\rho_1(u) = \\rho_1(v)$  iff  $u = v$  iff  $\\rho_2(u) = \\rho_2(v)$ . (ii) Let  $i \\in \\{1, 2\\}$ . For any  $u, v \\in V_F$ ,  $\\{\\rho_i(u), \\rho_i(v)\\} \\in E_{\\tilde{F}_i}$  iff  $\\{u, v\\} \\in E_F$  or  $u, v$  are in the same bag of  $T^r$  and  $\\{g_i(\\rho_i(u)), g(\\rho_i(v))\\} \\in E_G$ . Since  $g_1 \\circ \\rho_1 = g_2 \\circ \\rho_2$ , we have  $\\{\\rho_1(u), \\rho_1(v)\\} \\in E_{\\tilde{F}_1}$  iff  $\\{\\rho_2(u), \\rho_2(v)\\} \\in E_{\\tilde{F}_2}$ . (iii) Finally, noting that  $\\rho_1$  and  $\\rho_2$  are bijective and  $\\ell_{\\tilde{F}_1}(\\rho_1(u)) = \\ell_F(u) = \\ell_{\\tilde{F}_2}(\\rho_2(u))$  for all  $u \\in V_F$ , we obtain that  $\\tilde{F}_1 \\simeq \\tilde{F}_2$ . On the other hand,  $\\tilde{T}_1^{s_1} \\simeq \\tilde{T}_2^{s_2}$  trivially holds. The remaining procedure is almost the same as in the previous lemma.",
+        "bbox": [
+            169,
+            126,
+            826,
+            272
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "text",
+        "text": "We next show that the mapping bStrHom can be further decomposed as shown in Lemma C.31. We need an auxiliary concept:",
+        "bbox": [
+            169,
+            287,
+            823,
+            316
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "text",
+        "text": "Definition C.30. Given two tree-decomposed graphs  $(F,T^r)$  and  $(\\tilde{F},\\tilde{T}^{s})$  , a homomorphism  $(\\rho ,\\tau)$  from  $(F,T^{r})$  to  $(\\tilde{F},\\tilde{T}^{s})$  is called bag-strong surjective (abbreviated as \"bStrSurj\") if  $\\rho$  is a bagstrong homomorphism from  $(F,T^{r})$  to  $\\tilde{F}$  and is surjective on both vertices and edges, and  $\\tau$  is an isomorphism from  $T^r$  to  $\\tilde{T}^s$  . Denote BStrSurj  $((F,T^r),(\\tilde{F},\\tilde{T}^{s}))$  to be the set of all bStrSurj from  $(F,T^{r})$  to  $(\\tilde{F},\\tilde{T}^{s})$  , and denote bStrSurj  $((F,T^r),(\\tilde{F},\\tilde{T}^{s})) = |\\mathsf{BStrSurj}((F,T^r),(\\tilde{F},\\tilde{T}^{s}))|$",
+        "bbox": [
+            169,
+            321,
+            823,
+            402
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "text",
+        "text": "Lemma C.31. Let  $M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}$  be any model. For any graph  $G$  and tree-decomposed graph  $(F, T^r) \\in S^M$ ,",
+        "bbox": [
+            169,
+            404,
+            826,
+            434
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathsf{bStrHom}((F,T^{r}),G) = \\sum_{(\\tilde{F},\\tilde{T}^{s})\\in \\mathcal{S}^{M}}\\frac{\\mathsf{bStrSurj}\\left((F,T^{r}),( \\tilde{F},\\tilde{T}^{s})\\right)\\cdot\\mathsf{blso}\\left((\\tilde{F},\\tilde{T}^{s}),G\\right)}{\\mathsf{aut}(\\tilde{F},\\tilde{T}^{s})},\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            210,
+            440,
+            784,
+            491
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "text",
+        "text": "where  $\\operatorname{aut}(\\tilde{F},\\tilde{T}^s)$  denotes the number of automorphisms of  $(\\tilde{F},\\tilde{T}^{s})$  . Here, the summation ranges over all non-isomorphic (tree-decomposed) graphs in  $S^M$  and is well-defined as there are only a finite number of graphs making the value in the summation non-zero.",
+        "bbox": [
+            169,
+            500,
+            823,
+            544
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "text",
+        "text": "Proof. The proof has a similar structure to the previous lemma. We define the following set of three-tuples:",
+        "bbox": [
+            169,
+            559,
+            826,
+            588
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nS = \\left\\{\\left(\\left(\\tilde {F}, \\tilde {T} ^ {s}\\right), (\\rho , \\tau), g\\right): \\left(\\tilde {F}, \\tilde {T} ^ {s}\\right) \\in \\mathcal {S} ^ {M}, (\\rho , \\tau) \\in \\mathrm {B S t r S u r j} \\left(\\left(F, T ^ {r}\\right), \\left(\\tilde {F}, \\tilde {T} ^ {s}\\right)\\right), g \\in \\mathrm {B l s o} \\left(\\left(\\tilde {F}, \\tilde {T} ^ {s}\\right), G\\right) \\right\\}.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            171,
+            594,
+            851,
+            621
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "text",
+        "text": "Define a mappings  $\\sigma$  such that  $\\sigma\\left((\\tilde{F},\\tilde{T}^s),(\\rho,\\tau),g\\right) = g\\circ\\rho$  for all  $\\left((\\tilde{F},\\tilde{T}^s),(\\rho,\\tau),g\\right)\\in S$ . It suffices to prove the following three statements:",
+        "bbox": [
+            169,
+            628,
+            826,
+            665
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1.  $\\sigma$  is a mapping from  $S$  to  $\\mathsf{BStrHom}((F,T^r),G)$",
+            "2.  $\\sigma$  is surjective;",
+            "3.  $\\sigma\\left((\\tilde{F}_1, \\tilde{T}_1^{s_1}), (\\rho_1, \\tau_1), g_1\\right) = \\sigma\\left((\\tilde{F}_2, \\tilde{T}_2^{s_2}), (\\rho_2, \\tau_2), g_2\\right)$  iff there exists an isomorphism  $(\\tilde{\\rho}, \\tilde{\\tau})$  from  $(\\tilde{F}_1, \\tilde{T}_1^{s_1})$  to  $(\\tilde{F}_2, \\tilde{T}_2^{s_2})$  such that  $\\tilde{\\rho} \\circ \\rho_1 = \\rho_2$ ,  $\\tilde{\\tau} \\circ \\tau_1 = \\tau_2$ ,  $g_1 = g_2 \\circ \\tilde{\\rho}$ ."
+        ],
+        "bbox": [
+            189,
+            671,
+            823,
+            756
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "text",
+        "text": "We will prove these statements one by one.",
+        "bbox": [
+            171,
+            761,
+            457,
+            776
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1. We first prove that  $\\sigma$  is a mapping from  $S$  to  $\\mathsf{BStrHom}((F,T^r),G)$ . Pick any  $\\left((\\tilde{F},\\tilde{T}^s),(\\rho ,\\tau),g\\right)\\in S$ . Pick any  $t\\in V_T$  and  $u,v\\in \\beta_{T}(t)$ . Then,  $\\{u,v\\} \\in E_F$  iff  $\\{\\rho (u),\\rho (v)\\} \\in E_{\\tilde{F}}$  (since  $\\rho$  is a strong homomorphism from  $F[\\beta_T(t)]$  to  $\\tilde{F} [\\rho (\\beta_T(t))]$ ). Also,  $\\rho (u),\\rho (v)\\in \\beta_{\\tilde{F}}(\\tau (t))$  are in the same bag. Similarly,  $\\{\\rho (u),\\rho (v)\\} \\in E_{\\tilde{F}}$  iff  $\\{g(\\rho (u)),g(\\rho (v))\\} \\in E_G$  (since  $g$  is a bIso). Thus,  $g\\circ \\rho$  is a bag-strong homomorphism.",
+            "2. We next prove that  $\\sigma$  is surjective. Given  $h\\in \\mathsf{BStrHom}((F,T^r),G)$  , define  $(\\tilde{F},\\tilde{T}^{s}),(\\rho ,\\tau)$  and  $g$  as follows. First define a relation  $\\sim$  on set  $V_{F}$  such that  $u\\sim v$  iff the following hold:"
+        ],
+        "bbox": [
+            189,
+            782,
+            826,
+            902
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "text",
+        "text": "a)  $h(u) = h(v)$",
+        "bbox": [
+            230,
+            909,
+            346,
+            924
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "page_number",
+        "text": "35",
+        "bbox": [
+            488,
+            946,
+            506,
+            959
+        ],
+        "page_idx": 34
+    },
+    {
+        "type": "text",
+        "text": "b) There exists a path  $P$  in  $T^r$  with endpoints  $t_1, t_2 \\in V_T$  such that  $u \\in \\beta_T(t_1)$ ,  $v \\in \\beta_T(t_2)$ , and all node  $t$  on path  $P$  satisfies that  $h(u) \\in h(\\beta_T(t))$ .",
+        "bbox": [
+            228,
+            103,
+            823,
+            133
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "It is easy to see that  $\\sim$  is an equivalence relation on  $V_{F}$ . We can thus define a mapping  $\\rho$  that respects the relation, i.e.,  $\\rho(u) = \\rho(v)$  iff  $u \\sim v$  for all  $u, v \\in V_{F}$ . Moreover, for any edge  $\\{u, v\\} \\in E_{F}$ ,  $\\rho(u) \\neq \\rho(v)$  (since  $h$  is a homomorphism and  $h(u) \\neq h(v)$ ). This implies that we can define  $\\tilde{F}$  to be the homomorphic image of  $F$  such that  $\\rho$  is the surjective homomorphism on both vertices and edges.",
+        "bbox": [
+            210,
+            138,
+            826,
+            212
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "We then define the mapping  $g: V_{\\tilde{F}} \\to V_G$  such that  $g(\\rho(u)) = h(u)$  for all  $u \\in V_F$ . Note that  $g$  is well-defined since  $\\rho(u) = \\rho(v)$  implies  $h(u) = h(v)$  for all  $u, v \\in V_F$ , and  $\\rho: V_F \\to V_{\\tilde{F}}$  is surjective. It follows that  $h = g \\circ \\rho$ . To prove that  $g$  is a homomorphism, note that for all  $\\{x, y\\} \\in E_{\\tilde{F}}$ , there exists an edge  $\\{u, v\\} \\in E_F$  with  $\\rho(u) = x$ ,  $\\rho(v) = y$ , which implies that  $\\{h(u), h(v)\\} \\in E_G$  (since  $h$  is a homomorphism), namely,  $\\{g(x), g(y)\\} \\in E_G$ .",
+        "bbox": [
+            210,
+            217,
+            823,
+            289
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "We next define tree  $\\tilde{T}^s = (V_T, E_T, \\beta_{\\tilde{T}})$ ,  $s = t$ , and identity mapping  $\\tau$  so that  $\\tau$  is an isomorphism from  $T_r$  to  $T_s$  (ignoring the labels). Set  $\\beta_{\\tilde{T}}(t) = \\rho(\\beta_T(t))$  for all  $t \\in V_T$ . We will prove that  $(\\tilde{F}, \\tilde{T}^s) \\in S^M$  is a valid tree decomposition. It suffices to prove that Definition C.1(c) holds. Pick any vertex  $x \\in V_{\\tilde{F}}$  and tree node  $t_1, t_2 \\in B_{\\tilde{T}}(x)$ . Then, there exists  $u \\in \\beta_T(t_1), v \\in \\beta_T(t_2)$  such that  $\\rho(u) = x$ ,  $\\rho(v) = x$ . Therefore,  $u \\sim v$ . As such, there exists a path  $P$  in  $T^r$  such that all node  $t$  on  $P$  satisfies that there exists  $w \\in \\beta_T(t)$  with  $h(w) = h(u)$ , namely,  $w \\sim u$ . Consequently,  $\\rho(u) \\in \\beta_{\\tilde{T}}(t)$ , implying that  $\\tilde{T}^s[B_{\\tilde{T}}(x)]$  is connected. This proves that  $(\\tilde{F}, \\tilde{T}^s) \\in S^M$ . Also,  $(\\rho, \\tau)$  is clearly a homomorphism from  $(F, T^r)$  to  $(\\tilde{F}, \\tilde{T}^s)$  according to Definition C.20.",
+        "bbox": [
+            210,
+            296,
+            823,
+            434
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "It remains to prove that  $\\rho$  is a bag-strong homomorphism and  $g$  is a bIso. Pick any  $t \\in V_T$  and  $u, v \\in \\beta_T(t)$ . If  $\\{u, v\\} \\notin E_F$ , then  $\\{h(u), h(v)\\} \\notin E_G$  (since  $h$  is a bag-strong homomorphism). Therefore,  $\\{\\rho(u), \\rho(v)\\} \\notin E_{\\tilde{F}}$  (since  $g$  is a homomorphism), namely,  $\\rho$  is a bag-strong homomorphism. Since  $\\rho$  is surjective,  $\\{\\rho(u), \\rho(v)\\}$  ranges over all vertices in the same bag of  $\\hat{T}^s$  when  $t \\in V_T$  and  $u, v \\in \\beta_T(t)$  are arbitrary. Therefore,  $g$  is a bIso because  $\\{\\rho(u), \\rho(v)\\} \\notin E_{\\tilde{F}}$  iff  $\\{h(u), h(v)\\} \\notin E_G$ .",
+        "bbox": [
+            210,
+            439,
+            825,
+            526
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "3. We finally prove that  $\\sigma\\left((\\tilde{F}_1, \\tilde{T}_1^{s_1}), (\\rho_1, \\tau_1), g_1\\right) = \\sigma\\left((\\tilde{F}_2, \\tilde{T}_2^{s_2}), (\\rho_2, \\tau_2), g_2\\right)$  implies there exists an isomorphism  $(\\tilde{\\rho}, \\tilde{\\tau})$  from  $(\\tilde{F}_1, \\tilde{T}_1^{s_1})$  to  $(\\tilde{F}_2, \\tilde{T}_2^{s_2})$  such that  $\\tilde{\\rho} \\circ \\rho_1 = \\rho_2$ ,  $\\tilde{\\tau} \\circ \\tau_1 = \\tau_2$ ,  $g_1 = g_2 \\circ \\tilde{\\rho}$ . Let  $h = g_1 \\circ \\rho_1 = g_2 \\circ \\rho_2$ . Here, we will only prove that  $\\tilde{F}_1 \\simeq \\tilde{F}_2$  since the remaining procedure is almost the same as previous proofs. Since both  $\\tilde{F}_1$  and  $\\tilde{F}_2$  are homomorphic images of  $F$ , it suffices to prove that, for all  $u, v \\in V_F$ ,  $\\rho_1(u) = \\rho_1(v)$  iff the following hold:",
+        "bbox": [
+            191,
+            532,
+            823,
+            632
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a)  $h(u) = h(v)$",
+            "b) There exists a path  $P$  in  $T^r$  with endpoints  $t_1, t_2 \\in V_T$  such that  $u \\in \\beta_T(t_1)$ ,  $v \\in \\beta_T(t_2)$ , and all node  $t$  on path  $P$  satisfies that  $h(u) \\in h(\\beta_T(t))$ ."
+        ],
+        "bbox": [
+            230,
+            638,
+            823,
+            686
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "On one hand, if  $\\rho_{1}(u) = \\rho_{1}(v)$ , we clearly have  $h(u) = h(v)$  and there exists  $t_1 \\in B_T(u), t_2 \\in B_T(v)$  such that  $t_1, t_2 \\in B_{\\tilde{T}_1}(\\rho_1(u))$ . Since  $\\tilde{T}_1[B_{\\tilde{T}_1}(\\rho_1(u))]$  is connected, there is a path  $P$  with endpoints  $t_1, t_2$  such that all nodes  $t$  on  $P$  satisfies  $\\rho_{1}(u) \\in \\rho_{1}(\\beta_{T}(t))$  and thus  $h(u) \\in h(\\beta_{T}(t))$ .",
+        "bbox": [
+            210,
+            691,
+            823,
+            751
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "On the other hand, if  $\\rho_1(u) \\neq \\rho_1(v)$  but the above items (a) and (b) hold, consider two cases:",
+        "bbox": [
+            210,
+            751,
+            823,
+            766
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "-  $u$  and  $v$  are in the same bag of  $T$ . Then,  $\\rho_1(u)$  and  $\\rho_1(v)$  are in the same bag of  $\\tilde{T}_1$ . Since  $g_1$  is a bIso,  $g_1(\\rho_1(u)) \\neq g_1(\\rho_1(v))$ , which contradicts item (a).",
+            "-  $u$  and  $v$  are not in the same bag of  $T$ . Then, there exist two adjacent nodes  $t_1, t_2$  on  $P$  such that  $\\rho_1(u) \\in \\beta_{\\tilde{T}_1}(t_1), \\rho_1(u) \\notin \\beta_{\\tilde{T}_1}(t_2)$ . By Definition C.4(c),  $\\beta_{\\tilde{T}_1}(t_2) \\subset \\beta_{\\tilde{T}_1}(t_1)$ . Also, item (b) implies that there exists  $w \\in \\beta_T(t_2)$  such that  $h(w) = h(u)$ . Therefore,  $\\rho_1(u)$  and  $\\rho_1(w)$  are two different nodes in  $\\beta_{\\tilde{T}_1}(t_1)$  with  $g_1(\\rho_1(u)) = h(u) = h(w) = g_1(\\rho_1(w))$ . This contradicts the condition that  $g_1$  is a bIso."
+        ],
+        "bbox": [
+            236,
+            773,
+            823,
+            880
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "This yields the desired result that  $\\tilde{F}_1\\simeq \\tilde{F}_2$",
+        "bbox": [
+            210,
+            886,
+            498,
+            902
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "Combining the above three items concludes the proof.",
+        "bbox": [
+            171,
+            909,
+            529,
+            924
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "page_number",
+        "text": "36",
+        "bbox": [
+            488,
+            946,
+            509,
+            960
+        ],
+        "page_idx": 35
+    },
+    {
+        "type": "text",
+        "text": "Let  $M \\in \\{\\mathrm{Sub}, \\mathrm{L}, \\mathrm{LF}, \\mathrm{F}\\}$  be any model. We can list all non-isomorphic tree-decomposed (labeled) graphs in  $S^M$  into an infinite sequence  $(F_1, T_1^{r_1}), (F_2, T_2^{r_2}), \\dots$ . Consider two types of ordering:",
+        "bbox": [
+            169,
+            103,
+            823,
+            133
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- Ordered by the size of the graphs  $F_{i}$ . We denote this ordering as 1st. This ordering requires that for any  $i < j$ , either  $|V_{F_i}| < |V_{F_j}|$  or  $|V_{F_i}| = |V_{F_j}|$  and  $|E_{F_i}| \\geq |E_{F_j}|$ . Note that when two graphs have an equal number of vertices, we place the graph with more edges to the front. When the number of edges is also the same, they can be arranged in any fixed order.",
+            "- Ordered by the size of the trees  $T_{i}$ . We denote this ordering as 2nd. This ordering requires that  $|V_{T_i}| \\leq |V_{T_j}|$  for any  $i < j$ . When the number of tree nodes is the same, they can be arranged in any fixed order."
+        ],
+        "bbox": [
+            197,
+            138,
+            825,
+            239
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "Without loss of generality, we assume that the labels of each graph  $F_{i}$  are integers in the range of  $[1, |V_{F_i}|]$ . Note that using integer labels of a bounded range is already sufficient to represent all non-isomorphic labeled graphs up to a bijective label transformation. This ensures that the index  $i$  is countable.",
+        "bbox": [
+            169,
+            244,
+            826,
+            301
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "For the 1st ordering, define the following notations:",
+        "bbox": [
+            171,
+            308,
+            514,
+            324
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a) Let  $f: \\mathcal{S}^M \\times \\mathcal{S}^M \\to \\mathbb{N}$  be any mapping. Define the associated (infinite) matrix  $A^{f,M,1\\mathrm{st}} \\in \\mathbb{N}^{\\mathbb{N}_+ \\times \\mathbb{N}_+}$  such that  $A_{i,j}^{f,M,1\\mathrm{st}} = f((F_i, T_i^{r_i}), (F_j, T_j^{r_j}))$ .",
+            "b) Let  $g: \\mathcal{S}^M \\times \\mathcal{G} \\to \\mathbb{N}$  be any mapping. Given a graph  $G \\in \\mathcal{G}$ , define the (infinite) vector  $\\pmb{p}_G^{g,M,1\\mathrm{st}} \\in \\mathbb{N}^{\\mathbb{N}_+}$  such that  $p_{G,i}^{f,M,1\\mathrm{st}} = g((F_i,T_i^{r_i}),G)$ .",
+            "c) Let  $h: \\mathcal{G} \\times \\mathcal{G} \\to \\mathbb{N}$  be any mapping. Given a graph  $G \\in \\mathcal{G}$ , define the (infinite) vector  $p_{G}^{h,M,1\\mathrm{st}} \\in \\mathbb{N}^{\\mathbb{N}_{+}}$  such that  $p_{G,i}^{h,M,1\\mathrm{st}} = h(F_i, G)$ ."
+        ],
+        "bbox": [
+            189,
+            328,
+            825,
+            433
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "We can similarly define  $\\pmb{A}^{f,M,2\\mathrm{nd}}$ ,  $\\pmb{p}_G^{g,M,2\\mathrm{nd}}$ ,  $\\pmb{p}_G^{h,M,2\\mathrm{nd}}$  for the ordering 2nd.",
+        "bbox": [
+            169,
+            438,
+            666,
+            458
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "Corollary C.32. Let  $M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}$  be any model and  $G, H$  be two graphs. Then,  $\\operatorname{hom}(F, G) = \\operatorname{hom}(F, H)$  for all  $(F, T^r) \\in S^M$  iff  $\\text{blso}((F, T^r), G) = \\text{blso}((F, T^r), H)$  for all  $(F, T^r) \\in S^M$ .",
+        "bbox": [
+            169,
+            459,
+            826,
+            502
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "Proof. We separately consider each direction.",
+        "bbox": [
+            171,
+            517,
+            477,
+            532
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "1. We first prove that if  $\\mathsf{blso}((F,T^r),G) = \\mathsf{blso}((F,T^r),H)$  for all  $(F,T^{r})\\in S^{M}$ , then  $\\mathsf{hom}(F,G) = \\mathsf{hom}(F,H)$  for all  $(F,T^{r})\\in S^{M}$ . According to Lemmas C.29 and C.31, we can rewrite the corresponding equations into matrix forms for any  $F\\in \\mathcal{G}$ :",
+        "bbox": [
+            191,
+            537,
+            825,
+            580
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {p} _ {F} ^ {\\text {h o m}, M, 1 \\mathrm {s t}} = \\boldsymbol {A} ^ {\\text {b E x t}, M, 1 \\mathrm {s t}} \\left(\\boldsymbol {A} ^ {\\text {a u t}, M, 1 \\mathrm {s t}}\\right) ^ {- 1} \\boldsymbol {p} _ {F} ^ {\\text {b S t r H o m}, M, 1 \\mathrm {s t}} \\tag {19}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            346,
+            585,
+            823,
+            604
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {p} _ {F} ^ {\\mathrm {b S t r H o m}, M, 1 \\mathrm {s t}} = \\boldsymbol {A} ^ {\\mathrm {b S t r S u r j}, M, 1 \\mathrm {s t}} \\left(\\boldsymbol {A} ^ {\\mathrm {a u t}, M, 1 \\mathrm {s t}}\\right) ^ {- 1} \\boldsymbol {p} _ {F} ^ {\\mathrm {b l s o}, M, 1 \\mathrm {s t}}. \\tag {20}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            326,
+            606,
+            823,
+            627
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "From the above equations, we immediately obtain that  $p_{G}^{\\mathrm{blso},M,1\\mathrm{st}} = p_{H}^{\\mathrm{blso},M,1\\mathrm{st}}$  implies  $p_{G}^{\\mathrm{hom},M,1\\mathrm{st}} = p_{H}^{\\mathrm{hom},M,1\\mathrm{st}}$ .",
+        "bbox": [
+            210,
+            631,
+            823,
+            667
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "2. We next prove that if  $\\mathsf{hom}(F,G) = \\mathsf{hom}(F,H)$  for all  $(F,T^r)\\in S^M$ , then  $\\mathsf{blso}((F,T^r),G) = \\mathsf{blso}((F,T^r),H)$  for all  $(F,T^{r})\\in S^{M}$ . This can be seen from the following facts:",
+        "bbox": [
+            189,
+            672,
+            825,
+            715
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a)  $A^{\\mathrm{out},M,1\\mathrm{st}}$  is a diagonal matrix and all diagonal elements are positive integers.",
+            "b)  $A^{\\mathrm{bExt}, M, 1\\mathrm{st}}$  is a lower triangular matrix and all diagonal elements are positive integers. This is because for any two tree-composed graphs  $(F_i, T_i^{r_i})$  and  $(F_j, T_j^{r_j})$ ,  $\\mathsf{bExt}((F_i, T_i^{r_i}), (F_j, T_j^{r_j})) > 0$  only if  $|V_{F_i}| = |V_{F_j}|$  and  $|E_{F_i}| \\leq |E_{F_j}|$ .",
+            "c)  $A^{\\mathrm{bStrSurj}, M, 1\\mathrm{st}}$  is also a lower triangular matrix and all diagonal elements are positive integers. This is because for any two tree-composed graphs  $(F_i, T_i^{r_i})$  and  $(F_j, T_j^{r_j})$ ,  $\\mathsf{bStrSurj}((F_i, T_i^{r_i}), (F_j, T_j^{r_j})) > 0$  only if  $|V_{F_i}| > |V_{F_j}|$  or  $(|V_{F_i}| = |V_{F_j}|$  and  $|E_{F_i}| = |E_{F_j}|$ )."
+        ],
+        "bbox": [
+            230,
+            722,
+            825,
+            849
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "Therefore, the composition  $A^{\\mathrm{bExt},M,1\\mathrm{st}}(A^{\\mathrm{aut},M,1\\mathrm{st}})^{-1}A^{\\mathrm{bStrSurj},M,1\\mathrm{st}}(A^{\\mathrm{aut},M,1\\mathrm{st}})^{-1}$  is lower triangular and is invertible (although it is an infinite matrix). We thus arrive at the desired conclusion that  $p_{G}^{\\mathrm{hom},M,1\\mathrm{st}} = p_{H}^{\\mathrm{hom},M,1\\mathrm{st}}$  implies  $p_{G}^{\\mathrm{blso},M,1\\mathrm{st}} = p_{H}^{\\mathrm{blso},M,1\\mathrm{st}}$ .",
+        "bbox": [
+            210,
+            854,
+            825,
+            904
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "image",
+        "img_path": "images/d51fa9c9fbdd38a72f32f1eb85b9a93490904c19b1a58b1587d2c50ae029aaec.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            910,
+            823,
+            922
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "page_number",
+        "text": "37",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 36
+    },
+    {
+        "type": "text",
+        "text": "Corollary C.33. Let  $M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}$  be any model and  $G, H$  be two graphs. Then,  $\\text{cnt}^M((F, T^r), G) = \\text{cnt}^M((F, T^r), H)$  for all  $(F, T^r) \\in \\mathcal{S}^M$  iff  $\\text{blso}((F, T^r), G) = \\text{blso}((F, T^r), H)$  for all  $(F, T^r) \\in \\mathcal{S}^M$ .",
+        "bbox": [
+            169,
+            103,
+            823,
+            148
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "Proof. We separately consider each direction.",
+        "bbox": [
+            171,
+            161,
+            475,
+            176
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "1. We first prove that if  $\\mathsf{cnt}^M((F, T^r), G) = \\mathsf{cnt}^M((F, T^r), H)$  for all  $(F, T^r) \\in S^M$ , then  $\\mathsf{blso}((F, T^r), G) = \\mathsf{blso}((F, T^r), H)$  for all  $(F, T^r) \\in S^M$ . According to Lemma C.23, we can rewrite Equation (18) into the matrix form for all  $F \\in \\mathcal{G}$ :",
+        "bbox": [
+            191,
+            181,
+            823,
+            226
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {p} _ {F} ^ {\\text {b l s o}, M, 2 \\text {n d}} = \\boldsymbol {A} ^ {\\text {b l s o H o m}, M, 2 \\text {n d}} \\boldsymbol {p} _ {F} ^ {\\text {c n t}, M, 2 \\text {n d}}. \\tag {21}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            383,
+            232,
+            823,
+            252
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "This immediately obtains that  $\\pmb{p}_G^{\\mathrm{cnt},M,2\\mathrm{nd}} = \\pmb{p}_H^{\\mathrm{cnt},M,2\\mathrm{nd}}$  implies  $\\pmb{p}_G^{\\mathrm{blso},M,2\\mathrm{nd}} = \\pmb{p}_H^{\\mathrm{blso},M,2\\mathrm{nd}}$ .",
+        "bbox": [
+            210,
+            258,
+            792,
+            280
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "2. We next prove that  $\\pmb{p}_G^{\\mathrm{blo},M,2\\mathrm{nd}} = \\pmb{p}_H^{\\mathrm{blo},M,2\\mathrm{nd}}$  implies  $\\pmb{p}_G^{\\mathrm{cnt},M,2\\mathrm{nd}} = \\pmb{p}_H^{\\mathrm{cnt},M,2\\mathrm{nd}}$ . According to Lemma C.25, we have",
+        "bbox": [
+            191,
+            282,
+            823,
+            315
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {A} ^ {\\text {b l s o H o m}, M, 2 \\text {n d}} = \\boldsymbol {A} ^ {\\text {b l s o S u r j}, M, 2 \\text {n d}} \\left(\\boldsymbol {A} ^ {\\text {a u t}, M, 2 \\text {n d}}\\right) ^ {- 1} \\boldsymbol {A} ^ {\\text {b l s o l n j}, M, 2 \\text {n d}}. \\tag {22}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            310,
+            319,
+            823,
+            339
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "Moreover, we have the following facts:",
+        "bbox": [
+            210,
+            345,
+            472,
+            359
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a)  $A^{\\mathrm{out},M,2\\mathrm{nd}}$  is a diagonal matrix and all diagonal elements are positive integers.",
+            "b)  $A^{\\mathrm{blsohnj},M,2\\mathrm{nd}}$  is an upper triangular matrix and all diagonal elements are positive integers. This is because for any two tree-composed graphs  $(F_i,T_i^{r_i})$  and  $(F_j,T_j^{r_j})$ ,  $\\mathsf{blsohnj}((F_i,T_i^{r_i}),(F_j,T_j^{r_j})) > 0$  only if  $|V_{T_i}| \\leq |V_{T_j}|$ .",
+            "c) Similarly,  $\\mathbf{A}^{\\mathrm{bIsoSurj},M,2\\mathrm{nd}}$  is a lower triangular matrix and all diagonal elements are positive integers."
+        ],
+        "bbox": [
+            230,
+            364,
+            823,
+            460
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "Unfortunately, since  $A^{\\mathrm{blsolnj},M,2\\mathrm{nd}}$  is an infinite upper triangular matrix, the inverse matrix is not well-defined. Nevertheless, we can use a special property of bIsoHom to complete our proof, namely,  $\\mathrm{bIsOHom}((F_i,T_i^{r_i}),(F_j,T_j^{r_j})) > 0$  implies that the depth of  $T_{i}^{r_{i}}$  is equal to the depth of  $T_{j}^{r_{j}}$ . Therefore, denoting by  $A^{\\mathrm{dep},d}$  the diagonal matrix where  $A_{ii}^{\\mathrm{dep},d} = \\mathbb{I}[\\mathrm{the~depth~of~}T_i^{r_i}\\mathrm{~is~}d]$ , we have",
+        "bbox": [
+            210,
+            465,
+            825,
+            544
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {A} ^ {\\text {d e p}, d} \\boldsymbol {p} _ {F} ^ {\\text {b l s o}, M, 2 \\text {n d}} = \\boldsymbol {A} ^ {\\text {b l s o S u r j}, M, 2 \\text {n d}} \\left(\\boldsymbol {A} ^ {\\text {a u t}, M, 2 \\text {n d}}\\right) ^ {- 1} \\boldsymbol {A} ^ {\\text {b l s o l n j}, M, 2 \\text {n d}} \\boldsymbol {A} ^ {\\text {d e p}, d} \\boldsymbol {p} _ {F} ^ {\\text {c n t}, M, 2 \\text {n d}} \\tag {23}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            238,
+            551,
+            823,
+            571
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "for all  $F \\in \\mathcal{G}$ . Fix any integer  $d \\geq 0$ , and assume that  $A^{\\mathrm{dep},d}p_{G}^{\\mathrm{blso},M,2\\mathrm{nd}} = A^{\\mathrm{dep},d}p_{H}^{\\mathrm{blso},M,2\\mathrm{nd}}$ . We will prove that  $A^{\\mathrm{dep},d}p_{G}^{\\mathrm{cnt},M,2\\mathrm{nd}} = A^{\\mathrm{dep},d}p_{H}^{\\mathrm{cnt},M,2\\mathrm{nd}}$ . Since  $A^{\\mathrm{blsoSurj},M,2\\mathrm{nd}}$  is lower triangular with positive diagonal elements, it is invertible and thus",
+        "bbox": [
+            210,
+            578,
+            823,
+            627
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {A} ^ {\\text {b l s o l n j}, M, 2 \\text {n d}} \\boldsymbol {A} ^ {\\text {d e p}, d} \\boldsymbol {p} _ {G} ^ {\\text {c n t}, M, 2 \\text {n d}} = \\boldsymbol {A} ^ {\\text {b l s o l n j}, M, 2 \\text {n d}} \\boldsymbol {A} ^ {\\text {d e p}, d} \\boldsymbol {p} _ {H} ^ {\\text {c n t}, M, 2 \\text {n d}}.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            305,
+            633,
+            730,
+            654
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "Moreover, by definition of unfolding tree, there are only finite non-zero elements in both  $A^{\\mathrm{dep},d}p_{G}^{\\mathrm{cnt},M,2\\mathrm{nd}}$  and  $A^{\\mathrm{dep},d}p_{H}^{\\mathrm{cnt},M,2\\mathrm{nd}}$ , and the corresponding non-zero indices can only be in a fixed (finite) set. In this case, the upper triangular matrix  $A^{\\mathrm{blosnj},M,2\\mathrm{nd}}$  is reduced to a finite-dimensional matrix and thus  $A^{\\mathrm{dep},d}p_{G}^{\\mathrm{cnt},M,2\\mathrm{nd}} = A^{\\mathrm{dep},d}p_{H}^{\\mathrm{cnt},M,2\\mathrm{nd}}$ . By enumerating all  $d \\geq 0$ , we obtain the desired result.",
+        "bbox": [
+            210,
+            659,
+            823,
+            736
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "image",
+        "img_path": "images/3284e7d445ad611e884904d67281b951c3a22d1e19d1ad06f234e93520c4ec12.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            743,
+            825,
+            755
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "Combined with all previous results, we have arrived at the concluding corollary:",
+        "bbox": [
+            169,
+            772,
+            700,
+            787
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "Corollary C.34. Let  $M \\in \\{\\mathrm{Sub},\\mathsf{L},\\mathsf{LF},\\mathsf{F}\\}$  be any model. For any two graphs  $G, H$ ,  $\\chi_G^M (G) = \\chi_H^M (H)$  iff  $\\operatorname {hom}(F,G) = \\operatorname {hom}(F,H)$  for all  $(F,T^{r})\\in S^{M}$ .",
+        "bbox": [
+            169,
+            790,
+            826,
+            824
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "Proof. According to Corollary C.16, for any two graphs  $G,H$ , we have  $\\chi_G^M (G) = \\chi_H^M (H)$  iff  $\\mathsf{cnt}^M ((F,T^r),G) = \\mathsf{cnt}^M ((F,T^r),H)$  for all  $(F,T^{r})\\in S^{M}$ . Then, Corollary C.33 implies that  $\\mathsf{cnt}^M ((F,T^r),G) = \\mathsf{cnt}^M ((F,T^r),H)$  for all  $(F,T^{r})\\in S^{M}$  iff  $\\mathsf{blos}((F,T^r),G) = \\mathsf{blos}((F,T^r),H)$  for all  $(F,T^{r})\\in S^{M}$ . Finally, Corollary C.32 implies that  $\\mathsf{blos}((F,T^r),G) = \\mathsf{blos}((F,T^r),H)$  for all  $(F,T^{r})\\in S^{M}$  iff  $\\mathsf{hom}(F,G) = \\mathsf{hom}(F,H)$  for all  $(F,T^{r})\\in S^{M}$ . We thus conclude the proof.",
+        "bbox": [
+            169,
+            837,
+            826,
+            925
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "page_number",
+        "text": "38",
+        "bbox": [
+            488,
+            946,
+            508,
+            960
+        ],
+        "page_idx": 37
+    },
+    {
+        "type": "text",
+        "text": "C.3 PART 2: NESTED EAR DECOMPOSITION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            103,
+            486,
+            118
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "In this part, we give equivalent formulations of the set  $S^M$  for any model  $M \\in \\{\\mathrm{Sub}, \\mathrm{L}, \\mathrm{LF}, \\mathrm{F}\\}$  using the concept of NED defined in Definition 3.3.",
+        "bbox": [
+            169,
+            128,
+            823,
+            159
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "We first present some technical lemmas that will be used to deal with the tree decomposition of disconnected graphs.",
+        "bbox": [
+            169,
+            164,
+            826,
+            194
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "Lemma C.35. Let  $M \\in \\{\\mathrm{Sub}, \\mathrm{L}, \\mathrm{LF}, \\mathrm{F}\\}$  be any model, and let  $(F, T^r) \\in S^M$  be a tree-decomposed graph with  $\\beta_T(t) = \\{u, v\\}$  for some  $u, v \\in V_F$ . Assume that  $F$  has two connected components  $S_1, S_2 \\subset V_F$  and  $u \\in S_1$  and  $v \\in S_2$  are in different components. Then, there exist tree decompositions  $(F[S_1], T_1^{r_1}), (F[S_2], T_2^{r_2}) \\in S^M$  such that  $\\beta_{T_1}(r_1) = \\{u, u\\}$  and  $\\beta_{T_2}(r_2) = \\{v, v\\}$ .",
+        "bbox": [
+            169,
+            196,
+            825,
+            257
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "Proof. We can simply define  $T_{1} = (V_{T_{1}}, E_{T_{1}}, \\beta_{T_{1}})$  with  $V_{T_{1}} = V_{T}$ ,  $E_{T_{1}} = E_{T}$ ,  $r_{1} = r$ , and  $\\beta_{T_{1}}(t) = \\{\\{\\phi_{1}(w): w \\in \\beta_{T}(t)\\}\\}$  with",
+        "bbox": [
+            169,
+            270,
+            823,
+            301
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\phi_ {1} (w) = \\left\\{ \\begin{array}{l l} w & \\text {i f} w \\in S _ {1}, \\\\ u & \\text {i f} w \\in S _ {2}. \\end{array} \\right.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            398,
+            306,
+            586,
+            340
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "In other words,  $T_{1}$  has the same structure as  $T$  but with different bags such that all vertices in the same connected component as  $v$  are replaced by  $u$ . It is easy to see that  $T_{1}^{r_{1}}$  is a canonical tree decomposition of  $F[S_{1}]$  following Definition C.1. Moreover, we clearly have that for all  $M \\in \\{\\mathrm{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}$ ,  $(F[S_{1}], T_{1}^{r_{1}}) \\in S^{M}$ . We can construct  $T_{2} = (V_{T_{2}}, E_{T_{2}}, \\beta_{T_{2}})$  by symmetry, which concludes the proof.",
+        "bbox": [
+            169,
+            347,
+            825,
+            417
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "The above lemma can be immediately generalized into the following one:",
+        "bbox": [
+            169,
+            433,
+            656,
+            450
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "Corollary C.36. Let  $M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}$  be any model, and let  $(F, T^r) \\in S^M$  be a tree-decomposed graph. For each connected component  $S \\in V_F$  of  $F$ , pick any  $t \\in V_T$  with the minimum depth such that  $S \\cap \\beta_T(t) \\neq \\emptyset$ . Then, there exists a tree decomposition  $\\tilde{T}^s$  of  $F[S]$  satisfying that  $(F[S], \\tilde{T}^s) \\in S^M$  and  $S \\cap \\beta_T(t) \\subset \\beta_{\\tilde{T}}(s)$ .",
+        "bbox": [
+            169,
+            452,
+            823,
+            516
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "Lemma C.37. Let  $M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}$  be any model, and let  $(F, T^r) \\in S^M$  be a tree-decomposed graph such that  $F$  is connected. Then, for each child node  $t$  of  $r$  in  $T^r$  and any vertex  $x$  in  $F[T^r[t]]$ , there is a path in  $F[T^r[t]]$  from  $x$  to some vertex in  $\\beta_T(r)$ . Consequently, the number of connected components of  $F[T^r[t]]$  is bounded by the number of different elements in  $\\beta_T(r)$ .",
+        "bbox": [
+            169,
+            518,
+            825,
+            578
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "Proof. Assume the above statement does not hold, and let  $S \\subset V_{F[T^r[t]]}$  be the connected component of  $F[T^r[t]]$  such that  $S \\cap \\beta_T(r) = \\emptyset$ . Note that  $F[S] = F[T^r[t]][S]$ . Since  $F$  is connected, there exists an edge  $\\{v, w\\} \\in E_F$  such that  $v \\in S$  and  $w \\notin S$ . Since  $S$  is a connected component of  $F[T^r[t]]$ , we have  $w \\notin V_{F[T^r[t]]}$ , and thus both  $v, w \\notin \\beta_T(r)$ . This yields a contradiction since  $\\{v, w\\}$  should be contained in a bag in  $T^r$  but  $B_T(v) \\cap B_T(w) = \\emptyset$ .",
+        "bbox": [
+            169,
+            592,
+            825,
+            667
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "To reduce duplication, below we only give proofs for Local 2-GNN and Local 2-FGNN. One can easily write a proof for Subgrapp HNN based on the proof of Local 2-GNN, and write a proof for 2-FGNN based on the proof of Local 2-FGNN.",
+        "bbox": [
+            169,
+            681,
+            823,
+            724
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "Lemma C.38. For any tree-decomposed graph  $(F, T^r) \\in S^{\\mathsf{L}}$ ,  $F$  has a strong NED.",
+        "bbox": [
+            169,
+            727,
+            723,
+            744
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "Proof. Based on Corollary C.36, we can assume that  $F$  is connected without loss of generality. We will prove the following stronger result: for any connected  $(F,T^r)\\in S^{\\mathsf{L}}$  with  $\\beta_T(r) = \\{\\{u,v\\}\\}$ ,  $F$  has a strong NED where  $u,v$  are endpoints of the first ear. (For the case of  $u = v$ , the other endpoint can be arbitrary.)",
+        "bbox": [
+            169,
+            758,
+            825,
+            816
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "text",
+        "text": "The proof is based on induction over the number the vertices in  $T^r$ . The above statement obviously holds for the base case of  $|V_T| = 1$ . Now assume that the statement holds when  $|V_T| \\leq m$ , and consider the case of  $|V_T| = m + 1$ . Note that for any two different children  $t, t'$  of  $r$ ,  $F[T^r[t]]$  and  $F[T^r[t']]$  can only share vertices  $u, v$ . For each child node  $t$  of  $r$ , denote its unique child node as  $\\tilde{t}$ . It is easy to see that  $T^r[\\tilde{t}]$  is a canonical tree decomposition of  $F[T^r[\\tilde{t}]]$  and  $(F[T^r[\\tilde{t}]], T^r[\\tilde{t}]) \\in S^\\perp$ . However, one needs to be cautious as  $F[T^r[\\tilde{t}]]$  may not be connected (unlike the original graph  $F$ ). Below, we separately consider the following cases:",
+        "bbox": [
+            169,
+            821,
+            825,
+            925
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "page_number",
+        "text": "39",
+        "bbox": [
+            488,
+            946,
+            508,
+            960
+        ],
+        "page_idx": 38
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1. First consider the case when  $u = v$ . In this case, Lemma C.37 implies that  $F[T^r[\\tilde{t}]]$  is connected. According to the induction hypothesis,  $F[T^r[\\tilde{t}]]$  has a strong NED (denoted as  $\\mathcal{P}_{\\tilde{t}}$ ) such that  $u$  is an endpoint of the first ear. We then merge the ear decompositions  $\\mathcal{P}_{\\tilde{t}}$  for all  $\\tilde{t}$  into a whole  $\\mathcal{P}$ , specify a root ear  $P_1$  in any  $\\mathcal{P}_{\\tilde{t}}$ , and let the first ear of other  $\\mathcal{P}_{\\tilde{t}}$  nest on  $P_1$  (with empty nested interval). It is easy to see that  $\\mathcal{P}$  is a strong NED of  $F$  and  $u$  is an endpoint of the first ear.",
+            "2. Next consider the case when  $u \\neq v$ . In this case, without loss of generality, assume  $\\beta_T(\\tilde{t}) = \\{u, w\\}$  for some  $w \\in V_F$ ."
+        ],
+        "bbox": [
+            191,
+            103,
+            823,
+            227
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "text",
+        "text": "- Subcase 1:  $F[T^r[\\tilde{t}]]$  is connected. Then,  $F[T^r[\\tilde{t}]]$  has a strong NED  $\\mathcal{P}_{\\hat{t}}$  such that  $u$  and  $w$  are endpoints of the first ear.",
+        "bbox": [
+            230,
+            234,
+            823,
+            263
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a) If  $\\bar{w} = v$ , then  $F[T^r[t]] = F[T^r[\\hat{t}]]$  and  $F[T^r[t]]$  clearly has a strong NED.",
+            "b) If  $w \\notin N_F[v]$  or  $w = u$ , there are two additional subcases depending on whether  $\\{u, v\\} \\in E_F$ . If  $\\{u, v\\} \\in E_F$ , then  $F[T^r[t]]$  has a strong NED (which can be constructed from  $\\mathcal{P}_{\\tilde{t}}$  by adding an ear  $\\{\\{u, v\\}\\}$  and letting the first ear in  $\\mathcal{P}_{\\tilde{t}}$  nest on  $\\{\\{u, v\\}\\}$  (with empty nested interval). Otherwise,  $F[T^r[t]] \\backslash \\{v\\}$  has a strong NED and  $v$  is an isolated vertex in  $F[T^r[t]]$ .",
+            "c) If  $w \\in N_F(v)$  and  $w \\neq u$ , then  $F[T^r[t]]$  also has a strong NED, which can be constructed from  $\\mathcal{P}_{\\tilde{t}}$  by extending the first ear to incorporate the edge  $\\{w, v\\}$  (we still need an additional ear  $\\{\\{u, v\\}\\}$  if  $\\{u, v\\} \\in E_F$ )."
+        ],
+        "bbox": [
+            263,
+            263,
+            823,
+            390
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "text",
+        "text": "- Subcase 2:  $F[T^r[\\tilde{t}]]$  is disconnected. In this subcase, Lemma C.37 implies that  $F[T^r[\\tilde{t}]]$  has exactly two connected components, and  $u$  and  $w$  are in different connected components (which can be easily proved by noting that  $v$  can only link to  $u$  or  $w$  in  $F[T^r[\\tilde{t}]]$ ). We can thus invoke Lemma C.35, which shows that both connected components of  $F[T^r[\\tilde{t}]]$ , denoted as  $\\widehat{F}_1$  and  $\\widehat{F}_2$ , admits tree decompositions  $\\widehat{T}_1^{s_1}$  and  $\\widehat{T}_2^{s_2}$  such that  $(\\widehat{F}_1, \\widehat{T}_1^{s_1}), (\\widehat{F}_2, \\widehat{T}_2^{s_2}) \\in S^\\mathsf{L}$ ,  $\\beta_{\\widehat{T}_1}(s_1) = \\{\\{u, u\\}\\}$  and  $\\beta_{\\widehat{T}_2}(s_2) = \\{\\{w, w\\}\\}$ . Moreover,  $|V_{\\widehat{T}_1}| = |V_{\\widehat{T}_2}| = |V_{T^r[\\tilde{t}]}| < m + 1$ . Therefore, according to the induction hypothesis,  $\\widehat{F}_1$  has a strong NED  $\\mathcal{P}_{\\tilde{t},1}$  with  $u$  as an endpoint of the first ear, and  $\\widehat{F}_2$  has a strong NED  $\\mathcal{P}_{\\tilde{t},2}$  with  $w$  as an endpoint of the first ear. If  $\\{w, v\\} \\in E_F$ , we can extend the first ear in  $\\mathcal{P}_{\\tilde{t},2}$  to include the edge  $\\{w, v\\}$ . Finally, if  $\\{u, v\\} \\in E_F$ , by setting the first ear to be  $\\{\\{u, v\\}\\}$ , we can merge the two NEDs  $\\mathcal{P}_{\\tilde{t},1}$  and  $\\mathcal{P}_{\\tilde{t},2}$  to obtain the NED of  $F[T^r[t]]$  such that  $u, v$  are endpoints of the first ear.",
+        "bbox": [
+            230,
+            391,
+            825,
+            580
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "text",
+        "text": "Overall, we always have that:",
+        "bbox": [
+            212,
+            589,
+            410,
+            603
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a) If  $F[T^r[t]]$  is connected, then it admits a strong NED such that  $u, v$  are endpoints of the first ear.",
+            "b) If  $F[T^r[t]]$  is disconnected, then it has two connected components each admitting a strong NED such that  $u, v$  belong to an endpoint of the first ear for each of the two NEDs, respectively."
+        ],
+        "bbox": [
+            230,
+            609,
+            823,
+            681
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "text",
+        "text": "Finally, noting that  $F = \\bigcup_{t,\\mathfrak{pa}_{Tr}(t) = r}F[T^r [t]]$ , we can merge all NEDs of  $F[T^{r}[t]]$  to obtain a strong NED of  $F$  satisfying that  $u,v$  are endpoints of the first ear.",
+        "bbox": [
+            210,
+            686,
+            823,
+            719
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "text",
+        "text": "We thus conclude the proof of the induction step.",
+        "bbox": [
+            171,
+            724,
+            496,
+            739
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "text",
+        "text": "Lemma C.39. For any graph  $F$ , if  $F$  admits a strong NED, then there exists a tree decomposition  $T^r$  of  $F$  such that  $(F,T^{r})\\in S^{\\mathsf{L}}$ .",
+        "bbox": [
+            169,
+            747,
+            823,
+            776
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "text",
+        "text": "Proof. We can assume that  $F$  is connected without loss of generality, as it is easy to merge the tree decompositions of different connected components to form a single tree which is a tree decomposition of the whole graph. We will prove the following stronger result: for any connected graph  $F$ , if  $F$  admits a strong NED where  $u$  and  $v$  are two different endpoints of the first ear, then there exists a tree decomposition  $T^r$  of  $F$  such that  $(F, T^r) \\in S^\\mathbb{L}$  and  $\\beta_T(r) = \\{(u, v)\\}$ .",
+        "bbox": [
+            169,
+            790,
+            823,
+            862
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "text",
+        "text": "The proof is based on induction over the number the edges in  $F$ . The above statement obviously holds for the base case of  $|E_F| = 1$ . Now assume that the statement holds when  $|E_F| \\leq m$ , and consider the case of  $|E_F| = m + 1$ . Denote the  $i$ -th ear as  $P_i$  and denote  $P_1 = \\{\\{w_0, w_1\\}, \\dots, \\{w_{l-1}, w_l\\}\\}$  where  $w_0 = u$ ,  $w_l = v$ . Consider the following three cases:",
+        "bbox": [
+            169,
+            867,
+            823,
+            925
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "page_number",
+        "text": "40",
+        "bbox": [
+            488,
+            948,
+            509,
+            959
+        ],
+        "page_idx": 39
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1. Either  $u$  or  $v$  is a cut vertex in  $F$ . Without loss of generality, assume that  $u$  is a cut vertex. In this case, there must exist a child ear  $P_i$  nested on  $P_1$  such that  $u$  is an endpoint of  $P_i$  and  $I(P_i) = \\emptyset$ . We can thus split all ears into two parts  $\\mathcal{P}_1$  and  $\\mathcal{P}_2$ , where  $\\mathcal{P}_1$  contains the ear  $P_i$  and its descendant ears in the NED tree, and  $\\mathcal{P}_2$  contains other ears. Denote by  $F[\\mathcal{P}_k]$  the connected graph induced by all edges in  $\\mathcal{P}_k$  ( $k \\in \\{1, 2\\}$ ). Then,  $\\mathcal{P}_k$  is a strong NED of the graph  $F[\\mathcal{P}_k]$ . Since both  $\\mathcal{P}_1$  and  $\\mathcal{P}_2$  contain no more than  $m$  edges, according to the induction hypothesis, there are tree decompositions  $T_k^{r_k}$  of  $F[\\mathcal{P}_k]$  ( $k \\in \\{1, 2\\}$ ) such that  $\\beta_{T_1}(r_1) = \\{\\{u, w\\}\\}$  for some  $w \\in V_F$ ,  $\\beta_{T_2}(r_2) = \\{\\{u, v\\}\\}$ , and  $(F[\\mathcal{P}_k], T_k^{r_k}) \\in S^{\\mathsf{L}}$  ( $k \\in \\{1, 2\\}$ ). We can glue the tree  $T_1^{r_1}$  and  $T_2^{r_2}$  into a larger tree  $T^{r_2}$  by adding a new node  $t$  with  $\\beta_T(t) = \\{\\{u, v, w\\}\\}$ , setting  $\\mathsf{pa}_{T^r}(t) = r_2$  and  $\\mathsf{pa}_{T^r}(r_1) = t$ . Clearly,  $T^{r_2}$  is a valid tree decomposition of  $F$ ,  $\\beta_T(r_2) = \\{\\{u, v\\}\\}$ , and  $(F, T^{r_2}) \\in S^{\\mathsf{L}}$ .",
+            "2. There is an ear  $P_{i}$  nested on  $P_{1}$  with  $I(P_{i}) = P_{1}$ . In this case, we can split all ears into two parts  $\\mathcal{P}_1$  and  $\\mathcal{P}_2$ , where  $\\mathcal{P}_1$  contains the ear  $P_{i}$  and its descendant ears in the NED tree, and  $\\mathcal{P}_2$  contains other ears. Then,  $\\mathcal{P}_k$  is a strong NED of the graph  $F[\\mathcal{P}_k]$ . Since both  $\\mathcal{P}_1$  and  $\\mathcal{P}_2$  contain no more than  $m$  edges, according to the induction hypothesis, there is a tree decomposition  $T_k^{r_k}$  of  $F[\\mathcal{P}_k]$  such that  $\\beta_{T_k}(r_k) = \\{\\{u,v\\}\\}$  and  $(F[\\mathcal{P}_k], T_k^{r_k}) \\in S^{\\mathsf{L}}$ . By merging the root node, we can glue the tree  $T_1^{r_1}$  and  $T_2^{r_2}$  into a larger tree  $T^r$ . Clearly,  $T^r$  is a valid tree decomposition of  $F$ ,  $\\beta_T(r) = \\{\\{u,v\\}\\}$ , and  $(F,T^r) \\in S^{\\mathsf{L}}$ .",
+            "3. Neither  $u$  nor  $v$  is a cut vertex in  $F$  and all ears  $P_i$  nested on  $P_1$  satisfies  $I(P_i) \\subsetneq P_1$ . In this case, we have either  $\\{w_0, w_1\\} \\notin I(P_i)$  for all ear  $P_i$  nested on  $P_1$  or  $\\{w_{l-1}, w_l\\} \\notin I(P_i)$  for all ear  $P_i$  nested on  $P_1$  (otherwise, it would contradict the definition of strong NED). Without loss of generality, assume  $\\{w_0, w_1\\} \\notin I(P_i)$  for all ears  $P_i$  nested on  $P_1$ . Then, it is clear that  $l > 1$ , and the subgraph  $F \\setminus \\{u\\}$  is connected and also admits a strong NED where  $w_1$  and  $v$  are two endpoints of the first ear. Therefore, according to the induction hypothesis, there is a tree decomposition  $\\tilde{T}^s$  of  $F \\setminus \\{u\\}$  satisfying  $(F \\setminus \\{u\\}, \\tilde{T}^s) \\in S^\\mathsf{L}$  and  $\\beta_{\\tilde{T}}(s) = \\{w_1, v\\}$ . We can then construct a tree  $T^r$  from  $\\tilde{T}^s$  by adding two fresh nodes  $r$  and  $r'$  and setting  $\\mathsf{pa}_{T^r}(s) = r'$  and  $\\mathsf{pa}_{T^r}(r') = r$ . Set  $\\beta_T(r) = \\{u, v\\}$ ,  $\\beta_T(r') = \\{u, v, w_1\\}$ , and  $\\beta_T(t) = \\beta_{\\tilde{T}}(t)$  for all  $t \\in V_{\\tilde{T}}$ . It is easy to see that the constructed  $T^r$  is a tree decomposition of  $F$  and  $(F, T^r) \\in S^\\mathsf{L}$ ."
+        ],
+        "bbox": [
+            191,
+            103,
+            823,
+            518
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "text",
+        "text": "Combining the above three cases concludes the induction step.",
+        "bbox": [
+            171,
+            523,
+            584,
+            537
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "image",
+        "img_path": "images/3706a2b0f9b29c7dff4d686d5f1d771438cbdc11234be0e289fc186a18f43089.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            523,
+            823,
+            534
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "text",
+        "text": "Theorem C.40. For any graph  $F$ ,  $F$  has a strong NED iff there is a tree decomposition  $T^r$  of  $F$  such that  $(F, T^r) \\in S^\\mathsf{L}$ .",
+        "bbox": [
+            169,
+            546,
+            823,
+            575
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "text",
+        "text": "Proof. This is a direct consequence of Lemmas C.38 and C.39.",
+        "bbox": [
+            171,
+            590,
+            589,
+            606
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "image",
+        "img_path": "images/7e164b3e67d390003e8e6b05ccb28558864c7e056100271d6404a714d427a1d9.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            592,
+            823,
+            603
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "text",
+        "text": "We next turn to Local 2-FGNN, where the proof has a similar structure as that of Local 2-GNN.",
+        "bbox": [
+            171,
+            622,
+            800,
+            638
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "text",
+        "text": "Lemma C.41. For any tree-decomposed graph  $(F, T^r) \\in S^{\\mathrm{LF}}$ ,  $F$  has an almost-strong NED.",
+        "bbox": [
+            171,
+            641,
+            787,
+            657
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "text",
+        "text": "Proof. Based on Corollary C.36, we can assume that  $F$  is connected without loss of generality. We will prove the following stronger result: for any connected  $(F,T^r)\\in S^{\\mathrm{LF}}$  with  $\\beta_T(r) = \\{\\{u,v\\}\\}$ ,  $F$  has an almost-strong NED where  $u,v$  are endpoints of the first ear. (For the case of  $u = v$ , the other endpoint can be arbitrary.)",
+        "bbox": [
+            169,
+            672,
+            823,
+            729
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "text",
+        "text": "Similar to the proof of Lemma C.38, assume that the statement holds when  $|V_T| \\leq m$ , and consider the case of  $|V_T| = m + 1$ . For each child node  $t$  of  $r$ , if  $t$  only has one child, the proof exactly follows the one in Lemma C.38. Therefore, it suffices to consider the case where  $t$  has two children  $q_1$  and  $q_2$ . Denote  $\\beta_T(t) = \\{\\{u,v,w\\}\\}$ ,  $\\beta_T(q_1) = \\{\\{u,w\\}\\}$ , and  $\\beta_T(q_2) = \\{\\{v,w\\}\\}$ . Since  $w \\in N_F[u] \\cup N_F[v]$  (by definition of  $S^{\\mathsf{LF}}$ ), we can assume that  $w \\in N_F[v]$  without loss of generality. Recall that  $(F[T^r [q_1]], T^r [q_1])$ ,  $(F[T^r [q_2]], T^r [q_2]) \\in S^{\\mathsf{LF}}$ . Below, we separately consider the following cases:",
+        "bbox": [
+            169,
+            734,
+            823,
+            821
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "text",
+        "text": "1. First consider the case when  $u = v$ . In this case, Lemma C.37 implies that both  $F[T^r[q_1]]$  and  $F[T^r[q_2]]$  are connected (since either  $u = w$  or  $\\{u, w\\} \\in E_F$ ). According to the induction hypothesis, both  $F[T^r[q_1]]$  and  $F[T^r[q_2]]$  admit an almost-strong NED such that  $u$  and  $w$  are endpoints of the first ear. If  $w = u$ , we can merge the two almost-strong NEDs so that the first ear in one NED is nested on the first ear of the other (with an empty nested interval). Otherwise, the first ears in the two almost-strong NEDs share both endpoints and we can clearly merge them (in case of  $\\{u, w\\} \\in E_F$ , there is a common ear  $\\{\\{u, w\\}\\}$  in both NEDs,",
+        "bbox": [
+            192,
+            825,
+            825,
+            925
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "page_number",
+        "text": "41",
+        "bbox": [
+            488,
+            948,
+            506,
+            959
+        ],
+        "page_idx": 40
+    },
+    {
+        "type": "text",
+        "text": "which is taken only once). In both subcases, we obtain an almost-strong NED of  $F[T^r[t]]$  such that  $u$  is an endpoint of the first ear.",
+        "bbox": [
+            210,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "text",
+        "text": "2. Next consider the case when  $u \\neq v$ . In this case,  $F[T^r[q_1]]$  and  $F[T^r[q_2]]$  share only one vertex  $w$ . Note that  $F[T^r[q_2]]$  is connected by Lemma C.37. We separately consider two subcases:",
+        "bbox": [
+            191,
+            138,
+            826,
+            180
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- Subcase 1:  $F[T^r[q_1]]$  is connected. Then, according to the induction hypothesis,  $F[T^r[q_1]]$  admits an almost-strong NED  $\\mathcal{P}_1$  such that  $u$  and  $w$  are endpoints of the first ear, and  $F[T^r[q_2]]$  admits an almost-strong NED  $\\mathcal{P}_2$  such that  $w$  and  $v$  are endpoints of the first ear. (i) If  $w = v$ , we can merge the two almost-strong NEDs so that the first ear of  $\\mathcal{P}_2$  is nested on the first ear of  $\\mathcal{P}_1$  (with empty nested interval). (ii) If  $w = u$ , we can merge the two almost-strong NEDs so that the first ear of  $\\mathcal{P}_1$  is nested on the first ear of  $\\mathcal{P}_2$  (with empty nested interval). (iii) If  $w \\in N_F(v)$  and  $w \\neq u$ , the first ear of  $\\mathcal{P}_2$  can be chosen as  $\\{\\{w, v\\}\\}$ . Then, we can merge the two almost-strong NEDs by gluing the first year in  $\\mathcal{P}_1$  with the ear  $\\{\\{w, v\\}\\}$  in  $\\mathcal{P}_2$ . One can see that the resulting NED is almost-strong. Overall, we always have that  $F[T^r[t]]$  admits an almost-strong NED such that  $u, v$  are endpoints of the first ear.",
+            "- Subcase 2:  $F[T^r[q_1]]$  is disconnected. In this subcase,  $w \\neq u$ . Similar to the proof of Lemma C.38, we obtain that  $F[T^r[q_1]]$  has exactly two connected components, and  $u$  and  $w$  are in different connected components. We can thus invoke Lemma C.35, which shows that both connected components of  $F[T^r[q_1]]$ , denoted as  $\\widehat{F}_1$  and  $\\widehat{F}_2$ , admit almost-strong NEDs  $\\mathcal{P}_1$  and  $\\mathcal{P}_2$ , respectively. Moreover,  $u$  is an endpoint of the first ear in  $\\mathcal{P}_1$ , and  $w$  is an endpoint of the first ear in  $\\mathcal{P}_2$ . Besides,  $F[T^r[q_2]]$  admits an almost-strong NED  $\\mathcal{P}_3$  with  $w, v$  as the endpoints of the first ear. By letting the first ear of  $\\mathcal{P}_2$  nest on the first ear of  $\\mathcal{P}_3$  (with an empty nested interval), we can merge  $\\mathcal{P}_2$  and  $\\mathcal{P}_3$ . Then, we can merge  $\\mathcal{P}_1$  and  $\\mathcal{P}_2 \\cup \\mathcal{P}_3$  following the same procedure as Subcase 2 in the proof of Lemma C.38."
+        ],
+        "bbox": [
+            238,
+            186,
+            826,
+            483
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "text",
+        "text": "Overall, we always have that:",
+        "bbox": [
+            210,
+            489,
+            410,
+            503
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a) If  $F[T^r[t]]$  is connected, then it admits an almost-strong NED such that  $u, v$  are endpoints of the first ear.",
+            "b) If  $F[T^r[t]]$  is disconnected, then it has two connected components each admitting an almost-strong NED such that  $u, v$  belong to an endpoint of the first ear for each of the two NEDs, respectively."
+        ],
+        "bbox": [
+            230,
+            510,
+            825,
+            580
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "text",
+        "text": "In both subcases, it follows that  $F[T^{r}[t]]$  admits an almost-strong NED with  $u, v$  as two endpoints. Noting that  $F = \\bigcup_{t, \\mathfrak{pa}_{Tr}(t) = r} F[T^{r}[t]]$ , we can merge all NEDs of  $F[T^{r}[t]]$  to obtain an almost-strong NED of  $F$  with  $u, v$  as two endpoints.",
+        "bbox": [
+            210,
+            588,
+            825,
+            633
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "text",
+        "text": "We thus conclude the proof of the induction step.",
+        "bbox": [
+            171,
+            638,
+            496,
+            654
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "text",
+        "text": "Lemma C.42. For any graph  $F$ , if  $F$  admits an almost-strong NED, then there exists a tree decomposition  $T^r$  of  $F$  such that  $(F,T^{r})\\in S^{\\mathrm{LF}}$ .",
+        "bbox": [
+            169,
+            661,
+            823,
+            691
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "text",
+        "text": "Proof. We can assume that  $F$  is connected without loss of generality. We will prove the following stronger result: for any graph  $F$ , if  $F$  admits an almost-strong NED where  $u$  and  $v$  are two different endpoints of the first ear, then there exists a tree decomposition  $T^r$  of  $F$  such that  $(F, T^r) \\in S^{\\mathsf{LF}}$  and  $\\beta_T(r) = \\{\\{u, v\\}\\}$ .",
+        "bbox": [
+            169,
+            705,
+            823,
+            763
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "text",
+        "text": "Similar to the proof of Lemma C.39, assume that the statement holds when  $|E_F| \\leq m$ , and consider the case of  $|E_F| = m + 1$ . Denote the first ear as  $P_1 = \\{\\{w_0, w_1\\}, \\dots, \\{w_{l-1}, w_l\\}\\}$  where  $w_0 = u$ ,  $w_l = v$ . Consider the following three cases:",
+        "bbox": [
+            169,
+            768,
+            825,
+            811
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1. Either  $u$  or  $v$  is a cut vertex in  $F$ . This case is exactly the same as in Lemma C.39.",
+            "2. There is an ear  $P_{i}$  nested on  $P_{1}$  with  $I(P_{i}) = P_{1}$ . This case is also the same as in Lemma C.39.",
+            "3. Otherwise,  $l \\geq 2$ ."
+        ],
+        "bbox": [
+            189,
+            816,
+            823,
+            875
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "text",
+        "text": "- If there is an ear  $P_i$  nested on  $P_1$  with  $\\{w_0, w_1\\}, \\{w_1, w_2\\} \\in I(P_i)$ , then  $l \\geq 3$ . By definition of almost-strong NED, there does not exist an ear  $P_j$  nested on  $P_1$  with  $\\{w_{l-2}, w_{l-1}\\}, \\{w_{l-1}, w_l\\} \\in I(P_i)$ . In this case, we can split  $P_1$  into two parts:",
+        "bbox": [
+            238,
+            881,
+            826,
+            926
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "page_number",
+        "text": "42",
+        "bbox": [
+            488,
+            948,
+            509,
+            959
+        ],
+        "page_idx": 41
+    },
+    {
+        "type": "text",
+        "text": "$P_{1,u} = P_1\\backslash \\{\\{w_{l - 1},w_l\\}\\}$ , and  $P_{1,v} = \\{\\{w_{l - 1},w_l\\}\\}$ . Then, we can rearrange any ear nested on  $P_{1}$  so that it is nested on either  $P_{1,u}$  or  $P_{1,v}$ . In this way, we can split all ears into two sets  $\\mathcal{P}_u$  and  $\\mathcal{P}_v$ , one corresponding to  $P_{1,u}$  and its descendant ears, and the other corresponding to  $P_{1,v}$  and its descendant ears. Therefore, according to the induction hypothesis, there is a tree decomposition  $\\tilde{T}_u^s$  of  $F[\\mathcal{P}_u]$  satisfying  $(F[\\mathcal{P}_u],\\tilde{T}_u^s)\\in S^{\\mathrm{LF}}$  and  $\\beta_{\\tilde{T}_u}(s) = \\{u,w_{l - 1}\\}$ , and similarly, there is a tree decomposition  $\\tilde{T}_v^{s'}$  of  $F[\\mathcal{P}_v]$  satisfying  $(F[\\mathcal{P}_v],\\tilde{T}_v^{s'})\\in S^{\\mathrm{LF}}$  and  $\\beta_{\\tilde{T}_v}(s) = \\{w_{l - 1},v\\}$ . We can then construct a tree  $T^r$  by merging  $\\tilde{T}_u^s$  and  $\\tilde{T}_v^{s'}$  and adding two fresh nodes  $r,r'$ , where  $r$  is the root node,  $\\mathsf{pa}_{T^r}(r') = r$  and  $\\mathsf{pa}_{T^r}(s) = \\mathsf{pa}_{T^r}(s') = r'$ . Set  $\\beta_T(r) = \\{u,v\\}$  and  $\\beta_T(r') = \\{u,v,w_{l - 1}\\}$ . It is easy to see that the constructed  $T^r$  is a tree decomposition of  $F$  and  $(F,T^r)\\in S^{\\mathrm{LF}}$ .",
+        "bbox": [
+            251,
+            102,
+            823,
+            268
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "text",
+        "text": "- If there does not exist an ear  $P_{i}$  nested on  $P_{1}$  with  $\\{w_{0}, w_{1}\\}, \\{w_{1}, w_{2}\\} \\in I(P_{i})$ , then we follow exactly the same analysis as in the previous item, except that now we split  $P_{1}$  into two parts:  $P_{1,u} = \\{\\{w_{0}, w_{1}\\}\\}$ , and  $P_{1,v} = P_{1} \\backslash \\{\\{w_{0}, w_{1}\\}\\}$ . We can still construct a tree decomposition  $T^{r}$  of  $F$  such that  $(F, T^{r}) \\in S^{\\mathsf{LF}}$ .",
+        "bbox": [
+            238,
+            270,
+            823,
+            329
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "text",
+        "text": "Combining the above three cases concludes the proof.",
+        "bbox": [
+            169,
+            334,
+            527,
+            349
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "image",
+        "img_path": "images/f30cfda95c20f021405fe098e02b8544b38c7e149fb3a7dbe403bfc85e4ba0ae.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            335,
+            823,
+            347
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "text",
+        "text": "Theorem C.43. For any graph  $F$ ,  $F$  has an almost-strong NED iff there is a tree decomposition  $T^r$  of  $F$  such that  $(F, T^r) \\in S^{\\mathsf{LF}}$ .",
+        "bbox": [
+            169,
+            357,
+            823,
+            386
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "text",
+        "text": "Proof. This is a direct consequence of Lemmas C.41 and C.42.",
+        "bbox": [
+            171,
+            400,
+            589,
+            416
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "image",
+        "img_path": "images/dc7856b177da299e7a9b26c6c4c4cfe5734d2fe1f098f50df076484b30cb27c0.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            402,
+            823,
+            412
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "text",
+        "text": "C.4 PART 3: PEBBLE GAME",
+        "text_level": 1,
+        "bbox": [
+            171,
+            431,
+            377,
+            446
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "text",
+        "text": "In this part, we prove that  $S^M$  is maximal (Definition 3.1(b)) for any  $M \\in \\{\\mathrm{Sub}, \\mathrm{L}, \\mathrm{LF}, \\mathrm{F}\\}$ . To achieve this, we first introduce a general class of graphs which we call Fürer graphs (Fürer, 2001). Intuitions and illustrations of Fürer graphs can be found in Zhang et al. (2023a).",
+        "bbox": [
+            169,
+            457,
+            823,
+            501
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "text",
+        "text": "Definition C.44 (Fürer graphs). Given any connected graph  $F = (V_F, E_F, \\ell_F)$ , the Fürer graph  $G(F) = (V_{G(F)}, E_{G(F)}, \\ell_{G(F)})$  is constructed as follows:",
+        "bbox": [
+            169,
+            503,
+            823,
+            535
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nV _ {G (F)} = \\{(x, X): x \\in V _ {F}, X \\subset N _ {F} (x), | X | \\bmod 2 = 0 \\},\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            261,
+            540,
+            656,
+            556
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nE _ {G (F)} = \\left\\{\\{(x, X), (y, Y) \\} \\subset V _ {G}: \\{x, y \\} \\in E _ {F}, (x \\in Y \\leftrightarrow y \\in X) \\right\\},\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            261,
+            559,
+            733,
+            577
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\ell_ {G (F)} (x, X) = \\ell_ {F} (x) \\quad \\forall (x, X) \\in V _ {G (F)}.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            263,
+            579,
+            545,
+            597
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "text",
+        "text": "Here,  $x \\in Y \\leftrightarrow y \\in X$  holds when either  $(x \\in Y \\text{ and } y \\in X)$  or  $(x \\notin Y \\text{ and } y \\notin X)$  holds. For each  $x \\in V_F$ , denote the set",
+        "bbox": [
+            169,
+            602,
+            823,
+            631
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\operatorname {M e t a} _ {F} (x) := \\{(x, X): X \\subset N _ {F} (x), | X | \\bmod 2 = 0 \\}, \\tag {24}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            312,
+            638,
+            823,
+            655
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "text",
+        "text": "which is called the meta vertices of  $G(F)$  associated to  $x$ . Note that  $V_{G(F)} = \\bigcup_{x\\in V_F}\\mathsf{Meta}_F(x)$ .",
+        "bbox": [
+            169,
+            659,
+            805,
+            678
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "text",
+        "text": "We next define an operation called \"twist\":",
+        "bbox": [
+            171,
+            686,
+            455,
+            700
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "text",
+        "text": "Definition C.45 (Twist). Let  $G(F) = (V_{G(F)}, E_{G(F)}, \\ell_{G(F)})$  be the Fürer graph of  $F = (V_F, E_F, \\ell_F)$ , and let  $\\{x, y\\} \\in E_F$  be an edge of  $F$ . The twisted Fürer graph of  $G(F)$  for edge  $\\{x, y\\}$ , is constructed as follows:  $\\text{twist}(G(F), \\{x, y\\}) := (V_{G(F)}, E_{\\text{twist}(G(F), \\{x, y\\})}, \\ell_{G(F)})$ , where",
+        "bbox": [
+            169,
+            704,
+            823,
+            750
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nE _ {\\operatorname {w i s t} (G (F), \\{x, y \\})} := E _ {G (F)} \\triangle \\{\\{\\xi , \\eta \\}: \\xi \\in \\operatorname {M e t a} _ {F} (x), \\eta \\in \\operatorname {M e t a} _ {F} (y) \\},\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            259,
+            756,
+            736,
+            773
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "text",
+        "text": "and  $\\triangle$  is the symmetric difference operator, i.e.,  $A\\triangle B = (A\\backslash B)\\cup (B\\backslash A)$ . For an edge set  $S = \\{e_1,\\dots ,e_k\\} \\subset E_F$ , we further define",
+        "bbox": [
+            169,
+            779,
+            823,
+            809
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\operatorname {t w i s t} (G (F), S) := \\operatorname {t w i s t} (\\dots \\operatorname {t w i s t} (G (F), e _ {1}) \\dots , e _ {k}). \\tag {25}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            815,
+            823,
+            830
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "text",
+        "text": "Note that Equation (25) is well-defined as the resulting graph does not depend on the order of edges  $e_1, \\dots, e_k$  for twisting.",
+        "bbox": [
+            169,
+            837,
+            823,
+            866
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "text",
+        "text": "The following result is well-known (see e.g., Zhang et al., 2023a, Corollary I.5 and Lemma I.7)):",
+        "bbox": [
+            169,
+            876,
+            810,
+            892
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "text",
+        "text": "Theorem C.46. For any connected graph  $F$  and any set  $S_1, S_2 \\subset E_F$ ,  $\\operatorname{twist}(G(F), S_1) \\simeq \\operatorname{twist}(G(F), S_2)$  iff  $|S_1| \\mod 2 = |S_2| \\mod 2$ .",
+        "bbox": [
+            169,
+            895,
+            823,
+            925
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "page_number",
+        "text": "43",
+        "bbox": [
+            488,
+            948,
+            506,
+            959
+        ],
+        "page_idx": 42
+    },
+    {
+        "type": "text",
+        "text": "This result shows that the structure of twisted Fürer graph does not depend on which edge is twisted. Therefore, we can pick any edge  $\\{u,v\\} \\in E_F$  and denote  $H(F) = \\mathrm{twist}(G(F),\\{u,v\\})$ . We now show that  $G(F),H(F)$  can be distinguished via homomorphism information:",
+        "bbox": [
+            169,
+            103,
+            823,
+            147
+        ],
+        "page_idx": 43
+    },
+    {
+        "type": "text",
+        "text": "Theorem C.47. For any graph  $F$ , let (twisted) Fürer graphs  $G(F)$ ,  $H(F)$  be defined above. Then,  $\\operatorname{hom}(F, G(F)) \\neq \\operatorname{hom}(F, H(F))$ .",
+        "bbox": [
+            169,
+            148,
+            823,
+            180
+        ],
+        "page_idx": 43
+    },
+    {
+        "type": "text",
+        "text": "Proof. In the proof below, we use  $g$  and  $h$  to denote mappings where  $g: V_F \\to V_{G(F)}$  and  $h: V_F \\to V_{H(F)}$ . The proof is divided into the following parts.",
+        "bbox": [
+            169,
+            191,
+            825,
+            222
+        ],
+        "page_idx": 43
+    },
+    {
+        "type": "text",
+        "text": "1. We first prove that there is a homomorphism  $g \\in \\operatorname{Hom}(F, G(F))$  satisfying  $g(w) \\in \\operatorname{Meta}_F(w)$  for all  $w \\in V_F$ , but there is no homomorphism  $h \\in \\operatorname{Hom}(F, H(F))$  satisfying  $h(w) \\in \\operatorname{Meta}_F(w)$  for all  $w \\in V_F$ .",
+        "bbox": [
+            191,
+            227,
+            825,
+            272
+        ],
+        "page_idx": 43
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a) Define a mapping  $g: V_F \\to V_{G(F)}$  such that  $g(w) = (w, \\emptyset)$  for all  $w \\in V_F$ . We clearly have  $\\{w, x\\} \\in E_F$  implies that  $\\{g(w), g(x)\\} \\in E_{G(F)}$ . Moreover,  $\\ell_{G(F)}(g(w)) = \\ell_F(w)$ . Therefore,  $g \\in \\operatorname{Hom}(F, G(F))$  is indeed a homomorphism.",
+            "b) If we similarly define  $h: V_F \\to V_{H(F)}$  such that  $h(w) = (w, \\emptyset)$  for all  $w \\in V_F$ , then for all  $\\{w, x\\} \\in E_F \\backslash \\{\\{u, v\\}\\}$  we have  $\\{h(w), h(x)\\} \\in E_{H(F)}$ , but  $\\{h(u), h(v)\\} \\notin E_{H(F)}$  since the edge  $\\{u, v\\}$  is twisted."
+        ],
+        "bbox": [
+            230,
+            277,
+            823,
+            368
+        ],
+        "page_idx": 43
+    },
+    {
+        "type": "text",
+        "text": "It remains to prove that for all  $h: V_F \\to V_{H(F)}$  of the form  $h(w) = (w, U_w)$  (for each  $w \\in V_F$ ),  $h \\notin \\operatorname{Hom}(F, H(F))$ . It suffices to prove that there is an odd number of edges  $\\{w, x\\} \\in E_F$  such that  $\\{h(w), h(x)\\} \\notin E_{H(F)}$ . Let  $h, \\tilde{h}$  be two such mappings that differ in only one vertex  $z$ , i.e.,  $h(w) = \\tilde{h}(w)$  for all  $w \\neq z$  but  $h(z) = (z, U_z) \\neq (z, \\tilde{U}_z) = \\tilde{h}(z)$ . Denote  $D_z = \\tilde{U}_z \\triangle U_z$ . Based on the definition of Fürer graph, it follows that",
+        "bbox": [
+            210,
+            375,
+            823,
+            455
+        ],
+        "page_idx": 43
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- for all  $\\{w, x\\} \\in E_F$  with  $w \\neq z, x \\neq z$ , we have  $\\{h(w), h(x)\\} \\in E_{H(F)}$  iff  $\\{\\tilde{h}(w), \\tilde{h}(x)\\} \\in E_{H(F)}$ ;",
+            "- for all  $\\{w, z\\} \\in E_F$  with  $w \\notin D_z$ , we also have  $\\{h(w), h(z)\\} \\in E_{H(F)}$  iff  $\\{\\tilde{h}(w), \\tilde{h}(z)\\} \\in E_{H(F)}$ ;",
+            "- for all  $\\{w, z\\} \\in E_F$  with  $w \\in D_z$ , we have  $\\{h(w), h(z)\\} \\in E_{H(F)}$  iff  $\\{\\tilde{h}(w), \\tilde{h}(z)\\} \\notin E_{H(F)}$ ."
+        ],
+        "bbox": [
+            236,
+            460,
+            825,
+            560
+        ],
+        "page_idx": 43
+    },
+    {
+        "type": "text",
+        "text": "Since  $|U_z| \\bmod 2 = 0$  and  $|\\tilde{U}_z| \\bmod 2 = 0$ , we have  $|D_z| \\bmod 2 = 0$  and thus the number of edges  $\\{w, x\\} \\in E_F$  such that  $\\{h(w), h(x)\\} \\notin E_{H(F)}$  has the same parity as the number of edges such that  $\\{\\tilde{h}(w), \\tilde{h}(x)\\} \\notin E_{H(F)}$ . Finally, noting that all mappings  $h$  can be obtained from the one in (b) by continually modifying  $U_w$  for each  $w \\in V_F$  and the parity remains unchanged, we have concluded the proof of this part.",
+        "bbox": [
+            210,
+            568,
+            825,
+            643
+        ],
+        "page_idx": 43
+    },
+    {
+        "type": "text",
+        "text": "2. We next prove that for any permutation  $\\pi : V_F \\to V_F$ , there does not exist a homomorphism  $h \\in \\operatorname{Hom}(F, H(F))$  satisfying  $h(w) \\in \\operatorname{Meta}_F(\\pi(w))$  for all  $w \\in V_F$ . Assume the conclusion does not hold and pick any  $h$  satisfying the above condition. Consider the following cases:",
+        "bbox": [
+            191,
+            648,
+            823,
+            691
+        ],
+        "page_idx": 43
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a) Case 1: if  $\\pi$  is an automorphism of  $F$ , then it is easy to see that  $h \\circ \\pi^{-1} \\in \\operatorname{Hom}(F, H(F))$ , because  $\\{w, x\\} \\in E_F \\Rightarrow \\{\\pi^{-1}(w), \\pi^{-1}(x)\\} \\in E_F \\Rightarrow \\{h(\\pi^{-1}(w)), h(\\pi^{-1}(x))\\} \\in E_F$  for all  $w, x \\in V_F$ , and  $\\ell_F(w) = \\ell_F(\\pi^{-1}(w)) = \\ell_{G(F)}(h(\\pi^{-1}(w)))$  for all  $w \\in V_F$ . Moreover,  $h \\circ \\pi^{-1}$  satisfies that  $h(\\pi^{-1}(w)) \\in \\operatorname{Meta}_F(w)$ , yielding a contradiction to point 1.",
+            "b) Case 2: if  $\\pi$  is not an automorphism of  $F$ , then there exists an edge  $\\{w, x\\} \\in E_F$  such that  $\\{\\pi(w), \\pi(x)\\} \\notin E_F$ . In this case, we must have  $\\{h(w), h(x)\\} \\notin E_{H(F)}$  since by definition  $h(w) \\in \\mathsf{Meta}_F(\\pi(w))$ ,  $h(x) \\in \\mathsf{Meta}_F(\\pi(x))$ , and  $\\{\\pi(w), \\pi(x)\\}$  is not an edge of  $F$ ."
+        ],
+        "bbox": [
+            230,
+            698,
+            823,
+            828
+        ],
+        "page_idx": 43
+    },
+    {
+        "type": "text",
+        "text": "In both cases,  $h$  is invalid and thus there is no homomorphism  $h \\in \\operatorname{Hom}(F, H(F))$  satisfying  $h(w) \\in \\operatorname{Meta}_F(\\pi(w))$  for all  $w \\in V_F$ .",
+        "bbox": [
+            210,
+            833,
+            823,
+            863
+        ],
+        "page_idx": 43
+    },
+    {
+        "type": "text",
+        "text": "3. We finally prove that the following two sets have equal size (i.e.,  $|S_G| = |S_H|$ ):",
+        "bbox": [
+            191,
+            868,
+            735,
+            883
+        ],
+        "page_idx": 43
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nS _ {G} := \\left\\{g \\in \\operatorname {H o m} (F, G (F)): \\exists w, x, y \\in V _ {F} \\text {s . t .} x \\neq y, g (x), g (y) \\in \\operatorname {M e t a} _ {F} (w) \\right\\},\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            241,
+            887,
+            787,
+            902
+        ],
+        "page_idx": 43
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nS _ {H} := \\left\\{h \\in \\operatorname {H o m} (F, H (F)): \\exists w, x, y \\in V _ {F} \\text {s . t .} x \\neq y, h (x), h (y) \\in \\operatorname {M e t a} _ {F} (w) \\right\\}.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            241,
+            904,
+            792,
+            921
+        ],
+        "page_idx": 43
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 43
+    },
+    {
+        "type": "page_number",
+        "text": "44",
+        "bbox": [
+            488,
+            948,
+            508,
+            959
+        ],
+        "page_idx": 43
+    },
+    {
+        "type": "text",
+        "text": "It suffices to prove that, for any proper subset  $U \\subsetneq V_F$ , we have  $|S_G^U| = |S_H^U|$ , where  $S_G$  and  $S_H$  are defined as follows:",
+        "bbox": [
+            210,
+            102,
+            823,
+            132
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nS _ {G} ^ {U} := \\left\\{g \\in \\operatorname {H o m} (F, G (F)): g (x) \\in \\bigcup_ {w \\in U} \\operatorname {M e t a} _ {F} (w) \\forall x \\in V _ {F} \\right\\},\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            292,
+            137,
+            740,
+            179
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nS _ {H} ^ {U} := \\left\\{h \\in \\operatorname {H o m} (F, H (F)): h (x) \\in \\bigcup_ {w \\in U} \\operatorname {M e t a} _ {F} (w) \\forall x \\in V _ {F} \\right\\}.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            294,
+            181,
+            741,
+            223
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "text",
+        "text": "Fix  $U \\subsetneq V_F$  and pick  $z \\in V_F \\backslash U$ . Let  $P$  be a simple path from  $u$  to  $z$  of the form  $P = \\{\\{w_0, w_1\\}, \\dots, \\{w_{k-1}, w_k\\}\\} \\subset E_F$  where  $\\{w_0, w_1\\} = \\{u, v\\}$ ,  $w_k = z$ . Define a mapping  $\\sigma$  that takes  $g \\in S_G^U$  as input and outputs a mapping  $h: V_F \\to V_{H(F)}$ :",
+        "bbox": [
+            210,
+            229,
+            823,
+            273
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nh (x) = \\left\\{ \\begin{array}{l l} g (x) & \\text {i f} [ g (x) ] _ {0} \\notin \\{w _ {1}, \\dots , w _ {k - 1} \\}, \\\\ ([ g (x) ] _ {0}, [ g (x) ] _ {1} \\bigtriangleup \\{w _ {i - 1}, w _ {i + 1} \\}) & \\text {i f} [ g (x) ] _ {0} = w _ {i}, i \\in [ k - 1 ], \\end{array} \\right.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            256,
+            281,
+            767,
+            314
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "text",
+        "text": "where we write  $g(x) = ([g(x)]_0, [g(x)]_1)$ . We will prove that  $h \\in S_H^U$ . Since we clearly have  $h(w) \\in U$  for all  $w \\in V_F$ , it suffices to prove that  $h \\in \\mathsf{Hom}(F, H(F))$ . Let  $\\{x, y\\} \\in E_F$  be any edge in  $F$ .",
+        "bbox": [
+            210,
+            321,
+            823,
+            364
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "text",
+        "text": "- If  $[g(x)]_0, [g(y)]_0 \\notin \\{w_1, \\dots, w_{k-1}\\}$ , then  $\\{h(x), h(y)\\} = \\{g(x), g(y)\\} \\in E_{G(F)}$ . Also, since  $\\{[g(x)]_0, [g(y)]_0\\} \\neq \\{u, v\\}$ ,  $\\{[g(x)]_0, [g(y)]_0\\}$  is not twisted and thus  $\\{h(x), h(y)\\} \\in E_{H(F)}$ .",
+        "bbox": [
+            238,
+            369,
+            823,
+            415
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "text",
+        "text": "- If  $[g(x)]_0 \\notin \\{w_1, \\dots, w_{k-1}\\}$ ,  $[g(y)]_0 = w_i$  for some  $i \\in [k-1]$ , and  $\\{[g(x)]_0, [g(y)]_0\\} \\neq \\{\\{u, v\\}, \\{w_{k-1}, w_k\\}\\}$ , then",
+        "bbox": [
+            238,
+            416,
+            823,
+            446
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\{h (x), h (y) \\} = \\left\\{\\left([ g (x) ] _ {0}, [ g (x) ] _ {1}\\right), \\left([ g (y) ] _ {0}, [ g (y) ] _ {1} \\triangle \\left\\{w _ {i - 1}, w _ {i + 1} \\right\\}\\right) \\right\\}.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            303,
+            450,
+            772,
+            468
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "text",
+        "text": "We have",
+        "bbox": [
+            253,
+            474,
+            313,
+            486
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\{g (x), g (y) \\} \\in E _ {G (F)} \\iff ([ g (x) ] _ {0} \\in [ g (y) ] _ {1}) \\leftrightarrow ([ g (y) ] _ {0} \\in [ g (x) ] _ {1}) \\\\ \\Longleftrightarrow ([ g (x) ] _ {0} \\in [ g (y) ] _ {1} \\triangle \\left\\{w _ {i - 1}, w _ {i + 1} \\right\\}) \\leftrightarrow ([ g (y) ] _ {0} \\in [ g (x) ] _ {1}) \\\\ \\iff \\{h (x), h (y) \\} \\in E _ {G (F)}. \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            254,
+            494,
+            833,
+            549
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "text",
+        "text": "Also, since  $\\{[g(x)]_0, [g(y)]_0\\}$  is not twisted,  $\\{h(x), h(y)\\} \\in E_{H(F)}$ .",
+        "bbox": [
+            251,
+            555,
+            702,
+            571
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- If  $[g(x)]_0, [g(y)]_0 \\in \\{w_1, \\dots, w_{k-1}\\}$ , the analysis is similar to the above one and we can still prove that  $\\{h(x), h(y)\\} \\in E_{G(F)}$  and thus  $\\{h(x), h(y)\\} \\in E_{G(F)}$ .",
+            "- If  $\\{[g(x)]_0, [g(y)]_0\\} = \\{u, v\\}$ , without loss of generality assume  $w_0 = u = [g(x)]_0$  and  $w_1 = v = [g(y)]_0$ . We have"
+        ],
+        "bbox": [
+            238,
+            571,
+            823,
+            632
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\{g (x), g (y) \\} \\in E _ {G (F)} \\iff (u \\in [ g (y) ] _ {1}) \\leftrightarrow (v \\in [ g (x) ] _ {1}) \\\\ \\Longleftrightarrow (u \\notin [ g (y) ] _ {1} \\triangle \\{u, w _ {2} \\}) \\leftrightarrow (v \\in [ g (x) ] _ {1}) \\\\ \\Longleftrightarrow \\{h (x), h (y) \\} \\notin E _ {G (F)}. \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            305,
+            637,
+            769,
+            691
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "text",
+        "text": "However, since  $\\{u,v\\}$  is twisted, we still have  $\\{h(x),h(y)\\} \\in E_{H(F)}$",
+        "bbox": [
+            250,
+            698,
+            717,
+            714
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "text",
+        "text": "- Finally, we do not need to consider the case  $\\{[g(x)]_0, [g(y)]_0\\} = \\{w_{k-1}, w_k\\}$  since  $w_k = z \\notin U$ .",
+        "bbox": [
+            238,
+            714,
+            823,
+            742
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "text",
+        "text": "This proves that  $h \\in S_H^U$ . Moreover, it is straightforward to see that the mapping  $\\sigma(g) = h$  is a bijection from  $S_G^U$  to  $S_H^U$ . We have thus proved that  $|S_G^U| = |S_H^U|$ .",
+        "bbox": [
+            210,
+            750,
+            823,
+            781
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "text",
+        "text": "Combining the above three items, we obtain  $\\mathsf{Hom}(F,G(F)) > \\mathsf{Hom}(F,H(F))$ , concluding the proof.",
+        "bbox": [
+            169,
+            786,
+            825,
+            815
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "text",
+        "text": "In the subsequent analysis, we will prove that for all model  $M$  considered in Theorem 3.4 and any connected graph  $F$ ,  $\\chi_{G(F)}^{M}(G(F)) = \\chi_{H(F)}^{M}(H(F))$  if  $F \\notin S^{M}$ . The proof is based on an important technique developed in Cai et al. (1992), called the pebble game. When restricting our analysis on Fürer graphs, the pebble game can be greatly simplified as shown in Fürer (2001); Zhang et al. (2023a). Below, we separately describe the corresponding pebble game for each model  $M$ .",
+        "bbox": [
+            169,
+            830,
+            823,
+            902
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "text",
+        "text": "We first define a key concept called the connected component.",
+        "bbox": [
+            171,
+            909,
+            581,
+            924
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "page_number",
+        "text": "45",
+        "bbox": [
+            488,
+            946,
+            506,
+            959
+        ],
+        "page_idx": 44
+    },
+    {
+        "type": "text",
+        "text": "Definition C.48 (Connected components). Let  $F = (V_F, E_F)$  be a connected graph and let  $U \\subset V_F$  be a vertex set, called separation vertices. We say two edges  $\\{u, v\\}, \\{x, y\\} \\in E_F$  are in the same connected component if there is a simple path  $\\{\\{y_0, y_1\\}, \\dots, \\{y_{k-1}, y_k\\}\\}$  satisfying that  $\\{y_0, y_1\\} = \\{u, v\\}, \\{y_{k-1}, y_k\\} = \\{x, y\\}$  and  $y_i \\notin U$  for all  $i \\in [k-1]$ . It is easy to see that the above relation between edges forms an equivalence relation. Therefore, we can define a partition over the edge set, denoted by  $\\mathbb{C}\\mathbb{C}_F(U) = \\{P_i : i \\in [m]\\}$  for some  $m$ , where each  $P_i \\subset E_F$  is called a connected component.",
+        "bbox": [
+            169,
+            103,
+            826,
+            202
+        ],
+        "page_idx": 45
+    },
+    {
+        "type": "text",
+        "text": "We are now ready to describe the game rule. There are two players (named Alice and Bob), a graph  $F$ , and several pebbles. At the beginning, all pebbles lie outside the graph. Through the game process, some pebbles will be placed on the vertices of  $F$  and thus separate the edges  $E_{F}$  into connected components according to Definition C.48. In each game round, Alice updates the location of pebbles, while Bob maintains a subset of connected components, ensuring that the number of selected components is odd. There are three major types of operations:",
+        "bbox": [
+            169,
+            213,
+            826,
+            299
+        ],
+        "page_idx": 45
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1. Add a pebble  $\\mathfrak{p}$ . Alice places a pebble  $\\mathfrak{p}$  (previously outside the graph) on some vertex of  $F$ . If introducing this new pebble does not change the connected components, then Bob does nothing. Otherwise, there must be a connected component  $P$  separated by  $\\mathfrak{p}$  into several components  $P = \\bigcup_{i \\in [m]} P_i$  for some  $m$ . Bob will update his selected components as follows: if  $P$  is selected, then remove  $P$  and add a subset of connected components in  $\\{P_1, \\dots, P_m\\}$  while ensuring that the number of selected components in total is odd.",
+            "2. Remove a pebble  $\\mathfrak{p}$ . Alice removes a pebble  $\\mathfrak{p}$  (previously on some vertex) outside the graph. If introducing this new pebble does not change the connected components, then Bob does nothing. Otherwise, there are multiple connected components  $P_{1},\\dots ,P_{m}$  getting merged into a whole  $P = \\bigcup_{i\\in [m]}P_{i}$ . Bob will update his selected components by removing all  $P_{i}$ ,  $i\\in [m]$  (if selected) and optionally adding  $P$ , while ensuring that the number of selected components in total is odd.",
+            "3. Swap two pebbles  $\\mathsf{p}$  and  $\\mathsf{p}^{\\prime}$ . Alice swaps the position of two pebbles  $\\mathsf{p}$  and  $\\mathsf{p}^{\\prime}$ . This operation does not change the connected components and thus Bob does nothing."
+        ],
+        "bbox": [
+            191,
+            304,
+            825,
+            510
+        ],
+        "page_idx": 45
+    },
+    {
+        "type": "text",
+        "text": "At any time, if there is an edge  $\\{x,y\\}$  such that both endpoints hold pebbles and the connected component  $\\{\\{x,y\\}\\}$  is selected by Bob, then Bob loses the game and Alice wins. If Alice cannot win through the game process, then Bob wins.",
+        "bbox": [
+            169,
+            516,
+            825,
+            559
+        ],
+        "page_idx": 45
+    },
+    {
+        "type": "text",
+        "text": "We now define the concrete pebble game for each model  $M$  considered in this paper. In cases of Subgraph GNN, Local 2-GNN, Local 2-FGNN, and 2-FGNN, there are three pebbles  $\\mathsf{p}_u$ ,  $\\mathsf{p}_v$ ,  $\\mathsf{p}_w$ . As described before, all pebbles lie outside the graph at the beginning. Alice first adds the pebble  $\\mathsf{p}_u$  (operation 1) and then adds the pebble  $\\mathsf{p}_v$  (operation 1). Next, the game cyclically executes the following process:",
+        "bbox": [
+            169,
+            566,
+            826,
+            637
+        ],
+        "page_idx": 45
+    },
+    {
+        "type": "text",
+        "text": "- Subgraph GNN. Alice can choose either one of the following ways to play:",
+        "bbox": [
+            197,
+            642,
+            712,
+            657
+        ],
+        "page_idx": 45
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- Remove the pebble  $\\mathsf{p}_v$  (operation 2), and re-add the pebble  $\\mathsf{p}_v$  (operation 1).",
+            "- Add the pebble  $\\mathsf{p}_w$  (operation 1) adjacent to the pebble  $\\mathsf{p}_v$ , swap pebble  $\\mathsf{p}_v$  with  $\\mathsf{p}_w$  (operation 3), and remove the pebble  $\\mathsf{p}_w$  (operation 2)."
+        ],
+        "bbox": [
+            235,
+            660,
+            821,
+            703
+        ],
+        "page_idx": 45
+    },
+    {
+        "type": "text",
+        "text": "- Local 2-GNN. Alice can choose either one of the following ways to play:",
+        "bbox": [
+            197,
+            705,
+            697,
+            720
+        ],
+        "page_idx": 45
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- Remove the pebble  $\\mathsf{p}_u$  (operation 2), and re-add the pebble  $\\mathsf{p}_u$  (operation 1).",
+            "- Remove the pebble  $\\mathsf{p}_v$  (operation 2), and re-add the pebble  $\\mathsf{p}_v$  (operation 1).",
+            "- Add the pebble  $\\mathsf{p}_w$  (operation 1) adjacent to the pebble  $\\mathsf{p}_u$ , swap pebble  $\\mathsf{p}_u$  with  $\\mathsf{p}_w$  (operation 3), and remove the pebble  $\\mathsf{p}_w$  (operation 2).",
+            "- Add the pebble  $\\mathsf{p}_w$  (operation 1) adjacent to the pebble  $\\mathsf{p}_v$ , swap pebble  $\\mathsf{p}_v$  with  $\\mathsf{p}_w$  (operation 3), and remove the pebble  $\\mathsf{p}_w$  (operation 2)."
+        ],
+        "bbox": [
+            235,
+            723,
+            821,
+            811
+        ],
+        "page_idx": 45
+    },
+    {
+        "type": "text",
+        "text": "- Local 2-FGNN. Alice can choose either one of the following ways to play:",
+        "bbox": [
+            197,
+            814,
+            707,
+            829
+        ],
+        "page_idx": 45
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- Remove the pebble  $\\mathsf{p}_u$  (operation 2), and re-add the pebble  $\\mathsf{p}_u$  (operation 1).",
+            "- Remove the pebble  $\\mathfrak{p}_v$  (operation 2), and re-add the pebble  $\\mathfrak{p}_v$  (operation 1).",
+            "- Add the pebble  $\\mathsf{p}_w$  (operation 1) adjacent to either pebble  $\\mathsf{p}_u$  or pebble  $\\mathsf{p}_v$ , swap pebble  $\\mathsf{p}_w$  with any adjacent pebble (operation 3), and remove the pebble  $\\mathsf{p}_w$  (operation 2)."
+        ],
+        "bbox": [
+            235,
+            830,
+            821,
+            890
+        ],
+        "page_idx": 45
+    },
+    {
+        "type": "text",
+        "text": "- 2-FGNN. Alice adds the pebble  $\\mathsf{p}_w$  (operation 1), swap the pebbles  $\\mathsf{p}_w$  with either  $\\mathsf{p}_u$  or  $\\mathsf{p}_v$  (operation 3), and remove the pebble  $\\mathsf{p}_w$  (operation 2).",
+        "bbox": [
+            197,
+            893,
+            821,
+            921
+        ],
+        "page_idx": 45
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 45
+    },
+    {
+        "type": "page_number",
+        "text": "46",
+        "bbox": [
+            488,
+            948,
+            509,
+            959
+        ],
+        "page_idx": 45
+    },
+    {
+        "type": "text",
+        "text": "Proposition C.49. For any model  $M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}$  and any graph  $F$ , if Alice cannot win the pebble game associated to model  $M$  on graph  $F$ , then  $\\chi_{G(F)}^{M}(G(F)) = \\chi_{H(F)}^{M}(H(F))$ .",
+        "bbox": [
+            169,
+            103,
+            823,
+            136
+        ],
+        "page_idx": 46
+    },
+    {
+        "type": "text",
+        "text": "Proof. Note that the pebble game does not depend on the vertex labels  $\\ell_F$  in  $F$ . We first assume that the vertex labels are all different, i.e.,  $\\ell_F(x) \\neq \\ell_F(y)$  for all  $x, y \\in V_F$ . Then, the labels of vertices  $\\xi \\in V_{G(F)}, \\eta \\in V_{H(F)}$  are the same iff  $\\xi$  and  $\\eta$  belong to the same meta vertex, i.e.,  $\\xi, \\eta \\in \\mathsf{Meta}_F(x)$  for some  $x \\in V_F$ . Based on Proposition C.9,  $\\chi_{G(F)}^M(G(F)) = \\chi_{H(F)}^M(H(F))$  iff  $\\tilde{\\chi}_{G(F)}^M(G(F)) = \\tilde{\\chi}_{H(F)}^M(H(F))$  where  $\\tilde{\\chi}^M$  is defined in Equations (7) to (10). In this case, the pebble games above exactly correspond to the aggregation formulas in Equations (7) to (10), and we can invoke the results in Zhang et al. (2023a, Theorem I.17) to show that  $\\tilde{\\chi}_{G(F)}^M(G(F)) = \\tilde{\\chi}_{H(F)}^M(H(F))$  (note that we do not need to consider the augmented Fürer graphs defined in their paper based on our label assignment). Therefore,  $\\chi_{G(F)}^M(G(F)) = \\chi_{H(F)}^M(H(F))$ .",
+        "bbox": [
+            169,
+            150,
+            826,
+            291
+        ],
+        "page_idx": 46
+    },
+    {
+        "type": "text",
+        "text": "We next consider the general case when multiple vertices can have the same label in  $F$ . However, this can only make it harder to distinguish between  $G(F)$  and  $H(F)$ , and we clearly have  $\\chi_{G(F)}^{M}(G(F)) = \\chi_{H(F)}^{M}(H(F))$ .",
+        "bbox": [
+            169,
+            296,
+            825,
+            342
+        ],
+        "page_idx": 46
+    },
+    {
+        "type": "text",
+        "text": "Based on the above result, in the subsequent proof we will analyze the players' strategy in the pebble game. Surprisingly, it turns out that given a graph  $F$ , if Alice can win the game, then her strategy can be described using the tree decomposition of  $F$  defined in Definition C.6.",
+        "bbox": [
+            169,
+            356,
+            823,
+            398
+        ],
+        "page_idx": 46
+    },
+    {
+        "type": "text",
+        "text": "To illustrate this point, we need the concept of game state graph. Given a graph  $F$ , a game state is a three-tuple  $(u, v, Q)$  where  $u, v \\in V_F \\cup \\{\\emptyset\\}$  and  $Q \\subset \\mathbb{C}\\mathbb{C}_F(\\{u, v\\})$ , denoting the vertex that holds pebble  $\\mathfrak{p}_u$ , the vertex that holds pebble  $\\mathfrak{p}_v$ , and a subset of connected component selected by Bob, respectively. Here, the symbol  $\\emptyset$  means that a pebble is left outside the graph. One can see that after any round, whether Alice can win the remaining game purely depends on this tuple. Now fix Alice's strategy. In each round, each game state will be transited to a finite number of states depending on how Bob plays, and all states and transitions form a directed graph, which we call the game state graph. The state  $(u, v, Q)$  is called a terminal state if  $\\min_{P \\in Q} |P| = 1$ . It is straightforward to see that Alice wins the game at any terminal state, as stated in the following result:",
+        "bbox": [
+            169,
+            405,
+            825,
+            531
+        ],
+        "page_idx": 46
+    },
+    {
+        "type": "text",
+        "text": "Proposition C.50. Given graph  $F$  and model  $M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}$ , let  $G^S$  be the game state graph defined above corresponding to an Alice's strategy. Then, Alice can win the game if there is an integer  $t$  such that any path in  $G^S$  of length  $t$  starting from the initial state  $(\\emptyset, \\emptyset, \\{E_F\\})$  goes through a terminal state.",
+        "bbox": [
+            169,
+            534,
+            825,
+            590
+        ],
+        "page_idx": 46
+    },
+    {
+        "type": "text",
+        "text": "Proof. It suffices to prove that, at any terminal state, Alice can win the game. Let  $(u,v,Q)$  be a terminal state with  $\\{x,y\\} \\in Q$ . If  $\\{u,v\\} = \\{x,y\\}$ , Alice already wins. If  $u \\notin \\{x,y\\}$  and  $v \\in \\{x,y\\}$ , Alice also wins in the next round since she can add the pebble  $\\mathfrak{p}_w$  on some vertex  $w$  adjacent to  $\\mathfrak{p}_v$  such that  $\\{v,w\\} = \\{x,y\\}$ . This is a valid game rule for all model  $M$ . Next, if  $u \\in \\{x,y\\}$  and  $v \\notin \\{x,y\\}$ , Alice also wins in the next round since she can first remove the pebble  $\\mathfrak{p}_v$  and then place it on the unique vertex in  $\\{x,y\\} \\backslash \\{u\\}$ . This is also a valid game rule for all model  $M$ . Note the removing  $\\mathfrak{p}_v$  does not merge the connected component  $\\{\\{x,y\\}\\}$  since  $v \\notin \\{x,y\\}$ . Finally, if both  $u \\notin \\{x,y\\}$  and  $v \\notin \\{x,y\\}$ ,  $F$  only has one edge and Alice can clearly win. Combining these cases, we conclude that Alice can always win.",
+        "bbox": [
+            169,
+            606,
+            826,
+            733
+        ],
+        "page_idx": 46
+    },
+    {
+        "type": "text",
+        "text": "Based on the above proposition, a game state  $(u,v,Q)$  is called unreachable if any path starting from the initial state  $(\\emptyset ,\\emptyset ,\\{E_F\\})$  and ending at  $(u,v,Q)$  goes through some terminal state. We do not need to consider unreachable states since Alice always wins before reaching it. We next introduce an important technical concept:",
+        "bbox": [
+            169,
+            748,
+            823,
+            805
+        ],
+        "page_idx": 46
+    },
+    {
+        "type": "text",
+        "text": "Definition C.51. Given a game state graph  $G^{\\mathsf{S}}$ , a state  $(u, v, \\{P\\})$  is termed as \"contracted\" if for any transition  $((u, v, \\{P\\}), (u', v', \\{P'\\})) \\in E_{G^{\\mathsf{s}}}, P' \\subset P$ . It is called strictly contracted if for any transition  $((u, v, \\{P\\}), (u', v', \\{P'\\})) \\in E_{G^{\\mathsf{s}}}, P' \\subsetneq P$ .",
+        "bbox": [
+            169,
+            808,
+            823,
+            853
+        ],
+        "page_idx": 46
+    },
+    {
+        "type": "text",
+        "text": "We have the following result:",
+        "bbox": [
+            171,
+            862,
+            369,
+            878
+        ],
+        "page_idx": 46
+    },
+    {
+        "type": "text",
+        "text": "Lemma C.52. For any model  $M \\in \\{\\text{Sub}, \\text{L}, \\text{LF}, \\text{F}\\}$  and any graph  $F$ , if Alice can win the pebble game associated to model  $M$  on graph  $F$ , then there exists a game state graph  $G^S$  corresponding to a winning strategy of Alice such that any reachable and non-terminal state is strictly contracted.",
+        "bbox": [
+            169,
+            881,
+            823,
+            925
+        ],
+        "page_idx": 46
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 46
+    },
+    {
+        "type": "page_number",
+        "text": "47",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 46
+    },
+    {
+        "type": "text",
+        "text": "Proof. We first prove that there is a strategy for Alice such that any reachable and non-terminal state is contracted. We will assume that for any reachable and non-terminal state  $(u, v, \\{P\\})$ , if there is only one transition of the form  $((u, v, \\{P\\}),(u', v', \\{P'\\}))$ , then  $P \\notin P'$ , since making the connected component larger should be a bad strategy. (Otherwise, it is straightforward to construct a new strategy for Alice satisfying the assumption.) Now suppose that there exists a reachable state  $(u, v, \\{P\\})$  that is not contracted. Note that the game state graph induced by all reachable states is a Directed Acyclic Graph (DAG), so we can choose the state  $(u, v, \\{P\\})$  such that any path from the initial state  $(\\emptyset, \\emptyset, \\{E_F\\})$  to  $(u, v, \\{P\\})$  does not pass any intermediate state that is not contracted. Below, we will construct a new strategy that removes the state  $(u, v, \\{P\\})$  (making it unreachable).",
+        "bbox": [
+            169,
+            103,
+            826,
+            231
+        ],
+        "page_idx": 47
+    },
+    {
+        "type": "text",
+        "text": "Note that we clearly have  $u \\neq \\emptyset$  and  $v \\neq \\emptyset$ . Without loss of generality, assume that  $((u,v,\\{P\\}),(u,v',\\{P'\\}))$  is a transition such that  $P' \\notin P$  (the case of transition  $((u,v,\\{P\\}),(u',v,\\{P'\\}))$  is the same by symmetry). Moreover, we can assume that  $P \\notin \\mathbb{C}\\mathbb{C}_F(\\{u\\})$  and  $P \\notin \\mathbb{C}\\mathbb{C}_F(\\{v\\})$  (in other words, both  $u$  and  $v$  are at the boundary of the connected component  $P$ ). It would be easier to analyze the case where either  $u$  or  $v$  are not at the boundary. We separately consider the following cases:",
+        "bbox": [
+            169,
+            234,
+            826,
+            321
+        ],
+        "page_idx": 47
+    },
+    {
+        "type": "text",
+        "text": "1. The transition  $((u,v,\\{P\\}),(u,v^{\\prime},\\{P^{\\prime}\\}))$  corresponds to Alice removing pebble  $\\mathfrak{p}_v$  and placing it on  $v^{\\prime}$ . Let  $(u_0,v_0,Q_0),\\dots ,(u_T,v_T,Q_T)$  be any path from the initial state  $(\\emptyset ,\\emptyset ,\\{E_F\\})$  to  $(u,v,\\{P\\})$ , and let  $t\\leq T$  be the maximal number such that  $v_{t}\\neq v$ . We will construct a new strategy for Alice as follows.",
+        "bbox": [
+            192,
+            325,
+            825,
+            383
+        ],
+        "page_idx": 47
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- At state  $(u_t, v_t, Q_t)$ , she removes pebble  $\\mathfrak{p}_v$  and places it on  $v'$ , yielding (possibly multiple) states of the form  $(u_t, v', \\tilde{Q}_t)$ .",
+            "- For all  $t < \\tilde{t} < T$ , we have  $v_{t} = v$ . For each  $\\tilde{t}$  and possible  $\\tilde{Q}_{\\tilde{t}}$ , she will apply the original strategy at  $(u_{\\tilde{t}}, v_{\\tilde{t}}, Q_{\\tilde{t}})$  to the state  $(u_{\\tilde{t}}, v', \\tilde{Q}_{\\tilde{t}})$ , namely, placing pebble  $\\mathsf{p}_w$  on vertex  $u_{\\tilde{t} + 1}$ , swapping pebble  $\\mathsf{p}_u$  with  $\\mathsf{p}_w$ , and leaving  $\\mathsf{p}_w$  outside the graph. (Note that she cannot remove pebble  $\\mathsf{p}_u$  first; otherwise the state  $(u_{\\tilde{t}}, v_{\\tilde{t}}, Q_{\\tilde{t}})$  will not be contracted since  $u$  is at the boundary of  $Q_{\\tilde{t}}$ .)"
+        ],
+        "bbox": [
+            238,
+            388,
+            826,
+            501
+        ],
+        "page_idx": 47
+    },
+    {
+        "type": "text",
+        "text": "It follows that  $((u_{\\tilde{t}},v^{\\prime},\\tilde{Q}_{\\tilde{t}}),(u_{\\tilde{t} +1},v^{\\prime},\\tilde{Q}_{\\tilde{t} +1}))$  is a transition of the new strategy,  $(u_{\\tilde{t}},v^{\\prime},\\tilde{Q}_{\\tilde{t}})$  is contracted, and all states of the form  $(u,v^{\\prime},\\tilde{Q}_{\\tilde{T}})$  is reachable in the original strategy.",
+        "bbox": [
+            210,
+            506,
+            823,
+            542
+        ],
+        "page_idx": 47
+    },
+    {
+        "type": "text",
+        "text": "We can repeat the above procedure for all paths from the initial state  $(\\emptyset, \\emptyset, \\{E_F\\})$  to state  $(u, v, \\{P\\})$ . Then, in the new strategy  $(u, v, \\{P\\})$  will be unreachable. However, the state  $(u_t, v_t, Q_t)$  now may violate the condition in Lemma C.52. Note that the assumption in the first paragraph is still satisfied for state  $(u_t, v_t, Q_t)$ . In this case, we can recursively apply the above procedure for state  $(u_t, v_t, Q_t)$ . Note that the procedure will only repeat a finite number of times, as the length of the path from the initial state to the state  $(u_t, v_t, Q_t)$  is strictly less than the length of the path from the initial state to the state  $(u, v, \\{P\\})$ .",
+        "bbox": [
+            210,
+            546,
+            825,
+            647
+        ],
+        "page_idx": 47
+    },
+    {
+        "type": "text",
+        "text": "2. The transition  $((u,v,\\{P\\}),(u,v^{\\prime},\\{P^{\\prime}\\}))$  corresponds to Alice placing pebble  $\\mathfrak{p}_w$  on  $v^{\\prime}$ , swapping  $\\mathfrak{p}_v$  and  $\\mathfrak{p}_w$ , and removing pebble  $\\mathfrak{p}_w$ . In this case, we must have  $P\\notin \\mathbb{C}\\mathbb{C}_F(\\{u,v,v'\\})$  (i.e.,  $v^{\\prime}$  further splits the component  $P$ ). Otherwise, the state  $(u,v,\\{P\\})$  must have only one transition and  $P\\subset P^{\\prime}$ , a contradiction to the first paragraph. Since  $P\\notin \\mathbb{C}\\mathbb{C}_F(\\{u,v,v'\\})$ , it is thus easy to see that, if Alice changes her strategy by just removing pebble  $\\mathfrak{p}_v$  and placing it on vertex  $v^{\\prime}$ , all transitions starting from  $(u,v,\\{P\\})$  does not change. Therefore, we can just invoke the previous item to construct a desired strategy.",
+        "bbox": [
+            191,
+            650,
+            825,
+            751
+        ],
+        "page_idx": 47
+    },
+    {
+        "type": "text",
+        "text": "Combining the two cases, we conclude that there is a strategy for Alice such that any reachable and non-terminal state is contracted. We next prove that any reachable and non-terminal state can be strictly contracted. Assume the result does not hold and the state  $(u,v,\\{P\\})$  is reachable, non-terminal, but not strictly contracted. Then, there is a transition  $((u,v,\\{P\\}),(u',v',\\{P\\})) \\in E_{G^s}$ . Consider the following two cases:",
+        "bbox": [
+            169,
+            755,
+            826,
+            828
+        ],
+        "page_idx": 47
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1.  $P \\notin \\mathbb{C}C_F(\\{u\\})$  and  $P \\notin \\mathbb{C}C_F(\\{v\\})$  (i.e., both  $u$  and  $v$  are at the boundary of the connected component  $P$ ). It follows that  $u' = u$  and  $v' = v$ . This implies that the game state graph is not acyclic, a contradiction.",
+            "2.  $P \\in \\mathbb{C}\\mathbb{C}_F(\\{u\\})$  or  $P \\in \\mathbb{C}\\mathbb{C}_F(\\{v\\})$ . Without loss of generality, assume that  $P \\in \\mathbb{C}\\mathbb{C}_F(\\{u\\})$ . Since  $(u,v,\\{P\\})$  is non-terminal and not strictly contracted, there is a reachable and non-terminal state  $(\\tilde{u},\\tilde{v},\\{P\\})$  with either  $\\tilde{u} = u$  or  $\\tilde{v} = u$  such that there is a path from  $(u,v,\\{P\\})$"
+        ],
+        "bbox": [
+            191,
+            832,
+            825,
+            926
+        ],
+        "page_idx": 47
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 47
+    },
+    {
+        "type": "page_number",
+        "text": "48",
+        "bbox": [
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+        ],
+        "page_idx": 47
+    },
+    {
+        "type": "text",
+        "text": "to  $(\\tilde{u},\\tilde{v},\\{P\\})$  and  $(\\tilde{u},\\tilde{v},\\{P\\})$  is strictly contracted. We can then change the strategy at state  $(u,v,\\{P\\})$  to make it strictly contracted. Concretely, Alice can remove pebble  $\\mathsf{p}_v$  (which does not merge connected components selected by Bob since  $P\\in \\mathbb{C}\\mathbb{C}_F(\\{u\\}))$ , and place pebble  $\\mathsf{p}_v$  on the vertex that corresponds to the strategy at state  $(\\tilde{u},\\tilde{v},\\{P\\})$  (possibly with a difference that  $u$  and  $v$  become exchanged). This makes the state  $(u,v,\\{P\\})$  strictly contracted.",
+        "bbox": [
+            210,
+            103,
+            823,
+            175
+        ],
+        "page_idx": 48
+    },
+    {
+        "type": "text",
+        "text": "Combining the two cases concludes the proof.",
+        "bbox": [
+            171,
+            180,
+            478,
+            196
+        ],
+        "page_idx": 48
+    },
+    {
+        "type": "image",
+        "img_path": "images/a72e207dcf82b05fef066d22c7af4a5edebfb8a1c197ac154832756dc7f72c6d.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            181,
+            823,
+            193
+        ],
+        "page_idx": 48
+    },
+    {
+        "type": "text",
+        "text": "We are now ready to state the main theorem:",
+        "bbox": [
+            171,
+            213,
+            467,
+            227
+        ],
+        "page_idx": 48
+    },
+    {
+        "type": "text",
+        "text": "Theorem C.53. Let  $M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}$  be any model. Given any connected graph  $F$ , if Alice can win the pebble game associated with model  $M$  on graph  $F$ , then there is a tree decomposition  $T^r$  of  $F$  such that  $(F, T^r) \\in S^M$ , where  $S^M$  is defined in Definition C.6.",
+        "bbox": [
+            169,
+            232,
+            823,
+            275
+        ],
+        "page_idx": 48
+    },
+    {
+        "type": "text",
+        "text": "Proof. Let  $G^{\\mathsf{S}}$  be the game state graph satisfying Lemma C.52. For each game state  $s$ , denote by  $\\mathrm{next}_{G^{\\mathsf{S}}}(\\bar{s})$  the set of states  $s'$  such that  $(s, s')$  is a transition in  $G^{\\mathsf{S}}$  and  $s'$  contains only a single connected component, i.e.,  $s'$  has the form  $(u, v, \\{P\\})$ . For a terminal state  $s$  of the form  $(u, v, \\{\\{x, y\\}\\})$ , define  $\\mathrm{next}_{G^{\\mathsf{S}}}(\\bar{s}) = \\{(x, y, \\{\\{x, y\\}\\})\\}$ . By definition,  $\\mathrm{next}_{G^{\\mathsf{S}}}(\\emptyset, \\emptyset, \\{E_F\\}) = \\{(u, \\emptyset, Q_1), \\dots, (u, \\emptyset, Q_m)\\}$  for some  $u \\in V_F$  and  $Q_1, \\dots, Q_m$  is the finest partition of  $\\mathbb{C}\\mathbb{C}_F(\\{u\\})$ .",
+        "bbox": [
+            169,
+            292,
+            823,
+            364
+        ],
+        "page_idx": 48
+    },
+    {
+        "type": "text",
+        "text": "The tree  $T^r$  will be recursively constructed as follows. First create the tree root  $r$  with  $\\beta_T(r) = \\{u, u\\}$ . As we will see later, the root node will be associated with the set of states  $S(r) := \\mathrm{next}_{G^s}(\\emptyset, \\emptyset, \\{E_F\\})$ . We then do the following procedure:",
+        "bbox": [
+            169,
+            369,
+            823,
+            412
+        ],
+        "page_idx": 48
+    },
+    {
+        "type": "text",
+        "text": "Let  $t$  be a leaf node in the current tree associated with a non-empty set of game states  $S(t)$  such that  $|\\bigcup_{(u,v,\\{P\\}) \\in S(t)} P| > 1$ . For each state  $(u,v,\\{P\\}) \\in S(t)$ , create a new node  $\\tilde{t}$  and set its parent to be  $t$ . Pick any state  $(u',v',\\{P'\\}) \\in \\mathrm{next}_{G^s}(u,v,\\{P\\})$ . Then, there must be a unique vertex  $w \\in \\{\\{u',v'\\}\\} \\setminus \\{\\{u,v\\}\\}$  and  $w$  does not depend on which  $(u',v',\\{P'\\})$  is picked (by definition of the game rule). Set  $\\beta_T(\\tilde{t}) = \\beta_T(t) \\cup \\{\\{w\\}\\}$ . Then, do the following constructions:",
+        "bbox": [
+            169,
+            419,
+            823,
+            494
+        ],
+        "page_idx": 48
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- If there is a state of the form  $(u, w, \\{P'\\}) \\in \\mathrm{next}_{G^s}(u, v, \\{P\\})$ , then create a new node  $t'$  and connect it to the parent  $\\tilde{t}$ . Set  $\\beta_T(t') = \\{(u, w)\\}$ , and the node  $t'$  will be associated to the set of states  $S(t') = \\{(u, w, \\{P'\\}): (u, w, \\{P'\\}) \\in \\mathrm{next}_{G^s}(u, v, \\{P\\})\\}$ .",
+            "- If there is a state of the form  $(w, v, \\{P''\\}) \\in \\mathsf{next}_{G^{\\mathsf{s}}}(u, v, \\{P\\})$ , then create a new node  $t''$  and connect it to the parent  $\\tilde{t}$ . Set  $\\beta_{T}(t'') = \\{(w, v)\\}$ , and the node  $t''$  will be associated to the set of states  $S(t'') = \\{(w, v, \\{P''\\}): (w, v, \\{P''\\}) \\in \\mathsf{next}_{G^{\\mathsf{s}}}(u, v, \\{P\\})\\}$ ."
+        ],
+        "bbox": [
+            196,
+            500,
+            823,
+            597
+        ],
+        "page_idx": 48
+    },
+    {
+        "type": "text",
+        "text": "Note that (i) either  $w \\neq u$  or  $w \\neq v$  (since the game state graph is a DAG); (ii) both items can be used for one state  $(u, v, \\{P\\})$ , which happens for FWL-type GNNs. The construction of the tree completes when each leaf node is associated with only one game state of the form  $(x, y, \\{\\{x, y\\}\\})$ . It is easy to see that the above procedure terminates after adding a finite number of tree nodes.",
+        "bbox": [
+            169,
+            601,
+            823,
+            657
+        ],
+        "page_idx": 48
+    },
+    {
+        "type": "text",
+        "text": "We now prove that  $T^r$  is a canonical tree decomposition of  $F$  and  $(F, T^r) \\in S^M$ .",
+        "bbox": [
+            171,
+            664,
+            709,
+            680
+        ],
+        "page_idx": 48
+    },
+    {
+        "type": "text",
+        "text": "1. We first prove that any edge in  $F$  is contained in some bag of  $T^r$ . Pick any non-leaf tree node  $t$  of even depth and any of its child  $\\tilde{t}$ , denote  $(u, v, P) \\in S(t)$  be the state associated with  $\\tilde{t}$  in the above construction, and denote  $\\beta_T(\\tilde{t}) = \\{\\{u, v, w\\}\\}$ . We have",
+        "bbox": [
+            191,
+            686,
+            823,
+            731
+        ],
+        "page_idx": 48
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nP = \\left(\\bigcup_{\\substack{t^{\\prime},\\mathsf{pa}_{T^{r}}(t^{\\prime}) = \\tilde{t},\\\\ (x,y,\\{P^{\\prime}\\})\\in S(t^{\\prime})}}P^{\\prime}\\right)\\cup \\{\\{u,w\\} \\in E_{F}\\} \\cup \\{\\{v,w\\} \\in E_{F}\\} .\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            300,
+            738,
+            733,
+            810
+        ],
+        "page_idx": 48
+    },
+    {
+        "type": "text",
+        "text": "Here,  $P' \\subset P$  is a consequence of Lemma C.52, and the remaining edges  $\\{u, w\\}$  or  $\\{v, w\\}$  are in  $P$  because  $(u, v, P)$  is strictly contracted and  $w$  must be in the interior of  $P$ . Therefore,",
+        "bbox": [
+            210,
+            816,
+            823,
+            847
+        ],
+        "page_idx": 48
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nP\\cup \\{\\{u,v\\} \\in E_{F}\\} = \\left(\\bigcup_{\\substack{t^{\\prime},\\mathsf{pa}_{Tr}(t^{\\prime}) = \\tilde{t},\\\\ (x,y,\\{P^{\\prime}\\})\\in S(t^{\\prime})}}(P^{\\prime}\\cup \\{\\{x,y\\} \\in E_{F}\\})\\right)\\cup \\{\\{x,y\\} \\in E_{F}:x,y\\in \\beta_{T}(\\tilde{t})\\} .\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            210,
+            854,
+            864,
+            926
+        ],
+        "page_idx": 48
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 48
+    },
+    {
+        "type": "page_number",
+        "text": "49",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 48
+    },
+    {
+        "type": "text",
+        "text": "Recursively applying the above equation yields the desired result",
+        "bbox": [
+            210,
+            103,
+            643,
+            119
+        ],
+        "page_idx": 49
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nE_{F} = \\bigcup_{\\substack{\\tilde{t},\\mathsf{dep}_{T}(\\tilde{t})\\text{is odd}}}\\{\\{x,y\\} \\in E_{F}:x,y\\in \\beta_{T}(\\tilde{t})\\} ,\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            349,
+            127,
+            686,
+            162
+        ],
+        "page_idx": 49
+    },
+    {
+        "type": "text",
+        "text": "because (i)  $\\bigcup_{(u,\\emptyset,P)\\in S(r)}P = E_F$ , and (ii) all leaf node  $t$  with  $S(t) = (x,y,\\{\\{x,y\\}\\})$  contains an edge  $\\{x,y\\}$  in its bag.",
+        "bbox": [
+            209,
+            169,
+            823,
+            203
+        ],
+        "page_idx": 49
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "2. We next prove that  $T^r$  satisfies the condition of Definition C.1(c). Fix any vertex  $w \\in V_F$ , and let  $t$  be the tree node with minimal depth that contains  $w$ . Without loss of generality, assume that  $t$  is not the root. In this case, the depth of  $t$  is odd and  $\\beta_T(t) = \\{ \\{u, v, w\\} \\}$  for some  $u \\neq w, v \\neq w$ . Let  $t'$  be a child node of  $t$  and we have  $w \\in \\beta_T(t')$ . It remains to prove that for any descendant  $\\tilde{t} \\in \\mathsf{Desc}_T(t')$ , if  $w \\in \\beta_T(\\tilde{t})$ , then  $w \\in \\beta_T(\\hat{t})$  for all  $\\hat{t}$  on the path between  $t'$  and  $\\tilde{t}$ . This is actually a direct consequence of Lemma C.52, because when a pebble originally placed on  $w$  is removed, all edges linked to  $w$  will not be selected by Bob and thus any pebble can never be placed on  $w$  again.",
+            "3.  $T^r$  is canonical as Definition C.4 is clearly satisfied.",
+            "4. Finally, it is also easy to see that  $(F,T^{r})$  satisfies Definition C.6."
+        ],
+        "bbox": [
+            191,
+            208,
+            826,
+            368
+        ],
+        "page_idx": 49
+    },
+    {
+        "type": "text",
+        "text": "We thus conclude the proof.",
+        "bbox": [
+            171,
+            372,
+            359,
+            388
+        ],
+        "page_idx": 49
+    },
+    {
+        "type": "image",
+        "img_path": "images/4d53758aaeab6f723fd0adf83db534c477b1e660bc87bf25007e8c1d9d904a52.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            373,
+            823,
+            385
+        ],
+        "page_idx": 49
+    },
+    {
+        "type": "text",
+        "text": "Corollary C.54. Let  $M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}$  be any model. For any connected graph  $F \\notin \\mathcal{F}^M$ , let  $G(F)$  and  $H(F)$  be the Fürer graph and twisted Fürer graph with respect to  $F$ . Then,  $\\text{hom}(F, G(F)) \\neq \\text{hom}(F, H(F))$  and  $\\chi_{G(F)}^M(G(F)) = \\chi_{H(F)}^M(H(F))$ .",
+        "bbox": [
+            169,
+            396,
+            823,
+            444
+        ],
+        "page_idx": 49
+    },
+    {
+        "type": "text",
+        "text": "Proof. The proof directly follows from applying Theorems C.47 and C.53 and Proposition C.49.  $\\square$",
+        "bbox": [
+            169,
+            459,
+            795,
+            474
+        ],
+        "page_idx": 49
+    },
+    {
+        "type": "image",
+        "img_path": "images/efca4d81cb4eb459488c47a94aea190aa1e1b3a11b88ab86f90a692f33d4efc8.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            460,
+            823,
+            472
+        ],
+        "page_idx": 49
+    },
+    {
+        "type": "text",
+        "text": "Finally, we remark that our construction can be easily generalized for disconnected graphs  $F \\notin \\mathcal{F}^M$ . Let  $F$  be the disjoint union of graphs  $\\{F_i : i \\in [m]\\}$  where each  $F_i$  is the graph corresponding to a connected component of  $F$ . Assume that  $F_1$  is the connected component with the most number of edges (in case of a tie, pick the graph with the most number of vertices). Define  $\\tilde{G}(F)$  be the disjoint union of  $G(F_1), F_2, \\dots, F_m$  and  $\\tilde{H}(F)$  to be the disjoint union of  $H(F_1), F_2, \\dots, F_m$ . It follows that  $\\chi_{\\tilde{G}(F)}\\tilde{G}(F) = \\chi_{\\tilde{H}(F)}\\tilde{H}(F)$  and",
+        "bbox": [
+            169,
+            489,
+            826,
+            585
+        ],
+        "page_idx": 49
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\operatorname {h o m} (F, \\tilde {G} (F)) = \\prod_ {i \\in [ m ]} \\operatorname {h o m} (F _ {i}, \\tilde {G} (F)) = \\prod_ {i \\in [ m ]} (\\operatorname {h o m} (F _ {i}, G (F _ {1})) + \\operatorname {h o m} (F _ {i}, F _ {2} \\cup \\dots \\cup F _ {m})) \\\\ > \\prod_ {i \\in [ m ]} \\left(\\operatorname {h o m} \\left(F _ {i}, H \\left(F _ {1}\\right)\\right) + \\operatorname {h o m} \\left(F _ {i}, F _ {2} \\cup \\dots \\cup F _ {m}\\right)\\right) = \\operatorname {h o m} (F, \\tilde {H} (F)), \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            184,
+            592,
+            808,
+            665
+        ],
+        "page_idx": 49
+    },
+    {
+        "type": "text",
+        "text": "where we use the fact that  $\\operatorname{hom}(F_i, G(F_1)) = \\operatorname{hom}(F_i, H(F_1))$  when  $F_1 \\notin \\operatorname{Spasm}(F_i)$  (which can be easily proved following Theorem C.47). This concludes the proof of the general case.",
+        "bbox": [
+            169,
+            672,
+            823,
+            703
+        ],
+        "page_idx": 49
+    },
+    {
+        "type": "text",
+        "text": "D NODE/EDGE-LEVEL EXPRESSIVITY",
+        "text_level": 1,
+        "bbox": [
+            171,
+            722,
+            503,
+            738
+        ],
+        "page_idx": 49
+    },
+    {
+        "type": "text",
+        "text": "This section aims to prove Theorem 3.7. For the ease of reading, we first restate it below:",
+        "bbox": [
+            169,
+            753,
+            758,
+            770
+        ],
+        "page_idx": 49
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.7 For all model  $M$  defined in Section 2,  $\\mathcal{F}_{\\mathfrak{n}}^{M}$  and  $\\mathcal{F}_{\\mathrm{e}}^{M}$  (except MPNN) exist. Moreover,",
+        "bbox": [
+            169,
+            775,
+            823,
+            792
+        ],
+        "page_idx": 49
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- MPNN:  $\\mathcal{F}_{\\mathfrak{n}}^{\\mathsf{MP}} = \\{F^{w}: F \\text{ is a tree}\\}$ ;",
+            "- Subgraph GNN:"
+        ],
+        "bbox": [
+            197,
+            795,
+            455,
+            830
+        ],
+        "page_idx": 49
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {F} _ {\\mathfrak {n}} ^ {\\mathrm {S u b}} = \\left\\{F ^ {w}: F \\text {h a s a N E D w i t h s h a r d e n p o i n t} w \\right\\} = \\left\\{F ^ {w}: F \\backslash \\{w \\} \\text {i s a f o r e s t} \\right\\},\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            210,
+            829,
+            769,
+            845
+        ],
+        "page_idx": 49
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {F} _ {\\mathrm {e}} ^ {\\mathrm {S} _ {\\mathrm {u b}}} = \\left\\{F ^ {w x}: F \\text {h a s a N E D w i t h s h a r e d e n p o i n t} w \\right\\} = \\left\\{F ^ {w x}: F \\backslash \\{w \\} \\text {i s a f o r e s t} \\right\\};\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            210,
+            844,
+            769,
+            859
+        ],
+        "page_idx": 49
+    },
+    {
+        "type": "text",
+        "text": "- 2-FGNN:  $\\mathcal{F}_{\\mathfrak{n}}^{\\mathsf{F}} = \\{F^{w}: F$  has a NED where  $w$  is an endpoint of the first ear},",
+        "bbox": [
+            197,
+            863,
+            725,
+            878
+        ],
+        "page_idx": 49
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {F} _ {\\mathrm {e}} ^ {\\mathrm {F}} = \\left\\{F ^ {w x}: F \\text {h a s a N E D w h e r e} w \\text {a n d} x \\text {a r e e x i p o n t s o f t h e f i r s t e a r} \\right\\}.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            212,
+            878,
+            702,
+            893
+        ],
+        "page_idx": 49
+    },
+    {
+        "type": "text",
+        "text": "The cases of Local 2-GNN and Local 2-FGNN are similar to 2-FGNN by replacing \"NED\" with \"strong NED\" and \"almost-strong NED\", respectively.",
+        "bbox": [
+            169,
+            895,
+            823,
+            925
+        ],
+        "page_idx": 49
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 49
+    },
+    {
+        "type": "page_number",
+        "text": "50",
+        "bbox": [
+            488,
+            946,
+            509,
+            960
+        ],
+        "page_idx": 49
+    },
+    {
+        "type": "text",
+        "text": "We remark that the definition of node/edge-level homomorphism expressivity involves only connected graphs for sake of simplicity. Similar to the proof of the graph-level expressivity, the proof of Theorem 3.7 consists of three parts. Among them, the proof related to tree decomposition and ear decomposition is quite similar to that of graph-level expressivity, so we only illustrate the proof sketch for clarity (Appendix D.1). However, the proof related to pebble game and counterexample graphs will require additional techniques, which is detailed in Appendix D.2.",
+        "bbox": [
+            169,
+            103,
+            826,
+            189
+        ],
+        "page_idx": 50
+    },
+    {
+        "type": "text",
+        "text": "D.1 RELATED TO TREE DECOMPOSITION AND EAR DECOMPOSITION",
+        "text_level": 1,
+        "bbox": [
+            169,
+            204,
+            656,
+            218
+        ],
+        "page_idx": 50
+    },
+    {
+        "type": "text",
+        "text": "We first extend several notations that are used in Appendix C.2.",
+        "bbox": [
+            169,
+            229,
+            591,
+            244
+        ],
+        "page_idx": 50
+    },
+    {
+        "type": "text",
+        "text": "Definition D.1 (Tree decomposition for rooted graphs). Given a rooted graph  $G^u$  and tree  $T^r$ , we say  $T^r$  is a tree decomposition of  $G^u$  if  $T^r$  is a tree decomposition of  $G$  and all elements in  $u$  belong to the root bag, i.e.,  $u_i \\in \\beta_T(r)$  for all  $i$ .",
+        "bbox": [
+            169,
+            247,
+            823,
+            291
+        ],
+        "page_idx": 50
+    },
+    {
+        "type": "text",
+        "text": "Based on this definition, given model  $M$ , we can define  $\\mathcal{S}_{\\mathfrak{n}}^{M}$  to be the family of tree-decomposed graphs  $(F^u,T^r)$  such that  $(F,T^r)\\in S^M$ ; similarly, we define  $\\mathcal{S}_{\\mathrm{e}}^{M}$  to be the family of tree-decomposed graphs  $(F^{uv},T^{r})$  such that  $(F,T^{r})\\in S^{M}$ .",
+        "bbox": [
+            169,
+            300,
+            823,
+            344
+        ],
+        "page_idx": 50
+    },
+    {
+        "type": "text",
+        "text": "Definition D.2 (Bag isomorphism for rooted graphs). Given tree-decomposed graph  $(F^{\\mathbf{u}}, T^{r})$  and rooted graph  $G^{\\mathbf{v}}$  where  $\\mathbf{u}$  and  $\\mathbf{v}$  have equal length, a bag isomorphism from  $(F^{\\mathbf{u}}, T^{r})$  to  $G^{\\mathbf{v}}$  is a homomorphism  $f$  from  $F^{\\mathbf{u}}$  to  $G^{\\mathbf{v}}$  such that  $f$  is a bag isomorphism from  $(F, T^{r})$  to  $G$ .",
+        "bbox": [
+            169,
+            345,
+            825,
+            390
+        ],
+        "page_idx": 50
+    },
+    {
+        "type": "text",
+        "text": "We can similarly define bag-isomorphism homomorphism (bIsoHom, bIsoSurj, bIsoInj), bag extension (bExt), and bag-strong surjective (bStrSurj) from tree-decomposed graph  $(F^u, T^r)$  to tree-decomposed graph  $(\\tilde{F}^v, \\tilde{T}^s)$ .",
+        "bbox": [
+            169,
+            398,
+            823,
+            445
+        ],
+        "page_idx": 50
+    },
+    {
+        "type": "text",
+        "text": "Definition D.3 (Generalization of Definition C.15 for rooted graphs). Let  $M \\in \\{\\mathrm{Sub}, \\mathrm{L}, \\mathrm{LF}, \\mathrm{F}\\}$  be any model. Given a rooted graph  $G^{u}$  and a tree-decomposed graph  $(F^w, T^r)$ , define",
+        "bbox": [
+            169,
+            446,
+            823,
+            477
+        ],
+        "page_idx": 50
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathsf {c n t} ^ {M} \\left(\\left(F ^ {w}, T ^ {r}\\right), G ^ {u}\\right) := \\left| \\left\\{v \\in V _ {G}: \\exists D \\in \\mathbb {N} _ {+} \\text {s . t .} \\left(\\left[ F _ {G} ^ {M, (D)} (u, v) \\right] ^ {u}, T _ {G} ^ {M, (D)} (u, v)\\right) \\simeq \\left(F ^ {w}, T ^ {r}\\right) \\right\\} \\right|.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            169,
+            481,
+            859,
+            508
+        ],
+        "page_idx": 50
+    },
+    {
+        "type": "text",
+        "text": "Given a rooted graph  $G^{uv}$  and a tree-decomposed graph  $(F^{wx}, T^r)$ , define",
+        "bbox": [
+            169,
+            512,
+            666,
+            527
+        ],
+        "page_idx": 50
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathsf {c n t} ^ {M} \\left(\\left(F ^ {w x}, T ^ {r}\\right), G ^ {u v}\\right) := \\mathbb {I} \\left[ \\exists D \\in \\mathbb {N} _ {+} \\text {s . t .} \\left(\\left[ F _ {G} ^ {M, (D)} (u, v) \\right] ^ {u v}, T _ {G} ^ {M, (D)} (u, v)\\right) \\simeq \\left(F ^ {w x}, T ^ {r}\\right) \\right].\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            171,
+            532,
+            818,
+            560
+        ],
+        "page_idx": 50
+    },
+    {
+        "type": "text",
+        "text": "Here,  $\\left(F_G^{M,(D)}(u,v),T_G^{M,(D)}(u,v)\\right)$  is the depth-2D unfolding tree of  $G$  at  $(u,v)$  for model  $M$ .",
+        "bbox": [
+            169,
+            563,
+            808,
+            589
+        ],
+        "page_idx": 50
+    },
+    {
+        "type": "text",
+        "text": "All the following lemmas are straightforward extensions of those in Appendix C.2.",
+        "bbox": [
+            169,
+            598,
+            715,
+            613
+        ],
+        "page_idx": 50
+    },
+    {
+        "type": "text",
+        "text": "Lemma D.4. Let  $M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}$  be any model. For any graph  $G^v$  and tree-decomposed graph  $(F^u, T^r) \\in S_{\\mathfrak{n}}^{M}$ , we have",
+        "bbox": [
+            169,
+            614,
+            823,
+            646
+        ],
+        "page_idx": 50
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\operatorname {b l s o} \\left(\\left(F ^ {u}, T ^ {r}\\right), G ^ {v}\\right) = \\sum_ {\\left(\\tilde {F} ^ {w}, \\tilde {T} ^ {s}\\right) \\in \\mathcal {S} _ {\\mathfrak {n}} ^ {M}} \\operatorname {b l s o H o m} \\left(\\left(F ^ {u}, T ^ {r}\\right), \\left(\\tilde {F} ^ {w}, \\tilde {T} ^ {s}\\right)\\right) \\cdot \\operatorname {c n t} ^ {M} \\left(\\left(\\tilde {F} ^ {w}, \\tilde {T} ^ {s}\\right), G ^ {v}\\right).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            176,
+            648,
+            818,
+            688
+        ],
+        "page_idx": 50
+    },
+    {
+        "type": "text",
+        "text": "The edge-level result is similar.",
+        "bbox": [
+            169,
+            691,
+            380,
+            705
+        ],
+        "page_idx": 50
+    },
+    {
+        "type": "text",
+        "text": "Lemma D.5. Let  $M \\in \\{\\mathrm{Sub},\\mathrm{L},\\mathrm{LF},\\mathrm{F}\\}$  be any model. For any tree-decomposed graphs  $(F^u,T^r),(\\tilde{F}^v,\\tilde{T}^s) \\in S_{\\mathfrak{n}}^{M}$ ,",
+        "bbox": [
+            169,
+            709,
+            823,
+            742
+        ],
+        "page_idx": 50
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathsf{b l s o H o m}((F^{u},T^{r}),(\\tilde{F}^{v},\\tilde{T}^{s})) = \\sum_{(\\widehat{F}^{w},\\widehat{T}^{t})\\in \\mathcal{S}_{n}^{M}}\\frac{\\mathsf{b l s o S u r j}\\left((F^{u},T^{r}),( \\widehat{F}^{w},\\widehat{T}^{t})\\right)\\cdot\\mathsf{b l s o I n j}\\left((\\widehat{F}^{w},\\widehat{T}^{t}),( \\tilde{F}^{v},\\tilde{T}^{s})\\right)}{\\mathsf{a u t}(\\widehat{F}^{w},\\widehat{T}^{t})}.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            169,
+            746,
+            854,
+            797
+        ],
+        "page_idx": 50
+    },
+    {
+        "type": "text",
+        "text": "The edge-level result is similar.",
+        "bbox": [
+            169,
+            801,
+            380,
+            816
+        ],
+        "page_idx": 50
+    },
+    {
+        "type": "text",
+        "text": "Lemma D.6. Let  $M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}$  be any model. For any graph  $G^v$  and tree-decomposed graph  $(F^u, T^r) \\in S_{\\mathfrak{n}}^{M}$ ,",
+        "bbox": [
+            169,
+            819,
+            825,
+            849
+        ],
+        "page_idx": 50
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\operatorname {h o m} (F ^ {u}, G ^ {v}) = \\sum_ {(\\tilde {F} ^ {w}, \\tilde {T} ^ {s}) \\in \\mathcal {S} _ {n} ^ {M}} \\frac {\\mathrm {b E x t} ((F ^ {u} , T ^ {r}) , (\\tilde {F} ^ {w} , \\tilde {T} ^ {s})) \\cdot \\mathrm {b S t r H o m} ((\\tilde {F} ^ {w} , \\tilde {T} ^ {s}) , G ^ {v})}{\\operatorname {a u t} (\\tilde {F} ^ {w} , \\tilde {T} ^ {s})}.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            207,
+            853,
+            785,
+            905
+        ],
+        "page_idx": 50
+    },
+    {
+        "type": "text",
+        "text": "The edge-level result is similar.",
+        "bbox": [
+            171,
+            909,
+            380,
+            924
+        ],
+        "page_idx": 50
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 50
+    },
+    {
+        "type": "page_number",
+        "text": "51",
+        "bbox": [
+            488,
+            946,
+            506,
+            960
+        ],
+        "page_idx": 50
+    },
+    {
+        "type": "text",
+        "text": "Lemma D.7. Let  $M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}$  be any model. For any graph  $G^v$  and tree-decomposed graph  $(F^u, T^r) \\in S_{\\mathfrak{n}}^{M}$ ,",
+        "bbox": [
+            169,
+            103,
+            823,
+            133
+        ],
+        "page_idx": 51
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathsf{bStrHom}((F^{u},T^{r}),G^{v}) = \\sum_{(\\tilde{F}^{w},\\tilde{T}^{s})\\in \\mathcal{S}_{n}^{M}}\\frac{\\mathsf{bStrSurj}\\left((F^{u},T^{r}),( \\tilde{F}^{w},\\tilde{T}^{s})\\right)\\cdot\\mathsf{blso}\\left((\\tilde{F}^{w},\\tilde{T}^{s}),G^{v}\\right)}{\\mathsf{aut}(\\tilde{F}^{w},\\tilde{T}^{s})},\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            178,
+            138,
+            818,
+            189
+        ],
+        "page_idx": 51
+    },
+    {
+        "type": "text",
+        "text": "The edge-level result is similar.",
+        "bbox": [
+            171,
+            195,
+            380,
+            210
+        ],
+        "page_idx": 51
+    },
+    {
+        "type": "text",
+        "text": "Corollary D.8. Let  $M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}$  be any model. For any two graphs  $G^u, H^v, \\chi_G^M(u) = \\chi_H^M(v)$  iff  $\\operatorname{hom}(F^w, G^u) = \\operatorname{hom}(F^w, H^v)$  for all  $(F^w, T^r) \\in S_{\\mathfrak{n}}^M$ . For any two graphs  $G^{uv}, H^{wx}, \\chi_G^M(u, v) = \\chi_H^M(w, x)$  iff  $\\operatorname{hom}(F^{yz}, G^{uv}) = \\operatorname{hom}(F^{yz}, H^{wx})$  for all  $(F^{yz}, T^r) \\in S_{\\mathfrak{e}}^M$ .",
+        "bbox": [
+            169,
+            212,
+            823,
+            260
+        ],
+        "page_idx": 51
+    },
+    {
+        "type": "text",
+        "text": "Lemma D.9. Let  $M \\in \\{\\mathrm{Sub}, \\mathrm{L}, \\mathrm{LF}, \\mathrm{F}\\}$  be any model.",
+        "bbox": [
+            171,
+            262,
+            527,
+            277
+        ],
+        "page_idx": 51
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- For any rooted connected graph  $F^u$ ,  $F^u \\in \\mathcal{F}_{\\mathfrak{n}}^{M}$  iff there is a tree decomposition  $T^{r}$  of  $F^u$  such that  $(F^u, T^r) \\in S_{\\mathfrak{n}}^{M}$ .",
+            "- For any rooted connected graph  $F^{uv}$ ,  $F^{uv} \\in \\mathcal{F}_{\\mathrm{e}}^{M}$  iff there is a tree decomposition  $T^{r}$  of  $F^{uv}$  such that  $(F^{uv}, T^{r}) \\in \\mathcal{S}_{\\mathrm{e}}^{M}$ ."
+        ],
+        "bbox": [
+            196,
+            282,
+            823,
+            348
+        ],
+        "page_idx": 51
+    },
+    {
+        "type": "text",
+        "text": "By combining the proof of previous lemmas and theorems, we can prove that both  $\\mathcal{F}_{\\mathrm{n}}^{M}$  and  $\\mathcal{F}_{\\mathrm{e}}^{M}$  satisfy Definition 3.6(a) for all  $M \\in \\{\\mathrm{Sub}, \\mathrm{L}, \\mathrm{LF}, \\mathrm{F}\\}$ .",
+        "bbox": [
+            169,
+            358,
+            823,
+            388
+        ],
+        "page_idx": 51
+    },
+    {
+        "type": "text",
+        "text": "D.2 COUNTEREXAMPLES",
+        "text_level": 1,
+        "bbox": [
+            171,
+            404,
+            364,
+            417
+        ],
+        "page_idx": 51
+    },
+    {
+        "type": "text",
+        "text": "Let  $F^{\\mathbf{w}}$  be a rooted graph that marks the special vertices  $w_{1},\\dots ,w_{m}$ . Assume that  $F^{\\mathbf{w}} \\notin \\mathcal{F}_{\\mathfrak{n}}^{M}$  ( $m = 1$ ) or  $F^{\\mathbf{w}} \\notin \\mathcal{F}_{\\mathfrak{e}}^{M}$  ( $m = 2$ ). In this subsection, we will construct a pair of graphs  $G^{\\mathbf{u}}$  and  $H^{\\mathbf{v}}$  such that  $\\chi_G^M (\\pmb {u}) = \\chi_H^M (\\pmb {v})$  and  $\\mathrm{hom}(F^{\\mathbf{w}},G^{\\mathbf{u}})\\neq \\mathrm{hom}(F^{\\mathbf{w}},H^{\\mathbf{v}})$ . However, it turns out that, if we naively follow the proof in Appendix C.4 by constructing the Fürer graph and twisted Fürer graph with respect to  $F$  without considering the marked vertices  $\\pmb{w}$ , then the graphs may no longer be counterexamples here. For example, for the edge-level expressivity,  $F^{w_1,w_2}\\notin \\mathcal{F}_{\\mathfrak{e}}^{M}$  does not imply that  $F\\notin \\mathcal{F}^{M}$  where  $M$  can be any model studied in this paper.",
+        "bbox": [
+            169,
+            429,
+            823,
+            529
+        ],
+        "page_idx": 51
+    },
+    {
+        "type": "text",
+        "text": "To address the problem, we instead introduce a novel construction of counterexample graphs defined as follows:",
+        "bbox": [
+            169,
+            534,
+            823,
+            563
+        ],
+        "page_idx": 51
+    },
+    {
+        "type": "text",
+        "text": "Definition D.10 (Clique-augmented Fürer graphs). Let  $F$  be any connected graph and  $w_{1}, \\dots, w_{m} \\in V_{F}$  be a sequence of vertices. Given an integer  $k \\geq m$ , the  $k$ -clique-augmented Fürer graph with respect to  $F^{\\mathbf{w}}$ , denoted by  $G_{k}(F^{\\mathbf{w}})$ , is the Fürer graph of  $\\tilde{F}$  where  $\\tilde{F}$  is the union of graph  $F$  and a  $k$ -clique that contains  $w_{1}, \\dots, w_{m}$  (and does not contain other vertices in  $F$ ). The twisted Fürer graph of  $G_{k}(F^{\\mathbf{w}})$  is denoted as  $H_{k}(F^{\\mathbf{w}})$ .",
+        "bbox": [
+            169,
+            566,
+            825,
+            640
+        ],
+        "page_idx": 51
+    },
+    {
+        "type": "text",
+        "text": "We have the following main result:",
+        "bbox": [
+            171,
+            648,
+            405,
+            664
+        ],
+        "page_idx": 51
+    },
+    {
+        "type": "text",
+        "text": "Theorem D.11. Let  $M \\in \\{\\mathrm{Sub}, \\mathrm{L}, \\mathrm{LF}, \\mathrm{F}\\}$  be any model defined in Section 2.",
+        "bbox": [
+            171,
+            667,
+            676,
+            683
+        ],
+        "page_idx": 51
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- For any rooted graph  $F^w$  marking vertex  $w$ , if  $F^w \\notin \\mathcal{F}_{\\mathfrak{n}}^{M}$ , then there is a vertex  $(w, U) \\in \\operatorname{Meta}_{\\tilde{F}}(w)$  such that  $U \\subset V_F$  and  $\\chi_{G_4(F^w)}^M((w, \\emptyset)) = \\chi_{H_4(F^w)}^M((w, U))$ ;",
+            "- For any rooted graph  $F^w$  marking two vertices  $w_1, w_2$ , if  $F^w \\notin \\mathcal{F}_{\\mathrm{e}}^{M}$ , then there are two vertices  $(w_1, U_1) \\in \\operatorname{Meta}_{\\tilde{F}}(w_1)$  and  $(w_2, U_2) \\in \\operatorname{Meta}_{\\tilde{F}}(w_2)$  such that  $U_1, U_2 \\subset V_F$  and  $\\chi_{G_4(F^w)}^{M}((w_1, \\emptyset), (w_2, \\emptyset)) = \\chi_{H_4(F^w)}^{M}((w_1, U_1), (w_2, U_2))$ ."
+        ],
+        "bbox": [
+            196,
+            688,
+            825,
+            773
+        ],
+        "page_idx": 51
+    },
+    {
+        "type": "text",
+        "text": "Proof. Below, we only give a proof of the second item. Let  $\\tilde{F}$  be the union of  $F$  and the 4-clique according to Definition D.10. Similar to the graph-level expressivity, we will extend the pebble game defined in Appendix C.4 to edge-level. For the edge-level pebble game, the two pebbles  $\\mathfrak{p}_u,\\mathfrak{p}_v$  are initially placed on two predefined vertices  $w_{1},w_{2}$ , and Bob initially selects a predefined set of connected components  $Q\\subset \\mathbb{C}\\mathbb{C}_{\\tilde{F}}(\\{w_1,w_2\\})$  (of odd size). Then, the game executes the same process as in the text above Proposition C.49. The winning criterion for the two players is also the same. We denote the above edge-level pebble game as  $\\mathsf{PG}_{\\tilde{F}}(w_1,w_2,Q)$ .",
+        "bbox": [
+            169,
+            790,
+            823,
+            890
+        ],
+        "page_idx": 51
+    },
+    {
+        "type": "text",
+        "text": "Based on the isomorphism property of Fürer graphs established in Zhang et al. (2023a, Theorem I.9 and Theorem I.17), it is straightforward to see that, for any vertices  $\\xi, \\xi' \\in \\mathsf{Meta}_{\\tilde{F}}(w_1)$  and",
+        "bbox": [
+            169,
+            895,
+            825,
+            926
+        ],
+        "page_idx": 51
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 51
+    },
+    {
+        "type": "page_number",
+        "text": "52",
+        "bbox": [
+            488,
+            946,
+            509,
+            960
+        ],
+        "page_idx": 51
+    },
+    {
+        "type": "text",
+        "text": "$\\eta, \\eta' \\in \\mathsf{Meta}_{\\tilde{F}}(w_2), \\chi_{G_4(F^w)}^M(\\xi, \\eta) = \\chi_{H_4(F^w)}^M(\\xi', \\eta')$  iff Alice cannot win the edge-level pebble game  $\\mathsf{PG}_{\\tilde{F}}(w_1, w_2, Q)$  for some  $Q \\subset \\mathbb{C}\\mathbb{C}_{\\tilde{F}}(\\{w_1, w_2\\})$  (depending on  $\\xi, \\xi', \\eta, \\eta'$ ). Moreover, let  $\\xi' = (w_1, \\emptyset)$ ,  $\\eta' = (w_2, \\emptyset)$  and let  $\\xi_2$  and  $\\eta_2$  range over all even-size subsets of  $N_F(w_1)$  and  $N_F(w_2)$ , respectively, that is to say,  $\\xi_2$  and  $\\eta_2$  does not contain vertices that are only in the 4-clique. Then, the equivalent pebble game ranges over all  $Q$  of odd size such that every connected component in  $Q$  contains only vertices in  $F$ , i.e.,  $Q \\subset \\mathbb{C}\\mathbb{C}_F(\\{w_1, w_2\\})$ . Therefore, if Alice cannot win the edge-level pebble game  $\\mathsf{PG}_{\\tilde{F}}(w_1, w_2, Q)$  for some  $Q \\subset \\mathbb{C}\\mathbb{C}_F(\\{w_1, w_2\\})$ , then there are two vertices  $\\xi \\in \\mathsf{Meta}_{\\tilde{F}}(w_1)$  and  $\\eta \\in \\mathsf{Meta}_{\\tilde{F}}(w_2)$  such that  $\\xi_2, \\eta_2 \\subset V_F$  and  $\\chi_{G_4(F^w)}^M((w_1, \\emptyset), (w_2, \\emptyset)) = \\chi_{H_4(F^w)}^M(\\xi, \\eta)$ .",
+        "bbox": [
+            169,
+            102,
+            826,
+            242
+        ],
+        "page_idx": 52
+    },
+    {
+        "type": "text",
+        "text": "It thus remains to prove that, if Alice can win the edge-level pebble game  $\\mathsf{PG}_{\\tilde{F}}(w_1,w_2,Q)$  for all  $Q\\subset \\mathbb{C}C_F(\\{w_1,w_2\\})$ , then  $F^{\\boldsymbol{w}}\\in \\mathcal{F}_{\\mathrm{e}}^{M}$ . To prove this result, we similarly define the game state graph as in Appendix C.4, with the only difference that the initial state is now  $(w_{1},w_{2},Q)$ . The proof is divided into the following steps:",
+        "bbox": [
+            169,
+            248,
+            823,
+            306
+        ],
+        "page_idx": 52
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1. Assume that  $Q$  contains a single connected component, i.e.,  $Q = \\{P\\}$ . We will prove that, if Alice can win the game  $\\mathsf{PG}_{\\tilde{F}}(w_1, w_2, \\{P\\})$ , then there is a game strategy for Alice such that for any reachable and non-terminal state  $(x, y, \\{P'\\})$ ,  $P' \\subset P$ . If it is not the case, pick any state  $(x, y, \\{P'\\})$  such that  $P' \\notin P$  and any path from  $(w_1, w_2, \\{P\\})$  to  $(x, y, \\{P'\\})$  does not pass any intermediate state that is not contracted. It follows that either  $w_1$  or  $w_2$  does not hold any pebble, and all edges connected to this vertex is in  $P'$ . Therefore,  $P'$  contains all vertices in the 4-clique. In this case, Alice cannot win as Bob can always keep the connected component containing all vertices in the 4-clique in subsequent rounds, i.e., Bob always selects a component containing a vertex in the 4-clique that does not hold a pebble.",
+            "2. Based on the above result, in the game  $\\mathsf{PG}_{\\tilde{F}}(w_1,w_2,\\{P\\})$ , the connected components selected by Bob are always in  $F$ , and thus the game process is the same as  $\\mathsf{PG}_F(w_1,w_2,\\{P\\})$ . We can then prove the same result as Lemma C.52. Concretely, if Alice can win the edge-level pebble game  $\\mathsf{PG}_{\\tilde{F}}(w_1,w_2,\\{P\\})$ , then there exists a game state graph  $G^{\\mathbb{S}}$  corresponding to a winning strategy, such that for any transition  $((x,y,\\{P'\\}), (x',y',\\{P''\\}))$  where  $(x,y,\\{P'\\})$  is a reachable and non-terminal state, we have  $P'' \\subsetneq P'$ .",
+            "3. Next, we can follow the same procedure in the proof of Theorem C.53 to construct a tree decomposition  $T^r$  for the subgraph  $F[P]$  containing all edges in  $P$ , such that  $(F[P], T^r) \\in S_{\\mathbf{e}}^{M}$  and  $\\beta_T(r) = \\{w_1, w_2\\}$ .",
+            "4. Finally, let  $P$  ranges over all connected components in  $\\mathbb{C}C_F(\\{w_1, w_2\\})$ , for each  $P$  we have a tree-decomposed graph. We can glue these tree-decomposed graphs by merging the root to obtain the tree decomposition of  $F$ , because the root bags of all  $T^r$  are the same and any vertex  $x \\notin \\{w_1, w_2\\}$  appears in the bag of only one  $T^r$ . Moreover, the glued tree  $\\tilde{T}^s$  clearly satisfies that  $(F, \\tilde{T}^s) \\in S_{\\mathrm{e}}^{M}$ ."
+        ],
+        "bbox": [
+            191,
+            311,
+            825,
+            660
+        ],
+        "page_idx": 52
+    },
+    {
+        "type": "text",
+        "text": "Combining these items shows that  $F^{\\boldsymbol{w}} \\in \\mathcal{F}_{\\mathrm{e}}^{M}$  and concludes the proof.",
+        "bbox": [
+            169,
+            664,
+            640,
+            680
+        ],
+        "page_idx": 52
+    },
+    {
+        "type": "text",
+        "text": "Corollary D.12. Let  $F^w$  any rooted graph marking  $m$  vertices such that  $F^w \\notin \\mathcal{F}_n^M$  ( $m = 1$ ) or  $F^w \\notin \\mathcal{F}_{\\mathrm{e}}^{M}$  ( $m = 2$ ), and let  $G_{k}(F^{w})$  and  $H_{k}(F^{w})$  be clique-augmented Fuirer graphs defined above. Denote by  $\\xi, \\eta \\in V_{G_k(F^w)}^m$  two vertex tuples of length  $m$  where  $\\xi_{i} = (w_{i}, \\emptyset)$  and  $\\eta_{i} \\in \\operatorname{Meta}_{\\tilde{F}}(w_{i})$  with  $\\eta_{i,2} \\subset V_{F}$  for  $i \\in [m]$ . Then,  $\\operatorname{hom}(F^w, [G_k(F^w)]^\\xi) \\neq \\operatorname{hom}(F^w, [H_k(F^w)]^\\eta)$ .",
+        "bbox": [
+            169,
+            693,
+            823,
+            756
+        ],
+        "page_idx": 52
+    },
+    {
+        "type": "text",
+        "text": "Proof. The proof exactly parallels that of Theorem C.47 by separately considering three cases. We omit the proof for clarity.",
+        "bbox": [
+            169,
+            776,
+            823,
+            805
+        ],
+        "page_idx": 52
+    },
+    {
+        "type": "text",
+        "text": "Corollary D.13. Let  $M \\in \\{\\mathrm{Sub}, \\mathrm{L}, \\mathrm{LF}, \\mathrm{F}\\}$  be any model.",
+        "bbox": [
+            171,
+            818,
+            549,
+            834
+        ],
+        "page_idx": 52
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- For any graph  $F^w \\notin \\mathcal{F}_{\\mathfrak{n}}^{M}$ , let  $G_4(F^w)$  and  $H_4(F^w)$  be the clique-augmented Fürer graph and the corresponding twisted Fürer graph. Then, there exists two vertices  $\\xi, \\xi'$  such that  $\\operatorname{hom}(F^w, [G_4(F^w)]^\\xi) \\neq \\operatorname{hom}(F^w, [H_4(F^w)]^{\\xi'})$  and  $\\chi_{G_4(F^w)}^M(\\xi) = \\chi_{H_4(F^w)}^M(\\xi')$ .",
+            "- For any graph  $F^{wx} \\notin \\mathcal{F}_{\\mathrm{e}}^{M}$ , let  $G_{4}(F^{wx})$  and  $H_{4}(F^{wx})$  be the clique-augmented Fürer graph and the corresponding twisted Fürer graph defined in Definition D.10. Then, there exists"
+        ],
+        "bbox": [
+            196,
+            839,
+            823,
+            925
+        ],
+        "page_idx": 52
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 52
+    },
+    {
+        "type": "page_number",
+        "text": "53",
+        "bbox": [
+            488,
+            946,
+            506,
+            959
+        ],
+        "page_idx": 52
+    },
+    {
+        "type": "text",
+        "text": "four vertices  $\\xi, \\eta, \\xi', \\eta'$  such that  $\\operatorname{hom}(F^{wx}, [G_4(F^{wx})]^{\\xi \\eta}) \\neq \\operatorname{hom}(F^{wx}, [H_4(F^{wx})]^{\\xi' \\eta'})$  and  $\\chi_{G_4(F^{wx})}^M(\\xi, \\eta) = \\chi_{H_4(F^{wx})}^M(\\xi', \\eta')$ .",
+        "bbox": [
+            207,
+            101,
+            826,
+            137
+        ],
+        "page_idx": 53
+    },
+    {
+        "type": "text",
+        "text": "Proof. The proof directly follows from Corollary D.12 and Theorem D.11.  $\\square$",
+        "bbox": [
+            171,
+            148,
+            666,
+            165
+        ],
+        "page_idx": 53
+    },
+    {
+        "type": "image",
+        "img_path": "images/9755f732146412f3b844f348a388921eb520c4d51b5e6eb103ddb710f7c75feb.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            150,
+            825,
+            162
+        ],
+        "page_idx": 53
+    },
+    {
+        "type": "text",
+        "text": "E HIGHER-ORDER GNNS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            184,
+            403,
+            200
+        ],
+        "page_idx": 53
+    },
+    {
+        "type": "text",
+        "text": "E.1 DEFINITION OF HIGHER-ORDER GNNS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            215,
+            485,
+            229
+        ],
+        "page_idx": 53
+    },
+    {
+        "type": "text",
+        "text": "In this subsection, we give formal definitions of the CR algorithms for various higher-order GNNs, each of which generalizes a model in Section 2.",
+        "bbox": [
+            169,
+            241,
+            823,
+            271
+        ],
+        "page_idx": 53
+    },
+    {
+        "type": "text",
+        "text": "- Subgraph  $k$ -GNN. In a Subgraph  $k$ -GNN, a graph  $G$  is treated as a set of subgraphs  $\\{G^u : u \\in V_G^k\\}$ , where each subgraph  $G^u$  is obtained from  $G$  by marking  $k$  special vertices  $u \\in V_G^k$ , and thus there are  $n^k$  subgraphs when  $G$  has  $n$  vertices. Subgraph GNN maintains a color  $\\chi_G^{\\mathrm{Sub}(k)}(\\pmb{u}, v)$  for each vertex  $v$  in graph  $G^u$ . Initially,  $\\chi_G^{\\mathrm{Sub}(k), (0)}(\\pmb{u}, v) = (\\ell_G(v), \\mathsf{atp}_G(\\pmb{u}), \\mathbb{I}[u_1 = v], \\dots, \\mathbb{I}[u_k = v])$ . It then runs MPNNs independently on each graph  $G^u$ :",
+        "bbox": [
+            196,
+            277,
+            825,
+            368
+        ],
+        "page_idx": 53
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\chi_ {G} ^ {\\operatorname {S u b} (k), (t + 1)} (\\boldsymbol {u}, v) = \\operatorname {h a s h} \\left(\\chi_ {G} ^ {\\operatorname {S u b} (k), (t)} (\\boldsymbol {u}, v), \\{\\left\\{\\chi_ {G} ^ {\\operatorname {S u b} (k), (t)} (\\boldsymbol {u}, w): w \\in N _ {G} (v) \\right\\} \\}\\right). \\tag {26}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            228,
+            375,
+            825,
+            400
+        ],
+        "page_idx": 53
+    },
+    {
+        "type": "text",
+        "text": "Denote the stable color as  $\\chi_G^{\\mathrm{Sub}(k)}(\\pmb{u},\\upsilon)$ . Define  $\\chi_G^{\\mathrm{Sub}(k)}(\\pmb{u})\\coloneqq \\{\\{\\chi_G^{\\mathrm{Sub}(k)}(\\pmb{u},\\upsilon):v\\in V_G\\}\\}$ . Then, the graph representation is defined as  $\\chi_G^{\\mathrm{Sub}(k)}(G) = \\{\\{\\chi_G^{\\mathrm{Sub}(k)}(\\pmb {u}):\\pmb {u}\\in V_G\\} \\}$ . We remark that Subgraph  $k$  -GNN is precisely the  $k$  -VSAN proposed in Qian et al. (2022).",
+        "bbox": [
+            210,
+            407,
+            823,
+            458
+        ],
+        "page_idx": 53
+    },
+    {
+        "type": "text",
+        "text": "- Local  $k$ -GNN. Local  $k$ -GNN maintains a color  $\\chi_G^{\\mathsf{L}(k)}(\\pmb{u})$  for each vertex  $k$ -tuple  $\\pmb{u} \\in V_G^k$ . Initially,  $\\chi_G^{\\mathsf{L}(k),(0)}(\\pmb{u}) = (\\ell_G(u_1), \\dots, \\ell_G(u_k), \\mathsf{atp}_G(\\pmb{u}))$ , called the isomorphism type of vertex  $k$ -tuple  $\\pmb{u}$ , where  $\\mathsf{atp}_G(\\pmb{u})$  is the atomic type of  $\\pmb{u}$ . Then, in each iteration  $t + 1$ ,",
+        "bbox": [
+            197,
+            460,
+            825,
+            510
+        ],
+        "page_idx": 53
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\chi_ {G} ^ {\\mathsf {L} (k), (t + 1)} (\\boldsymbol {u}) = \\operatorname {h a s h} \\left(\\chi_ {G} ^ {\\mathsf {L} (k), (t)} (\\boldsymbol {u}), \\{\\{\\chi_ {G} ^ {\\mathsf {L} (k), (t)} (\\boldsymbol {v}): \\boldsymbol {v} \\in N _ {G} ^ {(1)} (\\boldsymbol {u}) \\} \\}, \\dots , \\right. \\\\ \\left. \\left\\{\\left\\{\\chi_ {G} ^ {\\mathrm {L} (k), (t)} (\\boldsymbol {v}): \\boldsymbol {v} \\in N _ {G} ^ {(k)} (\\boldsymbol {u}) \\right\\} \\right\\}\\right), \\tag {27} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            272,
+            516,
+            823,
+            570
+        ],
+        "page_idx": 53
+    },
+    {
+        "type": "text",
+        "text": "where  $N_G^{(j)}(\\pmb{u}) = \\{(u_1, \\dots, u_{j-1}, w, u_{j+1}, \\dots, u_k) : w \\in N_G(u_j)\\}$ . Denote the stable color as  $\\chi_G^{\\mathsf{L}(k)}(\\pmb{u})$ . The graph representation is defined as  $\\chi_G^{\\mathsf{L}(k)}(G) := \\{\\chi_G^{\\mathsf{L}(k)}(\\pmb{u}) : \\pmb{u} \\in V_G\\}$ . We remark that Local  $k$ -GNN is precisely the  $\\delta$ - $k$ -LWL proposed in Morris et al. (2020).",
+        "bbox": [
+            209,
+            578,
+            823,
+            627
+        ],
+        "page_idx": 53
+    },
+    {
+        "type": "text",
+        "text": "- Local  $k$ -FGNN. Local  $k$ -FGNN is almost the same as Local  $k$ -GNN except that the update formula is replaced by the following one:",
+        "bbox": [
+            197,
+            628,
+            825,
+            657
+        ],
+        "page_idx": 53
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\chi_ {G} ^ {\\mathsf {L F} (k), (t + 1)} (\\boldsymbol {u}) = \\operatorname {h a s h} \\left(\\chi_ {G} ^ {\\mathsf {L F} (k), (t)} (\\boldsymbol {u}), \\{\\{\\left(\\chi_ {G} ^ {\\mathsf {L F} (k), (t)} (w, u _ {2}, \\dots , u _ {k}), \\chi_ {G} ^ {\\mathsf {L F} (k), (t)} (u _ {1}, w, u _ {3}, \\dots , u _ {k}), \\right. \\right. \\\\ \\left. \\dots , \\chi_ {G} ^ {\\mathsf {L F} (k), (t)} \\left(u _ {1}, \\dots , u _ {k - 1}, w\\right): w \\in N _ {G} \\left(u _ {1}\\right) \\cup \\dots \\cup N _ {G} \\left(u _ {k}\\right) \\right\\} \\Bigg). \\tag {28} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            210,
+            664,
+            828,
+            728
+        ],
+        "page_idx": 53
+    },
+    {
+        "type": "text",
+        "text": "We remark that Local  $k$ -FGNN is precisely the  $\\mathrm{SLFWL}(k)$  proposed in Zhang et al. (2023a).",
+        "bbox": [
+            210,
+            729,
+            820,
+            743
+        ],
+        "page_idx": 53
+    },
+    {
+        "type": "text",
+        "text": "-  $k$ -FGNN. It is just the standard  $k$ -FWL (Cai et al., 1992). Compared with Local  $k$ -FGNN, the update formula is now global:",
+        "bbox": [
+            197,
+            744,
+            823,
+            773
+        ],
+        "page_idx": 53
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\chi_ {G} ^ {\\mathsf {F} (k), (t + 1)} (\\boldsymbol {u}) = \\operatorname {h a s h} \\left(\\chi_ {G} ^ {\\mathsf {F} (k), (t)} (\\boldsymbol {u}), \\{\\{\\left(\\chi_ {G} ^ {\\mathsf {F} (k), (t)} (w, u _ {2}, \\dots , u _ {k}), \\chi_ {G} ^ {\\mathsf {F} (k), (t)} (u _ {1}, w, u _ {3}, \\dots , u _ {k}), \\right. \\right. \\\\ \\left. \\dots , \\chi_ {G} ^ {\\mathsf {F} (k), (t)} \\left(u _ {1}, \\dots , u _ {k - 1}, w\\right): w \\in V _ {G} \\right\\}. \\Bigg). \\tag {29} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            220,
+            780,
+            823,
+            845
+        ],
+        "page_idx": 53
+    },
+    {
+        "type": "text",
+        "text": "We note that the computational complexity of Subgraph  $(k - 1)$ -GNN, Local  $k$ -GNN, and Local  $k$ -FGNN is the same, i.e.,  $\\Theta(n^{k - 1}m)$  for a graph of  $n$  vertices and  $m$  edges. Moreover, it is easy to see that Subgraph 0-GNN, Local 1-GNN, and Local 1-FGNN all reduce to MPNN. For  $k$ -FGNN, the computational complexity is  $\\Theta(n^{k + 1})$ , which is strictly higher than Subgraph  $(k - 1)$ -GNN, Local  $k$ -GNN, and Local  $k$ -FGNN.",
+        "bbox": [
+            169,
+            854,
+            825,
+            924
+        ],
+        "page_idx": 53
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 53
+    },
+    {
+        "type": "page_number",
+        "text": "54",
+        "bbox": [
+            488,
+            946,
+            509,
+            959
+        ],
+        "page_idx": 53
+    },
+    {
+        "type": "text",
+        "text": "E.2 HIGHER-ORDER STRONG NED",
+        "text_level": 1,
+        "bbox": [
+            171,
+            103,
+            429,
+            118
+        ],
+        "page_idx": 54
+    },
+    {
+        "type": "text",
+        "text": "In this subsection, we generalize the strong NED into higher-order versions. We first define the concept of higher-order ear:",
+        "bbox": [
+            169,
+            128,
+            823,
+            157
+        ],
+        "page_idx": 54
+    },
+    {
+        "type": "text",
+        "text": "Definition E.1 ( $k$ -order ear). Given integer  $k \\geq 1$ , a  $k$ -order ear is a graph  $G$  formed by the union of  $k$  paths  $P_{1}, \\dots, P_{k}$  (possibly with zero length) plus an edge set  $Q$  satisfying the following conditions:",
+        "bbox": [
+            169,
+            160,
+            825,
+            200
+        ],
+        "page_idx": 54
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- For each path  $P_{i}$ , denote its two endpoints as  $u_{i}, v_{i}$ , called the outer endpoint and inner endpoint, respectively. Then, all edges in  $Q$  are linked between inner endpoints, i.e.,  $Q \\subset \\{\\{v_{i}, v_{j}\\} : 1 \\leq i, j \\leq k, v_{i} \\neq v_{j}\\}$ .",
+            "- Any different paths  $P_{i}, P_{j}$  do not intersect except at the inner endpoint (when  $v_{i} = v_{j}$ ).",
+            "-  $G$  is a connected graph."
+        ],
+        "bbox": [
+            196,
+            209,
+            823,
+            294
+        ],
+        "page_idx": 54
+    },
+    {
+        "type": "text",
+        "text": "The endpoints of the  $k$ -order ear is defined to be all outer endpoints  $u_{1}, \\dots, u_{k}$ .",
+        "bbox": [
+            171,
+            301,
+            696,
+            316
+        ],
+        "page_idx": 54
+    },
+    {
+        "type": "text",
+        "text": "It is easy to see that a 2-order ear is precisely a simple path, since linking two different paths (possibly with an additional edge) still yields a path. Below, we denote by inner  $(G)$  and outer  $(G)$  the set of inner endpoints and outer endpoints in ear  $G$ , respectively. We also denote by path  $(G)$  the set of paths in ear  $G$ . It follows that  $|\\text{inner}(G)| = |\\text{outer}(G)| = |\\text{path}(G)| = k$ . Finally, given a path  $P$  and two vertices  $w_{1}, w_{2}$  in  $P$ , denote by subpath  $_{P}(w_{1}, w_{2})$  the subpath in  $P$  such that the two endpoints are  $w_{1}, w_{2}$ .",
+        "bbox": [
+            169,
+            325,
+            823,
+            407
+        ],
+        "page_idx": 54
+    },
+    {
+        "type": "text",
+        "text": "Definition E.2 (Nested interval). Let  $G$  and  $H$  be two  $k$ -order ears with inner  $(G) = \\{v_{1},\\dots ,v_{k}\\}$ , outer  $(G) = \\{u_{1},\\dots ,u_{k}\\}$ , and outer  $(H) = \\{w_{1},\\dots ,w_{k}\\}$ , where each  $\\{u_i,v_i\\}$  corresponds to the endpoints of a path  $P_{i}\\in \\mathrm{path}(G)$ . We say  $H$  is nested on  $G$  if one or more endpoint  $w_{i}$  of  $H$  ( $i\\in [k]$ ) is in path  $P_{i}$ , and all other vertices in  $H$  are not in  $G$ . The nested interval is defined to be the union of subpaths subpath  $P_{i}(w_{i},v_{i})$  for all  $i\\in [k]$  satisfying that  $w_{i}$  is in  $P_{i}$ .",
+        "bbox": [
+            169,
+            410,
+            823,
+            482
+        ],
+        "page_idx": 54
+    },
+    {
+        "type": "text",
+        "text": "We give an illustration of the nested interval of two 3-order ears in Figure 2. Equipped with the above definition, we are ready to introduce the higher-order strong NED:",
+        "bbox": [
+            169,
+            489,
+            823,
+            518
+        ],
+        "page_idx": 54
+    },
+    {
+        "type": "text",
+        "text": "Definition E.3 ( $k$ -order strong NED). Given a graph  $G$ , a  $k$ -order strong NED  $\\mathcal{P}$  is a partition of the edge set  $E_G$  into a sequence of edge sets  $Q_1, \\dots, Q_m$ , which satisfies the following conditions:",
+        "bbox": [
+            169,
+            520,
+            825,
+            550
+        ],
+        "page_idx": 54
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "Each  $Q_{i}$  is a  $k$  -order ear.",
+            "- Any two ears  $Q_{i}$  and  $Q_{j}$  with indices  $1 \\leq i < j \\leq c$  do not intersect, where  $c$  is the number of connected components of  $G$ .",
+            "- For each  $Q_{j}$  with index  $j > c$ , it is nested on some  $k$ -order ear  $Q_{i}$  with index  $1 \\leq i < j$ . Moreover, except for the endpoints of  $Q_{j}$  on  $Q_{i}$ , no other vertices in  $Q_{j}$  are in any previous ear  $Q_{k}$  for  $1 \\leq k < i$ .",
+            "- Denote by  $I(Q_{j}) \\subset Q_{i}$  the nested interval of  $Q_{j}$  in  $Q_{i}$ . For all  $Q_{j}, Q_{k}$  with  $c < j < k \\leq m$ , if  $Q_{j}$  and  $Q_{k}$  are nested on the same ear, then  $I(Q_{j}) \\subset I(Q_{k})$ ."
+        ],
+        "bbox": [
+            197,
+            556,
+            823,
+            691
+        ],
+        "page_idx": 54
+    },
+    {
+        "type": "text",
+        "text": "E.3 PROOFS IN SECTION 3.4",
+        "text_level": 1,
+        "bbox": [
+            171,
+            705,
+            385,
+            720
+        ],
+        "page_idx": 54
+    },
+    {
+        "type": "text",
+        "text": "We first generalize the tree-decomposed graphs in Definition C.6 to higher-order versions:",
+        "bbox": [
+            169,
+            731,
+            763,
+            746
+        ],
+        "page_idx": 54
+    },
+    {
+        "type": "text",
+        "text": "Definition E.4. Define four families of tree-decomposed graphs  $\\mathcal{S}^{\\mathrm{Sub}(k)}, \\mathcal{S}^{\\mathrm{L}(k)}, \\mathcal{S}^{\\mathrm{LF}(k)}$ , and  $\\mathcal{S}^{\\mathrm{F}(k)}$  as follows:",
+        "bbox": [
+            169,
+            747,
+            823,
+            777
+        ],
+        "page_idx": 54
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "a)  $(F,T^{r})\\in S^{\\mathsf{F}(k)}$  iff  $(F,T^{r})$  satisfies Definition C.4 with width  $k$",
+            "b)  $(F,T^{r})\\in S^{\\mathsf{LF}(k)}$  iff  $(F,T^{r})$  satisfies Definition C.4 with width  $k$ , and for any tree node  $t$  of odd depth, it has only one child if  $w\\notin \\{v:v\\in N_G(u),u\\in \\beta_T(s)\\}$  where  $s$  is the parent node of  $t$  and  $w$  is the unique vertex in  $\\beta_T(t)\\backslash \\beta_T(s)$ ;",
+            "c)  $(F,T^{r})\\in S^{\\mathbb{L}(k)}$  iff  $(F,T^{r})$  satisfies Definition C.4 with width  $k$ , and any tree node  $t$  of odd depth has only one child;",
+            "d)  $(F,T^{r})\\in S^{\\mathrm{Sub}(k)}$  iff  $(F,T^{r})$  satisfies Definition C.4 with width  $k$ , and there exists a multiset  $U\\subset V_{G}$  of size  $|U| = k$  such that  $U\\subset \\beta_T(t)$  for all  $t\\in V_T$ ."
+        ],
+        "bbox": [
+            189,
+            782,
+            826,
+            925
+        ],
+        "page_idx": 54
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 54
+    },
+    {
+        "type": "page_number",
+        "text": "55",
+        "bbox": [
+            488,
+            946,
+            506,
+            959
+        ],
+        "page_idx": 54
+    },
+    {
+        "type": "text",
+        "text": "Then, we can analogously prove the following theorems. The proofs are almost the same as in Appendix C, so we omit them for clarity.",
+        "bbox": [
+            169,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 55
+    },
+    {
+        "type": "text",
+        "text": "Theorem E.5. Let  $M \\in \\{\\mathsf{Sub}(k), \\mathsf{L}(k), \\mathsf{LF}(k), \\mathsf{F}(k)\\}$ . Then, any graphs  $G$  and  $H$  have the same representation under model  $M$  (i.e.,  $\\chi_G^M(G) = \\chi_H^M(H)$ ) iff  $\\mathsf{hom}(F, G) = \\mathsf{hom}(F, H)$  for all  $(F, T^r) \\in S^M$ .",
+        "bbox": [
+            169,
+            133,
+            825,
+            180
+        ],
+        "page_idx": 55
+    },
+    {
+        "type": "text",
+        "text": "Theorem E.6. Let  $M \\in \\{\\mathrm{Sub}(k), \\mathsf{L}(k), \\mathsf{LF}(k), \\mathsf{F}(k)\\}$  be any model, and let  $F$  be any graph such that no tree decomposition  $(F, T^r) \\in S^M$ . Let  $G(F)$  and  $H(F)$  be the Fürer graph and twisted Fürer graph with respect to  $F$ . Then,  $\\mathrm{hom}(F, G(F)) \\neq \\mathrm{hom}(F, H(F))$  and  $\\chi_{G(F)}^{M}(G(F)) = \\chi_{H(F)}^{M}(H(F))$ .",
+        "bbox": [
+            169,
+            181,
+            826,
+            247
+        ],
+        "page_idx": 55
+    },
+    {
+        "type": "text",
+        "text": "Theorem E.7. For any graph  $F$ , there is a tree decomposition  $T^r$  of  $F$  such that  $(F, T^r) \\in S^{\\mathrm{Sub}(k)}$  iff there exists  $U \\subset V_F$  such that  $|U| \\leq k$  and  $F \\backslash U$  is a forest.",
+        "bbox": [
+            169,
+            250,
+            823,
+            281
+        ],
+        "page_idx": 55
+    },
+    {
+        "type": "text",
+        "text": "Theorem E.8. For any graph  $F$ , there is a tree decomposition  $T^r$  of  $F$  such that  $(F, T^r) \\in S^{\\mathsf{L}(k)}$  iff  $F$  has a  $k$ -order strong NED.",
+        "bbox": [
+            169,
+            284,
+            823,
+            314
+        ],
+        "page_idx": 55
+    },
+    {
+        "type": "text",
+        "text": "Theorem E.9. For any graph  $F$ , there is a tree decomposition  $T^r$  of  $F$  such that  $(F, T^r) \\in S^{\\mathsf{F}(k)}$  iff  $\\operatorname{tw}(F) \\leq k$ .",
+        "bbox": [
+            169,
+            315,
+            823,
+            347
+        ],
+        "page_idx": 55
+    },
+    {
+        "type": "text",
+        "text": "E.4 EXPRESSIVITY GAP BETWEEN HIGHER-ORDER GNNS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            362,
+            586,
+            376
+        ],
+        "page_idx": 55
+    },
+    {
+        "type": "text",
+        "text": "In this subsection, we show how homomorphism expressivity can be used to build a complete expressiveness hierarchy for higher-order GNNs as shown in Corollary 4.2.",
+        "bbox": [
+            169,
+            388,
+            823,
+            417
+        ],
+        "page_idx": 55
+    },
+    {
+        "type": "text",
+        "text": "Gap between Subgraph  $k$ -GNN and Local  $(k + 1)$ -GNN  $(k \\geq 1)$ . The counterexample graph is a  $(k + 1) \\times (2k + 2)$  grid consisting  $(k + 1) \\times (2k + 2)$  vertices. First, it is easy to see that the graph is not in  $\\mathcal{F}^{\\mathrm{Sub}(k)}$ , i.e., deleting  $k$  vertices of the graph cannot yield a forest. To see this, note that the graph consists of  $k \\times (2k + 1)$  \"squares\", and each vertex is related to at most four squares. Therefore, deleting  $k$  vertices cannot eliminate all squares when  $k \\geq 2$  (because  $4k < k \\times (2k + 1)$ ). For the case of  $k = 1$ , we clearly have that deleting one vertex cannot eliminate all squares.",
+        "bbox": [
+            169,
+            422,
+            825,
+            508
+        ],
+        "page_idx": 55
+    },
+    {
+        "type": "text",
+        "text": "We next show that the  $(k + 1)\\times (2k + 2)$  grid is in  $\\mathcal{F}^{\\mathsf{L}(k + 1)}$ . This is also simple as shown in Figure 8, where each color represents a  $k$ -order ear. It can be seen that the  $(k + 1)\\times (2k + 2)$  grid has a  $(k + 1)$ -order strong NED.",
+        "bbox": [
+            169,
+            515,
+            550,
+            574
+        ],
+        "page_idx": 55
+    },
+    {
+        "type": "image",
+        "img_path": "images/830e1ed6f48f52caf086dee6661a9f5214573c1207c8b1c416a1eb46510085f3.jpg",
+        "image_caption": [
+            "Figure 8: An 4-order strong NED of the  $4 \\times 8$  grid graph."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            568,
+            513,
+            816,
+            602
+        ],
+        "page_idx": 55
+    },
+    {
+        "type": "text",
+        "text": "Relation between Subgraph  $k$ -GNN and  $k$ -FGNN ( $k \\geq 2$ ). To show that  $\\mathcal{F}^{\\mathrm{Sub}(k)} \\not\\subsetneq \\mathcal{F}^{\\mathrm{F}(k)}$ , consider the  $(k + 2)$ -clique. Clearly, deleting  $k$  vertices from the  $(k + 2)$ -clique yields a graph consisting of two vertices linked by an edge, which is a tree. On the other hand, the",
+        "bbox": [
+            169,
+            579,
+            549,
+            648
+        ],
+        "page_idx": 55
+    },
+    {
+        "type": "text",
+        "text": "treewidth of a  $(k + 2)$ -clique is  $k + 1$  (Fact C.3). So the  $(k + 2)$ -clique is in  $\\mathcal{F}^{\\mathrm{Sub}(k)}$  but not in  $\\mathcal{F}^{\\mathsf{F}(k)}$ .",
+        "bbox": [
+            169,
+            648,
+            823,
+            667
+        ],
+        "page_idx": 55
+    },
+    {
+        "type": "text",
+        "text": "To show that  $\\mathcal{F}^{\\mathsf{F}(k)} \\not\\subset \\mathcal{F}^{\\mathrm{Sub}(k)}$ , we can again use the grid graph, but this time consider the  $k \\times (2k + 2)$  grid. On the one hand, a standard result in graph theory shows that the treewidth of a  $a \\times b$  grid graph is  $\\min(a, b)$ . On the other hand, following the similar analysis above, we can prove that deleting  $k$  vertices from the  $k \\times (2k + 2)$  grid cannot eliminate all squares.",
+        "bbox": [
+            169,
+            672,
+            825,
+            731
+        ],
+        "page_idx": 55
+    },
+    {
+        "type": "text",
+        "text": "Gap between Local  $k$ -GNN and  $k$ -FGNN ( $k \\geq 2$ ). The counterexample graph  $F$  is the union of the following graphs  $K_0 \\cup K_1 \\cup \\dots \\cup K_k$ , where  $K_0$  is a  $(k+1)$ -clique with vertex set  $\\{u_1, \\dots, u_{k+1}\\}$  and for  $i \\in [k+1]$ ,  $K_i$  is a  $(k+1)$ -clique with vertex set  $\\{u_1, \\dots, u_{i-1}, v_i, u_{i+1}, \\dots, u_{k+1}\\}$  where  $v_i$  is a new vertex (not in  $K_0$ ) and  $v_i \\neq v_j$  for  $i \\neq j$ . Namely, each  $K_i$  has  $k$  common vertices with  $K_0$ . It is easy to construct a tree decomposition  $T^r$  of  $F$  such that  $(F, T^r) \\in S^{\\mathsf{F}(k)}$ . On the other hand, it is easy to see that the graph does not have a  $k$ -order strong NED (equivalently, one can easily check that  $F$  does not admit a tree decomposition  $T^r$  satisfying  $(F, T^r) \\in S^{\\mathsf{L}(k)}$ ).",
+        "bbox": [
+            169,
+            736,
+            825,
+            839
+        ],
+        "page_idx": 55
+    },
+    {
+        "type": "text",
+        "text": "Gap between  $k$ -FGNN and Local  $(k + 1)$ -GNN  $(k \\geq 2)$ . The counterexample graph is again the  $(k + 1) \\times (2k + 2)$  grid. We have proved that the graph is in  $\\mathcal{F}^{\\mathsf{L}(k + 1)}$  but not in  $\\mathcal{F}^{\\mathsf{L}(k)}$ .",
+        "bbox": [
+            169,
+            844,
+            823,
+            875
+        ],
+        "page_idx": 55
+    },
+    {
+        "type": "text",
+        "text": "Regarding Local  $k$ -IGN and Frasca et al. (2022). Finally, we remark that based on the above results, we essentially proved an open question raised in Frasca et al. (2022) regarding the expressive power of Local  $k$ -IGN (i.e., the  $\\mathsf{RelGN}(k)$  proposed in their paper). Following Zhang et al. (2023a),",
+        "bbox": [
+            169,
+            881,
+            825,
+            925
+        ],
+        "page_idx": 55
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 55
+    },
+    {
+        "type": "page_number",
+        "text": "56",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 55
+    },
+    {
+        "type": "image",
+        "img_path": "images/7a6402c5f447a7ef59e95c70fd19c4236ad3df702d0cbce15941509b93644af3.jpg",
+        "image_caption": [
+            "(a)"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            196,
+            109,
+            359,
+            164
+        ],
+        "page_idx": 56
+    },
+    {
+        "type": "image",
+        "img_path": "images/85b2010bdf07964a946f085449200e5651ba79ad301064f30d516e29ed43346b.jpg",
+        "image_caption": [
+            "(b)",
+            "Figure 9: More illustration of NED. All NEDs in these example graphs are strong but not endpoint-shared. See Appendix F.2 for additional discussions."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            380,
+            118,
+            553,
+            164
+        ],
+        "page_idx": 56
+    },
+    {
+        "type": "image",
+        "img_path": "images/99fd93c219affb4cee6c3df9c6ccbd5677d623576126da9d89f8fdb09e6c3fd4.jpg",
+        "image_caption": [
+            "(c)"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            578,
+            104,
+            800,
+            162
+        ],
+        "page_idx": 56
+    },
+    {
+        "type": "text",
+        "text": "it is straightforward to see that Local  $k$ -IGN is as expressive as Local  $k$ -GNN. Therefore, Local  $k$ -IGN is strictly more expressive than  $(k - 1)$ -FGNN and strictly less expressive than  $k$ -FGNN.",
+        "bbox": [
+            169,
+            267,
+            823,
+            296
+        ],
+        "page_idx": 56
+    },
+    {
+        "type": "text",
+        "text": "F ADDITIONAL DISCUSSIONS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            335,
+            434,
+            351
+        ],
+        "page_idx": 56
+    },
+    {
+        "type": "text",
+        "text": "F.1 REGARDING THE DEFINITION OF HOMOMORPHISM EXPRESSIVITY",
+        "text_level": 1,
+        "bbox": [
+            171,
+            378,
+            666,
+            393
+        ],
+        "page_idx": 56
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we have shown that homomorphism expressivity exists for a variety of popular GNNs. Unfortunately, due to the \"iff\" statement in Definition 3.1, homomorphism expressivity may not always be well-defined in the general case. We remark in this subsection that, there do exist pathological, intentionally designed GNNs such that the homomorphism expressivity is not well-defined.",
+        "bbox": [
+            169,
+            412,
+            823,
+            470
+        ],
+        "page_idx": 56
+    },
+    {
+        "type": "text",
+        "text": "Consider a simple GNN  $M$  that outputs the representation of a graph  $G$  as follows. If  $G$  is a cycle of odd length, it outputs  $(1,L)$  where  $L$  is the length of the cycle. Otherwise, it outputs  $(0,\\chi_{G}^{\\mathrm{MP}}(G))$ , namely, running a MPNN on the graph. It follows that  $M$  is strictly more powerful than MPNN, e.g., it can distinguish between the 9-cycle and three triangles, which MPNN fails to distinguish. As a result,  $M$  can count all trees under homomorphism. Moreover, it cannot count other patterns under homomorphism, as the counterexample graphs for MPNN (i.e., Fürer graphs) are not cycles of odd length, so they are still counterexample graphs for model  $M$ . Therefore, the homomorphism expressivity of  $M$  should be exactly the family of forests if it exists. However, the homomorphism information of forests cannot determine the representation of model  $M$  since  $M$  is strictly more powerful than MPNN. So we conclude that  $\\mathcal{F}^{M}$  does not exist.",
+        "bbox": [
+            169,
+            476,
+            826,
+            617
+        ],
+        "page_idx": 56
+    },
+    {
+        "type": "text",
+        "text": "Note that the above GNN construction is inherently unnatural and hardly appears in practice. Actually, when a GNN is defined via the message-passing paradigm, we suspect that its homomorphism expressivity is always well-defined. As stated in Appendix B, we conjecture that, for any GNN characterized by a color refinement algorithm that outputs stable colors, the homomorphism expressivity always exists.",
+        "bbox": [
+            169,
+            622,
+            823,
+            694
+        ],
+        "page_idx": 56
+    },
+    {
+        "type": "text",
+        "text": "F.2 REGARDING THE DEFINITION OF NED",
+        "text_level": 1,
+        "bbox": [
+            171,
+            729,
+            480,
+            743
+        ],
+        "page_idx": 56
+    },
+    {
+        "type": "text",
+        "text": "In the main text, we have illustrated several types of NED with simple example graphs. To gain a deeper understanding of Definition 3.3, in this section we will present a few more complex examples (see Figure 9). Notably, unlike graphs in Figure 1(b), for all graphs in Figure 9 their NED contains ears such that only one endpoint is in its nested ear (e.g., ears 2 and 4 in Figure 9(a)). In other words, these ears have empty nested intervals.",
+        "bbox": [
+            169,
+            763,
+            823,
+            834
+        ],
+        "page_idx": 56
+    },
+    {
+        "type": "text",
+        "text": "The presence of ears with empty nested interval stems from the fact that the corresponding graph is not biconnected (Zhang et al., 2023b). Indeed, one can check that all graphs in Definition 3.3(b) do not have cut vertices, while all graphs in Figure 9 have cut vertices. Moreover, in these examples, the number of ears with empty nested interval always equals to the number of biconnected components minus one. In particular, for biconnected graphs, it can be proved that any NED does not contain an ear with empty nested interval.",
+        "bbox": [
+            169,
+            840,
+            826,
+            925
+        ],
+        "page_idx": 56
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 56
+    },
+    {
+        "type": "page_number",
+        "text": "57",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 56
+    },
+    {
+        "type": "text",
+        "text": "G HOMOMORPHISM AND SUBGRAPH COUNTING POWER",
+        "text_level": 1,
+        "bbox": [
+            169,
+            102,
+            658,
+            118
+        ],
+        "page_idx": 57
+    },
+    {
+        "type": "text",
+        "text": "G.1 PROOF OF THEOREM 4.5",
+        "text_level": 1,
+        "bbox": [
+            171,
+            132,
+            390,
+            146
+        ],
+        "page_idx": 57
+    },
+    {
+        "type": "text",
+        "text": "Our proof draws inspiration from a recent technique developed in Seppelt (2023). We note that the original results in Seppelt (2023) are described for unlabeled graphs, but it is easy to extend these results to labeled graphs. To begin with, we define a concept called the graph categorical product.",
+        "bbox": [
+            169,
+            159,
+            823,
+            202
+        ],
+        "page_idx": 57
+    },
+    {
+        "type": "text",
+        "text": "Definition G.1. The categorical product of two graphs  $G, H$ , denoted as  $G \\times H$ , is a graph where  $V_{G \\times H} = V_G \\times V_H$ ,  $\\ell_{G \\times H}(g, h) = (\\ell_G(g), \\ell_H(h))$  for all  $g \\in V_G$ ,  $h \\in V_H$ , and  $E_{G \\times H} = \\{ (g, h), (g', h') \\} : \\{ g, g' \\} \\in E_G, \\{ h, h' \\} \\in E_H$ .",
+        "bbox": [
+            171,
+            204,
+            823,
+            248
+        ],
+        "page_idx": 57
+    },
+    {
+        "type": "text",
+        "text": "Lemma G.2 (Seppelt (2023)). For any graphs  $F, G,$  and  $H$ ,  $\\operatorname{hom}(F,G\\times H) = \\operatorname{hom}(F,G)\\cdot$ $\\operatorname{hom}(F,H)$ .",
+        "bbox": [
+            169,
+            250,
+            823,
+            280
+        ],
+        "page_idx": 57
+    },
+    {
+        "type": "text",
+        "text": "Proof. We define a mapping  $\\tau$  from  $\\operatorname{Hom}(F, G) \\times \\operatorname{Hom}(F, H)$  to  $\\operatorname{Hom}(F, G \\times H)$  as follows. for all  $\\sigma_1 \\in \\operatorname{Hom}(F, G)$  and  $\\sigma_2 \\in \\operatorname{Hom}(F, H)$ , define  $\\rho = \\tau(\\sigma_1, \\sigma_2)$  where  $\\rho(f) = (\\sigma_1(f), \\sigma_2(f))$  for all  $f \\in V_F$ . It is easy to see that  $\\tau$  is a bijective mapping from  $\\operatorname{Hom}(F, G) \\times \\operatorname{Hom}(F, H)$  to  $\\operatorname{Hom}(F, G \\times H)$ .",
+        "bbox": [
+            169,
+            294,
+            825,
+            351
+        ],
+        "page_idx": 57
+    },
+    {
+        "type": "text",
+        "text": "Definition G.3. Given graphs  $F$  and  $G$ , denote by  $\\operatorname{Surj}(F, G)$  the set of all homomorphisms from  $F$  to  $G$  that are surjective on both the vertices and edges of  $G$ , and define  $\\operatorname{surj}(F, G) = |\\operatorname{Surj}(F, G)|$ .",
+        "bbox": [
+            169,
+            359,
+            823,
+            388
+        ],
+        "page_idx": 57
+    },
+    {
+        "type": "text",
+        "text": "The following proposition is straightforward (similar to Lemma C.25):",
+        "bbox": [
+            169,
+            398,
+            637,
+            412
+        ],
+        "page_idx": 57
+    },
+    {
+        "type": "text",
+        "text": "Proposition G.4.  $\\operatorname{hom}(G, H) = \\sum_{F} \\operatorname{surj}(G, F) \\cdot \\operatorname{sub}(F, H)$ , where  $F$  ranges over all non-isomorphic graphs.",
+        "bbox": [
+            169,
+            417,
+            823,
+            446
+        ],
+        "page_idx": 57
+    },
+    {
+        "type": "text",
+        "text": "Lemma G.5. Let  $M$  be a GNN model such that its homomorphism expressivity  $\\mathcal{F}^M$  exists. Given a finite set of graphs  $\\mathcal{L}$  and a function  $\\alpha : \\mathcal{L} \\to \\mathbb{R} / \\{0\\}$ , if",
+        "bbox": [
+            168,
+            448,
+            823,
+            479
+        ],
+        "page_idx": 57
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\chi_ {G} ^ {M} (G) = \\chi_ {H} ^ {M} (H) \\Longrightarrow \\sum_ {L \\in \\mathcal {L}} \\alpha (L) \\operatorname {h o m} (L, G) = \\sum_ {L \\in \\mathcal {L}} \\alpha (L) \\operatorname {h o m} (L, H),\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            259,
+            484,
+            736,
+            516
+        ],
+        "page_idx": 57
+    },
+    {
+        "type": "text",
+        "text": "holds for all graphs  $G$  and  $H$ , then  $\\mathcal{L} \\subset \\mathcal{F}^M$ .",
+        "bbox": [
+            169,
+            523,
+            473,
+            539
+        ],
+        "page_idx": 57
+    },
+    {
+        "type": "text",
+        "text": "Proof. Let  $n$  be the largest number of vertices for all graphs in  $\\mathcal{L}$ , and let  $\\tilde{\\mathcal{L}}$  be the set of all non-isomorphic graphs with no more than  $n$  vertices. We can arrange all graphs in  $\\tilde{\\mathcal{L}}$  into a sequence  $L_{1}, L_{2}, \\ldots, L_{N}$  satisfying the following property:  $|V_{L_{i}}| \\leq |V_{L_{i+1}}|$  or  $(|V_{L_{i}}| = |V_{L_{i+1}}|$  and  $|E_{L_{i}}| \\leq |E_{L_{i+1}}|)$  for all  $i = 1, 2, \\ldots, N-1$ . We then define the matrices  $A^{\\text{hom}}$ ,  $A^{\\text{surj}}$ , and  $A^{\\text{sub}}$ , where the elements in the  $i^{th}$  row and  $j^{th}$  column are  $\\text{hom}(L_{i}, L_{j})$ ,  $\\text{surj}(L_{i}, L_{j})$ , and  $\\text{sub}(L_{i}, L_{j})$ , respectively. Proposition G.4 implies that  $A^{\\text{hom}} = A^{\\text{surj}} \\cdot A^{\\text{sub}}$ . Since  $A^{\\text{surj}}$  is a lower triangular matrix with nonzero diagonal elements and  $A^{\\text{sub}}$  is an upper triangular matrix with non-zero diagonal elements, the matrix  $A^{\\text{hom}}$  is invertible.",
+        "bbox": [
+            169,
+            555,
+            825,
+            675
+        ],
+        "page_idx": 57
+    },
+    {
+        "type": "text",
+        "text": "We next extend  $\\alpha$  to a function  $\\tilde{\\alpha}:\\tilde{\\mathcal{L}}\\to \\mathbb{R}$  by setting  $\\tilde{\\alpha} (L) = \\alpha (L)$  for all  $L\\in \\mathcal{L}$  and  $\\tilde{\\alpha} (\\tilde{L}) = 0$  for all  $\\tilde{L}\\in \\tilde{\\mathcal{L}}\\backslash \\mathcal{L}$ . Additionally, if  $\\chi_G^M (G) = \\chi_H^M (H)$ , then  $\\mathrm{hom}(F,G) = \\mathrm{hom}(F,H)$  for all  $F\\in \\mathcal{F}^{M}$  (by definition of homomorphism expressivity). Then, given any graph  $K\\in \\tilde{\\mathcal{L}}$ , Lemma G.2 implies that  $\\mathrm{hom}(F,G\\times K) = \\mathrm{hom}(F,H\\times K)$  for all  $F\\in \\mathcal{F}^{M}$ . This further implies that  $\\chi_{G\\times K}^{M}(G\\times K) = \\chi_{H\\times K}^{M}(H\\times K)$  by definition of homomorphism expressivity. Therefore, Lemma G.2 implies that",
+        "bbox": [
+            169,
+            681,
+            825,
+            761
+        ],
+        "page_idx": 57
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\sum_ {L \\in \\mathcal {L}} \\alpha (L) \\operatorname {h o m} (L, G) \\cdot \\operatorname {h o m} (L, K) = \\sum_ {L \\in \\mathcal {L}} \\alpha (L) \\operatorname {h o m} (L, H) \\cdot \\operatorname {h o m} (L, K). \\tag {30}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            254,
+            767,
+            823,
+            800
+        ],
+        "page_idx": 57
+    },
+    {
+        "type": "text",
+        "text": "Namely,",
+        "bbox": [
+            171,
+            805,
+            230,
+            820
+        ],
+        "page_idx": 57
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\sum_ {L \\in \\tilde {\\mathcal {L}}} \\tilde {\\alpha} (L) \\operatorname {h o m} (L, G) \\cdot \\operatorname {h o m} (L, K) = \\sum_ {L \\in \\tilde {\\mathcal {L}}} \\tilde {\\alpha} (L) \\operatorname {h o m} (L, H) \\cdot \\operatorname {h o m} (L, K). \\tag {31}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            256,
+            825,
+            823,
+            859
+        ],
+        "page_idx": 57
+    },
+    {
+        "type": "text",
+        "text": "Now define vectors  $\\pmb{p}_G$  and  $\\pmb{p}_H$ , where the  $i^{th}$  element of vector  $\\pmb{p}_G$  and  $\\pmb{p}_H$  is  $\\tilde{\\alpha}(L_i) \\mathrm{hom}(L_i, G)$  and  $\\tilde{\\alpha}(L_i) \\mathrm{hom}(L_i, H)$ , respectively. We then have the following equation:",
+        "bbox": [
+            169,
+            867,
+            823,
+            897
+        ],
+        "page_idx": 57
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {A} ^ {\\text {h o m}} \\cdot \\boldsymbol {p} _ {G} = \\boldsymbol {A} ^ {\\text {h o m}} \\cdot \\boldsymbol {p} _ {H}.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            411,
+            902,
+            583,
+            920
+        ],
+        "page_idx": 57
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 57
+    },
+    {
+        "type": "page_number",
+        "text": "58",
+        "bbox": [
+            488,
+            946,
+            506,
+            959
+        ],
+        "page_idx": 57
+    },
+    {
+        "type": "text",
+        "text": "Since  $\\mathbf{A}^{\\mathrm{hom}}$  is invertible,  $\\pmb{p}_G^{\\mathrm{hom}} = \\pmb{p}_H^{\\mathrm{hom}}$ . Therefore,  $\\tilde{\\alpha}(L)\\mathrm{hom}(L,G) = \\tilde{\\alpha}(L)\\mathrm{hom}(L,H)$  for all  $L \\in \\tilde{\\mathcal{L}}$ , namely,  $\\mathrm{hom}(L,G) = \\mathrm{hom}(L,H)$  for all  $L \\in \\mathcal{L}$ . To sum up, we have proved that for all graph  $G,H$ ,  $\\chi_G^M (G) = \\chi_H^M (H)$  implies that  $\\mathrm{hom}(L,G) = \\mathrm{hom}(L,H)$  for all  $L \\in \\mathcal{L}$ . By definition of homomorphism expressivity, we conclude that  $\\mathcal{L} \\subset \\mathcal{F}^M$ .",
+        "bbox": [
+            169,
+            102,
+            823,
+            164
+        ],
+        "page_idx": 58
+    },
+    {
+        "type": "text",
+        "text": "Theorem G.6. Let  $M \\in \\{\\mathrm{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}$  be any model. For any graph  $F$ , if  $\\mathrm{Spasm}(F) \\backslash \\mathcal{F}^M \\neq \\emptyset$ , then there exists a pair of graphs  $G, H$  such that  $\\chi_G^M(G) = \\chi_H^M(H)$  and  $\\mathrm{sub}(F, G) \\neq \\mathrm{sub}(F, H)$ .",
+        "bbox": [
+            169,
+            170,
+            823,
+            202
+        ],
+        "page_idx": 58
+    },
+    {
+        "type": "text",
+        "text": "Proof. Recall that  $\\operatorname{sub}(F, G) = \\sum_{\\tilde{F} \\in \\operatorname{Spasm}^{\\neq}(F)} \\alpha(F, \\tilde{F}) \\cdot \\operatorname{hom}(\\tilde{F}, G)$  where  $\\alpha(F, \\tilde{F}) \\neq 0$  for all  $\\tilde{F} \\in \\operatorname{Spasm}^{\\neq}(F)$ . If the above theorem does not hold, then for all graphs  $G, H$ ,  $\\chi_G^M(G) = \\chi_H^M(H)$  implies that",
+        "bbox": [
+            169,
+            215,
+            823,
+            265
+        ],
+        "page_idx": 58
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\sum_ {\\tilde {F} \\in \\operatorname {S p a s m} ^ {\\mathcal {Z}} (F)} \\alpha (F, \\tilde {F}) \\cdot \\operatorname {h o m} (\\tilde {F}, G) = \\sum_ {\\tilde {F} \\in \\operatorname {S p a s m} ^ {\\mathcal {Z}} (F)} \\alpha (F, \\tilde {F}) \\cdot \\operatorname {h o m} (\\tilde {F}, G).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            256,
+            268,
+            740,
+            306
+        ],
+        "page_idx": 58
+    },
+    {
+        "type": "text",
+        "text": "Then, Lemma G.5 implies that  $\\operatorname{Spasm}(F) \\subset \\mathcal{F}^M$ , yielding a contradiction.",
+        "bbox": [
+            171,
+            311,
+            666,
+            327
+        ],
+        "page_idx": 58
+    },
+    {
+        "type": "image",
+        "img_path": "images/57ebffe1e90b2e72ff9a5e60fade4ea4cb2a805920423b6c74c3c01178cff917.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            807,
+            311,
+            823,
+            321
+        ],
+        "page_idx": 58
+    },
+    {
+        "type": "text",
+        "text": "We next extend our analysis to the node/edge-level subgraph counting. Since the proof techniques are almost the same as the graph-level setting, we only present key definitions and lemmas below while omitting the detailed proofs for clarity.",
+        "bbox": [
+            169,
+            340,
+            823,
+            383
+        ],
+        "page_idx": 58
+    },
+    {
+        "type": "text",
+        "text": "Definition G.7. Given two rooted graphs  $G^{\\mathbf{u}}$  and  $H^{\\mathbf{v}}$  with  $\\mathbf{u} \\in V_G^m$  and  $\\mathbf{v} \\in V_H^m$  for some  $m \\in \\mathbb{N}_+$ , the categorical product of  $G^{\\mathbf{u}}$  and  $H^{\\mathbf{v}}$ , denoted as  $(G \\times H)^{(u_1, v_1), \\dots, (u_m, v_m)}$ , is a rooted graph obtained from  $G \\times H$  by marking vertices  $(u_1, v_1), \\dots, (u_m, v_m)$ .",
+        "bbox": [
+            169,
+            386,
+            823,
+            431
+        ],
+        "page_idx": 58
+    },
+    {
+        "type": "text",
+        "text": "Lemma G.8 (Extension of Lemma G.2). For any rooted graphs  $F^{\\mathbf{w}}$ ,  $G^{\\mathbf{u}}$ ,  $H^{\\mathbf{v}}$  where  $|\\mathbf{u}| = |\\mathbf{v}| = | \\mathbf{w} | = m$ ,",
+        "bbox": [
+            169,
+            433,
+            823,
+            463
+        ],
+        "page_idx": 58
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\operatorname {h o m} \\left(F ^ {\\boldsymbol {w}}, (G \\times H) ^ {(u _ {1}, v _ {1}), \\dots , (u _ {m}, v _ {m})}\\right) = \\operatorname {h o m} \\left(F ^ {\\boldsymbol {w}}, G ^ {\\boldsymbol {u}}\\right) \\cdot \\operatorname {h o m} \\left(F ^ {\\boldsymbol {w}}, H ^ {\\boldsymbol {v}}\\right).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            254,
+            467,
+            740,
+            484
+        ],
+        "page_idx": 58
+    },
+    {
+        "type": "text",
+        "text": "Proposition G.9 (Extension of Proposition G.4). For any rooted graphs  $G^u$ ,  $H^v$  where  $|\\boldsymbol{u}| = |\\boldsymbol{v}| = m$ ,  $\\operatorname{hom}(G^u, H^v) = \\sum_{F^w} \\operatorname{surj}(G^u, F^w) \\cdot \\operatorname{sub}(F^w, H^v)$ , where  $F^w$  ranges over all non-isomorphic rooted graphs marking  $m$  vertices.",
+        "bbox": [
+            169,
+            488,
+            823,
+            532
+        ],
+        "page_idx": 58
+    },
+    {
+        "type": "text",
+        "text": "We next present the main lemma for node-level subgraph counting. We omit the edge-level result for clarity.",
+        "bbox": [
+            169,
+            542,
+            823,
+            570
+        ],
+        "page_idx": 58
+    },
+    {
+        "type": "text",
+        "text": "Lemma G.10 (Extension of Lemma G.5). Let  $M$  be a GNN model such that its node-level homomorphism expressivity  $\\mathcal{F}_{\\mathfrak{n}}^{M}$  exists. Given a finite set of rooted graphs  $\\mathcal{L}_{\\mathfrak{n}}$  and a function  $\\alpha : \\mathcal{L}_{\\mathfrak{n}} \\to \\mathbb{R} / \\{0\\}$ , if",
+        "bbox": [
+            169,
+            573,
+            823,
+            616
+        ],
+        "page_idx": 58
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\chi_ {G} ^ {M} (u) = \\chi_ {H} ^ {M} (v) \\Longrightarrow \\sum_ {L ^ {w} \\in \\mathcal {L} _ {\\mathrm {n}}} \\alpha (L ^ {w}) \\operatorname {h o m} (L ^ {w}, G ^ {u}) = \\sum_ {L ^ {w} \\in \\mathcal {L} _ {\\mathrm {n}}} \\alpha (L ^ {w}) \\operatorname {h o m} (L ^ {w}, H ^ {v}),\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            218,
+            619,
+            774,
+            654
+        ],
+        "page_idx": 58
+    },
+    {
+        "type": "text",
+        "text": "holds for all rooted graphs  $G^{u}$  and  $H^{v}$ , then  $\\mathcal{L}_{\\mathfrak{n}} \\subset \\mathcal{F}_{\\mathfrak{n}}^{M}$ .",
+        "bbox": [
+            169,
+            657,
+            542,
+            674
+        ],
+        "page_idx": 58
+    },
+    {
+        "type": "text",
+        "text": "Theorem G.11. Let  $M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}$  be any model. For any rooted graph  $F^w$ , if  $\\text{Spasm}(F^w) \\backslash \\mathcal{F}_\\mathfrak{n}^M \\neq \\emptyset$ , then there exist a pair of graphs  $G, H$  and vertices  $u \\in V_G, v \\in V_H$ , such that  $\\chi_G^M(u) = \\chi_H^M(v)$  and  $\\text{sub}(F^w, G^u) \\neq \\text{sub}(F^w, H^v)$ .",
+        "bbox": [
+            169,
+            676,
+            826,
+            720
+        ],
+        "page_idx": 58
+    },
+    {
+        "type": "text",
+        "text": "G.2 GRAPH STATISTICS AND EXAMPLES",
+        "text_level": 1,
+        "bbox": [
+            171,
+            734,
+            465,
+            750
+        ],
+        "page_idx": 58
+    },
+    {
+        "type": "text",
+        "text": "In this section, we list the statistics of all moderate-size graphs that can/cannot be counted for each model in Section 2 at graph/node/edge-level. Table 4 presents the statistics under homomorphism count, while Table 5 presents the statistics under subgraph count. These tables offer a clear picture into how large the expressivity gaps are between different models. Several important findings are discussed below:",
+        "bbox": [
+            169,
+            761,
+            823,
+            830
+        ],
+        "page_idx": 58
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- For homomorphism counting, graphs of 8 edges suffice to reveal the expressivity gaps between each pair of architectures (at edge-level).",
+            "- However, for subgraph counting, moderate-size graphs cannot reveal the gap between Local 2-GNN, Local 2-FGNN, and 2-FGNN. Moreover, even Subgraph GNN already matches the power of 2-FWL in counting small subgraphs at graph-level, but it is quite weak in counting subgraphs at node/edge-level."
+        ],
+        "bbox": [
+            197,
+            838,
+            823,
+            925
+        ],
+        "page_idx": 58
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 58
+    },
+    {
+        "type": "page_number",
+        "text": "59",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 58
+    },
+    {
+        "type": "text",
+        "text": "- Subgraph counting is much more challenging than homomorphism counting. Intuitively, this is because the homomorphism image of a graph usually has 4-cliques, and any graph that contains a 4 clique as subgraph cannot be counted under homomorphism.",
+        "bbox": [
+            200,
+            103,
+            823,
+            147
+        ],
+        "page_idx": 59
+    },
+    {
+        "type": "text",
+        "text": "In Table 6, we list all subgraphs at moderate size (no more than 6 vertices or 8 edges) that can/cannot be counted by each GNN model. Here, we only list the graph-level expressivity as the node/edge-level expressivity involves too many non-isomorphic rooted graphs and cannot be fully presented (see Tables 4 and 5). We believe these results is comprehensive enough to cover most substructures of interest in the GNN community.",
+        "bbox": [
+            174,
+            152,
+            823,
+            223
+        ],
+        "page_idx": 59
+    },
+    {
+        "type": "table",
+        "img_path": "images/3997a051c0d5748ee917b19c1e4c7d3266dedf9becf59a0f0129bee4a0c802de.jpg",
+        "table_caption": [
+            "Table 4: The number of (connected) graphs (or rooted graphs) of  $n$  vertices or  $m$  edges that can be counted under homomorphism by different models. These statistics can be viewed as a quantitative expressivity comparison between models."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\" colspan=\"2\"></td><td colspan=\"5\">Number of vertices n</td><td colspan=\"7\">Number of edges m</td><td></td></tr><tr><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td></tr><tr><td rowspan=\"6\">Graph level</td><td>MPNN</td><td>1</td><td>1</td><td>2</td><td>3</td><td>6</td><td>1</td><td>1</td><td>2</td><td>3</td><td>6</td><td>11</td><td>23</td><td>47</td></tr><tr><td>Subgraph GNN</td><td>1</td><td>2</td><td>5</td><td>15</td><td>51</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>29</td><td>76</td><td>210</td></tr><tr><td>Local 2-GNN</td><td>1</td><td>2</td><td>5</td><td>15</td><td>55</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>29</td><td>77</td><td>216</td></tr><tr><td>Local 2-FGNN</td><td>1</td><td>2</td><td>5</td><td>15</td><td>56</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>29</td><td>77</td><td>216</td></tr><tr><td>2-FGNN</td><td>1</td><td>2</td><td>5</td><td>15</td><td>56</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>29</td><td>77</td><td>216</td></tr><tr><td>All</td><td>1</td><td>2</td><td>6</td><td>21</td><td>112</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>30</td><td>79</td><td>227</td></tr><tr><td rowspan=\"6\">Node level</td><td>MPNN</td><td>1</td><td>2</td><td>4</td><td>9</td><td>20</td><td>1</td><td>2</td><td>4</td><td>9</td><td>20</td><td>48</td><td>115</td><td>286</td></tr><tr><td>Subgraph GNN</td><td>1</td><td>3</td><td>8</td><td>27</td><td>88</td><td>1</td><td>2</td><td>5</td><td>12</td><td>31</td><td>83</td><td>228</td><td>640</td></tr><tr><td>Local 2-GNN</td><td>1</td><td>3</td><td>10</td><td>44</td><td>215</td><td>1</td><td>2</td><td>5</td><td>13</td><td>37</td><td>113</td><td>361</td><td>1210</td></tr><tr><td>Local 2-FGNN</td><td>1</td><td>3</td><td>10</td><td>44</td><td>217</td><td>1</td><td>2</td><td>5</td><td>13</td><td>37</td><td>113</td><td>361</td><td>1210</td></tr><tr><td>2-FGNN</td><td>1</td><td>3</td><td>10</td><td>44</td><td>217</td><td>1</td><td>2</td><td>5</td><td>13</td><td>37</td><td>113</td><td>361</td><td>1210</td></tr><tr><td>All</td><td>1</td><td>3</td><td>11</td><td>58</td><td>407</td><td>1</td><td>2</td><td>5</td><td>13</td><td>37</td><td>114</td><td>367</td><td>1248</td></tr><tr><td rowspan=\"5\">Edge level</td><td>Subgraph GNN</td><td>1</td><td>4</td><td>18</td><td>77</td><td>340</td><td>1</td><td>3</td><td>10</td><td>33</td><td>107</td><td>347</td><td>1126</td><td>3664</td></tr><tr><td>Local 2-GNN</td><td>1</td><td>4</td><td>21</td><td>116</td><td>693</td><td>1</td><td>3</td><td>10</td><td>35</td><td>124</td><td>450</td><td>1665</td><td>6267</td></tr><tr><td>Local 2-FGNN</td><td>1</td><td>4</td><td>21</td><td>118</td><td>735</td><td>1</td><td>3</td><td>10</td><td>35</td><td>124</td><td>451</td><td>1678</td><td>6373</td></tr><tr><td>2-FGNN</td><td>1</td><td>4</td><td>21</td><td>118</td><td>735</td><td>1</td><td>3</td><td>10</td><td>35</td><td>124</td><td>451</td><td>1678</td><td>6374</td></tr><tr><td>All</td><td>1</td><td>4</td><td>23</td><td>162</td><td>1549</td><td>1</td><td>3</td><td>10</td><td>35</td><td>125</td><td>460</td><td>1747</td><td>6830</td></tr></table>",
+        "bbox": [
+            173,
+            305,
+            826,
+            549
+        ],
+        "page_idx": 59
+    },
+    {
+        "type": "table",
+        "img_path": "images/813d80c4da3bfefa40ff2abf9e6e6fa921d94f815f1cef7348f8bb651abab6dc.jpg",
+        "table_caption": [
+            "Table 5: The number of (connected) graphs (or rooted graphs) of  $n$  vertices or  $m$  edges that can be subgraph-counted by different models. These statistics can be viewed as a quantitative expressivity comparison between models."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\" colspan=\"2\"></td><td colspan=\"5\">Number of vertices n</td><td colspan=\"7\">Number of edges m</td><td></td></tr><tr><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td></tr><tr><td rowspan=\"6\">Graph level</td><td>MPNN</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Subgraph GNN</td><td>1</td><td>2</td><td>5</td><td>13</td><td>24</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>20</td><td>21</td><td>22</td></tr><tr><td>Local 2-GNN</td><td>1</td><td>2</td><td>5</td><td>13</td><td>24</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>20</td><td>21</td><td>22</td></tr><tr><td>Local 2-FGNN</td><td>1</td><td>2</td><td>5</td><td>13</td><td>24</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>20</td><td>21</td><td>22</td></tr><tr><td>2-FGNN</td><td>1</td><td>2</td><td>5</td><td>13</td><td>24</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>20</td><td>21</td><td>22</td></tr><tr><td>All</td><td>1</td><td>2</td><td>6</td><td>21</td><td>112</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>30</td><td>79</td><td>227</td></tr><tr><td rowspan=\"6\">Node level</td><td>MPNN</td><td>1</td><td>2</td><td>2</td><td>2</td><td>2</td><td>1</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td></tr><tr><td>Subgraph GNN</td><td>1</td><td>3</td><td>8</td><td>18</td><td>27</td><td>1</td><td>2</td><td>5</td><td>10</td><td>16</td><td>23</td><td>30</td><td>38</td></tr><tr><td>Local 2-GNN</td><td>1</td><td>3</td><td>10</td><td>37</td><td>84</td><td>1</td><td>2</td><td>5</td><td>13</td><td>37</td><td>72</td><td>75</td><td>86</td></tr><tr><td>Local 2-FGNN</td><td>1</td><td>3</td><td>10</td><td>37</td><td>84</td><td>1</td><td>2</td><td>5</td><td>13</td><td>37</td><td>72</td><td>75</td><td>86</td></tr><tr><td>2-FGNN</td><td>1</td><td>3</td><td>10</td><td>37</td><td>84</td><td>1</td><td>2</td><td>5</td><td>13</td><td>37</td><td>72</td><td>75</td><td>86</td></tr><tr><td>All</td><td>1</td><td>3</td><td>11</td><td>58</td><td>407</td><td>1</td><td>2</td><td>5</td><td>13</td><td>37</td><td>114</td><td>367</td><td>1248</td></tr><tr><td rowspan=\"5\">Edge level</td><td>Subgraph GNN</td><td>1</td><td>4</td><td>18</td><td>47</td><td>81</td><td>1</td><td>3</td><td>10</td><td>25</td><td>46</td><td>69</td><td>95</td><td>124</td></tr><tr><td>Local 2-GNN</td><td>1</td><td>4</td><td>21</td><td>92</td><td>208</td><td>1</td><td>3</td><td>10</td><td>35</td><td>105</td><td>171</td><td>179</td><td>216</td></tr><tr><td>Local 2-FGNN</td><td>1</td><td>4</td><td>21</td><td>92</td><td>208</td><td>1</td><td>3</td><td>10</td><td>35</td><td>105</td><td>171</td><td>179</td><td>216</td></tr><tr><td>2-FGNN</td><td>1</td><td>4</td><td>21</td><td>92</td><td>208</td><td>1</td><td>3</td><td>10</td><td>35</td><td>105</td><td>171</td><td>179</td><td>216</td></tr><tr><td>All</td><td>1</td><td>4</td><td>23</td><td>162</td><td>1549</td><td>1</td><td>3</td><td>10</td><td>35</td><td>125</td><td>460</td><td>1747</td><td>6830</td></tr></table>",
+        "bbox": [
+            173,
+            654,
+            826,
+            898
+        ],
+        "page_idx": 59
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            173,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 59
+    },
+    {
+        "type": "page_number",
+        "text": "60",
+        "bbox": [
+            488,
+            948,
+            508,
+            959
+        ],
+        "page_idx": 59
+    },
+    {
+        "type": "table",
+        "img_path": "images/cb7551d9d7670e8cc23bde3e307b21ea923da88950875d69e72b9a1911b101d8.jpg",
+        "table_caption": [
+            "Table 6: The ability of GNNs to homomorphism-count and subgraph-count different graphs  $F$  within a bounded size of  $n \\leq  6$  vertices or  $m \\leq  8$  edges. When one or more GNNs fail to subgraphcount  $F$  ,this table also gives a homomorphic image  $\\widetilde{F} \\in  \\operatorname{Spasm}\\left( F\\right)$  that can be used to construct counterexample graphs (see Section 4.2)."
+        ],
+        "table_footnote": [
+            "(see the next page)"
+        ],
+        "table_body": "<table><tr><td>nmF</td><td>Homomorphism MP Sub L LF F</td><td>Subgraph MP Sub L LF F</td><td>F</td><td>nmF</td><td>Homomorphism MP Sub L LF F</td><td>Subgraph MP Sub L LF F</td><td>F</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>2 1←</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>-</td><td>5 7</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>3 2←</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>-</td><td>5 8</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>3 3</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>4 3</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>4 3</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>-</td><td>5 10</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>4 4</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 5</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>4 4</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 5</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>4 5</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 5</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>4 6</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>6 5</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 4</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 5</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 4</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 5</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 4</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>-</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 5</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 5</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 5</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 5</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>√</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 5</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 5</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✗</td><td>x</td></tr><tr><td>5 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr></table>",
+        "bbox": [
+            179,
+            132,
+            816,
+            911
+        ],
+        "page_idx": 60
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            173,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 60
+    },
+    {
+        "type": "page_number",
+        "text": "61",
+        "bbox": [
+            488,
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+        ],
+        "page_idx": 60
+    },
+    {
+        "type": "table",
+        "img_path": "images/0039710d63fef7e0d91a9a9f84e700ae76934c4fec46b09424917c462f9266c8.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "(see the next page)"
+        ],
+        "table_body": "<table><tr><td>n m</td><td>F</td><td>Homomorphism</td><td>Subgraph</td><td>MP Sub L F</td><td>MP Sub L F</td><td>F</td><td>n m</td><td>F</td><td>Homomorphism</td><td>Subgraph</td><td>MP Sub L F</td><td>MP Sub L F</td><td>F</td><td>n m</td><td>F</td><td>Homomorphism</td><td>Subgraph</td><td>MP Sub L F</td><td>MP Sub L F</td><td>F</td></tr><tr><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>X</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>X</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>X</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td></tr><tr><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td></tr><tr><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td></tr><tr><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td></tr><tr><td>X</td><td>X</td><td>X</td><td>X</td><td>x</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td></tr><tr><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td></tr><tr><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>x</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td></tr></table>",
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+        "page_idx": 61
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            173,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 61
+    },
+    {
+        "type": "page_number",
+        "text": "62",
+        "bbox": [
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+        "page_idx": 61
+    },
+    {
+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [
+            "(see the next page)"
+        ],
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+        "type": "header",
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+            "(see the next page)"
+        ],
+        "table_body": "<table><tr><td>n m</td><td>F</td><td>Homomorphism Subgraph MP Sub L F</td><td>MP Sub L F</td><td>F</td><td>n m</td><td>F</td><td>Homomorphism Subgraph MP Sub L F</td><td>MP Sub L F</td><td>F</td><td>n m</td><td>F</td><td>Homomorphism Subgraph MP Sub L F</td><td>MP Sub L F</td><td>F</td><td>n m</td><td>F</td><td>Homomorphism Subgraph MP Sub L F</td><td>MP Sub L F</td><td>F</td></tr><tr><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td></tr><tr><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td></td></tr><tr><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td></td></tr><tr><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td></td></tr><tr><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td></td></tr><tr><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td></td></tr><tr><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td></td></tr><tr><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td></td></tr><tr><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td></td></tr></table>",
+        "bbox": [
+            181,
+            99,
+            815,
+            910
+        ],
+        "page_idx": 65
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            173,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 65
+    },
+    {
+        "type": "page_number",
+        "text": "66",
+        "bbox": [
+            488,
+            948,
+            508,
+            959
+        ],
+        "page_idx": 65
+    },
+    {
+        "type": "table",
+        "img_path": "images/290474986d627a58da6126bf55449ab79bd368cdeb95f2d749235d86c5f4a236.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "(see the next page)"
+        ],
+        "table_body": "<table><tr><td>n m</td><td>F</td><td>Homomorphism</td><td>Subgraph</td><td>MP Sub L F F</td><td>MP Sub L F F</td></tr><tr><td>x</td><td>√</td><td>√</td><td>√</td><td>x</td><td>√</td></tr><tr><td>x</td><td>√</td><td>√</td><td>√</td><td>x</td><td>√</td></tr><tr><td>x</td><td>√</td><td>√</td><td>√</td><td>x</td><td>√</td></tr><tr><td>x</td><td>√</td><td>√</td><td>√</td><td>x</td><td>√</td></tr><tr><td>x</td><td>√</td><td>√</td><td>√</td><td>x</td><td>√</td></tr><tr><td>x</td><td>√</td><td>√</td><td>√</td><td>x</td><td>√</td></tr><tr><td>x</td><td>√</td><td>√</td><td>√</td><td>x</td><td>√</td></tr><tr><td>x</td><td>√</td><td>√</td><td>√</td><td>x</td><td>√</td></tr></table>",
+        "bbox": [
+            181,
+            99,
+            815,
+            910
+        ],
+        "page_idx": 66
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            173,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 66
+    },
+    {
+        "type": "page_number",
+        "text": "67",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 66
+    },
+    {
+        "type": "table",
+        "img_path": "images/363789b35f253122d6b4338a02327fc097b6b836fbd0dff0fabacab310df8bec.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>98</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td>86</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>X</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>y</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000</td><td>F</td><td>n m</td><td>F</td><td colspan=\"101\">Homonorphism Subgraph MP Sub L F F</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>",
+        "bbox": [
+            181,
+            104,
+            815,
+            916
+        ],
+        "page_idx": 67
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            173,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 67
+    },
+    {
+        "type": "page_number",
+        "text": "68",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 67
+    },
+    {
+        "type": "text",
+        "text": "H POLYNOMIAL EXPRESSIVITY",
+        "text_level": 1,
+        "bbox": [
+            171,
+            102,
+            454,
+            118
+        ],
+        "page_idx": 68
+    },
+    {
+        "type": "text",
+        "text": "Puny et al. (2023) proposed the equivariant graph polynomials, which are polynomials  $P$  that take real squared matrices  $\\mathbf{X} \\in \\mathbb{R}^{n \\times n}$  as input and outputs  $P(\\mathbf{X}) \\in \\mathbb{R}^{n \\times n}$ , such that  $P$  is equivariant under permutations, i.e.,  $P(\\pi \\cdot \\mathbf{X}) = \\pi \\cdot P(\\mathbf{X})$  for all permutation  $\\pi \\in S_n$ . The authors gave a concrete basis of equivariant polynomials, where each basis element  $P_{F^{uv}}$  corresponds to a rooted multi-graph  $F^{uv}$  marking two vertices  $u, v$ . Puny et al. (2023) showed that when  $\\mathbf{X}$  is restricted to be the adjacency matrix of an undirected simple graph  $G$ , each  $F^{uv}$  will also reduce to an undirected simple graph, and  $P_{F^{uv}}(\\mathbf{X})$  precisely computes the (unlabeled) homomorphism count  $\\mathrm{hom}(F^{uv}, G^{wx})$  for all  $w, x \\in V_G$ . Besides the original definition, Puny et al. (2023) also proposed the invariant graph polynomials and node-level equivariant graph polynomials, which are similarly related to the graph-level and node-level homomorphism count.",
+        "bbox": [
+            169,
+            133,
+            826,
+            273
+        ],
+        "page_idx": 68
+    },
+    {
+        "type": "text",
+        "text": "As a direct consequence, if a GNN model  $M$  cannot count graph  $F^{uv} / F^u /F$  under homomorphism, it then cannot compute the equivariant/invariant graph polynomial  $P_{F^{uv}} / P_{F^u} / P_F$ . Based on these connections, our results can be directly used to provide insights into which equivariant graph polynomials cannot be computed by model  $M$ . This recovers several results in their paper and answers an open problem shown below.",
+        "bbox": [
+            169,
+            279,
+            823,
+            349
+        ],
+        "page_idx": 68
+    },
+    {
+        "type": "text",
+        "text": "Corollary H.1. MPNN and  $PPGN++$  are bounded by the Prototypical node-based model and Prototypical edge-based model (defined in Puny et al. (2023)) for computing node-level and edge-level equivariant graph polynomials, respectively.",
+        "bbox": [
+            169,
+            354,
+            826,
+            397
+        ],
+        "page_idx": 68
+    },
+    {
+        "type": "text",
+        "text": "Proof. Without loss of generality, we assume that the corresponding graphs of all equivariant polynomials are connected. According to Puny et al. (2023, Proposition H.2), the Prototypical node-based model can compute all  $P_{F^u}$  where  $F^u$  is a (rooted) tree and cannot compute other graph polynomials. If MPNN is not bounded by the Prototypical node-based model, then it can compute some  $P_{\\tilde{F}^u}$  where  $\\tilde{F}$  is not a tree. However, this is impossible since MPNN can only count forests under homomorphism according to Theorem 3.7. Note that the MPNN defined in their paper is equivalent to our definition when only considering connected graphs (the extra global aggregation  $11^\\top X$  in their definition (9) does not improve the homomorphism expressivity).",
+        "bbox": [
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+        "page_idx": 68
+    },
+    {
+        "type": "text",
+        "text": "We next turn to  $\\mathrm{PPGN}++$ , and the proof is similar (but more involved). We first show that the Prototypical edge-based model can compute any  $P_{F^{uv}}$  satisfying that the treewidth of the graph  $\\tilde{F} := (V_F, E_F \\cup \\{\\{u, v\\}\\}, \\ell_F)$  is no more than 2. If  $\\mathsf{tw}(\\tilde{F}) \\leq 2$ , it is a partial 2-tree. Thus, there is an ordering  $w_1, \\dots, w_n$  of the vertex set  $V_F$  such that when deleting each vertex  $w_i$  and all incident edges in turn, we only ever delete vertices of degree at most 2. Now we claim that we can always order the two vertices  $u, v$  at the end, i.e.,  $w_{n-1} = u$  and  $w_n = v$ . Otherwise, there is a subset  $U \\subset V_F$  such that all vertices in the induced subgraph  $\\tilde{F}[U]$  are of degree at least 3 except  $u, v$ . It follows that the  $\\mathsf{tw}(\\tilde{F}[U]) \\geq \\mathsf{tw}(H) \\geq 3$  where graph  $H$  is the graph obtained from  $\\tilde{F}[U]$  by contracting  $u$  (or  $v$ ) if  $\\deg_{\\tilde{F}[U]}(u) \\leq 2$  (or  $\\deg_{\\tilde{F}[U]}(v) \\leq 2$ ). This yields a contradiction and verifies the claim that we can always set  $w_{n-1} = u$  and  $w_n = v$ .",
+        "bbox": [
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+        ],
+        "page_idx": 68
+    },
+    {
+        "type": "text",
+        "text": "Now, following the proof in Puny et al. (2023, Proposition H.3), the Prototypical edge-based model can contract  $\\tilde{F}^{uv}$  to a graph with only two vertices  $u, v$  and thus can compute the edge-level polynomial  $P_{\\tilde{F}^{uv}}$ . If PPGN++ is not bounded by the Prototypical edge-based model, then it can compute some  $P_{\\tilde{F}^{uv}}$  where  $\\mathrm{tw}(\\tilde{F}) \\geq 3$ . Therefore, it can count the graph  $\\tilde{F}$  under homomorphism at graph-level (since it can already count  $\\tilde{F}$  at edge-level). This implies that PPGN++ is strictly more expressive than 2-FGNN (2-FWL) because we have proved that all graphs in  $\\mathcal{F}^{\\mathrm{F}}$  have a treewidth no more than 2. This yields a contradiction since PPGN++ is still bounded by 2-FWL in distinguishing non-isomorphic graphs.",
+        "bbox": [
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+        "page_idx": 68
+    },
+    {
+        "type": "text",
+        "text": "We also provide insights into the following results in their paper:",
+        "bbox": [
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+        ],
+        "page_idx": 68
+    },
+    {
+        "type": "text",
+        "text": "Corollary H.2. The Prototypical node-based model is not 3-node-polynomial-exact. The Prototypical edge-based model is not 6-node-polynomial-exact and not 5-edge-polynomial-exact.",
+        "bbox": [
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+        ],
+        "page_idx": 68
+    },
+    {
+        "type": "text",
+        "text": "This is simply because the triangle is not a tree and the 4-clique does not have a NED (or equivalently, the treewidth of a 4-clique is 3). It is also clear why the degree of the edge-based polynomial is 6, which is one less than that of the node-based polynomial using the concept of NED.",
+        "bbox": [
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+        ],
+        "page_idx": 68
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 68
+    },
+    {
+        "type": "page_number",
+        "text": "69",
+        "bbox": [
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+        ],
+        "page_idx": 68
+    },
+    {
+        "type": "text",
+        "text": "I EXPERIMENTAL DETAILS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            102,
+            415,
+            118
+        ],
+        "page_idx": 69
+    },
+    {
+        "type": "text",
+        "text": "In this section, we provide all the experimental details in Section 5.",
+        "bbox": [
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+        ],
+        "page_idx": 69
+    },
+    {
+        "type": "text",
+        "text": "I.1 DATASETS",
+        "text_level": 1,
+        "bbox": [
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+            166,
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+            181
+        ],
+        "page_idx": 69
+    },
+    {
+        "type": "text",
+        "text": "We conduct experiments on tive tasks: (i) graph homomorphism counting, (ii) subgraph counting, (iii) ZINC-subset (Dwivedi et al., 2020), (iii) ZINC-full (Dwivedi et al., 2020), and (iv) Alchemy (Chen et al., 2019a).",
+        "bbox": [
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+        ],
+        "page_idx": 69
+    },
+    {
+        "type": "text",
+        "text": "Homomorphism/Subgraph Counting. For both homomorphism and subgraph counting tasks, we use the standard synthetic graph dataset constructed in Zhao et al. (2022a) (which has been used in a number of papers, see e.g., Frasca et al., 2022; Huang et al., 2023; Zhang et al., 2023a). For homomorphism counting, we count the number of graph/node/edge-level homomorphisms for each pattern in Table 1 and normalize the value by the mean and variance across all graphs in the dataset. The evaluation metric of graph-level expressivity is chosen as the Mean Absolute Error (MAE). For node/edge-level expressivity, the error on each graph is defined to be the sum of absolute error over all vertices/edges. We then report the MAE across all graphs in the dataset. This ensures that graph/node/edge-level errors are roughly at the same scale (since  $\\mathrm{hom}(F,G) = \\sum_{w\\in V_G}\\mathrm{hom}(F^u,G^w) = \\sum_{w,x\\in V_G}\\mathrm{hom}(F^{uv},G^{wx})$  for all  $u,v\\in V_F$ ). For subgraph counting, the data processing and evaluation metric is similar to homomorphism counting, but there is a slight difference in the node/edge-level setting: there are no marked vertices in the pattern graph  $F$  (see Table 3). Instead, given a graph  $G$  in the dataset and a vertex  $w\\in V_G$ , we count the number of subgraphs containing  $w$  that are isomorphic to  $F$  and  $w$  can be mapped to any vertex in  $F$ . Due to this difference, in the node/edge-level setting, the error on each graph is defined to be the average of absolute error over all vertices/edges.",
+        "bbox": [
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+        ],
+        "page_idx": 69
+    },
+    {
+        "type": "text",
+        "text": "ZINC. ZINC (Dwivedi et al., 2020) is a standard real-world dataset for benchmarking molecular property prediction. The dataset consists of 250K molecular graphs, and the task is to predict the constrained solubility of the given molecule. In addition to the full dataset (denoted as ZINC-full), ZINC-subset is a sampled dataset with 12k molecules from the ZINC-full dataset. We train and test our models on both datasets following the standard protocol from Dwivedi et al. (2020).",
+        "bbox": [
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+        ],
+        "page_idx": 69
+    },
+    {
+        "type": "text",
+        "text": "Alchemy. Alchemy (Chen et al., 2019a) is another real-world dataset with 12 graph-level quantum mechanical properties. We follow the sampling and training protocol from Lim et al. (2023); Puny et al. (2023), using 100K samples for training, 10K samples for testing, and 10K samples for validation.",
+        "bbox": [
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+        ],
+        "page_idx": 69
+    },
+    {
+        "type": "text",
+        "text": "I.2 MODEL DETAILS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            623,
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+            636
+        ],
+        "page_idx": 69
+    },
+    {
+        "type": "text",
+        "text": "All models are implemented using the PyTorch (Paszke et al., 2019) framework and the PyTorch Geometric library (Fey & Lenssen, 2019). We consider four types of GNNs defined in Section 2: MPNN, Subgraph GNN, Local 2-GNN, and Local 2-FGNN. For each GNN model, the feature initialization, message-passing layers, and final pooling operation are separately defined below.",
+        "bbox": [
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+        ],
+        "page_idx": 69
+    },
+    {
+        "type": "text",
+        "text": "Initialization. On both ZINC and Alchemy datasets, each graph node is an atom. We maintain a learnable atom embedding for each type of atom and use it to initialize features in GNN models. For MPNN, the initial feature  $h^{(0)}(u)$  of node  $u$  is simply the atom embedding, denoted as  $h^{(0)}(u) = e_{\\mathrm{atom}(u)}^{\\mathrm{N}}$ . For other models, the initial feature  $h^{(0)}(u,v)$  of node pair  $(u,v)$  consists of two parts: the first part is the node embedding of  $v$ , and the second part is a distance encoding that embeds the shortest path distance between  $u$  and  $v$ , as adopted in Zhang et al. (2023a). We note that while incorporating distance encoding does not increase the models' theoretical expressive power (see Zhang et al. (2023a)), it may add an inductive bias that can be helpful in real-world tasks. Formally, the initial feature can be written as  $h^{(0)}(u,v) = [e_{\\mathrm{atom}(v)}^{\\mathrm{N}}, e_{\\mathrm{clip}(\\mathrm{dis}(u,v))}^{\\mathrm{D}}]$ . Here, we clip the distance to a predefined value max_dis so that there are a finite number of distance embeddings, and distances greater than the hyper-parameter max_dis (including the disconnected case) share the embedding.",
+        "bbox": [
+            169,
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+        ],
+        "page_idx": 69
+    },
+    {
+        "type": "text",
+        "text": "Propagation. On both ZINC and Alchemy datasets, each edge in a graph corresponds to a chemical bond and has a bond type. We maintain a learnable edge embedding for each type of edges in each layer and denote the embedding of edge  $\\{u,v\\}$  in layer  $l$  as  $g^{(l)}(u,v)$ . For MPNN, we use the",
+        "bbox": [
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+        ],
+        "page_idx": 69
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 69
+    },
+    {
+        "type": "page_number",
+        "text": "70",
+        "bbox": [
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+        ],
+        "page_idx": 69
+    },
+    {
+        "type": "text",
+        "text": "standard GIN architecture proposed in Xu et al. (2019), which has the following form:",
+        "bbox": [
+            169,
+            103,
+            740,
+            119
+        ],
+        "page_idx": 70
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nh ^ {(l + 1)} (u) = \\operatorname {R e L U} \\left(\\mathsf {B N} ^ {(l)} \\left(f ^ {(l)} (u)\\right)\\right), \\tag {32}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            240,
+            125,
+            823,
+            143
+        ],
+        "page_idx": 70
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nf ^ {(l)} (u) = \\mathsf {G I N} ^ {(l)} \\left(h ^ {(l)} (u), \\sum_ {v \\in N _ {G} (u)} \\operatorname {R e L U} \\left(\\mathsf {F C} ^ {(l)} \\left(h ^ {(l)} (v)\\right) + g ^ {(l)} (u, v)\\right)\\right), \\tag {33}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            256,
+            147,
+            825,
+            196
+        ],
+        "page_idx": 70
+    },
+    {
+        "type": "text",
+        "text": "where",
+        "bbox": [
+            171,
+            203,
+            217,
+            215
+        ],
+        "page_idx": 70
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\operatorname {G I N} ^ {(l)} (\\boldsymbol {x}, \\boldsymbol {y}) = \\operatorname {M L P} ^ {(l)} \\left((1 + \\epsilon^ {(l)}) \\boldsymbol {x} + \\boldsymbol {y}\\right). \\tag {34}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            351,
+            212,
+            825,
+            239
+        ],
+        "page_idx": 70
+    },
+    {
+        "type": "text",
+        "text": "Here,  $\\mathsf{FC}^{(l)}$  is a parameterized linear transformation,  $\\epsilon^{(l)}$  is a learnable parameter,  $\\mathsf{BN}^{(l)}$  is the batch normalization (Ioffe & Szegedy, 2015), and  $\\mathsf{MLP}^{(l)}$  is a two-layer feed-forward network with another batch normalization in the hidden layer.",
+        "bbox": [
+            169,
+            243,
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+        ],
+        "page_idx": 70
+    },
+    {
+        "type": "text",
+        "text": "For other architectures, the  $l$ -th GNN layer analogously has the following form:",
+        "bbox": [
+            169,
+            296,
+            696,
+            311
+        ],
+        "page_idx": 70
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nh ^ {(l + 1)} (u, v) = \\operatorname {R e L U} \\left(\\mathsf {B N} ^ {(l)} \\left(f ^ {(l)} (u, v)\\right)\\right), \\tag {35}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            318,
+            825,
+            338
+        ],
+        "page_idx": 70
+    },
+    {
+        "type": "text",
+        "text": "where the term  $f^{(l)}(u,v)$  is defined separately for each model:",
+        "bbox": [
+            169,
+            345,
+            584,
+            363
+        ],
+        "page_idx": 70
+    },
+    {
+        "type": "text",
+        "text": "- Subgraph GNN:",
+        "bbox": [
+            197,
+            369,
+            331,
+            385
+        ],
+        "page_idx": 70
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nf ^ {(l)} (u, v) = \\mathsf {G I N} ^ {(l)} \\left(h ^ {(l)} (u, v), \\sum_ {w \\in N _ {G} (v)} \\operatorname {R e L U} \\left(\\mathsf {F C} ^ {(l)} \\left(h ^ {(l)} (u, w)\\right) + g ^ {(l)} (w, v)\\right)\\right). \\tag {36}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            225,
+            391,
+            825,
+            441
+        ],
+        "page_idx": 70
+    },
+    {
+        "type": "text",
+        "text": "- Local 2-GNN:",
+        "bbox": [
+            197,
+            449,
+            315,
+            463
+        ],
+        "page_idx": 70
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} f ^ {(l)} (u, v) = \\mathsf {G I N} ^ {(l, 1)} \\left(h ^ {(l)} (u, v), \\sum_ {w \\in N _ {G} (u)} \\operatorname {R e L U} \\left(\\mathsf {F C} ^ {(l, 1)} \\left(h ^ {(l)} (w, v)\\right) + g ^ {(l)} (u, w)\\right)\\right) \\\\ + \\operatorname {G I N} ^ {(l, 2)} \\left(h ^ {(l)} (u, v), \\sum_ {w \\in N _ {G} (v)} \\operatorname {R e L U} \\left(\\mathsf {F C} ^ {(l, 2)} \\left(h ^ {(l)} (u, w)\\right) + g ^ {(l)} (w, v)\\right)\\right). \\tag {37} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            222,
+            468,
+            823,
+            580
+        ],
+        "page_idx": 70
+    },
+    {
+        "type": "text",
+        "text": "- Local 2-FGNN:",
+        "bbox": [
+            197,
+            580,
+            323,
+            594
+        ],
+        "page_idx": 70
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} f ^ {(l)} (u, v) \\\\ = \\mathsf {G I N} ^ {(l, 1)} \\left(h ^ {(l)} (u, v), \\sum_ {w \\in N _ {G} (u)} \\mathsf {R e L U} \\left(\\mathsf {F C} ^ {(l, 1)} \\left(h ^ {(l)} (u, w)\\right) + \\mathsf {F C} ^ {(l, 2)} \\left(h ^ {(l)} (w, v)\\right) + g ^ {(l)} (u, w)\\right)\\right) \\\\ + \\mathsf {G I N} ^ {(l, 2)} \\left(h ^ {(l)} (u, v), \\sum_ {w \\in N _ {G} (v)} \\operatorname {R e L U} \\left(\\mathsf {F C} ^ {(l, 1)} \\left(h ^ {(l)} (w, v)\\right) + \\mathsf {F C} ^ {(l, 2)} \\left(h ^ {(l)} (u, w)\\right) + g ^ {(l)} (w, v)\\right)\\right). \\tag {38} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            217,
+            601,
+            864,
+            734
+        ],
+        "page_idx": 70
+    },
+    {
+        "type": "text",
+        "text": "For all the above GNN architectures, it can be seen that each layer only aggregates the local neighborhood of vertices or vertex pairs. This design will have shortcomings for disconnected graphs since a vertex cannot aggregate information from other connected components no matter how deep the model is (Barceló et al., 2020). Note that there do exist disconnected graphs in real-world datasets like ZINC. Therefore, on real-world datasets like ZINC and Alchemy, we also incorporate a global aggregation with the following form for each layer (similar to the global aggregation in Frasca et al. (2022); Zhang et al. (2023a)):",
+        "bbox": [
+            169,
+            741,
+            826,
+            839
+        ],
+        "page_idx": 70
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\operatorname {G I N} ^ {(l, G)} \\left(h ^ {(l)} (u, v), \\sum_ {w \\in V _ {G}} h ^ {(l)} (u, w)\\right) \\tag {39}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            366,
+            845,
+            825,
+            888
+        ],
+        "page_idx": 70
+    },
+    {
+        "type": "text",
+        "text": "Note that the global aggregation does not increase model's theoretical expressive power according to Zhang et al. (2023a).",
+        "bbox": [
+            169,
+            895,
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+            924
+        ],
+        "page_idx": 70
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 70
+    },
+    {
+        "type": "page_number",
+        "text": "71",
+        "bbox": [
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+            946,
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+        ],
+        "page_idx": 70
+    },
+    {
+        "type": "text",
+        "text": "Pooling. Except for edge-level tasks, a final pooling layer is used to produce node-level features  $h(u)$  for all nodes  $u$ . It is implemented as follows:",
+        "bbox": [
+            169,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 71
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nh (u) = \\operatorname {M L P} \\left(\\sum_ {v \\in \\mathcal {V}} h ^ {(L)} (u, v)\\right), \\tag {40}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            385,
+            140,
+            825,
+            181
+        ],
+        "page_idx": 71
+    },
+    {
+        "type": "text",
+        "text": "where MLP is a 2-layer perceptron. For graph-level tasks, we further use a mean pooling layer to aggregate all  $h(u)$  and obtain the graph representation.",
+        "bbox": [
+            169,
+            188,
+            823,
+            217
+        ],
+        "page_idx": 71
+    },
+    {
+        "type": "text",
+        "text": "I.3 TRAINING DETAILS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            234,
+            346,
+            247
+        ],
+        "page_idx": 71
+    },
+    {
+        "type": "text",
+        "text": "All experiments are run on a single NVIDIA Tesla V100 GPU. For all tasks, we use the distance encoding hyper-parameter max_dis = 5. To enable a fair comparison between models, for each task we keep the same depth for different models while varying the hidden dimension so that the number of model parameters is roughly the same. Note that the dimensions are chosen such that all models roughly obey the 500K parameter budget in ZINC and Alchemy. The hidden dimension size and the number of model parameters are listed as follows. All models are trained using the Adam optimizer.",
+        "bbox": [
+            169,
+            258,
+            823,
+            357
+        ],
+        "page_idx": 71
+    },
+    {
+        "type": "table",
+        "img_path": "images/ef175cb5ac67476bb9e25aaf8ec0f80d867843808100f2fdb94f597ed5f0e52a.jpg",
+        "table_caption": [
+            "Table 7: Model size in different tasks."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Task</td><td colspan=\"3\">Hidden dimension</td><td colspan=\"3\"># Parameters</td></tr><tr><td>Counting</td><td>ZINC</td><td>Alchemy</td><td>Counting</td><td>ZINC</td><td>Alchemy</td></tr><tr><td>MPNN</td><td>128</td><td>150</td><td>150</td><td>314,119</td><td>510,158</td><td>509,719</td></tr><tr><td>Subgraph GNN</td><td>128</td><td>120</td><td>120</td><td>314,759</td><td>503,774</td><td>503,425</td></tr><tr><td>Local 2-GNN</td><td>96</td><td>96</td><td>96</td><td>317,388</td><td>495,188</td><td>494,911</td></tr><tr><td>Local 2-FGNN</td><td>96</td><td>96</td><td>96</td><td>317,388</td><td>495,188</td><td>494,911</td></tr></table>",
+        "bbox": [
+            232,
+            396,
+            764,
+            477
+        ],
+        "page_idx": 71
+    },
+    {
+        "type": "text",
+        "text": "Homomorphism/Subgraph Counting. We use a model depth of  $L = 5$  in all experiments. Following prior work (Huang et al., 2023), we remove all BN layers in all models. The initial learning rate is chosen as 0.001 and is decayed by a factor of 0.9 once the MAE on the validation set plateaus for 10 epochs. Each model is trained for 1200 epochs with a batch size of 512. We ran each experiment 4 times independently with different seeds and reported the average performance at the last epoch. We found that the standard deviation among different seeds is negligible.",
+        "bbox": [
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+        ],
+        "page_idx": 71
+    },
+    {
+        "type": "text",
+        "text": "ZINC. We use a model depth of  $L = 6$  in all experiments for both ZINC-subset and ZINC-full. Following prior work (Zhang et al., 2023a; Frasca et al., 2022), The initial learning rate is chosen as 0.001 and is decayed by a factor of 0.5 once the MAE on the validation set plateaus for 20 epochs. Each model is trained for 400 epochs on ZINC-subset and 500 epochs on ZINC-full, both with a batch size of 128. We report the MAE for the model checkpoint with the best validation performance. We ran each experiment 10 times independently with different seeds and reported the average performance as well as the standard deviation.",
+        "bbox": [
+            169,
+            589,
+            823,
+            686
+        ],
+        "page_idx": 71
+    },
+    {
+        "type": "text",
+        "text": "Alchemy. We use a model depth of  $L = 6$  in all experiments. Following prior work (Lim et al., 2023; Puny et al., 2023), The initial learning rate is chosen as 0.002 and is decayed by a factor of 0.5 once the MAE on the validation set plateaus for 20 epochs. Each model is trained for 500 epochs with a batch size of 128. We report the MAE for the model checkpoint with the best validation performance. We ran each experiment 10 times independently with different seeds and reported the average performance as well as the standard deviation.",
+        "bbox": [
+            169,
+            693,
+            823,
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+        ],
+        "page_idx": 71
+    },
+    {
+        "type": "text",
+        "text": "I.4 PERFORMANCE OF BASELINE MODELS IN LITERATURE",
+        "text_level": 1,
+        "bbox": [
+            171,
+            794,
+            586,
+            808
+        ],
+        "page_idx": 71
+    },
+    {
+        "type": "text",
+        "text": "For completeness, in this subsection we give a comprehensive list of the performance of GNN models in the literature on ZINC and Alchemy datasets. The numbers in each table below are directly taken from the original papers.",
+        "bbox": [
+            169,
+            820,
+            823,
+            863
+        ],
+        "page_idx": 71
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 71
+    },
+    {
+        "type": "page_number",
+        "text": "72",
+        "bbox": [
+            488,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 71
+    },
+    {
+        "type": "table",
+        "img_path": "images/c0a9194b1d854e8d21ff031a59aafb978db1e5f05c6bdccba10a1b61bfe2597a.jpg",
+        "table_caption": [
+            "Table 8: Performance of different GNN models on ZINC dataset reported in the literature."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Model</td><td>Reference</td><td>Test MAE</td></tr><tr><td rowspan=\"8\">MPNN</td><td>GIN</td><td>(Xu et al., 2019)</td><td>0.526±0.051</td></tr><tr><td>GraphSAGE</td><td>(Hamilton et al., 2017)</td><td>0.398±0.002</td></tr><tr><td>GAT</td><td>(Veličković et al., 2018)</td><td>0.384±0.007</td></tr><tr><td>GCN</td><td>(Kipf &amp; Welling, 2017)</td><td>0.367±0.011</td></tr><tr><td>MoNet</td><td>(Monti et al., 2017)</td><td>0.292±0.006</td></tr><tr><td>GatedGCN-PE</td><td>(Bresson &amp; Laurent, 2017)</td><td>0.214±0.006</td></tr><tr><td>MPNN(sum)</td><td>(Gilmer et al., 2017)</td><td>0.145±0.007</td></tr><tr><td>PNA</td><td>(Corso et al., 2020)</td><td>0.142±0.010</td></tr><tr><td rowspan=\"4\">Higher-order GNN</td><td>RingGNN</td><td>(Chen et al., 2019b)</td><td>0.353±0.019</td></tr><tr><td>PPGN</td><td>(Maron et al., 2019a)</td><td>0.303±0.068</td></tr><tr><td>PPGN</td><td>(Puny et al., 2023)</td><td>0.079±0.005</td></tr><tr><td>PPGN++</td><td>(Puny et al., 2023)</td><td>0.076±0.003</td></tr><tr><td rowspan=\"7\">Subgraph GNN</td><td>NGNN</td><td>(Zhang &amp; Li, 2021)</td><td>0.111±0.003</td></tr><tr><td>GNN-AK</td><td>(Zhao et al., 2022a)</td><td>0.105±0.010</td></tr><tr><td>GNN-AK+</td><td>(Zhao et al., 2022a)</td><td>0.091±0.002</td></tr><tr><td>ESAN</td><td>(Bevilacqua et al., 2022)</td><td>0.102±0.003</td></tr><tr><td>SUN</td><td>(Frasca et al., 2022)</td><td>0.083±0.003</td></tr><tr><td>I2-GNN</td><td>(Huang et al., 2023)</td><td>0.083±0.001</td></tr><tr><td>ID-MPNN</td><td>(Zhou et al., 2023a)</td><td>0.083±0.003</td></tr><tr><td rowspan=\"4\">Local (F)GNN</td><td>SetGNN</td><td>(Zhao et al., 2022b)</td><td>0.075±0.003</td></tr><tr><td>GNN-SSWL</td><td>(Zhang et al., 2023a)</td><td>0.082±0.010</td></tr><tr><td>GNN-SSWL+</td><td>(Zhang et al., 2023a)</td><td>0.070±0.005</td></tr><tr><td>N2-GNN</td><td>(Feng et al., 2023)</td><td>0.059±0.002</td></tr><tr><td rowspan=\"4\">Substructure-based GNN</td><td>GSN</td><td>(Bouritsas et al., 2022)</td><td>0.101±0.010</td></tr><tr><td>CIN (Small)</td><td>(Bodnar et al., 2021a)</td><td>0.094±0.004</td></tr><tr><td>CIN</td><td>(Bodnar et al., 2021a)</td><td>0.079±0.006</td></tr><tr><td>CIN++</td><td>(Giusti et al., 2023)</td><td>0.077±0.004</td></tr><tr><td rowspan=\"6\">Graph Transformer</td><td>SAN</td><td>(Kreuzer et al., 2021)</td><td>0.139±0.006</td></tr><tr><td>K-Subgraph SAT</td><td>(Chen et al., 2022)</td><td>0.094±0.008</td></tr><tr><td>Graphormer</td><td>(Ying et al., 2021)</td><td>0.122±0.006</td></tr><tr><td>URPE</td><td>(Luo et al., 2022)</td><td>0.086±0.007</td></tr><tr><td>Graphormer-GD</td><td>(Zhang et al., 2023b)</td><td>0.081±0.009</td></tr><tr><td>GPS</td><td>(Rampasek et al., 2022)</td><td>0.070±0.004</td></tr><tr><td rowspan=\"6\">Other</td><td>PF-GNN</td><td>(Dupty et al., 2021)</td><td>0.122±0.010</td></tr><tr><td>KP-GIN</td><td>(Feng et al., 2022)</td><td>0.093±0.007</td></tr><tr><td>SignNet</td><td>(Lim et al., 2023)</td><td>0.084±0.006</td></tr><tr><td>PathNN</td><td>(Michel et al., 2023)</td><td>0.090±0.004</td></tr><tr><td>PPGN++(6)</td><td>(Puny et al., 2023)</td><td>0.071±0.001</td></tr><tr><td>PlanE</td><td>(Dimitrov et al., 2023)</td><td>0.076±0.003</td></tr></table>",
+        "bbox": [
+            241,
+            156,
+            754,
+            671
+        ],
+        "page_idx": 72
+    },
+    {
+        "type": "table",
+        "img_path": "images/3928926ab09920a6189b90389dbdffd3021c5980983a1f1dcae5080207e4d00c.jpg",
+        "table_caption": [
+            "Table 9: Performance of different GNN models on Alchemy dataset reported in the literature."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Reference</td><td>Test MAE</td></tr><tr><td>GIN</td><td>(Xu et al., 2019)</td><td>0.180±0.006</td></tr><tr><td>PF-GNN</td><td>(Dupty et al., 2021)</td><td>0.111±0.010</td></tr><tr><td>δ-2-GNN</td><td>(Morris et al., 2020)</td><td>0.118±0.001</td></tr><tr><td>Recon-GNN</td><td>(Cotta et al., 2021)</td><td>0.125±0.001</td></tr><tr><td>SpeqNet</td><td>(Morris et al., 2022)</td><td>0.115±0.001</td></tr><tr><td>SignNet</td><td>(Lim et al., 2023)</td><td>0.113±0.002</td></tr><tr><td>PPGN</td><td>(Puny et al., 2023)</td><td>0.113±0.001</td></tr><tr><td>PPGN++</td><td>(Puny et al., 2023)</td><td>0.111±0.002</td></tr><tr><td>PPGN++(6)</td><td>(Puny et al., 2023)</td><td>0.109±0.001</td></tr></table>",
+        "bbox": [
+            323,
+            762,
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+        ],
+        "page_idx": 72
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            173,
+            32,
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+            47
+        ],
+        "page_idx": 72
+    },
+    {
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+    }
+]
\ No newline at end of file
diff --git a/2024/Beyond Weisfeiler-Lehman_ A Quantitative Framework for GNN Expressiveness/fa29bbb1-c854-4aeb-a13c-5bf1509f8f38_model.json b/2024/Beyond Weisfeiler-Lehman_ A Quantitative Framework for GNN Expressiveness/fa29bbb1-c854-4aeb-a13c-5bf1509f8f38_model.json
new file mode 100644
index 0000000000000000000000000000000000000000..4e033918733f908dae386fd3f024dfb7dc5e2a9f
--- /dev/null
+++ b/2024/Beyond Weisfeiler-Lehman_ A Quantitative Framework for GNN Expressiveness/fa29bbb1-c854-4aeb-a13c-5bf1509f8f38_model.json	
@@ -0,0 +1,14943 @@
+[
+    [
+        {
+            "type": "header",
+            "bbox": [
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+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.1,
+                0.79,
+                0.147
+            ],
+            "angle": 0,
+            "content": "BEYOND WEISFEILER-LEHMAN: A QUANTITATIVE FRAMEWORK FOR GNN EXPRESSIVENESS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.182,
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+                0.821,
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+            ],
+            "angle": 0,
+            "content": "Bohang Zhang\\(^{1* \\dagger}\\) Jingchu Gai\\(^{2*}\\) Yiheng Du\\(^{3}\\) Qiwei Ye\\(^{4}\\) Di He\\(^{1 \\boxtimes}\\) Liwei Wang\\(^{1,5 \\boxtimes}\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.184,
+                0.186,
+                0.686,
+                0.199
+            ],
+            "angle": 0,
+            "content": "\\(^{1}\\)National Key Laboratory of General Artificial Intelligence, SIST, Peking University"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.184,
+                0.199,
+                0.729,
+                0.213
+            ],
+            "angle": 0,
+            "content": "\\(^{2}\\)School of Mathematical Science, Peking University \\(^{3}\\)Yuanpei College, Peking University"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.184,
+                0.213,
+                0.805,
+                0.241
+            ],
+            "angle": 0,
+            "content": "\\(^{4}\\)Beijing Academy of Artificial Intelligence \\(^{5}\\)Center for Machine Learning Research, Peking University zhangbohang@pku.edu.cn, {gaijingchu,duyiheng}@stu.pku.edu.cn"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.184,
+                0.242,
+                0.545,
+                0.255
+            ],
+            "angle": 0,
+            "content": "qwye@baai.ac.cn, {dihe, wanglw}@pku.edu.cn"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.451,
+                0.273,
+                0.547,
+                0.287
+            ],
+            "angle": 0,
+            "content": "ABSTRACT"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.23,
+                0.301,
+                0.769,
+                0.567
+            ],
+            "angle": 0,
+            "content": "Designing expressive Graph Neural Networks (GNNs) is a fundamental topic in the graph learning community. So far, GNN expressiveness has been primarily assessed via the Weisfeiler-Lehman (WL) hierarchy. However, such an expressivity measure has notable limitations: it is inherently coarse, qualitative, and may not well reflect practical requirements (e.g., the ability to encode substructures). In this paper, we introduce a novel framework for quantitatively studying the expressiveness of GNN architectures, addressing all the above limitations. Specifically, we identify a fundamental expressivity measure termed homomorphism expressivity, which quantifies the ability of GNN models to count graphs under homomorphism. Homomorphism expressivity offers a complete and practical assessment tool: the completeness enables direct expressivity comparisons between GNN models, while the practicality allows for understanding concrete GNN abilities such as subgraph counting. By examining four classes of prominent GNNs as case studies, we derive simple, unified, and elegant descriptions of their homomorphism expressivity for both invariant and equivariant settings. Our results provide novel insights into a series of previous work, unify the landscape of different subareas in the community, and settle several open questions. Empirically, extensive experiments on both synthetic and real-world tasks verify our theory, showing that the practical performance of GNN models aligns well with the proposed metric."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.174,
+                0.588,
+                0.338,
+                0.603
+            ],
+            "angle": 0,
+            "content": "1 INTRODUCTION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.619,
+                0.827,
+                0.732
+            ],
+            "angle": 0,
+            "content": "Owing to the ubiquity of graph-structured data in numerous applications, Graph Neural Networks (GNNs) have achieved enormous success in the field of machine learning over the past few years. However, one of the most prominent drawbacks of popular GNNs lies in the limited expressive power. In particular, Morris et al. (2019); Xu et al. (2019) showed that Message Passing GNNs (MPNNs) are intrinsically bounded by the 1-dimensional Weisfeiler-Lehman test (1-WL) in distinguishing non-isomorphic graphs (Weisfeiler & Lehman, 1968). Since then, the Weisfeiler-Lehman hierarchy has become a yardstick to measure the expressiveness and guide designing more powerful GNN architectures (see Appendix A.1 for an overview of representative approaches in this area)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.737,
+                0.827,
+                0.892
+            ],
+            "angle": 0,
+            "content": "However, as more and more architectures have been proposed, the limitations of the WL hierarchy are becoming increasingly evident. First, the WL hierarchy is arguably too coarse to evaluate the expressive power of practical GNN models (Morris et al., 2022; Puny et al., 2023). On one hand, architectures inspired by higher-order WL tests (Maron et al., 2019b;a; Morris et al., 2019) often suffer from substantial computation/memory costs. On the other hand, most practical and efficient GNNs are only proved to be strictly more expressive than 1-WL by leveraging toy example graphs (e.g., Zhang & Li, 2021; Bevilacqua et al., 2022; Wijesinghe & Wang, 2022a). Such a qualitative characterization may provide little insight into the models' true expressiveness. Besides, the expressive power brought from the WL hierarchy often does not align well with the one required in practice (Veličković, 2022). Hence, how to study the expressiveness of GNN models in a quantitative, systematic, and practical way remains a central research direction for the GNN community."
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.191,
+                0.897,
+                0.378,
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+            ],
+            "angle": 0,
+            "content": "*Equal technical contributions."
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.193,
+                0.911,
+                0.277,
+                0.924
+            ],
+            "angle": 0,
+            "content": "\\(\\dagger\\) Project lead."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.191,
+                0.897,
+                0.378,
+                0.924
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.495,
+                0.949,
+                0.504,
+                0.96
+            ],
+            "angle": 0,
+            "content": "1"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.104,
+                0.827,
+                0.258
+            ],
+            "angle": 0,
+            "content": "To address the above limitations, this paper takes a different approach by studying GNN expressivity from the following practical angle: What structural information can a GNN model encode? Since the ability to detect/count graph substructures is crucial in various real-world applications (Chen et al., 2020; Huang et al., 2023; Tahmasebi et al., 2023), many expressive GNNs have been proposed based on preprocessing substructure information (Bouritsas et al., 2022; Barceló et al., 2021; Bodnar et al., 2021b;a). However, instead of augmenting GNNs by manually preprocessed (task-specific) substructures, it is nowadays more desirable to design generic, domain-agnostic GNNs that can end-to-end learn different structural information suitable for diverse applications. This naturally gives rise to the fundamental question of characterizing the complete set of substructures prevalent GNN models can encode. Unfortunately, this problem is widely recognized as challenging even when examining simple structures like cycles (Fürer, 2017; Arvind et al., 2020; Huang et al., 2023)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.264,
+                0.828,
+                0.378
+            ],
+            "angle": 0,
+            "content": "Our contributions. Motivated by GNNs' ability to encode substructures, this paper presents a novel framework for quantitatively analyzing the expressive power of GNN models. Our approach is rooted in a critical discovery: given a GNN model \\( M \\), the model's output representation for any graph \\( G \\) can be fully determined by the structural information of \\( G \\) over some pattern family \\( \\mathcal{F}^M \\), where \\( \\mathcal{F}^M \\) corresponds to precisely all (and only) those substructures that can be \"coded\" by model \\( M \\). In this way, the set \\( \\mathcal{F}^M \\) can be naturally viewed as an expressivity description of \\( M \\): after identifying \\( \\mathcal{F}^M \\) for each model \\( M \\), the expressivity of different models can then be qualitatively/quantitatively compared by simply looking at their set inclusion relation and set difference."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.382,
+                0.827,
+                0.482
+            ],
+            "angle": 0,
+            "content": "The crux here is to define an appropriate notion of \"encodability\" so that \\(\\mathcal{F}^M\\) can admit a simple description. We identify that a good candidate is the homomorphism expressivity: i.e., \\(\\mathcal{F}^M\\) consists of all substructures that can be counted by model \\(M\\) under homomorphism (see Section 2 for a formal definition). Homomorphism is a foundational concept in graph theory (Lovasz, 2012) and is linked to many important topics such as graph coloring, graph matching, and subgraph counting. With this concept, we are able to give complete, unified, and surprisingly elegant descriptions of the pattern family \\(\\mathcal{F}^M\\) for a wide range of mainstream GNN architectures listed below:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.19,
+                0.486,
+                0.825,
+                0.501
+            ],
+            "angle": 0,
+            "content": "- MPNN (e.g., Gilmer et al., 2017; Hamilton et al., 2017; Kipf & Welling, 2017; Xu et al., 2019);"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.19,
+                0.505,
+                0.825,
+                0.52
+            ],
+            "angle": 0,
+            "content": "- Subgraph GNN (You et al., 2021; Zhang & Li, 2021; Bevilacqua et al., 2022; Qian et al., 2022);"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.19,
+                0.523,
+                0.733,
+                0.538
+            ],
+            "angle": 0,
+            "content": "- Local GNN (Morris et al., 2020; 2022; Zhang et al., 2023a; Frasca et al., 2022);"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.189,
+                0.541,
+                0.74,
+                0.557
+            ],
+            "angle": 0,
+            "content": "- Folklore-type GNN (Maron et al., 2019a; Zhang et al., 2023a; Feng et al., 2023)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.189,
+                0.486,
+                0.825,
+                0.557
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.563,
+                0.827,
+                0.662
+            ],
+            "angle": 0,
+            "content": "Technically, the descriptions are based on a novel application and extension of the concept of nested ear decomposition (NED) in graph theory (Eppstein, 1992). We prove that: (i) (necessity) each model \\(M\\) above can count (under homomorphism) a specific family of patterns \\(\\mathcal{F}^M\\), characterized by a specific type of NED; (ii) (sufficiency) any pattern \\(F \\notin \\mathcal{F}^M\\) cannot be counted under homomorphism by model \\(M\\); (iii) (completeness) for any graph, information collected from the homomorphism count in pattern family \\(\\mathcal{F}^M\\) determines its representation computed by model \\(M\\). Therefore, homomorphism expressivity is well-defined and is a complete expressivity measure for GNN models."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.667,
+                0.827,
+                0.752
+            ],
+            "angle": 0,
+            "content": "Our theory can be generalized in various aspects. One significant extension is the node-level and edge-level expressivity for equivariant GNNs (Azizian & Lelarge, 2021; Geerts & Reutter, 2022), which can be naturally tackled by a fine-grained analysis of NED. As another non-trivial generalization, we study higher-order GNN variants for several of the above architectures and derive results by defining higher-order NED. Both aspects demonstrate the flexibility of our proposed framework, suggesting it as a general recipe for analyzing future architectures."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.758,
+                0.828,
+                0.926
+            ],
+            "angle": 0,
+            "content": "Implications. Homomorphism expressivity serves as a powerful toolbox for bridging different sub-areas in the GNN community, providing fresh understandings of a series of known results that were previously proved in complex ways, and answering a set of unresolved open problems. First, our results can readily establish a complete expressiveness hierarchy among all the aforementioned architectures and their higher-order extensions. This recovers and extends a number of results in Morris et al. (2020); Qian et al. (2022); Zhang et al. (2023a); Frasca et al. (2022) and answers their open problems (Section 4.1). In fact, our results go far beyond revealing the expressivity gap between models: we essentially answer how large the gap is and establish a systematic approach to constructing counterexample graphs. Second, based on the relation between homomorphism and subgraph count, we are able to characterize the subgraph counting power of GNN models for all patterns at graph, node, and edge levels, significantly advancing an open direction initiated in Fürer (2017); Arvind et al. (2020) (Section 4.2). As a special case, our results extend recent findings in"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.948,
+                0.506,
+                0.96
+            ],
+            "angle": 0,
+            "content": "2"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.104,
+                0.827,
+                0.19
+            ],
+            "angle": 0,
+            "content": "Huang et al. (2023) about the cycle counting power of GNN models, highlighting that Local 2-GNN can already subgraph-count all cycles/paths within 7 nodes (even at edge-level). Third, our results provide a new toolbox for studying the polynomial expressivity proposed recently in Puny et al. (2023), extending it to various practical architectures and answering an open question (Section 4.3). Empirically, an extensive set of experiments verifies our theory, showing that the homomorphism expressivity of different models matches well with their practical performance in diverse tasks."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.209,
+                0.327,
+                0.225
+            ],
+            "angle": 0,
+            "content": "2 PRELIMINARY"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.24,
+                0.828,
+                0.382
+            ],
+            "angle": 0,
+            "content": "Notations. We use \\(\\{\\}\\) and \\(\\{\\}\\) to denote sets and multisets, respectively. Given a (multi)set \\(S\\), its cardinality is denoted as \\(|S|\\). In this paper, we consider finite, undirected, vertex-labeled graphs with no self-loops or repeated edges. Let \\(G = (V_G, E_G, \\ell_G)\\) be a graph with vertex set \\(V_G\\), edge set \\(E_G\\), and label function \\(\\ell_G\\), where each edge in \\(E_G\\) is a set \\(\\{u, v\\} \\subset V_G\\) of cardinality two, and \\(\\ell_G(u)\\) is the label of vertex \\(u\\). The rooted graph \\(G^u\\) is a graph obtained from \\(G\\) by marking the special vertex \\(u \\in V_G\\); we can similarly consider marking two special vertices \\(u, v \\in V_G\\) (denote by \\(G^{uv}\\)). The neighbors of vertex \\(u\\) is denoted as \\(N_G(u) := \\{v \\in V_G : \\{u, v\\} \\in E_G\\}\\). A graph \\(F = (V_F, E_F, \\ell_F)\\) is a subgraph of \\(G\\) if \\(V_F \\subset V_G\\), \\(E_F \\subset E_G\\), and \\(\\ell_F(u) = \\ell_G(u)\\) for all \\(u \\in V_F\\). A simple path \\(P\\) in \\(G\\) is an edge set of the form \\(\\{\\{w_0, w_1\\}, \\dots, \\{w_{k-1}, w_k\\}\\} \\subset E_G\\) where \\(w_i \\neq w_j\\) for all \\(i \\neq j\\). Here, \\(w_0\\) and \\(w_k\\) are called endpoints of \\(P\\) and other vertices are called internal points."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.387,
+                0.826,
+                0.513
+            ],
+            "angle": 0,
+            "content": "Homomorphism, isomorphism, and subgraph count. Given two graphs \\( F \\) and \\( G \\), a homomorphism from \\( F \\) to \\( G \\) is a mapping \\( f: V_F \\to V_G \\) that preserves edges and labels, i.e., \\( \\ell_F(u) = \\ell_G(f(u)) \\) for all \\( u \\in V_F \\), and \\( \\{f(u), f(v)\\} \\in E_G \\) for all \\( \\{u, v\\} \\in E_F \\). When the mapping \\( f \\) exists, we say \\( F \\) is homomorphic to \\( G \\). We denote by \\( \\operatorname{Hom}(F, G) \\) the set of all homomorphisms from \\( F \\) to \\( G \\) and define \\( \\operatorname{hom}(F, G) = |\\operatorname{Hom}(F, G)| \\), which counts the number of homomorphisms for pattern \\( F \\) in graph \\( G \\). If \\( f \\) is further surjective on both vertices and edges, we call \\( G \\) a homomorphic image of \\( F \\). Denote by \\( \\operatorname{Spasm}(F) \\) the set of all homomorphic images of \\( F \\), called the spasm of \\( F \\). For rooted graphs, homomorphism should additionally preserve vertex marking: i.e., if \\( f \\) is a homomorphism from \\( F^{uv} \\) to \\( G^{xy} \\), then \\( f(u) = x \\) and \\( f(v) = y \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.518,
+                0.826,
+                0.577
+            ],
+            "angle": 0,
+            "content": "A mapping \\( f: V_F \\to V_G \\) is called an isomorphism if \\( f \\) is a bijection and both \\( f \\) and its inverse \\( f^{-1} \\) are homomorphisms. We denote by \\( \\operatorname{Sub}(F, G) \\) the set of all subgraphs of \\( G \\) isomorphic to \\( F \\) and define \\( \\operatorname{sub}(F, G) = |\\operatorname{Sub}(F, G)| \\), which counts the number of patterns \\( F \\) occurred in graph \\( G \\) as a subgraph. We note that a similar definition holds for rooted graphs (e.g., \\( \\operatorname{sub}(F^{uv}, G^{xy}) \\))."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.582,
+                0.826,
+                0.666
+            ],
+            "angle": 0,
+            "content": "Graph neural networks. GNNs can be generally described as graph functions that are invariant under isomorphism. To achieve such invariance, most popular GNN models follow a color refinement (CR) paradigm: they maintain a feature representation (color) for each vertex or vertex tuples and iteratively refine these features through equivariant aggregation layers. Finally, there is a global pooling layer to merge all features and obtain the graph representation. Below, we separately define the corresponding CR algorithms for four mainstream classes of GNNs studied in this paper."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.197,
+                0.669,
+                0.825,
+                0.716
+            ],
+            "angle": 0,
+            "content": "- MPNN. Given a graph \\(G\\), MPNN maintains a color \\(\\chi_G^{\\mathrm{MP}}(u)\\) for each vertex \\(u \\in V_G\\). Initially, the color only depends on the vertex label, i.e., \\(\\chi_G^{\\mathrm{MP},(0)}(u) = \\ell_G(u)\\). Then, in each iteration, the color is refined by the following update formula (where hash is a perfect hash function):"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.301,
+                0.717,
+                0.825,
+                0.742
+            ],
+            "angle": 0,
+            "content": "\\[\n\\chi_ {G} ^ {\\mathsf {M P}, (t + 1)} (u) = \\operatorname {h a s h} \\left(\\chi_ {G} ^ {\\mathsf {M P}, (t)} (u), \\{\\{\\chi_ {G} ^ {\\mathsf {M P}, (t)} (v): v \\in N _ {G} (u) \\} \\}\\right). \\tag {1}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.744,
+                0.826,
+                0.789
+            ],
+            "angle": 0,
+            "content": "After a sufficient number of iterations, the colors become stable. We denote by \\(\\chi_G^{\\mathrm{MP}}(u)\\) the stable color of \\(u\\), which is also the node feature of \\(u\\) computed by the MPNN. The graph representation is defined as the multiset of node colors, i.e., \\(\\chi_G^{\\mathrm{MP}}(G) = \\{\\{\\chi_G^{\\mathrm{MP}}(u): u \\in V_G\\}\\}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.2,
+                0.789,
+                0.825,
+                0.849
+            ],
+            "angle": 0,
+            "content": "- Subgraph GNN. It treats a graph \\(G\\) as a set of subgraphs \\(\\{\\{G^u : u \\in V_G\\}\\}\\), each obtained from \\(G\\) by marking a special vertex \\(u \\in V_G\\). Subgraph GNN maintains a color \\(\\chi_G^{\\mathrm{Sub}}(u, v)\\) for each vertex \\(v\\) in graph \\(G^u\\). Initially, \\(\\chi_G^{\\mathrm{Sub}, (0)}(u, v) = (\\ell_G(v), \\mathbb{I}[u = v])\\), where the latter term distinguishes the special mark. It then runs MPNNs independently on each graph \\(G^u\\):"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.273,
+                0.85,
+                0.825,
+                0.876
+            ],
+            "angle": 0,
+            "content": "\\[\n\\chi_ {G} ^ {\\text {S u b}, (t + 1)} (u, v) = \\operatorname {h a s h} \\left(\\chi_ {G} ^ {\\text {S u b}, (t)} (u, v), \\{\\{\\chi_ {G} ^ {\\text {S u b}, (t)} (u, w): w \\in N _ {G} (v) \\} \\}\\right). \\tag {2}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.878,
+                0.826,
+                0.927
+            ],
+            "angle": 0,
+            "content": "Denote the stable color of \\((u,v)\\) as \\(\\chi_G^{\\mathrm{Sub}}(u,v)\\). The node feature of \\(u\\) computed by Subgraph GNN is defined by merging all colors in \\(G^{u}\\), i.e., \\(\\chi_G^{\\mathrm{Sub}}(u) \\coloneqq \\mathsf{hash}\\left(\\{\\chi_G^{\\mathrm{Sub}}(u,v):v\\in V_G\\}\\right)\\). Finally, the graph representation is defined as \\(\\chi_G^{\\mathrm{Sub}}(G) = \\{\\{\\chi_G^{\\mathrm{Sub}}(u):u\\in V_G\\}\\}\\)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.505,
+                0.96
+            ],
+            "angle": 0,
+            "content": "3"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.104,
+                0.825,
+                0.161
+            ],
+            "angle": 0,
+            "content": "- Local GNN. Inspired by the \\(k\\)-WL test (Grohe, 2017), Local \\(k\\)-GNN is defined by replacing all global aggregations in \\(k\\)-WL by sparse ones that only aggregate local neighbors, yielding a much more efficient CR algorithm. As an example, the iteration of Local 2-GNN has the following form and enjoys the same computational complexity as a Subgraph GNN."
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.214,
+                0.162,
+                0.825,
+                0.2
+            ],
+            "angle": 0,
+            "content": "\\[\n\\chi_ {G} ^ {\\mathrm {L}, (t + 1)} (u, v) = \\operatorname {h a s h} \\left(\\chi_ {G} ^ {\\mathrm {L}, (t)} (u, v), \\{\\{\\chi_ {G} ^ {\\mathrm {L}, (t)} (w, v): w \\in N _ {G} (u) \\} \\}, \\{\\{\\chi_ {G} ^ {\\mathrm {L}, (t)} (u, w): w \\in N _ {G} (v) \\} \\}\\right). \\tag {3}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.209,
+                0.2,
+                0.825,
+                0.248
+            ],
+            "angle": 0,
+            "content": "Initially, \\(\\chi_G^{\\mathsf{L},(0)}(u,v) = (\\ell_G(u),\\ell_G(v),\\mathbb{I}[u = v],\\mathbb{I}[\\{u,v\\} \\in E_G])\\), which is called the isomorphism type of vertex pair \\((u,v)\\). We similarly denote the stable color as \\(\\chi_G^{\\mathsf{L}}(u,v)\\) and define the node feature \\(\\chi_G^{\\mathsf{L}}(u)\\) and graph representation \\(\\chi_G^{\\mathsf{L}}(G)\\) as in the Subgraph GNN."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.199,
+                0.248,
+                0.825,
+                0.277
+            ],
+            "angle": 0,
+            "content": "- Folklore-type GNN. The Folklore GNN (FGNN) is inspired by the standard \\(k\\)-FWL test (Cai et al., 1992). As an example, the iteration formula of 2-FGNN is written as follows:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.255,
+                0.278,
+                0.825,
+                0.303
+            ],
+            "angle": 0,
+            "content": "\\[\n\\chi_ {G} ^ {\\mathsf {F}, (t + 1)} (u, v) = \\operatorname {h a s h} \\left(\\chi_ {G} ^ {\\mathsf {F}, (t)} (u, v), \\{\\left(\\chi_ {G} ^ {\\mathsf {F}, (t)} (w, v), \\chi_ {G} ^ {\\mathsf {F}, (t)} (u, w)\\right): w \\in V _ {G} \\} \\}\\right). \\tag {4}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.304,
+                0.825,
+                0.332
+            ],
+            "angle": 0,
+            "content": "One can similarly consider the more efficient Local 2-FGNN by only aggregating local neighbors, which has the same computational complexity as Local 2-GNN and Subgraph GNN:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.212,
+                0.333,
+                0.825,
+                0.377
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\chi_ {G} ^ {\\mathrm {L F}, (t + 1)} (u, v) = \\operatorname {h a s h} \\left(\\chi_ {G} ^ {\\mathrm {L F}, (t)} (u, v), \\{\\{\\left(\\chi_ {G} ^ {\\mathrm {L F}, (t)} (w, v), \\chi_ {G} ^ {\\mathrm {L F}, (t)} (u, w)\\right): w \\in N _ {G} (u) \\cup N _ {G} (v) \\} \\}\\right). \\\\ \\text {T h e s t a b l e c o l o r , n o d e f e a t u r e , a n d g r a p h r e p r e s e n t a t i o n c a n b e s i m a r l y d e f i n e d .} \\end{array} \\tag {5}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.384,
+                0.826,
+                0.427
+            ],
+            "angle": 0,
+            "content": "Finally, we note that the latter three types of GNNs can be naturally generalized into higher-order variants. We give a general definition of all these architectures in Appendix E.1. For the base case of \\( k = 1 \\), Subgraph \\( (k - 1) \\)-GNN, Local \\( k \\)-GNN, and Local \\( k \\)-FGNN all reduce to the MPNN."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.445,
+                0.753,
+                0.46
+            ],
+            "angle": 0,
+            "content": "3 HOMOMORPHISM EXPRESSIVITY OF GRAPH NEURAL NETWORKS"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.474,
+                0.443,
+                0.487
+            ],
+            "angle": 0,
+            "content": "3.1 HOMOMORPHISM EXPRESSIVITY"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.498,
+                0.825,
+                0.569
+            ],
+            "angle": 0,
+            "content": "Given a GNN model \\(M\\) and a substructure \\(F\\), we say \\(M\\) can count graph \\(F\\) under homomorphism if, for any graph \\(G\\), the graph representation \\(\\chi_G^M (G)\\) determines the homomorphism count \\(\\mathrm{hom}(F,G)\\). In other words, \\(\\chi_G^M (G) = \\chi_H^M (H)\\) implies \\(\\mathrm{hom}(F,G) = \\mathrm{hom}(F,H)\\) for any graphs \\(G,H\\). The central question studied in this paper is, what substructures \\(F\\) can a GNN model \\(M\\) count under homomorphism? This gives rise to the notion of homomorphism expressivity defined below:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.571,
+                0.825,
+                0.601
+            ],
+            "angle": 0,
+            "content": "Definition 3.1. The homomorphism expressivity of a GNN model \\(M\\), denoted by \\(\\mathcal{F}^M\\), is a family of (labeled) graphs satisfying the following conditions<sup>1</sup>:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.19,
+                0.602,
+                0.821,
+                0.62
+            ],
+            "angle": 0,
+            "content": "a) For any two graphs \\(G,H,\\chi_G^M (G) = \\chi_H^M (H)\\) iff \\(\\operatorname {hom}(F,G) = \\operatorname {hom}(F,H)\\) for all \\(F\\in \\mathcal{F}^{M}\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.19,
+                0.62,
+                0.825,
+                0.651
+            ],
+            "angle": 0,
+            "content": "b) \\(\\mathcal{F}^M\\) is maximal, i.e., for any graph \\(F \\notin \\mathcal{F}^M\\), there exists a pair of graphs \\(G, H\\) such that \\(\\chi_G^M(G) = \\chi_H^M(H)\\) and \\(\\operatorname{hom}(F, G) \\neq \\operatorname{hom}(F, H)\\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.19,
+                0.602,
+                0.825,
+                0.651
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.652,
+                0.825,
+                0.695
+            ],
+            "angle": 0,
+            "content": "Example 3.2. As a simple example, consider a maximally expressive GNN \\(M\\) that can solve the graph isomorphism problem, i.e., it computes the same representation for two graphs iff they are isomorphic. Then, \\(\\mathcal{F}^M\\) contains all graphs. This is a classic result proved in Lovász (1967)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.701,
+                0.826,
+                0.842
+            ],
+            "angle": 0,
+            "content": "The significance of homomorphism expressivity can be justified in the following aspects. First, it is a complete expressivity measure. Based on item (a), the homomorphism count within \\(\\mathcal{F}^M\\) essentially captures all information embedded in the graph representation computed by model \\(M\\). This contrasts with previously studied metrics such as the ability to compute biconnectivity properties (Zhang et al., 2023b) or count cycles (Huang et al., 2023), which only reflects restricted aspects of expressivity. Second, homomorphism expressivity is a quantitative measure and is much finer than qualitative expressivity results obtained from the graph isomorphism test. Specifically, by item (a), a GNN model \\(M_1\\) is more expressive than another model \\(M_2\\) in distinguishing non-isomorphic graphs iff \\(\\mathcal{F}^{M_2} \\subset \\mathcal{F}^{M_1}\\). Furthermore, by item (b), \\(M_1\\) is strictly more expressive than \\(M_2\\) iff \\(\\mathcal{F}^{M_2} \\subsetneq \\mathcal{F}^{M_1}\\), and the expressivity gap can be quantitatively understood via the set difference \\(\\mathcal{F}^{M_1} \\setminus \\mathcal{F}^{M_2}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.847,
+                0.825,
+                0.89
+            ],
+            "angle": 0,
+            "content": "Consequently, by deriving which graphs are encompassed in the graph family \\(\\mathcal{F}^M\\), homomorphism expressivity provides a novel way to analyze and compare the expressivity of GNN models. In the next subsection, we will give exact characterizations of \\(\\mathcal{F}^M\\) for all models \\(M\\) defined in Section 2."
+        },
+        {
+            "type": "page_footnote",
+            "bbox": [
+                0.171,
+                0.898,
+                0.825,
+                0.925
+            ],
+            "angle": 0,
+            "content": "<sup>1</sup>While homomorphism expressivity exists for all common GNNs such as the ones in Section 2, we note that it may not be well-defined for certain pathological GNNs. See Appendix F.1 for a deep discussion on it."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.505,
+                0.96
+            ],
+            "angle": 0,
+            "content": "4"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.184,
+                0.072,
+                0.312,
+                0.14
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.225,
+                0.144,
+                0.364,
+                0.157
+            ],
+            "angle": 0,
+            "content": "(a) Illustration of NED"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.319,
+                0.073,
+                0.413,
+                0.138
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.441,
+                0.072,
+                0.528,
+                0.138
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.536,
+                0.07,
+                0.622,
+                0.141
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.631,
+                0.07,
+                0.716,
+                0.141
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.725,
+                0.07,
+                0.81,
+                0.141
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.42,
+                0.144,
+                0.823,
+                0.158
+            ],
+            "angle": 0,
+            "content": "(b) Examples of endpoint-shared/strong/almost-strong/general NED"
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.171,
+                0.161,
+                0.825,
+                0.19
+            ],
+            "angle": 0,
+            "content": "Figure 1: Illustration of NED and its variants. The number \\( j \\) next to each edge indicates that the edge belongs to the ear \\( P_{j} \\). Different colors represent different ears. See Figure 9 for more examples."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.203,
+                0.322,
+                0.216
+            ],
+            "angle": 0,
+            "content": "3.2 MAIN RESULTS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.23,
+                0.825,
+                0.258
+            ],
+            "angle": 0,
+            "content": "To derive our main results, we leverage a concept in graph theory known as nested ear decomposition (NED), which is originally introduced in Eppstein (1992). Here, we adapt the definition as follows:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.261,
+                0.826,
+                0.289
+            ],
+            "angle": 0,
+            "content": "Definition 3.3. Given a graph \\( G \\), a NED \\( \\mathcal{P} \\) is a partition of the edge set \\( E_G \\) into a sequence of simple paths \\( P_1, \\dots, P_m \\) (called ears), which satisfies the following conditions:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.197,
+                0.292,
+                0.826,
+                0.32
+            ],
+            "angle": 0,
+            "content": "- Any two ears \\( P_{i} \\) and \\( P_{j} \\) with indices \\( 1 \\leq i < j \\leq c \\) do not intersect, where \\( c \\) is the number of connected components of \\( G \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.323,
+                0.826,
+                0.395
+            ],
+            "angle": 0,
+            "content": "- For each ear \\( P_{j} \\) with index \\( j > c \\), there is an ear \\( P_{i} \\) with index \\( 1 \\leq i < j \\) such that one or two endpoints of \\( P_{j} \\) lie in ear \\( P_{i} \\) (we say \\( P_{j} \\) is nested on \\( P_{i} \\)). Moreover, except for the endpoints lying in ear \\( P_{i} \\), no other vertices in \\( P_{j} \\) are in any previous ear \\( P_{k} \\) for \\( 1 \\leq k < j \\). If both endpoints of \\( P_{j} \\) lie in \\( P_{i} \\), the subpath in \\( P_{i} \\) that shares the endpoints of \\( P_{j} \\) is called the nested interval of \\( P_{j} \\) in \\( P_{i} \\), denoted as \\( I(P_{j}) \\subset P_{i} \\). If only one endpoint lies in \\( P_{i} \\), define \\( I(P_{j}) = \\emptyset \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.395,
+                0.78,
+                0.413
+            ],
+            "angle": 0,
+            "content": "- For all ears \\(P_{j}, P_{k}\\) with \\(c < j < k \\leq m\\), either \\(I(P_{j}) \\cap I(P_{k}) = \\emptyset\\) or \\(I(P_{j}) \\subset I(P_{k})\\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.197,
+                0.292,
+                0.826,
+                0.413
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.417,
+                0.825,
+                0.46
+            ],
+            "angle": 0,
+            "content": "Intuitively, Definition 3.3 states that the relation between different ears forms a forest, in that each ear is nested on its parent. Moreover, the nested intervals either do not intersect or have inclusion relations for different children of the same parent ear. We give illustrations of NED in Figure 1."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.466,
+                0.774,
+                0.481
+            ],
+            "angle": 0,
+            "content": "In this paper, we considerably extend the concept of NED to several variants defined below:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.487,
+                0.825,
+                0.517
+            ],
+            "angle": 0,
+            "content": "- Endpoint-shared NED: a NED is called endpoint-shared if all ears with non-empty nested intervals share a common endpoint (see Figure 1(b,1))."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.517,
+                0.825,
+                0.547
+            ],
+            "angle": 0,
+            "content": "- **Strong NED:** a NED is called strong if for any two children \\( P_j \\), \\( P_k \\) (\\( j < k \\)) nested on the same parent ear, we have \\( I(P_j) \\subset I(P_k) \\) (see Figure 1(b,2))."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.547,
+                0.825,
+                0.577
+            ],
+            "angle": 0,
+            "content": "- Almost-strong NED: a NED is called almost-strong if for any children \\( P_j \\), \\( P_k \\) (\\( j < k \\)) nested on the same parent ear and \\( |I(P_j)| > 1 \\), we have \\( I(P_j) \\subset I(P_k) \\) (see Figure 1(b,3))."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.198,
+                0.487,
+                0.825,
+                0.577
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.583,
+                0.476,
+                0.597
+            ],
+            "angle": 0,
+            "content": "We are now ready to present our main results:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.599,
+                0.825,
+                0.628
+            ],
+            "angle": 0,
+            "content": "Theorem 3.4. For all GNN models \\(M\\) defined in Section 2, the graph family \\(\\mathcal{F}^M\\) satisfying Definition 3.1 exists (and is unique). Moreover, each \\(\\mathcal{F}^M\\) can be separately described below:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.629,
+                0.458,
+                0.645
+            ],
+            "angle": 0,
+            "content": "- MPNN: \\(\\mathcal{F}^{\\mathrm{MP}} = \\{F:F\\) is a forest\\};"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.645,
+                0.642,
+                0.661
+            ],
+            "angle": 0,
+            "content": "- Subgraph GNN: \\(\\mathcal{F}^{\\mathrm{Sub}} = \\{F:F\\) has an endpoint-shared NED\\};"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.199,
+                0.662,
+                0.543,
+                0.677
+            ],
+            "angle": 0,
+            "content": "- Local 2-GNN: \\(\\mathcal{F}^{\\mathsf{L}} = \\{F:F\\) has a strong NED\\};"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.199,
+                0.678,
+                0.616,
+                0.694
+            ],
+            "angle": 0,
+            "content": "- Local 2-FGNN: \\(\\mathcal{F}^{\\mathrm{LF}} = \\{F:F\\) has an almost-strong NED\\};"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.199,
+                0.695,
+                0.463,
+                0.71
+            ],
+            "angle": 0,
+            "content": "- 2-FGNN: \\(\\mathcal{F}^{\\mathrm{F}} = \\{F:F\\) has a NED\\}."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.198,
+                0.629,
+                0.642,
+                0.71
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.716,
+                0.827,
+                0.926
+            ],
+            "angle": 0,
+            "content": "Theorem 3.4 gives a unified description of the homomorphism expressivity for all popular GNN models defined in Section 2. Despite the elegant conclusion, the proof process is actually involved and represents a major technical contribution, so we present a proof sketch below. Our proof is divided into three parts, presented in Appendices C.2 to C.4. First, we show the existence of \\(\\mathcal{F}^M\\) for each model \\(M\\) based on a beautiful theory developed in Dell et al. (2018). Using the technique of unfolding tree, we prove that \\(\\mathcal{F}^M\\) at least contains all graphs \\(F\\) that allow a specific type of tree decomposition (Diestel, 2017), and the homomorphism information of these graphs determines the representation of any graph \\(G\\) computed by \\(M\\) (i.e., Definition 3.1(a) holds). However, characterizing \\(\\mathcal{F}^M\\) in terms of tree decomposition is sophisticated and not intuitive for most models \\(M\\). In the next step, we give an equivalent description of \\(\\mathcal{F}^M\\) based on novel extensions of NED proposed in Definition 3.3, which is simpler and more elegant. In the last step, we prove that \\(\\mathcal{F}^M\\) does not contain other graphs. This is achieved by building non-trivial relations between three distinct theoretical tools: tree decomposition, pebble game (Cai et al., 1992), and Fürer graph (Fürer, 2001). Through a fine-grained analysis of the Fürer graphs expanded by \\(F \\notin \\mathcal{F}^M\\) (see Theorems C.47 and C.53), we show they are precisely a pair of graphs satisfying Definition 3.1(b), thus concluding the proof."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.948,
+                0.504,
+                0.96
+            ],
+            "angle": 0,
+            "content": "5"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.827,
+                0.314
+            ],
+            "angle": 0,
+            "content": "Discussions with Dvořák (2010); Dell et al. (2018). Our work significantly extends a beautiful theory developed in Dvořák (2010); Dell et al. (2018), which showed that a pair of graphs \\( G \\), \\( H \\) are indistinguishable by 1-WL iff \\( \\operatorname{hom}(F, G) = \\operatorname{hom}(F, H) \\) for all trees \\( F \\), and more generally, they are indistinguishable by \\( k \\)-FWL iff \\( \\operatorname{hom}(F, G) = \\operatorname{hom}(F, H) \\) for all graphs \\( F \\) of bounded treewidth \\( k \\). In this paper, we successfully generalize these results to a broad range of practical GNN models. Moreover, two distinct contributions are worth discussing. First, we highlight a key insight that homomorphism can serve as a fundamental expressivity measure, which has far-reaching consequences as will be elaborated in Section 4. To show that \\( \\mathcal{F}^M \\) is a valid expressivity measure, we prove an extra and non-trivial result that \\( \\mathcal{F}^M \\) is maximal (Definition 3.1(b)). Without this crucial property, \\( \\mathcal{F}^{M_1} \\supsetneq \\mathcal{F}^{M_2} \\) will not necessarily mean that model \\( M_1 \\) is strictly more expressive than \\( M_2 \\), thus preventing any quantitative comparison between models. Second, Dell et al. (2018) leveraged treewidth to describe results, which, unfortunately, cannot be applied to most GNN models studied here. Instead, we resort to the novel concept of NED, by which we successfully derive unified and elegant descriptions for all models. Moreover, as will be shown later, NED is quite flexible and can be naturally generalized to node/edge-level expressivity, which is not studied in prior work."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.319,
+                0.825,
+                0.362
+            ],
+            "angle": 0,
+            "content": "Finally, we remark that one can derive an equivalent (perhaps simpler) description of \\(\\mathcal{F}^{\\mathrm{Sub}}\\), based on the fact that a graph \\(F\\) has an endpoint-shared NED iff \\(F\\) becomes a forest when deleting the shared endpoint. Formally, denoting by \\(F \\backslash \\{u\\}\\) the induced subgraph of \\(F\\) over \\(V_F \\backslash \\{u\\}\\), we have"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.364,
+                0.585,
+                0.381
+            ],
+            "angle": 0,
+            "content": "Corollary 3.5. \\(\\mathcal{F}^{\\mathrm{Sub}} = \\{F:\\exists u\\in V_F\\) s.t. \\(F\\backslash \\{u\\}\\) is a forest}."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.393,
+                0.567,
+                0.407
+            ],
+            "angle": 0,
+            "content": "3.3 EXTENDING TO NODE/EDGE-LEVEL EXPRESSIVITY"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.419,
+                0.825,
+                0.476
+            ],
+            "angle": 0,
+            "content": "So far, this paper mainly focuses on the graph-level expressivity, i.e., what information is encoded in the graph representation. In this subsection, we extend all results in Theorem 3.4 to the more fine-grained node/edge-level expressivity by answering what information is encoded in the node/edge features of a GNN (i.e., \\(\\chi_G^M(u)\\) or \\(\\chi_G^M(u,v)\\) in Section 2). This yields the following definition:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.479,
+                0.825,
+                0.508
+            ],
+            "angle": 0,
+            "content": "Definition 3.6. The node-level homomorphism expressivity of a GNN model \\(M\\), denoted by \\(\\mathcal{F}_{\\mathrm{n}}^{M}\\), is a family of connected rooted graphs satisfying the following conditions:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.19,
+                0.51,
+                0.827,
+                0.541
+            ],
+            "angle": 0,
+            "content": "a) For any connected graphs \\(G,H\\) and vertices \\(u\\in V_G,v\\in V_H,\\chi_G^M (u) = \\chi_H^M (v)\\) iff \\(\\operatorname {hom}(F^w,G^u) = \\operatorname {hom}(F^w,H^v)\\) for all \\(F^w\\in \\mathcal{F}_{\\mathfrak{n}}^{M}\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.189,
+                0.543,
+                0.825,
+                0.574
+            ],
+            "angle": 0,
+            "content": "b) For any connected rooted graph \\( F^w \\notin \\mathcal{F}_{\\mathfrak{n}}^{M} \\), there exists a pair of connected graphs \\( G, H \\) and vertices \\( u \\in V_G, v \\in V_H \\) such that \\( \\chi_G^M(u) = \\chi_H^M(v) \\) and \\( \\operatorname{hom}(F^w, G^u) \\neq \\operatorname{hom}(F^w, H^v) \\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.189,
+                0.51,
+                0.827,
+                0.574
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.578,
+                0.825,
+                0.622
+            ],
+            "angle": 0,
+            "content": "One can similarly define the edge-level homomorphism expressivity \\(\\mathcal{F}_{\\mathrm{e}}^{M}\\) to be a family of connected rooted graphs, each marking two special vertices (we omit the definition for clarity). The following result exactly characterizes \\(\\mathcal{F}_{\\mathrm{n}}^{M}\\) and \\(\\mathcal{F}_{\\mathrm{e}}^{M}\\) for all models \\(M\\) considered in this paper:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.624,
+                0.825,
+                0.64
+            ],
+            "angle": 0,
+            "content": "Theorem 3.7. For all model \\(M\\) defined in Section 2, \\(\\mathcal{F}_{\\mathfrak{n}}^{M}\\) and \\(\\mathcal{F}_{\\mathrm{e}}^{M}\\) (except MPNN) exist. Moreover,"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.199,
+                0.642,
+                0.456,
+                0.658
+            ],
+            "angle": 0,
+            "content": "- MPNN: \\(\\mathcal{F}_{\\mathfrak{n}}^{\\mathsf{MP}} = \\{F^{w}: F \\text{ is a tree}\\}\\);"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.199,
+                0.662,
+                0.329,
+                0.676
+            ],
+            "angle": 0,
+            "content": "- Subgraph GNN:"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.199,
+                0.642,
+                0.456,
+                0.676
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.212,
+                0.675,
+                0.771,
+                0.706
+            ],
+            "angle": 0,
+            "content": "\\(\\mathcal{F}_{\\mathtt{n}}^{\\mathtt{Sub}} = \\{F^{w}:F\\) has a NED with shared endpoint \\(w\\} = \\{F^{w}:F\\backslash \\{w\\}\\) is a forest}, \\(\\mathcal{F}_{\\mathtt{e}}^{\\mathtt{Sub}} = \\{F^{wx}:F\\) has a NED with shared endpoint \\(w\\} = \\{F^{wx}:F\\backslash \\{w\\}\\) is a forest};"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.199,
+                0.708,
+                0.726,
+                0.738
+            ],
+            "angle": 0,
+            "content": "- 2-FGNN: \\(\\mathcal{F}_{\\mathfrak{n}}^{\\mathsf{F}} = \\{F^{w}: F\\) has a NED where \\(w\\) is an endpoint of the first ear}\\(, \\)\\mathcal{F}_{\\mathrm{e}}^{\\mathrm{F}} = \\{F^{wx}: F\\) has a NED where \\(w\\) and \\(x\\) are endpoints of the first ear\\}$."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.74,
+                0.825,
+                0.769
+            ],
+            "angle": 0,
+            "content": "The cases of Local 2-GNN and Local 2-FGNN are similar to 2-FGNN by replacing \"NED\" with \"strong NED\" and \"almost-strong NED\", respectively."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.775,
+                0.825,
+                0.804
+            ],
+            "angle": 0,
+            "content": "In summary, the node/edge-level homomorphism expressivity can be naturally described using NED by further specifying the endpoints of the first ear."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.817,
+                0.488,
+                0.831
+            ],
+            "angle": 0,
+            "content": "3.4 EXTENDING TO HIGHER-ORDER GNNS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.84,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "Finally, we discuss how our results can be naturally extended to higher-order GNNs, thus providing a complete picture of the homomorphism expressivity hierarchy for infinitely many architectures. We focus on three representative examples: Subgraph \\( k \\)-GNN (Qian et al., 2022), Local \\( k \\)-GNN (Morris et al., 2020), and \\( k \\)-FGNN (Azizian & Lelarge, 2021). Subgraph \\( k \\)-GNN extracts a graph \\( G^u \\) for each vertex \\( k \\)-tuple \\( u \\in V_G^k \\) and runs MPNNs independently, which recovers Subgraph GNN when \\( k = 1 \\). As the reader may have guessed, the following result exactly parallels Corollary 3.5:"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.505,
+                0.96
+            ],
+            "angle": 0,
+            "content": "6"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.825,
+                0.134
+            ],
+            "angle": 0,
+            "content": "Theorem 3.8. The homomorphism expressivity of Subgraph \\( k \\)-GNN exists and can be described as \\( \\mathcal{F}^{\\mathrm{Sub}(k)} = \\{F : \\exists U \\subset V_F \\text{ s.t. } |U| \\leq k \\text{ and } F \\backslash U \\text{ is a forest}\\} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.14,
+                0.545,
+                0.293
+            ],
+            "angle": 0,
+            "content": "We next turn to Local \\(k\\)-GNN. To describe the result, we introduce a novel extension of Definition 3.3, called the \\(k\\)-order ear. Intuitively, it is formed by a graph of no more than \\(k\\) vertices, plus \\(k\\) paths each linking to a vertex in the graph (see Figure 2(a) for an illustration). Note that a 2-order ear is exactly a simple path. Then, we can naturally define the nested \"interval\" (see the solid orange lines in Figure 2(b) for an illustration) and thus define the concept of \\(k\\)-order strong NED. Due to space limit, a formal definition is deferred to Definition E.3. We have the following main result:"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.56,
+                0.14,
+                0.822,
+                0.22
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.552,
+                0.223,
+                0.826,
+                0.28
+            ],
+            "angle": 0,
+            "content": "Figure 2: Illustration of higher-order ears. Each curve indicates a path (with possibly zero length) and each straight segment indicates an edge."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.297,
+                0.825,
+                0.327
+            ],
+            "angle": 0,
+            "content": "Theorem 3.9. The homomorphism expressivity of Local \\( k \\)-GNN exists and can be described as \\( \\mathcal{F}^{\\mathsf{L}(k)} = \\{F : F \\text{ has a } k \\text{-order strong NED}\\} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.332,
+                0.825,
+                0.39
+            ],
+            "angle": 0,
+            "content": "Finally, let us consider the standard \\(k\\)-FGNN (or equivalently, the \\(k\\)-FWL). Unfortunately, we cannot find a description of its homomorphism expressivity based on some form of higher-order NED; nevertheless, it is easy to describe the results using the notion of treewidth (see Definition C.2). Specifically, denoting \\(\\mathrm{tw}(F)\\) to be the treewidth of graph \\(F\\), we have the following result:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.393,
+                0.825,
+                0.424
+            ],
+            "angle": 0,
+            "content": "Theorem 3.10. The homomorphism expressivity of \\( k \\)-FGNN exists and can be described as \\( \\mathcal{F}^{\\mathsf{F}(k)} = \\{F : \\mathrm{tw}(F) \\leq k\\} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.435,
+                0.825,
+                0.466
+            ],
+            "angle": 0,
+            "content": "Interestingly, one can see that \\(\\mathcal{F}^{\\mathrm{Sub}(0)}\\), \\(\\mathcal{F}^{\\mathrm{L}(1)}\\), and \\(\\mathcal{F}^{\\mathrm{F}(1)}\\) all degenerate to the family of forests, which coincides with the fact that all these higher-order GNNs reduces to MPNN for the base case."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.486,
+                0.329,
+                0.501
+            ],
+            "angle": 0,
+            "content": "4 IMPLICATIONS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.517,
+                0.825,
+                0.575
+            ],
+            "angle": 0,
+            "content": "The previous section has provided a complete description of the homomorphism expressivity for a variety of GNN models. In this section, we highlight the significance of these results through three different contexts. We will show how homomorphism expressivity can be used to link different GNN subareas, provide new insights into various known results, and answer a number of open problems."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.591,
+                0.513,
+                0.605
+            ],
+            "angle": 0,
+            "content": "4.1 QUALITATIVE EXPRESSIVITY COMPARISON"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.616,
+                0.825,
+                0.644
+            ],
+            "angle": 0,
+            "content": "One direct corollary of Theorem 3.4 is that it readily enables expressivity comparison among all models in Section 2. This can be summarized below:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.648,
+                0.825,
+                0.692
+            ],
+            "angle": 0,
+            "content": "Corollary 4.1. Under the notation of Theorem 3.4, \\(\\mathcal{F}^{\\mathrm{MP}}\\subsetneq \\mathcal{F}^{\\mathrm{Sub}}\\subsetneq \\mathcal{F}^{\\mathrm{L}}\\subsetneq \\mathcal{F}^{\\mathrm{LF}}\\subsetneq \\mathcal{F}^{\\mathrm{F}}\\). Thus, the expressive power of the following GNN models strictly increases in order (in terms of distinguishing non-isomorphic graphs): MPNN, Subgraph GNN, Local 2-GNN, Local 2-FGNN, and 2-FGNN."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.699,
+                0.825,
+                0.77
+            ],
+            "angle": 0,
+            "content": "Proof. \\(\\mathcal{F}^{\\mathrm{MP}}\\subset \\mathcal{F}^{\\mathrm{Sub}}\\) follows from Corollary 3.5 and the fact that deleting any vertex of a forest yields a forest. \\(\\mathcal{F}^{\\mathrm{Sub}}\\subset \\mathcal{F}^{\\mathrm{L}}\\) follows by the fact that any endpoint-shared NED is a strong NED. \\(\\mathcal{F}^{\\mathrm{L}}\\subset \\mathcal{F}^{\\mathrm{LF}}\\subset \\mathcal{F}^{\\mathrm{F}}\\) follows similarly since any strong NED is an almost-strong NED and any almost-strong NED is a NED. To prove strict separation results, one can check that the four graphs in Figure 1(b) precisely reveal the gap between each pair of graph families, thus concluding the proof."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.778,
+                0.825,
+                0.876
+            ],
+            "angle": 0,
+            "content": "Corollary 4.1 recovers a series of results recently proved in Zhang et al. (2023a); Frasca et al. (2022). Compared to their results, our approach draws a much clearer picture of the expressivity gap between different architectures and essentially answers how large the gaps are. Moreover, we provide systematic guidance for finding counterexample graphs that unveil the expressivity gap: as shown in Corollary C.54, any graph \\( F \\in \\mathcal{F}^{M_2} \\backslash \\mathcal{F}^{M_1} \\) immediately gives a pair of non-isomorphic graphs that reveals the gap between models \\( M_1 \\) and \\( M_2 \\). We note that this readily recovers the counterexamples constructed in Zhang et al. (2023a) and greatly simplifies their sophisticated case-by-case analysis."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.882,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "We next turn to three types of higher-order GNNs studied in Section 3.4, for which we can establish a complete expressiveness hierarchy, as presented in Corollary 4.2. A graphical illustration of these results is given in Figure 3."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.504,
+                0.96
+            ],
+            "angle": 0,
+            "content": "7"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.21,
+                0.09,
+                0.787,
+                0.169
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.207,
+                0.171,
+                0.788,
+                0.185
+            ],
+            "angle": 0,
+            "content": "Figure 3: Expressiveness hierarchy of MPNN, Subgraph GNN, Local GNN, and FGNN."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.202,
+                0.736,
+                0.216
+            ],
+            "angle": 0,
+            "content": "Corollary 4.2. Under the notations in Section 3.4, for any \\( k > 0 \\), the following hold:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.19,
+                0.216,
+                0.825,
+                0.231
+            ],
+            "angle": 0,
+            "content": "a) \\(\\mathcal{F}^{\\mathrm{Sub}(k - 1)}\\subsetneq \\mathcal{F}^{\\mathrm{Sub}(k)}\\) . I.e., the expressive power of Subgraph \\(k\\) GNN strictly increases with \\(k\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.19,
+                0.231,
+                0.781,
+                0.246
+            ],
+            "angle": 0,
+            "content": "b) \\(\\mathcal{F}^{\\mathsf{L}(k)}\\subsetneq \\mathcal{F}^{\\mathsf{L}(k + 1)}\\) . I.e., the expressive power of Local \\(k\\) -GNN strictly increases with \\(k\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.19,
+                0.246,
+                0.825,
+                0.261
+            ],
+            "angle": 0,
+            "content": "c) \\(\\mathcal{F}^{\\mathrm{Sub}(k)}\\subsetneq \\mathcal{F}^{\\mathrm{L}(k + 1)}\\) . I.e., Local \\((k + 1)\\) -GNN is strictly more expressive than Subgraph \\(k\\) GNN;"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.19,
+                0.262,
+                0.825,
+                0.292
+            ],
+            "angle": 0,
+            "content": "d) \\(\\mathcal{F}^{\\mathsf{F}(k)}\\subsetneq \\mathcal{F}^{\\mathsf{L}(k + 1)}\\subsetneq \\mathcal{F}^{\\mathsf{F}(k + 1)}\\) . I.e., the expressive power of Local \\((k + 1)\\) -GNN lies strictly between \\(k\\) -FWL and \\((k + 1)\\) -FWL;"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.191,
+                0.293,
+                0.825,
+                0.337
+            ],
+            "angle": 0,
+            "content": "e) \\(\\mathcal{F}^{\\mathrm{Sub}(k)}\\subsetneq \\mathcal{F}^{\\mathrm{F}(k + 1)}\\), and for all \\(k > 1\\), \\(\\mathcal{F}^{\\mathrm{Sub}(k)}\\backslash \\mathcal{F}^{\\mathrm{F}(k + 1)}\\neq \\emptyset\\) and \\(\\mathcal{F}^{\\mathrm{F}(k + 1)}\\backslash \\mathcal{F}^{\\mathrm{Sub}(k)}\\neq \\emptyset\\). In other words, the expressive power of Subgraph \\(k\\)-GNN lies strictly within \\((k + 1)\\)-FWL, but it is incomparable to \\(k\\)-FWL when \\(k > 1\\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.19,
+                0.216,
+                0.825,
+                0.337
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.343,
+                0.827,
+                0.442
+            ],
+            "angle": 0,
+            "content": "Corollary 4.2 recovers results in Morris et al. (2020); Qian et al. (2022) and further answers two open problems. First, Corollary 4.2(c) is a new result that bridges Morris et al. (2020) with Qian et al. (2022) and partially answers an open question in Zhang et al. (2023a, Appendix C). Another new result is Corollary 4.2(d), which essentially answers a fundamental open problem raised in Frasca et al. (2022, Appendix E), showing that their proposed \\(\\mathsf{RelGN}(k)\\) model is bounded by \\(k\\)-FWL with an inherent expressivity gap (see Appendix E.4 for a detailed discussion). To sum up, all these challenging open problems become straightforward through the lens of homomorphism expressivity."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.458,
+                0.427,
+                0.471
+            ],
+            "angle": 0,
+            "content": "4.2 SUBGRAPH COUNTING POWER"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.483,
+                0.825,
+                0.571
+            ],
+            "angle": 0,
+            "content": "The significance of homomorphism expressivity can go much beyond qualitative comparisons between models. As another implication, it provides a systematic way to study GNNs' ability to encode structural information such as subgraph count, which has been found crucial in numerous practical applications. Specifically, a well-known result in graph theory states that, for any graphs \\( F, G \\), the subgraph count \\( \\operatorname{sub}(F, G) \\) can be determined by \\( \\operatorname{hom}(\\tilde{F}, G) \\) where \\( \\tilde{F} \\) ranges over all homomorphic images of \\( F \\) (i.e., \\( \\operatorname{Spasm}(F) \\), see Section 2) (Lovász, 2012; Curticapean et al., 2017)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.578,
+                0.825,
+                0.621
+            ],
+            "angle": 0,
+            "content": "Mathematically, given any graph \\( F \\), let \\( \\operatorname{Spasm}^{\\mathcal{Z}}(F) \\) be any maximal set of pairwise non-isomorphic graphs chosen from \\( \\operatorname{Spasm}(F) \\) (see Figure 4(a) for an illustration). Then, we have the following linear relation for all graph \\( G \\):"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.335,
+                0.624,
+                0.825,
+                0.66
+            ],
+            "angle": 0,
+            "content": "\\[\n\\operatorname {s u b} (F, G) = \\sum_ {\\tilde {F} \\in \\operatorname {S p a s m} ^ {\\neq} (F)} \\alpha (F, \\tilde {F}) \\cdot \\operatorname {h o m} (\\tilde {F}, G), \\tag {6}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.666,
+                0.825,
+                0.696
+            ],
+            "angle": 0,
+            "content": "where \\(\\alpha (F,\\tilde{F})\\neq 0\\) is a constant scalar coefficient independent of \\(G\\). Based on this formula, we can easily study the subgraph counting power of GNN models as shown in Proposition 4.4."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.698,
+                0.825,
+                0.77
+            ],
+            "angle": 0,
+            "content": "Definition 4.3. Given a GNN model \\(M\\), we say \\(M\\) can subgraph-count graph \\(F\\) at graph-level if \\(\\chi_G^M (G) = \\chi_H^M (H)\\) implies \\(\\operatorname {sub}(F,G) = \\operatorname {sub}(F,H)\\) for any graphs \\(G,H\\). We say \\(M\\) can subgraphcount rooted graph \\(F^{w}\\) at node-level if \\(\\chi_G^M (u) = \\chi_H^M (v)\\) implies \\(\\operatorname {sub}(F^w,G^u) = \\operatorname {sub}(F^w,H^v)\\) for any graphs \\(G,H\\) and vertices \\(u\\in V_{G},v\\in V_{H}\\). We can similarly define the edge-level subgraph counting ability for rooted graphs marking two special vertices."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.772,
+                0.827,
+                0.821
+            ],
+            "angle": 0,
+            "content": "Proposition 4.4. For any GNN model \\(M\\) defined in Section 2, it can subgraph-count graph \\(F\\) (at graph-level) if \\(\\tilde{F} \\in \\mathcal{F}^M\\) for all \\(\\tilde{F} \\in \\operatorname{Spasm}(F)\\). It can subgraph-count \\(F^w\\) (at node-level) if \\(\\tilde{F}^w \\in \\mathcal{F}_{\\mathfrak{n}}^M\\) for all \\(\\tilde{F}^w \\in \\operatorname{Spasm}(F^w)\\). A similar result holds for edge-level subgraph counting."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.825,
+                0.826,
+                0.926
+            ],
+            "angle": 0,
+            "content": "The above proposition offers a simple way to affirm the ability of a GNN model \\(M\\) to subgraph-count any pattern at graph/node/edge-level. On the other hand, one may wonder whether the converse direction also holds, i.e., \\(M\\) cannot subgraph-count \\(F\\) if there exists a homomorphic image \\(\\tilde{F} \\in \\mathsf{Spasm}(F)\\) such that \\(\\tilde{F} \\notin \\mathcal{F}^M\\). We find that it is indeed the case. Specifically, if the set \\(\\mathsf{Spasm}(F) \\backslash \\mathcal{F}^M\\) is not empty, then one can always find a pair of counterexample graphs \\(G, H\\) such that \\(\\chi_G^M(G) = \\chi_H^M(H)\\) but \\(\\mathsf{sub}(F, G) \\neq \\mathsf{sub}(F, H)\\). We eventually arrive at the following main theorem (see Appendix G.1 for a proof):"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.504,
+                0.96
+            ],
+            "angle": 0,
+            "content": "8"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.104,
+                0.825,
+                0.149
+            ],
+            "angle": 0,
+            "content": "Theorem 4.5. For any GNN model \\(M\\) such that their homomorphism expressivity \\(\\mathcal{F}^M\\) exists, \\(M\\) can subgraph-count \\(F\\) iff \\(\\operatorname{Spasm}(F) \\subset \\mathcal{F}^M\\). Similar results hold for rooted graphs \\(F^u / F^{uv}\\) by replacing \\(\\mathcal{F}^M\\) with node/edge-level homomorphism expressivity \\(\\mathcal{F}_{\\mathfrak{n}}^M / \\mathcal{F}_{\\mathfrak{e}}^M\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.15,
+                0.825,
+                0.207
+            ],
+            "angle": 0,
+            "content": "Example 4.6. As an example, we can readily characterize the cycle/path counting power of various GNNs. Denote by \\( C_n / P_n \\) the simple cycle/path of \\( n \\) vertices. Let \\( \\{u, v\\} \\in E_{C_n} \\) be any edge in \\( C_n \\), and \\( \\{w, x\\} \\in E_{P_n} \\) be any edge in \\( P_n \\) where \\( w \\) is an endpoint of \\( P_n \\). The following table lists exactly all cycles/paths each model can count at graph/node/edge-level."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.174,
+                0.212,
+                0.53,
+                0.307
+            ],
+            "angle": 0,
+            "content": "<table><tr><td rowspan=\"2\">Structure Model</td><td colspan=\"3\">Cycle</td><td colspan=\"3\">Path</td></tr><tr><td>\\( C_n \\)</td><td>\\( C_n^u \\)</td><td>\\( C_n^{uv} \\)</td><td>\\( P_n \\)</td><td>\\( P_n^w \\)</td><td>\\( P_n^{wx} \\)</td></tr><tr><td>MPNN</td><td>None</td><td>None</td><td>None</td><td>n≤3</td><td>n≤3</td><td>n≤3</td></tr><tr><td>Subgraph GNN</td><td>n≤7</td><td>n≤4</td><td>n≤4</td><td>n≤7</td><td>n≤4</td><td>n≤4</td></tr><tr><td>Local 2-GNN</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Local 2-FGNN</td><td colspan=\"3\">n≤7</td><td colspan=\"3\">n≤7</td></tr><tr><td>2-FGNN</td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.56,
+                0.209,
+                0.715,
+                0.265
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.539,
+                0.268,
+                0.832,
+                0.313
+            ],
+            "angle": 0,
+            "content": "(a) \\(\\mathsf{Spasm}^{\\neq}(C_6)\\) has 10 graphs. (b) Rooted \\(C_6\\) Figure 4: Illustration of homomorphic images of the 6-cycle and rooted 6-cycle."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.755,
+                0.209,
+                0.82,
+                0.264
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.744,
+                0.268,
+                0.831,
+                0.281
+            ],
+            "angle": 0,
+            "content": "(b) Rooted \\(C_6\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.319,
+                0.825,
+                0.432
+            ],
+            "angle": 0,
+            "content": "**Discussions with prior work.** Our results significantly extend Huang et al. (2023) in several aspects. First, we show Subgraph GNN can count 6-cycle at graph-level by simply enumerating its spasm (see Figure 4(a)). However, it cannot count rooted 5/6-cycle at node-level because the homomorphic image can contain cycles that do not pass the marked vertex (see Figure 4(b)). This provides novel insights into Huang et al. (2023) and extends their results (albeit with a simpler analysis). Second, we reveal that Local 2-GNN can already count all cycles/pathways that 2-FWL can count (even at edge-level). This identifies a new architecture with both efficiency and strong expressiveness in subgraph counting, considerably extending the finding in the concurrent work of Zhou et al. (2023b)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.437,
+                0.825,
+                0.495
+            ],
+            "angle": 0,
+            "content": "In Appendix G.2 (Tables 4 and 5), we summarize the statistics of all moderate-size patterns each model can count under homomorphisms/subgraphs, which enables quantitative expressivity comparisons of different models in a clear and exact manner. We also comprehensively list the counting ability of all moderate-size patterns in Table 6, which we believe can be helpful for future research."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.51,
+                0.416,
+                0.525
+            ],
+            "angle": 0,
+            "content": "4.3 POLYNOMIAL EXPRESSIVITY"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.536,
+                0.825,
+                0.648
+            ],
+            "angle": 0,
+            "content": "As the third implication, homomorphism expressivity is closely related to the polynomial expressivity recently proposed in Puny et al. (2023). Concretely, given a model \\(M\\), a graph \\(F\\) is in \\(\\mathcal{F}^M\\) if \\(M\\) can express the invariant graph polynomial \\(P_F\\) (defined in Puny et al. (2023), Section 2.2), and a rooted graph \\(F^{uv}\\) is in \\(\\mathcal{F}_{\\mathrm{e}}^{M}\\) if \\(M\\) can express the equivariant graph polynomial \\(P_{F^{uv}}\\). Based on this connection, our work introduces a novel toolbox for studying polynomial expressivity via the NED framework and offers new insights into which graph polynomials can be computed for a variety of practical GNNs. Moreover, we readily settle an open question in Puny et al. (2023), which upper bounds the polynomial expressivity for their proposed PPGN++."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.65,
+                0.825,
+                0.68
+            ],
+            "angle": 0,
+            "content": "Corollary 4.7. \\( PPGN++ \\) is bounded by (and thus as expressive as) the Prototypical edge-based model defined in Puny et al. (2023) for computing equivariant graph polynomials."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.687,
+                0.69,
+                0.703
+            ],
+            "angle": 0,
+            "content": "Due to space limit, please refer to Appendix H for proof and more discussions."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.722,
+                0.329,
+                0.737
+            ],
+            "angle": 0,
+            "content": "5 EXPERIMENTS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.75,
+                0.825,
+                0.863
+            ],
+            "angle": 0,
+            "content": "This section aims to verify our theory through a comprehensive set of experiments. In each experiment, we implement four types of GNN models listed in Section 2, i.e., MPNN, Subgraph GNN, Local 2-GNN, and Local 2-FGNN. Note that all of these models are much more efficient than 2-FWL. Our primary objective here is not to produce SOTA results, but rather to provide a unified and equitable empirical comparison among these models. To ensure fairness, we employ the same GIN-based design (Xu et al., 2019) for all models and control their model sizes and training budgets to be roughly the same on each task. Details of model configurations are given in Appendix I. Our code is available at https://github.com/subgraph23/homomorphism-expressivity."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.868,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "Synthetic task. We first test whether these GNN models can easily learn homomorphism information from data as our theory predicts. We use the benchmark dataset from Zhao et al. (2022a) and comprehensively test the homomorphism expressivity at graph/node/edge-level by carefully selecting 8 substructures shown in Table 1. The reported performance is measured by the normalized"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.494,
+                0.949,
+                0.504,
+                0.96
+            ],
+            "angle": 0,
+            "content": "9"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.175,
+                0.104,
+                0.508,
+                0.133
+            ],
+            "angle": 0,
+            "content": "Table 1: Experimental results on homomorphism counting. Red/blue nodes indicate marked vertices."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.179,
+                0.133,
+                0.508,
+                0.209
+            ],
+            "angle": 0,
+            "content": "<table><tr><td rowspan=\"2\">Task Model</td><td colspan=\"3\">Graph-level</td><td colspan=\"2\">Node-level</td><td colspan=\"3\">Edge-level</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>MPNN</td><td>.300</td><td>.233</td><td>.254</td><td>.505</td><td>.478</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Subgraph GNN</td><td>.011</td><td>.015</td><td>.012</td><td>.004</td><td>.058</td><td>.003</td><td>.058</td><td>.048</td></tr><tr><td>Local 2-GNN</td><td>.008</td><td>.008</td><td>.010</td><td>.003</td><td>.004</td><td>.005</td><td>.006</td><td>.008</td></tr><tr><td>Local 2-FGNN</td><td>.003</td><td>.005</td><td>.004</td><td>.005</td><td>.005</td><td>.007</td><td>.007</td><td>.008</td></tr></table>"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.513,
+                0.1,
+                0.821,
+                0.142
+            ],
+            "angle": 0,
+            "content": "Table 2: Experimental results on ZINC and Alchemy datasets. See Appendix I.4 for comparisons of more GNN models in literature."
+        },
+        {
+            "type": "table",
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+            "angle": 0,
+            "content": "<table><tr><td rowspan=\"2\">Task \nModel</td><td colspan=\"2\">ZINC</td><td rowspan=\"2\">Alchemy</td></tr><tr><td>Subset</td><td>Full</td></tr><tr><td>MPNN</td><td>.138 ± .006</td><td>.030 ± .002</td><td>.122 ± .002</td></tr><tr><td>Subgraph GNN</td><td>.110 ± .007</td><td>.028 ± .002</td><td>.116 ± .001</td></tr><tr><td>Local 2-GNN</td><td>.069 ± .001</td><td>.024 ± .002</td><td>.114 ± .001</td></tr><tr><td>Local 2-FGNN</td><td>.064 ± .002</td><td>.023 ± .001</td><td>.111 ± .001</td></tr></table>"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
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+            "angle": 0,
+            "content": "Table 3: Experimental results on the (Chordal) Cycle Counting task."
+        },
+        {
+            "type": "table",
+            "bbox": [
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+                0.237,
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+            "angle": 0,
+            "content": "<table><tr><td rowspan=\"2\">Task Model</td><td colspan=\"6\">Graph-level</td><td colspan=\"6\">Node-level</td><td colspan=\"6\">Edge-level</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>MPNN</td><td>.358</td><td>.208</td><td>.188</td><td>.146</td><td>.261</td><td>.205</td><td>.600</td><td>.413</td><td>.300</td><td>.207</td><td>.318</td><td>.237</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Subgraph GNN</td><td>.010</td><td>.020</td><td>.024</td><td>.046</td><td>.007</td><td>.027</td><td>.003</td><td>.005</td><td>.092</td><td>.082</td><td>.050</td><td>.073</td><td>.001</td><td>.003</td><td>.090</td><td>.096</td><td>.038</td><td>.065</td></tr><tr><td>Local 2-GNN</td><td>.008</td><td>.011</td><td>.017</td><td>.034</td><td>.007</td><td>.016</td><td>.002</td><td>.005</td><td>.010</td><td>.023</td><td>.004</td><td>.015</td><td>.001</td><td>.005</td><td>.010</td><td>.019</td><td>.005</td><td>.014</td></tr><tr><td>Local 2-FGNN</td><td>.003</td><td>.004</td><td>.010</td><td>.020</td><td>.003</td><td>.010</td><td>.004</td><td>.006</td><td>.012</td><td>.021</td><td>.004</td><td>.014</td><td>.003</td><td>.006</td><td>.012</td><td>.022</td><td>.005</td><td>.012</td></tr></table>"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
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+            ],
+            "angle": 0,
+            "content": "Mean Absolute Error (MAE) on the test dataset. It can be seen that the model performance indeed correlates to our theoretical predictions: (i) MPNN cannot encode any substructure under homomorphism; (ii) Subgraph GNN cannot encode the 2th, 3rd, 5th, 7th, 8th substructures; (iii) Local 2-GNN cannot encode the 3rd and 8th substructures; (iv) Local 2-FGNN can encode all substructures."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.401,
+                0.825,
+                0.527
+            ],
+            "angle": 0,
+            "content": "Cycle counting power. Cycles are important structures in numerous graph learning tasks, yet encoding them is notoriously hard for GNNs. We next test the ability of different GNN models to subgraph-count (chordal) cycles at graph/node/edge-level. We follow the setting in Frasca et al. (2022); Zhang et al. (2023a); Huang et al. (2023) and present results in Table 3 (measured by the normalized test MAE). Remarkably, despite the same computational cost and model size, Local 2-(F)GNN performs significantly better than Subgraph GNN and achieves good performance for counting all \\(3/4/5/6\\)-cycles as well as chordal \\(4/5\\)-cycles (even at edge-level). These results match Example 4.6 and may suggest Local 2-(F)GNN as generic, efficient, yet powerful architectures in solving chemical and biological tasks where counting cycles is essential (e.g., benzene rings)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.532,
+                0.825,
+                0.618
+            ],
+            "angle": 0,
+            "content": "Real-world tasks. We finally test these GNN models on three real-world benchmarks: ZINC-subset, ZINC-full (Dwivedi et al., 2020), and Alchemy (Chen et al., 2019a). Following the standard configuration, all models obey a 500K parameter budget. The results are shown in Table 2. It can be seen that the performance continues to improve when a more expressive model is used. In particular, Local 2-FGNN achieves the best performance on all tasks, suggesting that its theoretical expressivity guarantee can translate to practical performance in real-world settings."
+        },
+        {
+            "type": "title",
+            "bbox": [
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+            ],
+            "angle": 0,
+            "content": "6 CONCLUSION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.671,
+                0.827,
+                0.852
+            ],
+            "angle": 0,
+            "content": "In this paper, we present a new framework for systematically and quantitatively studying the expressive power of various GNN architectures. Through the lens of homomorphism expressivity, we give exact descriptions of the graph family each model can encode in terms of homomorphism counting. Our framework stands as a valuable toolbox to unify the landscape between different subareas in the GNN community, providing deep insights into a number of prior works and answering their open problems. In particular, one can establish a complete expressiveness hierarchy between models, determine the subgraph counting capabilities of GNNs at graph/node/edge-level, and understand their polynomial expressivity. On the theoretical side, our results establish deep connections with a series of fundamental topics in graph theory (see Appendix A.2); On the practical side, these results closely correlate with the empirical performance of GNN models, as demonstrated through extensive experiments. Finally, Appendix B outlines several open directions for further exploration, and we believe that the homomorphism expressivity framework paves a fresh way for future study of more expressive GNNs."
+        },
+        {
+            "type": "text",
+            "bbox": [
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+            "angle": 0,
+            "content": "Acknowledgement Bohang Zhang would like to thank Shengjie Luo for helpful discussions. We sincerely appreciate all reviewers for the valuable suggestions. This work is supported by National Science and Technology Major Project (2022ZD0114902) and National Science Foundation of China (NSFC62276005, NSFC62376007)."
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+        },
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+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.173,
+                0.103,
+                0.826,
+                0.528
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.491,
+                0.948,
+                0.51,
+                0.96
+            ],
+            "angle": 0,
+            "content": "16"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.131,
+                0.318,
+                0.16
+            ],
+            "angle": 0,
+            "content": "Appendix"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.173,
+                0.183,
+                0.354,
+                0.202
+            ],
+            "angle": 0,
+            "content": "Table of Contents"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.214,
+                0.21,
+                0.786,
+                0.224
+            ],
+            "angle": 0,
+            "content": "A More Related Work 18"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.235,
+                0.226,
+                0.786,
+                0.24
+            ],
+            "angle": 0,
+            "content": "A.1 Expressive Graph Neural Networks 18"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.235,
+                0.241,
+                0.786,
+                0.255
+            ],
+            "angle": 0,
+            "content": "A.2 Broader impacts and additional discussions 20"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.235,
+                0.226,
+                0.786,
+                0.255
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.214,
+                0.267,
+                0.786,
+                0.281
+            ],
+            "angle": 0,
+            "content": "B Limitations and Open Directions 21"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.214,
+                0.293,
+                0.786,
+                0.307
+            ],
+            "angle": 0,
+            "content": "C Proof of Theorem 3.4 21"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.214,
+                0.267,
+                0.786,
+                0.307
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.235,
+                0.309,
+                0.786,
+                0.323
+            ],
+            "angle": 0,
+            "content": "C.1 Preliminary 22"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.235,
+                0.324,
+                0.786,
+                0.337
+            ],
+            "angle": 0,
+            "content": "C.2 Part 1: tree decomposition 25"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.235,
+                0.338,
+                0.786,
+                0.352
+            ],
+            "angle": 0,
+            "content": "C.3 Part 2: nested ear decomposition 39"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.235,
+                0.354,
+                0.786,
+                0.368
+            ],
+            "angle": 0,
+            "content": "C.4 Part 3: pebble game 43"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.235,
+                0.309,
+                0.786,
+                0.368
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.214,
+                0.38,
+                0.786,
+                0.394
+            ],
+            "angle": 0,
+            "content": "D Node/edge-level Expressivity 50"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.235,
+                0.395,
+                0.786,
+                0.409
+            ],
+            "angle": 0,
+            "content": "D.1 Related to tree decomposition and ear decomposition 51"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.235,
+                0.41,
+                0.786,
+                0.424
+            ],
+            "angle": 0,
+            "content": "D.2 Counterexamples 52"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.235,
+                0.395,
+                0.786,
+                0.424
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.214,
+                0.437,
+                0.786,
+                0.45
+            ],
+            "angle": 0,
+            "content": "E Higher-order GNNs 54"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.235,
+                0.452,
+                0.786,
+                0.466
+            ],
+            "angle": 0,
+            "content": "E.1 Definition of higher-order GNNs 54"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.235,
+                0.467,
+                0.786,
+                0.481
+            ],
+            "angle": 0,
+            "content": "E.2 Higher-order strong NED 55"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.235,
+                0.482,
+                0.786,
+                0.495
+            ],
+            "angle": 0,
+            "content": "E.3 Proofs in Section 3.4 55"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.235,
+                0.497,
+                0.786,
+                0.511
+            ],
+            "angle": 0,
+            "content": "E.4 Expressivity gap between higher-order GNNs 56"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.235,
+                0.452,
+                0.786,
+                0.511
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.214,
+                0.524,
+                0.786,
+                0.537
+            ],
+            "angle": 0,
+            "content": "F Additional Discussions 57"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.235,
+                0.539,
+                0.786,
+                0.552
+            ],
+            "angle": 0,
+            "content": "F.1 Regarding the definition of homomorphism expressivity 57"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.235,
+                0.554,
+                0.786,
+                0.568
+            ],
+            "angle": 0,
+            "content": "F.2 Regarding the definition of NED 57"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.235,
+                0.539,
+                0.786,
+                0.568
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.214,
+                0.58,
+                0.786,
+                0.594
+            ],
+            "angle": 0,
+            "content": "G Homomorphism and Subgraph Counting Power 58"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.235,
+                0.595,
+                0.786,
+                0.608
+            ],
+            "angle": 0,
+            "content": "G.1 Proof of Theorem 4.5 58"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.235,
+                0.61,
+                0.786,
+                0.624
+            ],
+            "angle": 0,
+            "content": "G.2 Graph statistics and examples 59"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.235,
+                0.595,
+                0.786,
+                0.624
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.214,
+                0.637,
+                0.786,
+                0.651
+            ],
+            "angle": 0,
+            "content": "H Polynomial Expressivity 69"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.214,
+                0.663,
+                0.786,
+                0.677
+            ],
+            "angle": 0,
+            "content": "I Experimental Details 70"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.235,
+                0.679,
+                0.786,
+                0.691
+            ],
+            "angle": 0,
+            "content": "I.1 Datasets 70"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.235,
+                0.693,
+                0.786,
+                0.706
+            ],
+            "angle": 0,
+            "content": "I.2 Model details 70"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.235,
+                0.709,
+                0.786,
+                0.722
+            ],
+            "angle": 0,
+            "content": "I.3 Training details 72"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.235,
+                0.724,
+                0.786,
+                0.737
+            ],
+            "angle": 0,
+            "content": "I.4 Performance of baseline models in literature 72"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.235,
+                0.679,
+                0.786,
+                0.737
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.491,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "17"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.103,
+                0.409,
+                0.119
+            ],
+            "angle": 0,
+            "content": "A MORE RELATED WORK"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.134,
+                0.512,
+                0.149
+            ],
+            "angle": 0,
+            "content": "A.1 EXPRESSIVE GRAPH NEURAL NETWORKS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.16,
+                0.825,
+                0.218
+            ],
+            "angle": 0,
+            "content": "Since Morris et al. (2019); Xu et al. (2019) discovered the limited expressive power of MPNNs in distinguishing non-isomorphic graphs, a large amount of work has been devoted to developing GNNs with better expressiveness. Below, we briefly review representative approaches in this area. For a comprehensive survey on expressive GNNs, we refer readers to Morris et al. (2023)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.223,
+                0.827,
+                0.378
+            ],
+            "angle": 0,
+            "content": "Higher-order GNNs. Inspired by the relation between MPNNs and the 1-WL test, a natural approach to designing provably more expressive GNNs is to mimic the higher-order WL tests. This gives rise to two fundamental types of higher-order GNNs. One type of GNNs mimics the \\( k \\)-WL test (Grohe, 2017), and representative architectures include \\( k \\)-GNN (Morris et al., 2019) and \\( k \\)-IGN (Maron et al., 2019b;c); the other type of GNNs mimics the \\( k \\)-FWL (Folklore WL) test (Cai et al., 1992) and is referred to as the the \\( k \\)-FGNN (Maron et al., 2019a). Azizian & Lelarge (2021); Geerts & Reutter (2022) proved that each of these architectures is exactly as expressive as the corresponding higher-order WL/FWL test. Therefore, the expressiveness grows strictly as the order \\( k \\) increases; when \\( k \\) approaches infinity, they can universally approximate any continuous graph functions (Chen et al., 2019b; Keriven & Peyré, 2019). However, due to the inherent computation/memory complexity, these architectures are generally not practical in real-world applications."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.383,
+                0.827,
+                0.606
+            ],
+            "angle": 0,
+            "content": "Local GNNs. To improve computational efficiency, a subsequent line of work seeks to develop more scalable and practical GNN architectures by taking into account the local/sparse nature of graphs. Locality/sparsity is also an important inductive bias for graphs but is not well-exploited in higher-order GNNs, since their layer aggregation is inherently global and the graph adjacency information is only encoded in initial node features. To address these shortcomings, Morris et al. (2020) proposed the Local \\(k\\)-GNN (and several variants) as a replacement of \\(k\\)-GNN, which directly incorporates graph adjacency into network layers and only aggregates neighboring information instead of the global one. The authors further proved that Local \\(k\\)-GNN is strictly more expressive than \\(k\\)-GNN. Building upon Local \\(k\\)-GNN, Morris et al. (2022) proposed the \\((k,s)\\)-SpeqNet that further reduces the computational cost by considering a subset of \\(k\\)-tuples whose vertices can be grouped into no more than \\(s\\) connected components. A similar idea appeared in Zhao et al. (2022b), in which the authors proposed the \\((k,s)\\)-SetGNN by considering \\(k\\)-sets instead of \\(k\\)-tuples. Besides Local \\(k\\)-GNN, recent architectures proposed in Frasca et al. (2022) and Zhang et al. (2023a) can be analogously understood as Local \\(k\\)-IGN and Local \\(k\\)-FGNN, respectively. Very recently, Feng et al. (2023); Zhou et al. (2023b) generalized the Local \\(k\\)-FGNN to a broad class of Folklore-type GNNs and achieved good performance on several benchmark datasets."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.612,
+                0.827,
+                0.864
+            ],
+            "angle": 0,
+            "content": "Subgraph GNNs. Graphs that are indistinguishable by WL tests typically possess a high degree of symmetry. In light of this observation, Subgraph GNNs have recently emerged as a compelling approach to designing expressive GNNs. The basic idea is to break symmetry by transforming the original graph into a collection of slightly modified subgraphs and feeding these subgraphs into a GNN model. The earliest forms of Subgraph GNNs may track back to Cotta et al. (2021); Papp et al. (2021) (albeit with a different motivation), where the authors proposed to feed node-deleted subgraphs into an MPNN. Papp & Wattenhofer (2022) later argued to use node marking instead of node deletion for better expressive power, resulting in the standard Subgraph GNN studied in this paper. Zhang & Li (2021); You et al. (2021) proposed the Nested GNN and Identity-aware GNN, both of which can be treated as variants of Subgraph GNNs that use ego networks as subgraphs. In particular, the heterogeneous message passing proposed in You et al. (2021) can also be seen as a form of node marking. We note that the model proposed in Vignac et al. (2020) can also be interpreted as a Subgraph GNN. Qian et al. (2022) proposed the higher-order Subgraph GNN by marking \\( k \\) nodes per subgraph, resulting in \\( n^k \\) different subgraphs when the original graph has \\( n \\) vertices. We call this architecture Subgraph \\( k \\)-GNN in this paper. The authors proved that Subgraph \\( k \\)-GNN is strictly bounded by \\( (k + 1) \\)-FWL and is incomparable to \\( k \\)-FWL when \\( k > 1 \\). Zhou et al. (2023a) further generalized Subgraph \\( k \\)-GNN to \\( (l, k) \\)-GNN by using \\( l \\)-GNN instead of MPNN to process each subgraph. It was proved that \\( (l, k) \\)-GNN is bounded by \\( (k + l) \\)-GNN for \\( l \\geq 2 \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.868,
+                0.825,
+                0.927
+            ],
+            "angle": 0,
+            "content": "Recently, Subgraph GNNs have been greatly extended by further allowing interactions between subgraphs. This is achieved by designing cross-subgraph aggregation layers (rather than feeding each subgraph independently into a GNN). Bevilacqua et al. (2022) developed the Equivariant Subgraph Aggregation Network that introduces a global aggregation between subgraphs. A similar design is"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.491,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "18"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.104,
+                0.827,
+                0.218
+            ],
+            "angle": 0,
+            "content": "proposed in the concurrent work of Zhao et al. (2022a). These architectures were later proved to strictly improve the expressivity of the original Subgraph GNNs (Zhang et al., 2023b;a). Frasca et al. (2022) built a general design space of Subgraph GNNs that unifies prior work and showed that all models in this space are bounded by a variant of 2-IGN (dubbed the Local 2-IGN in this paper), which is then bounded by 2-FWL. Zhang et al. (2023a) later proved that Local 2-IGN is as expressive as Local 2-GNN and strictly less expressive than 2-FWL (2-FGNN). In this paper, we still use the term \"Subgraph GNN\" to refer to the original architecture in the previous paragraph, while using \"Local 2-GNN\" to refer to the general architecture in Frasca et al. (2022); Zhang et al. (2023a)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.174,
+                0.222,
+                0.827,
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+            ],
+            "angle": 0,
+            "content": "Substructure-based GNNs. Another line of work sought to develop expressive GNNs from practical considerations. In particular, Chen et al. (2020) pointed out that the ability of GNNs to detect/count graph substructures like path, cycle, and clique is crucial in numerous applications. Yet, MPNNs cannot subgraph-count any cycles/cliques. While higher-order WL tests can be more powerful in counting cycles (Fürer, 2017; Arvind et al., 2020), they suffer from substantial computational cost. As such, several works proposed to directly incorporate substructure/homomorphism counting into the node features as a preprocessing step to boost the expressiveness of MPNNs (Bouritsas et al., 2022; Barceló et al., 2021) or serve as an expressive graph kernel (Grohe, 2020; Nguyen & Maehara, 2020; Welke et al., 2023). Going beyond node features, Bodnar et al. (2021b;a); Giusti et al. (2023) further proposed a message-passing framework that enables interaction between nodes, edges, and higher-order substructures like cycles and cliques. We note that the Autobahn, TOGL, and Cy2C-GNN proposed in Thiede et al. (2021); Horn et al. (2022); Choi et al. (2022) can also be viewed as Substructure-based GNNs. However, most of the above approaches consider a fixed, predefined set of substructures rather than designing generic architectures that can learn substructures in an end-to-end fashion. Recently, Tahmasebi et al. (2023) proposed the RNP-GNN, an architecture that can count any substructure by recursively splitting a graph into a collection of vertex-marked subgraphs. We note that this design shares interesting similarities to higher-order subgraph GNNs and also the SpeqNet (Morris et al., 2022). Huang et al. (2023) proposed a generic model called \\(\\mathrm{I}^2\\)-GNN based on a variant of Subgraph 2-GNN, which can count 6-cycle at node-level. In this paper, we show the Local 2-GNN can already count all cycles/paths within 7 nodes even at edge-level while being more efficient than \\(\\mathrm{I}^2\\)-GNN (when using a similar ego network design). Finally, we remark that the polynomial expressivity proposed in Puny et al. (2023) can also be seen as a generalization of substructure counting, which further takes into account the real-valued node/edge features."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.548,
+                0.828,
+                0.744
+            ],
+            "angle": 0,
+            "content": "Distance-based GNNs. Besides structural information, distance serves as another fundamental attribute of a graph, which, again, is not captured by MPNNs and the 1-WL test. Li et al. (2020) first proposed to improve the expressive power of GNNs by augmenting node features with Distance Encoding (DE). Related to DE, another approach to injecting distance information is the \\(k\\)-hop MPNN, which aggregates \\(k\\)-hop neighbors in a message-passing layer (Feng et al., 2022; Abboud et al., 2022; Wang et al., 2023). Feng et al. (2022); Zhang et al. (2023a) proved that the expressive power of \\(k\\)-hop MPNN is strictly bounded by 2-FWL. Distance can also be naturally incorporated in Graph Transformers through relative positional encoding, yielding the Graphormer architecture that has achieved remarkable performance across various benchmarks (Ying et al., 2021). Recently, Zhang et al. (2023b) built an interesting connection between distance and biconnectivity properties, showing that distance-enhanced GNNs can detect cut vertices and cut edges of a graph. This provides insights into the practical superiority of these models as biconnectivity is closely linked to real applications in chemistry and social network analysis. Zhang et al. (2023a) proved that Local 2-GNN can provably encode the distance (and thus biconnectivity) of a graph."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.75,
+                0.828,
+                0.849
+            ],
+            "angle": 0,
+            "content": "Spectral-based GNNs. Graph spectra are also a class of fundamental properties and have long been used to design GNN models (Bruna et al., 2014; Defferrard et al., 2016). Balcilar et al. (2021b;a) showed that designing GNNs in the spectral domain can easily break the 1-WL expressivity. For Graph Transformers, Kreuzer et al. (2021); Dwivedi & Bresson (2020); Dwivedi et al. (2022) proposed to incorporate the spectra of the graph Laplacian matrix to boost the expressive power beyond the 1-WL test. Lim et al. (2023) further designed a principled equivariant architecture that takes the Laplacian eigenvalues and eigenvectors as inputs, which generalizes prior work."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.854,
+                0.828,
+                0.927
+            ],
+            "angle": 0,
+            "content": "Other approaches. Murphy et al. (2019); Chen et al. (2020) proposed Relational Pooling as a general approach to designing expressive GNN architectures, whose basic idea is to implement a permutation-invariant GNN by symmetrizing permutation-sensitive base models. Wijesinghe & Wang (2022b) proposed the GraphSNN, which improves the expressive power of MPNNs by using more distinguishing edge features. Specifically, each edge feature encodes the structure of the over-"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.491,
+                0.948,
+                0.511,
+                0.961
+            ],
+            "angle": 0,
+            "content": "19"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.825,
+                0.149
+            ],
+            "angle": 0,
+            "content": "lap subgraph of two 1-hop ego networks centered on the two endpoints of the edge. Very recently, Dimitrov et al. (2023) designed a GNN model that can distinguish all planar graphs, thus achieving a strong expressivity in chemical applications since molecular graphs are often planar."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.163,
+                0.579,
+                0.178
+            ],
+            "angle": 0,
+            "content": "A.2 BROADER IMPACTS AND ADDITIONAL DISCUSSIONS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.188,
+                0.827,
+                0.355
+            ],
+            "angle": 0,
+            "content": "Broader impact in graph theory. Due to the fundamental nature of GNN architectures studied in this paper, our theoretical results may potentially have broader impacts on the graph theory community. Specifically, we study the color refinement (CR) algorithms corresponding to four types of (higher-order) GNNs: Subgraph \\((k - 1)\\)-GNN, Local \\(k\\)-GNN, Local \\(k\\)-FGNN, and \\(k\\)-FGNN. All algorithms can be seen as natural extensions of the 1-WL test since they all reduce to 1-WL when \\(k = 1\\). In the graph theory community, Subgraph GNN has another name known as the vertex-individualized CR algorithm, which appears widely in literature (Babai, 2016; Rattan & Seppelt, 2023; Neuen & Schweitzer, 2018) and has become part of the core algorithm for fast graph isomorphism testing software (e.g., McKay & Piperno, 2014). On the other hand, Local \\(k\\)-GNN and Local \\(k\\)-FGNN are surprisingly related to the guarded logic (Barceló et al., 2020; De Rijke, 2000; Baader, 2003), since the aggregations are purely local (guarded by the edge). From this perspective, these CR algorithms can be seen as natural extensions of guarded logic in higher-order scenarios."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.362,
+                0.825,
+                0.432
+            ],
+            "angle": 0,
+            "content": "Besides these CR algorithms, our new extensions of NED may also have implications in graph theory. In particular, the strong NED (as well as its higher-order version) is elegant and may serve as a descriptive tool to characterize certain graph families. In addition, we establish intrinsic connections between NED and tree decomposition, which may have value in understanding other graph topics related to tree decomposition."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.438,
+                0.827,
+                0.593
+            ],
+            "angle": 0,
+            "content": "Finally, to our knowledge, the node/edge-level homomorphism and the corresponding subgraph counting abilities of different CR algorithms do not seem to have been systematically investigated before. Whereas in this paper, all graph/node/edge-level expressivity is studied in a unified manner. To achieve this, we introduce additional proof techniques which we believe may facilitate future study in related areas. For example, the original technique for constructing counterexample graphs satisfying Definition 3.1(b) does not apply to node/edge-level settings, since they are no longer counterexample graphs satisfying Definition 3.6(b) (no matter which vertices \\( u \\in V_G, v \\in V_H \\) are marked). To address the problem, we propose clique-augmented Fürer graphs, a novel class of counterexample graphs that extend several prior works (e.g., Cai et al., 1992; Fürer, 2001), and conduct a fine-grained analysis of their automorphism property (see Appendix D.2). We believe this new technique can be used to generalize other results from graph-level to node/edge-level settings."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.598,
+                0.825,
+                0.738
+            ],
+            "angle": 0,
+            "content": "Discussions with Barceló et al. (2021). In the GNN community, Barceló et al. (2021) first proposed to incorporate the homomorphism count of predefined substructures into node features as an approach to enhancing the expressivity of MPNNs. They systematically investigated the questions of what substructures are useful and how the homomorphism information of these substructures can boost the model expressivity to even break out \\( k \\)-FWL. Yet, they only gave a partial (incomplete) characterization of the substructures that can be counted by the specific \\( \\mathcal{F} \\)-MPNN architecture and did not answer what substructures cannot be encoded, whereas our paper fully addresses both questions for a variety of popular GNN models. Note that these aspects are crucial to ensure that homomorphism expressivity is well-defined. As such, our paper first identifies that homomorphism expressivity is a complete, quantitative expressivity measure to compare different GNN models."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.744,
+                0.827,
+                0.925
+            ],
+            "angle": 0,
+            "content": "Discussions with the concurrent work of Neuen (2023). After the initial submission, we became aware of a concurrent work (Neuen, 2023), which proved that \\( k \\)-FWL cannot count any graph with treewidth larger than \\( k \\) under homomorphism. In our context, this result exactly shows that Definition 3.1(b) is satisfied, and thus the homomorphism expressivity of \\( k \\)-FWL is well-defined. Notably, their construction of counterexample graphs is also based on Fürer graphs. Nevertheless, the proof technique between the two works is quite different: the proof in Neuen (2023) is built upon a key concept called oddomorphism, while our proof is based on the relation between tree decomposition and the simplified pebble game developed in Zhang et al. (2023a). It is essential to underscore that our results and proof technique are more general and go beyond the standard \\( k \\)-FWL, in that (i) it applies to a broad range of color refinement algorithms related to practical GNN architectures and (ii) it further extends to node/edge-level homomorphism expressivity. Our theoretical results thus strictly incorporate the results in Neuen (2023). The approach in Neuen (2023) (based on oddomorphism) cannot be easily generalized to these settings."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.51,
+                0.961
+            ],
+            "angle": 0,
+            "content": "20"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.103,
+                0.526,
+                0.119
+            ],
+            "angle": 0,
+            "content": "B LIMITATIONS AND OPEN DIRECTIONS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.136,
+                0.825,
+                0.166
+            ],
+            "angle": 0,
+            "content": "There are still several open questions that are not fully explored in this paper. We list them below as promising directions for future study."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.171,
+                0.827,
+                0.298
+            ],
+            "angle": 0,
+            "content": "Existence of homomorphism expressivity for refinement-based GNN architectures. In this paper, we prove that homomorphism expressivity exists for a wide range of architectures defined in Section 2. On the other hand, we also note that it may not be well-defined for certain pathological GNNs, as illustrated in Appendix F.1. Given this observation, a fundamental question is: what conditions can guarantee that the homomorphism expressivity of a GNN exists? Here, we hypothesize that a very mild condition can suffice. Specifically, we conjecture that as long as a GNN architecture is defined following a general form of color refinement procedure that outputs stable color mappings, its homomorphism expressivity always exists. We leave this conjecture as an important open problem for future study."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.304,
+                0.825,
+                0.416
+            ],
+            "angle": 0,
+            "content": "Regarding higher-order Local FGNN. This paper characterizes the homomorphism expressivity for three classes of higher-order GNNs: Subgraph \\( k \\)-GNN, Local \\( k \\)-GNN, and \\( k \\)-FGNN. In particular, we introduce the \\( k \\)-order ear and \\( k \\)-order strong NED as a way to describe the homomorphism expressivity of Local \\( k \\)-GNN. However, it remains unclear how to give a simple description of the homomorphism expressivity for Local \\( k \\)-FGNN that can generalize the concept of almost-strong NED for \\( k = 2 \\). As such, our current expressiveness hierarchy (Figure 3) does not support Local \\( k \\)-FGNN yet. We leave this as an open problem and make the following conjecture below. We note that a similar open question has been informally raised in Zhang et al. (2023a)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.421,
+                0.827,
+                0.451
+            ],
+            "angle": 0,
+            "content": "Conjecture B.1. For all \\( k \\geq 2 \\), Local \\( k \\)-FGNN is strictly more expressive than Local \\( k \\)-GNN and strictly less expressive than \\( k \\)-FGNN."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.463,
+                0.827,
+                0.548
+            ],
+            "angle": 0,
+            "content": "Expressivity gap between Local 2-(F)GNN and 2-FGNN in practical aspects. We have proved that 2-FGNN is strictly more expressive than Local 2-(F)GNN. However, from a practical perspective, we surprisingly find that the subgraph counting ability of Local 2-(F)GNN matches that of 2-FGNN for all structures within a moderate size (see Table 5), although the former is much more efficient. This leads to the intriguing question of what fundamental gaps exist between the two models in practical aspects, or is the efficiency gain free?"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.554,
+                0.827,
+                0.709
+            ],
+            "angle": 0,
+            "content": "Other architectures. In this paper, we comprehensively study a variety of popular GNN architectures ranging from Subgraph GNNs and Local GNNs to higher-order GNNs, and further link these architectures to Substructure-based GNNs (see Appendix A.1). Yet, we still do not cover all popular GNN architectures, such as the GSWL-based Subgraph GNN (Zhang et al., 2023a; Bevilacqua et al., 2022), SpeqNet (Morris et al., 2022), and \\(\\mathrm{I}^2\\)-GNN (Huang et al., 2023). Moreover, the classes of Distance-based GNNs and Spectral-based GNNs (Appendix A.1) are also widely used in practice, which deserve future study. We would like to raise the question of characterizing the homomorphism expressivity of Distance-based GNNs and Spectral-based GNNs as an important open question. In this way, one can gain deep insights into these models' true expressivity and enable quantitative comparisons between all mainstream architectures. Moreover, it will become clear to what extent other GNN models can encode distance and spectral information about a graph."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.731,
+                0.418,
+                0.747
+            ],
+            "angle": 0,
+            "content": "C PROOF OF THEOREM 3.4"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.765,
+                0.822,
+                0.78
+            ],
+            "angle": 0,
+            "content": "This section gives the proof of the main theorem. For ease of reading, we first restate Theorem 3.4:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.786,
+                0.825,
+                0.816
+            ],
+            "angle": 0,
+            "content": "Theorem 3.4. For all GNN models \\(M\\) defined in Section 2, the graph family \\(\\mathcal{F}^M\\) satisfying Definition 3.1 exists (and is unique). Moreover, each \\(\\mathcal{F}^M\\) can be separately described below:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.197,
+                0.82,
+                0.642,
+                0.837
+            ],
+            "angle": 0,
+            "content": "- Subgraph GNN: \\(\\mathcal{F}^{\\mathrm{Sub}} = \\{F:F\\) has an endpoint-shared NED\\};"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.843,
+                0.543,
+                0.859
+            ],
+            "angle": 0,
+            "content": "- Local 2-GNN: \\(\\mathcal{F}^{\\mathsf{L}} = \\{F:F\\) has a strong NED\\};"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.865,
+                0.616,
+                0.881
+            ],
+            "angle": 0,
+            "content": "- Local 2-FGNN: \\(\\mathcal{F}^{\\mathrm{LF}} = \\{F:F\\) has an almost-strong NED\\};"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.887,
+                0.463,
+                0.903
+            ],
+            "angle": 0,
+            "content": "- 2-FGNN: \\(\\mathcal{F}^{\\mathrm{F}} = \\{F:F\\) has a NED\\}."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.197,
+                0.82,
+                0.642,
+                0.903
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.91,
+                0.822,
+                0.926
+            ],
+            "angle": 0,
+            "content": "For MPNN, since it is a special case of Subgraph \\(k\\)-GNN, the proof can be found in Appendix E.3."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "21"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.104,
+                0.318,
+                0.119
+            ],
+            "angle": 0,
+            "content": "C.1 PRELIMINARY"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.13,
+                0.825,
+                0.215
+            ],
+            "angle": 0,
+            "content": "Additional notations and concepts. Besides the notations defined in Section 2, we further define the following notations. We use the symbol \\(\\mathcal{G}\\) to denote the set of all finite, simple, undirected, labeled graphs. Let \\(G = (V_G,E_G,\\ell_G)\\) be a graph in \\(\\mathcal{G}\\). When the label is the same for all vertices, we can omit the term \\(\\ell_G\\) and write \\(G = (V_G,E_G)\\). Given vertex \\(u\\in V_{G}\\), denote the degree of \\(u\\) as \\(\\deg_G(u) = |N_G(u)|\\), and denote the closed neighborhood of \\(u\\) as \\(N_G[u] = N_G(u)\\cup \\{u\\}\\). The shortest path distance between vertices \\(u\\) and \\(v\\) is denoted by \\(\\mathrm{dis}_G(u,v)\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.22,
+                0.827,
+                0.363
+            ],
+            "angle": 0,
+            "content": "Given a vertex set \\( S \\subset V_G \\), the induced subgraph of \\( G \\) over \\( S \\), denoted as \\( G[S] \\), is the subgraph of \\( G \\) with vertex set \\( S \\) and edge set \\( \\{\\{u, v\\} \\in E_G : u, v \\in S\\} \\). Similarly, without abuse of notation, given an edge set \\( R \\subset E_G \\), the induced subgraph of \\( G \\) over \\( R \\), denoted as \\( G[R] \\), is the subgraph of \\( G \\) with vertex set \\( \\bigcup_{\\{u, v\\} \\in R} \\{u, v\\} \\) and edge set \\( R \\). Given a vertex tuple \\( \\mathbf{u} = (u_1, \\dots, u_k) \\), denote by \\( G^{\\mathbf{u}} = G^{u_1, \\dots, u_k} \\) the rooted graph obtained from \\( G \\) by marking vertices \\( u_1, \\dots, u_k \\). The atomic type of \\( G \\) over \\( \\mathbf{u} \\), denoted by \\( \\mathrm{atp}_G(\\mathbf{u}) \\), is a \\( k \\times k \\) matrix where the element at position \\( (i, j) \\) is the tuple \\( (\\mathbb{I}[u_i = u_j], \\mathbb{I}[\\{u_i, u_j\\} \\in E_G]) \\). Given two graphs \\( G, H \\), the graph union \\( G \\cup H \\) is defined to be the graph \\( (V_G \\cup V_H, E_G \\cup E_H, \\ell_{G \\cup H}) \\), where \\( \\ell_{G \\cup H}(u) := \\ell_G(u) \\) for all \\( u \\in V_G \\) and \\( \\ell_{G \\cup H}(u) := \\ell_H(u) \\) for all \\( u \\in V_H \\). It is well-defined iff \\( \\ell_G(u) = \\ell_H(u) \\) for all \\( u \\in V_G \\cap V_H \\). Finally, we use the notation \\( G \\simeq H \\) to denote that \\( G \\) and \\( H \\) are isomorphic graphs."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.369,
+                0.825,
+                0.455
+            ],
+            "angle": 0,
+            "content": "A graph \\( T = (V_T, E_T, \\ell_T) \\) is called a tree if it does not contain cycles. Let \\( T^r \\) be a rooted tree where \\( r \\) is the root vertex. For each vertex \\( t \\in V_T \\), define its depth \\( \\mathsf{dep}_{T^r}(t) := \\mathsf{dis}_T(t, r) \\) to be the distance to the root, and denote by \\( \\mathsf{Desc}_{T^r}(t) \\) the set of descendants of \\( t \\), namely, \\( s \\in \\mathsf{Desc}_{T^r}(t) \\) iff \\( \\mathsf{dep}_{T^r}(s) = \\mathsf{dep}_{T^r}(t) + \\mathsf{dis}_T(t, s) \\). For each \\( t \\in V_T \\setminus \\{r\\} \\), denote by \\( \\mathsf{pa}_{T^r}(t) \\) the parent vertex of \\( t \\), i.e., the unique vertex \\( s \\in N_T(t) \\) satisfying \\( \\mathsf{dep}_{T^r}(t) = \\mathsf{dep}_{T^r}(s) + 1 \\). Define the subtree of \\( T^r \\) rooted at node \\( t \\) by \\( T^r[t] \\), which is exactly the induced subgraph \\( T[\\mathsf{Desc}_{T^r}(t)]^t \\) with root \\( t \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.46,
+                0.825,
+                0.488
+            ],
+            "angle": 0,
+            "content": "Tree decomposition. Our proof is based on a central concept in graph theory, called tree decomposition. It can be formally defined below:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.491,
+                0.825,
+                0.521
+            ],
+            "angle": 0,
+            "content": "Definition C.1 (Tree decomposition). Given a graph \\( G = (V_G, E_G, \\ell_G) \\), its tree decomposition is a tree \\( T = (V_T, E_T, \\beta_T) \\), where the label function \\( \\beta_T : V_T \\to 2^{V_G} \\) satisfies the following conditions:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.191,
+                0.526,
+                0.822,
+                0.553
+            ],
+            "angle": 0,
+            "content": "a) Each tree node \\( t \\in V_T \\) is associated to a non-empty subset of vertices \\( \\beta_T(t) \\subset V_G \\) in \\( G \\), called a bag. We say tree node \\( t \\) contains vertex \\( u \\) if \\( u \\in \\beta_T(t) \\);"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.19,
+                0.559,
+                0.822,
+                0.586
+            ],
+            "angle": 0,
+            "content": "b) For each edge \\(\\{u,v\\} \\in V_G\\) , there exists at least one tree node \\(t\\in V_T\\) that contains the edge, i.e., \\(\\{u,v\\} \\subset \\beta_T(t)\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.19,
+                0.592,
+                0.822,
+                0.62
+            ],
+            "angle": 0,
+            "content": "c) For each vertex \\( u \\in V_G \\), all tree nodes \\( t \\) containing \\( u \\) form a (non-empty) connected subtree. Formally, denoting \\( B_T(u) = \\{t \\in V_T : u \\in \\beta_T(t)\\} \\), then \\( T[B_T(u)] \\) is connected."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.19,
+                0.526,
+                0.822,
+                0.62
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.627,
+                0.72,
+                0.643
+            ],
+            "angle": 0,
+            "content": "If \\( T \\) is a tree decomposition of \\( G \\), we call the pair \\( (G, T) \\) a tree-decomposed graph."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.653,
+                0.825,
+                0.682
+            ],
+            "angle": 0,
+            "content": "We remark that given a graph \\( G \\), there are multiple ways to decompose it and thus its tree decomposition is not unique. Several examples of tree decomposition are given in Figure 5."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.685,
+                0.825,
+                0.728
+            ],
+            "angle": 0,
+            "content": "Definition C.2 (Treewidth). The width of a tree decomposition is defined as one less than the maximum bag size, i.e., \\(\\max_{t\\in T}|\\beta_T(t)| - 1\\). The treewidth of a graph \\(G\\), denoted as \\(\\mathrm{tw}(G)\\), is the minimum positive integer \\(k\\) such that there exists a tree decomposition of width \\(k\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.738,
+                0.533,
+                0.752
+            ],
+            "angle": 0,
+            "content": "Some important facts about treewidth are listed below:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.756,
+                0.49,
+                0.771
+            ],
+            "angle": 0,
+            "content": "Fact C.3. For any graph \\( G \\), the following hold:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.197,
+                0.777,
+                0.822,
+                0.804
+            ],
+            "angle": 0,
+            "content": "- The treewidth of \\( G \\) is at most \\( |V_G| - 1 \\), i.e., a trivial tree decomposition that only has one node \\( t \\) and \\( \\beta_T(t) = V_G \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.81,
+                0.449,
+                0.824
+            ],
+            "angle": 0,
+            "content": "- \\(\\operatorname{tw}(G) = |V_G| - 1\\) iff \\(G\\) is a clique."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.199,
+                0.829,
+                0.396,
+                0.843
+            ],
+            "angle": 0,
+            "content": "- \\(\\operatorname{tw}(G) = 1\\) iff \\(G\\) is a forest."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.197,
+                0.777,
+                0.822,
+                0.843
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.855,
+                0.827,
+                0.926
+            ],
+            "angle": 0,
+            "content": "The above definition of tree decomposition is quite flexible without constraints on the structure of the tree or the size of each bag. Below, we define several restricted variants of tree decomposition, which (we will later see) are closely related to the GNN architectures studied in this paper. To begin with, we first define a general concept that slight modifies the original definition (Definition C.1) such that the tree becomes rooted and each bag is a multiset of vertices rather than a set."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "22"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.182,
+                0.087,
+                0.825,
+                0.31
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.352,
+                0.32,
+                0.644,
+                0.336
+            ],
+            "angle": 0,
+            "content": "Figure 5: Illustration of tree decomposition."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.359,
+                0.825,
+                0.39
+            ],
+            "angle": 0,
+            "content": "Definition C.4 (Canonical tree decomposition). Given a graph \\( G = (V_G, E_G, \\ell_G) \\), a canonical tree decomposition of width \\( k \\) is a rooted tree \\( T^r = (V_T, E_T, \\beta_T) \\) satisfying the following conditions:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.191,
+                0.395,
+                0.577,
+                0.41
+            ],
+            "angle": 0,
+            "content": "a) The depth of \\(T\\) is even, i.e. \\(\\max_{t\\in V_T}\\mathsf{dep}_{T^r}(t)\\) is even;"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.19,
+                0.413,
+                0.825,
+                0.442
+            ],
+            "angle": 0,
+            "content": "b) Each tree node \\( t \\in V_T \\) is associated to a multiset of vertices \\( \\beta_T(t) \\subset V_G \\), called a bag. Moreover, \\( |\\beta_T(t)| = k \\) if \\( \\mathsf{dep}_{T^r}(t) \\) is even and \\( |\\beta_T(t)| = k + 1 \\) if \\( \\mathsf{dep}_{T^r}(t) \\) is odd;"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.19,
+                0.446,
+                0.825,
+                0.474
+            ],
+            "angle": 0,
+            "content": "c) For all tree edges \\(\\{s,t\\} \\in E_T\\) where \\(\\mathrm{dep}_{T^r}(s)\\) is even and \\(\\mathrm{dep}_{T^r}(t)\\) is odd, \\(\\beta_{T}(s)\\subset \\beta_{T}(t)\\) (where “\\(\\subset\\)” denotes the multiset inclusion relation);"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.191,
+                0.479,
+                0.588,
+                0.493
+            ],
+            "angle": 0,
+            "content": "d) The conditions (b) and (c) in Definition C.1 are satisfied."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.19,
+                0.395,
+                0.825,
+                0.493
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.505,
+                0.825,
+                0.547
+            ],
+            "angle": 0,
+            "content": "As examples, one can check that the tree decomposition of all graphs in Figure 5(b) is canonical, but the tree decomposition in Figure 5(a) is not. An important observation about canonical tree decomposition is shown below:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.551,
+                0.825,
+                0.581
+            ],
+            "angle": 0,
+            "content": "Proposition C.5. Let \\((F, T^r)\\) be any tree-decomposed graph where \\(T^r\\) is a canonical tree decomposition of \\(F\\). For any vertices \\(u, v \\in V_F\\), either of the following holds:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.197,
+                0.586,
+                0.801,
+                0.601
+            ],
+            "angle": 0,
+            "content": "- \\( u \\) and \\( v \\) are in the same bag of \\( T^r \\), i.e., there is a node \\( t \\in V_T \\) such that \\( \\{\\{u, v\\}\\} \\subset \\beta_T(t) \\);"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.607,
+                0.602,
+                0.623
+            ],
+            "angle": 0,
+            "content": "- The induced subgraph \\( T[B_T(u) \\cup B_T(v)] \\) is disconnected."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.197,
+                0.586,
+                0.801,
+                0.623
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.637,
+                0.827,
+                0.737
+            ],
+            "angle": 0,
+            "content": "Proof. Assume that \\( u \\) and \\( v \\) are not in the same bag, i.e., \\( B_{T}(u) \\cap B_{T}(v) = \\emptyset \\). Pick \\( s \\in B_{T}(u) \\) and \\( t \\in B_{T}(v) \\) such that \\( \\mathsf{dep}_{T^r}(s) \\) and \\( \\mathsf{dep}_{T^r}(t) \\) are minimal, respectively. Without loss of generality, assume that \\( \\mathsf{dep}_{T^r}(s) \\leq \\mathsf{dep}_{T^r}(t) \\). Then, \\( t \\neq r \\) is not the root node and thus we can pick its parent \\( \\mathsf{pa}_{T^r}(t) \\). It follows that \\( \\mathsf{dep}_{T^r}(t) \\) is odd and \\( \\beta_{T}(\\mathsf{pa}_{T^{r}}(t)) \\subset \\beta_{T}(t) \\) by definition of canonical tree decomposition. Therefore, \\( u \\notin \\beta_{T}(\\mathsf{pa}_{T^{r}}(t)) \\). Moreover, by the assumption that \\( \\mathsf{dep}_{T^r}(s) \\leq \\mathsf{dep}_{T^r}(t) \\), any node in \\( B_{T}(u) \\) is not a descendant of \\( t \\). We thus conclude that there does not exist a tree edge such that the two endpoints contain \\( u \\) and \\( v \\), respectively."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.751,
+                0.744,
+                0.765
+            ],
+            "angle": 0,
+            "content": "Now we are ready to define several restricted variants of canonical tree decomposition:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.768,
+                0.822,
+                0.784
+            ],
+            "angle": 0,
+            "content": "Definition C.6. Define four families of tree-decomposed graphs \\(S^{\\mathrm{Sub}}\\), \\(S^{\\mathrm{L}}\\), \\(S^{\\mathrm{LF}}\\), and \\(S^{\\mathrm{F}}\\) as follows:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.191,
+                0.789,
+                0.654,
+                0.806
+            ],
+            "angle": 0,
+            "content": "a) \\((F,T^{r})\\in S^{\\mathsf{F}}\\) iff \\((F,T^{r})\\) satisfies Definition C.4 with width \\(k = 2\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.19,
+                0.811,
+                0.826,
+                0.855
+            ],
+            "angle": 0,
+            "content": "b) \\((F,T^{r})\\in S^{\\mathrm{LF}}\\) iff \\((F,T^{r})\\) satisfies Definition C.4 with width \\(k = 2\\), and for any tree node \\(t\\) of odd depth, it has only one child if \\(w\\notin \\{v:v\\in N_G[u],u\\in \\beta_T(s)\\}\\) where \\(s\\) is the parent node of \\(t\\) and \\(w\\) is the unique vertex in \\(\\beta_T(t)\\backslash \\beta_T(s)\\);"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.191,
+                0.86,
+                0.825,
+                0.89
+            ],
+            "angle": 0,
+            "content": "c) \\((F,T^r)\\in S^{\\mathsf{L}}\\) iff \\((F,T^{r})\\) satisfies Definition C.4 with width \\(k = 2\\), and any tree node \\(t\\) of odd depth has only one child;"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.19,
+                0.895,
+                0.825,
+                0.925
+            ],
+            "angle": 0,
+            "content": "d) \\((F, T^r) \\in S^{\\mathrm{Sub}}\\) iff \\((F, T^r)\\) satisfies Definition C.4 with width \\(k = 2\\), and there exists a vertex \\(u \\in V_G\\) such that \\(u \\in \\beta_T(t)\\) for all \\(t \\in V_T\\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.19,
+                0.789,
+                0.826,
+                0.925
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "23"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.744,
+                0.12
+            ],
+            "angle": 0,
+            "content": "Examples of tree-decomposed graphs in the four families are illustrated in Figure 5(b)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.126,
+                0.748,
+                0.141
+            ],
+            "angle": 0,
+            "content": "Before closing this subsection, we define several notations for tree-decomposed graphs:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.143,
+                0.825,
+                0.174
+            ],
+            "angle": 0,
+            "content": "Definition C.7. Given canonical tree-decomposed graph \\((G,T^r)\\) and node \\(t\\in V_T\\) , denote by \\(G[T^{r}[t]]\\) the subgraph of \\(G\\) induced by the vertex set \\(\\{u:u\\in \\beta_{T^r}(s),s\\in \\mathsf{Desc}_{T^r}(t)\\}\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.177,
+                0.827,
+                0.223
+            ],
+            "angle": 0,
+            "content": "Definition C.8. Given two canonical tree-decomposed graphs \\((G, T^r)\\) and \\((\\tilde{G}, \\tilde{T}^s)\\), a pair of mappings \\((\\rho, \\tau)\\) is called an isomorphism from \\((G, T^r)\\) to \\((\\tilde{G}, \\tilde{T}^s)\\), denoted by \\((G, T^r) \\simeq (\\tilde{G}, \\tilde{T}^s)\\), if the following hold:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.191,
+                0.23,
+                0.442,
+                0.246
+            ],
+            "angle": 0,
+            "content": "a) \\(\\rho\\) is an isomorphism from \\(G\\) to \\(\\tilde{G}\\);"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.191,
+                0.253,
+                0.581,
+                0.269
+            ],
+            "angle": 0,
+            "content": "b) \\(\\tau\\) is an isomorphism from \\(T^r\\) to \\(\\tilde{T}^s\\) (ignoring labels \\(\\beta\\));"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.191,
+                0.275,
+                0.468,
+                0.291
+            ],
+            "angle": 0,
+            "content": "c) For any \\(t\\in T^r\\) \\(\\rho (\\beta_T(t)) = \\beta_{\\bar{T}}(\\tau (t))\\)"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.191,
+                0.23,
+                0.581,
+                0.291
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.3,
+                0.827,
+                0.425
+            ],
+            "angle": 0,
+            "content": "Equivalent formulation of GNN architectures. Below, we give equivalent definitions for several GNN architectures presented in Section 2, which will be used in subsequent analysis. Let \\(G\\) be a graph and \\(u,v\\in V_G\\). For all models \\(M\\) including Subgraph GNN, Local 2-GNN, Local 2-FGNN, and 2-FGNN, we define the initial color \\(\\tilde{\\chi}_G^{M,(0)}(u,v)\\) to be the isomorphism type of vertex pair \\((u,v)\\) (i.e., \\(\\mathsf{atp}_G(u,v)\\) plus labels of each vertex). Note that this matches the original definition except for Subgraph GNN. Then in each iteration \\(t\\), the color is updated according to the following formula. Here, for clarity, we denote \\(\\tilde{\\chi}_G^{M,(t)}(u,S) = \\{\\{\\tilde{\\chi}_G^{M,(t)}(u,v):v\\in S\\}\\}\\) and \\(\\tilde{\\chi}_G^{M,(t)}(S,v) = \\{\\{\\tilde{\\chi}_G^{M,(t)}(u,v):u\\in S\\}\\}\\) for any model \\(M\\) and set \\(S\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.43,
+                0.324,
+                0.445
+            ],
+            "angle": 0,
+            "content": "- Subgraph GNN:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.261,
+                0.449,
+                0.826,
+                0.475
+            ],
+            "angle": 0,
+            "content": "\\[\n\\tilde {\\chi} _ {G} ^ {\\text {S u b}, (t + 1)} (u, v) = \\operatorname {h a s h} \\left(\\tilde {\\chi} _ {G} ^ {\\text {S u b}, (t)} (u, v), \\tilde {\\chi} _ {G} ^ {\\text {S u b}, (t)} (u, N _ {G} (v)), \\tilde {\\chi} _ {G} ^ {\\text {S u b}, (t)} (u, V _ {G})\\right). \\tag {7}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.481,
+                0.314,
+                0.495
+            ],
+            "angle": 0,
+            "content": "- Local 2-GNN:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.277,
+                0.499,
+                0.825,
+                0.552
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\tilde {\\chi} _ {G} ^ {\\mathsf {L}, (t + 1)} (u, v) = \\operatorname {h a s h} \\left(\\tilde {\\chi} _ {G} ^ {\\mathsf {L}, (t)} (u, v), \\tilde {\\chi} _ {G} ^ {\\mathsf {L}, (t)} (u, N _ {G} (v)), \\tilde {\\chi} _ {G} ^ {\\mathsf {L}, (t)} \\left(N _ {G} (u), v\\right), \\right. \\tag {8} \\\\ \\left. \\tilde {\\chi} _ {G} ^ {\\mathsf {L}, (t)} (u, V _ {G}), \\tilde {\\chi} _ {G} ^ {\\mathsf {L}, (t)} (V _ {G}, v)\\right). \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.558,
+                0.321,
+                0.572
+            ],
+            "angle": 0,
+            "content": "- Local 2-FGNN:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.214,
+                0.575,
+                0.824,
+                0.641
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\tilde {\\chi} _ {G} ^ {\\mathsf {L F}, (t + 1)} (u, v) = \\mathsf {h a s h} \\left(\\tilde {\\chi} _ {G} ^ {\\mathsf {L F}, (t)} (u, v), \\{\\{\\left(\\tilde {\\chi} _ {G} ^ {\\mathsf {L F}, (t)} (w, v), \\tilde {\\chi} _ {G} ^ {\\mathsf {L F}, (t)} (u, w)\\right): w \\in N _ {G} [ u ] \\cup N _ {G} [ v ] \\}, \\right. \\\\ \\left. \\tilde {\\chi} _ {G} ^ {\\mathrm {L F}, (t)} \\left(u, V _ {G}\\right), \\tilde {\\chi} _ {G} ^ {\\mathrm {L F}, (t)} \\left(V _ {G}, v\\right)\\right). \\tag {9} \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.199,
+                0.643,
+                0.281,
+                0.655
+            ],
+            "angle": 0,
+            "content": "2-FGNN:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.246,
+                0.659,
+                0.825,
+                0.713
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\tilde {\\chi} _ {G} ^ {\\mathsf {F}, (t + 1)} (u, v) = \\operatorname {h a s h} \\left(\\tilde {\\chi} _ {G} ^ {\\mathsf {F}, (t)} (u, v), \\{\\{\\left(\\tilde {\\chi} _ {G} ^ {\\mathsf {F}, (t)} (w, v), \\tilde {\\chi} _ {G} ^ {\\mathsf {F}, (t)} (u, w)\\right): w \\in V _ {G} \\} \\}, \\right. \\tag {10} \\\\ \\left. \\tilde {\\chi} _ {G} ^ {\\mathsf {F}, (t)} (u, V _ {G}), \\tilde {\\chi} _ {G} ^ {\\mathsf {F}, (t)} (V _ {G}, v)\\right). \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.72,
+                0.827,
+                0.775
+            ],
+            "angle": 0,
+            "content": "It can be seen that we additionally add global aggregations for these architectures. Moreover, in Local 2-FGNN we replace the neighbors by closed neighbors \\(N_G[u] \\cup N_G[v]\\). The stable color of \\((u,v)\\) for different models is denoted by \\(\\tilde{\\chi}_G^{\\mathrm{Sub}}(u,v)\\), \\(\\tilde{\\chi}_G^{\\mathrm{L}}(u,v)\\), \\(\\tilde{\\chi}_G^{\\mathrm{LF}}(u,v)\\), \\(\\tilde{\\chi}_G^{\\mathrm{F}}(u,v)\\), respectively. We have the following result:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.778,
+                0.827,
+                0.84
+            ],
+            "angle": 0,
+            "content": "Proposition C.9. Let \\( M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\} \\) be any model. For any graphs \\( G, H \\), \\( \\tilde{\\chi}_G^M(G) = \\tilde{\\chi}_H^M(H) \\) iff \\( \\chi_G^M(G) = \\chi_H^M(H) \\). Furthermore, if \\( G \\) and \\( H \\) are connected, then for any vertices \\( u, v \\in V_G \\) and \\( x, y \\in V_H \\), \\( \\tilde{\\chi}_G^M(u, v) = \\tilde{\\chi}_H^M(x, y) \\) iff \\( \\chi_G^M(u, v) = \\chi_H^M(x, y) \\). In other words, the color mapping \\( \\tilde{\\chi}^M \\) is as fine as the original one \\( \\chi^M \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.854,
+                0.827,
+                0.924
+            ],
+            "angle": 0,
+            "content": "Proof. The proof simply follows from Zhang et al. (2023a, Proposition 4.2 and Theorem 4.4), because (i) node marking in the initial color of Subgraph GNN is as expressive as using the isomorphism type, (ii) the global aggregation does not improve the expressivity when the corresponding local aggregation is presented, (iii) the single-point aggregation does not improve the expressivity in Local 2-FGNN. \\(\\square\\)"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.51,
+                0.96
+            ],
+            "angle": 0,
+            "content": "24"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.104,
+                0.436,
+                0.119
+            ],
+            "angle": 0,
+            "content": "C.2 PART 1: TREE DECOMPOSITION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.13,
+                0.825,
+                0.159
+            ],
+            "angle": 0,
+            "content": "We first define the unfolding tree of different CR algorithms, which is a standard tool in analyzing GNN expressivity."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.16,
+                0.827,
+                0.225
+            ],
+            "angle": 0,
+            "content": "Definition C.10 (Unfolding tree of Subgraph GNN). Given a graph \\(G\\), vertices \\(u, v \\in V_G\\), and a non-negative integer \\(D\\), the depth-2D Subgraph GNN unfolding tree of graph \\(G\\) at \\((u, v)\\), denoted as \\(\\left(F_G^{\\mathrm{Sub},(D)}(u, v), T_G^{\\mathrm{Sub},(D)}(u, v)\\right)\\), is a tree-decomposed graph \\((F, T^r) \\in S^{\\mathrm{Sub}}\\) constructed as follows:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.232,
+                0.826,
+                0.276
+            ],
+            "angle": 0,
+            "content": "1. Initialization. At the beginning, \\( F = G[\\{u,v\\}] \\) (if \\( u = v \\), \\( F \\) only has one vertex), and \\( T \\) only has a root node \\( r \\) with \\( \\beta_T(r) = \\{\\{u,v\\}\\} \\). Define a mapping \\( \\pi : V_F \\to V_G \\) as \\( \\pi(u) = u \\) and \\( \\pi(v) = v \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.191,
+                0.281,
+                0.718,
+                0.296
+            ],
+            "angle": 0,
+            "content": "2. Loop for \\( D \\) rounds. For each leaf node \\( t \\) in \\( T^r \\), do the following procedure:"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.191,
+                0.232,
+                0.826,
+                0.296
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.296,
+                0.825,
+                0.325
+            ],
+            "angle": 0,
+            "content": "Let \\(\\beta_T(t) = \\{\\{u,x\\}\\}\\). For each \\(w \\in V_G\\), add a fresh child node \\(t_w\\) to \\(T^r\\) and designate \\(t\\) as its parent. Then, consider the following three cases:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.232,
+                0.33,
+                0.826,
+                0.387
+            ],
+            "angle": 0,
+            "content": "a) If \\( w \\neq \\pi(u) \\) and \\( w \\neq \\pi(x) \\), then add a fresh vertex \\( z \\) to \\( F \\) and extend \\( \\pi \\) with \\( \\pi(z) = w \\). The label of \\( z \\) in \\( F \\) is set by \\( \\ell_F(z) = \\ell_G(w) \\). Define \\( \\beta_T(t_w) = \\beta_T(t) \\cup \\{\\{z\\}\\} \\). Then, we add edges between \\( z \\) and \\( \\beta_T(t) \\), so that \\( \\pi \\) is an isomorphism from \\( F[\\beta_T(t_w)] \\) to \\( G[\\pi(\\beta_T(t_w))] \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.386,
+                0.82,
+                0.402
+            ],
+            "angle": 0,
+            "content": "b) If \\( w = \\pi(u) \\), then we simply set \\( \\beta_T(t_w) = \\beta_T(t) \\cup \\{\\{u\\}\\} \\) without modifying graph \\( F \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.402,
+                0.82,
+                0.417
+            ],
+            "angle": 0,
+            "content": "c) If \\( w = \\pi(x) \\), then we simply set \\( \\beta_T(t_w) = \\beta_T(t) \\cup \\{x\\} \\) without modifying graph \\( F \\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.231,
+                0.33,
+                0.826,
+                0.417
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.212,
+                0.422,
+                0.825,
+                0.451
+            ],
+            "angle": 0,
+            "content": "Finally, add a fresh child node \\( t_w' \\) to \\( T^r \\), designate \\( t_w \\) as its parent, and set \\( \\beta_T(t_w') \\) based on the following three cases:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.232,
+                0.457,
+                0.599,
+                0.473
+            ],
+            "angle": 0,
+            "content": "a) If \\( w \\neq \\pi(u) \\) and \\( w \\neq \\pi(x) \\), then \\( \\beta_T(t_w') = \\{\\{u, z\\}\\} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.232,
+                0.473,
+                0.503,
+                0.488
+            ],
+            "angle": 0,
+            "content": "b) If \\( w = \\pi(u) \\), then \\( \\beta_T(t_w') = \\{\\{u, u\\}\\} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.232,
+                0.488,
+                0.503,
+                0.502
+            ],
+            "angle": 0,
+            "content": "c) If \\( w = \\pi(x) \\), then \\( \\beta_T(t_w') = \\{\\{u, x\\}\\} \\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.232,
+                0.457,
+                0.599,
+                0.502
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.508,
+                0.826,
+                0.552
+            ],
+            "angle": 0,
+            "content": "It is easy to see that the depth of tree \\( T^r \\) increases by 2 after each round, \\( T^r \\) is always a canonical tree decomposition of \\( F \\), and \\( (F, T^r) \\in S^{\\mathsf{Sub}} \\). An illustration of the construction of unfolding tree is given in Figure 6(a)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.56,
+                0.825,
+                0.589
+            ],
+            "angle": 0,
+            "content": "We next define the unfolding tree of Local 2-GNN, which differs in the loop part such that the bags \\(\\beta_{T}(t_{w})\\) and \\(\\beta_{T}(t_{w}^{\\prime})\\) now do not necessarily contain \\(u\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.59,
+                0.826,
+                0.643
+            ],
+            "angle": 0,
+            "content": "Definition C.11 (Unfolding tree of Local 2-GNN). Given a graph \\(G\\), vertices \\(u, v \\in V_G\\), and a non-negative integer \\(D\\), the depth-2D Local 2-GNN unfolding tree of graph \\(G\\) at \\((u, v)\\), denoted as \\(\\left(F_G^{\\mathsf{L},(D)}(u, v), T_G^{\\mathsf{L},(D)}(u, v)\\right)\\), is a tree-decomposed graph \\((F, T^r) \\in S^\\mathsf{L}\\) constructed as follows:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.648,
+                0.783,
+                0.664
+            ],
+            "angle": 0,
+            "content": "1. Initialization. The procedure is exactly the same as Subgraph GNN (Definition C.10)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.669,
+                0.718,
+                0.684
+            ],
+            "angle": 0,
+            "content": "2. Loop for \\( D \\) rounds. For each leaf node \\( t \\) in \\( T^r \\), do the following procedure:"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.192,
+                0.648,
+                0.783,
+                0.684
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.684,
+                0.825,
+                0.712
+            ],
+            "angle": 0,
+            "content": "Let \\(\\beta_T(t) = \\{\\{x,y\\}\\}\\). For each \\(w \\in V_G\\), add a fresh child node \\(t_w\\) to \\(T\\) and designate \\(t\\) as its parent. Then, consider the following three cases:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.232,
+                0.719,
+                0.826,
+                0.775
+            ],
+            "angle": 0,
+            "content": "a) If \\( w \\neq \\pi(x) \\) and \\( w \\neq \\pi(y) \\), then add a fresh vertex \\( z \\) to \\( F \\) and extend \\( \\pi \\) with \\( \\pi(z) = w \\). The label of \\( z \\) in \\( F \\) is set by \\( \\ell_F(z) = \\ell_G(w) \\). Define \\( \\beta_T(t_w) = \\beta_T(t) \\cup \\{\\{z\\}\\} \\). Then, we add edges between \\( z \\) and \\( \\beta_T(t) \\), so that \\( \\pi \\) is an isomorphism from \\( F[\\beta_T(t_w)] \\) to \\( G[\\pi(\\beta_T(t_w))] \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.775,
+                0.82,
+                0.79
+            ],
+            "angle": 0,
+            "content": "b) If \\( w = \\pi(x) \\), then we simply set \\( \\beta_T(t_w) = \\beta_T(t) \\cup \\{\\{x\\}\\} \\) without modifying graph \\( F \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.79,
+                0.819,
+                0.805
+            ],
+            "angle": 0,
+            "content": "c) If \\( w = \\pi(y) \\), then we simply set \\( \\beta_T(t_w) = \\beta_T(t) \\cup \\{\\{y\\}\\} \\) without modifying graph \\( F \\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.231,
+                0.719,
+                0.826,
+                0.805
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.212,
+                0.81,
+                0.825,
+                0.84
+            ],
+            "angle": 0,
+            "content": "Next, add a fresh child node \\( t_w' \\) in \\( T^r \\), designate \\( t_w \\) as its parent, and set \\( \\beta_T(t_w') \\) based on the following three cases:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.845,
+                0.598,
+                0.861
+            ],
+            "angle": 0,
+            "content": "a) If \\( w \\neq \\pi(x) \\) and \\( w \\neq \\pi(y) \\), then \\( \\beta_T(t'_w) = \\{\\{x, z\\}\\} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.861,
+                0.503,
+                0.876
+            ],
+            "angle": 0,
+            "content": "b) If \\( w = \\pi(x) \\), then \\( \\beta_T(t'_w) = \\{ \\{x, x\\} \\} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.876,
+                0.502,
+                0.89
+            ],
+            "angle": 0,
+            "content": "c) If \\( w = \\pi(y) \\), then \\( \\beta_T(t'_w) = \\{ \\{x, y\\} \\} \\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.231,
+                0.845,
+                0.598,
+                0.89
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.212,
+                0.896,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "Finally, we repeat the above procedure (point 2) once more, but this time the bag \\(\\beta_{T}(t_{w}^{\\prime})\\) is replaced by the following three cases (changing \\(x\\) to \\(y\\)):"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.961
+            ],
+            "angle": 0,
+            "content": "25"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.177,
+                0.13,
+                0.199,
+                0.145
+            ],
+            "angle": 0,
+            "content": "(a)"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.243,
+                0.101,
+                0.34,
+                0.171
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.283,
+                0.186,
+                0.3,
+                0.198
+            ],
+            "angle": 0,
+            "content": "G"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.382,
+                0.109,
+                0.484,
+                0.164
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.391,
+                0.183,
+                0.476,
+                0.201
+            ],
+            "angle": 0,
+            "content": "\\(F_{G}^{\\mathrm{Sub},(1)}(1,2)\\)"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.244,
+                0.212,
+                0.337,
+                0.281
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.283,
+                0.297,
+                0.299,
+                0.308
+            ],
+            "angle": 0,
+            "content": "G"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.383,
+                0.203,
+                0.484,
+                0.292
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.397,
+                0.294,
+                0.469,
+                0.311
+            ],
+            "angle": 0,
+            "content": "\\(F_{G}^{\\mathrm{L},(1)}(1,2)\\)"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.55,
+                0.09,
+                0.775,
+                0.183
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.619,
+                0.184,
+                0.703,
+                0.201
+            ],
+            "angle": 0,
+            "content": "\\(T_{G}^{\\mathrm{Sub},(1)}(1,2)\\)"
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.178,
+                0.351,
+                0.199,
+                0.364
+            ],
+            "angle": 0,
+            "content": "(c)"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.244,
+                0.323,
+                0.337,
+                0.391
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.283,
+                0.407,
+                0.299,
+                0.418
+            ],
+            "angle": 0,
+            "content": "G"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.383,
+                0.321,
+                0.483,
+                0.373
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.424,
+                0.381,
+                0.443,
+                0.395
+            ],
+            "angle": 0,
+            "content": "\\(④^{\\prime \\prime}\\)"
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.394,
+                0.404,
+                0.472,
+                0.422
+            ],
+            "angle": 0,
+            "content": "\\(F_{G}^{\\mathrm{LF},(1)}(1,2)\\)"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.528,
+                0.202,
+                0.796,
+                0.293
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.626,
+                0.295,
+                0.698,
+                0.31
+            ],
+            "angle": 0,
+            "content": "\\(T_{G}^{\\mathbf{L},(1)}(1,2)\\)"
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.178,
+                0.462,
+                0.199,
+                0.475
+            ],
+            "angle": 0,
+            "content": "(d)"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.244,
+                0.434,
+                0.337,
+                0.502
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.283,
+                0.518,
+                0.299,
+                0.529
+            ],
+            "angle": 0,
+            "content": "G"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.383,
+                0.442,
+                0.484,
+                0.495
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.396,
+                0.515,
+                0.469,
+                0.532
+            ],
+            "angle": 0,
+            "content": "\\(F_{G}^{\\mathsf{F},(1)}(1,2)\\)"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.528,
+                0.313,
+                0.796,
+                0.403
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.623,
+                0.405,
+                0.701,
+                0.422
+            ],
+            "angle": 0,
+            "content": "\\(T_{G}^{\\mathrm{LF},(1)}\\) (1,2)"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.528,
+                0.423,
+                0.796,
+                0.513
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.626,
+                0.515,
+                0.698,
+                0.532
+            ],
+            "angle": 0,
+            "content": "\\(T_{G}^{\\mathsf{F},(1)}(1,2)\\)"
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.171,
+                0.542,
+                0.825,
+                0.573
+            ],
+            "angle": 0,
+            "content": "Figure 6: The depth-2 unfolding tree of graph \\( G \\) at vertex pair (1,2) for Subgraph GNN, Local 2-GNN, Local 2-FGNN, and 2-FGNN, respectively."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.597,
+                0.597,
+                0.613
+            ],
+            "angle": 0,
+            "content": "a) If \\( w \\neq \\pi(x) \\) and \\( w \\neq \\pi(y) \\), then \\( \\beta_T(t'_w) = \\{\\{y, z\\}\\} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.613,
+                0.501,
+                0.628
+            ],
+            "angle": 0,
+            "content": "b) If \\( w = \\pi(y) \\), then \\( \\beta_T(t'_w) = \\{\\{y, y\\}\\} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.628,
+                0.501,
+                0.644
+            ],
+            "angle": 0,
+            "content": "c) If \\( w = \\pi(x) \\), then \\( \\beta_T(t_w') = \\{\\{x, y\\}\\} \\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.231,
+                0.597,
+                0.597,
+                0.644
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.649,
+                0.664,
+                0.665
+            ],
+            "angle": 0,
+            "content": "An illustration of the construction of unfolding tree is given in Figure 6(b)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.676,
+                0.825,
+                0.704
+            ],
+            "angle": 0,
+            "content": "We next define the unfolding tree of Local 2-FGNN, which differs in the loop part such that the node \\( t_w \\) can have two children under certain conditions."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.709,
+                0.825,
+                0.762
+            ],
+            "angle": 0,
+            "content": "Definition C.12 (Unfolding tree of Local 2-FGNN). Given a graph \\( G \\), vertices \\( u, v \\in V_G \\), and a non-negative integer \\( D \\), the depth-2D Local 2-FGNN unfolding tree of graph \\( G \\) at \\( (u, v) \\), denoted as \\( \\left(F_G^{\\mathsf{LF},(D)}(u, v), T_G^{\\mathsf{LF},(D)}(u, v)\\right) \\), is a tree-decomposed graph \\( (F, T^r) \\in S^{\\mathsf{LF}} \\) constructed as follows:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.191,
+                0.767,
+                0.783,
+                0.783
+            ],
+            "angle": 0,
+            "content": "1. Initialization. The procedure is exactly the same as Subgraph GNN (Definition C.10)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.191,
+                0.789,
+                0.718,
+                0.803
+            ],
+            "angle": 0,
+            "content": "2. Loop for \\(D\\) rounds. For each leaf node \\(t\\) in \\(T^r\\), do the following procedure:"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.191,
+                0.767,
+                0.783,
+                0.803
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.803,
+                0.825,
+                0.832
+            ],
+            "angle": 0,
+            "content": "Let \\(\\beta_T(t) = \\{\\{x,y\\}\\}\\). For each \\(w \\in N_G[\\pi(x)] \\cup N_G[\\pi(y)]\\), add a fresh child node \\(t_w\\) to \\(T\\) and designate \\(t\\) as its parent. Then, consider the following three cases:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.232,
+                0.838,
+                0.826,
+                0.895
+            ],
+            "angle": 0,
+            "content": "a) If \\( w \\neq \\pi(x) \\) and \\( w \\neq \\pi(y) \\), then add a fresh vertex \\( z \\) to \\( F \\) and extend \\( \\pi \\) with \\( \\pi(z) = w \\). The label of \\( z \\) in \\( F \\) is set by \\( \\ell_F(z) = \\ell_G(w) \\). Define \\( \\beta_T(t_w) = \\beta_T(t) \\cup \\{\\{z\\}\\} \\). Then, we add edges between \\( z \\) and \\( \\beta_T(t) \\), so that \\( \\pi \\) is an isomorphism from \\( F[\\beta_T(t_w)] \\) to \\( G[\\pi(\\beta_T(t_w))] \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.895,
+                0.82,
+                0.91
+            ],
+            "angle": 0,
+            "content": "b) If \\( w = \\pi(x) \\), then we simply set \\( \\beta_T(t_w) = \\beta_T(t) \\cup \\{\\{x\\}\\} \\) without modifying graph \\( F \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.91,
+                0.819,
+                0.926
+            ],
+            "angle": 0,
+            "content": "c) If \\( w = \\pi(y) \\), then we simply set \\( \\beta_T(t_w) = \\beta_T(t) \\cup \\{\\{y\\}\\} \\) without modifying graph \\( F \\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.231,
+                0.838,
+                0.826,
+                0.926
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "26"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.104,
+                0.825,
+                0.135
+            ],
+            "angle": 0,
+            "content": "Next, add two fresh children \\(t_w'\\) and \\(t_w''\\) in \\(T^r\\), designate \\(t_w\\) as their parent, and set \\(\\beta_T(t_w')\\) and \\(\\beta_T(t_w'')\\) based on the following three cases:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.139,
+                0.747,
+                0.155
+            ],
+            "angle": 0,
+            "content": "a) If \\( w \\neq \\pi(x) \\) and \\( w \\neq \\pi(y) \\), then \\( \\beta_T(t_w') = \\{\\{x, z\\}\\} \\) and \\( \\beta_T(t_w'') = \\{\\{y, z\\}\\} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.155,
+                0.653,
+                0.169
+            ],
+            "angle": 0,
+            "content": "b) If \\( w = \\pi(x) \\), then \\( \\beta_T(t_w') = \\{ \\{x, x\\} \\} \\) and \\( \\beta_T(t_w'') = \\{ \\{x, y\\} \\} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.17,
+                0.651,
+                0.185
+            ],
+            "angle": 0,
+            "content": "c) If \\( w = \\pi(y) \\), then \\( \\beta_T(t_w') = \\{ \\{x, y\\} \\} \\) and \\( \\beta_T(t_w'') = \\{ \\{y, x\\} \\} \\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.231,
+                0.139,
+                0.747,
+                0.185
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.19,
+                0.825,
+                0.219
+            ],
+            "angle": 0,
+            "content": "For each \\(w \\notin N_G[\\pi(x)] \\cup N_G[\\pi(y)]\\), follow the same procedure as Local 2-GNN (Definition C.11)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.226,
+                0.664,
+                0.241
+            ],
+            "angle": 0,
+            "content": "An illustration of the construction of unfolding tree is given in Figure 6(c)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.252,
+                0.825,
+                0.279
+            ],
+            "angle": 0,
+            "content": "We finally define the unfolding tree of 2-FGNN, which differs in the loop part such that all nodes \\( t_w \\) have two children."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.283,
+                0.826,
+                0.336
+            ],
+            "angle": 0,
+            "content": "Definition C.13 (Unfolding tree of 2-FGNN). Given a graph \\( G \\), vertices \\( u, v \\in V_G \\), and a non-negative integer \\( D \\), the depth-2D 2-FGNN unfolding tree of graph \\( G \\) at \\( (u, v) \\), denoted as \\( \\left(F_G^{\\mathrm{Sub},(D)}(u, v), T_G^{\\mathrm{Sub},(D)}(u, v)\\right) \\), is a tree-decomposed graph \\( (F, T^r) \\in S^{\\mathsf{F}} \\) constructed as follows:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.341,
+                0.778,
+                0.357
+            ],
+            "angle": 0,
+            "content": "1. Initialization. The procedure is exactly the same as Local 2-FGNN (Definition C.12)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.363,
+                0.825,
+                0.392
+            ],
+            "angle": 0,
+            "content": "2. Loop for \\( D \\) rounds. The procedure is similar to Local 2-FGNN (Definition C.12) except that the condition \\( w \\in N_G[\\pi(x)] \\cup N_G[\\pi(y)] \\) is relaxed to all vertices."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.192,
+                0.341,
+                0.825,
+                0.392
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.398,
+                0.664,
+                0.413
+            ],
+            "angle": 0,
+            "content": "An illustration of the construction of unfolding tree is given in Figure 6(d)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.424,
+                0.492,
+                0.438
+            ],
+            "angle": 0,
+            "content": "We are now ready to present the first core result:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.441,
+                0.825,
+                0.513
+            ],
+            "angle": 0,
+            "content": "Lemma C.14. Let \\( M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\} \\) be any model. For any two graphs \\( G, H \\), any vertices \\( u, v \\in V_G, x, y \\in V_H \\), and any non-negative integer \\( D \\), \\( \\tilde{\\chi}_G^{M,(D)}(u,v) = \\tilde{\\chi}_H^{M,(D)}(x,y) \\) iff there exists an isomorphism \\( (\\rho, \\tau) \\) from \\( \\left(F_G^{M,(D)}(u,v), T_G^{M,(D)}(u,v)\\right) \\) to \\( \\left(F_H^{M,(D)}(x,y), T_H^{M,(D)}(x,y)\\right) \\) such that \\( \\rho(u) = x, \\rho(v) = y \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.527,
+                0.825,
+                0.557
+            ],
+            "angle": 0,
+            "content": "Proof. Here, we only give the proof for Local 2-GNN, and the proofs for Subgraph GNN, Local 2-FGNN and 2-FGNN are almost the same so we omit them for clarity."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.563,
+                0.825,
+                0.606
+            ],
+            "angle": 0,
+            "content": "Proof for Local 2-GNN. The proof is based on induction over \\( D \\). When \\( D = 0 \\), the theorem obviously holds. Now assume that the theorem holds for \\( D \\leq d \\), and consider \\( D = d + 1 \\). Below, we omit \\( \\mathsf{L} \\) in the corner mark for clarity."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.612,
+                0.825,
+                0.674
+            ],
+            "angle": 0,
+            "content": "1. We first prove that \\(\\tilde{\\chi}_G^{(d + 1)}(u,v) = \\tilde{\\chi}_H^{(d + 1)}(x,y)\\) implies that there exists an isomorphism \\((\\rho ,\\tau)\\) from \\(\\left(F_G^{(d + 1)}(u,v),T_G^{(d + 1)}(u,v)\\right)\\) to \\(\\left(F_H^{(d + 1)}(x,y),T_H^{(d + 1)}(x,y)\\right)\\) such that \\(\\rho (u) = x,\\rho (v) = y\\) . If \\(\\tilde{\\chi}_G^{(d + 1)}(u,v) = \\tilde{\\chi}_H^{(d + 1)}(x,y)\\) , then"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.226,
+                0.677,
+                0.825,
+                0.721
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\left\\{\\left\\{\\left(\\tilde {\\chi} _ {G} ^ {(d)} (u, w), \\mathsf {a t p} _ {G} (u, v, w)\\right): w \\in V _ {G} \\right\\} \\right\\} = \\left\\{\\left\\{\\left(\\tilde {\\chi} _ {H} ^ {(d)} (x, z), \\mathsf {a t p} _ {H} (x, y, z)\\right): z \\in V _ {H} \\right\\}, \\right. (11) \\\\ \\{\\{(\\tilde {\\chi} _ {G} ^ {(d)} (w, v), \\mathsf {a t p} _ {G} (u, v, w)): w \\in V _ {G} \\} \\} = \\{\\{(\\tilde {\\chi} _ {H} ^ {(d)} (z, y), \\mathsf {a t p} _ {H} (x, y, z)): z \\in V _ {H} \\}. (12) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.212,
+                0.724,
+                0.825,
+                0.755
+            ],
+            "angle": 0,
+            "content": "Let \\( n = |V_G| = |V_H| \\). Thus, we can denote \\( V_G = \\{w_1, \\dots, w_n\\} = \\{w_1', \\dots, w_n'\\} \\) and \\( V_H = \\{z_1, \\dots, z_n\\} = \\{z_1', \\dots, z_n'\\} \\) such that"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.238,
+                0.759,
+                0.748,
+                0.779
+            ],
+            "angle": 0,
+            "content": "- \\((\\tilde{\\chi}_G^{(d)}(u, w_i), \\mathsf{atp}_G(u, v, w_i)) = (\\tilde{\\chi}_H^{(d)}(x, z_i), \\mathsf{atp}_H(x, y, z_i))\\) for all \\(i \\in [n]\\);"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.238,
+                0.779,
+                0.748,
+                0.799
+            ],
+            "angle": 0,
+            "content": "- \\((\\tilde{\\chi}_G^{(d)}(w_i', v), \\mathsf{atp}_G(u, v, w_i')) = (\\tilde{\\chi}_H^{(d)}(z_i', y), \\mathsf{atp}_H(x, y, z_i'))\\) for all \\(i \\in [n]\\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.238,
+                0.759,
+                0.748,
+                0.799
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.212,
+                0.804,
+                0.602,
+                0.818
+            ],
+            "angle": 0,
+            "content": "On the other hand, by definition of tree unfolding, we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.284,
+                0.823,
+                0.753,
+                0.871
+            ],
+            "angle": 0,
+            "content": "\\[\nF _ {G} ^ {(d + 1)} (u, v) = \\left(\\bigcup_ {w _ {i}} F _ {G} ^ {(d)} (u, w _ {i})\\right) \\cup \\left(\\bigcup_ {w _ {i} ^ {\\prime}} F _ {G} ^ {(d)} (w _ {i} ^ {\\prime}, v)\\right) \\cup F _ {G} ^ {(1)} (u, v),\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.285,
+                0.873,
+                0.745,
+                0.922
+            ],
+            "angle": 0,
+            "content": "\\[\nF _ {H} ^ {(d + 1)} (x, y) = \\left(\\bigcup_ {z _ {i}} F _ {H} ^ {(d)} (x, z _ {i})\\right) \\cup \\left(\\bigcup_ {z _ {i} ^ {\\prime}} F _ {H} ^ {(d)} (z _ {i} ^ {\\prime}, y)\\right) \\cup F _ {H} ^ {(1)} (x, y),\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "27"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.189,
+                0.101,
+                0.46,
+                0.239
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.472,
+                0.101,
+                0.811,
+                0.238
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.331,
+                0.253,
+                0.666,
+                0.269
+            ],
+            "angle": 0,
+            "content": "Figure 7: Illustration of the proof of Lemma C.14."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.288,
+                0.825,
+                0.332
+            ],
+            "angle": 0,
+            "content": "where \\(\\cup\\) represents the graph union. Here, all \\(w_{i}, w_{j}^{\\prime} \\notin \\{u, v\\}\\) in different graphs are treated as different vertices when taking the union, while \\(u, v\\) in different graphs are shared. See Figure 7 for an illustration of the above equations."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.338,
+                0.827,
+                0.539
+            ],
+            "angle": 0,
+            "content": "By induction, there exists an isomorphism \\((\\rho_i,\\tau_i)\\) from \\(\\left(F_G^{(d)}(u,w_i),T_G^{(d)}(u,w_i)\\right)\\) to \\(\\left(F_H^{(d)}(x,z_i),T_H^{(d)}(x,z_i)\\right)\\) such that \\(\\rho_{i}(u) = x\\), \\(\\rho_{i}(w_{i}) = z_{i}\\) \\((i\\in [n])\\), and there exist an isomorphism \\(\\rho_{i}^{\\prime}\\) from \\(\\left(F_G^{(d)}(w_i',v),T_G^{(d)}(w_i',v)\\right)\\) to \\(\\left(F_H^{(d)}(z_i',y),T_H^{(d)}(z_i',y)\\right)\\) such that \\(\\rho_{i}^{\\prime}(v) = y\\), \\(\\rho_{i}^{\\prime}(w_{i}^{\\prime}) = z_{i}^{\\prime}\\) \\((i\\in [n])\\). Moreover, we have \\(\\mathsf{atp}_G(u,v,w_i) = \\mathsf{atp}_H(x,y,z_i)\\) and \\(\\mathsf{atp}_G(u,v,w_i') = \\mathsf{atp}_H(x,y,z_i')\\) for \\(i\\in [n]\\), which implies that \\(F_{G}^{(1)}(u,v)\\) is isomorphic to \\(F_H^{(1)}(x,y)\\). Therefore, if we construct \\(\\tilde{\\rho}\\) by merging all \\(\\rho_{i}\\) and \\(\\rho_{i}'\\) \\((i\\in [n])\\), and construct \\(\\tilde{\\tau}\\) by merging all \\(\\tau_{i}\\) and \\(\\tau_{i}'\\) and further specifying an appropriate mapping between tree nodes of depth no more than 1 in \\(T_G^{(d + 1)}(u,v)\\) and \\(T_H^{(d + 1)}(x,y)\\), then it is straightforward to see that \\((\\tilde{\\rho},\\tilde{\\tau})\\) is well-defined and is an isomorphism from \\(\\left(F_G^{(d + 1)}(u,v),T_G^{(d + 1)}(u,v)\\right)\\) to \\(\\left(F_H^{(d + 1)}(x,y),T_H^{(d + 1)}(x,y)\\right)\\) such that \\(\\tilde{\\rho}(u) = x\\), \\(\\tilde{\\rho}(v) = y\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.545,
+                0.825,
+                0.603
+            ],
+            "angle": 0,
+            "content": "2. We next prove that if there exists an isomorphism \\((\\rho, \\tau)\\) from the tree-decomposed graph \\(\\left(F_G^{(d+1)}(u, v), T_G^{(d+1)}(u, v)\\right)\\) to \\(\\left(F_H^{(d+1)}(x, y), T_H^{(d+1)}(x, y)\\right)\\) such that \\(\\rho(u) = x\\), \\(\\rho(v) = y\\), then \\(\\tilde{\\chi}_G^{(d+1)}(u, v) = \\tilde{\\chi}_H^{(d+1)}(x, y)\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.602,
+                0.826,
+                0.72
+            ],
+            "angle": 0,
+            "content": "Without loss of generality, assume \\( u \\neq v \\) and \\( x \\neq y \\). Since \\( \\tau \\) is an isomorphism from \\( T_G^{(d+1)}(u,v) \\) to \\( T_H^{(d+1)}(x,y) \\), \\( \\tau \\) maps all tree nodes of depth 1 in \\( T_G^{(d+1)}(u,v) \\) to all tree nodes of depth 1 in \\( T_H^{(d+1)}(x,y) \\). Let \\( s_1, \\dots, s_n \\) be all nodes of depth 2 in \\( T_G^{(d+1)}(u,v) \\) such that \\( u \\in \\beta_{T_G^{(d+1)}(u,v)}(s_i) \\) (it follows that \\( n = |V_G| \\)), and let \\( s_i' \\) be the parent of \\( s_i \\). Similarly, let \\( t_1, \\dots, t_n \\) be all nodes of depth 2 in \\( T_H^{(d+1)}(x,y) \\) such that \\( x \\in \\beta_{T_H^{(d+1)}(x,y)}(t_i) \\), and let \\( t_i' \\) be the parent of \\( t_i \\). Moreover, we can arrange the order so that the following are satisfied (for each \\( i \\in [n] \\)):"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.726,
+                0.815,
+                0.745
+            ],
+            "angle": 0,
+            "content": "a) \\(\\tau\\) is an isomorphism from the subtree \\(T_{G}^{(d + 1)}(u,v)[s_i]\\) to the subtree \\(T_{H}^{(d + 1)}(x,y)[t_i]\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.745,
+                0.733,
+                0.765
+            ],
+            "angle": 0,
+            "content": "b) For all \\(s \\in \\mathsf{Desc}_{T_G^{(d + 1)}(u,v)}(s_i)\\), \\(\\rho(\\beta_{T_G^{(d + 1)}(u,v)}(s)) = \\beta_{T_H^{(d + 1)}(x,y)}(\\tau(s))\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.764,
+                0.825,
+                0.809
+            ],
+            "angle": 0,
+            "content": "c) By definition of the unfolding tree, \\(\\rho\\) is an isomorphism from the induced subgraph \\(F_{G}^{(d + 1)}(u,v)[T_{G}^{(d + 1)}(u,v)[s_{i}]]\\) to the induced subgraph \\(F_{H}^{(d + 1)}(x,y)[T_{H}^{(d + 1)}(x,y)[t_{i}]]\\) (see Definition C.7)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.81,
+                0.825,
+                0.851
+            ],
+            "angle": 0,
+            "content": "d) Let \\(\\beta_{T_G^{(d + 1)}(u,v)}(s_i') = \\{\\{u,v,\\tilde{w}_i\\}\\}\\) and \\(\\beta_{T_H^{(d + 1)}(x,y)}(t_i') = \\{\\{x,y,\\tilde{z}_i\\}\\}\\). Then, \\(\\rho (\\tilde{w}_i) = \\tilde{z}_i\\), and thus \\(\\{v,\\tilde{w}_i\\} \\in E_{F_G^{(d + 1)}(u,v)}\\) iff \\(\\{y,\\tilde{z}_i\\} \\in E_{F_H^{(d + 1)}(x,y)}\\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.231,
+                0.726,
+                0.825,
+                0.851
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.212,
+                0.856,
+                0.827,
+                0.929
+            ],
+            "angle": 0,
+            "content": "By items (a) to (c), \\(\\left(F_{G}^{(d + 1)}(u,v)\\left[T_{G}^{(d + 1)}(u,v)[s_{i}]\\right],T_{G}^{(d + 1)}(u,v)[s_{i}]\\right)\\) is isomorphic to \\(\\left(F_H^{(d + 1)}(x,y)\\left[T_H^{(d + 1)}(x,y)[t_i]\\right],T_H^{(d + 1)}(x,y)[t_i]\\right)\\). On the other hand, by definition of the unfolding tree, \\(\\left(F_G^{(d + 1)}(u,v)\\left[T_G^{(d + 1)}(u,v)[s_i]\\right],T_G^{(d + 1)}(u,v)[s_i]\\right)\\) is isomorphic to"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "28"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.212,
+                0.102,
+                0.825,
+                0.145
+            ],
+            "angle": 0,
+            "content": "the depth-2d unfolding tree \\(\\left(F_{G}^{(d)}(u,w_{i}),T_{G}^{(d)}(u,w_{i})\\right)\\) for some \\(w_{i}\\in V_{G}\\) satisfying that \\(\\{w_i,v\\} \\in E_G\\) iff \\(\\{\\tilde{w}_i,v\\} \\in E_{F_G^{(d + 1)}(u,v)}\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.212,
+                0.144,
+                0.825,
+                0.252
+            ],
+            "angle": 0,
+            "content": "Similarly, \\(\\left(F_H^{(d + 1)}(x,y)\\left[T_H^{(d + 1)}(x,y)[t_i]\\right],T_H^{(d + 1)}(x,y)[t_i]\\right)\\) is isomorphic to \\(\\left(F_{H}^{(d)}(x,z_{i}),T_{H}^{(d)}(x,z_{i})\\right)\\) for some \\(z_{i}\\in V_{H}\\) satisfying that \\(\\{z_i,y\\} \\in E_H\\) iff \\(\\{\\tilde{z}_i,y\\} \\in E_{F_H^{(d + 1)}(x,y)}\\). Combining all the above equivalence yields that \\(\\left(F_G^{(d)}(u,w_i),T_G^{(d)}(u,w_i)\\right)\\) is isomorphic to \\(\\left(F_H^{(d)}(x,z_i),T_H^{(d)}(x,z_i)\\right)\\), and \\(\\{w_i,v\\} \\in E_G\\) iff \\(\\{z_i,y\\} \\in E_H\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.212,
+                0.256,
+                0.825,
+                0.289
+            ],
+            "angle": 0,
+            "content": "By induction, we have \\(\\tilde{\\chi}_G^{(d)}(u,w_i) = \\tilde{\\chi}_H^{(d)}(x,z_i)\\). Moreover, we clearly have that \\(\\{u,w_i\\} \\in E_G\\) iff \\(\\{x,z_i\\} \\in E_H\\), and \\(\\{u,v\\} \\in E_G\\) iff \\(\\{x,y\\} \\in E_H\\). Therefore,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.318,
+                0.294,
+                0.824,
+                0.317
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left(\\tilde {\\chi} _ {G} ^ {(d)} (u, w _ {i}), \\mathsf {a t p} _ {G} (u, v, w _ {i})\\right) = \\left(\\tilde {\\chi} _ {H} ^ {(d)} (x, z _ {i}), \\mathsf {a t p} _ {H} (x, y, z _ {i})\\right). \\tag {13}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.212,
+                0.32,
+                0.825,
+                0.35
+            ],
+            "angle": 0,
+            "content": "Next, note that \\(\\tilde{w}_i\\) are different from each other for \\(i\\in [n]\\) by definition of unfolding tree. Thus, \\(w_{i}\\) are also different from each other. It follows that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.225,
+                0.355,
+                0.825,
+                0.378
+            ],
+            "angle": 0,
+            "content": "\\[\n\\{\\{(\\tilde {\\chi} _ {G} ^ {(d)} (u, w), \\mathsf {a t p} _ {G} (u, v, w)): w \\in V _ {G} \\} \\} = \\{\\{(\\tilde {\\chi} _ {H} ^ {(d)} (x, z), \\mathsf {a t p} _ {H} (x, y, z)): z \\in V _ {H} \\} \\}. \\tag {14}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.213,
+                0.381,
+                0.553,
+                0.396
+            ],
+            "angle": 0,
+            "content": "Again using the same analysis as before, we obtain"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.226,
+                0.402,
+                0.825,
+                0.424
+            ],
+            "angle": 0,
+            "content": "\\[\n\\{\\{(\\tilde {\\chi} _ {G} ^ {(d)} (w, v), \\mathsf {a t p} _ {G} (u, v, w)): w \\in V _ {G} \\} \\} = \\{\\{(\\tilde {\\chi} _ {H} ^ {(d)} (z, y), \\mathsf {a t p} _ {H} (x, y, z)): z \\in V _ {H} \\} \\}. \\tag {15}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.212,
+                0.429,
+                0.825,
+                0.46
+            ],
+            "angle": 0,
+            "content": "It remains to prove that \\(\\tilde{\\chi}_G^{(d)}(u,v) = \\tilde{\\chi}_H^{(d)}(x,y)\\). To prove this, note that Equation (14) implies that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.223,
+                0.465,
+                0.825,
+                0.487
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left\\{\\left\\{\\left(\\tilde {\\chi} _ {G} ^ {(d ^ {\\prime})} (u, w), \\mathsf {a t p} _ {G} (u, v, w)\\right): w \\in V _ {G} \\right\\} \\right\\} = \\left\\{\\left\\{\\left(\\tilde {\\chi} _ {H} ^ {(d ^ {\\prime})} (x, z), \\mathsf {a t p} _ {H} (x, y, z)\\right): z \\in V _ {H} \\right\\} \\right\\} \\tag {16}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.212,
+                0.491,
+                0.581,
+                0.507
+            ],
+            "angle": 0,
+            "content": "holds for all \\(0 \\leq d' \\leq d\\), and Equation (15) implies that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.223,
+                0.513,
+                0.825,
+                0.535
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left\\{\\left\\{\\left(\\tilde {\\chi} _ {G} ^ {(d ^ {\\prime})} (w, v), \\mathsf {a t p} _ {G} (u, v, w)\\right): w \\in V _ {G} \\right\\}\\right\\} = \\left\\{ \\right.\\left\\{\\left(\\tilde {\\chi} _ {H} ^ {(d)} (z, y), \\mathsf {a t p} _ {H} (x, y, z)\\right): z \\in V _ {H} \\right\\}. \\tag {17}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.212,
+                0.539,
+                0.825,
+                0.586
+            ],
+            "angle": 0,
+            "content": "holds for all \\(0 \\leq d' \\leq d\\). Combined with Equations (16) and (17) and the fact that \\(\\tilde{\\chi}_{G}^{(0)}(u,v) = \\tilde{\\chi}_{H}^{(0)}(x,y)\\), we can incrementally prove that \\(\\tilde{\\chi}_{G}^{(d')}(u,v) = \\tilde{\\chi}_{H}^{(d')}(x,y)\\) for all \\(d' \\leq d + 1\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.592,
+                0.402,
+                0.607
+            ],
+            "angle": 0,
+            "content": "We have thus concluded the proof."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.615,
+                0.825,
+                0.645
+            ],
+            "angle": 0,
+            "content": "Definition C.15. Let \\( M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\} \\) be any model. Given a graph \\( G \\) and a tree-decomposed graph \\( (F, T^r) \\), define"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.171,
+                0.65,
+                0.83,
+                0.677
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathsf {c n t} ^ {M} \\left(\\left(F, T ^ {r}\\right), G\\right) := \\left| \\left\\{\\left(u, v\\right) \\in V _ {G} ^ {2}: \\exists D \\in \\mathbb {N} _ {+} \\text {s . t .} \\left(F _ {G} ^ {M, (D)} (u, v), T _ {G} ^ {M, (D)} (u, v)\\right) \\simeq \\left(F, T ^ {r}\\right) \\right\\} \\right|,\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.684,
+                0.814,
+                0.708
+            ],
+            "angle": 0,
+            "content": "where \\(\\left(F_G^{M,(D)}(u,v),T_G^{M,(D)}(u,v)\\right)\\) is the depth-2D unfolding tree of \\(G\\) at \\((u,v)\\) for model \\(M\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.711,
+                0.825,
+                0.744
+            ],
+            "angle": 0,
+            "content": "Corollary C.16. Let \\( M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\} \\) be any model. For any graphs \\( G, H \\), \\( \\chi_G^M(G) = \\chi_H^M(H) \\) iff \\( \\mathsf{cnt}^M((F, T^r), G) = \\mathsf{cnt}^M((F, T^r), H) \\) holds for all \\( (F, T^r) \\in S^M \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.756,
+                0.826,
+                0.856
+            ],
+            "angle": 0,
+            "content": "Proof. “\\(\\Rightarrow\\)”. If \\(\\chi_G^M (G) = \\chi_H^M (H)\\), then \\(\\{\\{\\chi_G^M (u,v):u,v\\in V_G\\} \\} = \\{\\{\\chi_H^M (x,y):x,y\\in V_H\\}\\}\\). For each color \\(c\\) in the above multiset, pick \\(u,v\\in V_{G}\\) with \\(\\chi_G^M (u,v) = c\\). It follows that if \\((F,T^{r})\\simeq (F_{G}^{M,(D)}(u,v),T_{G}^{M,(D)}(u,v))\\in S^{M}\\) for some \\(D\\), then \\(\\mathsf{cnt}^M ((F,T^r),G) = |\\{(u,v)\\in V_G^2:\\chi_G^M (u,v) = c\\} | = |\\{(x,y)\\in V_H^2:\\chi_H^M (x,y) = c\\} | = \\mathsf{cnt}^M ((F,T^r),H)\\) by Lemma C.14. On the other hand, if \\((F,T^{r})\\neq (F_{G}^{M,(D)}(u,v),T_{G}^{M,(D)}(u,v))\\) for all \\(u,v\\in V_{G}\\) and all \\(D\\), then clearly \\(\\mathsf{cnt}^M ((F,T^r),G) = \\mathsf{cnt}^M ((F,T^r),H) = 0\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.86,
+                0.826,
+                0.927
+            ],
+            "angle": 0,
+            "content": "\\(\\Longleftarrow\\) . If \\(\\mathsf{cnt}^M ((F,T^r),G) = \\mathsf{cnt}^M ((F,T^r),H)\\) holds for all \\((F,T^{r})\\in S^{M}\\) , it clearly holds for all \\((F_M^{(D)}(u,v),T_M^{(D)}(u,v))\\) with \\(u,v\\in V_G\\) and a sufficiently large \\(D\\) . This guarantees that for all color \\(c,\\left|\\{(u,v)\\in V_G^2:\\chi_G^M (u,v) = c\\} \\right| = \\left|\\{(x,y)\\in V_H^2:\\chi_H^M (x,y) = c\\} \\right|\\) by Lemma C.14. Therefore, \\(\\{\\{\\chi_G^M (u,v):u,v\\in V_G\\} \\} = \\{\\{\\chi_H^M (x,y):x,y\\in V_H\\} \\}\\) , concluding the proof."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.961
+            ],
+            "angle": 0,
+            "content": "29"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.7,
+                0.119
+            ],
+            "angle": 0,
+            "content": "We next define an important concept called bag isomorphism (Dell et al., 2018)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.123,
+                0.827,
+                0.181
+            ],
+            "angle": 0,
+            "content": "Definition C.17. Given a tree-decomposed graph \\((F, T^r)\\) and a graph \\(G\\), a bag isomorphism from \\((F, T^r)\\) to \\(G\\) (abbreviated as \"bIso\") is a homomorphism \\(f\\) from \\(F\\) to \\(G\\) such that, for all \\(t \\in V_T\\), \\(f\\) is an isomorphism from \\(F[\\beta_T(t)]\\) to \\(G[f(\\beta_T(t))]\\). Denote \\(\\mathsf{BIso}((F, T^r), G)\\) to be the set of all bag isomorphisms from \\((F, T^r)\\) to \\(G\\), and denote \\(\\mathsf{bIso}((F, T^r), G) = |\\mathsf{BIso}((F, T^r), G)|\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.185,
+                0.825,
+                0.213
+            ],
+            "angle": 0,
+            "content": "Remark C.18. To prove that a mapping \\( f \\) from \\( (F, T^r) \\) to \\( G \\) is a bIso, it suffices to prove the following conditions:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.217,
+                0.684,
+                0.233
+            ],
+            "angle": 0,
+            "content": "1. For any two different vertices \\( u, v \\in V_G \\) in the same bag, \\( f(u) \\neq f(v) \\);"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.191,
+                0.235,
+                0.775,
+                0.252
+            ],
+            "angle": 0,
+            "content": "2. For any two vertices \\( u, v \\in V_G \\) in the same bag, \\( \\{u, v\\} \\in E_F \\) iff \\( \\{f(u), f(v)\\} \\in E_G \\);"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.255,
+                0.457,
+                0.271
+            ],
+            "angle": 0,
+            "content": "3. For any \\(u\\in V_G,\\ell_F(u) = \\ell_G(f(u))\\)"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.191,
+                0.217,
+                0.775,
+                0.271
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.282,
+                0.704,
+                0.297
+            ],
+            "angle": 0,
+            "content": "The following fact is straightforward from the construction of the unfolding tree:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.301,
+                0.825,
+                0.356
+            ],
+            "angle": 0,
+            "content": "Fact C.19. Let \\( M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\} \\) be any model considered above. For any graph \\( G \\), any vertex pair \\( (u, v) \\in V_G^2 \\), and any non-negative integer \\( D \\), there is a bIso \\( \\pi \\) from \\( \\left(F_G^{M,(D)}(u, v), T_G^{M,(D)}(u, v)\\right) \\) to \\( G \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.366,
+                0.825,
+                0.395
+            ],
+            "angle": 0,
+            "content": "Similarly, we need the following concept to describe the relation between two tree-decomposed graphs. We note that these technical concepts also appeared in Dell et al. (2018)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.4,
+                0.825,
+                0.432
+            ],
+            "angle": 0,
+            "content": "Definition C.20. Given two tree-decomposed graphs \\((F, T^r)\\) and \\((\\tilde{F}, \\tilde{T}^s)\\), a pair of mappings \\((\\rho, \\tau)\\) is called homomorphism from \\((F, T^r)\\) to \\((\\tilde{F}, \\tilde{T}^s)\\) if it satisfies the following conditions:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.19,
+                0.439,
+                0.825,
+                0.47
+            ],
+            "angle": 0,
+            "content": "a) \\(\\tau\\) is a homomorphism from \\(T\\) to \\(\\tilde{T}\\) (ignoring labels \\(\\beta\\)) and is depth-preserving, i.e., \\(\\mathsf{dep}_{T^r}(t) = \\mathsf{dep}_{\\tilde{T}^s}(\\tau(t))\\) for all \\(t \\in V_T\\);"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.19,
+                0.474,
+                0.825,
+                0.507
+            ],
+            "angle": 0,
+            "content": "b) For all \\(t \\in V_T\\), \\(\\rho\\) is a homomorphism from \\(F[\\beta_T(t)]\\) to \\(\\tilde{F}[\\beta_{\\tilde{T}}(\\tau(t))]\\). Note that this implies that \\(\\rho\\) is a homomorphism from \\(F\\) to \\(\\tilde{F}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.191,
+                0.512,
+                0.503,
+                0.529
+            ],
+            "angle": 0,
+            "content": "c) The depth of \\( T^r \\) is equal to the depth of \\( \\hat{T}^s \\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.19,
+                0.439,
+                0.825,
+                0.529
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.542,
+                0.827,
+                0.699
+            ],
+            "angle": 0,
+            "content": "Definition C.21. Under Definition C.20, \\((\\rho, \\tau)\\) is further called a bag-isomorphism homomorphism (abbreviated as \"bIsoHom\") from \\((F, T^r)\\) to \\((\\tilde{F}, \\tilde{T}^s)\\) if it is a homomorphism satisfying that, for all \\(t \\in V_T\\), \\(\\rho\\) is an isomorphism from \\(F[\\beta_T(t)]\\) to \\(\\tilde{F}[\\beta_{\\tilde{T}}(\\tau(t))]\\). Furthermore, \\((\\rho, \\tau)\\) is called a bIsoSurj if \\(\\tau\\) is surjective; and \\((\\rho, \\tau)\\) is called a bIsoInj if \\(\\tau\\) is injective. We use BIsoHom \\(\\left((F, T^r), (\\tilde{F}, \\tilde{T}^s)\\right)\\) to denote the set of bIsoHoms from \\((F, T^r)\\) to \\((\\tilde{F}, \\tilde{T}^s)\\), and let bIsoHom \\(\\left((F, T^r), (\\tilde{F}, \\tilde{T}^s)\\right) = \\left|\\text{BIsoHom}\\left((F, T^r), (\\tilde{F}, \\tilde{T}^s)\\right)\\right|\\). The notations BIsoSurj \\(\\left((F, T^r), (\\tilde{F}, \\tilde{T}^s)\\right)\\), bIsoSurj \\(\\left((F, T^r), (\\tilde{F}, \\tilde{T}^s)\\right)\\), BIsolnj \\(\\left((F, T^r), (\\tilde{F}, \\tilde{T}^s)\\right)\\), and bIsolnj \\(\\left((F, T^r), (\\tilde{F}, \\tilde{T}^s)\\right)\\) are defined accordingly."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.703,
+                0.825,
+                0.733
+            ],
+            "angle": 0,
+            "content": "Remark C.22. In the above definition, the depth of a tree \\( T^r \\) is the maximal depth among all tree nodes in \\( T^r \\). Note that we do not require that all leaf nodes have the same depth in \\( T^r \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.744,
+                0.514,
+                0.759
+            ],
+            "angle": 0,
+            "content": "We are now ready to present the second core result:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.763,
+                0.827,
+                0.793
+            ],
+            "angle": 0,
+            "content": "Lemma C.23. Let \\( M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\} \\) be any model. For any graph \\( G \\) and tree-decomposed graph \\( (F, T^r) \\in S^M \\),"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.197,
+                0.801,
+                0.825,
+                0.839
+            ],
+            "angle": 0,
+            "content": "\\[\n\\operatorname {b l s o} \\left(\\left(F, T ^ {r}\\right), G\\right) = \\sum_ {\\left(\\tilde {F}, \\tilde {T} ^ {s}\\right) \\in \\mathcal {S} ^ {M}} \\operatorname {b l s o H o m} \\left(\\left(F, T ^ {r}\\right), \\left(\\tilde {F}, \\tilde {T} ^ {s}\\right)\\right) \\cdot \\operatorname {c n t} ^ {M} \\left(\\left(\\tilde {F}, \\tilde {T} ^ {s}\\right), G\\right). \\tag {18}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.848,
+                0.825,
+                0.878
+            ],
+            "angle": 0,
+            "content": "Here, the summation ranges over all non-isomorphic (tree-decomposed) graphs in \\( S^M \\) and is well-defined as there are only a finite number of graphs making the value in the summation non-zero."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.896,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "Proof. Here, we only give the proof for Local 2-GNN, and the proofs for Subgraph GNN, Local 2-FGNN and 2-FGNN are almost the same so we omit them for clarity."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.51,
+                0.96
+            ],
+            "angle": 0,
+            "content": "30"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.825,
+                0.152
+            ],
+            "angle": 0,
+            "content": "Proof for Local 2-GNN. We assume that the root bag of \\((F, T^r)\\) is \\(\\{\\{u, v\\}\\}\\), and the depth of \\((F, T^r)\\) is \\(2d\\). Let \\(y, z \\in V_G\\) be any vertices in \\(G\\), and denote \\((F_G^{(d)}(y, z), T_G^{(d)}(y, z))\\) as the depth-2d Local 2-GNN unfolding tree at \\((y, z)\\). Define the following two sets:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.172,
+                0.16,
+                0.823,
+                0.206
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} S _ {1} (y, z) = \\left\\{g: g \\in \\mathsf {B l s o} \\left((F, T ^ {r}), G\\right), g (u) = y, g (v) = z \\right\\}, \\\\ S _ {2} (y, z) = \\left\\{(\\rho , \\tau): (\\rho , \\tau) \\in \\mathsf {B l s o H o m} \\left((F, T ^ {r}), \\left(F _ {G} ^ {(d)} (y, z), T _ {G} ^ {(d)} (y, z)\\right)\\right), \\rho (u) = y, \\rho (v) = z \\right\\}. \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.214,
+                0.565,
+                0.23
+            ],
+            "angle": 0,
+            "content": "Then, Lemma C.23 is equivalent to the following equation:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.372,
+                0.239,
+                0.624,
+                0.272
+            ],
+            "angle": 0,
+            "content": "\\[\n\\sum_ {y, z \\in V _ {G}} | S _ {1} (y, z) | = \\sum_ {y, z \\in V _ {G}} | S _ {2} (y, z) |.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.282,
+                0.566,
+                0.299
+            ],
+            "angle": 0,
+            "content": "We will prove that \\( |S_{1}(y,z)| = |S_{2}(y,z)| \\) for all \\( y,z\\in V_G \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.306,
+                0.825,
+                0.352
+            ],
+            "angle": 0,
+            "content": "Given \\(y, z \\in V_G\\), according to Fact C.19, there exists a bIso \\(\\pi\\) from \\((F_G^{(d)}(y,z), T_G^{(d)}(y,z))\\) to graph \\(G\\). Define a mapping \\(\\sigma\\) such that \\(\\sigma(\\rho, \\tau) = \\pi \\circ \\rho\\) for all \\((\\rho, \\tau) \\in S_2(y,z)\\). It suffices to prove that \\(\\sigma\\) is a bijection from \\(S_2(y,z)\\) to \\(S_1(y,z)\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.356,
+                0.826,
+                0.4
+            ],
+            "angle": 0,
+            "content": "1. We first prove that \\(\\sigma\\) is a mapping from \\(S_{2}(y,z)\\) to \\(S_{1}(y,z)\\), i.e., \\(\\pi \\circ \\rho \\in S_{1}(y,z)\\) for all \\((\\rho, \\tau) \\in S_{2}(y,z)\\). First, we clearly have \\((\\pi \\circ \\rho)(u) = \\pi(y) = y\\), \\((\\pi \\circ \\rho)(v) = \\pi(z) = z\\). We next prove that \\(\\pi \\circ \\rho \\in \\mathsf{BlsO}((F, T^{r}), G)\\). The proof is based on Remark C.18."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.405,
+                0.826,
+                0.451
+            ],
+            "angle": 0,
+            "content": "a) Let \\( w, x \\in V_F \\), \\( w \\neq x \\) be any vertices in the same bag of \\( T^r \\). Since \\( (\\rho, \\tau) \\) is a bIsoHom, \\( \\rho(w) \\neq \\rho(x) \\) and \\( \\rho(w) \\) and \\( \\rho(x) \\) are in the same bag of \\( T_G^{(d)}(y, z) \\). Again, since \\( \\pi \\) is a bIso, we have \\( \\pi(\\rho(w)) \\neq \\pi(\\rho(x)) \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.453,
+                0.826,
+                0.517
+            ],
+            "angle": 0,
+            "content": "b) Let \\( w, x \\in V_F \\) be any vertices in the same bag of \\( T^r \\). Since \\( (\\rho, \\tau) \\) is a bIso-Hom, \\( \\rho(w) \\) and \\( \\rho(x) \\) are in the same bag of \\( T_G^{(d)}(y,z) \\), and \\( \\{w,x\\} \\in E_F \\) iff \\( \\{\\rho(w),\\rho(x)\\} \\in E_{F_G^{(d)}(y,z)} \\). Again, since \\( \\pi \\) is a bIso, \\( \\{\\rho(w),\\rho(x)\\} \\in E_{F_G^{(d)}(y,z)} \\) iff \\( \\{\\pi(\\rho(w)),\\pi(\\rho(x))\\} \\in E_G \\). Therefore, \\( \\{w,x\\} \\in E_F \\) iff \\( \\{\\pi(\\rho(w)),\\pi(\\rho(x))\\} \\in E_G \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.232,
+                0.518,
+                0.65,
+                0.538
+            ],
+            "angle": 0,
+            "content": "c) We clearly have \\(\\ell_F(w) = \\ell_{F_G^{(d)}(y,z)}(\\rho (w)) = \\ell_G(\\pi (\\rho (w)))\\)"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.231,
+                0.405,
+                0.826,
+                0.538
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.212,
+                0.544,
+                0.531,
+                0.56
+            ],
+            "angle": 0,
+            "content": "We have proved that \\(\\pi \\circ \\rho \\in \\mathrm{BlsO}((F,T^r),G)\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.566,
+                0.826,
+                0.694
+            ],
+            "angle": 0,
+            "content": "2. We then prove that \\(\\sigma\\) is a surjection. For all \\(g\\in S_1(y,z)\\), we define a mapping \\((\\rho ,\\tau)\\) from \\((F,T^{r})\\) to \\((F_{G}^{(d)}(y,z),T_{G}^{(d)}(y,z))\\) as follows. First define \\(\\rho (u) = y\\), \\(\\rho (v) = z\\), and set \\(\\tau (r)\\) to be the root of \\((F_{G}^{(d)}(y,z),T_{G}^{(d)}(y,z))\\). Let \\(w_{1},\\dots ,w_{m}\\in V_{F}\\) and \\(w_1^{\\prime},\\dots ,w_{m^{\\prime}}^{\\prime}\\in V_{F}\\) be vertices such that all \\(\\{u,w_i\\}\\) and \\(\\{w_i',v\\}\\) correspond to bags of \\(T^{r}\\) associated to all tree nodes of depth 2. Similarly, by definition of the Local 2-GNN unfolding tree, let \\(x_{1},\\dots ,x_{n}\\in V_{F_{G}^{(d)}(y,z)}\\) be different vertices and \\(x_1^\\prime ,\\dots ,x_n^\\prime \\in V_{F_G^{(d)}(y,z)}\\) be different vertices such that all \\(\\{y,x_i\\}\\) and \\(\\{x_i',z\\}\\) correspond to bags of \\(T_{G}^{(d)}(y,z)\\) associated to all tree nodes of depth 2. Since \\(g\\) and \\(\\pi\\) are bIsos, we have:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.238,
+                0.7,
+                0.825,
+                0.734
+            ],
+            "angle": 0,
+            "content": "- For every \\( w_{i} \\) (\\( i \\in [m] \\)), there exists \\( x_{j} \\) (\\( j \\in [n] \\)), such that \\( g(w_{i}) = \\pi(x_{j}) = \\tilde{x}_{j} \\) for some \\( \\tilde{x}_{j} \\in V_{G} \\) and \\( F[\\{\\{u, v, w_{i}\\}\\}] \\simeq F_{G}^{(d)}(y, z)[\\{\\{y, z, x_{j}\\}\\}] \\simeq G[\\{\\{y, z, \\tilde{x}_{j}\\}\\} \\);"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.238,
+                0.735,
+                0.825,
+                0.771
+            ],
+            "angle": 0,
+            "content": "- For every \\( w_{i}^{\\prime} \\) (\\( i \\in [m'] \\)), there exists \\( x_{j}^{\\prime} \\) (\\( j \\in [n] \\)), such that \\( g(w_{i}^{\\prime}) = \\pi(x_{j}^{\\prime}) = \\tilde{x}_{j}^{\\prime} \\) for some \\( \\tilde{x}_{j}^{\\prime} \\in V_{G} \\) and \\( F[\\{\\{u, v, w_{i}^{\\prime}\\}\\}] \\simeq F_{G}^{(d)}(y, z)[\\{\\{y, z, x_{j}^{\\prime}\\}\\}] \\simeq G[\\{\\{y, z, \\tilde{x}_{j}^{\\prime}\\}] \\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.238,
+                0.7,
+                0.825,
+                0.771
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.776,
+                0.826,
+                0.852
+            ],
+            "angle": 0,
+            "content": "We then define \\(\\rho(w_i) = x_j\\) for each \\(i \\in [m]\\) and \\(\\rho(w_i') = x_j'\\) for each \\(i \\in [m']\\). Based on the above two items, one can easily define \\(\\tau\\) such that each node \\(s\\) in \\(T^r\\) of depth 1 or 2 is mapped by \\(\\tau\\) to a node \\(t\\) in \\(T_G^{(d)}(y,z)\\) of the same depth, \\(\\rho(\\beta_T(s)) = \\beta_{T_G^{(d)}}(t)\\), and \\(\\rho\\) is an isomorphism from \\(F[\\beta_T(s)]\\) to \\(F_G^{(d)}(y,z)[\\beta_{T_G^{(d)}(y,z)}(t)]\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.856,
+                0.827,
+                0.929
+            ],
+            "angle": 0,
+            "content": "Next, we can recursively define \\(\\rho\\)'s image on \\(F[T^r[s]]\\) for each tree node \\(s\\) of depth 2 following the same construction above. This is because \\(g\\) is still a bIso from \\((F[T^r[s]], T^r[s])\\) to \\(G\\), \\(\\pi\\) is still a bIso from \\(\\left(F_G^{(d)}(y,z)[T_G^{(d)}(y,z)[\\tau(s)]], T_G^{(d)}(y,z)[\\tau(s)]\\right)\\) to \\(G\\), and \\(g(\\beta_T(s)) = \\pi(\\beta_{T_G^{(d)}}(\\tau(s)))\\). Recursively applying this procedure, we can construct \\((\\rho, \\tau)\\) such that it is a"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "31"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.21,
+                0.103,
+                0.825,
+                0.141
+            ],
+            "angle": 0,
+            "content": "bIsoHom from \\((F, T^r)\\) to \\(\\left(F_G^{(d)}(y, z), T_G^{(d)}(y, z)\\right)\\). That is to say, we have proved that for all \\(g \\in S_1(u, v)\\), there is a preimage \\((\\rho, \\tau) \\in S_2(y, z)\\) such that \\(\\sigma(\\rho, \\tau) = g\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.146,
+                0.827,
+                0.287
+            ],
+            "angle": 0,
+            "content": "3. We finally prove that \\(\\sigma\\) is an injection. Let \\((\\rho_1, \\tau_1), (\\rho_2, \\tau_2) \\in S_2(y, z)\\) such that \\(\\pi \\circ \\rho_1 = \\pi \\circ \\rho_2\\). Let \\(w_1, \\dots, w_m \\in V_F\\), \\(w_1', \\dots, w_{m'}' \\in V_F\\), \\(x_1, \\dots, x_n \\in V_{F_G^{(d)}(y,z)}\\), and \\(x_1', \\dots, x_n' \\in V_{F_G^{(d)}(y,z)}\\) be defined as in the previous item. For each \\(i \\in [m]\\), let \\(j_1(i)\\) and \\(j_2(i)\\) be indices satisfying \\(\\rho_1(w_i) = x_{j_1(i)}\\) and \\(\\rho_2(w_i) = x_{j_2(i)}\\). It follows that \\(\\pi(x_{j_1(i)}) = \\pi(x_{j_2(i)})\\). By definition of the Local 2-GNN unfolding tree, we must have \\(x_{j_1(i)} = x_{j_2(i)}\\), and thus \\(\\rho_1(w_i) = \\rho_2(w_i)\\). Using a similar approach, we can prove that \\(\\rho_1(w_i') = \\rho_2(w_i')\\) for each \\(i \\in [m']\\). Next, we can recursively apply the above procedure to the subtree \\(T^r[s]\\) for each tree node of depth 2 following the previous item, and finally prove that \\(\\rho_1 = \\rho_2\\). Therefore, \\(\\sigma\\) is an injection."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.292,
+                0.531,
+                0.308
+            ],
+            "angle": 0,
+            "content": "Combining the above three items completes the proof."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.314,
+                0.825,
+                0.36
+            ],
+            "angle": 0,
+            "content": "Proposition C.24. Under Definition C.21, (i) if \\((\\rho, \\tau)\\) is a bIsoSurj, then \\(\\rho\\) is a surjection from \\(F\\) to \\(\\tilde{F}\\) on both vertices and edges; (ii) if \\((\\rho, \\tau)\\) is a bIsoInj, then \\(\\rho\\) is an injection from \\(F\\) to \\(\\tilde{F}\\) on both vertices and edges."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.374,
+                0.825,
+                0.453
+            ],
+            "angle": 0,
+            "content": "Proof. We first prove that \\(\\rho\\) is surjective if \\((\\rho, \\tau)\\) is a bIsoSurj. We will only prove that \\(\\rho\\) is surjective on edges, as proving that \\(\\rho\\) is surjective on vertices is almost the same. For any \\(\\{x, y\\} \\in E_{\\tilde{F}}\\), by Definition C.1(b) we can pick \\(\\tilde{t} \\in V_{\\tilde{T}}\\) such that \\(\\{x, y\\} \\in \\beta_{\\tilde{T}}(\\tilde{t})\\). Since \\(\\tau\\) is surjective, there exists \\(t \\in V_T\\) such that \\(\\tau(t) = \\tilde{t}\\). By definition of bag isomorphism, there exists \\(u, v \\in \\beta_T(t)\\) such that \\(\\rho(u) = x\\), \\(\\rho(v) = y\\), and \\(F[\\{u, v\\}] \\simeq \\tilde{F}[\\{x, y\\}]\\). Therefore, \\(\\{u, v\\} \\in E_F\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.457,
+                0.825,
+                0.487
+            ],
+            "angle": 0,
+            "content": "We next prove that \\(\\rho\\) is injective if \\((\\rho, \\tau)\\) is a bIsoInj. Pick any \\(u \\in V_F\\). It suffices to prove that \\(\\rho(u) = \\rho(v)\\) iff \\(u = v\\) for any \\(v \\in V_F\\). If the result does not hold, consider two cases:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.197,
+                0.492,
+                0.825,
+                0.522
+            ],
+            "angle": 0,
+            "content": "- There exists \\( v \\in V_F, v \\neq u \\) such that \\( \\rho(u) = \\rho(v) \\) and \\( \\{u, v\\} \\) are in the same bag of \\( F \\). This contradicts the definition of bag isomorphism."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.197,
+                0.527,
+                0.827,
+                0.662
+            ],
+            "angle": 0,
+            "content": "- For all \\( v \\in V_F \\) such that \\( v \\neq u \\) and \\( \\rho(u) = \\rho(v) \\), \\( \\{u, v\\} \\) are not in the same bag of \\( F \\). By Proposition C.5, \\( T\\left[\\bigcup_{v \\in V_F: \\rho(u) = \\rho(v)} B_T(v)\\right] \\) is disconnected. We can thus pick a path \\( P \\) in \\( T \\) such that the endpoints \\( t_1 \\) and \\( t_2 \\) are in different connected components of \\( T\\left[\\bigcup_{v \\in V_F: \\rho(u) = \\rho(v)} B_T(v)\\right] \\). This implies that there is a tree node \\( t_3 \\) in \\( P \\) such that \\( \\beta_T(t_3) \\cap \\{v \\in V_F: \\rho(u) = \\rho(v)\\} = \\emptyset \\). Consequently, \\( \\rho(u) \\in \\rho(\\beta_T(t_1)) = \\beta_{\\tilde{T}}(\\tau(t_1)) \\), \\( \\rho(u) \\in \\rho(\\beta_T(t_2)) = \\beta_{\\tilde{T}}(\\tau(t_2)) \\), but \\( \\rho(u) \\notin \\beta_{\\tilde{T}}(\\tau(t_3)) \\). On the other hand, since \\( \\tau \\) is injective, \\( \\tilde{T}\\left[\\tau(V_P)\\right] \\) is also a path and \\( \\tau(t_3) \\) is on the path between \\( \\tau(t_1) \\) and \\( \\tau(t_2) \\) in \\( \\tilde{T} \\). This contradicts the definition of tree decomposition (Definition C.1(c))."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.197,
+                0.492,
+                0.827,
+                0.662
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.668,
+                0.478,
+                0.684
+            ],
+            "angle": 0,
+            "content": "Combining the two cases concludes the proof."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.69,
+                0.825,
+                0.723
+            ],
+            "angle": 0,
+            "content": "Lemma C.25. Let \\( M \\in \\{\\mathrm{Sub},\\mathrm{L},\\mathrm{LF},\\mathrm{F}\\} \\) be any model. For any tree-decomposed graphs \\( (F,T^r),(\\tilde{F},\\tilde{T}^s) \\in S^M \\),"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.182,
+                0.728,
+                0.815,
+                0.78
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathsf {b l s o H o m} ((F, T ^ {r}), (\\tilde {F}, \\tilde {T} ^ {s})) = \\sum_ {(\\widehat {F}, \\widehat {T} ^ {t}) \\in \\mathcal {S} ^ {M}} \\frac {\\mathsf {b l s o S u r j} \\left((F , T ^ {r}) , (\\widehat {F} , \\widehat {T} ^ {t})\\right) \\cdot \\mathsf {b l s o l n j} \\left((\\widehat {F} , \\widehat {T} ^ {t}) , (\\tilde {F} , \\tilde {T} ^ {s})\\right)}{\\mathsf {a u t} (\\widehat {F} , \\widehat {T} ^ {t})},\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.169,
+                0.788,
+                0.825,
+                0.833
+            ],
+            "angle": 0,
+            "content": "where \\(\\operatorname{aut}(\\widehat{F},\\widehat{T}^t)\\) denotes the number of automorphisms of \\((\\widehat{F},\\widehat{T}^t)\\). Here, the summation ranges over all non-isomorphic (tree-decomposed) graphs in \\(S^M\\) and is well-defined as there are only a finite number of graphs making the value in the summation non-zero."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.847,
+                0.508,
+                0.863
+            ],
+            "angle": 0,
+            "content": "Proof. We define the following set of three-tuples:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.216,
+                0.868,
+                0.779,
+                0.923
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} S = \\left\\{\\left((\\widehat {F}, \\widehat {T} ^ {t}), (\\rho^ {\\mathsf {S}}, \\tau^ {\\mathsf {S}}), (\\rho^ {\\mathsf {I}}, \\tau^ {\\mathsf {I}})\\right): (\\widehat {F}, \\widehat {T} ^ {t}) \\in \\mathcal {S} ^ {M}, \\right. \\\\ \\left. \\left(\\rho^ {\\mathsf {S}}, \\tau^ {\\mathsf {S}}\\right) \\in \\mathsf {B l s o S u r j} \\left(\\left(F, T ^ {r}\\right), \\left(\\widehat {F}, \\widehat {T} ^ {t}\\right)\\right), \\left(\\rho^ {\\mathsf {I}}, \\tau^ {\\mathsf {I}}\\right) \\in \\mathsf {B l s o l n j} \\left(\\left(\\widehat {F}, \\widehat {T} ^ {t}\\right), \\left(\\tilde {F}, \\tilde {T} ^ {s}\\right)\\right) \\right\\}. \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "32"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.105,
+                0.377,
+                0.12
+            ],
+            "angle": 0,
+            "content": "Define a mappings \\(\\sigma\\) such that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.332,
+                0.123,
+                0.664,
+                0.149
+            ],
+            "angle": 0,
+            "content": "\\[\n\\sigma \\left(\\left(\\widehat {F}, \\widehat {T} ^ {t}\\right), \\left(\\rho^ {\\mathsf {S}}, \\tau^ {\\mathsf {S}}\\right), \\left(\\rho^ {\\mathsf {I}}, \\tau^ {\\mathsf {I}}\\right)\\right) = \\left(\\rho^ {\\mathsf {I}} \\circ \\rho^ {\\mathsf {S}}, \\tau^ {\\mathsf {I}} \\circ \\tau^ {\\mathsf {S}}\\right)\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.153,
+                0.768,
+                0.178
+            ],
+            "angle": 0,
+            "content": "for all \\(\\left((\\widehat{F},\\widehat{T}^t),(\\rho^\\mathsf{S},\\tau^\\mathsf{S}),(\\rho^\\mathsf{I},\\tau^\\mathsf{I})\\right)\\in S.\\) It suffices to prove the following three statements:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.184,
+                0.576,
+                0.201
+            ],
+            "angle": 0,
+            "content": "1. \\(\\sigma\\) is a mapping from \\(S\\) to \\(\\mathsf{B}\\mathsf{I}\\mathsf{o}\\mathsf{s}\\mathsf{H}\\mathsf{o}\\mathsf{m}((F,T^r),(\\tilde{F},\\tilde{T}^s))\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.207,
+                0.316,
+                0.222
+            ],
+            "angle": 0,
+            "content": "2. \\(\\sigma\\) is surjective;"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.191,
+                0.228,
+                0.827,
+                0.285
+            ],
+            "angle": 0,
+            "content": "3. \\(\\sigma\\left((\\widehat{F}_1, \\widehat{T}_1^{t_1}), (\\rho_1^{\\mathsf{S}}, \\tau_1^{\\mathsf{S}}), (\\rho_1^{\\mathsf{I}}, \\tau_1^{\\mathsf{I}})\\right) = \\sigma\\left((\\widehat{F}_2, \\widehat{T}_2^{t_2}), (\\rho_2^{\\mathsf{S}}, \\tau_2^{\\mathsf{S}}), (\\rho_2^{\\mathsf{I}}, \\tau_2^{\\mathsf{I}})\\right)\\) iff there exists an isomorphism \\((\\widehat{\\rho}, \\widehat{\\tau})\\) from \\((\\widehat{F}_1, \\widehat{T}_1^{t_1})\\) to \\((\\widehat{F}_2, \\widehat{T}_2^{t_2})\\) such that \\(\\widehat{\\rho} \\circ \\rho_1^{\\mathsf{S}} = \\rho_2^{\\mathsf{S}}, \\widehat{\\tau} \\circ \\tau_1^{\\mathsf{S}} = \\tau_2^{\\mathsf{S}}, \\rho_1^{\\mathsf{I}} = \\rho_2^{\\mathsf{I}} \\circ \\widehat{\\rho}, \\tau_1^{\\mathsf{I}} = \\tau_2^{\\mathsf{I}} \\circ \\widehat{\\tau}\\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.191,
+                0.184,
+                0.827,
+                0.285
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.291,
+                0.459,
+                0.307
+            ],
+            "angle": 0,
+            "content": "We will prove these statements one by one."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.313,
+                0.825,
+                0.357
+            ],
+            "angle": 0,
+            "content": "1. We first prove that \\(\\sigma\\) is a mapping from \\(S\\) to \\(\\mathsf{BIsoHom}((F,T^r),(\\tilde{F},\\tilde{T}^s))\\). This simply follows from the fact that both bIsoSurj and bIsoInj are bIsoHom, and the composition of two bIsoHoms are still a bIsoHom."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.191,
+                0.362,
+                0.826,
+                0.396
+            ],
+            "angle": 0,
+            "content": "2. We next prove that \\(\\sigma\\) is surjective. Given \\((\\rho^{\\mathsf{H}},\\tau^{\\mathsf{H}})\\in \\mathrm{BlsoHom}((F,T^{r}),(\\tilde{F},\\tilde{T}^{s}))\\), we define \\((\\widehat{F},\\widehat{T}^{t})\\), \\((\\rho^{\\mathsf{S}},\\tau^{\\mathsf{S}})\\), and \\((\\rho^{\\mathsf{I}},\\tau^{\\mathsf{I}})\\) as follows:"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.191,
+                0.313,
+                0.826,
+                0.396
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.401,
+                0.758,
+                0.419
+            ],
+            "angle": 0,
+            "content": "a) Let \\(\\widehat{F} = \\tilde{F}[\\rho^{\\mathsf{H}}(V_F)]\\) and \\(\\widehat{T}^t = \\tilde{T}^s[\\tau^{\\mathsf{H}}(V_T)]\\). We clearly have \\((\\widehat{F}, \\widehat{T}^t) \\in S^M\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.419,
+                0.814,
+                0.437
+            ],
+            "angle": 0,
+            "content": "b) Let \\(\\rho^{\\mathsf{S}} = \\rho^{\\mathsf{H}}\\) and \\(\\tau^{\\mathsf{S}} = \\tau^{\\mathsf{H}}\\). Obviously, \\((\\rho^{\\mathsf{S}},\\tau^{\\mathsf{S}})\\) is a bIsoSurj from \\((F,T^{r})\\) to \\((\\widehat{F},\\widehat{T}^{t})\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.437,
+                0.823,
+                0.47
+            ],
+            "angle": 0,
+            "content": "c) Define identity mappings \\(\\rho^{\\mathsf{I}}(u) = u\\) for all \\(u\\in V_{\\widehat{F}}\\) and \\(\\tau^1 (t) = t\\) for all \\(t\\in V_{\\widehat{T}}\\). Obviously, \\((\\rho^{\\mathsf{I}},\\tau^{\\mathsf{I}})\\) is a bIsoInj from \\((\\widehat{F},\\widehat{T}^{t})\\) to \\((\\tilde{F},\\tilde{T}^{s})\\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.231,
+                0.401,
+                0.823,
+                0.47
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.212,
+                0.474,
+                0.684,
+                0.491
+            ],
+            "angle": 0,
+            "content": "We clearly have \\(\\rho^{\\mathsf{H}} = \\rho^{\\mathsf{I}}\\circ \\rho^{\\mathsf{S}}\\) and \\(\\tau^{\\mathrm{H}} = \\tau^{\\mathrm{I}}\\circ \\tau^{\\mathrm{S}}\\). Thus, \\(\\sigma\\) is a surjection."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.497,
+                0.825,
+                0.569
+            ],
+            "angle": 0,
+            "content": "3. We finally prove the aforementioned item 3. It suffices to prove only one direction, namely, \\(\\sigma\\left((\\widehat{F}_1,\\widehat{T}_1^{t_1}),(p_1^{\\mathsf{S}},\\tau_1^{\\mathsf{S}}),(p_1^{\\mathsf{l}},\\tau_1^{\\mathsf{l}})\\right) = \\sigma\\left((\\widehat{F}_2,\\widehat{T}_2^{t_2}),(p_2^{\\mathsf{S}},\\tau_2^{\\mathsf{S}}),(p_2^{\\mathsf{l}},\\tau_2^{\\mathsf{l}})\\right)\\) implies that there exists an isomorphism \\((\\widehat{\\rho},\\widehat{\\tau})\\) from \\((\\widehat{F}_1,\\widehat{T}_1^{t_1})\\) to \\((\\widehat{F}_2,\\widehat{T}_2^{t_2})\\) such that \\(\\widehat{\\rho}\\circ \\rho_{1}^{\\mathsf{S}} = \\rho_{2}^{\\mathsf{S}},\\widehat{\\tau}\\circ \\tau_{1}^{\\mathsf{S}} = \\tau_{2}^{\\mathsf{S}},\\) \\(\\rho_1^{\\mathrm{l}} = \\rho_2^{\\mathrm{l}}\\circ \\widehat{\\rho},\\tau_1^{\\mathrm{l}} = \\tau_2^{\\mathrm{l}}\\circ \\widehat{\\tau}\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.232,
+                0.574,
+                0.545,
+                0.592
+            ],
+            "angle": 0,
+            "content": "a) We first prove that \\(\\widehat{F}_1\\simeq \\widehat{F}_2\\) and \\(\\widehat{T}_1^{t_1}\\simeq \\widehat{T}_2^{t_2}\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.251,
+                0.591,
+                0.825,
+                0.649
+            ],
+            "angle": 0,
+            "content": "For any \\( u, v \\in V_F \\), if \\( \\rho_1^{\\mathsf{S}}(u) \\neq \\rho_1^{\\mathsf{S}}(v) \\), then \\( \\rho_1^{\\mathsf{I}}(\\rho_1^{\\mathsf{S}}(u)) \\neq \\rho_1^{\\mathsf{I}}(\\rho_1^{\\mathsf{S}}(v)) \\) since \\( \\rho_1^{\\mathsf{I}} \\) is an injection (by Proposition C.24). Therefore, \\( \\rho_2^{\\mathsf{I}}(\\rho_2^{\\mathsf{S}}(u)) \\neq \\rho_2^{\\mathsf{I}}(\\rho_2^{\\mathsf{S}}(v)) \\), and thus \\( \\rho_2^{\\mathsf{S}}(u) \\neq \\rho_2^{\\mathsf{S}}(v) \\). By symmetry, we also have that \\( \\rho_2^{\\mathsf{S}}(u) \\neq \\rho_2^{\\mathsf{S}}(v) \\) implies \\( \\rho_1^{\\mathsf{S}}(u) \\neq \\rho_1^{\\mathsf{S}}(v) \\). This proves that \\( \\rho_1^{\\mathsf{S}}(u) = \\rho_1^{\\mathsf{S}}(v) \\) iff \\( \\rho_2^{\\mathsf{S}}(u) = \\rho_2^{\\mathsf{S}}(v) \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.251,
+                0.647,
+                0.825,
+                0.766
+            ],
+            "angle": 0,
+            "content": "For any \\(u, v \\in V_F\\), if \\(\\{\\rho_1^S(u), \\rho_1^S(v)\\} \\in E_{\\widehat{F}_1}\\), then \\(\\{\\rho_1^1(\\rho_1^S(u)), \\rho_1^1(\\rho_1^S(v))\\} \\in E_{\\widehat{F}}\\) since \\(\\rho_1^1\\) is a homomorphism. Therefore, \\(\\{\\rho_2^1(\\rho_2^S(u)), \\rho_2^1(\\rho_2^S(v))\\} \\in E_{\\widehat{F}}\\). This implies that \\(\\rho_2^1(\\rho_2^S(u))\\) and \\(\\rho_2^1(\\rho_2^S(v))\\) are in the same bag of \\(\\tilde{T}^s\\). Therefore, there are vertices \\(x, y \\in V_{\\widehat{F}_2}\\) in the same bag of \\(\\widehat{T}_2^{s_2}\\) such that \\(\\rho_2^1(x) = \\rho_2^1(\\rho_2^S(u))\\) and \\(\\rho_2^1(y) = \\rho_2^1(\\rho_2^S(v))\\), and by definition of bag isomorphism we have \\(\\{x, y\\} \\in E_{\\widehat{F}_2}\\). Since \\(\\rho_2\\) is injective, \\(x = \\rho_2^S(u)\\) and \\(y = \\rho_2^S(v)\\), namely, \\(\\{\\rho_2^S(u), \\rho_2^S(v)\\} \\in E_{\\widehat{F}_2}\\). By symmetry, we can prove that \\(\\{\\rho_1^S(u), \\rho_1^S(v)\\} \\in E_{\\widehat{F}_1}\\) iff \\(\\{\\rho_2^S(u), \\rho_2^S(v)\\} \\in E_{\\widehat{F}_2}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.252,
+                0.765,
+                0.825,
+                0.821
+            ],
+            "angle": 0,
+            "content": "Finally, noting that \\(\\rho_1^{\\mathsf{S}}\\) and \\(\\rho_2^{\\mathsf{S}}\\) are surjective (by Proposition C.24) and \\(\\ell_{\\widehat{F}_1}(\\rho_1^{\\mathsf{S}}(u)) = \\ell_F(u) = \\ell_{\\widehat{F}_2}(\\rho_2^{\\mathsf{S}}(u))\\) for all \\(u \\in V_F\\), we obtain that \\(\\widehat{F}_1 \\simeq \\widehat{F}_2\\). Following the same procedure, we can prove that \\(\\widehat{T}_1^{t_1} \\simeq \\widehat{T}_2^{t_2}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.819,
+                0.825,
+                0.851
+            ],
+            "angle": 0,
+            "content": "b) Consequently, there exist isomorphisms \\(\\widehat{\\rho}\\) and \\(\\widehat{\\tau}\\) such that \\(\\widehat{\\rho} \\circ \\rho_1^{\\mathsf{S}} = \\rho_2^{\\mathsf{S}}, \\widehat{\\tau} \\circ \\tau_1^{\\mathsf{S}} = \\tau_2^{\\mathsf{S}}\\). For any node \\(q \\in V_T\\),"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.275,
+                0.852,
+                0.802,
+                0.874
+            ],
+            "angle": 0,
+            "content": "\\[\n\\widehat {\\rho} (\\beta_ {\\widehat {T} _ {1}} (\\tau_ {1} ^ {\\mathsf {S}} (q))) = \\widehat {\\rho} (\\rho_ {1} ^ {\\mathsf {S}} (\\beta_ {T} (q))) = \\rho_ {2} ^ {\\mathsf {S}} (\\beta_ {T} (q))) = \\beta_ {\\widehat {T} _ {2}} (\\tau_ {2} ^ {\\mathsf {S}} (q)) = \\beta_ {\\widehat {T} _ {2}} (\\widehat {\\tau} (\\tau_ {1} ^ {\\mathsf {S}} (q))).\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.252,
+                0.877,
+                0.825,
+                0.925
+            ],
+            "angle": 0,
+            "content": "Since \\(\\tau_1^S\\) is surjective, \\(\\tau_1^S (q)\\) ranges over all nodes in \\(\\widehat{T}_1^{s_1}\\) when \\(q\\) ranges over \\(V_{T}\\). We thus conclude that \\((\\rho ,\\tau)\\) is an isomorphism from \\((\\widehat{F}_1,\\widehat{T}_1^{t_1})\\) to \\((\\widehat{F}_2,\\widehat{T}_2^{t_2})\\) (see Definition C.8)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "33"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.232,
+                0.103,
+                0.825,
+                0.165
+            ],
+            "angle": 0,
+            "content": "c) We finally prove that \\(\\rho_1^{\\mathsf{I}} = \\rho_2^{\\mathsf{I}}\\circ \\widehat{\\rho}\\) and \\(\\tau_1^{\\mathsf{I}} = \\tau_2^{\\mathsf{I}}\\circ \\widehat{\\tau}\\). Pick any \\(u\\in V_F\\), we have \\(\\rho_2^{\\mathsf{I}}(\\widehat{\\rho} (\\rho_1^{\\mathsf{S}}(u))) = \\rho_2^{\\mathsf{I}}(\\rho_2^{\\mathsf{S}}(u)) = \\rho_1^{\\mathsf{I}}(\\rho_1^{\\mathsf{S}}(u))\\). Since \\(\\rho_{1}^{\\mathsf{S}}\\) is surjective, \\(\\rho_{1}^{\\mathsf{S}}(u)\\) ranges over all nodes in \\(\\widehat{F}_1\\) when \\(u\\) ranges over \\(V_{F}\\). This proves that \\(\\rho_1^{\\mathsf{I}} = \\rho_2^{\\mathsf{I}}\\circ \\widehat{\\rho}\\). Following the same procedure, we can prove that \\(\\tau_1^{\\mathsf{I}} = \\tau_2^{\\mathsf{I}}\\circ \\widehat{\\tau}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.17,
+                0.53,
+                0.186
+            ],
+            "angle": 0,
+            "content": "Combining the above three items concludes the proof."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.171,
+                0.825,
+                0.183
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.203,
+                0.825,
+                0.233
+            ],
+            "angle": 0,
+            "content": "In the following, we further define two technical concepts that will be used to describe the next result. We note that these technical concepts also appeared in Dell et al. (2018)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.238,
+                0.825,
+                0.331
+            ],
+            "angle": 0,
+            "content": "Definition C.26. A tree-decomposed graph \\((\\tilde{F},\\tilde{T}^{s})\\) is called a bag extension of another tree-decomposed graph \\((F,T^r)\\) if there is a graph \\(H\\) and a mapping \\((\\rho ,\\tau)\\) such that \\(F\\) is a subgraph of \\(H\\) and \\((\\rho ,\\tau)\\) is an isomorphism from the tree-decomposed graph \\((H,T^r)\\) to \\((\\tilde{F},\\tilde{T}^{s})\\). Define \\(\\mathsf{BExt}\\left((F,T^r),(\\tilde{F},\\tilde{T}^s)\\right)\\) to be the set of all mappings \\((\\rho ,\\tau)\\) that satisfies the above conditions, and define \\(\\mathsf{bExt}\\left((F,T^r),(\\tilde{F},\\tilde{T}^s)\\right) = \\left|\\mathsf{BExt}\\left((F,T^r),(\\tilde{F},\\tilde{T}^s)\\right)\\right|\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.335,
+                0.825,
+                0.381
+            ],
+            "angle": 0,
+            "content": "Remark C.27. In other words, a bag extension of a tree-decomposed graph \\((F,T^r)\\) can be obtained by adding an arbitrary number of edges to \\(F\\) while ensuring that each added edge is contained in a tree node in \\(T^r\\). A immediate fact is that \\((\\rho ,\\tau)\\) is a homomorphism from \\((F,T^{r})\\) to \\((\\tilde{F},\\tilde{T}^{s})\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.384,
+                0.825,
+                0.459
+            ],
+            "angle": 0,
+            "content": "Definition C.28. Given a tree-decomposed graph \\((F,T^r)\\) and a graph \\(G\\), a bag-strong homomorphism from \\((F,T^r)\\) to \\(G\\) (abbreviated as \"bStrHom\") is a homomorphism \\(f\\) from \\(F\\) to \\(G\\) such that, for all \\(t\\in V_T\\), \\(f\\) is a strong homomorphism from \\(F[\\beta_T(t)]\\) to \\(G[f(\\beta_T(t))]\\), i.e., \\(\\{u,v\\} \\in E_{F[\\beta_T(t)]}\\) iff \\(\\{f(u),f(v)\\} \\in E_{G[f(\\beta_T(t))]}\\). Denote BStrHom \\(((F,T^r),G)\\) to be the set of all bStrHom from \\((F,T^r)\\) to \\(G\\), and denote bStrHom \\(((F,T^r),G) = |\\mathsf{BStrHom}((F,T^r),G)|\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.469,
+                0.455,
+                0.485
+            ],
+            "angle": 0,
+            "content": "The following equation is straightforward:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.488,
+                0.827,
+                0.518
+            ],
+            "angle": 0,
+            "content": "Lemma C.29. Let \\( M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\} \\) be any model. For any graph \\( G \\) and tree-decomposed graph \\( (F, T^r) \\in S^M \\),"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.242,
+                0.525,
+                0.754,
+                0.577
+            ],
+            "angle": 0,
+            "content": "\\[\n\\operatorname {h o m} (F, G) = \\sum_ {(\\tilde {F}, \\tilde {T} ^ {s}) \\in \\mathcal {S} ^ {M}} \\frac {\\mathrm {b E x t} ((F , T ^ {r}) , (\\tilde {F} , \\tilde {T} ^ {s})) \\cdot \\mathrm {b S t r H o m} ((\\tilde {F} , \\tilde {T} ^ {s}) , G)}{\\operatorname {a u t} (\\tilde {F} , \\tilde {T} ^ {s})},\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.586,
+                0.825,
+                0.631
+            ],
+            "angle": 0,
+            "content": "where \\(\\operatorname{aut}(\\tilde{F},\\tilde{T}^{s})\\) denotes the number of automorphisms of \\((\\tilde{F},\\tilde{T}^{s})\\). Here, the summation ranges over all non-isomorphic (tree-decomposed) graphs in \\(S^M\\) and is well-defined as there are only a finite number of graphs making the value in the summation non-zero."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.648,
+                0.827,
+                0.678
+            ],
+            "angle": 0,
+            "content": "Proof. The proof has a similar structure to the previous lemma. We define the following set of three-tuples:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.172,
+                0.685,
+                0.864,
+                0.713
+            ],
+            "angle": 0,
+            "content": "\\[\nS = \\left\\{\\left((\\tilde {F}, \\tilde {T} ^ {s}), (\\rho , \\tau), g\\right): (\\tilde {F}, \\tilde {T} ^ {s}) \\in \\mathcal {S} ^ {M}, (\\rho , \\tau) \\in \\mathsf {B E x t} \\Big ((F, T ^ {r}), (\\tilde {F}, \\tilde {T} ^ {s}) \\Big), g \\in \\mathsf {B S t r H o m} ((\\tilde {F}, \\tilde {T} ^ {s}), G) \\right\\}.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.72,
+                0.825,
+                0.786
+            ],
+            "angle": 0,
+            "content": "Define a mappings \\(\\sigma\\) such that \\(\\sigma\\left((\\tilde{F},\\tilde{T}^s),(\\rho,\\tau),g\\right)=g\\circ\\rho\\) for all \\(\\left((\\tilde{F},\\tilde{T}^s),(\\rho,\\tau),g\\right)\\in S\\). It is straightforward to see that \\(\\sigma\\) is a mapping from \\(S\\) to \\(\\mathsf{BlsoHom}((F,T^r),(\\tilde{F},\\tilde{T}^s))\\), namely, \\(g\\circ\\rho\\) is a homomorphism from \\(F\\) to \\(G\\) for all \\(\\left((\\tilde{F},\\tilde{T}^s),(\\rho,\\tau),g\\right)\\in S\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.791,
+                0.822,
+                0.81
+            ],
+            "angle": 0,
+            "content": "We then prove that \\(\\sigma\\) is surjective. Given \\(h\\in \\mathrm{Hom}(F,G)\\), define \\((\\tilde{F},\\tilde{T}^{s})\\), \\((\\rho ,\\tau)\\), and \\(g\\) as follows:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.191,
+                0.815,
+                0.826,
+                0.849
+            ],
+            "angle": 0,
+            "content": "a) Define \\(\\tilde{F}\\) be the graph obtained from \\(F\\) by adding edges \\(\\{\\{u,v\\} : \\exists t \\in V_T \\text{s.t.} u,v \\in \\beta_T(t), \\{h(u),h(v)\\} \\in E_G\\}\\), and let \\(\\tilde{T}^s = T^r\\). Clearly, \\((\\tilde{F},\\tilde{T}^s)\\) is a bag extension of \\((F,T^r)\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.19,
+                0.853,
+                0.825,
+                0.887
+            ],
+            "angle": 0,
+            "content": "b) Define identity mappings \\(\\rho(u) = u\\) for all \\(u \\in V_F\\) and \\(\\tau(t) = t\\) for all \\(t \\in V_T\\). Clearly, \\((\\rho, \\tau) \\in \\mathsf{BExt}((F, T^r), (\\tilde{F}, \\tilde{T}^s))\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.191,
+                0.892,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "c) Let \\( g = h \\). It is easy to see that \\( g \\) is a strong homomorphism from \\( \\tilde{F}[\\beta_{\\tilde{T}}(t)] \\) to \\( G[g(\\beta_{\\tilde{T}}(t))] \\) for each \\( t \\in V_{\\tilde{T}} \\). Thus, \\( g \\in \\mathsf{BStrHom}((\\tilde{F}, \\tilde{T}^s), G) \\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.19,
+                0.815,
+                0.826,
+                0.926
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "34"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.6,
+                0.12
+            ],
+            "angle": 0,
+            "content": "Noting that \\( h = g = g \\circ \\rho \\), we have proved that \\( \\sigma \\) is a surjection."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.127,
+                0.827,
+                0.273
+            ],
+            "angle": 0,
+            "content": "We finally prove that \\(\\sigma\\left((\\tilde{F}_1, \\tilde{T}_1^{s_1}), (\\rho_1, \\tau_1), g_1\\right) = \\sigma\\left((\\tilde{F}_2, \\tilde{T}_2^{s_2}), (\\rho_2, \\tau_2), g_2\\right)\\) implies that there exists an isomorphism \\((\\tilde{\\rho}, \\tilde{\\tau})\\) from \\((\\tilde{F}_1, \\tilde{T}_1^{s_1})\\) to \\((\\tilde{F}_2, \\tilde{T}_2^{s_2})\\) such that \\(\\tilde{\\rho} \\circ \\rho_1 = \\rho_2\\), \\(\\tilde{\\tau} \\circ \\tau_1 = \\tau_2\\), \\(g_1 = g_2 \\circ \\tilde{\\rho}\\). We first prove that \\(\\tilde{F}_1 \\simeq \\tilde{F}_2\\) and \\(\\tilde{T}_1^{s_1} \\simeq \\tilde{T}_2^{s_2}\\). (i) For any \\(u, v \\in V_F\\), we obviously have \\(\\rho_1(u) = \\rho_1(v)\\) iff \\(u = v\\) iff \\(\\rho_2(u) = \\rho_2(v)\\). (ii) Let \\(i \\in \\{1, 2\\}\\). For any \\(u, v \\in V_F\\), \\(\\{\\rho_i(u), \\rho_i(v)\\} \\in E_{\\tilde{F}_i}\\) iff \\(\\{u, v\\} \\in E_F\\) or \\(u, v\\) are in the same bag of \\(T^r\\) and \\(\\{g_i(\\rho_i(u)), g(\\rho_i(v))\\} \\in E_G\\). Since \\(g_1 \\circ \\rho_1 = g_2 \\circ \\rho_2\\), we have \\(\\{\\rho_1(u), \\rho_1(v)\\} \\in E_{\\tilde{F}_1}\\) iff \\(\\{\\rho_2(u), \\rho_2(v)\\} \\in E_{\\tilde{F}_2}\\). (iii) Finally, noting that \\(\\rho_1\\) and \\(\\rho_2\\) are bijective and \\(\\ell_{\\tilde{F}_1}(\\rho_1(u)) = \\ell_F(u) = \\ell_{\\tilde{F}_2}(\\rho_2(u))\\) for all \\(u \\in V_F\\), we obtain that \\(\\tilde{F}_1 \\simeq \\tilde{F}_2\\). On the other hand, \\(\\tilde{T}_1^{s_1} \\simeq \\tilde{T}_2^{s_2}\\) trivially holds. The remaining procedure is almost the same as in the previous lemma."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.289,
+                0.825,
+                0.318
+            ],
+            "angle": 0,
+            "content": "We next show that the mapping bStrHom can be further decomposed as shown in Lemma C.31. We need an auxiliary concept:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.322,
+                0.825,
+                0.403
+            ],
+            "angle": 0,
+            "content": "Definition C.30. Given two tree-decomposed graphs \\((F,T^r)\\) and \\((\\tilde{F},\\tilde{T}^{s})\\) , a homomorphism \\((\\rho ,\\tau)\\) from \\((F,T^{r})\\) to \\((\\tilde{F},\\tilde{T}^{s})\\) is called bag-strong surjective (abbreviated as \"bStrSurj\") if \\(\\rho\\) is a bagstrong homomorphism from \\((F,T^{r})\\) to \\(\\tilde{F}\\) and is surjective on both vertices and edges, and \\(\\tau\\) is an isomorphism from \\(T^r\\) to \\(\\tilde{T}^s\\) . Denote BStrSurj \\(((F,T^r),(\\tilde{F},\\tilde{T}^{s}))\\) to be the set of all bStrSurj from \\((F,T^{r})\\) to \\((\\tilde{F},\\tilde{T}^{s})\\) , and denote bStrSurj \\(((F,T^r),(\\tilde{F},\\tilde{T}^{s})) = |\\mathsf{BStrSurj}((F,T^r),(\\tilde{F},\\tilde{T}^{s}))|\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.405,
+                0.827,
+                0.435
+            ],
+            "angle": 0,
+            "content": "Lemma C.31. Let \\( M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\} \\) be any model. For any graph \\( G \\) and tree-decomposed graph \\( (F, T^r) \\in S^M \\),"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.212,
+                0.441,
+                0.785,
+                0.492
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathsf{bStrHom}((F,T^{r}),G) = \\sum_{(\\tilde{F},\\tilde{T}^{s})\\in \\mathcal{S}^{M}}\\frac{\\mathsf{bStrSurj}\\left((F,T^{r}),( \\tilde{F},\\tilde{T}^{s})\\right)\\cdot\\mathsf{blso}\\left((\\tilde{F},\\tilde{T}^{s}),G\\right)}{\\mathsf{aut}(\\tilde{F},\\tilde{T}^{s})},\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.501,
+                0.825,
+                0.545
+            ],
+            "angle": 0,
+            "content": "where \\(\\operatorname{aut}(\\tilde{F},\\tilde{T}^s)\\) denotes the number of automorphisms of \\((\\tilde{F},\\tilde{T}^{s})\\) . Here, the summation ranges over all non-isomorphic (tree-decomposed) graphs in \\(S^M\\) and is well-defined as there are only a finite number of graphs making the value in the summation non-zero."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.56,
+                0.827,
+                0.589
+            ],
+            "angle": 0,
+            "content": "Proof. The proof has a similar structure to the previous lemma. We define the following set of three-tuples:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.172,
+                0.595,
+                0.852,
+                0.622
+            ],
+            "angle": 0,
+            "content": "\\[\nS = \\left\\{\\left(\\left(\\tilde {F}, \\tilde {T} ^ {s}\\right), (\\rho , \\tau), g\\right): \\left(\\tilde {F}, \\tilde {T} ^ {s}\\right) \\in \\mathcal {S} ^ {M}, (\\rho , \\tau) \\in \\mathrm {B S t r S u r j} \\left(\\left(F, T ^ {r}\\right), \\left(\\tilde {F}, \\tilde {T} ^ {s}\\right)\\right), g \\in \\mathrm {B l s o} \\left(\\left(\\tilde {F}, \\tilde {T} ^ {s}\\right), G\\right) \\right\\}.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.629,
+                0.827,
+                0.666
+            ],
+            "angle": 0,
+            "content": "Define a mappings \\(\\sigma\\) such that \\(\\sigma\\left((\\tilde{F},\\tilde{T}^s),(\\rho,\\tau),g\\right) = g\\circ\\rho\\) for all \\(\\left((\\tilde{F},\\tilde{T}^s),(\\rho,\\tau),g\\right)\\in S\\). It suffices to prove the following three statements:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.672,
+                0.538,
+                0.688
+            ],
+            "angle": 0,
+            "content": "1. \\(\\sigma\\) is a mapping from \\(S\\) to \\(\\mathsf{BStrHom}((F,T^r),G)\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.694,
+                0.314,
+                0.709
+            ],
+            "angle": 0,
+            "content": "2. \\(\\sigma\\) is surjective;"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.191,
+                0.715,
+                0.825,
+                0.757
+            ],
+            "angle": 0,
+            "content": "3. \\(\\sigma\\left((\\tilde{F}_1, \\tilde{T}_1^{s_1}), (\\rho_1, \\tau_1), g_1\\right) = \\sigma\\left((\\tilde{F}_2, \\tilde{T}_2^{s_2}), (\\rho_2, \\tau_2), g_2\\right)\\) iff there exists an isomorphism \\((\\tilde{\\rho}, \\tilde{\\tau})\\) from \\((\\tilde{F}_1, \\tilde{T}_1^{s_1})\\) to \\((\\tilde{F}_2, \\tilde{T}_2^{s_2})\\) such that \\(\\tilde{\\rho} \\circ \\rho_1 = \\rho_2\\), \\(\\tilde{\\tau} \\circ \\tau_1 = \\tau_2\\), \\(g_1 = g_2 \\circ \\tilde{\\rho}\\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.191,
+                0.672,
+                0.825,
+                0.757
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.762,
+                0.459,
+                0.777
+            ],
+            "angle": 0,
+            "content": "We will prove these statements one by one."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.783,
+                0.827,
+                0.868
+            ],
+            "angle": 0,
+            "content": "1. We first prove that \\(\\sigma\\) is a mapping from \\(S\\) to \\(\\mathsf{BStrHom}((F,T^r),G)\\). Pick any \\(\\left((\\tilde{F},\\tilde{T}^s),(\\rho ,\\tau),g\\right)\\in S\\). Pick any \\(t\\in V_T\\) and \\(u,v\\in \\beta_{T}(t)\\). Then, \\(\\{u,v\\} \\in E_F\\) iff \\(\\{\\rho (u),\\rho (v)\\} \\in E_{\\tilde{F}}\\) (since \\(\\rho\\) is a strong homomorphism from \\(F[\\beta_T(t)]\\) to \\(\\tilde{F} [\\rho (\\beta_T(t))]\\)). Also, \\(\\rho (u),\\rho (v)\\in \\beta_{\\tilde{F}}(\\tau (t))\\) are in the same bag. Similarly, \\(\\{\\rho (u),\\rho (v)\\} \\in E_{\\tilde{F}}\\) iff \\(\\{g(\\rho (u)),g(\\rho (v))\\} \\in E_G\\) (since \\(g\\) is a bIso). Thus, \\(g\\circ \\rho\\) is a bag-strong homomorphism."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.191,
+                0.873,
+                0.825,
+                0.904
+            ],
+            "angle": 0,
+            "content": "2. We next prove that \\(\\sigma\\) is surjective. Given \\(h\\in \\mathsf{BStrHom}((F,T^r),G)\\) , define \\((\\tilde{F},\\tilde{T}^{s}),(\\rho ,\\tau)\\) and \\(g\\) as follows. First define a relation \\(\\sim\\) on set \\(V_{F}\\) such that \\(u\\sim v\\) iff the following hold:"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.191,
+                0.783,
+                0.827,
+                0.904
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.232,
+                0.91,
+                0.347,
+                0.925
+            ],
+            "angle": 0,
+            "content": "a) \\(h(u) = h(v)\\)"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "35"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.23,
+                0.104,
+                0.825,
+                0.135
+            ],
+            "angle": 0,
+            "content": "b) There exists a path \\(P\\) in \\(T^r\\) with endpoints \\(t_1, t_2 \\in V_T\\) such that \\(u \\in \\beta_T(t_1)\\), \\(v \\in \\beta_T(t_2)\\), and all node \\(t\\) on path \\(P\\) satisfies that \\(h(u) \\in h(\\beta_T(t))\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.14,
+                0.827,
+                0.213
+            ],
+            "angle": 0,
+            "content": "It is easy to see that \\(\\sim\\) is an equivalence relation on \\(V_{F}\\). We can thus define a mapping \\(\\rho\\) that respects the relation, i.e., \\(\\rho(u) = \\rho(v)\\) iff \\(u \\sim v\\) for all \\(u, v \\in V_{F}\\). Moreover, for any edge \\(\\{u, v\\} \\in E_{F}\\), \\(\\rho(u) \\neq \\rho(v)\\) (since \\(h\\) is a homomorphism and \\(h(u) \\neq h(v)\\)). This implies that we can define \\(\\tilde{F}\\) to be the homomorphic image of \\(F\\) such that \\(\\rho\\) is the surjective homomorphism on both vertices and edges."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.212,
+                0.218,
+                0.825,
+                0.29
+            ],
+            "angle": 0,
+            "content": "We then define the mapping \\( g: V_{\\tilde{F}} \\to V_G \\) such that \\( g(\\rho(u)) = h(u) \\) for all \\( u \\in V_F \\). Note that \\( g \\) is well-defined since \\( \\rho(u) = \\rho(v) \\) implies \\( h(u) = h(v) \\) for all \\( u, v \\in V_F \\), and \\( \\rho: V_F \\to V_{\\tilde{F}} \\) is surjective. It follows that \\( h = g \\circ \\rho \\). To prove that \\( g \\) is a homomorphism, note that for all \\( \\{x, y\\} \\in E_{\\tilde{F}} \\), there exists an edge \\( \\{u, v\\} \\in E_F \\) with \\( \\rho(u) = x \\), \\( \\rho(v) = y \\), which implies that \\( \\{h(u), h(v)\\} \\in E_G \\) (since \\( h \\) is a homomorphism), namely, \\( \\{g(x), g(y)\\} \\in E_G \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.297,
+                0.825,
+                0.435
+            ],
+            "angle": 0,
+            "content": "We next define tree \\(\\tilde{T}^s = (V_T, E_T, \\beta_{\\tilde{T}})\\), \\(s = t\\), and identity mapping \\(\\tau\\) so that \\(\\tau\\) is an isomorphism from \\(T_r\\) to \\(T_s\\) (ignoring the labels). Set \\(\\beta_{\\tilde{T}}(t) = \\rho(\\beta_T(t))\\) for all \\(t \\in V_T\\). We will prove that \\((\\tilde{F}, \\tilde{T}^s) \\in S^M\\) is a valid tree decomposition. It suffices to prove that Definition C.1(c) holds. Pick any vertex \\(x \\in V_{\\tilde{F}}\\) and tree node \\(t_1, t_2 \\in B_{\\tilde{T}}(x)\\). Then, there exists \\(u \\in \\beta_T(t_1), v \\in \\beta_T(t_2)\\) such that \\(\\rho(u) = x\\), \\(\\rho(v) = x\\). Therefore, \\(u \\sim v\\). As such, there exists a path \\(P\\) in \\(T^r\\) such that all node \\(t\\) on \\(P\\) satisfies that there exists \\(w \\in \\beta_T(t)\\) with \\(h(w) = h(u)\\), namely, \\(w \\sim u\\). Consequently, \\(\\rho(u) \\in \\beta_{\\tilde{T}}(t)\\), implying that \\(\\tilde{T}^s[B_{\\tilde{T}}(x)]\\) is connected. This proves that \\((\\tilde{F}, \\tilde{T}^s) \\in S^M\\). Also, \\((\\rho, \\tau)\\) is clearly a homomorphism from \\((F, T^r)\\) to \\((\\tilde{F}, \\tilde{T}^s)\\) according to Definition C.20."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.44,
+                0.826,
+                0.527
+            ],
+            "angle": 0,
+            "content": "It remains to prove that \\(\\rho\\) is a bag-strong homomorphism and \\(g\\) is a bIso. Pick any \\(t \\in V_T\\) and \\(u, v \\in \\beta_T(t)\\). If \\(\\{u, v\\} \\notin E_F\\), then \\(\\{h(u), h(v)\\} \\notin E_G\\) (since \\(h\\) is a bag-strong homomorphism). Therefore, \\(\\{\\rho(u), \\rho(v)\\} \\notin E_{\\tilde{F}}\\) (since \\(g\\) is a homomorphism), namely, \\(\\rho\\) is a bag-strong homomorphism. Since \\(\\rho\\) is surjective, \\(\\{\\rho(u), \\rho(v)\\}\\) ranges over all vertices in the same bag of \\(\\hat{T}^s\\) when \\(t \\in V_T\\) and \\(u, v \\in \\beta_T(t)\\) are arbitrary. Therefore, \\(g\\) is a bIso because \\(\\{\\rho(u), \\rho(v)\\} \\notin E_{\\tilde{F}}\\) iff \\(\\{h(u), h(v)\\} \\notin E_G\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.534,
+                0.825,
+                0.633
+            ],
+            "angle": 0,
+            "content": "3. We finally prove that \\(\\sigma\\left((\\tilde{F}_1, \\tilde{T}_1^{s_1}), (\\rho_1, \\tau_1), g_1\\right) = \\sigma\\left((\\tilde{F}_2, \\tilde{T}_2^{s_2}), (\\rho_2, \\tau_2), g_2\\right)\\) implies there exists an isomorphism \\((\\tilde{\\rho}, \\tilde{\\tau})\\) from \\((\\tilde{F}_1, \\tilde{T}_1^{s_1})\\) to \\((\\tilde{F}_2, \\tilde{T}_2^{s_2})\\) such that \\(\\tilde{\\rho} \\circ \\rho_1 = \\rho_2\\), \\(\\tilde{\\tau} \\circ \\tau_1 = \\tau_2\\), \\(g_1 = g_2 \\circ \\tilde{\\rho}\\). Let \\(h = g_1 \\circ \\rho_1 = g_2 \\circ \\rho_2\\). Here, we will only prove that \\(\\tilde{F}_1 \\simeq \\tilde{F}_2\\) since the remaining procedure is almost the same as previous proofs. Since both \\(\\tilde{F}_1\\) and \\(\\tilde{F}_2\\) are homomorphic images of \\(F\\), it suffices to prove that, for all \\(u, v \\in V_F\\), \\(\\rho_1(u) = \\rho_1(v)\\) iff the following hold:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.232,
+                0.639,
+                0.348,
+                0.655
+            ],
+            "angle": 0,
+            "content": "a) \\(h(u) = h(v)\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.656,
+                0.825,
+                0.687
+            ],
+            "angle": 0,
+            "content": "b) There exists a path \\(P\\) in \\(T^r\\) with endpoints \\(t_1, t_2 \\in V_T\\) such that \\(u \\in \\beta_T(t_1)\\), \\(v \\in \\beta_T(t_2)\\), and all node \\(t\\) on path \\(P\\) satisfies that \\(h(u) \\in h(\\beta_T(t))\\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.231,
+                0.639,
+                0.825,
+                0.687
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.692,
+                0.825,
+                0.752
+            ],
+            "angle": 0,
+            "content": "On one hand, if \\(\\rho_{1}(u) = \\rho_{1}(v)\\), we clearly have \\(h(u) = h(v)\\) and there exists \\(t_1 \\in B_T(u), t_2 \\in B_T(v)\\) such that \\(t_1, t_2 \\in B_{\\tilde{T}_1}(\\rho_1(u))\\). Since \\(\\tilde{T}_1[B_{\\tilde{T}_1}(\\rho_1(u))]\\) is connected, there is a path \\(P\\) with endpoints \\(t_1, t_2\\) such that all nodes \\(t\\) on \\(P\\) satisfies \\(\\rho_{1}(u) \\in \\rho_{1}(\\beta_{T}(t))\\) and thus \\(h(u) \\in h(\\beta_{T}(t))\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.212,
+                0.752,
+                0.825,
+                0.767
+            ],
+            "angle": 0,
+            "content": "On the other hand, if \\(\\rho_1(u) \\neq \\rho_1(v)\\) but the above items (a) and (b) hold, consider two cases:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.238,
+                0.774,
+                0.822,
+                0.804
+            ],
+            "angle": 0,
+            "content": "- \\( u \\) and \\( v \\) are in the same bag of \\( T \\). Then, \\( \\rho_1(u) \\) and \\( \\rho_1(v) \\) are in the same bag of \\( \\tilde{T}_1 \\). Since \\( g_1 \\) is a bIso, \\( g_1(\\rho_1(u)) \\neq g_1(\\rho_1(v)) \\), which contradicts item (a)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.238,
+                0.806,
+                0.825,
+                0.881
+            ],
+            "angle": 0,
+            "content": "- \\( u \\) and \\( v \\) are not in the same bag of \\( T \\). Then, there exist two adjacent nodes \\( t_1, t_2 \\) on \\( P \\) such that \\( \\rho_1(u) \\in \\beta_{\\tilde{T}_1}(t_1), \\rho_1(u) \\notin \\beta_{\\tilde{T}_1}(t_2) \\). By Definition C.4(c), \\( \\beta_{\\tilde{T}_1}(t_2) \\subset \\beta_{\\tilde{T}_1}(t_1) \\). Also, item (b) implies that there exists \\( w \\in \\beta_T(t_2) \\) such that \\( h(w) = h(u) \\). Therefore, \\( \\rho_1(u) \\) and \\( \\rho_1(w) \\) are two different nodes in \\( \\beta_{\\tilde{T}_1}(t_1) \\) with \\( g_1(\\rho_1(u)) = h(u) = h(w) = g_1(\\rho_1(w)) \\). This contradicts the condition that \\( g_1 \\) is a bIso."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.238,
+                0.774,
+                0.825,
+                0.881
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.212,
+                0.887,
+                0.499,
+                0.903
+            ],
+            "angle": 0,
+            "content": "This yields the desired result that \\(\\tilde{F}_1\\simeq \\tilde{F}_2\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.91,
+                0.53,
+                0.925
+            ],
+            "angle": 0,
+            "content": "Combining the above three items concludes the proof."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.51,
+                0.961
+            ],
+            "angle": 0,
+            "content": "36"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.104,
+                0.825,
+                0.135
+            ],
+            "angle": 0,
+            "content": "Let \\( M \\in \\{\\mathrm{Sub}, \\mathrm{L}, \\mathrm{LF}, \\mathrm{F}\\} \\) be any model. We can list all non-isomorphic tree-decomposed (labeled) graphs in \\( S^M \\) into an infinite sequence \\( (F_1, T_1^{r_1}), (F_2, T_2^{r_2}), \\dots \\). Consider two types of ordering:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.14,
+                0.825,
+                0.196
+            ],
+            "angle": 0,
+            "content": "- Ordered by the size of the graphs \\( F_{i} \\). We denote this ordering as 1st. This ordering requires that for any \\( i < j \\), either \\( |V_{F_i}| < |V_{F_j}| \\) or \\( |V_{F_i}| = |V_{F_j}| \\) and \\( |E_{F_i}| \\geq |E_{F_j}| \\). Note that when two graphs have an equal number of vertices, we place the graph with more edges to the front. When the number of edges is also the same, they can be arranged in any fixed order."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.197,
+                0.826,
+                0.24
+            ],
+            "angle": 0,
+            "content": "- Ordered by the size of the trees \\( T_{i} \\). We denote this ordering as 2nd. This ordering requires that \\( |V_{T_i}| \\leq |V_{T_j}| \\) for any \\( i < j \\). When the number of tree nodes is the same, they can be arranged in any fixed order."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.198,
+                0.14,
+                0.826,
+                0.24
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.246,
+                0.827,
+                0.303
+            ],
+            "angle": 0,
+            "content": "Without loss of generality, we assume that the labels of each graph \\( F_{i} \\) are integers in the range of \\( [1, |V_{F_i}|] \\). Note that using integer labels of a bounded range is already sufficient to represent all non-isomorphic labeled graphs up to a bijective label transformation. This ensures that the index \\( i \\) is countable."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.309,
+                0.515,
+                0.325
+            ],
+            "angle": 0,
+            "content": "For the 1st ordering, define the following notations:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.191,
+                0.329,
+                0.825,
+                0.365
+            ],
+            "angle": 0,
+            "content": "a) Let \\( f: \\mathcal{S}^M \\times \\mathcal{S}^M \\to \\mathbb{N} \\) be any mapping. Define the associated (infinite) matrix \\( A^{f,M,1\\mathrm{st}} \\in \\mathbb{N}^{\\mathbb{N}_+ \\times \\mathbb{N}_+} \\) such that \\( A_{i,j}^{f,M,1\\mathrm{st}} = f((F_i, T_i^{r_i}), (F_j, T_j^{r_j})) \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.19,
+                0.365,
+                0.826,
+                0.399
+            ],
+            "angle": 0,
+            "content": "b) Let \\( g: \\mathcal{S}^M \\times \\mathcal{G} \\to \\mathbb{N} \\) be any mapping. Given a graph \\( G \\in \\mathcal{G} \\), define the (infinite) vector \\( \\pmb{p}_G^{g,M,1\\mathrm{st}} \\in \\mathbb{N}^{\\mathbb{N}_+} \\) such that \\( p_{G,i}^{f,M,1\\mathrm{st}} = g((F_i,T_i^{r_i}),G) \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.19,
+                0.399,
+                0.825,
+                0.434
+            ],
+            "angle": 0,
+            "content": "c) Let \\( h: \\mathcal{G} \\times \\mathcal{G} \\to \\mathbb{N} \\) be any mapping. Given a graph \\( G \\in \\mathcal{G} \\), define the (infinite) vector \\( p_{G}^{h,M,1\\mathrm{st}} \\in \\mathbb{N}^{\\mathbb{N}_{+}} \\) such that \\( p_{G,i}^{h,M,1\\mathrm{st}} = h(F_i, G) \\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.19,
+                0.329,
+                0.826,
+                0.434
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.439,
+                0.668,
+                0.459
+            ],
+            "angle": 0,
+            "content": "We can similarly define \\( \\pmb{A}^{f,M,2\\mathrm{nd}} \\), \\( \\pmb{p}_G^{g,M,2\\mathrm{nd}} \\), \\( \\pmb{p}_G^{h,M,2\\mathrm{nd}} \\) for the ordering 2nd."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.46,
+                0.827,
+                0.503
+            ],
+            "angle": 0,
+            "content": "Corollary C.32. Let \\( M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\} \\) be any model and \\( G, H \\) be two graphs. Then, \\( \\operatorname{hom}(F, G) = \\operatorname{hom}(F, H) \\) for all \\( (F, T^r) \\in S^M \\) iff \\( \\text{blso}((F, T^r), G) = \\text{blso}((F, T^r), H) \\) for all \\( (F, T^r) \\in S^M \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.518,
+                0.478,
+                0.534
+            ],
+            "angle": 0,
+            "content": "Proof. We separately consider each direction."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.539,
+                0.826,
+                0.582
+            ],
+            "angle": 0,
+            "content": "1. We first prove that if \\(\\mathsf{blso}((F,T^r),G) = \\mathsf{blso}((F,T^r),H)\\) for all \\((F,T^{r})\\in S^{M}\\), then \\(\\mathsf{hom}(F,G) = \\mathsf{hom}(F,H)\\) for all \\((F,T^{r})\\in S^{M}\\). According to Lemmas C.29 and C.31, we can rewrite the corresponding equations into matrix forms for any \\(F\\in \\mathcal{G}\\):"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.348,
+                0.586,
+                0.825,
+                0.606
+            ],
+            "angle": 0,
+            "content": "\\[\n\\boldsymbol {p} _ {F} ^ {\\text {h o m}, M, 1 \\mathrm {s t}} = \\boldsymbol {A} ^ {\\text {b E x t}, M, 1 \\mathrm {s t}} \\left(\\boldsymbol {A} ^ {\\text {a u t}, M, 1 \\mathrm {s t}}\\right) ^ {- 1} \\boldsymbol {p} _ {F} ^ {\\text {b S t r H o m}, M, 1 \\mathrm {s t}} \\tag {19}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.328,
+                0.607,
+                0.825,
+                0.628
+            ],
+            "angle": 0,
+            "content": "\\[\n\\boldsymbol {p} _ {F} ^ {\\mathrm {b S t r H o m}, M, 1 \\mathrm {s t}} = \\boldsymbol {A} ^ {\\mathrm {b S t r S u r j}, M, 1 \\mathrm {s t}} \\left(\\boldsymbol {A} ^ {\\mathrm {a u t}, M, 1 \\mathrm {s t}}\\right) ^ {- 1} \\boldsymbol {p} _ {F} ^ {\\mathrm {b l s o}, M, 1 \\mathrm {s t}}. \\tag {20}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.632,
+                0.825,
+                0.669
+            ],
+            "angle": 0,
+            "content": "From the above equations, we immediately obtain that \\( p_{G}^{\\mathrm{blso},M,1\\mathrm{st}} = p_{H}^{\\mathrm{blso},M,1\\mathrm{st}} \\) implies \\( p_{G}^{\\mathrm{hom},M,1\\mathrm{st}} = p_{H}^{\\mathrm{hom},M,1\\mathrm{st}} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.191,
+                0.674,
+                0.826,
+                0.717
+            ],
+            "angle": 0,
+            "content": "2. We next prove that if \\(\\mathsf{hom}(F,G) = \\mathsf{hom}(F,H)\\) for all \\((F,T^r)\\in S^M\\), then \\(\\mathsf{blso}((F,T^r),G) = \\mathsf{blso}((F,T^r),H)\\) for all \\((F,T^{r})\\in S^{M}\\). This can be seen from the following facts:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.723,
+                0.766,
+                0.738
+            ],
+            "angle": 0,
+            "content": "a) \\(A^{\\mathrm{out},M,1\\mathrm{st}}\\) is a diagonal matrix and all diagonal elements are positive integers."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.738,
+                0.825,
+                0.786
+            ],
+            "angle": 0,
+            "content": "b) \\( A^{\\mathrm{bExt}, M, 1\\mathrm{st}} \\) is a lower triangular matrix and all diagonal elements are positive integers. This is because for any two tree-composed graphs \\( (F_i, T_i^{r_i}) \\) and \\( (F_j, T_j^{r_j}) \\), \\( \\mathsf{bExt}((F_i, T_i^{r_i}), (F_j, T_j^{r_j})) > 0 \\) only if \\( |V_{F_i}| = |V_{F_j}| \\) and \\( |E_{F_i}| \\leq |E_{F_j}| \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.232,
+                0.787,
+                0.826,
+                0.851
+            ],
+            "angle": 0,
+            "content": "c) \\( A^{\\mathrm{bStrSurj}, M, 1\\mathrm{st}} \\) is also a lower triangular matrix and all diagonal elements are positive integers. This is because for any two tree-composed graphs \\( (F_i, T_i^{r_i}) \\) and \\( (F_j, T_j^{r_j}) \\), \\( \\mathsf{bStrSurj}((F_i, T_i^{r_i}), (F_j, T_j^{r_j})) > 0 \\) only if \\( |V_{F_i}| > |V_{F_j}| \\) or \\( (|V_{F_i}| = |V_{F_j}| \\) and \\( |E_{F_i}| = |E_{F_j}| \\))."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.231,
+                0.723,
+                0.826,
+                0.851
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.856,
+                0.826,
+                0.905
+            ],
+            "angle": 0,
+            "content": "Therefore, the composition \\( A^{\\mathrm{bExt},M,1\\mathrm{st}}(A^{\\mathrm{aut},M,1\\mathrm{st}})^{-1}A^{\\mathrm{bStrSurj},M,1\\mathrm{st}}(A^{\\mathrm{aut},M,1\\mathrm{st}})^{-1} \\) is lower triangular and is invertible (although it is an infinite matrix). We thus arrive at the desired conclusion that \\( p_{G}^{\\mathrm{hom},M,1\\mathrm{st}} = p_{H}^{\\mathrm{hom},M,1\\mathrm{st}} \\) implies \\( p_{G}^{\\mathrm{blso},M,1\\mathrm{st}} = p_{H}^{\\mathrm{blso},M,1\\mathrm{st}} \\)."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.911,
+                0.825,
+                0.923
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "37"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.825,
+                0.149
+            ],
+            "angle": 0,
+            "content": "Corollary C.33. Let \\( M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\} \\) be any model and \\( G, H \\) be two graphs. Then, \\( \\text{cnt}^M((F, T^r), G) = \\text{cnt}^M((F, T^r), H) \\) for all \\( (F, T^r) \\in \\mathcal{S}^M \\) iff \\( \\text{blso}((F, T^r), G) = \\text{blso}((F, T^r), H) \\) for all \\( (F, T^r) \\in \\mathcal{S}^M \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.162,
+                0.477,
+                0.178
+            ],
+            "angle": 0,
+            "content": "Proof. We separately consider each direction."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.182,
+                0.825,
+                0.227
+            ],
+            "angle": 0,
+            "content": "1. We first prove that if \\(\\mathsf{cnt}^M((F, T^r), G) = \\mathsf{cnt}^M((F, T^r), H)\\) for all \\((F, T^r) \\in S^M\\), then \\(\\mathsf{blso}((F, T^r), G) = \\mathsf{blso}((F, T^r), H)\\) for all \\((F, T^r) \\in S^M\\). According to Lemma C.23, we can rewrite Equation (18) into the matrix form for all \\(F \\in \\mathcal{G}\\):"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.384,
+                0.233,
+                0.825,
+                0.253
+            ],
+            "angle": 0,
+            "content": "\\[\n\\boldsymbol {p} _ {F} ^ {\\text {b l s o}, M, 2 \\text {n d}} = \\boldsymbol {A} ^ {\\text {b l s o H o m}, M, 2 \\text {n d}} \\boldsymbol {p} _ {F} ^ {\\text {c n t}, M, 2 \\text {n d}}. \\tag {21}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.259,
+                0.794,
+                0.281
+            ],
+            "angle": 0,
+            "content": "This immediately obtains that \\( \\pmb{p}_G^{\\mathrm{cnt},M,2\\mathrm{nd}} = \\pmb{p}_H^{\\mathrm{cnt},M,2\\mathrm{nd}} \\) implies \\( \\pmb{p}_G^{\\mathrm{blso},M,2\\mathrm{nd}} = \\pmb{p}_H^{\\mathrm{blso},M,2\\mathrm{nd}} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.284,
+                0.825,
+                0.316
+            ],
+            "angle": 0,
+            "content": "2. We next prove that \\( \\pmb{p}_G^{\\mathrm{blo},M,2\\mathrm{nd}} = \\pmb{p}_H^{\\mathrm{blo},M,2\\mathrm{nd}} \\) implies \\( \\pmb{p}_G^{\\mathrm{cnt},M,2\\mathrm{nd}} = \\pmb{p}_H^{\\mathrm{cnt},M,2\\mathrm{nd}} \\). According to Lemma C.25, we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.312,
+                0.32,
+                0.825,
+                0.34
+            ],
+            "angle": 0,
+            "content": "\\[\n\\boldsymbol {A} ^ {\\text {b l s o H o m}, M, 2 \\text {n d}} = \\boldsymbol {A} ^ {\\text {b l s o S u r j}, M, 2 \\text {n d}} \\left(\\boldsymbol {A} ^ {\\text {a u t}, M, 2 \\text {n d}}\\right) ^ {- 1} \\boldsymbol {A} ^ {\\text {b l s o l n j}, M, 2 \\text {n d}}. \\tag {22}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.212,
+                0.346,
+                0.473,
+                0.361
+            ],
+            "angle": 0,
+            "content": "Moreover, we have the following facts:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.366,
+                0.771,
+                0.381
+            ],
+            "angle": 0,
+            "content": "a) \\(A^{\\mathrm{out},M,2\\mathrm{nd}}\\) is a diagonal matrix and all diagonal elements are positive integers."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.381,
+                0.825,
+                0.431
+            ],
+            "angle": 0,
+            "content": "b) \\( A^{\\mathrm{blsohnj},M,2\\mathrm{nd}} \\) is an upper triangular matrix and all diagonal elements are positive integers. This is because for any two tree-composed graphs \\( (F_i,T_i^{r_i}) \\) and \\( (F_j,T_j^{r_j}) \\), \\( \\mathsf{blsohnj}((F_i,T_i^{r_i}),(F_j,T_j^{r_j})) > 0 \\) only if \\( |V_{T_i}| \\leq |V_{T_j}| \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.431,
+                0.825,
+                0.461
+            ],
+            "angle": 0,
+            "content": "c) Similarly, \\( \\mathbf{A}^{\\mathrm{bIsoSurj},M,2\\mathrm{nd}} \\) is a lower triangular matrix and all diagonal elements are positive integers."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.231,
+                0.366,
+                0.825,
+                0.461
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.467,
+                0.826,
+                0.545
+            ],
+            "angle": 0,
+            "content": "Unfortunately, since \\( A^{\\mathrm{blsolnj},M,2\\mathrm{nd}} \\) is an infinite upper triangular matrix, the inverse matrix is not well-defined. Nevertheless, we can use a special property of bIsoHom to complete our proof, namely, \\( \\mathrm{bIsOHom}((F_i,T_i^{r_i}),(F_j,T_j^{r_j})) > 0 \\) implies that the depth of \\( T_{i}^{r_{i}} \\) is equal to the depth of \\( T_{j}^{r_{j}} \\). Therefore, denoting by \\( A^{\\mathrm{dep},d} \\) the diagonal matrix where \\( A_{ii}^{\\mathrm{dep},d} = \\mathbb{I}[\\mathrm{the~depth~of~}T_i^{r_i}\\mathrm{~is~}d] \\), we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.239,
+                0.552,
+                0.825,
+                0.573
+            ],
+            "angle": 0,
+            "content": "\\[\n\\boldsymbol {A} ^ {\\text {d e p}, d} \\boldsymbol {p} _ {F} ^ {\\text {b l s o}, M, 2 \\text {n d}} = \\boldsymbol {A} ^ {\\text {b l s o S u r j}, M, 2 \\text {n d}} \\left(\\boldsymbol {A} ^ {\\text {a u t}, M, 2 \\text {n d}}\\right) ^ {- 1} \\boldsymbol {A} ^ {\\text {b l s o l n j}, M, 2 \\text {n d}} \\boldsymbol {A} ^ {\\text {d e p}, d} \\boldsymbol {p} _ {F} ^ {\\text {c n t}, M, 2 \\text {n d}} \\tag {23}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.579,
+                0.825,
+                0.628
+            ],
+            "angle": 0,
+            "content": "for all \\(F \\in \\mathcal{G}\\). Fix any integer \\(d \\geq 0\\), and assume that \\(A^{\\mathrm{dep},d}p_{G}^{\\mathrm{blso},M,2\\mathrm{nd}} = A^{\\mathrm{dep},d}p_{H}^{\\mathrm{blso},M,2\\mathrm{nd}}\\). We will prove that \\(A^{\\mathrm{dep},d}p_{G}^{\\mathrm{cnt},M,2\\mathrm{nd}} = A^{\\mathrm{dep},d}p_{H}^{\\mathrm{cnt},M,2\\mathrm{nd}}\\). Since \\(A^{\\mathrm{blsoSurj},M,2\\mathrm{nd}}\\) is lower triangular with positive diagonal elements, it is invertible and thus"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.306,
+                0.634,
+                0.731,
+                0.655
+            ],
+            "angle": 0,
+            "content": "\\[\n\\boldsymbol {A} ^ {\\text {b l s o l n j}, M, 2 \\text {n d}} \\boldsymbol {A} ^ {\\text {d e p}, d} \\boldsymbol {p} _ {G} ^ {\\text {c n t}, M, 2 \\text {n d}} = \\boldsymbol {A} ^ {\\text {b l s o l n j}, M, 2 \\text {n d}} \\boldsymbol {A} ^ {\\text {d e p}, d} \\boldsymbol {p} _ {H} ^ {\\text {c n t}, M, 2 \\text {n d}}.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.66,
+                0.825,
+                0.737
+            ],
+            "angle": 0,
+            "content": "Moreover, by definition of unfolding tree, there are only finite non-zero elements in both \\( A^{\\mathrm{dep},d}p_{G}^{\\mathrm{cnt},M,2\\mathrm{nd}} \\) and \\( A^{\\mathrm{dep},d}p_{H}^{\\mathrm{cnt},M,2\\mathrm{nd}} \\), and the corresponding non-zero indices can only be in a fixed (finite) set. In this case, the upper triangular matrix \\( A^{\\mathrm{blosnj},M,2\\mathrm{nd}} \\) is reduced to a finite-dimensional matrix and thus \\( A^{\\mathrm{dep},d}p_{G}^{\\mathrm{cnt},M,2\\mathrm{nd}} = A^{\\mathrm{dep},d}p_{H}^{\\mathrm{cnt},M,2\\mathrm{nd}} \\). By enumerating all \\( d \\geq 0 \\), we obtain the desired result."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.744,
+                0.826,
+                0.756
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.773,
+                0.701,
+                0.789
+            ],
+            "angle": 0,
+            "content": "Combined with all previous results, we have arrived at the concluding corollary:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.791,
+                0.827,
+                0.825
+            ],
+            "angle": 0,
+            "content": "Corollary C.34. Let \\( M \\in \\{\\mathrm{Sub},\\mathsf{L},\\mathsf{LF},\\mathsf{F}\\} \\) be any model. For any two graphs \\( G, H \\), \\( \\chi_G^M (G) = \\chi_H^M (H) \\) iff \\( \\operatorname {hom}(F,G) = \\operatorname {hom}(F,H) \\) for all \\( (F,T^{r})\\in S^{M} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.838,
+                0.828,
+                0.926
+            ],
+            "angle": 0,
+            "content": "Proof. According to Corollary C.16, for any two graphs \\(G,H\\), we have \\(\\chi_G^M (G) = \\chi_H^M (H)\\) iff \\(\\mathsf{cnt}^M ((F,T^r),G) = \\mathsf{cnt}^M ((F,T^r),H)\\) for all \\((F,T^{r})\\in S^{M}\\). Then, Corollary C.33 implies that \\(\\mathsf{cnt}^M ((F,T^r),G) = \\mathsf{cnt}^M ((F,T^r),H)\\) for all \\((F,T^{r})\\in S^{M}\\) iff \\(\\mathsf{blos}((F,T^r),G) = \\mathsf{blos}((F,T^r),H)\\) for all \\((F,T^{r})\\in S^{M}\\). Finally, Corollary C.32 implies that \\(\\mathsf{blos}((F,T^r),G) = \\mathsf{blos}((F,T^r),H)\\) for all \\((F,T^{r})\\in S^{M}\\) iff \\(\\mathsf{hom}(F,G) = \\mathsf{hom}(F,H)\\) for all \\((F,T^{r})\\in S^{M}\\). We thus conclude the proof."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.961
+            ],
+            "angle": 0,
+            "content": "38"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.104,
+                0.487,
+                0.119
+            ],
+            "angle": 0,
+            "content": "C.3 PART 2: NESTED EAR DECOMPOSITION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.129,
+                0.825,
+                0.16
+            ],
+            "angle": 0,
+            "content": "In this part, we give equivalent formulations of the set \\(S^M\\) for any model \\(M \\in \\{\\mathrm{Sub}, \\mathrm{L}, \\mathrm{LF}, \\mathrm{F}\\}\\) using the concept of NED defined in Definition 3.3."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.165,
+                0.827,
+                0.195
+            ],
+            "angle": 0,
+            "content": "We first present some technical lemmas that will be used to deal with the tree decomposition of disconnected graphs."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.197,
+                0.826,
+                0.258
+            ],
+            "angle": 0,
+            "content": "Lemma C.35. Let \\( M \\in \\{\\mathrm{Sub}, \\mathrm{L}, \\mathrm{LF}, \\mathrm{F}\\} \\) be any model, and let \\( (F, T^r) \\in S^M \\) be a tree-decomposed graph with \\( \\beta_T(t) = \\{u, v\\} \\) for some \\( u, v \\in V_F \\). Assume that \\( F \\) has two connected components \\( S_1, S_2 \\subset V_F \\) and \\( u \\in S_1 \\) and \\( v \\in S_2 \\) are in different components. Then, there exist tree decompositions \\( (F[S_1], T_1^{r_1}), (F[S_2], T_2^{r_2}) \\in S^M \\) such that \\( \\beta_{T_1}(r_1) = \\{u, u\\} \\) and \\( \\beta_{T_2}(r_2) = \\{v, v\\} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.271,
+                0.825,
+                0.302
+            ],
+            "angle": 0,
+            "content": "Proof. We can simply define \\( T_{1} = (V_{T_{1}}, E_{T_{1}}, \\beta_{T_{1}}) \\) with \\( V_{T_{1}} = V_{T} \\), \\( E_{T_{1}} = E_{T} \\), \\( r_{1} = r \\), and \\( \\beta_{T_{1}}(t) = \\{\\{\\phi_{1}(w): w \\in \\beta_{T}(t)\\}\\} \\) with"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.4,
+                0.308,
+                0.587,
+                0.342
+            ],
+            "angle": 0,
+            "content": "\\[\n\\phi_ {1} (w) = \\left\\{ \\begin{array}{l l} w & \\text {i f} w \\in S _ {1}, \\\\ u & \\text {i f} w \\in S _ {2}. \\end{array} \\right.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.348,
+                0.826,
+                0.419
+            ],
+            "angle": 0,
+            "content": "In other words, \\( T_{1} \\) has the same structure as \\( T \\) but with different bags such that all vertices in the same connected component as \\( v \\) are replaced by \\( u \\). It is easy to see that \\( T_{1}^{r_{1}} \\) is a canonical tree decomposition of \\( F[S_{1}] \\) following Definition C.1. Moreover, we clearly have that for all \\( M \\in \\{\\mathrm{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\} \\), \\( (F[S_{1}], T_{1}^{r_{1}}) \\in S^{M} \\). We can construct \\( T_{2} = (V_{T_{2}}, E_{T_{2}}, \\beta_{T_{2}}) \\) by symmetry, which concludes the proof."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.434,
+                0.658,
+                0.451
+            ],
+            "angle": 0,
+            "content": "The above lemma can be immediately generalized into the following one:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.453,
+                0.825,
+                0.517
+            ],
+            "angle": 0,
+            "content": "Corollary C.36. Let \\( M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\} \\) be any model, and let \\( (F, T^r) \\in S^M \\) be a tree-decomposed graph. For each connected component \\( S \\in V_F \\) of \\( F \\), pick any \\( t \\in V_T \\) with the minimum depth such that \\( S \\cap \\beta_T(t) \\neq \\emptyset \\). Then, there exists a tree decomposition \\( \\tilde{T}^s \\) of \\( F[S] \\) satisfying that \\( (F[S], \\tilde{T}^s) \\in S^M \\) and \\( S \\cap \\beta_T(t) \\subset \\beta_{\\tilde{T}}(s) \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.519,
+                0.826,
+                0.579
+            ],
+            "angle": 0,
+            "content": "Lemma C.37. Let \\( M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\} \\) be any model, and let \\( (F, T^r) \\in S^M \\) be a tree-decomposed graph such that \\( F \\) is connected. Then, for each child node \\( t \\) of \\( r \\) in \\( T^r \\) and any vertex \\( x \\) in \\( F[T^r[t]] \\), there is a path in \\( F[T^r[t]] \\) from \\( x \\) to some vertex in \\( \\beta_T(r) \\). Consequently, the number of connected components of \\( F[T^r[t]] \\) is bounded by the number of different elements in \\( \\beta_T(r) \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.593,
+                0.826,
+                0.669
+            ],
+            "angle": 0,
+            "content": "Proof. Assume the above statement does not hold, and let \\( S \\subset V_{F[T^r[t]]} \\) be the connected component of \\( F[T^r[t]] \\) such that \\( S \\cap \\beta_T(r) = \\emptyset \\). Note that \\( F[S] = F[T^r[t]][S] \\). Since \\( F \\) is connected, there exists an edge \\( \\{v, w\\} \\in E_F \\) such that \\( v \\in S \\) and \\( w \\notin S \\). Since \\( S \\) is a connected component of \\( F[T^r[t]] \\), we have \\( w \\notin V_{F[T^r[t]]} \\), and thus both \\( v, w \\notin \\beta_T(r) \\). This yields a contradiction since \\( \\{v, w\\} \\) should be contained in a bag in \\( T^r \\) but \\( B_T(v) \\cap B_T(w) = \\emptyset \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.682,
+                0.825,
+                0.725
+            ],
+            "angle": 0,
+            "content": "To reduce duplication, below we only give proofs for Local 2-GNN and Local 2-FGNN. One can easily write a proof for Subgrapp HNN based on the proof of Local 2-GNN, and write a proof for 2-FGNN based on the proof of Local 2-FGNN."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.728,
+                0.724,
+                0.745
+            ],
+            "angle": 0,
+            "content": "Lemma C.38. For any tree-decomposed graph \\((F, T^r) \\in S^{\\mathsf{L}}\\), \\(F\\) has a strong NED."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.76,
+                0.826,
+                0.817
+            ],
+            "angle": 0,
+            "content": "Proof. Based on Corollary C.36, we can assume that \\(F\\) is connected without loss of generality. We will prove the following stronger result: for any connected \\((F,T^r)\\in S^{\\mathsf{L}}\\) with \\(\\beta_T(r) = \\{\\{u,v\\}\\}\\), \\(F\\) has a strong NED where \\(u,v\\) are endpoints of the first ear. (For the case of \\(u = v\\), the other endpoint can be arbitrary.)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.823,
+                0.826,
+                0.926
+            ],
+            "angle": 0,
+            "content": "The proof is based on induction over the number the vertices in \\( T^r \\). The above statement obviously holds for the base case of \\( |V_T| = 1 \\). Now assume that the statement holds when \\( |V_T| \\leq m \\), and consider the case of \\( |V_T| = m + 1 \\). Note that for any two different children \\( t, t' \\) of \\( r \\), \\( F[T^r[t]] \\) and \\( F[T^r[t']] \\) can only share vertices \\( u, v \\). For each child node \\( t \\) of \\( r \\), denote its unique child node as \\( \\tilde{t} \\). It is easy to see that \\( T^r[\\tilde{t}] \\) is a canonical tree decomposition of \\( F[T^r[\\tilde{t}]] \\) and \\( (F[T^r[\\tilde{t}]], T^r[\\tilde{t}]) \\in S^\\perp \\). However, one needs to be cautious as \\( F[T^r[\\tilde{t}]] \\) may not be connected (unlike the original graph \\( F \\)). Below, we separately consider the following cases:"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.961
+            ],
+            "angle": 0,
+            "content": "39"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.193,
+                0.104,
+                0.825,
+                0.19
+            ],
+            "angle": 0,
+            "content": "1. First consider the case when \\( u = v \\). In this case, Lemma C.37 implies that \\( F[T^r[\\tilde{t}]] \\) is connected. According to the induction hypothesis, \\( F[T^r[\\tilde{t}]] \\) has a strong NED (denoted as \\( \\mathcal{P}_{\\tilde{t}} \\)) such that \\( u \\) is an endpoint of the first ear. We then merge the ear decompositions \\( \\mathcal{P}_{\\tilde{t}} \\) for all \\( \\tilde{t} \\) into a whole \\( \\mathcal{P} \\), specify a root ear \\( P_1 \\) in any \\( \\mathcal{P}_{\\tilde{t}} \\), and let the first ear of other \\( \\mathcal{P}_{\\tilde{t}} \\) nest on \\( P_1 \\) (with empty nested interval). It is easy to see that \\( \\mathcal{P} \\) is a strong NED of \\( F \\) and \\( u \\) is an endpoint of the first ear."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.197,
+                0.825,
+                0.228
+            ],
+            "angle": 0,
+            "content": "2. Next consider the case when \\( u \\neq v \\). In this case, without loss of generality, assume \\( \\beta_T(\\tilde{t}) = \\{u, w\\} \\) for some \\( w \\in V_F \\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.192,
+                0.104,
+                0.825,
+                0.228
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.235,
+                0.825,
+                0.264
+            ],
+            "angle": 0,
+            "content": "- Subcase 1: \\( F[T^r[\\tilde{t}]] \\) is connected. Then, \\( F[T^r[\\tilde{t}]] \\) has a strong NED \\( \\mathcal{P}_{\\hat{t}} \\) such that \\( u \\) and \\( w \\) are endpoints of the first ear."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.264,
+                0.264,
+                0.78,
+                0.28
+            ],
+            "angle": 0,
+            "content": "a) If \\( \\bar{w} = v \\), then \\( F[T^r[t]] = F[T^r[\\hat{t}]] \\) and \\( F[T^r[t]] \\) clearly has a strong NED."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.264,
+                0.28,
+                0.825,
+                0.349
+            ],
+            "angle": 0,
+            "content": "b) If \\( w \\notin N_F[v] \\) or \\( w = u \\), there are two additional subcases depending on whether \\( \\{u, v\\} \\in E_F \\). If \\( \\{u, v\\} \\in E_F \\), then \\( F[T^r[t]] \\) has a strong NED (which can be constructed from \\( \\mathcal{P}_{\\tilde{t}} \\) by adding an ear \\( \\{\\{u, v\\}\\} \\) and letting the first ear in \\( \\mathcal{P}_{\\tilde{t}} \\) nest on \\( \\{\\{u, v\\}\\} \\) (with empty nested interval). Otherwise, \\( F[T^r[t]] \\backslash \\{v\\} \\) has a strong NED and \\( v \\) is an isolated vertex in \\( F[T^r[t]] \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.264,
+                0.348,
+                0.825,
+                0.391
+            ],
+            "angle": 0,
+            "content": "c) If \\( w \\in N_F(v) \\) and \\( w \\neq u \\), then \\( F[T^r[t]] \\) also has a strong NED, which can be constructed from \\( \\mathcal{P}_{\\tilde{t}} \\) by extending the first ear to incorporate the edge \\( \\{w, v\\} \\) (we still need an additional ear \\( \\{\\{u, v\\}\\} \\) if \\( \\{u, v\\} \\in E_F \\))."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.264,
+                0.264,
+                0.825,
+                0.391
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.232,
+                0.392,
+                0.826,
+                0.581
+            ],
+            "angle": 0,
+            "content": "- Subcase 2: \\( F[T^r[\\tilde{t}]] \\) is disconnected. In this subcase, Lemma C.37 implies that \\( F[T^r[\\tilde{t}]] \\) has exactly two connected components, and \\( u \\) and \\( w \\) are in different connected components (which can be easily proved by noting that \\( v \\) can only link to \\( u \\) or \\( w \\) in \\( F[T^r[\\tilde{t}]] \\)). We can thus invoke Lemma C.35, which shows that both connected components of \\( F[T^r[\\tilde{t}]] \\), denoted as \\( \\widehat{F}_1 \\) and \\( \\widehat{F}_2 \\), admits tree decompositions \\( \\widehat{T}_1^{s_1} \\) and \\( \\widehat{T}_2^{s_2} \\) such that \\( (\\widehat{F}_1, \\widehat{T}_1^{s_1}), (\\widehat{F}_2, \\widehat{T}_2^{s_2}) \\in S^\\mathsf{L} \\), \\( \\beta_{\\widehat{T}_1}(s_1) = \\{\\{u, u\\}\\} \\) and \\( \\beta_{\\widehat{T}_2}(s_2) = \\{\\{w, w\\}\\} \\). Moreover, \\( |V_{\\widehat{T}_1}| = |V_{\\widehat{T}_2}| = |V_{T^r[\\tilde{t}]}| < m + 1 \\). Therefore, according to the induction hypothesis, \\( \\widehat{F}_1 \\) has a strong NED \\( \\mathcal{P}_{\\tilde{t},1} \\) with \\( u \\) as an endpoint of the first ear, and \\( \\widehat{F}_2 \\) has a strong NED \\( \\mathcal{P}_{\\tilde{t},2} \\) with \\( w \\) as an endpoint of the first ear. If \\( \\{w, v\\} \\in E_F \\), we can extend the first ear in \\( \\mathcal{P}_{\\tilde{t},2} \\) to include the edge \\( \\{w, v\\} \\). Finally, if \\( \\{u, v\\} \\in E_F \\), by setting the first ear to be \\( \\{\\{u, v\\}\\} \\), we can merge the two NEDs \\( \\mathcal{P}_{\\tilde{t},1} \\) and \\( \\mathcal{P}_{\\tilde{t},2} \\) to obtain the NED of \\( F[T^r[t]] \\) such that \\( u, v \\) are endpoints of the first ear."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.213,
+                0.59,
+                0.411,
+                0.604
+            ],
+            "angle": 0,
+            "content": "Overall, we always have that:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.232,
+                0.61,
+                0.825,
+                0.638
+            ],
+            "angle": 0,
+            "content": "a) If \\( F[T^r[t]] \\) is connected, then it admits a strong NED such that \\( u, v \\) are endpoints of the first ear."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.639,
+                0.825,
+                0.682
+            ],
+            "angle": 0,
+            "content": "b) If \\( F[T^r[t]] \\) is disconnected, then it has two connected components each admitting a strong NED such that \\( u, v \\) belong to an endpoint of the first ear for each of the two NEDs, respectively."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.231,
+                0.61,
+                0.825,
+                0.682
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.212,
+                0.688,
+                0.825,
+                0.72
+            ],
+            "angle": 0,
+            "content": "Finally, noting that \\( F = \\bigcup_{t,\\mathfrak{pa}_{Tr}(t) = r}F[T^r [t]] \\), we can merge all NEDs of \\( F[T^{r}[t]] \\) to obtain a strong NED of \\( F \\) satisfying that \\( u,v \\) are endpoints of the first ear."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.726,
+                0.497,
+                0.741
+            ],
+            "angle": 0,
+            "content": "We thus conclude the proof of the induction step."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.748,
+                0.825,
+                0.777
+            ],
+            "angle": 0,
+            "content": "Lemma C.39. For any graph \\(F\\), if \\(F\\) admits a strong NED, then there exists a tree decomposition \\(T^r\\) of \\(F\\) such that \\((F,T^{r})\\in S^{\\mathsf{L}}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.791,
+                0.825,
+                0.863
+            ],
+            "angle": 0,
+            "content": "Proof. We can assume that \\( F \\) is connected without loss of generality, as it is easy to merge the tree decompositions of different connected components to form a single tree which is a tree decomposition of the whole graph. We will prove the following stronger result: for any connected graph \\( F \\), if \\( F \\) admits a strong NED where \\( u \\) and \\( v \\) are two different endpoints of the first ear, then there exists a tree decomposition \\( T^r \\) of \\( F \\) such that \\( (F, T^r) \\in S^\\mathbb{L} \\) and \\( \\beta_T(r) = \\{(u, v)\\} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.868,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "The proof is based on induction over the number the edges in \\( F \\). The above statement obviously holds for the base case of \\( |E_F| = 1 \\). Now assume that the statement holds when \\( |E_F| \\leq m \\), and consider the case of \\( |E_F| = m + 1 \\). Denote the \\( i \\)-th ear as \\( P_i \\) and denote \\( P_1 = \\{\\{w_0, w_1\\}, \\dots, \\{w_{l-1}, w_l\\}\\} \\) where \\( w_0 = u \\), \\( w_l = v \\). Consider the following three cases:"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.949,
+                0.51,
+                0.96
+            ],
+            "angle": 0,
+            "content": "40"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.193,
+                0.104,
+                0.825,
+                0.26
+            ],
+            "angle": 0,
+            "content": "1. Either \\( u \\) or \\( v \\) is a cut vertex in \\( F \\). Without loss of generality, assume that \\( u \\) is a cut vertex. In this case, there must exist a child ear \\( P_i \\) nested on \\( P_1 \\) such that \\( u \\) is an endpoint of \\( P_i \\) and \\( I(P_i) = \\emptyset \\). We can thus split all ears into two parts \\( \\mathcal{P}_1 \\) and \\( \\mathcal{P}_2 \\), where \\( \\mathcal{P}_1 \\) contains the ear \\( P_i \\) and its descendant ears in the NED tree, and \\( \\mathcal{P}_2 \\) contains other ears. Denote by \\( F[\\mathcal{P}_k] \\) the connected graph induced by all edges in \\( \\mathcal{P}_k \\) (\\( k \\in \\{1, 2\\} \\)). Then, \\( \\mathcal{P}_k \\) is a strong NED of the graph \\( F[\\mathcal{P}_k] \\). Since both \\( \\mathcal{P}_1 \\) and \\( \\mathcal{P}_2 \\) contain no more than \\( m \\) edges, according to the induction hypothesis, there are tree decompositions \\( T_k^{r_k} \\) of \\( F[\\mathcal{P}_k] \\) (\\( k \\in \\{1, 2\\} \\)) such that \\( \\beta_{T_1}(r_1) = \\{\\{u, w\\}\\} \\) for some \\( w \\in V_F \\), \\( \\beta_{T_2}(r_2) = \\{\\{u, v\\}\\} \\), and \\( (F[\\mathcal{P}_k], T_k^{r_k}) \\in S^{\\mathsf{L}} \\) (\\( k \\in \\{1, 2\\} \\)). We can glue the tree \\( T_1^{r_1} \\) and \\( T_2^{r_2} \\) into a larger tree \\( T^{r_2} \\) by adding a new node \\( t \\) with \\( \\beta_T(t) = \\{\\{u, v, w\\}\\} \\), setting \\( \\mathsf{pa}_{T^r}(t) = r_2 \\) and \\( \\mathsf{pa}_{T^r}(r_1) = t \\). Clearly, \\( T^{r_2} \\) is a valid tree decomposition of \\( F \\), \\( \\beta_T(r_2) = \\{\\{u, v\\}\\} \\), and \\( (F, T^{r_2}) \\in S^{\\mathsf{L}} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.266,
+                0.825,
+                0.367
+            ],
+            "angle": 0,
+            "content": "2. There is an ear \\( P_{i} \\) nested on \\( P_{1} \\) with \\( I(P_{i}) = P_{1} \\). In this case, we can split all ears into two parts \\( \\mathcal{P}_1 \\) and \\( \\mathcal{P}_2 \\), where \\( \\mathcal{P}_1 \\) contains the ear \\( P_{i} \\) and its descendant ears in the NED tree, and \\( \\mathcal{P}_2 \\) contains other ears. Then, \\( \\mathcal{P}_k \\) is a strong NED of the graph \\( F[\\mathcal{P}_k] \\). Since both \\( \\mathcal{P}_1 \\) and \\( \\mathcal{P}_2 \\) contain no more than \\( m \\) edges, according to the induction hypothesis, there is a tree decomposition \\( T_k^{r_k} \\) of \\( F[\\mathcal{P}_k] \\) such that \\( \\beta_{T_k}(r_k) = \\{\\{u,v\\}\\} \\) and \\( (F[\\mathcal{P}_k], T_k^{r_k}) \\in S^{\\mathsf{L}} \\). By merging the root node, we can glue the tree \\( T_1^{r_1} \\) and \\( T_2^{r_2} \\) into a larger tree \\( T^r \\). Clearly, \\( T^r \\) is a valid tree decomposition of \\( F \\), \\( \\beta_T(r) = \\{\\{u,v\\}\\} \\), and \\( (F,T^r) \\in S^{\\mathsf{L}} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.372,
+                0.825,
+                0.519
+            ],
+            "angle": 0,
+            "content": "3. Neither \\( u \\) nor \\( v \\) is a cut vertex in \\( F \\) and all ears \\( P_i \\) nested on \\( P_1 \\) satisfies \\( I(P_i) \\subsetneq P_1 \\). In this case, we have either \\( \\{w_0, w_1\\} \\notin I(P_i) \\) for all ear \\( P_i \\) nested on \\( P_1 \\) or \\( \\{w_{l-1}, w_l\\} \\notin I(P_i) \\) for all ear \\( P_i \\) nested on \\( P_1 \\) (otherwise, it would contradict the definition of strong NED). Without loss of generality, assume \\( \\{w_0, w_1\\} \\notin I(P_i) \\) for all ears \\( P_i \\) nested on \\( P_1 \\). Then, it is clear that \\( l > 1 \\), and the subgraph \\( F \\setminus \\{u\\} \\) is connected and also admits a strong NED where \\( w_1 \\) and \\( v \\) are two endpoints of the first ear. Therefore, according to the induction hypothesis, there is a tree decomposition \\( \\tilde{T}^s \\) of \\( F \\setminus \\{u\\} \\) satisfying \\( (F \\setminus \\{u\\}, \\tilde{T}^s) \\in S^\\mathsf{L} \\) and \\( \\beta_{\\tilde{T}}(s) = \\{w_1, v\\} \\). We can then construct a tree \\( T^r \\) from \\( \\tilde{T}^s \\) by adding two fresh nodes \\( r \\) and \\( r' \\) and setting \\( \\mathsf{pa}_{T^r}(s) = r' \\) and \\( \\mathsf{pa}_{T^r}(r') = r \\). Set \\( \\beta_T(r) = \\{u, v\\} \\), \\( \\beta_T(r') = \\{u, v, w_1\\} \\), and \\( \\beta_T(t) = \\beta_{\\tilde{T}}(t) \\) for all \\( t \\in V_{\\tilde{T}} \\). It is easy to see that the constructed \\( T^r \\) is a tree decomposition of \\( F \\) and \\( (F, T^r) \\in S^\\mathsf{L} \\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.192,
+                0.104,
+                0.825,
+                0.519
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.524,
+                0.586,
+                0.539
+            ],
+            "angle": 0,
+            "content": "Combining the above three cases concludes the induction step."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.525,
+                0.824,
+                0.535
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.547,
+                0.825,
+                0.577
+            ],
+            "angle": 0,
+            "content": "Theorem C.40. For any graph \\(F\\), \\(F\\) has a strong NED iff there is a tree decomposition \\(T^r\\) of \\(F\\) such that \\((F, T^r) \\in S^\\mathsf{L}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.592,
+                0.59,
+                0.607
+            ],
+            "angle": 0,
+            "content": "Proof. This is a direct consequence of Lemmas C.38 and C.39."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.593,
+                0.824,
+                0.604
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.623,
+                0.801,
+                0.639
+            ],
+            "angle": 0,
+            "content": "We next turn to Local 2-FGNN, where the proof has a similar structure as that of Local 2-GNN."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.642,
+                0.788,
+                0.658
+            ],
+            "angle": 0,
+            "content": "Lemma C.41. For any tree-decomposed graph \\((F, T^r) \\in S^{\\mathrm{LF}}\\), \\(F\\) has an almost-strong NED."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.673,
+                0.825,
+                0.731
+            ],
+            "angle": 0,
+            "content": "Proof. Based on Corollary C.36, we can assume that \\(F\\) is connected without loss of generality. We will prove the following stronger result: for any connected \\((F,T^r)\\in S^{\\mathrm{LF}}\\) with \\(\\beta_T(r) = \\{\\{u,v\\}\\}\\), \\(F\\) has an almost-strong NED where \\(u,v\\) are endpoints of the first ear. (For the case of \\(u = v\\), the other endpoint can be arbitrary.)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.736,
+                0.825,
+                0.822
+            ],
+            "angle": 0,
+            "content": "Similar to the proof of Lemma C.38, assume that the statement holds when \\( |V_T| \\leq m \\), and consider the case of \\( |V_T| = m + 1 \\). For each child node \\( t \\) of \\( r \\), if \\( t \\) only has one child, the proof exactly follows the one in Lemma C.38. Therefore, it suffices to consider the case where \\( t \\) has two children \\( q_1 \\) and \\( q_2 \\). Denote \\( \\beta_T(t) = \\{\\{u,v,w\\}\\} \\), \\( \\beta_T(q_1) = \\{\\{u,w\\}\\} \\), and \\( \\beta_T(q_2) = \\{\\{v,w\\}\\} \\). Since \\( w \\in N_F[u] \\cup N_F[v] \\) (by definition of \\( S^{\\mathsf{LF}} \\)), we can assume that \\( w \\in N_F[v] \\) without loss of generality. Recall that \\( (F[T^r [q_1]], T^r [q_1]) \\), \\( (F[T^r [q_2]], T^r [q_2]) \\in S^{\\mathsf{LF}} \\). Below, we separately consider the following cases:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.193,
+                0.827,
+                0.826,
+                0.926
+            ],
+            "angle": 0,
+            "content": "1. First consider the case when \\( u = v \\). In this case, Lemma C.37 implies that both \\( F[T^r[q_1]] \\) and \\( F[T^r[q_2]] \\) are connected (since either \\( u = w \\) or \\( \\{u, w\\} \\in E_F \\)). According to the induction hypothesis, both \\( F[T^r[q_1]] \\) and \\( F[T^r[q_2]] \\) admit an almost-strong NED such that \\( u \\) and \\( w \\) are endpoints of the first ear. If \\( w = u \\), we can merge the two almost-strong NEDs so that the first ear in one NED is nested on the first ear of the other (with an empty nested interval). Otherwise, the first ears in the two almost-strong NEDs share both endpoints and we can clearly merge them (in case of \\( \\{u, w\\} \\in E_F \\), there is a common ear \\( \\{\\{u, w\\}\\} \\) in both NEDs,"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.949,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "41"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.104,
+                0.825,
+                0.133
+            ],
+            "angle": 0,
+            "content": "which is taken only once). In both subcases, we obtain an almost-strong NED of \\( F[T^r[t]] \\) such that \\( u \\) is an endpoint of the first ear."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.139,
+                0.827,
+                0.181
+            ],
+            "angle": 0,
+            "content": "2. Next consider the case when \\( u \\neq v \\). In this case, \\( F[T^r[q_1]] \\) and \\( F[T^r[q_2]] \\) share only one vertex \\( w \\). Note that \\( F[T^r[q_2]] \\) is connected by Lemma C.37. We separately consider two subcases:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.239,
+                0.188,
+                0.827,
+                0.341
+            ],
+            "angle": 0,
+            "content": "- Subcase 1: \\( F[T^r[q_1]] \\) is connected. Then, according to the induction hypothesis, \\( F[T^r[q_1]] \\) admits an almost-strong NED \\( \\mathcal{P}_1 \\) such that \\( u \\) and \\( w \\) are endpoints of the first ear, and \\( F[T^r[q_2]] \\) admits an almost-strong NED \\( \\mathcal{P}_2 \\) such that \\( w \\) and \\( v \\) are endpoints of the first ear. (i) If \\( w = v \\), we can merge the two almost-strong NEDs so that the first ear of \\( \\mathcal{P}_2 \\) is nested on the first ear of \\( \\mathcal{P}_1 \\) (with empty nested interval). (ii) If \\( w = u \\), we can merge the two almost-strong NEDs so that the first ear of \\( \\mathcal{P}_1 \\) is nested on the first ear of \\( \\mathcal{P}_2 \\) (with empty nested interval). (iii) If \\( w \\in N_F(v) \\) and \\( w \\neq u \\), the first ear of \\( \\mathcal{P}_2 \\) can be chosen as \\( \\{\\{w, v\\}\\} \\). Then, we can merge the two almost-strong NEDs by gluing the first year in \\( \\mathcal{P}_1 \\) with the ear \\( \\{\\{w, v\\}\\} \\) in \\( \\mathcal{P}_2 \\). One can see that the resulting NED is almost-strong. Overall, we always have that \\( F[T^r[t]] \\) admits an almost-strong NED such that \\( u, v \\) are endpoints of the first ear."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.239,
+                0.342,
+                0.828,
+                0.484
+            ],
+            "angle": 0,
+            "content": "- Subcase 2: \\( F[T^r[q_1]] \\) is disconnected. In this subcase, \\( w \\neq u \\). Similar to the proof of Lemma C.38, we obtain that \\( F[T^r[q_1]] \\) has exactly two connected components, and \\( u \\) and \\( w \\) are in different connected components. We can thus invoke Lemma C.35, which shows that both connected components of \\( F[T^r[q_1]] \\), denoted as \\( \\widehat{F}_1 \\) and \\( \\widehat{F}_2 \\), admit almost-strong NEDs \\( \\mathcal{P}_1 \\) and \\( \\mathcal{P}_2 \\), respectively. Moreover, \\( u \\) is an endpoint of the first ear in \\( \\mathcal{P}_1 \\), and \\( w \\) is an endpoint of the first ear in \\( \\mathcal{P}_2 \\). Besides, \\( F[T^r[q_2]] \\) admits an almost-strong NED \\( \\mathcal{P}_3 \\) with \\( w, v \\) as the endpoints of the first ear. By letting the first ear of \\( \\mathcal{P}_2 \\) nest on the first ear of \\( \\mathcal{P}_3 \\) (with an empty nested interval), we can merge \\( \\mathcal{P}_2 \\) and \\( \\mathcal{P}_3 \\). Then, we can merge \\( \\mathcal{P}_1 \\) and \\( \\mathcal{P}_2 \\cup \\mathcal{P}_3 \\) following the same procedure as Subcase 2 in the proof of Lemma C.38."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.239,
+                0.188,
+                0.828,
+                0.484
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.212,
+                0.49,
+                0.411,
+                0.505
+            ],
+            "angle": 0,
+            "content": "Overall, we always have that:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.232,
+                0.511,
+                0.825,
+                0.539
+            ],
+            "angle": 0,
+            "content": "a) If \\( F[T^r[t]] \\) is connected, then it admits an almost-strong NED such that \\( u, v \\) are endpoints of the first ear."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.54,
+                0.826,
+                0.582
+            ],
+            "angle": 0,
+            "content": "b) If \\( F[T^r[t]] \\) is disconnected, then it has two connected components each admitting an almost-strong NED such that \\( u, v \\) belong to an endpoint of the first ear for each of the two NEDs, respectively."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.231,
+                0.511,
+                0.826,
+                0.582
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.589,
+                0.826,
+                0.634
+            ],
+            "angle": 0,
+            "content": "In both subcases, it follows that \\( F[T^{r}[t]] \\) admits an almost-strong NED with \\( u, v \\) as two endpoints. Noting that \\( F = \\bigcup_{t, \\mathfrak{pa}_{Tr}(t) = r} F[T^{r}[t]] \\), we can merge all NEDs of \\( F[T^{r}[t]] \\) to obtain an almost-strong NED of \\( F \\) with \\( u, v \\) as two endpoints."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.64,
+                0.498,
+                0.655
+            ],
+            "angle": 0,
+            "content": "We thus conclude the proof of the induction step."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.662,
+                0.825,
+                0.692
+            ],
+            "angle": 0,
+            "content": "Lemma C.42. For any graph \\(F\\), if \\(F\\) admits an almost-strong NED, then there exists a tree decomposition \\(T^r\\) of \\(F\\) such that \\((F,T^{r})\\in S^{\\mathrm{LF}}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.706,
+                0.825,
+                0.764
+            ],
+            "angle": 0,
+            "content": "Proof. We can assume that \\(F\\) is connected without loss of generality. We will prove the following stronger result: for any graph \\(F\\), if \\(F\\) admits an almost-strong NED where \\(u\\) and \\(v\\) are two different endpoints of the first ear, then there exists a tree decomposition \\(T^r\\) of \\(F\\) such that \\((F, T^r) \\in S^{\\mathsf{LF}}\\) and \\(\\beta_T(r) = \\{\\{u, v\\}\\}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.769,
+                0.826,
+                0.812
+            ],
+            "angle": 0,
+            "content": "Similar to the proof of Lemma C.39, assume that the statement holds when \\( |E_F| \\leq m \\), and consider the case of \\( |E_F| = m + 1 \\). Denote the first ear as \\( P_1 = \\{\\{w_0, w_1\\}, \\dots, \\{w_{l-1}, w_l\\}\\} \\) where \\( w_0 = u \\), \\( w_l = v \\). Consider the following three cases:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.818,
+                0.755,
+                0.832
+            ],
+            "angle": 0,
+            "content": "1. Either \\( u \\) or \\( v \\) is a cut vertex in \\( F \\). This case is exactly the same as in Lemma C.39."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.191,
+                0.839,
+                0.825,
+                0.855
+            ],
+            "angle": 0,
+            "content": "2. There is an ear \\( P_{i} \\) nested on \\( P_{1} \\) with \\( I(P_{i}) = P_{1} \\). This case is also the same as in Lemma C.39."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.861,
+                0.333,
+                0.875
+            ],
+            "angle": 0,
+            "content": "3. Otherwise, \\( l \\geq 2 \\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.191,
+                0.818,
+                0.825,
+                0.875
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.239,
+                0.882,
+                0.827,
+                0.927
+            ],
+            "angle": 0,
+            "content": "- If there is an ear \\( P_i \\) nested on \\( P_1 \\) with \\( \\{w_0, w_1\\}, \\{w_1, w_2\\} \\in I(P_i) \\), then \\( l \\geq 3 \\). By definition of almost-strong NED, there does not exist an ear \\( P_j \\) nested on \\( P_1 \\) with \\( \\{w_{l-2}, w_{l-1}\\}, \\{w_{l-1}, w_l\\} \\in I(P_i) \\). In this case, we can split \\( P_1 \\) into two parts:"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.949,
+                0.51,
+                0.96
+            ],
+            "angle": 0,
+            "content": "42"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.252,
+                0.103,
+                0.825,
+                0.27
+            ],
+            "angle": 0,
+            "content": "\\(P_{1,u} = P_1\\backslash \\{\\{w_{l - 1},w_l\\}\\}\\), and \\(P_{1,v} = \\{\\{w_{l - 1},w_l\\}\\}\\). Then, we can rearrange any ear nested on \\(P_{1}\\) so that it is nested on either \\(P_{1,u}\\) or \\(P_{1,v}\\). In this way, we can split all ears into two sets \\(\\mathcal{P}_u\\) and \\(\\mathcal{P}_v\\), one corresponding to \\(P_{1,u}\\) and its descendant ears, and the other corresponding to \\(P_{1,v}\\) and its descendant ears. Therefore, according to the induction hypothesis, there is a tree decomposition \\(\\tilde{T}_u^s\\) of \\(F[\\mathcal{P}_u]\\) satisfying \\((F[\\mathcal{P}_u],\\tilde{T}_u^s)\\in S^{\\mathrm{LF}}\\) and \\(\\beta_{\\tilde{T}_u}(s) = \\{u,w_{l - 1}\\}\\), and similarly, there is a tree decomposition \\(\\tilde{T}_v^{s'}\\) of \\(F[\\mathcal{P}_v]\\) satisfying \\((F[\\mathcal{P}_v],\\tilde{T}_v^{s'})\\in S^{\\mathrm{LF}}\\) and \\(\\beta_{\\tilde{T}_v}(s) = \\{w_{l - 1},v\\}\\). We can then construct a tree \\(T^r\\) by merging \\(\\tilde{T}_u^s\\) and \\(\\tilde{T}_v^{s'}\\) and adding two fresh nodes \\(r,r'\\), where \\(r\\) is the root node, \\(\\mathsf{pa}_{T^r}(r') = r\\) and \\(\\mathsf{pa}_{T^r}(s) = \\mathsf{pa}_{T^r}(s') = r'\\). Set \\(\\beta_T(r) = \\{u,v\\}\\) and \\(\\beta_T(r') = \\{u,v,w_{l - 1}\\}\\). It is easy to see that the constructed \\(T^r\\) is a tree decomposition of \\(F\\) and \\((F,T^r)\\in S^{\\mathrm{LF}}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.239,
+                0.271,
+                0.825,
+                0.33
+            ],
+            "angle": 0,
+            "content": "- If there does not exist an ear \\( P_{i} \\) nested on \\( P_{1} \\) with \\( \\{w_{0}, w_{1}\\}, \\{w_{1}, w_{2}\\} \\in I(P_{i}) \\), then we follow exactly the same analysis as in the previous item, except that now we split \\( P_{1} \\) into two parts: \\( P_{1,u} = \\{\\{w_{0}, w_{1}\\}\\} \\), and \\( P_{1,v} = P_{1} \\backslash \\{\\{w_{0}, w_{1}\\}\\} \\). We can still construct a tree decomposition \\( T^{r} \\) of \\( F \\) such that \\( (F, T^{r}) \\in S^{\\mathsf{LF}} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.335,
+                0.529,
+                0.351
+            ],
+            "angle": 0,
+            "content": "Combining the above three cases concludes the proof."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.336,
+                0.824,
+                0.348
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.358,
+                0.825,
+                0.387
+            ],
+            "angle": 0,
+            "content": "Theorem C.43. For any graph \\( F \\), \\( F \\) has an almost-strong NED iff there is a tree decomposition \\( T^r \\) of \\( F \\) such that \\( (F, T^r) \\in S^{\\mathsf{LF}} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.401,
+                0.591,
+                0.417
+            ],
+            "angle": 0,
+            "content": "Proof. This is a direct consequence of Lemmas C.41 and C.42."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.403,
+                0.824,
+                0.414
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.433,
+                0.378,
+                0.447
+            ],
+            "angle": 0,
+            "content": "C.4 PART 3: PEBBLE GAME"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.458,
+                0.825,
+                0.502
+            ],
+            "angle": 0,
+            "content": "In this part, we prove that \\( S^M \\) is maximal (Definition 3.1(b)) for any \\( M \\in \\{\\mathrm{Sub}, \\mathrm{L}, \\mathrm{LF}, \\mathrm{F}\\} \\). To achieve this, we first introduce a general class of graphs which we call Fürer graphs (Fürer, 2001). Intuitions and illustrations of Fürer graphs can be found in Zhang et al. (2023a)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.505,
+                0.825,
+                0.536
+            ],
+            "angle": 0,
+            "content": "Definition C.44 (Fürer graphs). Given any connected graph \\( F = (V_F, E_F, \\ell_F) \\), the Fürer graph \\( G(F) = (V_{G(F)}, E_{G(F)}, \\ell_{G(F)}) \\) is constructed as follows:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.263,
+                0.541,
+                0.658,
+                0.558
+            ],
+            "angle": 0,
+            "content": "\\[\nV _ {G (F)} = \\{(x, X): x \\in V _ {F}, X \\subset N _ {F} (x), | X | \\bmod 2 = 0 \\},\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.263,
+                0.56,
+                0.734,
+                0.578
+            ],
+            "angle": 0,
+            "content": "\\[\nE _ {G (F)} = \\left\\{\\{(x, X), (y, Y) \\} \\subset V _ {G}: \\{x, y \\} \\in E _ {F}, (x \\in Y \\leftrightarrow y \\in X) \\right\\},\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.264,
+                0.58,
+                0.546,
+                0.598
+            ],
+            "angle": 0,
+            "content": "\\[\n\\ell_ {G (F)} (x, X) = \\ell_ {F} (x) \\quad \\forall (x, X) \\in V _ {G (F)}.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.603,
+                0.825,
+                0.632
+            ],
+            "angle": 0,
+            "content": "Here, \\( x \\in Y \\leftrightarrow y \\in X \\) holds when either \\( (x \\in Y \\text{ and } y \\in X) \\) or \\( (x \\notin Y \\text{ and } y \\notin X) \\) holds. For each \\( x \\in V_F \\), denote the set"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.313,
+                0.639,
+                0.825,
+                0.656
+            ],
+            "angle": 0,
+            "content": "\\[\n\\operatorname {M e t a} _ {F} (x) := \\{(x, X): X \\subset N _ {F} (x), | X | \\bmod 2 = 0 \\}, \\tag {24}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.66,
+                0.807,
+                0.679
+            ],
+            "angle": 0,
+            "content": "which is called the meta vertices of \\( G(F) \\) associated to \\( x \\). Note that \\( V_{G(F)} = \\bigcup_{x\\in V_F}\\mathsf{Meta}_F(x) \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.687,
+                0.457,
+                0.702
+            ],
+            "angle": 0,
+            "content": "We next define an operation called \"twist\":"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.705,
+                0.825,
+                0.751
+            ],
+            "angle": 0,
+            "content": "Definition C.45 (Twist). Let \\( G(F) = (V_{G(F)}, E_{G(F)}, \\ell_{G(F)}) \\) be the Fürer graph of \\( F = (V_F, E_F, \\ell_F) \\), and let \\( \\{x, y\\} \\in E_F \\) be an edge of \\( F \\). The twisted Fürer graph of \\( G(F) \\) for edge \\( \\{x, y\\} \\), is constructed as follows: \\( \\text{twist}(G(F), \\{x, y\\}) := (V_{G(F)}, E_{\\text{twist}(G(F), \\{x, y\\})}, \\ell_{G(F)}) \\), where"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.26,
+                0.757,
+                0.737,
+                0.775
+            ],
+            "angle": 0,
+            "content": "\\[\nE _ {\\operatorname {w i s t} (G (F), \\{x, y \\})} := E _ {G (F)} \\triangle \\{\\{\\xi , \\eta \\}: \\xi \\in \\operatorname {M e t a} _ {F} (x), \\eta \\in \\operatorname {M e t a} _ {F} (y) \\},\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.78,
+                0.825,
+                0.81
+            ],
+            "angle": 0,
+            "content": "and \\(\\triangle\\) is the symmetric difference operator, i.e., \\(A\\triangle B = (A\\backslash B)\\cup (B\\backslash A)\\). For an edge set \\(S = \\{e_1,\\dots ,e_k\\} \\subset E_F\\), we further define"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.317,
+                0.816,
+                0.825,
+                0.832
+            ],
+            "angle": 0,
+            "content": "\\[\n\\operatorname {t w i s t} (G (F), S) := \\operatorname {t w i s t} (\\dots \\operatorname {t w i s t} (G (F), e _ {1}) \\dots , e _ {k}). \\tag {25}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.838,
+                0.825,
+                0.867
+            ],
+            "angle": 0,
+            "content": "Note that Equation (25) is well-defined as the resulting graph does not depend on the order of edges \\( e_1, \\dots, e_k \\) for twisting."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.877,
+                0.812,
+                0.893
+            ],
+            "angle": 0,
+            "content": "The following result is well-known (see e.g., Zhang et al., 2023a, Corollary I.5 and Lemma I.7)):"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.896,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "Theorem C.46. For any connected graph \\( F \\) and any set \\( S_1, S_2 \\subset E_F \\), \\( \\operatorname{twist}(G(F), S_1) \\simeq \\operatorname{twist}(G(F), S_2) \\) iff \\( |S_1| \\mod 2 = |S_2| \\mod 2 \\)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.949,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "43"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.825,
+                0.148
+            ],
+            "angle": 0,
+            "content": "This result shows that the structure of twisted Fürer graph does not depend on which edge is twisted. Therefore, we can pick any edge \\(\\{u,v\\} \\in E_F\\) and denote \\(H(F) = \\mathrm{twist}(G(F),\\{u,v\\})\\). We now show that \\(G(F),H(F)\\) can be distinguished via homomorphism information:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.149,
+                0.825,
+                0.181
+            ],
+            "angle": 0,
+            "content": "Theorem C.47. For any graph \\( F \\), let (twisted) Fürer graphs \\( G(F) \\), \\( H(F) \\) be defined above. Then, \\( \\operatorname{hom}(F, G(F)) \\neq \\operatorname{hom}(F, H(F)) \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.193,
+                0.826,
+                0.223
+            ],
+            "angle": 0,
+            "content": "Proof. In the proof below, we use \\( g \\) and \\( h \\) to denote mappings where \\( g: V_F \\to V_{G(F)} \\) and \\( h: V_F \\to V_{H(F)} \\). The proof is divided into the following parts."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.228,
+                0.826,
+                0.273
+            ],
+            "angle": 0,
+            "content": "1. We first prove that there is a homomorphism \\( g \\in \\operatorname{Hom}(F, G(F)) \\) satisfying \\( g(w) \\in \\operatorname{Meta}_F(w) \\) for all \\( w \\in V_F \\), but there is no homomorphism \\( h \\in \\operatorname{Hom}(F, H(F)) \\) satisfying \\( h(w) \\in \\operatorname{Meta}_F(w) \\) for all \\( w \\in V_F \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.278,
+                0.825,
+                0.324
+            ],
+            "angle": 0,
+            "content": "a) Define a mapping \\( g: V_F \\to V_{G(F)} \\) such that \\( g(w) = (w, \\emptyset) \\) for all \\( w \\in V_F \\). We clearly have \\( \\{w, x\\} \\in E_F \\) implies that \\( \\{g(w), g(x)\\} \\in E_{G(F)} \\). Moreover, \\( \\ell_{G(F)}(g(w)) = \\ell_F(w) \\). Therefore, \\( g \\in \\operatorname{Hom}(F, G(F)) \\) is indeed a homomorphism."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.324,
+                0.825,
+                0.369
+            ],
+            "angle": 0,
+            "content": "b) If we similarly define \\( h: V_F \\to V_{H(F)} \\) such that \\( h(w) = (w, \\emptyset) \\) for all \\( w \\in V_F \\), then for all \\( \\{w, x\\} \\in E_F \\backslash \\{\\{u, v\\}\\} \\) we have \\( \\{h(w), h(x)\\} \\in E_{H(F)} \\), but \\( \\{h(u), h(v)\\} \\notin E_{H(F)} \\) since the edge \\( \\{u, v\\} \\) is twisted."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.231,
+                0.278,
+                0.825,
+                0.369
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.212,
+                0.375,
+                0.825,
+                0.456
+            ],
+            "angle": 0,
+            "content": "It remains to prove that for all \\( h: V_F \\to V_{H(F)} \\) of the form \\( h(w) = (w, U_w) \\) (for each \\( w \\in V_F \\)), \\( h \\notin \\operatorname{Hom}(F, H(F)) \\). It suffices to prove that there is an odd number of edges \\( \\{w, x\\} \\in E_F \\) such that \\( \\{h(w), h(x)\\} \\notin E_{H(F)} \\). Let \\( h, \\tilde{h} \\) be two such mappings that differ in only one vertex \\( z \\), i.e., \\( h(w) = \\tilde{h}(w) \\) for all \\( w \\neq z \\) but \\( h(z) = (z, U_z) \\neq (z, \\tilde{U}_z) = \\tilde{h}(z) \\). Denote \\( D_z = \\tilde{U}_z \\triangle U_z \\). Based on the definition of Fürer graph, it follows that"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.238,
+                0.461,
+                0.826,
+                0.495
+            ],
+            "angle": 0,
+            "content": "- for all \\(\\{w, x\\} \\in E_F\\) with \\(w \\neq z, x \\neq z\\), we have \\(\\{h(w), h(x)\\} \\in E_{H(F)}\\) iff \\(\\{\\tilde{h}(w), \\tilde{h}(x)\\} \\in E_{H(F)}\\);"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.238,
+                0.494,
+                0.826,
+                0.529
+            ],
+            "angle": 0,
+            "content": "- for all \\(\\{w, z\\} \\in E_F\\) with \\(w \\notin D_z\\), we also have \\(\\{h(w), h(z)\\} \\in E_{H(F)}\\) iff \\(\\{\\tilde{h}(w), \\tilde{h}(z)\\} \\in E_{H(F)}\\);"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.238,
+                0.529,
+                0.825,
+                0.561
+            ],
+            "angle": 0,
+            "content": "- for all \\(\\{w, z\\} \\in E_F\\) with \\(w \\in D_z\\), we have \\(\\{h(w), h(z)\\} \\in E_{H(F)}\\) iff \\(\\{\\tilde{h}(w), \\tilde{h}(z)\\} \\notin E_{H(F)}\\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.238,
+                0.461,
+                0.826,
+                0.561
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.569,
+                0.826,
+                0.645
+            ],
+            "angle": 0,
+            "content": "Since \\( |U_z| \\bmod 2 = 0 \\) and \\( |\\tilde{U}_z| \\bmod 2 = 0 \\), we have \\( |D_z| \\bmod 2 = 0 \\) and thus the number of edges \\( \\{w, x\\} \\in E_F \\) such that \\( \\{h(w), h(x)\\} \\notin E_{H(F)} \\) has the same parity as the number of edges such that \\( \\{\\tilde{h}(w), \\tilde{h}(x)\\} \\notin E_{H(F)} \\). Finally, noting that all mappings \\( h \\) can be obtained from the one in (b) by continually modifying \\( U_w \\) for each \\( w \\in V_F \\) and the parity remains unchanged, we have concluded the proof of this part."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.65,
+                0.825,
+                0.693
+            ],
+            "angle": 0,
+            "content": "2. We next prove that for any permutation \\(\\pi : V_F \\to V_F\\), there does not exist a homomorphism \\(h \\in \\operatorname{Hom}(F, H(F))\\) satisfying \\(h(w) \\in \\operatorname{Meta}_F(\\pi(w))\\) for all \\(w \\in V_F\\). Assume the conclusion does not hold and pick any \\(h\\) satisfying the above condition. Consider the following cases:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.232,
+                0.699,
+                0.825,
+                0.77
+            ],
+            "angle": 0,
+            "content": "a) Case 1: if \\(\\pi\\) is an automorphism of \\(F\\), then it is easy to see that \\(h \\circ \\pi^{-1} \\in \\operatorname{Hom}(F, H(F))\\), because \\(\\{w, x\\} \\in E_F \\Rightarrow \\{\\pi^{-1}(w), \\pi^{-1}(x)\\} \\in E_F \\Rightarrow \\{h(\\pi^{-1}(w)), h(\\pi^{-1}(x))\\} \\in E_F\\) for all \\(w, x \\in V_F\\), and \\(\\ell_F(w) = \\ell_F(\\pi^{-1}(w)) = \\ell_{G(F)}(h(\\pi^{-1}(w)))\\) for all \\(w \\in V_F\\). Moreover, \\(h \\circ \\pi^{-1}\\) satisfies that \\(h(\\pi^{-1}(w)) \\in \\operatorname{Meta}_F(w)\\), yielding a contradiction to point 1."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.231,
+                0.77,
+                0.825,
+                0.829
+            ],
+            "angle": 0,
+            "content": "b) Case 2: if \\(\\pi\\) is not an automorphism of \\(F\\), then there exists an edge \\(\\{w, x\\} \\in E_F\\) such that \\(\\{\\pi(w), \\pi(x)\\} \\notin E_F\\). In this case, we must have \\(\\{h(w), h(x)\\} \\notin E_{H(F)}\\) since by definition \\(h(w) \\in \\mathsf{Meta}_F(\\pi(w))\\), \\(h(x) \\in \\mathsf{Meta}_F(\\pi(x))\\), and \\(\\{\\pi(w), \\pi(x)\\}\\) is not an edge of \\(F\\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.231,
+                0.699,
+                0.825,
+                0.829
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.834,
+                0.825,
+                0.864
+            ],
+            "angle": 0,
+            "content": "In both cases, \\( h \\) is invalid and thus there is no homomorphism \\( h \\in \\operatorname{Hom}(F, H(F)) \\) satisfying \\( h(w) \\in \\operatorname{Meta}_F(\\pi(w)) \\) for all \\( w \\in V_F \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.869,
+                0.736,
+                0.885
+            ],
+            "angle": 0,
+            "content": "3. We finally prove that the following two sets have equal size (i.e., \\( |S_G| = |S_H| \\)):"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.242,
+                0.888,
+                0.789,
+                0.904
+            ],
+            "angle": 0,
+            "content": "\\[\nS _ {G} := \\left\\{g \\in \\operatorname {H o m} (F, G (F)): \\exists w, x, y \\in V _ {F} \\text {s . t .} x \\neq y, g (x), g (y) \\in \\operatorname {M e t a} _ {F} (w) \\right\\},\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.243,
+                0.905,
+                0.793,
+                0.922
+            ],
+            "angle": 0,
+            "content": "\\[\nS _ {H} := \\left\\{h \\in \\operatorname {H o m} (F, H (F)): \\exists w, x, y \\in V _ {F} \\text {s . t .} x \\neq y, h (x), h (y) \\in \\operatorname {M e t a} _ {F} (w) \\right\\}.\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.949,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "44"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.212,
+                0.103,
+                0.825,
+                0.133
+            ],
+            "angle": 0,
+            "content": "It suffices to prove that, for any proper subset \\( U \\subsetneq V_F \\), we have \\( |S_G^U| = |S_H^U| \\), where \\( S_G \\) and \\( S_H \\) are defined as follows:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.293,
+                0.138,
+                0.741,
+                0.18
+            ],
+            "angle": 0,
+            "content": "\\[\nS _ {G} ^ {U} := \\left\\{g \\in \\operatorname {H o m} (F, G (F)): g (x) \\in \\bigcup_ {w \\in U} \\operatorname {M e t a} _ {F} (w) \\forall x \\in V _ {F} \\right\\},\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.295,
+                0.183,
+                0.743,
+                0.224
+            ],
+            "angle": 0,
+            "content": "\\[\nS _ {H} ^ {U} := \\left\\{h \\in \\operatorname {H o m} (F, H (F)): h (x) \\in \\bigcup_ {w \\in U} \\operatorname {M e t a} _ {F} (w) \\forall x \\in V _ {F} \\right\\}.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.212,
+                0.231,
+                0.825,
+                0.275
+            ],
+            "angle": 0,
+            "content": "Fix \\(U \\subsetneq V_F\\) and pick \\(z \\in V_F \\backslash U\\). Let \\(P\\) be a simple path from \\(u\\) to \\(z\\) of the form \\(P = \\{\\{w_0, w_1\\}, \\dots, \\{w_{k-1}, w_k\\}\\} \\subset E_F\\) where \\(\\{w_0, w_1\\} = \\{u, v\\}\\), \\(w_k = z\\). Define a mapping \\(\\sigma\\) that takes \\(g \\in S_G^U\\) as input and outputs a mapping \\(h: V_F \\to V_{H(F)}\\):"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.258,
+                0.282,
+                0.769,
+                0.315
+            ],
+            "angle": 0,
+            "content": "\\[\nh (x) = \\left\\{ \\begin{array}{l l} g (x) & \\text {i f} [ g (x) ] _ {0} \\notin \\{w _ {1}, \\dots , w _ {k - 1} \\}, \\\\ ([ g (x) ] _ {0}, [ g (x) ] _ {1} \\bigtriangleup \\{w _ {i - 1}, w _ {i + 1} \\}) & \\text {i f} [ g (x) ] _ {0} = w _ {i}, i \\in [ k - 1 ], \\end{array} \\right.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.212,
+                0.322,
+                0.825,
+                0.366
+            ],
+            "angle": 0,
+            "content": "where we write \\( g(x) = ([g(x)]_0, [g(x)]_1) \\). We will prove that \\( h \\in S_H^U \\). Since we clearly have \\( h(w) \\in U \\) for all \\( w \\in V_F \\), it suffices to prove that \\( h \\in \\mathsf{Hom}(F, H(F)) \\). Let \\( \\{x, y\\} \\in E_F \\) be any edge in \\( F \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.239,
+                0.371,
+                0.825,
+                0.416
+            ],
+            "angle": 0,
+            "content": "- If \\([g(x)]_0, [g(y)]_0 \\notin \\{w_1, \\dots, w_{k-1}\\}\\), then \\(\\{h(x), h(y)\\} = \\{g(x), g(y)\\} \\in E_{G(F)}\\). Also, since \\(\\{[g(x)]_0, [g(y)]_0\\} \\neq \\{u, v\\}\\), \\(\\{[g(x)]_0, [g(y)]_0\\}\\) is not twisted and thus \\(\\{h(x), h(y)\\} \\in E_{H(F)}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.239,
+                0.417,
+                0.825,
+                0.447
+            ],
+            "angle": 0,
+            "content": "- If \\([g(x)]_0 \\notin \\{w_1, \\dots, w_{k-1}\\}\\), \\([g(y)]_0 = w_i\\) for some \\(i \\in [k-1]\\), and \\(\\{[g(x)]_0, [g(y)]_0\\} \\neq \\{\\{u, v\\}, \\{w_{k-1}, w_k\\}\\}\\), then"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.304,
+                0.452,
+                0.774,
+                0.469
+            ],
+            "angle": 0,
+            "content": "\\[\n\\{h (x), h (y) \\} = \\left\\{\\left([ g (x) ] _ {0}, [ g (x) ] _ {1}\\right), \\left([ g (y) ] _ {0}, [ g (y) ] _ {1} \\triangle \\left\\{w _ {i - 1}, w _ {i + 1} \\right\\}\\right) \\right\\}.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.254,
+                0.475,
+                0.315,
+                0.487
+            ],
+            "angle": 0,
+            "content": "We have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.255,
+                0.495,
+                0.834,
+                0.55
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\{g (x), g (y) \\} \\in E _ {G (F)} \\iff ([ g (x) ] _ {0} \\in [ g (y) ] _ {1}) \\leftrightarrow ([ g (y) ] _ {0} \\in [ g (x) ] _ {1}) \\\\ \\Longleftrightarrow ([ g (x) ] _ {0} \\in [ g (y) ] _ {1} \\triangle \\left\\{w _ {i - 1}, w _ {i + 1} \\right\\}) \\leftrightarrow ([ g (y) ] _ {0} \\in [ g (x) ] _ {1}) \\\\ \\iff \\{h (x), h (y) \\} \\in E _ {G (F)}. \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.253,
+                0.556,
+                0.704,
+                0.573
+            ],
+            "angle": 0,
+            "content": "Also, since \\(\\{[g(x)]_0, [g(y)]_0\\}\\) is not twisted, \\(\\{h(x), h(y)\\} \\in E_{H(F)}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.239,
+                0.573,
+                0.825,
+                0.602
+            ],
+            "angle": 0,
+            "content": "- If \\([g(x)]_0, [g(y)]_0 \\in \\{w_1, \\dots, w_{k-1}\\}\\), the analysis is similar to the above one and we can still prove that \\(\\{h(x), h(y)\\} \\in E_{G(F)}\\) and thus \\(\\{h(x), h(y)\\} \\in E_{G(F)}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.239,
+                0.603,
+                0.825,
+                0.633
+            ],
+            "angle": 0,
+            "content": "- If \\(\\{[g(x)]_0, [g(y)]_0\\} = \\{u, v\\}\\), without loss of generality assume \\(w_0 = u = [g(x)]_0\\) and \\(w_1 = v = [g(y)]_0\\). We have"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.239,
+                0.573,
+                0.825,
+                0.633
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.306,
+                0.638,
+                0.771,
+                0.692
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\{g (x), g (y) \\} \\in E _ {G (F)} \\iff (u \\in [ g (y) ] _ {1}) \\leftrightarrow (v \\in [ g (x) ] _ {1}) \\\\ \\Longleftrightarrow (u \\notin [ g (y) ] _ {1} \\triangle \\{u, w _ {2} \\}) \\leftrightarrow (v \\in [ g (x) ] _ {1}) \\\\ \\Longleftrightarrow \\{h (x), h (y) \\} \\notin E _ {G (F)}. \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.251,
+                0.699,
+                0.718,
+                0.715
+            ],
+            "angle": 0,
+            "content": "However, since \\(\\{u,v\\}\\) is twisted, we still have \\(\\{h(x),h(y)\\} \\in E_{H(F)}\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.239,
+                0.715,
+                0.825,
+                0.743
+            ],
+            "angle": 0,
+            "content": "- Finally, we do not need to consider the case \\(\\{[g(x)]_0, [g(y)]_0\\} = \\{w_{k-1}, w_k\\}\\) since \\(w_k = z \\notin U\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.75,
+                0.825,
+                0.782
+            ],
+            "angle": 0,
+            "content": "This proves that \\( h \\in S_H^U \\). Moreover, it is straightforward to see that the mapping \\( \\sigma(g) = h \\) is a bijection from \\( S_G^U \\) to \\( S_H^U \\). We have thus proved that \\( |S_G^U| = |S_H^U| \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.787,
+                0.826,
+                0.816
+            ],
+            "angle": 0,
+            "content": "Combining the above three items, we obtain \\(\\mathsf{Hom}(F,G(F)) > \\mathsf{Hom}(F,H(F))\\), concluding the proof."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.831,
+                0.825,
+                0.904
+            ],
+            "angle": 0,
+            "content": "In the subsequent analysis, we will prove that for all model \\(M\\) considered in Theorem 3.4 and any connected graph \\(F\\), \\(\\chi_{G(F)}^{M}(G(F)) = \\chi_{H(F)}^{M}(H(F))\\) if \\(F \\notin S^{M}\\). The proof is based on an important technique developed in Cai et al. (1992), called the pebble game. When restricting our analysis on Fürer graphs, the pebble game can be greatly simplified as shown in Fürer (2001); Zhang et al. (2023a). Below, we separately describe the corresponding pebble game for each model \\(M\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.91,
+                0.583,
+                0.925
+            ],
+            "angle": 0,
+            "content": "We first define a key concept called the connected component."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "45"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.827,
+                0.203
+            ],
+            "angle": 0,
+            "content": "Definition C.48 (Connected components). Let \\( F = (V_F, E_F) \\) be a connected graph and let \\( U \\subset V_F \\) be a vertex set, called separation vertices. We say two edges \\( \\{u, v\\}, \\{x, y\\} \\in E_F \\) are in the same connected component if there is a simple path \\( \\{\\{y_0, y_1\\}, \\dots, \\{y_{k-1}, y_k\\}\\} \\) satisfying that \\( \\{y_0, y_1\\} = \\{u, v\\}, \\{y_{k-1}, y_k\\} = \\{x, y\\} \\) and \\( y_i \\notin U \\) for all \\( i \\in [k-1] \\). It is easy to see that the above relation between edges forms an equivalence relation. Therefore, we can define a partition over the edge set, denoted by \\( \\mathbb{C}\\mathbb{C}_F(U) = \\{P_i : i \\in [m]\\} \\) for some \\( m \\), where each \\( P_i \\subset E_F \\) is called a connected component."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.214,
+                0.828,
+                0.3
+            ],
+            "angle": 0,
+            "content": "We are now ready to describe the game rule. There are two players (named Alice and Bob), a graph \\( F \\), and several pebbles. At the beginning, all pebbles lie outside the graph. Through the game process, some pebbles will be placed on the vertices of \\( F \\) and thus separate the edges \\( E_{F} \\) into connected components according to Definition C.48. In each game round, Alice updates the location of pebbles, while Bob maintains a subset of connected components, ensuring that the number of selected components is odd. There are three major types of operations:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.193,
+                0.305,
+                0.826,
+                0.392
+            ],
+            "angle": 0,
+            "content": "1. Add a pebble \\(\\mathfrak{p}\\). Alice places a pebble \\(\\mathfrak{p}\\) (previously outside the graph) on some vertex of \\(F\\). If introducing this new pebble does not change the connected components, then Bob does nothing. Otherwise, there must be a connected component \\(P\\) separated by \\(\\mathfrak{p}\\) into several components \\(P = \\bigcup_{i \\in [m]} P_i\\) for some \\(m\\). Bob will update his selected components as follows: if \\(P\\) is selected, then remove \\(P\\) and add a subset of connected components in \\(\\{P_1, \\dots, P_m\\}\\) while ensuring that the number of selected components in total is odd."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.394,
+                0.825,
+                0.479
+            ],
+            "angle": 0,
+            "content": "2. Remove a pebble \\(\\mathfrak{p}\\). Alice removes a pebble \\(\\mathfrak{p}\\) (previously on some vertex) outside the graph. If introducing this new pebble does not change the connected components, then Bob does nothing. Otherwise, there are multiple connected components \\(P_{1},\\dots ,P_{m}\\) getting merged into a whole \\(P = \\bigcup_{i\\in [m]}P_{i}\\). Bob will update his selected components by removing all \\(P_{i}\\), \\(i\\in [m]\\) (if selected) and optionally adding \\(P\\), while ensuring that the number of selected components in total is odd."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.482,
+                0.825,
+                0.511
+            ],
+            "angle": 0,
+            "content": "3. Swap two pebbles \\(\\mathsf{p}\\) and \\(\\mathsf{p}^{\\prime}\\). Alice swaps the position of two pebbles \\(\\mathsf{p}\\) and \\(\\mathsf{p}^{\\prime}\\). This operation does not change the connected components and thus Bob does nothing."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.192,
+                0.305,
+                0.826,
+                0.511
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.517,
+                0.826,
+                0.56
+            ],
+            "angle": 0,
+            "content": "At any time, if there is an edge \\(\\{x,y\\}\\) such that both endpoints hold pebbles and the connected component \\(\\{\\{x,y\\}\\}\\) is selected by Bob, then Bob loses the game and Alice wins. If Alice cannot win through the game process, then Bob wins."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.567,
+                0.827,
+                0.638
+            ],
+            "angle": 0,
+            "content": "We now define the concrete pebble game for each model \\(M\\) considered in this paper. In cases of Subgraph GNN, Local 2-GNN, Local 2-FGNN, and 2-FGNN, there are three pebbles \\(\\mathsf{p}_u\\), \\(\\mathsf{p}_v\\), \\(\\mathsf{p}_w\\). As described before, all pebbles lie outside the graph at the beginning. Alice first adds the pebble \\(\\mathsf{p}_u\\) (operation 1) and then adds the pebble \\(\\mathsf{p}_v\\) (operation 1). Next, the game cyclically executes the following process:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.643,
+                0.714,
+                0.658
+            ],
+            "angle": 0,
+            "content": "- Subgraph GNN. Alice can choose either one of the following ways to play:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.236,
+                0.661,
+                0.753,
+                0.675
+            ],
+            "angle": 0,
+            "content": "- Remove the pebble \\(\\mathsf{p}_v\\) (operation 2), and re-add the pebble \\(\\mathsf{p}_v\\) (operation 1)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.236,
+                0.676,
+                0.822,
+                0.704
+            ],
+            "angle": 0,
+            "content": "- Add the pebble \\(\\mathsf{p}_w\\) (operation 1) adjacent to the pebble \\(\\mathsf{p}_v\\), swap pebble \\(\\mathsf{p}_v\\) with \\(\\mathsf{p}_w\\) (operation 3), and remove the pebble \\(\\mathsf{p}_w\\) (operation 2)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.236,
+                0.661,
+                0.822,
+                0.704
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.199,
+                0.707,
+                0.698,
+                0.722
+            ],
+            "angle": 0,
+            "content": "- Local 2-GNN. Alice can choose either one of the following ways to play:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.236,
+                0.724,
+                0.754,
+                0.739
+            ],
+            "angle": 0,
+            "content": "- Remove the pebble \\(\\mathsf{p}_u\\) (operation 2), and re-add the pebble \\(\\mathsf{p}_u\\) (operation 1)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.236,
+                0.74,
+                0.753,
+                0.754
+            ],
+            "angle": 0,
+            "content": "- Remove the pebble \\(\\mathsf{p}_v\\) (operation 2), and re-add the pebble \\(\\mathsf{p}_v\\) (operation 1)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.236,
+                0.755,
+                0.822,
+                0.783
+            ],
+            "angle": 0,
+            "content": "- Add the pebble \\(\\mathsf{p}_w\\) (operation 1) adjacent to the pebble \\(\\mathsf{p}_u\\), swap pebble \\(\\mathsf{p}_u\\) with \\(\\mathsf{p}_w\\) (operation 3), and remove the pebble \\(\\mathsf{p}_w\\) (operation 2)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.236,
+                0.785,
+                0.822,
+                0.812
+            ],
+            "angle": 0,
+            "content": "- Add the pebble \\(\\mathsf{p}_w\\) (operation 1) adjacent to the pebble \\(\\mathsf{p}_v\\), swap pebble \\(\\mathsf{p}_v\\) with \\(\\mathsf{p}_w\\) (operation 3), and remove the pebble \\(\\mathsf{p}_w\\) (operation 2)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.236,
+                0.724,
+                0.822,
+                0.812
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.199,
+                0.815,
+                0.708,
+                0.83
+            ],
+            "angle": 0,
+            "content": "- Local 2-FGNN. Alice can choose either one of the following ways to play:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.236,
+                0.832,
+                0.754,
+                0.846
+            ],
+            "angle": 0,
+            "content": "- Remove the pebble \\(\\mathsf{p}_u\\) (operation 2), and re-add the pebble \\(\\mathsf{p}_u\\) (operation 1)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.236,
+                0.848,
+                0.753,
+                0.861
+            ],
+            "angle": 0,
+            "content": "- Remove the pebble \\(\\mathfrak{p}_v\\) (operation 2), and re-add the pebble \\(\\mathfrak{p}_v\\) (operation 1)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.236,
+                0.863,
+                0.822,
+                0.891
+            ],
+            "angle": 0,
+            "content": "- Add the pebble \\(\\mathsf{p}_w\\) (operation 1) adjacent to either pebble \\(\\mathsf{p}_u\\) or pebble \\(\\mathsf{p}_v\\), swap pebble \\(\\mathsf{p}_w\\) with any adjacent pebble (operation 3), and remove the pebble \\(\\mathsf{p}_w\\) (operation 2)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.236,
+                0.832,
+                0.822,
+                0.891
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.199,
+                0.894,
+                0.822,
+                0.922
+            ],
+            "angle": 0,
+            "content": "- 2-FGNN. Alice adds the pebble \\(\\mathsf{p}_w\\) (operation 1), swap the pebbles \\(\\mathsf{p}_w\\) with either \\(\\mathsf{p}_u\\) or \\(\\mathsf{p}_v\\) (operation 3), and remove the pebble \\(\\mathsf{p}_w\\) (operation 2)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.949,
+                0.51,
+                0.96
+            ],
+            "angle": 0,
+            "content": "46"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.104,
+                0.825,
+                0.137
+            ],
+            "angle": 0,
+            "content": "Proposition C.49. For any model \\( M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\} \\) and any graph \\( F \\), if Alice cannot win the pebble game associated to model \\( M \\) on graph \\( F \\), then \\( \\chi_{G(F)}^{M}(G(F)) = \\chi_{H(F)}^{M}(H(F)) \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.151,
+                0.827,
+                0.292
+            ],
+            "angle": 0,
+            "content": "Proof. Note that the pebble game does not depend on the vertex labels \\(\\ell_F\\) in \\(F\\). We first assume that the vertex labels are all different, i.e., \\(\\ell_F(x) \\neq \\ell_F(y)\\) for all \\(x, y \\in V_F\\). Then, the labels of vertices \\(\\xi \\in V_{G(F)}, \\eta \\in V_{H(F)}\\) are the same iff \\(\\xi\\) and \\(\\eta\\) belong to the same meta vertex, i.e., \\(\\xi, \\eta \\in \\mathsf{Meta}_F(x)\\) for some \\(x \\in V_F\\). Based on Proposition C.9, \\(\\chi_{G(F)}^M(G(F)) = \\chi_{H(F)}^M(H(F))\\) iff \\(\\tilde{\\chi}_{G(F)}^M(G(F)) = \\tilde{\\chi}_{H(F)}^M(H(F))\\) where \\(\\tilde{\\chi}^M\\) is defined in Equations (7) to (10). In this case, the pebble games above exactly correspond to the aggregation formulas in Equations (7) to (10), and we can invoke the results in Zhang et al. (2023a, Theorem I.17) to show that \\(\\tilde{\\chi}_{G(F)}^M(G(F)) = \\tilde{\\chi}_{H(F)}^M(H(F))\\) (note that we do not need to consider the augmented Fürer graphs defined in their paper based on our label assignment). Therefore, \\(\\chi_{G(F)}^M(G(F)) = \\chi_{H(F)}^M(H(F))\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.297,
+                0.826,
+                0.343
+            ],
+            "angle": 0,
+            "content": "We next consider the general case when multiple vertices can have the same label in \\( F \\). However, this can only make it harder to distinguish between \\( G(F) \\) and \\( H(F) \\), and we clearly have \\( \\chi_{G(F)}^{M}(G(F)) = \\chi_{H(F)}^{M}(H(F)) \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.357,
+                0.825,
+                0.4
+            ],
+            "angle": 0,
+            "content": "Based on the above result, in the subsequent proof we will analyze the players' strategy in the pebble game. Surprisingly, it turns out that given a graph \\( F \\), if Alice can win the game, then her strategy can be described using the tree decomposition of \\( F \\) defined in Definition C.6."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.406,
+                0.826,
+                0.532
+            ],
+            "angle": 0,
+            "content": "To illustrate this point, we need the concept of game state graph. Given a graph \\( F \\), a game state is a three-tuple \\( (u, v, Q) \\) where \\( u, v \\in V_F \\cup \\{\\emptyset\\} \\) and \\( Q \\subset \\mathbb{C}\\mathbb{C}_F(\\{u, v\\}) \\), denoting the vertex that holds pebble \\( \\mathfrak{p}_u \\), the vertex that holds pebble \\( \\mathfrak{p}_v \\), and a subset of connected component selected by Bob, respectively. Here, the symbol \\( \\emptyset \\) means that a pebble is left outside the graph. One can see that after any round, whether Alice can win the remaining game purely depends on this tuple. Now fix Alice's strategy. In each round, each game state will be transited to a finite number of states depending on how Bob plays, and all states and transitions form a directed graph, which we call the game state graph. The state \\( (u, v, Q) \\) is called a terminal state if \\( \\min_{P \\in Q} |P| = 1 \\). It is straightforward to see that Alice wins the game at any terminal state, as stated in the following result:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.535,
+                0.826,
+                0.592
+            ],
+            "angle": 0,
+            "content": "Proposition C.50. Given graph \\( F \\) and model \\( M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\} \\), let \\( G^S \\) be the game state graph defined above corresponding to an Alice's strategy. Then, Alice can win the game if there is an integer \\( t \\) such that any path in \\( G^S \\) of length \\( t \\) starting from the initial state \\( (\\emptyset, \\emptyset, \\{E_F\\}) \\) goes through a terminal state."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.607,
+                0.827,
+                0.734
+            ],
+            "angle": 0,
+            "content": "Proof. It suffices to prove that, at any terminal state, Alice can win the game. Let \\((u,v,Q)\\) be a terminal state with \\(\\{x,y\\} \\in Q\\). If \\(\\{u,v\\} = \\{x,y\\}\\), Alice already wins. If \\(u \\notin \\{x,y\\}\\) and \\(v \\in \\{x,y\\}\\), Alice also wins in the next round since she can add the pebble \\(\\mathfrak{p}_w\\) on some vertex \\(w\\) adjacent to \\(\\mathfrak{p}_v\\) such that \\(\\{v,w\\} = \\{x,y\\}\\). This is a valid game rule for all model \\(M\\). Next, if \\(u \\in \\{x,y\\}\\) and \\(v \\notin \\{x,y\\}\\), Alice also wins in the next round since she can first remove the pebble \\(\\mathfrak{p}_v\\) and then place it on the unique vertex in \\(\\{x,y\\} \\backslash \\{u\\}\\). This is also a valid game rule for all model \\(M\\). Note the removing \\(\\mathfrak{p}_v\\) does not merge the connected component \\(\\{\\{x,y\\}\\}\\) since \\(v \\notin \\{x,y\\}\\). Finally, if both \\(u \\notin \\{x,y\\}\\) and \\(v \\notin \\{x,y\\}\\), \\(F\\) only has one edge and Alice can clearly win. Combining these cases, we conclude that Alice can always win."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.749,
+                0.825,
+                0.806
+            ],
+            "angle": 0,
+            "content": "Based on the above proposition, a game state \\((u,v,Q)\\) is called unreachable if any path starting from the initial state \\((\\emptyset ,\\emptyset ,\\{E_F\\})\\) and ending at \\((u,v,Q)\\) goes through some terminal state. We do not need to consider unreachable states since Alice always wins before reaching it. We next introduce an important technical concept:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.809,
+                0.825,
+                0.854
+            ],
+            "angle": 0,
+            "content": "Definition C.51. Given a game state graph \\( G^{\\mathsf{S}} \\), a state \\( (u, v, \\{P\\}) \\) is termed as \"contracted\" if for any transition \\( ((u, v, \\{P\\}), (u', v', \\{P'\\})) \\in E_{G^{\\mathsf{s}}}, P' \\subset P \\). It is called strictly contracted if for any transition \\( ((u, v, \\{P\\}), (u', v', \\{P'\\})) \\in E_{G^{\\mathsf{s}}}, P' \\subsetneq P \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.863,
+                0.37,
+                0.879
+            ],
+            "angle": 0,
+            "content": "We have the following result:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.882,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "Lemma C.52. For any model \\( M \\in \\{\\text{Sub}, \\text{L}, \\text{LF}, \\text{F}\\} \\) and any graph \\( F \\), if Alice can win the pebble game associated to model \\( M \\) on graph \\( F \\), then there exists a game state graph \\( G^S \\) corresponding to a winning strategy of Alice such that any reachable and non-terminal state is strictly contracted."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "47"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.104,
+                0.827,
+                0.232
+            ],
+            "angle": 0,
+            "content": "Proof. We first prove that there is a strategy for Alice such that any reachable and non-terminal state is contracted. We will assume that for any reachable and non-terminal state \\((u, v, \\{P\\})\\), if there is only one transition of the form \\(((u, v, \\{P\\}),(u', v', \\{P'\\}))\\), then \\(P \\notin P'\\), since making the connected component larger should be a bad strategy. (Otherwise, it is straightforward to construct a new strategy for Alice satisfying the assumption.) Now suppose that there exists a reachable state \\((u, v, \\{P\\})\\) that is not contracted. Note that the game state graph induced by all reachable states is a Directed Acyclic Graph (DAG), so we can choose the state \\((u, v, \\{P\\})\\) such that any path from the initial state \\((\\emptyset, \\emptyset, \\{E_F\\})\\) to \\((u, v, \\{P\\})\\) does not pass any intermediate state that is not contracted. Below, we will construct a new strategy that removes the state \\((u, v, \\{P\\})\\) (making it unreachable)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.236,
+                0.828,
+                0.322
+            ],
+            "angle": 0,
+            "content": "Note that we clearly have \\( u \\neq \\emptyset \\) and \\( v \\neq \\emptyset \\). Without loss of generality, assume that \\( ((u,v,\\{P\\}),(u,v',\\{P'\\})) \\) is a transition such that \\( P' \\notin P \\) (the case of transition \\( ((u,v,\\{P\\}),(u',v,\\{P'\\})) \\) is the same by symmetry). Moreover, we can assume that \\( P \\notin \\mathbb{C}\\mathbb{C}_F(\\{u\\}) \\) and \\( P \\notin \\mathbb{C}\\mathbb{C}_F(\\{v\\}) \\) (in other words, both \\( u \\) and \\( v \\) are at the boundary of the connected component \\( P \\)). It would be easier to analyze the case where either \\( u \\) or \\( v \\) are not at the boundary. We separately consider the following cases:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.193,
+                0.327,
+                0.826,
+                0.385
+            ],
+            "angle": 0,
+            "content": "1. The transition \\(((u,v,\\{P\\}),(u,v^{\\prime},\\{P^{\\prime}\\}))\\) corresponds to Alice removing pebble \\(\\mathfrak{p}_v\\) and placing it on \\(v^{\\prime}\\). Let \\((u_0,v_0,Q_0),\\dots ,(u_T,v_T,Q_T)\\) be any path from the initial state \\((\\emptyset ,\\emptyset ,\\{E_F\\})\\) to \\((u,v,\\{P\\})\\), and let \\(t\\leq T\\) be the maximal number such that \\(v_{t}\\neq v\\). We will construct a new strategy for Alice as follows."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.239,
+                0.39,
+                0.825,
+                0.422
+            ],
+            "angle": 0,
+            "content": "- At state \\( (u_t, v_t, Q_t) \\), she removes pebble \\( \\mathfrak{p}_v \\) and places it on \\( v' \\), yielding (possibly multiple) states of the form \\( (u_t, v', \\tilde{Q}_t) \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.239,
+                0.427,
+                0.827,
+                0.502
+            ],
+            "angle": 0,
+            "content": "- For all \\(t < \\tilde{t} < T\\), we have \\(v_{t} = v\\). For each \\(\\tilde{t}\\) and possible \\(\\tilde{Q}_{\\tilde{t}}\\), she will apply the original strategy at \\((u_{\\tilde{t}}, v_{\\tilde{t}}, Q_{\\tilde{t}})\\) to the state \\((u_{\\tilde{t}}, v', \\tilde{Q}_{\\tilde{t}})\\), namely, placing pebble \\(\\mathsf{p}_w\\) on vertex \\(u_{\\tilde{t} + 1}\\), swapping pebble \\(\\mathsf{p}_u\\) with \\(\\mathsf{p}_w\\), and leaving \\(\\mathsf{p}_w\\) outside the graph. (Note that she cannot remove pebble \\(\\mathsf{p}_u\\) first; otherwise the state \\((u_{\\tilde{t}}, v_{\\tilde{t}}, Q_{\\tilde{t}})\\) will not be contracted since \\(u\\) is at the boundary of \\(Q_{\\tilde{t}}\\).)"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.239,
+                0.39,
+                0.827,
+                0.502
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.507,
+                0.825,
+                0.543
+            ],
+            "angle": 0,
+            "content": "It follows that \\(((u_{\\tilde{t}},v^{\\prime},\\tilde{Q}_{\\tilde{t}}),(u_{\\tilde{t} +1},v^{\\prime},\\tilde{Q}_{\\tilde{t} +1}))\\) is a transition of the new strategy, \\((u_{\\tilde{t}},v^{\\prime},\\tilde{Q}_{\\tilde{t}})\\) is contracted, and all states of the form \\((u,v^{\\prime},\\tilde{Q}_{\\tilde{T}})\\) is reachable in the original strategy."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.547,
+                0.826,
+                0.648
+            ],
+            "angle": 0,
+            "content": "We can repeat the above procedure for all paths from the initial state \\((\\emptyset, \\emptyset, \\{E_F\\})\\) to state \\((u, v, \\{P\\})\\). Then, in the new strategy \\((u, v, \\{P\\})\\) will be unreachable. However, the state \\((u_t, v_t, Q_t)\\) now may violate the condition in Lemma C.52. Note that the assumption in the first paragraph is still satisfied for state \\((u_t, v_t, Q_t)\\). In this case, we can recursively apply the above procedure for state \\((u_t, v_t, Q_t)\\). Note that the procedure will only repeat a finite number of times, as the length of the path from the initial state to the state \\((u_t, v_t, Q_t)\\) is strictly less than the length of the path from the initial state to the state \\((u, v, \\{P\\})\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.651,
+                0.826,
+                0.752
+            ],
+            "angle": 0,
+            "content": "2. The transition \\(((u,v,\\{P\\}),(u,v^{\\prime},\\{P^{\\prime}\\}))\\) corresponds to Alice placing pebble \\(\\mathfrak{p}_w\\) on \\(v^{\\prime}\\), swapping \\(\\mathfrak{p}_v\\) and \\(\\mathfrak{p}_w\\), and removing pebble \\(\\mathfrak{p}_w\\). In this case, we must have \\(P\\notin \\mathbb{C}\\mathbb{C}_F(\\{u,v,v'\\})\\) (i.e., \\(v^{\\prime}\\) further splits the component \\(P\\)). Otherwise, the state \\((u,v,\\{P\\})\\) must have only one transition and \\(P\\subset P^{\\prime}\\), a contradiction to the first paragraph. Since \\(P\\notin \\mathbb{C}\\mathbb{C}_F(\\{u,v,v'\\})\\), it is thus easy to see that, if Alice changes her strategy by just removing pebble \\(\\mathfrak{p}_v\\) and placing it on vertex \\(v^{\\prime}\\), all transitions starting from \\((u,v,\\{P\\})\\) does not change. Therefore, we can just invoke the previous item to construct a desired strategy."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.756,
+                0.827,
+                0.829
+            ],
+            "angle": 0,
+            "content": "Combining the two cases, we conclude that there is a strategy for Alice such that any reachable and non-terminal state is contracted. We next prove that any reachable and non-terminal state can be strictly contracted. Assume the result does not hold and the state \\((u,v,\\{P\\})\\) is reachable, non-terminal, but not strictly contracted. Then, there is a transition \\(((u,v,\\{P\\}),(u',v',\\{P\\})) \\in E_{G^s}\\). Consider the following two cases:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.833,
+                0.826,
+                0.876
+            ],
+            "angle": 0,
+            "content": "1. \\( P \\notin \\mathbb{C}C_F(\\{u\\}) \\) and \\( P \\notin \\mathbb{C}C_F(\\{v\\}) \\) (i.e., both \\( u \\) and \\( v \\) are at the boundary of the connected component \\( P \\)). It follows that \\( u' = u \\) and \\( v' = v \\). This implies that the game state graph is not acyclic, a contradiction."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.881,
+                0.826,
+                0.927
+            ],
+            "angle": 0,
+            "content": "2. \\( P \\in \\mathbb{C}\\mathbb{C}_F(\\{u\\}) \\) or \\( P \\in \\mathbb{C}\\mathbb{C}_F(\\{v\\}) \\). Without loss of generality, assume that \\( P \\in \\mathbb{C}\\mathbb{C}_F(\\{u\\}) \\). Since \\( (u,v,\\{P\\}) \\) is non-terminal and not strictly contracted, there is a reachable and non-terminal state \\( (\\tilde{u},\\tilde{v},\\{P\\}) \\) with either \\( \\tilde{u} = u \\) or \\( \\tilde{v} = u \\) such that there is a path from \\( (u,v,\\{P\\}) \\)"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.192,
+                0.833,
+                0.826,
+                0.927
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.961
+            ],
+            "angle": 0,
+            "content": "48"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.104,
+                0.825,
+                0.176
+            ],
+            "angle": 0,
+            "content": "to \\((\\tilde{u},\\tilde{v},\\{P\\})\\) and \\((\\tilde{u},\\tilde{v},\\{P\\})\\) is strictly contracted. We can then change the strategy at state \\((u,v,\\{P\\})\\) to make it strictly contracted. Concretely, Alice can remove pebble \\(\\mathsf{p}_v\\) (which does not merge connected components selected by Bob since \\(P\\in \\mathbb{C}\\mathbb{C}_F(\\{u\\}))\\), and place pebble \\(\\mathsf{p}_v\\) on the vertex that corresponds to the strategy at state \\((\\tilde{u},\\tilde{v},\\{P\\})\\) (possibly with a difference that \\(u\\) and \\(v\\) become exchanged). This makes the state \\((u,v,\\{P\\})\\) strictly contracted."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.181,
+                0.479,
+                0.197
+            ],
+            "angle": 0,
+            "content": "Combining the two cases concludes the proof."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.182,
+                0.825,
+                0.194
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.214,
+                0.468,
+                0.228
+            ],
+            "angle": 0,
+            "content": "We are now ready to state the main theorem:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.233,
+                0.825,
+                0.276
+            ],
+            "angle": 0,
+            "content": "Theorem C.53. Let \\( M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\} \\) be any model. Given any connected graph \\( F \\), if Alice can win the pebble game associated with model \\( M \\) on graph \\( F \\), then there is a tree decomposition \\( T^r \\) of \\( F \\) such that \\( (F, T^r) \\in S^M \\), where \\( S^M \\) is defined in Definition C.6."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.293,
+                0.825,
+                0.366
+            ],
+            "angle": 0,
+            "content": "Proof. Let \\( G^{\\mathsf{S}} \\) be the game state graph satisfying Lemma C.52. For each game state \\( s \\), denote by \\( \\mathrm{next}_{G^{\\mathsf{S}}}(\\bar{s}) \\) the set of states \\( s' \\) such that \\( (s, s') \\) is a transition in \\( G^{\\mathsf{S}} \\) and \\( s' \\) contains only a single connected component, i.e., \\( s' \\) has the form \\( (u, v, \\{P\\}) \\). For a terminal state \\( s \\) of the form \\( (u, v, \\{\\{x, y\\}\\}) \\), define \\( \\mathrm{next}_{G^{\\mathsf{S}}}(\\bar{s}) = \\{(x, y, \\{\\{x, y\\}\\})\\} \\). By definition, \\( \\mathrm{next}_{G^{\\mathsf{S}}}(\\emptyset, \\emptyset, \\{E_F\\}) = \\{(u, \\emptyset, Q_1), \\dots, (u, \\emptyset, Q_m)\\} \\) for some \\( u \\in V_F \\) and \\( Q_1, \\dots, Q_m \\) is the finest partition of \\( \\mathbb{C}\\mathbb{C}_F(\\{u\\}) \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.37,
+                0.825,
+                0.414
+            ],
+            "angle": 0,
+            "content": "The tree \\( T^r \\) will be recursively constructed as follows. First create the tree root \\( r \\) with \\( \\beta_T(r) = \\{u, u\\} \\). As we will see later, the root node will be associated with the set of states \\( S(r) := \\mathrm{next}_{G^s}(\\emptyset, \\emptyset, \\{E_F\\}) \\). We then do the following procedure:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.42,
+                0.825,
+                0.496
+            ],
+            "angle": 0,
+            "content": "Let \\( t \\) be a leaf node in the current tree associated with a non-empty set of game states \\( S(t) \\) such that \\( |\\bigcup_{(u,v,\\{P\\}) \\in S(t)} P| > 1 \\). For each state \\( (u,v,\\{P\\}) \\in S(t) \\), create a new node \\( \\tilde{t} \\) and set its parent to be \\( t \\). Pick any state \\( (u',v',\\{P'\\}) \\in \\mathrm{next}_{G^s}(u,v,\\{P\\}) \\). Then, there must be a unique vertex \\( w \\in \\{\\{u',v'\\}\\} \\setminus \\{\\{u,v\\}\\} \\) and \\( w \\) does not depend on which \\( (u',v',\\{P'\\}) \\) is picked (by definition of the game rule). Set \\( \\beta_T(\\tilde{t}) = \\beta_T(t) \\cup \\{\\{w\\}\\} \\). Then, do the following constructions:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.197,
+                0.501,
+                0.825,
+                0.547
+            ],
+            "angle": 0,
+            "content": "- If there is a state of the form \\((u, w, \\{P'\\}) \\in \\mathrm{next}_{G^s}(u, v, \\{P\\})\\), then create a new node \\(t'\\) and connect it to the parent \\(\\tilde{t}\\). Set \\(\\beta_T(t') = \\{(u, w)\\}\\), and the node \\(t'\\) will be associated to the set of states \\(S(t') = \\{(u, w, \\{P'\\}): (u, w, \\{P'\\}) \\in \\mathrm{next}_{G^s}(u, v, \\{P\\})\\}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.197,
+                0.551,
+                0.825,
+                0.598
+            ],
+            "angle": 0,
+            "content": "- If there is a state of the form \\((w, v, \\{P''\\}) \\in \\mathsf{next}_{G^{\\mathsf{s}}}(u, v, \\{P\\})\\), then create a new node \\(t''\\) and connect it to the parent \\(\\tilde{t}\\). Set \\(\\beta_{T}(t'') = \\{(w, v)\\}\\), and the node \\(t''\\) will be associated to the set of states \\(S(t'') = \\{(w, v, \\{P''\\}): (w, v, \\{P''\\}) \\in \\mathsf{next}_{G^{\\mathsf{s}}}(u, v, \\{P\\})\\}\\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.197,
+                0.501,
+                0.825,
+                0.598
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.602,
+                0.825,
+                0.659
+            ],
+            "angle": 0,
+            "content": "Note that (i) either \\( w \\neq u \\) or \\( w \\neq v \\) (since the game state graph is a DAG); (ii) both items can be used for one state \\( (u, v, \\{P\\}) \\), which happens for FWL-type GNNs. The construction of the tree completes when each leaf node is associated with only one game state of the form \\( (x, y, \\{\\{x, y\\}\\}) \\). It is easy to see that the above procedure terminates after adding a finite number of tree nodes."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.665,
+                0.71,
+                0.681
+            ],
+            "angle": 0,
+            "content": "We now prove that \\( T^r \\) is a canonical tree decomposition of \\( F \\) and \\( (F, T^r) \\in S^M \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.687,
+                0.825,
+                0.732
+            ],
+            "angle": 0,
+            "content": "1. We first prove that any edge in \\( F \\) is contained in some bag of \\( T^r \\). Pick any non-leaf tree node \\( t \\) of even depth and any of its child \\( \\tilde{t} \\), denote \\( (u, v, P) \\in S(t) \\) be the state associated with \\( \\tilde{t} \\) in the above construction, and denote \\( \\beta_T(\\tilde{t}) = \\{\\{u, v, w\\}\\} \\). We have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.301,
+                0.739,
+                0.735,
+                0.811
+            ],
+            "angle": 0,
+            "content": "\\[\nP = \\left(\\bigcup_{\\substack{t^{\\prime},\\mathsf{pa}_{T^{r}}(t^{\\prime}) = \\tilde{t},\\\\ (x,y,\\{P^{\\prime}\\})\\in S(t^{\\prime})}}P^{\\prime}\\right)\\cup \\{\\{u,w\\} \\in E_{F}\\} \\cup \\{\\{v,w\\} \\in E_{F}\\} .\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.818,
+                0.825,
+                0.848
+            ],
+            "angle": 0,
+            "content": "Here, \\( P' \\subset P \\) is a consequence of Lemma C.52, and the remaining edges \\( \\{u, w\\} \\) or \\( \\{v, w\\} \\) are in \\( P \\) because \\( (u, v, P) \\) is strictly contracted and \\( w \\) must be in the interior of \\( P \\). Therefore,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.212,
+                0.856,
+                0.865,
+                0.927
+            ],
+            "angle": 0,
+            "content": "\\[\nP\\cup \\{\\{u,v\\} \\in E_{F}\\} = \\left(\\bigcup_{\\substack{t^{\\prime},\\mathsf{pa}_{Tr}(t^{\\prime}) = \\tilde{t},\\\\ (x,y,\\{P^{\\prime}\\})\\in S(t^{\\prime})}}(P^{\\prime}\\cup \\{\\{x,y\\} \\in E_{F}\\})\\right)\\cup \\{\\{x,y\\} \\in E_{F}:x,y\\in \\beta_{T}(\\tilde{t})\\} .\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "49"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.212,
+                0.104,
+                0.644,
+                0.12
+            ],
+            "angle": 0,
+            "content": "Recursively applying the above equation yields the desired result"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.351,
+                0.128,
+                0.687,
+                0.164
+            ],
+            "angle": 0,
+            "content": "\\[\nE_{F} = \\bigcup_{\\substack{\\tilde{t},\\mathsf{dep}_{T}(\\tilde{t})\\text{is odd}}}\\{\\{x,y\\} \\in E_{F}:x,y\\in \\beta_{T}(\\tilde{t})\\} ,\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.21,
+                0.17,
+                0.825,
+                0.204
+            ],
+            "angle": 0,
+            "content": "because (i) \\(\\bigcup_{(u,\\emptyset,P)\\in S(r)}P = E_F\\), and (ii) all leaf node \\(t\\) with \\(S(t) = (x,y,\\{\\{x,y\\}\\})\\) contains an edge \\(\\{x,y\\}\\) in its bag."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.209,
+                0.827,
+                0.325
+            ],
+            "angle": 0,
+            "content": "2. We next prove that \\( T^r \\) satisfies the condition of Definition C.1(c). Fix any vertex \\( w \\in V_F \\), and let \\( t \\) be the tree node with minimal depth that contains \\( w \\). Without loss of generality, assume that \\( t \\) is not the root. In this case, the depth of \\( t \\) is odd and \\( \\beta_T(t) = \\{ \\{u, v, w\\} \\} \\) for some \\( u \\neq w, v \\neq w \\). Let \\( t' \\) be a child node of \\( t \\) and we have \\( w \\in \\beta_T(t') \\). It remains to prove that for any descendant \\( \\tilde{t} \\in \\mathsf{Desc}_T(t') \\), if \\( w \\in \\beta_T(\\tilde{t}) \\), then \\( w \\in \\beta_T(\\hat{t}) \\) for all \\( \\hat{t} \\) on the path between \\( t' \\) and \\( \\tilde{t} \\). This is actually a direct consequence of Lemma C.52, because when a pebble originally placed on \\( w \\) is removed, all edges linked to \\( w \\) will not be selected by Bob and thus any pebble can never be placed on \\( w \\) again."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.33,
+                0.561,
+                0.347
+            ],
+            "angle": 0,
+            "content": "3. \\( T^r \\) is canonical as Definition C.4 is clearly satisfied."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.352,
+                0.641,
+                0.369
+            ],
+            "angle": 0,
+            "content": "4. Finally, it is also easy to see that \\((F,T^{r})\\) satisfies Definition C.6."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.192,
+                0.209,
+                0.827,
+                0.369
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.373,
+                0.361,
+                0.389
+            ],
+            "angle": 0,
+            "content": "We thus conclude the proof."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.374,
+                0.825,
+                0.386
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.397,
+                0.825,
+                0.445
+            ],
+            "angle": 0,
+            "content": "Corollary C.54. Let \\( M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\} \\) be any model. For any connected graph \\( F \\notin \\mathcal{F}^M \\), let \\( G(F) \\) and \\( H(F) \\) be the Fürer graph and twisted Fürer graph with respect to \\( F \\). Then, \\( \\text{hom}(F, G(F)) \\neq \\text{hom}(F, H(F)) \\) and \\( \\chi_{G(F)}^M(G(F)) = \\chi_{H(F)}^M(H(F)) \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.46,
+                0.796,
+                0.476
+            ],
+            "angle": 0,
+            "content": "Proof. The proof directly follows from applying Theorems C.47 and C.53 and Proposition C.49. \\(\\square\\)"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.461,
+                0.825,
+                0.473
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.491,
+                0.827,
+                0.586
+            ],
+            "angle": 0,
+            "content": "Finally, we remark that our construction can be easily generalized for disconnected graphs \\( F \\notin \\mathcal{F}^M \\). Let \\( F \\) be the disjoint union of graphs \\( \\{F_i : i \\in [m]\\} \\) where each \\( F_i \\) is the graph corresponding to a connected component of \\( F \\). Assume that \\( F_1 \\) is the connected component with the most number of edges (in case of a tie, pick the graph with the most number of vertices). Define \\( \\tilde{G}(F) \\) be the disjoint union of \\( G(F_1), F_2, \\dots, F_m \\) and \\( \\tilde{H}(F) \\) to be the disjoint union of \\( H(F_1), F_2, \\dots, F_m \\). It follows that \\( \\chi_{\\tilde{G}(F)}\\tilde{G}(F) = \\chi_{\\tilde{H}(F)}\\tilde{H}(F) \\) and"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.186,
+                0.593,
+                0.809,
+                0.666
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\operatorname {h o m} (F, \\tilde {G} (F)) = \\prod_ {i \\in [ m ]} \\operatorname {h o m} (F _ {i}, \\tilde {G} (F)) = \\prod_ {i \\in [ m ]} (\\operatorname {h o m} (F _ {i}, G (F _ {1})) + \\operatorname {h o m} (F _ {i}, F _ {2} \\cup \\dots \\cup F _ {m})) \\\\ > \\prod_ {i \\in [ m ]} \\left(\\operatorname {h o m} \\left(F _ {i}, H \\left(F _ {1}\\right)\\right) + \\operatorname {h o m} \\left(F _ {i}, F _ {2} \\cup \\dots \\cup F _ {m}\\right)\\right) = \\operatorname {h o m} (F, \\tilde {H} (F)), \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.673,
+                0.825,
+                0.704
+            ],
+            "angle": 0,
+            "content": "where we use the fact that \\(\\operatorname{hom}(F_i, G(F_1)) = \\operatorname{hom}(F_i, H(F_1))\\) when \\(F_1 \\notin \\operatorname{Spasm}(F_i)\\) (which can be easily proved following Theorem C.47). This concludes the proof of the general case."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.723,
+                0.504,
+                0.739
+            ],
+            "angle": 0,
+            "content": "D NODE/EDGE-LEVEL EXPRESSIVITY"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.755,
+                0.759,
+                0.771
+            ],
+            "angle": 0,
+            "content": "This section aims to prove Theorem 3.7. For the ease of reading, we first restate it below:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.776,
+                0.825,
+                0.793
+            ],
+            "angle": 0,
+            "content": "Theorem 3.7 For all model \\(M\\) defined in Section 2, \\(\\mathcal{F}_{\\mathfrak{n}}^{M}\\) and \\(\\mathcal{F}_{\\mathrm{e}}^{M}\\) (except MPNN) exist. Moreover,"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.796,
+                0.457,
+                0.814
+            ],
+            "angle": 0,
+            "content": "- MPNN: \\(\\mathcal{F}_{\\mathfrak{n}}^{\\mathsf{MP}} = \\{F^{w}: F \\text{ is a tree}\\}\\);"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.199,
+                0.817,
+                0.33,
+                0.831
+            ],
+            "angle": 0,
+            "content": "- Subgraph GNN:"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.198,
+                0.796,
+                0.457,
+                0.831
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.212,
+                0.83,
+                0.771,
+                0.846
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathcal {F} _ {\\mathfrak {n}} ^ {\\mathrm {S u b}} = \\left\\{F ^ {w}: F \\text {h a s a N E D w i t h s h a r d e n p o i n t} w \\right\\} = \\left\\{F ^ {w}: F \\backslash \\{w \\} \\text {i s a f o r e s t} \\right\\},\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.212,
+                0.845,
+                0.771,
+                0.861
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathcal {F} _ {\\mathrm {e}} ^ {\\mathrm {S} _ {\\mathrm {u b}}} = \\left\\{F ^ {w x}: F \\text {h a s a N E D w i t h s h a r e d e n p o i n t} w \\right\\} = \\left\\{F ^ {w x}: F \\backslash \\{w \\} \\text {i s a f o r e s t} \\right\\};\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.864,
+                0.726,
+                0.879
+            ],
+            "angle": 0,
+            "content": "- 2-FGNN: \\(\\mathcal{F}_{\\mathfrak{n}}^{\\mathsf{F}} = \\{F^{w}: F\\) has a NED where \\(w\\) is an endpoint of the first ear},"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.214,
+                0.879,
+                0.703,
+                0.894
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathcal {F} _ {\\mathrm {e}} ^ {\\mathrm {F}} = \\left\\{F ^ {w x}: F \\text {h a s a N E D w h e r e} w \\text {a n d} x \\text {a r e e x i p o n t s o f t h e f i r s t e a r} \\right\\}.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.896,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "The cases of Local 2-GNN and Local 2-FGNN are similar to 2-FGNN by replacing \"NED\" with \"strong NED\" and \"almost-strong NED\", respectively."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.51,
+                0.961
+            ],
+            "angle": 0,
+            "content": "50"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.104,
+                0.827,
+                0.19
+            ],
+            "angle": 0,
+            "content": "We remark that the definition of node/edge-level homomorphism expressivity involves only connected graphs for sake of simplicity. Similar to the proof of the graph-level expressivity, the proof of Theorem 3.7 consists of three parts. Among them, the proof related to tree decomposition and ear decomposition is quite similar to that of graph-level expressivity, so we only illustrate the proof sketch for clarity (Appendix D.1). However, the proof related to pebble game and counterexample graphs will require additional techniques, which is detailed in Appendix D.2."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.171,
+                0.205,
+                0.658,
+                0.219
+            ],
+            "angle": 0,
+            "content": "D.1 RELATED TO TREE DECOMPOSITION AND EAR DECOMPOSITION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.231,
+                0.592,
+                0.246
+            ],
+            "angle": 0,
+            "content": "We first extend several notations that are used in Appendix C.2."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.248,
+                0.825,
+                0.292
+            ],
+            "angle": 0,
+            "content": "Definition D.1 (Tree decomposition for rooted graphs). Given a rooted graph \\( G^u \\) and tree \\( T^r \\), we say \\( T^r \\) is a tree decomposition of \\( G^u \\) if \\( T^r \\) is a tree decomposition of \\( G \\) and all elements in \\( u \\) belong to the root bag, i.e., \\( u_i \\in \\beta_T(r) \\) for all \\( i \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.301,
+                0.825,
+                0.345
+            ],
+            "angle": 0,
+            "content": "Based on this definition, given model \\(M\\), we can define \\(\\mathcal{S}_{\\mathfrak{n}}^{M}\\) to be the family of tree-decomposed graphs \\((F^u,T^r)\\) such that \\((F,T^r)\\in S^M\\); similarly, we define \\(\\mathcal{S}_{\\mathrm{e}}^{M}\\) to be the family of tree-decomposed graphs \\((F^{uv},T^{r})\\) such that \\((F,T^{r})\\in S^{M}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.347,
+                0.826,
+                0.391
+            ],
+            "angle": 0,
+            "content": "Definition D.2 (Bag isomorphism for rooted graphs). Given tree-decomposed graph \\((F^{\\mathbf{u}}, T^{r})\\) and rooted graph \\(G^{\\mathbf{v}}\\) where \\(\\mathbf{u}\\) and \\(\\mathbf{v}\\) have equal length, a bag isomorphism from \\((F^{\\mathbf{u}}, T^{r})\\) to \\(G^{\\mathbf{v}}\\) is a homomorphism \\(f\\) from \\(F^{\\mathbf{u}}\\) to \\(G^{\\mathbf{v}}\\) such that \\(f\\) is a bag isomorphism from \\((F, T^{r})\\) to \\(G\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.4,
+                0.825,
+                0.446
+            ],
+            "angle": 0,
+            "content": "We can similarly define bag-isomorphism homomorphism (bIsoHom, bIsoSurj, bIsoInj), bag extension (bExt), and bag-strong surjective (bStrSurj) from tree-decomposed graph \\((F^u, T^r)\\) to tree-decomposed graph \\((\\tilde{F}^v, \\tilde{T}^s)\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.447,
+                0.825,
+                0.478
+            ],
+            "angle": 0,
+            "content": "Definition D.3 (Generalization of Definition C.15 for rooted graphs). Let \\( M \\in \\{\\mathrm{Sub}, \\mathrm{L}, \\mathrm{LF}, \\mathrm{F}\\} \\) be any model. Given a rooted graph \\( G^{u} \\) and a tree-decomposed graph \\( (F^w, T^r) \\), define"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.171,
+                0.482,
+                0.861,
+                0.51
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathsf {c n t} ^ {M} \\left(\\left(F ^ {w}, T ^ {r}\\right), G ^ {u}\\right) := \\left| \\left\\{v \\in V _ {G}: \\exists D \\in \\mathbb {N} _ {+} \\text {s . t .} \\left(\\left[ F _ {G} ^ {M, (D)} (u, v) \\right] ^ {u}, T _ {G} ^ {M, (D)} (u, v)\\right) \\simeq \\left(F ^ {w}, T ^ {r}\\right) \\right\\} \\right|.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.513,
+                0.667,
+                0.529
+            ],
+            "angle": 0,
+            "content": "Given a rooted graph \\( G^{uv} \\) and a tree-decomposed graph \\( (F^{wx}, T^r) \\), define"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.173,
+                0.533,
+                0.82,
+                0.561
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathsf {c n t} ^ {M} \\left(\\left(F ^ {w x}, T ^ {r}\\right), G ^ {u v}\\right) := \\mathbb {I} \\left[ \\exists D \\in \\mathbb {N} _ {+} \\text {s . t .} \\left(\\left[ F _ {G} ^ {M, (D)} (u, v) \\right] ^ {u v}, T _ {G} ^ {M, (D)} (u, v)\\right) \\simeq \\left(F ^ {w x}, T ^ {r}\\right) \\right].\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.564,
+                0.81,
+                0.59
+            ],
+            "angle": 0,
+            "content": "Here, \\(\\left(F_G^{M,(D)}(u,v),T_G^{M,(D)}(u,v)\\right)\\) is the depth-2D unfolding tree of \\(G\\) at \\((u,v)\\) for model \\(M\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.599,
+                0.717,
+                0.614
+            ],
+            "angle": 0,
+            "content": "All the following lemmas are straightforward extensions of those in Appendix C.2."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.616,
+                0.825,
+                0.647
+            ],
+            "angle": 0,
+            "content": "Lemma D.4. Let \\( M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\} \\) be any model. For any graph \\( G^v \\) and tree-decomposed graph \\( (F^u, T^r) \\in S_{\\mathfrak{n}}^{M} \\), we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.177,
+                0.65,
+                0.819,
+                0.689
+            ],
+            "angle": 0,
+            "content": "\\[\n\\operatorname {b l s o} \\left(\\left(F ^ {u}, T ^ {r}\\right), G ^ {v}\\right) = \\sum_ {\\left(\\tilde {F} ^ {w}, \\tilde {T} ^ {s}\\right) \\in \\mathcal {S} _ {\\mathfrak {n}} ^ {M}} \\operatorname {b l s o H o m} \\left(\\left(F ^ {u}, T ^ {r}\\right), \\left(\\tilde {F} ^ {w}, \\tilde {T} ^ {s}\\right)\\right) \\cdot \\operatorname {c n t} ^ {M} \\left(\\left(\\tilde {F} ^ {w}, \\tilde {T} ^ {s}\\right), G ^ {v}\\right).\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.693,
+                0.381,
+                0.707
+            ],
+            "angle": 0,
+            "content": "The edge-level result is similar."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.71,
+                0.825,
+                0.743
+            ],
+            "angle": 0,
+            "content": "Lemma D.5. Let \\( M \\in \\{\\mathrm{Sub},\\mathrm{L},\\mathrm{LF},\\mathrm{F}\\} \\) be any model. For any tree-decomposed graphs \\( (F^u,T^r),(\\tilde{F}^v,\\tilde{T}^s) \\in S_{\\mathfrak{n}}^{M} \\),"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.171,
+                0.747,
+                0.855,
+                0.799
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathsf{b l s o H o m}((F^{u},T^{r}),(\\tilde{F}^{v},\\tilde{T}^{s})) = \\sum_{(\\widehat{F}^{w},\\widehat{T}^{t})\\in \\mathcal{S}_{n}^{M}}\\frac{\\mathsf{b l s o S u r j}\\left((F^{u},T^{r}),( \\widehat{F}^{w},\\widehat{T}^{t})\\right)\\cdot\\mathsf{b l s o I n j}\\left((\\widehat{F}^{w},\\widehat{T}^{t}),( \\tilde{F}^{v},\\tilde{T}^{s})\\right)}{\\mathsf{a u t}(\\widehat{F}^{w},\\widehat{T}^{t})}.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.803,
+                0.381,
+                0.817
+            ],
+            "angle": 0,
+            "content": "The edge-level result is similar."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.82,
+                0.826,
+                0.851
+            ],
+            "angle": 0,
+            "content": "Lemma D.6. Let \\( M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\} \\) be any model. For any graph \\( G^v \\) and tree-decomposed graph \\( (F^u, T^r) \\in S_{\\mathfrak{n}}^{M} \\),"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.209,
+                0.854,
+                0.787,
+                0.906
+            ],
+            "angle": 0,
+            "content": "\\[\n\\operatorname {h o m} (F ^ {u}, G ^ {v}) = \\sum_ {(\\tilde {F} ^ {w}, \\tilde {T} ^ {s}) \\in \\mathcal {S} _ {n} ^ {M}} \\frac {\\mathrm {b E x t} ((F ^ {u} , T ^ {r}) , (\\tilde {F} ^ {w} , \\tilde {T} ^ {s})) \\cdot \\mathrm {b S t r H o m} ((\\tilde {F} ^ {w} , \\tilde {T} ^ {s}) , G ^ {v})}{\\operatorname {a u t} (\\tilde {F} ^ {w} , \\tilde {T} ^ {s})}.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.91,
+                0.381,
+                0.925
+            ],
+            "angle": 0,
+            "content": "The edge-level result is similar."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.961
+            ],
+            "angle": 0,
+            "content": "51"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.104,
+                0.825,
+                0.135
+            ],
+            "angle": 0,
+            "content": "Lemma D.7. Let \\( M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\} \\) be any model. For any graph \\( G^v \\) and tree-decomposed graph \\( (F^u, T^r) \\in S_{\\mathfrak{n}}^{M} \\),"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.179,
+                0.14,
+                0.819,
+                0.19
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathsf{bStrHom}((F^{u},T^{r}),G^{v}) = \\sum_{(\\tilde{F}^{w},\\tilde{T}^{s})\\in \\mathcal{S}_{n}^{M}}\\frac{\\mathsf{bStrSurj}\\left((F^{u},T^{r}),( \\tilde{F}^{w},\\tilde{T}^{s})\\right)\\cdot\\mathsf{blso}\\left((\\tilde{F}^{w},\\tilde{T}^{s}),G^{v}\\right)}{\\mathsf{aut}(\\tilde{F}^{w},\\tilde{T}^{s})},\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.196,
+                0.381,
+                0.211
+            ],
+            "angle": 0,
+            "content": "The edge-level result is similar."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.213,
+                0.825,
+                0.261
+            ],
+            "angle": 0,
+            "content": "Corollary D.8. Let \\( M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\} \\) be any model. For any two graphs \\( G^u, H^v, \\chi_G^M(u) = \\chi_H^M(v) \\) iff \\( \\operatorname{hom}(F^w, G^u) = \\operatorname{hom}(F^w, H^v) \\) for all \\( (F^w, T^r) \\in S_{\\mathfrak{n}}^M \\). For any two graphs \\( G^{uv}, H^{wx}, \\chi_G^M(u, v) = \\chi_H^M(w, x) \\) iff \\( \\operatorname{hom}(F^{yz}, G^{uv}) = \\operatorname{hom}(F^{yz}, H^{wx}) \\) for all \\( (F^{yz}, T^r) \\in S_{\\mathfrak{e}}^M \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.263,
+                0.529,
+                0.279
+            ],
+            "angle": 0,
+            "content": "Lemma D.9. Let \\(M \\in \\{\\mathrm{Sub}, \\mathrm{L}, \\mathrm{LF}, \\mathrm{F}\\}\\) be any model."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.197,
+                0.284,
+                0.824,
+                0.314
+            ],
+            "angle": 0,
+            "content": "- For any rooted connected graph \\( F^u \\), \\( F^u \\in \\mathcal{F}_{\\mathfrak{n}}^{M} \\) iff there is a tree decomposition \\( T^{r} \\) of \\( F^u \\) such that \\( (F^u, T^r) \\in S_{\\mathfrak{n}}^{M} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.319,
+                0.825,
+                0.349
+            ],
+            "angle": 0,
+            "content": "- For any rooted connected graph \\( F^{uv} \\), \\( F^{uv} \\in \\mathcal{F}_{\\mathrm{e}}^{M} \\) iff there is a tree decomposition \\( T^{r} \\) of \\( F^{uv} \\) such that \\( (F^{uv}, T^{r}) \\in \\mathcal{S}_{\\mathrm{e}}^{M} \\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.197,
+                0.284,
+                0.825,
+                0.349
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.359,
+                0.825,
+                0.39
+            ],
+            "angle": 0,
+            "content": "By combining the proof of previous lemmas and theorems, we can prove that both \\(\\mathcal{F}_{\\mathrm{n}}^{M}\\) and \\(\\mathcal{F}_{\\mathrm{e}}^{M}\\) satisfy Definition 3.6(a) for all \\(M \\in \\{\\mathrm{Sub}, \\mathrm{L}, \\mathrm{LF}, \\mathrm{F}\\}\\)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.405,
+                0.365,
+                0.419
+            ],
+            "angle": 0,
+            "content": "D.2 COUNTEREXAMPLES"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.43,
+                0.825,
+                0.53
+            ],
+            "angle": 0,
+            "content": "Let \\( F^{\\mathbf{w}} \\) be a rooted graph that marks the special vertices \\( w_{1},\\dots ,w_{m} \\). Assume that \\( F^{\\mathbf{w}} \\notin \\mathcal{F}_{\\mathfrak{n}}^{M} \\) (\\( m = 1 \\)) or \\( F^{\\mathbf{w}} \\notin \\mathcal{F}_{\\mathfrak{e}}^{M} \\) (\\( m = 2 \\)). In this subsection, we will construct a pair of graphs \\( G^{\\mathbf{u}} \\) and \\( H^{\\mathbf{v}} \\) such that \\( \\chi_G^M (\\pmb {u}) = \\chi_H^M (\\pmb {v}) \\) and \\( \\mathrm{hom}(F^{\\mathbf{w}},G^{\\mathbf{u}})\\neq \\mathrm{hom}(F^{\\mathbf{w}},H^{\\mathbf{v}}) \\). However, it turns out that, if we naively follow the proof in Appendix C.4 by constructing the Fürer graph and twisted Fürer graph with respect to \\( F \\) without considering the marked vertices \\( \\pmb{w} \\), then the graphs may no longer be counterexamples here. For example, for the edge-level expressivity, \\( F^{w_1,w_2}\\notin \\mathcal{F}_{\\mathfrak{e}}^{M} \\) does not imply that \\( F\\notin \\mathcal{F}^{M} \\) where \\( M \\) can be any model studied in this paper."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.535,
+                0.825,
+                0.564
+            ],
+            "angle": 0,
+            "content": "To address the problem, we instead introduce a novel construction of counterexample graphs defined as follows:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.567,
+                0.826,
+                0.641
+            ],
+            "angle": 0,
+            "content": "Definition D.10 (Clique-augmented Fürer graphs). Let \\(F\\) be any connected graph and \\(w_{1}, \\dots, w_{m} \\in V_{F}\\) be a sequence of vertices. Given an integer \\(k \\geq m\\), the \\(k\\)-clique-augmented Fürer graph with respect to \\(F^{\\mathbf{w}}\\), denoted by \\(G_{k}(F^{\\mathbf{w}})\\), is the Fürer graph of \\(\\tilde{F}\\) where \\(\\tilde{F}\\) is the union of graph \\(F\\) and a \\(k\\)-clique that contains \\(w_{1}, \\dots, w_{m}\\) (and does not contain other vertices in \\(F\\)). The twisted Fürer graph of \\(G_{k}(F^{\\mathbf{w}})\\) is denoted as \\(H_{k}(F^{\\mathbf{w}})\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.65,
+                0.406,
+                0.665
+            ],
+            "angle": 0,
+            "content": "We have the following main result:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.668,
+                0.678,
+                0.684
+            ],
+            "angle": 0,
+            "content": "Theorem D.11. Let \\(M \\in \\{\\mathrm{Sub}, \\mathrm{L}, \\mathrm{LF}, \\mathrm{F}\\}\\) be any model defined in Section 2."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.197,
+                0.689,
+                0.825,
+                0.722
+            ],
+            "angle": 0,
+            "content": "- For any rooted graph \\( F^w \\) marking vertex \\( w \\), if \\( F^w \\notin \\mathcal{F}_{\\mathfrak{n}}^{M} \\), then there is a vertex \\( (w, U) \\in \\operatorname{Meta}_{\\tilde{F}}(w) \\) such that \\( U \\subset V_F \\) and \\( \\chi_{G_4(F^w)}^M((w, \\emptyset)) = \\chi_{H_4(F^w)}^M((w, U)) \\);"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.727,
+                0.826,
+                0.775
+            ],
+            "angle": 0,
+            "content": "- For any rooted graph \\( F^w \\) marking two vertices \\( w_1, w_2 \\), if \\( F^w \\notin \\mathcal{F}_{\\mathrm{e}}^{M} \\), then there are two vertices \\( (w_1, U_1) \\in \\operatorname{Meta}_{\\tilde{F}}(w_1) \\) and \\( (w_2, U_2) \\in \\operatorname{Meta}_{\\tilde{F}}(w_2) \\) such that \\( U_1, U_2 \\subset V_F \\) and \\( \\chi_{G_4(F^w)}^{M}((w_1, \\emptyset), (w_2, \\emptyset)) = \\chi_{H_4(F^w)}^{M}((w_1, U_1), (w_2, U_2)) \\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.197,
+                0.689,
+                0.826,
+                0.775
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.791,
+                0.825,
+                0.891
+            ],
+            "angle": 0,
+            "content": "Proof. Below, we only give a proof of the second item. Let \\(\\tilde{F}\\) be the union of \\(F\\) and the 4-clique according to Definition D.10. Similar to the graph-level expressivity, we will extend the pebble game defined in Appendix C.4 to edge-level. For the edge-level pebble game, the two pebbles \\(\\mathfrak{p}_u,\\mathfrak{p}_v\\) are initially placed on two predefined vertices \\(w_{1},w_{2}\\), and Bob initially selects a predefined set of connected components \\(Q\\subset \\mathbb{C}\\mathbb{C}_{\\tilde{F}}(\\{w_1,w_2\\})\\) (of odd size). Then, the game executes the same process as in the text above Proposition C.49. The winning criterion for the two players is also the same. We denote the above edge-level pebble game as \\(\\mathsf{PG}_{\\tilde{F}}(w_1,w_2,Q)\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.896,
+                0.826,
+                0.927
+            ],
+            "angle": 0,
+            "content": "Based on the isomorphism property of Fürer graphs established in Zhang et al. (2023a, Theorem I.9 and Theorem I.17), it is straightforward to see that, for any vertices \\(\\xi, \\xi' \\in \\mathsf{Meta}_{\\tilde{F}}(w_1)\\) and"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.51,
+                0.961
+            ],
+            "angle": 0,
+            "content": "52"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.103,
+                0.827,
+                0.243
+            ],
+            "angle": 0,
+            "content": "\\(\\eta, \\eta' \\in \\mathsf{Meta}_{\\tilde{F}}(w_2), \\chi_{G_4(F^w)}^M(\\xi, \\eta) = \\chi_{H_4(F^w)}^M(\\xi', \\eta')\\) iff Alice cannot win the edge-level pebble game \\(\\mathsf{PG}_{\\tilde{F}}(w_1, w_2, Q)\\) for some \\(Q \\subset \\mathbb{C}\\mathbb{C}_{\\tilde{F}}(\\{w_1, w_2\\})\\) (depending on \\(\\xi, \\xi', \\eta, \\eta'\\)). Moreover, let \\(\\xi' = (w_1, \\emptyset)\\), \\(\\eta' = (w_2, \\emptyset)\\) and let \\(\\xi_2\\) and \\(\\eta_2\\) range over all even-size subsets of \\(N_F(w_1)\\) and \\(N_F(w_2)\\), respectively, that is to say, \\(\\xi_2\\) and \\(\\eta_2\\) does not contain vertices that are only in the 4-clique. Then, the equivalent pebble game ranges over all \\(Q\\) of odd size such that every connected component in \\(Q\\) contains only vertices in \\(F\\), i.e., \\(Q \\subset \\mathbb{C}\\mathbb{C}_F(\\{w_1, w_2\\})\\). Therefore, if Alice cannot win the edge-level pebble game \\(\\mathsf{PG}_{\\tilde{F}}(w_1, w_2, Q)\\) for some \\(Q \\subset \\mathbb{C}\\mathbb{C}_F(\\{w_1, w_2\\})\\), then there are two vertices \\(\\xi \\in \\mathsf{Meta}_{\\tilde{F}}(w_1)\\) and \\(\\eta \\in \\mathsf{Meta}_{\\tilde{F}}(w_2)\\) such that \\(\\xi_2, \\eta_2 \\subset V_F\\) and \\(\\chi_{G_4(F^w)}^M((w_1, \\emptyset), (w_2, \\emptyset)) = \\chi_{H_4(F^w)}^M(\\xi, \\eta)\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.249,
+                0.825,
+                0.307
+            ],
+            "angle": 0,
+            "content": "It thus remains to prove that, if Alice can win the edge-level pebble game \\(\\mathsf{PG}_{\\tilde{F}}(w_1,w_2,Q)\\) for all \\(Q\\subset \\mathbb{C}C_F(\\{w_1,w_2\\})\\), then \\(F^{\\boldsymbol{w}}\\in \\mathcal{F}_{\\mathrm{e}}^{M}\\). To prove this result, we similarly define the game state graph as in Appendix C.4, with the only difference that the initial state is now \\((w_{1},w_{2},Q)\\). The proof is divided into the following steps:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.193,
+                0.313,
+                0.825,
+                0.44
+            ],
+            "angle": 0,
+            "content": "1. Assume that \\( Q \\) contains a single connected component, i.e., \\( Q = \\{P\\} \\). We will prove that, if Alice can win the game \\( \\mathsf{PG}_{\\tilde{F}}(w_1, w_2, \\{P\\}) \\), then there is a game strategy for Alice such that for any reachable and non-terminal state \\( (x, y, \\{P'\\}) \\), \\( P' \\subset P \\). If it is not the case, pick any state \\( (x, y, \\{P'\\}) \\) such that \\( P' \\notin P \\) and any path from \\( (w_1, w_2, \\{P\\}) \\) to \\( (x, y, \\{P'\\}) \\) does not pass any intermediate state that is not contracted. It follows that either \\( w_1 \\) or \\( w_2 \\) does not hold any pebble, and all edges connected to this vertex is in \\( P' \\). Therefore, \\( P' \\) contains all vertices in the 4-clique. In this case, Alice cannot win as Bob can always keep the connected component containing all vertices in the 4-clique in subsequent rounds, i.e., Bob always selects a component containing a vertex in the 4-clique that does not hold a pebble."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.446,
+                0.826,
+                0.53
+            ],
+            "angle": 0,
+            "content": "2. Based on the above result, in the game \\(\\mathsf{PG}_{\\tilde{F}}(w_1,w_2,\\{P\\})\\), the connected components selected by Bob are always in \\(F\\), and thus the game process is the same as \\(\\mathsf{PG}_F(w_1,w_2,\\{P\\})\\). We can then prove the same result as Lemma C.52. Concretely, if Alice can win the edge-level pebble game \\(\\mathsf{PG}_{\\tilde{F}}(w_1,w_2,\\{P\\})\\), then there exists a game state graph \\(G^{\\mathbb{S}}\\) corresponding to a winning strategy, such that for any transition \\(((x,y,\\{P'\\}), (x',y',\\{P''\\}))\\) where \\((x,y,\\{P'\\})\\) is a reachable and non-terminal state, we have \\(P'' \\subsetneq P'\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.536,
+                0.825,
+                0.58
+            ],
+            "angle": 0,
+            "content": "3. Next, we can follow the same procedure in the proof of Theorem C.53 to construct a tree decomposition \\( T^r \\) for the subgraph \\( F[P] \\) containing all edges in \\( P \\), such that \\( (F[P], T^r) \\in S_{\\mathbf{e}}^{M} \\) and \\( \\beta_T(r) = \\{w_1, w_2\\} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.192,
+                0.585,
+                0.825,
+                0.661
+            ],
+            "angle": 0,
+            "content": "4. Finally, let \\( P \\) ranges over all connected components in \\( \\mathbb{C}C_F(\\{w_1, w_2\\}) \\), for each \\( P \\) we have a tree-decomposed graph. We can glue these tree-decomposed graphs by merging the root to obtain the tree decomposition of \\( F \\), because the root bags of all \\( T^r \\) are the same and any vertex \\( x \\notin \\{w_1, w_2\\} \\) appears in the bag of only one \\( T^r \\). Moreover, the glued tree \\( \\tilde{T}^s \\) clearly satisfies that \\( (F, \\tilde{T}^s) \\in S_{\\mathrm{e}}^{M} \\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.192,
+                0.313,
+                0.826,
+                0.661
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.665,
+                0.641,
+                0.681
+            ],
+            "angle": 0,
+            "content": "Combining these items shows that \\( F^{\\boldsymbol{w}} \\in \\mathcal{F}_{\\mathrm{e}}^{M} \\) and concludes the proof."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.694,
+                0.825,
+                0.757
+            ],
+            "angle": 0,
+            "content": "Corollary D.12. Let \\( F^w \\) any rooted graph marking \\( m \\) vertices such that \\( F^w \\notin \\mathcal{F}_n^M \\) (\\( m = 1 \\)) or \\( F^w \\notin \\mathcal{F}_{\\mathrm{e}}^{M} \\) (\\( m = 2 \\)), and let \\( G_{k}(F^{w}) \\) and \\( H_{k}(F^{w}) \\) be clique-augmented Fuirer graphs defined above. Denote by \\( \\xi, \\eta \\in V_{G_k(F^w)}^m \\) two vertex tuples of length \\( m \\) where \\( \\xi_{i} = (w_{i}, \\emptyset) \\) and \\( \\eta_{i} \\in \\operatorname{Meta}_{\\tilde{F}}(w_{i}) \\) with \\( \\eta_{i,2} \\subset V_{F} \\) for \\( i \\in [m] \\). Then, \\( \\operatorname{hom}(F^w, [G_k(F^w)]^\\xi) \\neq \\operatorname{hom}(F^w, [H_k(F^w)]^\\eta) \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.777,
+                0.825,
+                0.806
+            ],
+            "angle": 0,
+            "content": "Proof. The proof exactly parallels that of Theorem C.47 by separately considering three cases. We omit the proof for clarity."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.819,
+                0.55,
+                0.835
+            ],
+            "angle": 0,
+            "content": "Corollary D.13. Let \\( M \\in \\{\\mathrm{Sub}, \\mathrm{L}, \\mathrm{LF}, \\mathrm{F}\\} \\) be any model."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.197,
+                0.84,
+                0.825,
+                0.888
+            ],
+            "angle": 0,
+            "content": "- For any graph \\( F^w \\notin \\mathcal{F}_{\\mathfrak{n}}^{M} \\), let \\( G_4(F^w) \\) and \\( H_4(F^w) \\) be the clique-augmented Fürer graph and the corresponding twisted Fürer graph. Then, there exists two vertices \\( \\xi, \\xi' \\) such that \\( \\operatorname{hom}(F^w, [G_4(F^w)]^\\xi) \\neq \\operatorname{hom}(F^w, [H_4(F^w)]^{\\xi'}) \\) and \\( \\chi_{G_4(F^w)}^M(\\xi) = \\chi_{H_4(F^w)}^M(\\xi') \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.895,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "- For any graph \\( F^{wx} \\notin \\mathcal{F}_{\\mathrm{e}}^{M} \\), let \\( G_{4}(F^{wx}) \\) and \\( H_{4}(F^{wx}) \\) be the clique-augmented Fürer graph and the corresponding twisted Fürer graph defined in Definition D.10. Then, there exists"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.197,
+                0.84,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "53"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.209,
+                0.102,
+                0.827,
+                0.138
+            ],
+            "angle": 0,
+            "content": "four vertices \\(\\xi, \\eta, \\xi', \\eta'\\) such that \\(\\operatorname{hom}(F^{wx}, [G_4(F^{wx})]^{\\xi \\eta}) \\neq \\operatorname{hom}(F^{wx}, [H_4(F^{wx})]^{\\xi' \\eta'})\\) and \\(\\chi_{G_4(F^{wx})}^M(\\xi, \\eta) = \\chi_{H_4(F^{wx})}^M(\\xi', \\eta')\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.15,
+                0.668,
+                0.166
+            ],
+            "angle": 0,
+            "content": "Proof. The proof directly follows from Corollary D.12 and Theorem D.11. \\(\\square\\)"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.151,
+                0.826,
+                0.163
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.185,
+                0.405,
+                0.202
+            ],
+            "angle": 0,
+            "content": "E HIGHER-ORDER GNNS"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.216,
+                0.486,
+                0.231
+            ],
+            "angle": 0,
+            "content": "E.1 DEFINITION OF HIGHER-ORDER GNNS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.242,
+                0.825,
+                0.272
+            ],
+            "angle": 0,
+            "content": "In this subsection, we give formal definitions of the CR algorithms for various higher-order GNNs, each of which generalizes a model in Section 2."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.197,
+                0.279,
+                0.826,
+                0.369
+            ],
+            "angle": 0,
+            "content": "- Subgraph \\( k \\)-GNN. In a Subgraph \\( k \\)-GNN, a graph \\( G \\) is treated as a set of subgraphs \\( \\{G^u : u \\in V_G^k\\} \\), where each subgraph \\( G^u \\) is obtained from \\( G \\) by marking \\( k \\) special vertices \\( u \\in V_G^k \\), and thus there are \\( n^k \\) subgraphs when \\( G \\) has \\( n \\) vertices. Subgraph GNN maintains a color \\( \\chi_G^{\\mathrm{Sub}(k)}(\\pmb{u}, v) \\) for each vertex \\( v \\) in graph \\( G^u \\). Initially, \\( \\chi_G^{\\mathrm{Sub}(k), (0)}(\\pmb{u}, v) = (\\ell_G(v), \\mathsf{atp}_G(\\pmb{u}), \\mathbb{I}[u_1 = v], \\dots, \\mathbb{I}[u_k = v]) \\). It then runs MPNNs independently on each graph \\( G^u \\):"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.23,
+                0.375,
+                0.826,
+                0.401
+            ],
+            "angle": 0,
+            "content": "\\[\n\\chi_ {G} ^ {\\operatorname {S u b} (k), (t + 1)} (\\boldsymbol {u}, v) = \\operatorname {h a s h} \\left(\\chi_ {G} ^ {\\operatorname {S u b} (k), (t)} (\\boldsymbol {u}, v), \\{\\left\\{\\chi_ {G} ^ {\\operatorname {S u b} (k), (t)} (\\boldsymbol {u}, w): w \\in N _ {G} (v) \\right\\} \\}\\right). \\tag {26}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.408,
+                0.825,
+                0.459
+            ],
+            "angle": 0,
+            "content": "Denote the stable color as \\(\\chi_G^{\\mathrm{Sub}(k)}(\\pmb{u},\\upsilon)\\). Define \\(\\chi_G^{\\mathrm{Sub}(k)}(\\pmb{u})\\coloneqq \\{\\{\\chi_G^{\\mathrm{Sub}(k)}(\\pmb{u},\\upsilon):v\\in V_G\\}\\}\\). Then, the graph representation is defined as \\(\\chi_G^{\\mathrm{Sub}(k)}(G) = \\{\\{\\chi_G^{\\mathrm{Sub}(k)}(\\pmb {u}):\\pmb {u}\\in V_G\\} \\}\\). We remark that Subgraph \\(k\\) -GNN is precisely the \\(k\\) -VSAN proposed in Qian et al. (2022)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.462,
+                0.826,
+                0.511
+            ],
+            "angle": 0,
+            "content": "- Local \\(k\\)-GNN. Local \\(k\\)-GNN maintains a color \\(\\chi_G^{\\mathsf{L}(k)}(\\pmb{u})\\) for each vertex \\(k\\)-tuple \\(\\pmb{u} \\in V_G^k\\). Initially, \\(\\chi_G^{\\mathsf{L}(k),(0)}(\\pmb{u}) = (\\ell_G(u_1), \\dots, \\ell_G(u_k), \\mathsf{atp}_G(\\pmb{u}))\\), called the isomorphism type of vertex \\(k\\)-tuple \\(\\pmb{u}\\), where \\(\\mathsf{atp}_G(\\pmb{u})\\) is the atomic type of \\(\\pmb{u}\\). Then, in each iteration \\(t + 1\\),"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.273,
+                0.517,
+                0.824,
+                0.571
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\chi_ {G} ^ {\\mathsf {L} (k), (t + 1)} (\\boldsymbol {u}) = \\operatorname {h a s h} \\left(\\chi_ {G} ^ {\\mathsf {L} (k), (t)} (\\boldsymbol {u}), \\{\\{\\chi_ {G} ^ {\\mathsf {L} (k), (t)} (\\boldsymbol {v}): \\boldsymbol {v} \\in N _ {G} ^ {(1)} (\\boldsymbol {u}) \\} \\}, \\dots , \\right. \\\\ \\left. \\left\\{\\left\\{\\chi_ {G} ^ {\\mathrm {L} (k), (t)} (\\boldsymbol {v}): \\boldsymbol {v} \\in N _ {G} ^ {(k)} (\\boldsymbol {u}) \\right\\} \\right\\}\\right), \\tag {27} \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.21,
+                0.579,
+                0.825,
+                0.628
+            ],
+            "angle": 0,
+            "content": "where \\(N_G^{(j)}(\\pmb{u}) = \\{(u_1, \\dots, u_{j-1}, w, u_{j+1}, \\dots, u_k) : w \\in N_G(u_j)\\}\\). Denote the stable color as \\(\\chi_G^{\\mathsf{L}(k)}(\\pmb{u})\\). The graph representation is defined as \\(\\chi_G^{\\mathsf{L}(k)}(G) := \\{\\chi_G^{\\mathsf{L}(k)}(\\pmb{u}) : \\pmb{u} \\in V_G\\}\\). We remark that Local \\(k\\)-GNN is precisely the \\(\\delta\\)-\\(k\\)-LWL proposed in Morris et al. (2020)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.629,
+                0.826,
+                0.658
+            ],
+            "angle": 0,
+            "content": "- Local \\( k \\)-FGNN. Local \\( k \\)-FGNN is almost the same as Local \\( k \\)-GNN except that the update formula is replaced by the following one:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.211,
+                0.665,
+                0.829,
+                0.729
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\chi_ {G} ^ {\\mathsf {L F} (k), (t + 1)} (\\boldsymbol {u}) = \\operatorname {h a s h} \\left(\\chi_ {G} ^ {\\mathsf {L F} (k), (t)} (\\boldsymbol {u}), \\{\\{\\left(\\chi_ {G} ^ {\\mathsf {L F} (k), (t)} (w, u _ {2}, \\dots , u _ {k}), \\chi_ {G} ^ {\\mathsf {L F} (k), (t)} (u _ {1}, w, u _ {3}, \\dots , u _ {k}), \\right. \\right. \\\\ \\left. \\dots , \\chi_ {G} ^ {\\mathsf {L F} (k), (t)} \\left(u _ {1}, \\dots , u _ {k - 1}, w\\right): w \\in N _ {G} \\left(u _ {1}\\right) \\cup \\dots \\cup N _ {G} \\left(u _ {k}\\right) \\right\\} \\Bigg). \\tag {28} \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.211,
+                0.73,
+                0.821,
+                0.744
+            ],
+            "angle": 0,
+            "content": "We remark that Local \\(k\\)-FGNN is precisely the \\(\\mathrm{SLFWL}(k)\\) proposed in Zhang et al. (2023a)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.745,
+                0.825,
+                0.774
+            ],
+            "angle": 0,
+            "content": "- \\( k \\)-FGNN. It is just the standard \\( k \\)-FWL (Cai et al., 1992). Compared with Local \\( k \\)-FGNN, the update formula is now global:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.221,
+                0.781,
+                0.824,
+                0.846
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\chi_ {G} ^ {\\mathsf {F} (k), (t + 1)} (\\boldsymbol {u}) = \\operatorname {h a s h} \\left(\\chi_ {G} ^ {\\mathsf {F} (k), (t)} (\\boldsymbol {u}), \\{\\{\\left(\\chi_ {G} ^ {\\mathsf {F} (k), (t)} (w, u _ {2}, \\dots , u _ {k}), \\chi_ {G} ^ {\\mathsf {F} (k), (t)} (u _ {1}, w, u _ {3}, \\dots , u _ {k}), \\right. \\right. \\\\ \\left. \\dots , \\chi_ {G} ^ {\\mathsf {F} (k), (t)} \\left(u _ {1}, \\dots , u _ {k - 1}, w\\right): w \\in V _ {G} \\right\\}. \\Bigg). \\tag {29} \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.855,
+                0.826,
+                0.925
+            ],
+            "angle": 0,
+            "content": "We note that the computational complexity of Subgraph \\((k - 1)\\)-GNN, Local \\(k\\)-GNN, and Local \\(k\\)-FGNN is the same, i.e., \\(\\Theta(n^{k - 1}m)\\) for a graph of \\(n\\) vertices and \\(m\\) edges. Moreover, it is easy to see that Subgraph 0-GNN, Local 1-GNN, and Local 1-FGNN all reduce to MPNN. For \\(k\\)-FGNN, the computational complexity is \\(\\Theta(n^{k + 1})\\), which is strictly higher than Subgraph \\((k - 1)\\)-GNN, Local \\(k\\)-GNN, and Local \\(k\\)-FGNN."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.51,
+                0.96
+            ],
+            "angle": 0,
+            "content": "54"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.104,
+                0.43,
+                0.119
+            ],
+            "angle": 0,
+            "content": "E.2 HIGHER-ORDER STRONG NED"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.13,
+                0.825,
+                0.159
+            ],
+            "angle": 0,
+            "content": "In this subsection, we generalize the strong NED into higher-order versions. We first define the concept of higher-order ear:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.161,
+                0.826,
+                0.202
+            ],
+            "angle": 0,
+            "content": "Definition E.1 (\\(k\\)-order ear). Given integer \\(k \\geq 1\\), a \\(k\\)-order ear is a graph \\(G\\) formed by the union of \\(k\\) paths \\(P_{1}, \\dots, P_{k}\\) (possibly with zero length) plus an edge set \\(Q\\) satisfying the following conditions:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.197,
+                0.21,
+                0.825,
+                0.255
+            ],
+            "angle": 0,
+            "content": "- For each path \\( P_{i} \\), denote its two endpoints as \\( u_{i}, v_{i} \\), called the outer endpoint and inner endpoint, respectively. Then, all edges in \\( Q \\) are linked between inner endpoints, i.e., \\( Q \\subset \\{\\{v_{i}, v_{j}\\} : 1 \\leq i, j \\leq k, v_{i} \\neq v_{j}\\} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.259,
+                0.785,
+                0.275
+            ],
+            "angle": 0,
+            "content": "- Any different paths \\( P_{i}, P_{j} \\) do not intersect except at the inner endpoint (when \\( v_{i} = v_{j} \\))."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.199,
+                0.28,
+                0.372,
+                0.295
+            ],
+            "angle": 0,
+            "content": "- \\(G\\) is a connected graph."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.197,
+                0.21,
+                0.825,
+                0.295
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.302,
+                0.697,
+                0.317
+            ],
+            "angle": 0,
+            "content": "The endpoints of the \\(k\\)-order ear is defined to be all outer endpoints \\(u_{1}, \\dots, u_{k}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.326,
+                0.825,
+                0.409
+            ],
+            "angle": 0,
+            "content": "It is easy to see that a 2-order ear is precisely a simple path, since linking two different paths (possibly with an additional edge) still yields a path. Below, we denote by inner \\((G)\\) and outer \\((G)\\) the set of inner endpoints and outer endpoints in ear \\(G\\), respectively. We also denote by path \\((G)\\) the set of paths in ear \\(G\\). It follows that \\(|\\text{inner}(G)| = |\\text{outer}(G)| = |\\text{path}(G)| = k\\). Finally, given a path \\(P\\) and two vertices \\(w_{1}, w_{2}\\) in \\(P\\), denote by subpath \\(_{P}(w_{1}, w_{2})\\) the subpath in \\(P\\) such that the two endpoints are \\(w_{1}, w_{2}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.411,
+                0.825,
+                0.483
+            ],
+            "angle": 0,
+            "content": "Definition E.2 (Nested interval). Let \\( G \\) and \\( H \\) be two \\( k \\)-order ears with inner \\( (G) = \\{v_{1},\\dots ,v_{k}\\} \\), outer \\( (G) = \\{u_{1},\\dots ,u_{k}\\} \\), and outer \\( (H) = \\{w_{1},\\dots ,w_{k}\\} \\), where each \\( \\{u_i,v_i\\} \\) corresponds to the endpoints of a path \\( P_{i}\\in \\mathrm{path}(G) \\). We say \\( H \\) is nested on \\( G \\) if one or more endpoint \\( w_{i} \\) of \\( H \\) (\\( i\\in [k] \\)) is in path \\( P_{i} \\), and all other vertices in \\( H \\) are not in \\( G \\). The nested interval is defined to be the union of subpaths subpath \\( P_{i}(w_{i},v_{i}) \\) for all \\( i\\in [k] \\) satisfying that \\( w_{i} \\) is in \\( P_{i} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.491,
+                0.825,
+                0.52
+            ],
+            "angle": 0,
+            "content": "We give an illustration of the nested interval of two 3-order ears in Figure 2. Equipped with the above definition, we are ready to introduce the higher-order strong NED:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.521,
+                0.826,
+                0.551
+            ],
+            "angle": 0,
+            "content": "Definition E.3 (\\(k\\)-order strong NED). Given a graph \\(G\\), a \\(k\\)-order strong NED \\(\\mathcal{P}\\) is a partition of the edge set \\(E_G\\) into a sequence of edge sets \\(Q_1, \\dots, Q_m\\), which satisfies the following conditions:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.557,
+                0.381,
+                0.571
+            ],
+            "angle": 0,
+            "content": "Each \\(Q_{i}\\) is a \\(k\\) -order ear."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.578,
+                0.825,
+                0.606
+            ],
+            "angle": 0,
+            "content": "- Any two ears \\( Q_{i} \\) and \\( Q_{j} \\) with indices \\( 1 \\leq i < j \\leq c \\) do not intersect, where \\( c \\) is the number of connected components of \\( G \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.613,
+                0.825,
+                0.655
+            ],
+            "angle": 0,
+            "content": "- For each \\( Q_{j} \\) with index \\( j > c \\), it is nested on some \\( k \\)-order ear \\( Q_{i} \\) with index \\( 1 \\leq i < j \\). Moreover, except for the endpoints of \\( Q_{j} \\) on \\( Q_{i} \\), no other vertices in \\( Q_{j} \\) are in any previous ear \\( Q_{k} \\) for \\( 1 \\leq k < i \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.661,
+                0.825,
+                0.692
+            ],
+            "angle": 0,
+            "content": "- Denote by \\( I(Q_{j}) \\subset Q_{i} \\) the nested interval of \\( Q_{j} \\) in \\( Q_{i} \\). For all \\( Q_{j}, Q_{k} \\) with \\( c < j < k \\leq m \\), if \\( Q_{j} \\) and \\( Q_{k} \\) are nested on the same ear, then \\( I(Q_{j}) \\subset I(Q_{k}) \\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.198,
+                0.557,
+                0.825,
+                0.692
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.706,
+                0.387,
+                0.721
+            ],
+            "angle": 0,
+            "content": "E.3 PROOFS IN SECTION 3.4"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.732,
+                0.764,
+                0.747
+            ],
+            "angle": 0,
+            "content": "We first generalize the tree-decomposed graphs in Definition C.6 to higher-order versions:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.748,
+                0.825,
+                0.779
+            ],
+            "angle": 0,
+            "content": "Definition E.4. Define four families of tree-decomposed graphs \\( \\mathcal{S}^{\\mathrm{Sub}(k)}, \\mathcal{S}^{\\mathrm{L}(k)}, \\mathcal{S}^{\\mathrm{LF}(k)} \\), and \\( \\mathcal{S}^{\\mathrm{F}(k)} \\) as follows:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.19,
+                0.784,
+                0.642,
+                0.801
+            ],
+            "angle": 0,
+            "content": "a) \\((F,T^{r})\\in S^{\\mathsf{F}(k)}\\) iff \\((F,T^{r})\\) satisfies Definition C.4 with width \\(k\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.19,
+                0.808,
+                0.827,
+                0.852
+            ],
+            "angle": 0,
+            "content": "b) \\((F,T^{r})\\in S^{\\mathsf{LF}(k)}\\) iff \\((F,T^{r})\\) satisfies Definition C.4 with width \\(k\\), and for any tree node \\(t\\) of odd depth, it has only one child if \\(w\\notin \\{v:v\\in N_G(u),u\\in \\beta_T(s)\\}\\) where \\(s\\) is the parent node of \\(t\\) and \\(w\\) is the unique vertex in \\(\\beta_T(t)\\backslash \\beta_T(s)\\);"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.19,
+                0.858,
+                0.825,
+                0.889
+            ],
+            "angle": 0,
+            "content": "c) \\((F,T^{r})\\in S^{\\mathbb{L}(k)}\\) iff \\((F,T^{r})\\) satisfies Definition C.4 with width \\(k\\), and any tree node \\(t\\) of odd depth has only one child;"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.19,
+                0.895,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "d) \\((F,T^{r})\\in S^{\\mathrm{Sub}(k)}\\) iff \\((F,T^{r})\\) satisfies Definition C.4 with width \\(k\\), and there exists a multiset \\(U\\subset V_{G}\\) of size \\(|U| = k\\) such that \\(U\\subset \\beta_T(t)\\) for all \\(t\\in V_T\\)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.19,
+                0.784,
+                0.827,
+                0.926
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "55"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.825,
+                0.133
+            ],
+            "angle": 0,
+            "content": "Then, we can analogously prove the following theorems. The proofs are almost the same as in Appendix C, so we omit them for clarity."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.135,
+                0.826,
+                0.181
+            ],
+            "angle": 0,
+            "content": "Theorem E.5. Let \\( M \\in \\{\\mathsf{Sub}(k), \\mathsf{L}(k), \\mathsf{LF}(k), \\mathsf{F}(k)\\} \\). Then, any graphs \\( G \\) and \\( H \\) have the same representation under model \\( M \\) (i.e., \\( \\chi_G^M(G) = \\chi_H^M(H) \\)) iff \\( \\mathsf{hom}(F, G) = \\mathsf{hom}(F, H) \\) for all \\( (F, T^r) \\in S^M \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.183,
+                0.827,
+                0.248
+            ],
+            "angle": 0,
+            "content": "Theorem E.6. Let \\( M \\in \\{\\mathrm{Sub}(k), \\mathsf{L}(k), \\mathsf{LF}(k), \\mathsf{F}(k)\\} \\) be any model, and let \\( F \\) be any graph such that no tree decomposition \\( (F, T^r) \\in S^M \\). Let \\( G(F) \\) and \\( H(F) \\) be the Fürer graph and twisted Fürer graph with respect to \\( F \\). Then, \\( \\mathrm{hom}(F, G(F)) \\neq \\mathrm{hom}(F, H(F)) \\) and \\( \\chi_{G(F)}^{M}(G(F)) = \\chi_{H(F)}^{M}(H(F)) \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.251,
+                0.825,
+                0.282
+            ],
+            "angle": 0,
+            "content": "Theorem E.7. For any graph \\(F\\), there is a tree decomposition \\(T^r\\) of \\(F\\) such that \\((F, T^r) \\in S^{\\mathrm{Sub}(k)}\\) iff there exists \\(U \\subset V_F\\) such that \\(|U| \\leq k\\) and \\(F \\backslash U\\) is a forest."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.285,
+                0.825,
+                0.315
+            ],
+            "angle": 0,
+            "content": "Theorem E.8. For any graph \\(F\\), there is a tree decomposition \\(T^r\\) of \\(F\\) such that \\((F, T^r) \\in S^{\\mathsf{L}(k)}\\) iff \\(F\\) has a \\(k\\)-order strong NED."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.316,
+                0.825,
+                0.348
+            ],
+            "angle": 0,
+            "content": "Theorem E.9. For any graph \\(F\\), there is a tree decomposition \\(T^r\\) of \\(F\\) such that \\((F, T^r) \\in S^{\\mathsf{F}(k)}\\) iff \\(\\operatorname{tw}(F) \\leq k\\)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.363,
+                0.588,
+                0.377
+            ],
+            "angle": 0,
+            "content": "E.4 EXPRESSIVITY GAP BETWEEN HIGHER-ORDER GNNS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.389,
+                0.825,
+                0.418
+            ],
+            "angle": 0,
+            "content": "In this subsection, we show how homomorphism expressivity can be used to build a complete expressiveness hierarchy for higher-order GNNs as shown in Corollary 4.2."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.424,
+                0.826,
+                0.51
+            ],
+            "angle": 0,
+            "content": "Gap between Subgraph \\( k \\)-GNN and Local \\( (k + 1) \\)-GNN \\( (k \\geq 1) \\). The counterexample graph is a \\( (k + 1) \\times (2k + 2) \\) grid consisting \\( (k + 1) \\times (2k + 2) \\) vertices. First, it is easy to see that the graph is not in \\( \\mathcal{F}^{\\mathrm{Sub}(k)} \\), i.e., deleting \\( k \\) vertices of the graph cannot yield a forest. To see this, note that the graph consists of \\( k \\times (2k + 1) \\) \"squares\", and each vertex is related to at most four squares. Therefore, deleting \\( k \\) vertices cannot eliminate all squares when \\( k \\geq 2 \\) (because \\( 4k < k \\times (2k + 1) \\)). For the case of \\( k = 1 \\), we clearly have that deleting one vertex cannot eliminate all squares."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.516,
+                0.551,
+                0.575
+            ],
+            "angle": 0,
+            "content": "We next show that the \\((k + 1)\\times (2k + 2)\\) grid is in \\(\\mathcal{F}^{\\mathsf{L}(k + 1)}\\). This is also simple as shown in Figure 8, where each color represents a \\(k\\)-order ear. It can be seen that the \\((k + 1)\\times (2k + 2)\\) grid has a \\((k + 1)\\)-order strong NED."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.569,
+                0.514,
+                0.818,
+                0.603
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.56,
+                0.611,
+                0.825,
+                0.641
+            ],
+            "angle": 0,
+            "content": "Figure 8: An 4-order strong NED of the \\(4 \\times 8\\) grid graph."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.58,
+                0.55,
+                0.65
+            ],
+            "angle": 0,
+            "content": "Relation between Subgraph \\( k \\)-GNN and \\( k \\)-FGNN (\\( k \\geq 2 \\)). To show that \\( \\mathcal{F}^{\\mathrm{Sub}(k)} \\not\\subsetneq \\mathcal{F}^{\\mathrm{F}(k)} \\), consider the \\( (k + 2) \\)-clique. Clearly, deleting \\( k \\) vertices from the \\( (k + 2) \\)-clique yields a graph consisting of two vertices linked by an edge, which is a tree. On the other hand, the"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.65,
+                0.825,
+                0.668
+            ],
+            "angle": 0,
+            "content": "treewidth of a \\((k + 2)\\)-clique is \\(k + 1\\) (Fact C.3). So the \\((k + 2)\\)-clique is in \\(\\mathcal{F}^{\\mathrm{Sub}(k)}\\) but not in \\(\\mathcal{F}^{\\mathsf{F}(k)}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.673,
+                0.826,
+                0.732
+            ],
+            "angle": 0,
+            "content": "To show that \\(\\mathcal{F}^{\\mathsf{F}(k)} \\not\\subset \\mathcal{F}^{\\mathrm{Sub}(k)}\\), we can again use the grid graph, but this time consider the \\(k \\times (2k + 2)\\) grid. On the one hand, a standard result in graph theory shows that the treewidth of a \\(a \\times b\\) grid graph is \\(\\min(a, b)\\). On the other hand, following the similar analysis above, we can prove that deleting \\(k\\) vertices from the \\(k \\times (2k + 2)\\) grid cannot eliminate all squares."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.737,
+                0.826,
+                0.84
+            ],
+            "angle": 0,
+            "content": "Gap between Local \\(k\\)-GNN and \\(k\\)-FGNN (\\(k \\geq 2\\)). The counterexample graph \\(F\\) is the union of the following graphs \\(K_0 \\cup K_1 \\cup \\dots \\cup K_k\\), where \\(K_0\\) is a \\((k+1)\\)-clique with vertex set \\(\\{u_1, \\dots, u_{k+1}\\}\\) and for \\(i \\in [k+1]\\), \\(K_i\\) is a \\((k+1)\\)-clique with vertex set \\(\\{u_1, \\dots, u_{i-1}, v_i, u_{i+1}, \\dots, u_{k+1}\\}\\) where \\(v_i\\) is a new vertex (not in \\(K_0\\)) and \\(v_i \\neq v_j\\) for \\(i \\neq j\\). Namely, each \\(K_i\\) has \\(k\\) common vertices with \\(K_0\\). It is easy to construct a tree decomposition \\(T^r\\) of \\(F\\) such that \\((F, T^r) \\in S^{\\mathsf{F}(k)}\\). On the other hand, it is easy to see that the graph does not have a \\(k\\)-order strong NED (equivalently, one can easily check that \\(F\\) does not admit a tree decomposition \\(T^r\\) satisfying \\((F, T^r) \\in S^{\\mathsf{L}(k)}\\))."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.845,
+                0.825,
+                0.876
+            ],
+            "angle": 0,
+            "content": "Gap between \\(k\\)-FGNN and Local \\((k + 1)\\)-GNN \\((k \\geq 2)\\). The counterexample graph is again the \\((k + 1) \\times (2k + 2)\\) grid. We have proved that the graph is in \\(\\mathcal{F}^{\\mathsf{L}(k + 1)}\\) but not in \\(\\mathcal{F}^{\\mathsf{L}(k)}\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.882,
+                0.826,
+                0.926
+            ],
+            "angle": 0,
+            "content": "Regarding Local \\(k\\)-IGN and Frasca et al. (2022). Finally, we remark that based on the above results, we essentially proved an open question raised in Frasca et al. (2022) regarding the expressive power of Local \\(k\\)-IGN (i.e., the \\(\\mathsf{RelGN}(k)\\) proposed in their paper). Following Zhang et al. (2023a),"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "56"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.197,
+                0.11,
+                0.36,
+                0.165
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.268,
+                0.171,
+                0.288,
+                0.184
+            ],
+            "angle": 0,
+            "content": "(a)"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.382,
+                0.119,
+                0.554,
+                0.165
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.458,
+                0.171,
+                0.478,
+                0.184
+            ],
+            "angle": 0,
+            "content": "(b)"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.579,
+                0.105,
+                0.801,
+                0.164
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.68,
+                0.171,
+                0.7,
+                0.184
+            ],
+            "angle": 0,
+            "content": "(c)"
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.171,
+                0.195,
+                0.825,
+                0.224
+            ],
+            "angle": 0,
+            "content": "Figure 9: More illustration of NED. All NEDs in these example graphs are strong but not endpoint-shared. See Appendix F.2 for additional discussions."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.268,
+                0.825,
+                0.297
+            ],
+            "angle": 0,
+            "content": "it is straightforward to see that Local \\( k \\)-IGN is as expressive as Local \\( k \\)-GNN. Therefore, Local \\( k \\)-IGN is strictly more expressive than \\( (k - 1) \\)-FGNN and strictly less expressive than \\( k \\)-FGNN."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.337,
+                0.436,
+                0.352
+            ],
+            "angle": 0,
+            "content": "F ADDITIONAL DISCUSSIONS"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.38,
+                0.668,
+                0.394
+            ],
+            "angle": 0,
+            "content": "F.1 REGARDING THE DEFINITION OF HOMOMORPHISM EXPRESSIVITY"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.414,
+                0.825,
+                0.471
+            ],
+            "angle": 0,
+            "content": "In this paper, we have shown that homomorphism expressivity exists for a variety of popular GNNs. Unfortunately, due to the \"iff\" statement in Definition 3.1, homomorphism expressivity may not always be well-defined in the general case. We remark in this subsection that, there do exist pathological, intentionally designed GNNs such that the homomorphism expressivity is not well-defined."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.477,
+                0.827,
+                0.618
+            ],
+            "angle": 0,
+            "content": "Consider a simple GNN \\(M\\) that outputs the representation of a graph \\(G\\) as follows. If \\(G\\) is a cycle of odd length, it outputs \\((1,L)\\) where \\(L\\) is the length of the cycle. Otherwise, it outputs \\((0,\\chi_{G}^{\\mathrm{MP}}(G))\\), namely, running a MPNN on the graph. It follows that \\(M\\) is strictly more powerful than MPNN, e.g., it can distinguish between the 9-cycle and three triangles, which MPNN fails to distinguish. As a result, \\(M\\) can count all trees under homomorphism. Moreover, it cannot count other patterns under homomorphism, as the counterexample graphs for MPNN (i.e., Fürer graphs) are not cycles of odd length, so they are still counterexample graphs for model \\(M\\). Therefore, the homomorphism expressivity of \\(M\\) should be exactly the family of forests if it exists. However, the homomorphism information of forests cannot determine the representation of model \\(M\\) since \\(M\\) is strictly more powerful than MPNN. So we conclude that \\(\\mathcal{F}^{M}\\) does not exist."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.623,
+                0.825,
+                0.695
+            ],
+            "angle": 0,
+            "content": "Note that the above GNN construction is inherently unnatural and hardly appears in practice. Actually, when a GNN is defined via the message-passing paradigm, we suspect that its homomorphism expressivity is always well-defined. As stated in Appendix B, we conjecture that, for any GNN characterized by a color refinement algorithm that outputs stable colors, the homomorphism expressivity always exists."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.73,
+                0.482,
+                0.744
+            ],
+            "angle": 0,
+            "content": "F.2 REGARDING THE DEFINITION OF NED"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.764,
+                0.825,
+                0.835
+            ],
+            "angle": 0,
+            "content": "In the main text, we have illustrated several types of NED with simple example graphs. To gain a deeper understanding of Definition 3.3, in this section we will present a few more complex examples (see Figure 9). Notably, unlike graphs in Figure 1(b), for all graphs in Figure 9 their NED contains ears such that only one endpoint is in its nested ear (e.g., ears 2 and 4 in Figure 9(a)). In other words, these ears have empty nested intervals."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.841,
+                0.827,
+                0.926
+            ],
+            "angle": 0,
+            "content": "The presence of ears with empty nested interval stems from the fact that the corresponding graph is not biconnected (Zhang et al., 2023b). Indeed, one can check that all graphs in Definition 3.3(b) do not have cut vertices, while all graphs in Figure 9 have cut vertices. Moreover, in these examples, the number of ears with empty nested interval always equals to the number of biconnected components minus one. In particular, for biconnected graphs, it can be proved that any NED does not contain an ear with empty nested interval."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "57"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.171,
+                0.103,
+                0.66,
+                0.119
+            ],
+            "angle": 0,
+            "content": "G HOMOMORPHISM AND SUBGRAPH COUNTING POWER"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.133,
+                0.391,
+                0.147
+            ],
+            "angle": 0,
+            "content": "G.1 PROOF OF THEOREM 4.5"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.16,
+                0.825,
+                0.203
+            ],
+            "angle": 0,
+            "content": "Our proof draws inspiration from a recent technique developed in Seppelt (2023). We note that the original results in Seppelt (2023) are described for unlabeled graphs, but it is easy to extend these results to labeled graphs. To begin with, we define a concept called the graph categorical product."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.205,
+                0.825,
+                0.249
+            ],
+            "angle": 0,
+            "content": "Definition G.1. The categorical product of two graphs \\(G, H\\), denoted as \\(G \\times H\\), is a graph where \\(V_{G \\times H} = V_G \\times V_H\\), \\(\\ell_{G \\times H}(g, h) = (\\ell_G(g), \\ell_H(h))\\) for all \\(g \\in V_G\\), \\(h \\in V_H\\), and \\(E_{G \\times H} = \\{ (g, h), (g', h') \\} : \\{ g, g' \\} \\in E_G, \\{ h, h' \\} \\in E_H \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.251,
+                0.825,
+                0.281
+            ],
+            "angle": 0,
+            "content": "Lemma G.2 (Seppelt (2023)). For any graphs \\(F, G,\\) and \\(H\\), \\(\\operatorname{hom}(F,G\\times H) = \\operatorname{hom}(F,G)\\cdot\\) \\(\\operatorname{hom}(F,H)\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.295,
+                0.826,
+                0.352
+            ],
+            "angle": 0,
+            "content": "Proof. We define a mapping \\(\\tau\\) from \\(\\operatorname{Hom}(F, G) \\times \\operatorname{Hom}(F, H)\\) to \\(\\operatorname{Hom}(F, G \\times H)\\) as follows. for all \\(\\sigma_1 \\in \\operatorname{Hom}(F, G)\\) and \\(\\sigma_2 \\in \\operatorname{Hom}(F, H)\\), define \\(\\rho = \\tau(\\sigma_1, \\sigma_2)\\) where \\(\\rho(f) = (\\sigma_1(f), \\sigma_2(f))\\) for all \\(f \\in V_F\\). It is easy to see that \\(\\tau\\) is a bijective mapping from \\(\\operatorname{Hom}(F, G) \\times \\operatorname{Hom}(F, H)\\) to \\(\\operatorname{Hom}(F, G \\times H)\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.36,
+                0.825,
+                0.39
+            ],
+            "angle": 0,
+            "content": "Definition G.3. Given graphs \\(F\\) and \\(G\\), denote by \\(\\operatorname{Surj}(F, G)\\) the set of all homomorphisms from \\(F\\) to \\(G\\) that are surjective on both the vertices and edges of \\(G\\), and define \\(\\operatorname{surj}(F, G) = |\\operatorname{Surj}(F, G)|\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.399,
+                0.638,
+                0.414
+            ],
+            "angle": 0,
+            "content": "The following proposition is straightforward (similar to Lemma C.25):"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.418,
+                0.825,
+                0.447
+            ],
+            "angle": 0,
+            "content": "Proposition G.4. \\(\\operatorname{hom}(G, H) = \\sum_{F} \\operatorname{surj}(G, F) \\cdot \\operatorname{sub}(F, H)\\), where \\(F\\) ranges over all non-isomorphic graphs."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.169,
+                0.449,
+                0.825,
+                0.48
+            ],
+            "angle": 0,
+            "content": "Lemma G.5. Let \\(M\\) be a GNN model such that its homomorphism expressivity \\(\\mathcal{F}^M\\) exists. Given a finite set of graphs \\(\\mathcal{L}\\) and a function \\(\\alpha : \\mathcal{L} \\to \\mathbb{R} / \\{0\\}\\), if"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.26,
+                0.485,
+                0.737,
+                0.517
+            ],
+            "angle": 0,
+            "content": "\\[\n\\chi_ {G} ^ {M} (G) = \\chi_ {H} ^ {M} (H) \\Longrightarrow \\sum_ {L \\in \\mathcal {L}} \\alpha (L) \\operatorname {h o m} (L, G) = \\sum_ {L \\in \\mathcal {L}} \\alpha (L) \\operatorname {h o m} (L, H),\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.524,
+                0.475,
+                0.54
+            ],
+            "angle": 0,
+            "content": "holds for all graphs \\(G\\) and \\(H\\), then \\(\\mathcal{L} \\subset \\mathcal{F}^M\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.556,
+                0.826,
+                0.676
+            ],
+            "angle": 0,
+            "content": "Proof. Let \\( n \\) be the largest number of vertices for all graphs in \\( \\mathcal{L} \\), and let \\( \\tilde{\\mathcal{L}} \\) be the set of all non-isomorphic graphs with no more than \\( n \\) vertices. We can arrange all graphs in \\( \\tilde{\\mathcal{L}} \\) into a sequence \\( L_{1}, L_{2}, \\ldots, L_{N} \\) satisfying the following property: \\( |V_{L_{i}}| \\leq |V_{L_{i+1}}| \\) or \\( (|V_{L_{i}}| = |V_{L_{i+1}}| \\) and \\( |E_{L_{i}}| \\leq |E_{L_{i+1}}|) \\) for all \\( i = 1, 2, \\ldots, N-1 \\). We then define the matrices \\( A^{\\text{hom}} \\), \\( A^{\\text{surj}} \\), and \\( A^{\\text{sub}} \\), where the elements in the \\( i^{th} \\) row and \\( j^{th} \\) column are \\( \\text{hom}(L_{i}, L_{j}) \\), \\( \\text{surj}(L_{i}, L_{j}) \\), and \\( \\text{sub}(L_{i}, L_{j}) \\), respectively. Proposition G.4 implies that \\( A^{\\text{hom}} = A^{\\text{surj}} \\cdot A^{\\text{sub}} \\). Since \\( A^{\\text{surj}} \\) is a lower triangular matrix with nonzero diagonal elements and \\( A^{\\text{sub}} \\) is an upper triangular matrix with non-zero diagonal elements, the matrix \\( A^{\\text{hom}} \\) is invertible."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.682,
+                0.826,
+                0.762
+            ],
+            "angle": 0,
+            "content": "We next extend \\(\\alpha\\) to a function \\(\\tilde{\\alpha}:\\tilde{\\mathcal{L}}\\to \\mathbb{R}\\) by setting \\(\\tilde{\\alpha} (L) = \\alpha (L)\\) for all \\(L\\in \\mathcal{L}\\) and \\(\\tilde{\\alpha} (\\tilde{L}) = 0\\) for all \\(\\tilde{L}\\in \\tilde{\\mathcal{L}}\\backslash \\mathcal{L}\\). Additionally, if \\(\\chi_G^M (G) = \\chi_H^M (H)\\), then \\(\\mathrm{hom}(F,G) = \\mathrm{hom}(F,H)\\) for all \\(F\\in \\mathcal{F}^{M}\\) (by definition of homomorphism expressivity). Then, given any graph \\(K\\in \\tilde{\\mathcal{L}}\\), Lemma G.2 implies that \\(\\mathrm{hom}(F,G\\times K) = \\mathrm{hom}(F,H\\times K)\\) for all \\(F\\in \\mathcal{F}^{M}\\). This further implies that \\(\\chi_{G\\times K}^{M}(G\\times K) = \\chi_{H\\times K}^{M}(H\\times K)\\) by definition of homomorphism expressivity. Therefore, Lemma G.2 implies that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.256,
+                0.768,
+                0.825,
+                0.801
+            ],
+            "angle": 0,
+            "content": "\\[\n\\sum_ {L \\in \\mathcal {L}} \\alpha (L) \\operatorname {h o m} (L, G) \\cdot \\operatorname {h o m} (L, K) = \\sum_ {L \\in \\mathcal {L}} \\alpha (L) \\operatorname {h o m} (L, H) \\cdot \\operatorname {h o m} (L, K). \\tag {30}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.806,
+                0.232,
+                0.821
+            ],
+            "angle": 0,
+            "content": "Namely,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.257,
+                0.827,
+                0.825,
+                0.861
+            ],
+            "angle": 0,
+            "content": "\\[\n\\sum_ {L \\in \\tilde {\\mathcal {L}}} \\tilde {\\alpha} (L) \\operatorname {h o m} (L, G) \\cdot \\operatorname {h o m} (L, K) = \\sum_ {L \\in \\tilde {\\mathcal {L}}} \\tilde {\\alpha} (L) \\operatorname {h o m} (L, H) \\cdot \\operatorname {h o m} (L, K). \\tag {31}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.868,
+                0.825,
+                0.898
+            ],
+            "angle": 0,
+            "content": "Now define vectors \\( \\pmb{p}_G \\) and \\( \\pmb{p}_H \\), where the \\( i^{th} \\) element of vector \\( \\pmb{p}_G \\) and \\( \\pmb{p}_H \\) is \\( \\tilde{\\alpha}(L_i) \\mathrm{hom}(L_i, G) \\) and \\( \\tilde{\\alpha}(L_i) \\mathrm{hom}(L_i, H) \\), respectively. We then have the following equation:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.413,
+                0.904,
+                0.584,
+                0.921
+            ],
+            "angle": 0,
+            "content": "\\[\n\\boldsymbol {A} ^ {\\text {h o m}} \\cdot \\boldsymbol {p} _ {G} = \\boldsymbol {A} ^ {\\text {h o m}} \\cdot \\boldsymbol {p} _ {H}.\n\\]"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "58"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.103,
+                0.825,
+                0.165
+            ],
+            "angle": 0,
+            "content": "Since \\( \\mathbf{A}^{\\mathrm{hom}} \\) is invertible, \\( \\pmb{p}_G^{\\mathrm{hom}} = \\pmb{p}_H^{\\mathrm{hom}} \\). Therefore, \\( \\tilde{\\alpha}(L)\\mathrm{hom}(L,G) = \\tilde{\\alpha}(L)\\mathrm{hom}(L,H) \\) for all \\( L \\in \\tilde{\\mathcal{L}} \\), namely, \\( \\mathrm{hom}(L,G) = \\mathrm{hom}(L,H) \\) for all \\( L \\in \\mathcal{L} \\). To sum up, we have proved that for all graph \\( G,H \\), \\( \\chi_G^M (G) = \\chi_H^M (H) \\) implies that \\( \\mathrm{hom}(L,G) = \\mathrm{hom}(L,H) \\) for all \\( L \\in \\mathcal{L} \\). By definition of homomorphism expressivity, we conclude that \\( \\mathcal{L} \\subset \\mathcal{F}^M \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.171,
+                0.825,
+                0.203
+            ],
+            "angle": 0,
+            "content": "Theorem G.6. Let \\( M \\in \\{\\mathrm{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\} \\) be any model. For any graph \\( F \\), if \\( \\mathrm{Spasm}(F) \\backslash \\mathcal{F}^M \\neq \\emptyset \\), then there exists a pair of graphs \\( G, H \\) such that \\( \\chi_G^M(G) = \\chi_H^M(H) \\) and \\( \\mathrm{sub}(F, G) \\neq \\mathrm{sub}(F, H) \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.217,
+                0.825,
+                0.266
+            ],
+            "angle": 0,
+            "content": "Proof. Recall that \\(\\operatorname{sub}(F, G) = \\sum_{\\tilde{F} \\in \\operatorname{Spasm}^{\\neq}(F)} \\alpha(F, \\tilde{F}) \\cdot \\operatorname{hom}(\\tilde{F}, G)\\) where \\(\\alpha(F, \\tilde{F}) \\neq 0\\) for all \\(\\tilde{F} \\in \\operatorname{Spasm}^{\\neq}(F)\\). If the above theorem does not hold, then for all graphs \\(G, H\\), \\(\\chi_G^M(G) = \\chi_H^M(H)\\) implies that"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.258,
+                0.27,
+                0.741,
+                0.307
+            ],
+            "angle": 0,
+            "content": "\\[\n\\sum_ {\\tilde {F} \\in \\operatorname {S p a s m} ^ {\\mathcal {Z}} (F)} \\alpha (F, \\tilde {F}) \\cdot \\operatorname {h o m} (\\tilde {F}, G) = \\sum_ {\\tilde {F} \\in \\operatorname {S p a s m} ^ {\\mathcal {Z}} (F)} \\alpha (F, \\tilde {F}) \\cdot \\operatorname {h o m} (\\tilde {F}, G).\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.312,
+                0.668,
+                0.328
+            ],
+            "angle": 0,
+            "content": "Then, Lemma G.5 implies that \\(\\operatorname{Spasm}(F) \\subset \\mathcal{F}^M\\), yielding a contradiction."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.808,
+                0.313,
+                0.824,
+                0.323
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.342,
+                0.825,
+                0.385
+            ],
+            "angle": 0,
+            "content": "We next extend our analysis to the node/edge-level subgraph counting. Since the proof techniques are almost the same as the graph-level setting, we only present key definitions and lemmas below while omitting the detailed proofs for clarity."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.387,
+                0.825,
+                0.433
+            ],
+            "angle": 0,
+            "content": "Definition G.7. Given two rooted graphs \\(G^{\\mathbf{u}}\\) and \\(H^{\\mathbf{v}}\\) with \\(\\mathbf{u} \\in V_G^m\\) and \\(\\mathbf{v} \\in V_H^m\\) for some \\(m \\in \\mathbb{N}_+\\), the categorical product of \\(G^{\\mathbf{u}}\\) and \\(H^{\\mathbf{v}}\\), denoted as \\((G \\times H)^{(u_1, v_1), \\dots, (u_m, v_m)}\\), is a rooted graph obtained from \\(G \\times H\\) by marking vertices \\((u_1, v_1), \\dots, (u_m, v_m)\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.434,
+                0.825,
+                0.464
+            ],
+            "angle": 0,
+            "content": "Lemma G.8 (Extension of Lemma G.2). For any rooted graphs \\( F^{\\mathbf{w}} \\), \\( G^{\\mathbf{u}} \\), \\( H^{\\mathbf{v}} \\) where \\( |\\mathbf{u}| = |\\mathbf{v}| = | \\mathbf{w} | = m \\),"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.255,
+                0.468,
+                0.741,
+                0.486
+            ],
+            "angle": 0,
+            "content": "\\[\n\\operatorname {h o m} \\left(F ^ {\\boldsymbol {w}}, (G \\times H) ^ {(u _ {1}, v _ {1}), \\dots , (u _ {m}, v _ {m})}\\right) = \\operatorname {h o m} \\left(F ^ {\\boldsymbol {w}}, G ^ {\\boldsymbol {u}}\\right) \\cdot \\operatorname {h o m} \\left(F ^ {\\boldsymbol {w}}, H ^ {\\boldsymbol {v}}\\right).\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.489,
+                0.825,
+                0.533
+            ],
+            "angle": 0,
+            "content": "Proposition G.9 (Extension of Proposition G.4). For any rooted graphs \\(G^u\\), \\(H^v\\) where \\(|\\boldsymbol{u}| = |\\boldsymbol{v}| = m\\), \\(\\operatorname{hom}(G^u, H^v) = \\sum_{F^w} \\operatorname{surj}(G^u, F^w) \\cdot \\operatorname{sub}(F^w, H^v)\\), where \\(F^w\\) ranges over all non-isomorphic rooted graphs marking \\(m\\) vertices."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.543,
+                0.825,
+                0.571
+            ],
+            "angle": 0,
+            "content": "We next present the main lemma for node-level subgraph counting. We omit the edge-level result for clarity."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.574,
+                0.825,
+                0.617
+            ],
+            "angle": 0,
+            "content": "Lemma G.10 (Extension of Lemma G.5). Let \\(M\\) be a GNN model such that its node-level homomorphism expressivity \\(\\mathcal{F}_{\\mathfrak{n}}^{M}\\) exists. Given a finite set of rooted graphs \\(\\mathcal{L}_{\\mathfrak{n}}\\) and a function \\(\\alpha : \\mathcal{L}_{\\mathfrak{n}} \\to \\mathbb{R} / \\{0\\}\\), if"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.22,
+                0.621,
+                0.776,
+                0.655
+            ],
+            "angle": 0,
+            "content": "\\[\n\\chi_ {G} ^ {M} (u) = \\chi_ {H} ^ {M} (v) \\Longrightarrow \\sum_ {L ^ {w} \\in \\mathcal {L} _ {\\mathrm {n}}} \\alpha (L ^ {w}) \\operatorname {h o m} (L ^ {w}, G ^ {u}) = \\sum_ {L ^ {w} \\in \\mathcal {L} _ {\\mathrm {n}}} \\alpha (L ^ {w}) \\operatorname {h o m} (L ^ {w}, H ^ {v}),\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.659,
+                0.544,
+                0.675
+            ],
+            "angle": 0,
+            "content": "holds for all rooted graphs \\( G^{u} \\) and \\( H^{v} \\), then \\( \\mathcal{L}_{\\mathfrak{n}} \\subset \\mathcal{F}_{\\mathfrak{n}}^{M} \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.677,
+                0.827,
+                0.722
+            ],
+            "angle": 0,
+            "content": "Theorem G.11. Let \\( M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\} \\) be any model. For any rooted graph \\( F^w \\), if \\( \\text{Spasm}(F^w) \\backslash \\mathcal{F}_\\mathfrak{n}^M \\neq \\emptyset \\), then there exist a pair of graphs \\( G, H \\) and vertices \\( u \\in V_G, v \\in V_H \\), such that \\( \\chi_G^M(u) = \\chi_H^M(v) \\) and \\( \\text{sub}(F^w, G^u) \\neq \\text{sub}(F^w, H^v) \\)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.736,
+                0.466,
+                0.751
+            ],
+            "angle": 0,
+            "content": "G.2 GRAPH STATISTICS AND EXAMPLES"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.762,
+                0.825,
+                0.831
+            ],
+            "angle": 0,
+            "content": "In this section, we list the statistics of all moderate-size graphs that can/cannot be counted for each model in Section 2 at graph/node/edge-level. Table 4 presents the statistics under homomorphism count, while Table 5 presents the statistics under subgraph count. These tables offer a clear picture into how large the expressivity gaps are between different models. Several important findings are discussed below:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.839,
+                0.825,
+                0.867
+            ],
+            "angle": 0,
+            "content": "- For homomorphism counting, graphs of 8 edges suffice to reveal the expressivity gaps between each pair of architectures (at edge-level)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.198,
+                0.869,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "- However, for subgraph counting, moderate-size graphs cannot reveal the gap between Local 2-GNN, Local 2-FGNN, and 2-FGNN. Moreover, even Subgraph GNN already matches the power of 2-FWL in counting small subgraphs at graph-level, but it is quite weak in counting subgraphs at node/edge-level."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.198,
+                0.839,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "59"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.033,
+                0.479,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.202,
+                0.104,
+                0.825,
+                0.148
+            ],
+            "angle": 0,
+            "content": "- Subgraph counting is much more challenging than homomorphism counting. Intuitively, this is because the homomorphism image of a graph usually has 4-cliques, and any graph that contains a 4 clique as subgraph cannot be counted under homomorphism."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.175,
+                0.153,
+                0.824,
+                0.224
+            ],
+            "angle": 0,
+            "content": "In Table 6, we list all subgraphs at moderate size (no more than 6 vertices or 8 edges) that can/cannot be counted by each GNN model. Here, we only list the graph-level expressivity as the node/edge-level expressivity involves too many non-isomorphic rooted graphs and cannot be fully presented (see Tables 4 and 5). We believe these results is comprehensive enough to cover most substructures of interest in the GNN community."
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.175,
+                0.262,
+                0.824,
+                0.304
+            ],
+            "angle": 0,
+            "content": "Table 4: The number of (connected) graphs (or rooted graphs) of \\( n \\) vertices or \\( m \\) edges that can be counted under homomorphism by different models. These statistics can be viewed as a quantitative expressivity comparison between models."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.174,
+                0.306,
+                0.827,
+                0.55
+            ],
+            "angle": 0,
+            "content": "<table><tr><td rowspan=\"2\" colspan=\"2\"></td><td colspan=\"5\">Number of vertices n</td><td colspan=\"7\">Number of edges m</td><td></td></tr><tr><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td></tr><tr><td rowspan=\"6\">Graph level</td><td>MPNN</td><td>1</td><td>1</td><td>2</td><td>3</td><td>6</td><td>1</td><td>1</td><td>2</td><td>3</td><td>6</td><td>11</td><td>23</td><td>47</td></tr><tr><td>Subgraph GNN</td><td>1</td><td>2</td><td>5</td><td>15</td><td>51</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>29</td><td>76</td><td>210</td></tr><tr><td>Local 2-GNN</td><td>1</td><td>2</td><td>5</td><td>15</td><td>55</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>29</td><td>77</td><td>216</td></tr><tr><td>Local 2-FGNN</td><td>1</td><td>2</td><td>5</td><td>15</td><td>56</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>29</td><td>77</td><td>216</td></tr><tr><td>2-FGNN</td><td>1</td><td>2</td><td>5</td><td>15</td><td>56</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>29</td><td>77</td><td>216</td></tr><tr><td>All</td><td>1</td><td>2</td><td>6</td><td>21</td><td>112</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>30</td><td>79</td><td>227</td></tr><tr><td rowspan=\"6\">Node level</td><td>MPNN</td><td>1</td><td>2</td><td>4</td><td>9</td><td>20</td><td>1</td><td>2</td><td>4</td><td>9</td><td>20</td><td>48</td><td>115</td><td>286</td></tr><tr><td>Subgraph GNN</td><td>1</td><td>3</td><td>8</td><td>27</td><td>88</td><td>1</td><td>2</td><td>5</td><td>12</td><td>31</td><td>83</td><td>228</td><td>640</td></tr><tr><td>Local 2-GNN</td><td>1</td><td>3</td><td>10</td><td>44</td><td>215</td><td>1</td><td>2</td><td>5</td><td>13</td><td>37</td><td>113</td><td>361</td><td>1210</td></tr><tr><td>Local 2-FGNN</td><td>1</td><td>3</td><td>10</td><td>44</td><td>217</td><td>1</td><td>2</td><td>5</td><td>13</td><td>37</td><td>113</td><td>361</td><td>1210</td></tr><tr><td>2-FGNN</td><td>1</td><td>3</td><td>10</td><td>44</td><td>217</td><td>1</td><td>2</td><td>5</td><td>13</td><td>37</td><td>113</td><td>361</td><td>1210</td></tr><tr><td>All</td><td>1</td><td>3</td><td>11</td><td>58</td><td>407</td><td>1</td><td>2</td><td>5</td><td>13</td><td>37</td><td>114</td><td>367</td><td>1248</td></tr><tr><td rowspan=\"5\">Edge level</td><td>Subgraph GNN</td><td>1</td><td>4</td><td>18</td><td>77</td><td>340</td><td>1</td><td>3</td><td>10</td><td>33</td><td>107</td><td>347</td><td>1126</td><td>3664</td></tr><tr><td>Local 2-GNN</td><td>1</td><td>4</td><td>21</td><td>116</td><td>693</td><td>1</td><td>3</td><td>10</td><td>35</td><td>124</td><td>450</td><td>1665</td><td>6267</td></tr><tr><td>Local 2-FGNN</td><td>1</td><td>4</td><td>21</td><td>118</td><td>735</td><td>1</td><td>3</td><td>10</td><td>35</td><td>124</td><td>451</td><td>1678</td><td>6373</td></tr><tr><td>2-FGNN</td><td>1</td><td>4</td><td>21</td><td>118</td><td>735</td><td>1</td><td>3</td><td>10</td><td>35</td><td>124</td><td>451</td><td>1678</td><td>6374</td></tr><tr><td>All</td><td>1</td><td>4</td><td>23</td><td>162</td><td>1549</td><td>1</td><td>3</td><td>10</td><td>35</td><td>125</td><td>460</td><td>1747</td><td>6830</td></tr></table>"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.175,
+                0.612,
+                0.824,
+                0.653
+            ],
+            "angle": 0,
+            "content": "Table 5: The number of (connected) graphs (or rooted graphs) of \\( n \\) vertices or \\( m \\) edges that can be subgraph-counted by different models. These statistics can be viewed as a quantitative expressivity comparison between models."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.174,
+                0.655,
+                0.827,
+                0.899
+            ],
+            "angle": 0,
+            "content": "<table><tr><td rowspan=\"2\" colspan=\"2\"></td><td colspan=\"5\">Number of vertices n</td><td colspan=\"7\">Number of edges m</td><td></td></tr><tr><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td></tr><tr><td rowspan=\"6\">Graph level</td><td>MPNN</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Subgraph GNN</td><td>1</td><td>2</td><td>5</td><td>13</td><td>24</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>20</td><td>21</td><td>22</td></tr><tr><td>Local 2-GNN</td><td>1</td><td>2</td><td>5</td><td>13</td><td>24</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>20</td><td>21</td><td>22</td></tr><tr><td>Local 2-FGNN</td><td>1</td><td>2</td><td>5</td><td>13</td><td>24</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>20</td><td>21</td><td>22</td></tr><tr><td>2-FGNN</td><td>1</td><td>2</td><td>5</td><td>13</td><td>24</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>20</td><td>21</td><td>22</td></tr><tr><td>All</td><td>1</td><td>2</td><td>6</td><td>21</td><td>112</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>30</td><td>79</td><td>227</td></tr><tr><td rowspan=\"6\">Node level</td><td>MPNN</td><td>1</td><td>2</td><td>2</td><td>2</td><td>2</td><td>1</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td></tr><tr><td>Subgraph GNN</td><td>1</td><td>3</td><td>8</td><td>18</td><td>27</td><td>1</td><td>2</td><td>5</td><td>10</td><td>16</td><td>23</td><td>30</td><td>38</td></tr><tr><td>Local 2-GNN</td><td>1</td><td>3</td><td>10</td><td>37</td><td>84</td><td>1</td><td>2</td><td>5</td><td>13</td><td>37</td><td>72</td><td>75</td><td>86</td></tr><tr><td>Local 2-FGNN</td><td>1</td><td>3</td><td>10</td><td>37</td><td>84</td><td>1</td><td>2</td><td>5</td><td>13</td><td>37</td><td>72</td><td>75</td><td>86</td></tr><tr><td>2-FGNN</td><td>1</td><td>3</td><td>10</td><td>37</td><td>84</td><td>1</td><td>2</td><td>5</td><td>13</td><td>37</td><td>72</td><td>75</td><td>86</td></tr><tr><td>All</td><td>1</td><td>3</td><td>11</td><td>58</td><td>407</td><td>1</td><td>2</td><td>5</td><td>13</td><td>37</td><td>114</td><td>367</td><td>1248</td></tr><tr><td rowspan=\"5\">Edge level</td><td>Subgraph GNN</td><td>1</td><td>4</td><td>18</td><td>47</td><td>81</td><td>1</td><td>3</td><td>10</td><td>25</td><td>46</td><td>69</td><td>95</td><td>124</td></tr><tr><td>Local 2-GNN</td><td>1</td><td>4</td><td>21</td><td>92</td><td>208</td><td>1</td><td>3</td><td>10</td><td>35</td><td>105</td><td>171</td><td>179</td><td>216</td></tr><tr><td>Local 2-FGNN</td><td>1</td><td>4</td><td>21</td><td>92</td><td>208</td><td>1</td><td>3</td><td>10</td><td>35</td><td>105</td><td>171</td><td>179</td><td>216</td></tr><tr><td>2-FGNN</td><td>1</td><td>4</td><td>21</td><td>92</td><td>208</td><td>1</td><td>3</td><td>10</td><td>35</td><td>105</td><td>171</td><td>179</td><td>216</td></tr><tr><td>All</td><td>1</td><td>4</td><td>23</td><td>162</td><td>1549</td><td>1</td><td>3</td><td>10</td><td>35</td><td>125</td><td>460</td><td>1747</td><td>6830</td></tr></table>"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.949,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "60"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.171,
+                0.075,
+                0.825,
+                0.132
+            ],
+            "angle": 0,
+            "content": "Table 6: The ability of GNNs to homomorphism-count and subgraph-count different graphs \\( F \\) within a bounded size of \\( n \\leq  6 \\) vertices or \\( m \\leq  8 \\) edges. When one or more GNNs fail to subgraphcount \\( F \\) ,this table also gives a homomorphic image \\( \\widetilde{F} \\in  \\operatorname{Spasm}\\left( F\\right) \\) that can be used to construct counterexample graphs (see Section 4.2)."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.18,
+                0.133,
+                0.818,
+                0.912
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>nmF</td><td>Homomorphism MP Sub L LF F</td><td>Subgraph MP Sub L LF F</td><td>F</td><td>nmF</td><td>Homomorphism MP Sub L LF F</td><td>Subgraph MP Sub L LF F</td><td>F</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>2 1←</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>-</td><td>5 7</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>3 2←</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>-</td><td>5 8</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>3 3</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>4 3</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>4 3</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>-</td><td>5 10</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>4 4</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 5</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>4 4</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 5</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>4 5</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 5</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>4 6</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>6 5</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 4</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 5</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 4</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 5</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 4</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>-</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 5</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 5</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 5</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 5</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>√</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 5</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 5</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✗</td><td>x</td></tr><tr><td>5 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr></table>"
+        },
+        {
+            "type": "table_footnote",
+            "bbox": [
+                0.443,
+                0.913,
+                0.555,
+                0.925
+            ],
+            "angle": 0,
+            "content": "(see the next page)"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.949,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "61"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.182,
+                0.101,
+                0.816,
+                0.911
+            ],
+            "angle": 270,
+            "content": "<table><tr><td>n m</td><td>F</td><td>Homomorphism</td><td>Subgraph</td><td>MP Sub L F</td><td>MP Sub L F</td><td>F</td><td>n m</td><td>F</td><td>Homomorphism</td><td>Subgraph</td><td>MP Sub L F</td><td>MP Sub L F</td><td>F</td><td>n m</td><td>F</td><td>Homomorphism</td><td>Subgraph</td><td>MP Sub L F</td><td>MP Sub L F</td><td>F</td></tr><tr><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>X</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>X</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>X</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td></tr><tr><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td></tr><tr><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td></tr><tr><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td></tr><tr><td>X</td><td>X</td><td>X</td><td>X</td><td>x</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td></tr><tr><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td></tr><tr><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>x</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td></tr></table>"
+        },
+        {
+            "type": "table_footnote",
+            "bbox": [
+                0.443,
+                0.912,
+                0.555,
+                0.924
+            ],
+            "angle": 0,
+            "content": "(see the next page)"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "62"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.182,
+                0.101,
+                0.815,
+                0.911
+            ],
+            "angle": 270,
+            "content": "<table><tr><td>n m F</td><td>Homomorphism Subgraph MPSubL F F</td><td>F</td><td>m F</td><td>Homomorphism Subgraph MPSubL F F</td><td>F</td></tr><tr><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>611</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>611</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>611</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>611</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>y</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>y</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>77</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>77</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>77</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>77</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>77</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>77</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>77</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr></table>"
+        },
+        {
+            "type": "table_footnote",
+            "bbox": [
+                0.443,
+                0.912,
+                0.555,
+                0.925
+            ],
+            "angle": 0,
+            "content": "(see the next page)"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "63"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.033,
+                0.479,
+                0.048
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.182,
+                0.101,
+                0.816,
+                0.911
+            ],
+            "angle": 270,
+            "content": "<table><tr><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>x</td><td>x</td><td>x</td><td>y</td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td></td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td></td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td></td><td></td><td></td></tr></table>"
+        },
+        {
+            "type": "table_footnote",
+            "bbox": [
+                0.443,
+                0.912,
+                0.555,
+                0.924
+            ],
+            "angle": 0,
+            "content": "(see the next page)"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "64"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.033,
+                0.479,
+                0.048
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.182,
+                0.101,
+                0.816,
+                0.911
+            ],
+            "angle": 270,
+            "content": "<table><tr><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>x</td><td>x</td><td>x</td><td></td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td></td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td></td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>x</td><td></td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td></td><td></td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td></td><td></td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td></td><td></td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td></td><td></td><td></td></tr></table>"
+        },
+        {
+            "type": "table_footnote",
+            "bbox": [
+                0.443,
+                0.912,
+                0.555,
+                0.924
+            ],
+            "angle": 0,
+            "content": "(see the next page)"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "65"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.182,
+                0.101,
+                0.816,
+                0.911
+            ],
+            "angle": 270,
+            "content": "<table><tr><td>n m</td><td>F</td><td>Homomorphism Subgraph MP Sub L F</td><td>MP Sub L F</td><td>F</td><td>n m</td><td>F</td><td>Homomorphism Subgraph MP Sub L F</td><td>MP Sub L F</td><td>F</td><td>n m</td><td>F</td><td>Homomorphism Subgraph MP Sub L F</td><td>MP Sub L F</td><td>F</td><td>n m</td><td>F</td><td>Homomorphism Subgraph MP Sub L F</td><td>MP Sub L F</td><td>F</td></tr><tr><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td></tr><tr><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td></td></tr><tr><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td></td></tr><tr><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td></td></tr><tr><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td></td></tr><tr><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td></td></tr><tr><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td></td></tr><tr><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td></td></tr><tr><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td></td></tr></table>"
+        },
+        {
+            "type": "table_footnote",
+            "bbox": [
+                0.443,
+                0.912,
+                0.555,
+                0.924
+            ],
+            "angle": 0,
+            "content": "(see the next page)"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.949,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "66"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.182,
+                0.101,
+                0.816,
+                0.911
+            ],
+            "angle": 270,
+            "content": "<table><tr><td>n m</td><td>F</td><td>Homomorphism</td><td>Subgraph</td><td>MP Sub L F F</td><td>MP Sub L F F</td></tr><tr><td>x</td><td>√</td><td>√</td><td>√</td><td>x</td><td>√</td></tr><tr><td>x</td><td>√</td><td>√</td><td>√</td><td>x</td><td>√</td></tr><tr><td>x</td><td>√</td><td>√</td><td>√</td><td>x</td><td>√</td></tr><tr><td>x</td><td>√</td><td>√</td><td>√</td><td>x</td><td>√</td></tr><tr><td>x</td><td>√</td><td>√</td><td>√</td><td>x</td><td>√</td></tr><tr><td>x</td><td>√</td><td>√</td><td>√</td><td>x</td><td>√</td></tr><tr><td>x</td><td>√</td><td>√</td><td>√</td><td>x</td><td>√</td></tr><tr><td>x</td><td>√</td><td>√</td><td>√</td><td>x</td><td>√</td></tr></table>"
+        },
+        {
+            "type": "table_footnote",
+            "bbox": [
+                0.443,
+                0.912,
+                0.555,
+                0.925
+            ],
+            "angle": 0,
+            "content": "(see the next page)"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "67"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.182,
+                0.106,
+                0.816,
+                0.917
+            ],
+            "angle": 270,
+            "content": 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m</td><td>F</td><td colspan=\"101\">Homonorphism Subgraph MP Sub L F F</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "68"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.103,
+                0.455,
+                0.119
+            ],
+            "angle": 0,
+            "content": "H POLYNOMIAL EXPRESSIVITY"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.17,
+                0.134,
+                0.827,
+                0.274
+            ],
+            "angle": 0,
+            "content": "Puny et al. (2023) proposed the equivariant graph polynomials, which are polynomials \\( P \\) that take real squared matrices \\( \\mathbf{X} \\in \\mathbb{R}^{n \\times n} \\) as input and outputs \\( P(\\mathbf{X}) \\in \\mathbb{R}^{n \\times n} \\), such that \\( P \\) is equivariant under permutations, i.e., \\( P(\\pi \\cdot \\mathbf{X}) = \\pi \\cdot P(\\mathbf{X}) \\) for all permutation \\( \\pi \\in S_n \\). The authors gave a concrete basis of equivariant polynomials, where each basis element \\( P_{F^{uv}} \\) corresponds to a rooted multi-graph \\( F^{uv} \\) marking two vertices \\( u, v \\). Puny et al. (2023) showed that when \\( \\mathbf{X} \\) is restricted to be the adjacency matrix of an undirected simple graph \\( G \\), each \\( F^{uv} \\) will also reduce to an undirected simple graph, and \\( P_{F^{uv}}(\\mathbf{X}) \\) precisely computes the (unlabeled) homomorphism count \\( \\mathrm{hom}(F^{uv}, G^{wx}) \\) for all \\( w, x \\in V_G \\). Besides the original definition, Puny et al. (2023) also proposed the invariant graph polynomials and node-level equivariant graph polynomials, which are similarly related to the graph-level and node-level homomorphism count."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.28,
+                0.825,
+                0.351
+            ],
+            "angle": 0,
+            "content": "As a direct consequence, if a GNN model \\( M \\) cannot count graph \\( F^{uv} / F^u /F \\) under homomorphism, it then cannot compute the equivariant/invariant graph polynomial \\( P_{F^{uv}} / P_{F^u} / P_F \\). Based on these connections, our results can be directly used to provide insights into which equivariant graph polynomials cannot be computed by model \\( M \\). This recovers several results in their paper and answers an open problem shown below."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.355,
+                0.827,
+                0.398
+            ],
+            "angle": 0,
+            "content": "Corollary H.1. MPNN and \\( PPGN++ \\) are bounded by the Prototypical node-based model and Prototypical edge-based model (defined in Puny et al. (2023)) for computing node-level and edge-level equivariant graph polynomials, respectively."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.414,
+                0.825,
+                0.527
+            ],
+            "angle": 0,
+            "content": "Proof. Without loss of generality, we assume that the corresponding graphs of all equivariant polynomials are connected. According to Puny et al. (2023, Proposition H.2), the Prototypical node-based model can compute all \\( P_{F^u} \\) where \\( F^u \\) is a (rooted) tree and cannot compute other graph polynomials. If MPNN is not bounded by the Prototypical node-based model, then it can compute some \\( P_{\\tilde{F}^u} \\) where \\( \\tilde{F} \\) is not a tree. However, this is impossible since MPNN can only count forests under homomorphism according to Theorem 3.7. Note that the MPNN defined in their paper is equivalent to our definition when only considering connected graphs (the extra global aggregation \\( 11^\\top X \\) in their definition (9) does not improve the homomorphism expressivity)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.533,
+                0.827,
+                0.682
+            ],
+            "angle": 0,
+            "content": "We next turn to \\(\\mathrm{PPGN}++\\), and the proof is similar (but more involved). We first show that the Prototypical edge-based model can compute any \\(P_{F^{uv}}\\) satisfying that the treewidth of the graph \\(\\tilde{F} := (V_F, E_F \\cup \\{\\{u, v\\}\\}, \\ell_F)\\) is no more than 2. If \\(\\mathsf{tw}(\\tilde{F}) \\leq 2\\), it is a partial 2-tree. Thus, there is an ordering \\(w_1, \\dots, w_n\\) of the vertex set \\(V_F\\) such that when deleting each vertex \\(w_i\\) and all incident edges in turn, we only ever delete vertices of degree at most 2. Now we claim that we can always order the two vertices \\(u, v\\) at the end, i.e., \\(w_{n-1} = u\\) and \\(w_n = v\\). Otherwise, there is a subset \\(U \\subset V_F\\) such that all vertices in the induced subgraph \\(\\tilde{F}[U]\\) are of degree at least 3 except \\(u, v\\). It follows that the \\(\\mathsf{tw}(\\tilde{F}[U]) \\geq \\mathsf{tw}(H) \\geq 3\\) where graph \\(H\\) is the graph obtained from \\(\\tilde{F}[U]\\) by contracting \\(u\\) (or \\(v\\)) if \\(\\deg_{\\tilde{F}[U]}(u) \\leq 2\\) (or \\(\\deg_{\\tilde{F}[U]}(v) \\leq 2\\)). This yields a contradiction and verifies the claim that we can always set \\(w_{n-1} = u\\) and \\(w_n = v\\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.687,
+                0.827,
+                0.807
+            ],
+            "angle": 0,
+            "content": "Now, following the proof in Puny et al. (2023, Proposition H.3), the Prototypical edge-based model can contract \\(\\tilde{F}^{uv}\\) to a graph with only two vertices \\(u, v\\) and thus can compute the edge-level polynomial \\(P_{\\tilde{F}^{uv}}\\). If PPGN++ is not bounded by the Prototypical edge-based model, then it can compute some \\(P_{\\tilde{F}^{uv}}\\) where \\(\\mathrm{tw}(\\tilde{F}) \\geq 3\\). Therefore, it can count the graph \\(\\tilde{F}\\) under homomorphism at graph-level (since it can already count \\(\\tilde{F}\\) at edge-level). This implies that PPGN++ is strictly more expressive than 2-FGNN (2-FWL) because we have proved that all graphs in \\(\\mathcal{F}^{\\mathrm{F}}\\) have a treewidth no more than 2. This yields a contradiction since PPGN++ is still bounded by 2-FWL in distinguishing non-isomorphic graphs."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.823,
+                0.6,
+                0.838
+            ],
+            "angle": 0,
+            "content": "We also provide insights into the following results in their paper:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.842,
+                0.825,
+                0.871
+            ],
+            "angle": 0,
+            "content": "Corollary H.2. The Prototypical node-based model is not 3-node-polynomial-exact. The Prototypical edge-based model is not 6-node-polynomial-exact and not 5-edge-polynomial-exact."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.882,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "This is simply because the triangle is not a tree and the 4-clique does not have a NED (or equivalently, the treewidth of a 4-clique is 3). It is also clear why the degree of the edge-based polynomial is 6, which is one less than that of the node-based polynomial using the concept of NED."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.51,
+                0.96
+            ],
+            "angle": 0,
+            "content": "69"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.103,
+                0.416,
+                0.119
+            ],
+            "angle": 0,
+            "content": "I EXPERIMENTAL DETAILS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.135,
+                0.615,
+                0.15
+            ],
+            "angle": 0,
+            "content": "In this section, we provide all the experimental details in Section 5."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.167,
+                0.287,
+                0.182
+            ],
+            "angle": 0,
+            "content": "I.1 DATASETS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.193,
+                0.825,
+                0.235
+            ],
+            "angle": 0,
+            "content": "We conduct experiments on tive tasks: (i) graph homomorphism counting, (ii) subgraph counting, (iii) ZINC-subset (Dwivedi et al., 2020), (iii) ZINC-full (Dwivedi et al., 2020), and (iv) Alchemy (Chen et al., 2019a)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.242,
+                0.827,
+                0.466
+            ],
+            "angle": 0,
+            "content": "Homomorphism/Subgraph Counting. For both homomorphism and subgraph counting tasks, we use the standard synthetic graph dataset constructed in Zhao et al. (2022a) (which has been used in a number of papers, see e.g., Frasca et al., 2022; Huang et al., 2023; Zhang et al., 2023a). For homomorphism counting, we count the number of graph/node/edge-level homomorphisms for each pattern in Table 1 and normalize the value by the mean and variance across all graphs in the dataset. The evaluation metric of graph-level expressivity is chosen as the Mean Absolute Error (MAE). For node/edge-level expressivity, the error on each graph is defined to be the sum of absolute error over all vertices/edges. We then report the MAE across all graphs in the dataset. This ensures that graph/node/edge-level errors are roughly at the same scale (since \\(\\mathrm{hom}(F,G) = \\sum_{w\\in V_G}\\mathrm{hom}(F^u,G^w) = \\sum_{w,x\\in V_G}\\mathrm{hom}(F^{uv},G^{wx})\\) for all \\(u,v\\in V_F\\)). For subgraph counting, the data processing and evaluation metric is similar to homomorphism counting, but there is a slight difference in the node/edge-level setting: there are no marked vertices in the pattern graph \\(F\\) (see Table 3). Instead, given a graph \\(G\\) in the dataset and a vertex \\(w\\in V_G\\), we count the number of subgraphs containing \\(w\\) that are isomorphic to \\(F\\) and \\(w\\) can be mapped to any vertex in \\(F\\). Due to this difference, in the node/edge-level setting, the error on each graph is defined to be the average of absolute error over all vertices/edges."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.472,
+                0.825,
+                0.543
+            ],
+            "angle": 0,
+            "content": "ZINC. ZINC (Dwivedi et al., 2020) is a standard real-world dataset for benchmarking molecular property prediction. The dataset consists of 250K molecular graphs, and the task is to predict the constrained solubility of the given molecule. In addition to the full dataset (denoted as ZINC-full), ZINC-subset is a sampled dataset with 12k molecules from the ZINC-full dataset. We train and test our models on both datasets following the standard protocol from Dwivedi et al. (2020)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.549,
+                0.827,
+                0.605
+            ],
+            "angle": 0,
+            "content": "Alchemy. Alchemy (Chen et al., 2019a) is another real-world dataset with 12 graph-level quantum mechanical properties. We follow the sampling and training protocol from Lim et al. (2023); Puny et al. (2023), using 100K samples for training, 10K samples for testing, and 10K samples for validation."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.624,
+                0.33,
+                0.637
+            ],
+            "angle": 0,
+            "content": "I.2 MODEL DETAILS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.649,
+                0.825,
+                0.707
+            ],
+            "angle": 0,
+            "content": "All models are implemented using the PyTorch (Paszke et al., 2019) framework and the PyTorch Geometric library (Fey & Lenssen, 2019). We consider four types of GNNs defined in Section 2: MPNN, Subgraph GNN, Local 2-GNN, and Local 2-FGNN. For each GNN model, the feature initialization, message-passing layers, and final pooling operation are separately defined below."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.712,
+                0.827,
+                0.876
+            ],
+            "angle": 0,
+            "content": "Initialization. On both ZINC and Alchemy datasets, each graph node is an atom. We maintain a learnable atom embedding for each type of atom and use it to initialize features in GNN models. For MPNN, the initial feature \\( h^{(0)}(u) \\) of node \\( u \\) is simply the atom embedding, denoted as \\( h^{(0)}(u) = e_{\\mathrm{atom}(u)}^{\\mathrm{N}} \\). For other models, the initial feature \\( h^{(0)}(u,v) \\) of node pair \\( (u,v) \\) consists of two parts: the first part is the node embedding of \\( v \\), and the second part is a distance encoding that embeds the shortest path distance between \\( u \\) and \\( v \\), as adopted in Zhang et al. (2023a). We note that while incorporating distance encoding does not increase the models' theoretical expressive power (see Zhang et al. (2023a)), it may add an inductive bias that can be helpful in real-world tasks. Formally, the initial feature can be written as \\( h^{(0)}(u,v) = [e_{\\mathrm{atom}(v)}^{\\mathrm{N}}, e_{\\mathrm{clip}(\\mathrm{dis}(u,v))}^{\\mathrm{D}}] \\). Here, we clip the distance to a predefined value max_dis so that there are a finite number of distance embeddings, and distances greater than the hyper-parameter max_dis (including the disconnected case) share the embedding."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.881,
+                0.825,
+                0.926
+            ],
+            "angle": 0,
+            "content": "Propagation. On both ZINC and Alchemy datasets, each edge in a graph corresponds to a chemical bond and has a bond type. We maintain a learnable edge embedding for each type of edges in each layer and denote the embedding of edge \\(\\{u,v\\}\\) in layer \\(l\\) as \\(g^{(l)}(u,v)\\). For MPNN, we use the"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.51,
+                0.96
+            ],
+            "angle": 0,
+            "content": "70"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.741,
+                0.12
+            ],
+            "angle": 0,
+            "content": "standard GIN architecture proposed in Xu et al. (2019), which has the following form:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.241,
+                0.126,
+                0.825,
+                0.145
+            ],
+            "angle": 0,
+            "content": "\\[\nh ^ {(l + 1)} (u) = \\operatorname {R e L U} \\left(\\mathsf {B N} ^ {(l)} \\left(f ^ {(l)} (u)\\right)\\right), \\tag {32}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.258,
+                0.148,
+                0.826,
+                0.197
+            ],
+            "angle": 0,
+            "content": "\\[\nf ^ {(l)} (u) = \\mathsf {G I N} ^ {(l)} \\left(h ^ {(l)} (u), \\sum_ {v \\in N _ {G} (u)} \\operatorname {R e L U} \\left(\\mathsf {F C} ^ {(l)} \\left(h ^ {(l)} (v)\\right) + g ^ {(l)} (u, v)\\right)\\right), \\tag {33}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.172,
+                0.204,
+                0.218,
+                0.216
+            ],
+            "angle": 0,
+            "content": "where"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.352,
+                0.213,
+                0.826,
+                0.24
+            ],
+            "angle": 0,
+            "content": "\\[\n\\operatorname {G I N} ^ {(l)} (\\boldsymbol {x}, \\boldsymbol {y}) = \\operatorname {M L P} ^ {(l)} \\left((1 + \\epsilon^ {(l)}) \\boldsymbol {x} + \\boldsymbol {y}\\right). \\tag {34}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.244,
+                0.825,
+                0.291
+            ],
+            "angle": 0,
+            "content": "Here, \\(\\mathsf{FC}^{(l)}\\) is a parameterized linear transformation, \\(\\epsilon^{(l)}\\) is a learnable parameter, \\(\\mathsf{BN}^{(l)}\\) is the batch normalization (Ioffe & Szegedy, 2015), and \\(\\mathsf{MLP}^{(l)}\\) is a two-layer feed-forward network with another batch normalization in the hidden layer."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.297,
+                0.697,
+                0.312
+            ],
+            "angle": 0,
+            "content": "For other architectures, the \\(l\\)-th GNN layer analogously has the following form:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.36,
+                0.319,
+                0.826,
+                0.339
+            ],
+            "angle": 0,
+            "content": "\\[\nh ^ {(l + 1)} (u, v) = \\operatorname {R e L U} \\left(\\mathsf {B N} ^ {(l)} \\left(f ^ {(l)} (u, v)\\right)\\right), \\tag {35}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.347,
+                0.586,
+                0.364
+            ],
+            "angle": 0,
+            "content": "where the term \\( f^{(l)}(u,v) \\) is defined separately for each model:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.199,
+                0.371,
+                0.332,
+                0.386
+            ],
+            "angle": 0,
+            "content": "- Subgraph GNN:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.227,
+                0.392,
+                0.826,
+                0.442
+            ],
+            "angle": 0,
+            "content": "\\[\nf ^ {(l)} (u, v) = \\mathsf {G I N} ^ {(l)} \\left(h ^ {(l)} (u, v), \\sum_ {w \\in N _ {G} (v)} \\operatorname {R e L U} \\left(\\mathsf {F C} ^ {(l)} \\left(h ^ {(l)} (u, w)\\right) + g ^ {(l)} (w, v)\\right)\\right). \\tag {36}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.199,
+                0.45,
+                0.316,
+                0.464
+            ],
+            "angle": 0,
+            "content": "- Local 2-GNN:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.223,
+                0.469,
+                0.825,
+                0.581
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} f ^ {(l)} (u, v) = \\mathsf {G I N} ^ {(l, 1)} \\left(h ^ {(l)} (u, v), \\sum_ {w \\in N _ {G} (u)} \\operatorname {R e L U} \\left(\\mathsf {F C} ^ {(l, 1)} \\left(h ^ {(l)} (w, v)\\right) + g ^ {(l)} (u, w)\\right)\\right) \\\\ + \\operatorname {G I N} ^ {(l, 2)} \\left(h ^ {(l)} (u, v), \\sum_ {w \\in N _ {G} (v)} \\operatorname {R e L U} \\left(\\mathsf {F C} ^ {(l, 2)} \\left(h ^ {(l)} (u, w)\\right) + g ^ {(l)} (w, v)\\right)\\right). \\tag {37} \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.199,
+                0.582,
+                0.325,
+                0.595
+            ],
+            "angle": 0,
+            "content": "- Local 2-FGNN:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.218,
+                0.602,
+                0.865,
+                0.735
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} f ^ {(l)} (u, v) \\\\ = \\mathsf {G I N} ^ {(l, 1)} \\left(h ^ {(l)} (u, v), \\sum_ {w \\in N _ {G} (u)} \\mathsf {R e L U} \\left(\\mathsf {F C} ^ {(l, 1)} \\left(h ^ {(l)} (u, w)\\right) + \\mathsf {F C} ^ {(l, 2)} \\left(h ^ {(l)} (w, v)\\right) + g ^ {(l)} (u, w)\\right)\\right) \\\\ + \\mathsf {G I N} ^ {(l, 2)} \\left(h ^ {(l)} (u, v), \\sum_ {w \\in N _ {G} (v)} \\operatorname {R e L U} \\left(\\mathsf {F C} ^ {(l, 1)} \\left(h ^ {(l)} (w, v)\\right) + \\mathsf {F C} ^ {(l, 2)} \\left(h ^ {(l)} (u, w)\\right) + g ^ {(l)} (w, v)\\right)\\right). \\tag {38} \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.742,
+                0.827,
+                0.84
+            ],
+            "angle": 0,
+            "content": "For all the above GNN architectures, it can be seen that each layer only aggregates the local neighborhood of vertices or vertex pairs. This design will have shortcomings for disconnected graphs since a vertex cannot aggregate information from other connected components no matter how deep the model is (Barceló et al., 2020). Note that there do exist disconnected graphs in real-world datasets like ZINC. Therefore, on real-world datasets like ZINC and Alchemy, we also incorporate a global aggregation with the following form for each layer (similar to the global aggregation in Frasca et al. (2022); Zhang et al. (2023a)):"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.367,
+                0.847,
+                0.826,
+                0.889
+            ],
+            "angle": 0,
+            "content": "\\[\n\\operatorname {G I N} ^ {(l, G)} \\left(h ^ {(l)} (u, v), \\sum_ {w \\in V _ {G}} h ^ {(l)} (u, w)\\right) \\tag {39}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.896,
+                0.825,
+                0.925
+            ],
+            "angle": 0,
+            "content": "Note that the global aggregation does not increase model's theoretical expressive power according to Zhang et al. (2023a)."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.508,
+                0.96
+            ],
+            "angle": 0,
+            "content": "71"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.173,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.104,
+                0.825,
+                0.133
+            ],
+            "angle": 0,
+            "content": "Pooling. Except for edge-level tasks, a final pooling layer is used to produce node-level features \\( h(u) \\) for all nodes \\( u \\). It is implemented as follows:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.386,
+                0.141,
+                0.826,
+                0.183
+            ],
+            "angle": 0,
+            "content": "\\[\nh (u) = \\operatorname {M L P} \\left(\\sum_ {v \\in \\mathcal {V}} h ^ {(L)} (u, v)\\right), \\tag {40}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.189,
+                0.825,
+                0.218
+            ],
+            "angle": 0,
+            "content": "where MLP is a 2-layer perceptron. For graph-level tasks, we further use a mean pooling layer to aggregate all \\( h(u) \\) and obtain the graph representation."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.235,
+                0.347,
+                0.248
+            ],
+            "angle": 0,
+            "content": "I.3 TRAINING DETAILS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.26,
+                0.825,
+                0.358
+            ],
+            "angle": 0,
+            "content": "All experiments are run on a single NVIDIA Tesla V100 GPU. For all tasks, we use the distance encoding hyper-parameter max_dis = 5. To enable a fair comparison between models, for each task we keep the same depth for different models while varying the hidden dimension so that the number of model parameters is roughly the same. Note that the dimensions are chosen such that all models roughly obey the 500K parameter budget in ZINC and Alchemy. The hidden dimension size and the number of model parameters are listed as follows. All models are trained using the Adam optimizer."
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.373,
+                0.383,
+                0.622,
+                0.396
+            ],
+            "angle": 0,
+            "content": "Table 7: Model size in different tasks."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.233,
+                0.397,
+                0.766,
+                0.478
+            ],
+            "angle": 0,
+            "content": "<table><tr><td rowspan=\"2\">Task</td><td colspan=\"3\">Hidden dimension</td><td colspan=\"3\"># Parameters</td></tr><tr><td>Counting</td><td>ZINC</td><td>Alchemy</td><td>Counting</td><td>ZINC</td><td>Alchemy</td></tr><tr><td>MPNN</td><td>128</td><td>150</td><td>150</td><td>314,119</td><td>510,158</td><td>509,719</td></tr><tr><td>Subgraph GNN</td><td>128</td><td>120</td><td>120</td><td>314,759</td><td>503,774</td><td>503,425</td></tr><tr><td>Local 2-GNN</td><td>96</td><td>96</td><td>96</td><td>317,388</td><td>495,188</td><td>494,911</td></tr><tr><td>Local 2-FGNN</td><td>96</td><td>96</td><td>96</td><td>317,388</td><td>495,188</td><td>494,911</td></tr></table>"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.499,
+                0.825,
+                0.584
+            ],
+            "angle": 0,
+            "content": "Homomorphism/Subgraph Counting. We use a model depth of \\( L = 5 \\) in all experiments. Following prior work (Huang et al., 2023), we remove all BN layers in all models. The initial learning rate is chosen as 0.001 and is decayed by a factor of 0.9 once the MAE on the validation set plateaus for 10 epochs. Each model is trained for 1200 epochs with a batch size of 512. We ran each experiment 4 times independently with different seeds and reported the average performance at the last epoch. We found that the standard deviation among different seeds is negligible."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.59,
+                0.825,
+                0.687
+            ],
+            "angle": 0,
+            "content": "ZINC. We use a model depth of \\( L = 6 \\) in all experiments for both ZINC-subset and ZINC-full. Following prior work (Zhang et al., 2023a; Frasca et al., 2022), The initial learning rate is chosen as 0.001 and is decayed by a factor of 0.5 once the MAE on the validation set plateaus for 20 epochs. Each model is trained for 400 epochs on ZINC-subset and 500 epochs on ZINC-full, both with a batch size of 128. We report the MAE for the model checkpoint with the best validation performance. We ran each experiment 10 times independently with different seeds and reported the average performance as well as the standard deviation."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.694,
+                0.825,
+                0.779
+            ],
+            "angle": 0,
+            "content": "Alchemy. We use a model depth of \\( L = 6 \\) in all experiments. Following prior work (Lim et al., 2023; Puny et al., 2023), The initial learning rate is chosen as 0.002 and is decayed by a factor of 0.5 once the MAE on the validation set plateaus for 20 epochs. Each model is trained for 500 epochs with a batch size of 128. We report the MAE for the model checkpoint with the best validation performance. We ran each experiment 10 times independently with different seeds and reported the average performance as well as the standard deviation."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.172,
+                0.795,
+                0.588,
+                0.809
+            ],
+            "angle": 0,
+            "content": "I.4 PERFORMANCE OF BASELINE MODELS IN LITERATURE"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.171,
+                0.821,
+                0.825,
+                0.864
+            ],
+            "angle": 0,
+            "content": "For completeness, in this subsection we give a comprehensive list of the performance of GNN models in the literature on ZINC and Alchemy datasets. The numbers in each table below are directly taken from the original papers."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.49,
+                0.948,
+                0.509,
+                0.96
+            ],
+            "angle": 0,
+            "content": "72"
+        }
+    ],
+    [
+        {
+            "type": "header",
+            "bbox": [
+                0.174,
+                0.033,
+                0.48,
+                0.049
+            ],
+            "angle": 0,
+            "content": "Published as a conference paper at ICLR 2024"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.203,
+                0.141,
+                0.794,
+                0.156
+            ],
+            "angle": 0,
+            "content": "Table 8: Performance of different GNN models on ZINC dataset reported in the literature."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.243,
+                0.157,
+                0.756,
+                0.672
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Method</td><td>Model</td><td>Reference</td><td>Test MAE</td></tr><tr><td rowspan=\"8\">MPNN</td><td>GIN</td><td>(Xu et al., 2019)</td><td>0.526±0.051</td></tr><tr><td>GraphSAGE</td><td>(Hamilton et al., 2017)</td><td>0.398±0.002</td></tr><tr><td>GAT</td><td>(Veličković et al., 2018)</td><td>0.384±0.007</td></tr><tr><td>GCN</td><td>(Kipf &amp; Welling, 2017)</td><td>0.367±0.011</td></tr><tr><td>MoNet</td><td>(Monti et al., 2017)</td><td>0.292±0.006</td></tr><tr><td>GatedGCN-PE</td><td>(Bresson &amp; Laurent, 2017)</td><td>0.214±0.006</td></tr><tr><td>MPNN(sum)</td><td>(Gilmer et al., 2017)</td><td>0.145±0.007</td></tr><tr><td>PNA</td><td>(Corso et al., 2020)</td><td>0.142±0.010</td></tr><tr><td rowspan=\"4\">Higher-order GNN</td><td>RingGNN</td><td>(Chen et al., 2019b)</td><td>0.353±0.019</td></tr><tr><td>PPGN</td><td>(Maron et al., 2019a)</td><td>0.303±0.068</td></tr><tr><td>PPGN</td><td>(Puny et al., 2023)</td><td>0.079±0.005</td></tr><tr><td>PPGN++</td><td>(Puny et al., 2023)</td><td>0.076±0.003</td></tr><tr><td rowspan=\"7\">Subgraph GNN</td><td>NGNN</td><td>(Zhang &amp; Li, 2021)</td><td>0.111±0.003</td></tr><tr><td>GNN-AK</td><td>(Zhao et al., 2022a)</td><td>0.105±0.010</td></tr><tr><td>GNN-AK+</td><td>(Zhao et al., 2022a)</td><td>0.091±0.002</td></tr><tr><td>ESAN</td><td>(Bevilacqua et al., 2022)</td><td>0.102±0.003</td></tr><tr><td>SUN</td><td>(Frasca et al., 2022)</td><td>0.083±0.003</td></tr><tr><td>I2-GNN</td><td>(Huang et al., 2023)</td><td>0.083±0.001</td></tr><tr><td>ID-MPNN</td><td>(Zhou et al., 2023a)</td><td>0.083±0.003</td></tr><tr><td rowspan=\"4\">Local (F)GNN</td><td>SetGNN</td><td>(Zhao et al., 2022b)</td><td>0.075±0.003</td></tr><tr><td>GNN-SSWL</td><td>(Zhang et al., 2023a)</td><td>0.082±0.010</td></tr><tr><td>GNN-SSWL+</td><td>(Zhang et al., 2023a)</td><td>0.070±0.005</td></tr><tr><td>N2-GNN</td><td>(Feng et al., 2023)</td><td>0.059±0.002</td></tr><tr><td rowspan=\"4\">Substructure-based GNN</td><td>GSN</td><td>(Bouritsas et al., 2022)</td><td>0.101±0.010</td></tr><tr><td>CIN (Small)</td><td>(Bodnar et al., 2021a)</td><td>0.094±0.004</td></tr><tr><td>CIN</td><td>(Bodnar et al., 2021a)</td><td>0.079±0.006</td></tr><tr><td>CIN++</td><td>(Giusti et al., 2023)</td><td>0.077±0.004</td></tr><tr><td rowspan=\"6\">Graph Transformer</td><td>SAN</td><td>(Kreuzer et al., 2021)</td><td>0.139±0.006</td></tr><tr><td>K-Subgraph SAT</td><td>(Chen et al., 2022)</td><td>0.094±0.008</td></tr><tr><td>Graphormer</td><td>(Ying et al., 2021)</td><td>0.122±0.006</td></tr><tr><td>URPE</td><td>(Luo et al., 2022)</td><td>0.086±0.007</td></tr><tr><td>Graphormer-GD</td><td>(Zhang et al., 2023b)</td><td>0.081±0.009</td></tr><tr><td>GPS</td><td>(Rampasek et al., 2022)</td><td>0.070±0.004</td></tr><tr><td rowspan=\"6\">Other</td><td>PF-GNN</td><td>(Dupty et al., 2021)</td><td>0.122±0.010</td></tr><tr><td>KP-GIN</td><td>(Feng et al., 2022)</td><td>0.093±0.007</td></tr><tr><td>SignNet</td><td>(Lim et al., 2023)</td><td>0.084±0.006</td></tr><tr><td>PathNN</td><td>(Michel et al., 2023)</td><td>0.090±0.004</td></tr><tr><td>PPGN++(6)</td><td>(Puny et al., 2023)</td><td>0.071±0.001</td></tr><tr><td>PlanE</td><td>(Dimitrov et al., 2023)</td><td>0.076±0.003</td></tr></table>"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.192,
+                0.746,
+                0.804,
+                0.761
+            ],
+            "angle": 0,
+            "content": "Table 9: Performance of different GNN models on Alchemy dataset reported in the literature."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.324,
+                0.763,
+                0.675,
+                0.895
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Model</td><td>Reference</td><td>Test MAE</td></tr><tr><td>GIN</td><td>(Xu et al., 2019)</td><td>0.180±0.006</td></tr><tr><td>PF-GNN</td><td>(Dupty et al., 2021)</td><td>0.111±0.010</td></tr><tr><td>δ-2-GNN</td><td>(Morris et al., 2020)</td><td>0.118±0.001</td></tr><tr><td>Recon-GNN</td><td>(Cotta et al., 2021)</td><td>0.125±0.001</td></tr><tr><td>SpeqNet</td><td>(Morris et al., 2022)</td><td>0.115±0.001</td></tr><tr><td>SignNet</td><td>(Lim et al., 2023)</td><td>0.113±0.002</td></tr><tr><td>PPGN</td><td>(Puny et al., 2023)</td><td>0.113±0.001</td></tr><tr><td>PPGN++</td><td>(Puny et al., 2023)</td><td>0.111±0.002</td></tr><tr><td>PPGN++(6)</td><td>(Puny et al., 2023)</td><td>0.109±0.001</td></tr></table>"
+        },
+        {
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+# BEYOND WEISFEILER-LEHMAN: A QUANTITATIVE FRAMEWORK FOR GNN EXPRESSIVENESS
+
+Bohang Zhang $^{1* \dagger}$  Jingchu Gai $^{2*}$  Yiheng Du $^{3}$  Qiwei Ye $^{4}$  Di He $^{1 \boxtimes}$  Liwei Wang $^{1,5 \boxtimes}$
+
+$^{1}$ National Key Laboratory of General Artificial Intelligence, SIST, Peking University
+
+$^{2}$ School of Mathematical Science, Peking University  $^{3}$ Yuanpei College, Peking University
+
+$^{4}$ Beijing Academy of Artificial Intelligence  $^{5}$ Center for Machine Learning Research, Peking University zhangbohang@pku.edu.cn, {gaijingchu,duyiheng}@stu.pku.edu.cn
+
+qwye@baai.ac.cn, {dihe, wanglw}@pku.edu.cn
+
+# ABSTRACT
+
+Designing expressive Graph Neural Networks (GNNs) is a fundamental topic in the graph learning community. So far, GNN expressiveness has been primarily assessed via the Weisfeiler-Lehman (WL) hierarchy. However, such an expressivity measure has notable limitations: it is inherently coarse, qualitative, and may not well reflect practical requirements (e.g., the ability to encode substructures). In this paper, we introduce a novel framework for quantitatively studying the expressiveness of GNN architectures, addressing all the above limitations. Specifically, we identify a fundamental expressivity measure termed homomorphism expressivity, which quantifies the ability of GNN models to count graphs under homomorphism. Homomorphism expressivity offers a complete and practical assessment tool: the completeness enables direct expressivity comparisons between GNN models, while the practicality allows for understanding concrete GNN abilities such as subgraph counting. By examining four classes of prominent GNNs as case studies, we derive simple, unified, and elegant descriptions of their homomorphism expressivity for both invariant and equivariant settings. Our results provide novel insights into a series of previous work, unify the landscape of different subareas in the community, and settle several open questions. Empirically, extensive experiments on both synthetic and real-world tasks verify our theory, showing that the practical performance of GNN models aligns well with the proposed metric.
+
+# 1 INTRODUCTION
+
+Owing to the ubiquity of graph-structured data in numerous applications, Graph Neural Networks (GNNs) have achieved enormous success in the field of machine learning over the past few years. However, one of the most prominent drawbacks of popular GNNs lies in the limited expressive power. In particular, Morris et al. (2019); Xu et al. (2019) showed that Message Passing GNNs (MPNNs) are intrinsically bounded by the 1-dimensional Weisfeiler-Lehman test (1-WL) in distinguishing non-isomorphic graphs (Weisfeiler & Lehman, 1968). Since then, the Weisfeiler-Lehman hierarchy has become a yardstick to measure the expressiveness and guide designing more powerful GNN architectures (see Appendix A.1 for an overview of representative approaches in this area).
+
+However, as more and more architectures have been proposed, the limitations of the WL hierarchy are becoming increasingly evident. First, the WL hierarchy is arguably too coarse to evaluate the expressive power of practical GNN models (Morris et al., 2022; Puny et al., 2023). On one hand, architectures inspired by higher-order WL tests (Maron et al., 2019b;a; Morris et al., 2019) often suffer from substantial computation/memory costs. On the other hand, most practical and efficient GNNs are only proved to be strictly more expressive than 1-WL by leveraging toy example graphs (e.g., Zhang & Li, 2021; Bevilacqua et al., 2022; Wijesinghe & Wang, 2022a). Such a qualitative characterization may provide little insight into the models' true expressiveness. Besides, the expressive power brought from the WL hierarchy often does not align well with the one required in practice (Veličković, 2022). Hence, how to study the expressiveness of GNN models in a quantitative, systematic, and practical way remains a central research direction for the GNN community.
+
+To address the above limitations, this paper takes a different approach by studying GNN expressivity from the following practical angle: What structural information can a GNN model encode? Since the ability to detect/count graph substructures is crucial in various real-world applications (Chen et al., 2020; Huang et al., 2023; Tahmasebi et al., 2023), many expressive GNNs have been proposed based on preprocessing substructure information (Bouritsas et al., 2022; Barceló et al., 2021; Bodnar et al., 2021b;a). However, instead of augmenting GNNs by manually preprocessed (task-specific) substructures, it is nowadays more desirable to design generic, domain-agnostic GNNs that can end-to-end learn different structural information suitable for diverse applications. This naturally gives rise to the fundamental question of characterizing the complete set of substructures prevalent GNN models can encode. Unfortunately, this problem is widely recognized as challenging even when examining simple structures like cycles (Fürer, 2017; Arvind et al., 2020; Huang et al., 2023).
+
+Our contributions. Motivated by GNNs' ability to encode substructures, this paper presents a novel framework for quantitatively analyzing the expressive power of GNN models. Our approach is rooted in a critical discovery: given a GNN model  $M$ , the model's output representation for any graph  $G$  can be fully determined by the structural information of  $G$  over some pattern family  $\mathcal{F}^M$ , where  $\mathcal{F}^M$  corresponds to precisely all (and only) those substructures that can be "coded" by model  $M$ . In this way, the set  $\mathcal{F}^M$  can be naturally viewed as an expressivity description of  $M$ : after identifying  $\mathcal{F}^M$  for each model  $M$ , the expressivity of different models can then be qualitatively/quantitatively compared by simply looking at their set inclusion relation and set difference.
+
+The crux here is to define an appropriate notion of "encodability" so that  $\mathcal{F}^M$  can admit a simple description. We identify that a good candidate is the homomorphism expressivity: i.e.,  $\mathcal{F}^M$  consists of all substructures that can be counted by model  $M$  under homomorphism (see Section 2 for a formal definition). Homomorphism is a foundational concept in graph theory (Lovasz, 2012) and is linked to many important topics such as graph coloring, graph matching, and subgraph counting. With this concept, we are able to give complete, unified, and surprisingly elegant descriptions of the pattern family  $\mathcal{F}^M$  for a wide range of mainstream GNN architectures listed below:
+
+- MPNN (e.g., Gilmer et al., 2017; Hamilton et al., 2017; Kipf & Welling, 2017; Xu et al., 2019);  
+- Subgraph GNN (You et al., 2021; Zhang & Li, 2021; Bevilacqua et al., 2022; Qian et al., 2022);  
+- Local GNN (Morris et al., 2020; 2022; Zhang et al., 2023a; Frasca et al., 2022);  
+- Folklore-type GNN (Maron et al., 2019a; Zhang et al., 2023a; Feng et al., 2023).
+
+Technically, the descriptions are based on a novel application and extension of the concept of nested ear decomposition (NED) in graph theory (Eppstein, 1992). We prove that: (i) (necessity) each model  $M$  above can count (under homomorphism) a specific family of patterns  $\mathcal{F}^M$ , characterized by a specific type of NED; (ii) (sufficiency) any pattern  $F \notin \mathcal{F}^M$  cannot be counted under homomorphism by model  $M$ ; (iii) (completeness) for any graph, information collected from the homomorphism count in pattern family  $\mathcal{F}^M$  determines its representation computed by model  $M$ . Therefore, homomorphism expressivity is well-defined and is a complete expressivity measure for GNN models.
+
+Our theory can be generalized in various aspects. One significant extension is the node-level and edge-level expressivity for equivariant GNNs (Azizian & Lelarge, 2021; Geerts & Reutter, 2022), which can be naturally tackled by a fine-grained analysis of NED. As another non-trivial generalization, we study higher-order GNN variants for several of the above architectures and derive results by defining higher-order NED. Both aspects demonstrate the flexibility of our proposed framework, suggesting it as a general recipe for analyzing future architectures.
+
+Implications. Homomorphism expressivity serves as a powerful toolbox for bridging different sub-areas in the GNN community, providing fresh understandings of a series of known results that were previously proved in complex ways, and answering a set of unresolved open problems. First, our results can readily establish a complete expressiveness hierarchy among all the aforementioned architectures and their higher-order extensions. This recovers and extends a number of results in Morris et al. (2020); Qian et al. (2022); Zhang et al. (2023a); Frasca et al. (2022) and answers their open problems (Section 4.1). In fact, our results go far beyond revealing the expressivity gap between models: we essentially answer how large the gap is and establish a systematic approach to constructing counterexample graphs. Second, based on the relation between homomorphism and subgraph count, we are able to characterize the subgraph counting power of GNN models for all patterns at graph, node, and edge levels, significantly advancing an open direction initiated in Fürer (2017); Arvind et al. (2020) (Section 4.2). As a special case, our results extend recent findings in
+
+Huang et al. (2023) about the cycle counting power of GNN models, highlighting that Local 2-GNN can already subgraph-count all cycles/paths within 7 nodes (even at edge-level). Third, our results provide a new toolbox for studying the polynomial expressivity proposed recently in Puny et al. (2023), extending it to various practical architectures and answering an open question (Section 4.3). Empirically, an extensive set of experiments verifies our theory, showing that the homomorphism expressivity of different models matches well with their practical performance in diverse tasks.
+
+# 2 PRELIMINARY
+
+Notations. We use  $\{\}$  and  $\{\}$  to denote sets and multisets, respectively. Given a (multi)set  $S$ , its cardinality is denoted as  $|S|$ . In this paper, we consider finite, undirected, vertex-labeled graphs with no self-loops or repeated edges. Let  $G = (V_G, E_G, \ell_G)$  be a graph with vertex set  $V_G$ , edge set  $E_G$ , and label function  $\ell_G$ , where each edge in  $E_G$  is a set  $\{u, v\} \subset V_G$  of cardinality two, and  $\ell_G(u)$  is the label of vertex  $u$ . The rooted graph  $G^u$  is a graph obtained from  $G$  by marking the special vertex  $u \in V_G$ ; we can similarly consider marking two special vertices  $u, v \in V_G$  (denote by  $G^{uv}$ ). The neighbors of vertex  $u$  is denoted as  $N_G(u) := \{v \in V_G : \{u, v\} \in E_G\}$ . A graph  $F = (V_F, E_F, \ell_F)$  is a subgraph of  $G$  if  $V_F \subset V_G$ ,  $E_F \subset E_G$ , and  $\ell_F(u) = \ell_G(u)$  for all  $u \in V_F$ . A simple path  $P$  in  $G$  is an edge set of the form  $\{\{w_0, w_1\}, \dots, \{w_{k-1}, w_k\}\} \subset E_G$  where  $w_i \neq w_j$  for all  $i \neq j$ . Here,  $w_0$  and  $w_k$  are called endpoints of  $P$  and other vertices are called internal points.
+
+Homomorphism, isomorphism, and subgraph count. Given two graphs  $F$  and  $G$ , a homomorphism from  $F$  to  $G$  is a mapping  $f: V_F \to V_G$  that preserves edges and labels, i.e.,  $\ell_F(u) = \ell_G(f(u))$  for all  $u \in V_F$ , and  $\{f(u), f(v)\} \in E_G$  for all  $\{u, v\} \in E_F$ . When the mapping  $f$  exists, we say  $F$  is homomorphic to  $G$ . We denote by  $\operatorname{Hom}(F, G)$  the set of all homomorphisms from  $F$  to  $G$  and define  $\operatorname{hom}(F, G) = |\operatorname{Hom}(F, G)|$ , which counts the number of homomorphisms for pattern  $F$  in graph  $G$ . If  $f$  is further surjective on both vertices and edges, we call  $G$  a homomorphic image of  $F$ . Denote by  $\operatorname{Spasm}(F)$  the set of all homomorphic images of  $F$ , called the spasm of  $F$ . For rooted graphs, homomorphism should additionally preserve vertex marking: i.e., if  $f$  is a homomorphism from  $F^{uv}$  to  $G^{xy}$ , then  $f(u) = x$  and  $f(v) = y$ .
+
+A mapping  $f: V_F \to V_G$  is called an isomorphism if  $f$  is a bijection and both  $f$  and its inverse  $f^{-1}$  are homomorphisms. We denote by  $\operatorname{Sub}(F, G)$  the set of all subgraphs of  $G$  isomorphic to  $F$  and define  $\operatorname{sub}(F, G) = |\operatorname{Sub}(F, G)|$ , which counts the number of patterns  $F$  occurred in graph  $G$  as a subgraph. We note that a similar definition holds for rooted graphs (e.g.,  $\operatorname{sub}(F^{uv}, G^{xy})$ ).
+
+Graph neural networks. GNNs can be generally described as graph functions that are invariant under isomorphism. To achieve such invariance, most popular GNN models follow a color refinement (CR) paradigm: they maintain a feature representation (color) for each vertex or vertex tuples and iteratively refine these features through equivariant aggregation layers. Finally, there is a global pooling layer to merge all features and obtain the graph representation. Below, we separately define the corresponding CR algorithms for four mainstream classes of GNNs studied in this paper.
+
+- MPNN. Given a graph  $G$ , MPNN maintains a color  $\chi_G^{\mathrm{MP}}(u)$  for each vertex  $u \in V_G$ . Initially, the color only depends on the vertex label, i.e.,  $\chi_G^{\mathrm{MP},(0)}(u) = \ell_G(u)$ . Then, in each iteration, the color is refined by the following update formula (where hash is a perfect hash function):
+
+$$
+\chi_ {G} ^ {\mathsf {M P}, (t + 1)} (u) = \operatorname {h a s h} \left(\chi_ {G} ^ {\mathsf {M P}, (t)} (u), \{\{\chi_ {G} ^ {\mathsf {M P}, (t)} (v): v \in N _ {G} (u) \} \}\right). \tag {1}
+$$
+
+After a sufficient number of iterations, the colors become stable. We denote by  $\chi_G^{\mathrm{MP}}(u)$  the stable color of  $u$ , which is also the node feature of  $u$  computed by the MPNN. The graph representation is defined as the multiset of node colors, i.e.,  $\chi_G^{\mathrm{MP}}(G) = \{\{\chi_G^{\mathrm{MP}}(u): u \in V_G\}\}$ .
+
+- Subgraph GNN. It treats a graph  $G$  as a set of subgraphs  $\{\{G^u : u \in V_G\}\}$ , each obtained from  $G$  by marking a special vertex  $u \in V_G$ . Subgraph GNN maintains a color  $\chi_G^{\mathrm{Sub}}(u, v)$  for each vertex  $v$  in graph  $G^u$ . Initially,  $\chi_G^{\mathrm{Sub}, (0)}(u, v) = (\ell_G(v), \mathbb{I}[u = v])$ , where the latter term distinguishes the special mark. It then runs MPNNs independently on each graph  $G^u$ :
+
+$$
+\chi_ {G} ^ {\text {S u b}, (t + 1)} (u, v) = \operatorname {h a s h} \left(\chi_ {G} ^ {\text {S u b}, (t)} (u, v), \{\{\chi_ {G} ^ {\text {S u b}, (t)} (u, w): w \in N _ {G} (v) \} \}\right). \tag {2}
+$$
+
+Denote the stable color of  $(u,v)$  as  $\chi_G^{\mathrm{Sub}}(u,v)$ . The node feature of  $u$  computed by Subgraph GNN is defined by merging all colors in  $G^{u}$ , i.e.,  $\chi_G^{\mathrm{Sub}}(u) \coloneqq \mathsf{hash}\left(\{\chi_G^{\mathrm{Sub}}(u,v):v\in V_G\}\right)$ . Finally, the graph representation is defined as  $\chi_G^{\mathrm{Sub}}(G) = \{\{\chi_G^{\mathrm{Sub}}(u):u\in V_G\}\}$ .
+
+- Local GNN. Inspired by the  $k$ -WL test (Grohe, 2017), Local  $k$ -GNN is defined by replacing all global aggregations in  $k$ -WL by sparse ones that only aggregate local neighbors, yielding a much more efficient CR algorithm. As an example, the iteration of Local 2-GNN has the following form and enjoys the same computational complexity as a Subgraph GNN.
+
+$$
+\chi_ {G} ^ {\mathrm {L}, (t + 1)} (u, v) = \operatorname {h a s h} \left(\chi_ {G} ^ {\mathrm {L}, (t)} (u, v), \{\{\chi_ {G} ^ {\mathrm {L}, (t)} (w, v): w \in N _ {G} (u) \} \}, \{\{\chi_ {G} ^ {\mathrm {L}, (t)} (u, w): w \in N _ {G} (v) \} \}\right). \tag {3}
+$$
+
+Initially,  $\chi_G^{\mathsf{L},(0)}(u,v) = (\ell_G(u),\ell_G(v),\mathbb{I}[u = v],\mathbb{I}[\{u,v\} \in E_G])$ , which is called the isomorphism type of vertex pair  $(u,v)$ . We similarly denote the stable color as  $\chi_G^{\mathsf{L}}(u,v)$  and define the node feature  $\chi_G^{\mathsf{L}}(u)$  and graph representation  $\chi_G^{\mathsf{L}}(G)$  as in the Subgraph GNN.
+
+- Folklore-type GNN. The Folklore GNN (FGNN) is inspired by the standard  $k$ -FWL test (Cai et al., 1992). As an example, the iteration formula of 2-FGNN is written as follows:
+
+$$
+\chi_ {G} ^ {\mathsf {F}, (t + 1)} (u, v) = \operatorname {h a s h} \left(\chi_ {G} ^ {\mathsf {F}, (t)} (u, v), \{\left(\chi_ {G} ^ {\mathsf {F}, (t)} (w, v), \chi_ {G} ^ {\mathsf {F}, (t)} (u, w)\right): w \in V _ {G} \} \}\right). \tag {4}
+$$
+
+One can similarly consider the more efficient Local 2-FGNN by only aggregating local neighbors, which has the same computational complexity as Local 2-GNN and Subgraph GNN:
+
+$$
+\begin{array}{l} \chi_ {G} ^ {\mathrm {L F}, (t + 1)} (u, v) = \operatorname {h a s h} \left(\chi_ {G} ^ {\mathrm {L F}, (t)} (u, v), \{\{\left(\chi_ {G} ^ {\mathrm {L F}, (t)} (w, v), \chi_ {G} ^ {\mathrm {L F}, (t)} (u, w)\right): w \in N _ {G} (u) \cup N _ {G} (v) \} \}\right). \\ \text {T h e s t a b l e c o l o r , n o d e f e a t u r e , a n d g r a p h r e p r e s e n t a t i o n c a n b e s i m a r l y d e f i n e d .} \end{array} \tag {5}
+$$
+
+Finally, we note that the latter three types of GNNs can be naturally generalized into higher-order variants. We give a general definition of all these architectures in Appendix E.1. For the base case of  $k = 1$ , Subgraph  $(k - 1)$ -GNN, Local  $k$ -GNN, and Local  $k$ -FGNN all reduce to the MPNN.
+
+# 3 HOMOMORPHISM EXPRESSIVITY OF GRAPH NEURAL NETWORKS
+
+# 3.1 HOMOMORPHISM EXPRESSIVITY
+
+Given a GNN model  $M$  and a substructure  $F$ , we say  $M$  can count graph  $F$  under homomorphism if, for any graph  $G$ , the graph representation  $\chi_G^M (G)$  determines the homomorphism count  $\mathrm{hom}(F,G)$ . In other words,  $\chi_G^M (G) = \chi_H^M (H)$  implies  $\mathrm{hom}(F,G) = \mathrm{hom}(F,H)$  for any graphs  $G,H$ . The central question studied in this paper is, what substructures  $F$  can a GNN model  $M$  count under homomorphism? This gives rise to the notion of homomorphism expressivity defined below:
+
+Definition 3.1. The homomorphism expressivity of a GNN model  $M$ , denoted by  $\mathcal{F}^M$ , is a family of (labeled) graphs satisfying the following conditions<sup>1</sup>:
+
+a) For any two graphs  $G,H,\chi_G^M (G) = \chi_H^M (H)$  iff  $\operatorname {hom}(F,G) = \operatorname {hom}(F,H)$  for all  $F\in \mathcal{F}^{M}$  
+b)  $\mathcal{F}^M$  is maximal, i.e., for any graph  $F \notin \mathcal{F}^M$ , there exists a pair of graphs  $G, H$  such that  $\chi_G^M(G) = \chi_H^M(H)$  and  $\operatorname{hom}(F, G) \neq \operatorname{hom}(F, H)$ .
+
+Example 3.2. As a simple example, consider a maximally expressive GNN  $M$  that can solve the graph isomorphism problem, i.e., it computes the same representation for two graphs iff they are isomorphic. Then,  $\mathcal{F}^M$  contains all graphs. This is a classic result proved in Lovász (1967).
+
+The significance of homomorphism expressivity can be justified in the following aspects. First, it is a complete expressivity measure. Based on item (a), the homomorphism count within  $\mathcal{F}^M$  essentially captures all information embedded in the graph representation computed by model  $M$ . This contrasts with previously studied metrics such as the ability to compute biconnectivity properties (Zhang et al., 2023b) or count cycles (Huang et al., 2023), which only reflects restricted aspects of expressivity. Second, homomorphism expressivity is a quantitative measure and is much finer than qualitative expressivity results obtained from the graph isomorphism test. Specifically, by item (a), a GNN model  $M_1$  is more expressive than another model  $M_2$  in distinguishing non-isomorphic graphs iff  $\mathcal{F}^{M_2} \subset \mathcal{F}^{M_1}$ . Furthermore, by item (b),  $M_1$  is strictly more expressive than  $M_2$  iff  $\mathcal{F}^{M_2} \subsetneq \mathcal{F}^{M_1}$ , and the expressivity gap can be quantitatively understood via the set difference  $\mathcal{F}^{M_1} \setminus \mathcal{F}^{M_2}$ .
+
+Consequently, by deriving which graphs are encompassed in the graph family  $\mathcal{F}^M$ , homomorphism expressivity provides a novel way to analyze and compare the expressivity of GNN models. In the next subsection, we will give exact characterizations of  $\mathcal{F}^M$  for all models  $M$  defined in Section 2.
+
+![](images/367ee691a84a43507936b5344ade34b24c0f77306722872c498684d810fe1472.jpg)  
+(a) Illustration of NED
+
+![](images/1f2dea9b8014da564ef24f81d834fcb7ed1b46ee56f322fed817aaea5b3cb53d.jpg)  
+(b) Examples of endpoint-shared/strong/almost-strong/general NED
+
+![](images/051c09048d4e3475808eae00d8f11038e6c45aa77d8b05caa0c23f267e40c336.jpg)
+
+![](images/6595a83d633e94cfae26f06d8ee5547d7af51a42b08bdbfe883a45d9e8a38b81.jpg)  
+Figure 1: Illustration of NED and its variants. The number  $j$  next to each edge indicates that the edge belongs to the ear  $P_{j}$ . Different colors represent different ears. See Figure 9 for more examples.
+
+![](images/17a6c6836659767f7a03becf75a5ef2a424251feddf08a5820e2a732475262dd.jpg)
+
+![](images/d008ae43680a676211cae9716cb6e496dc4f6290d0d995b316fac4ad9a4fb90e.jpg)
+
+# 3.2 MAIN RESULTS
+
+To derive our main results, we leverage a concept in graph theory known as nested ear decomposition (NED), which is originally introduced in Eppstein (1992). Here, we adapt the definition as follows:
+
+Definition 3.3. Given a graph  $G$ , a NED  $\mathcal{P}$  is a partition of the edge set  $E_G$  into a sequence of simple paths  $P_1, \dots, P_m$  (called ears), which satisfies the following conditions:
+
+- Any two ears  $P_{i}$  and  $P_{j}$  with indices  $1 \leq i < j \leq c$  do not intersect, where  $c$  is the number of connected components of  $G$ .  
+- For each ear  $P_{j}$  with index  $j > c$ , there is an ear  $P_{i}$  with index  $1 \leq i < j$  such that one or two endpoints of  $P_{j}$  lie in ear  $P_{i}$  (we say  $P_{j}$  is nested on  $P_{i}$ ). Moreover, except for the endpoints lying in ear  $P_{i}$ , no other vertices in  $P_{j}$  are in any previous ear  $P_{k}$  for  $1 \leq k < j$ . If both endpoints of  $P_{j}$  lie in  $P_{i}$ , the subpath in  $P_{i}$  that shares the endpoints of  $P_{j}$  is called the nested interval of  $P_{j}$  in  $P_{i}$ , denoted as  $I(P_{j}) \subset P_{i}$ . If only one endpoint lies in  $P_{i}$ , define  $I(P_{j}) = \emptyset$ .  
+- For all ears  $P_{j}, P_{k}$  with  $c < j < k \leq m$ , either  $I(P_{j}) \cap I(P_{k}) = \emptyset$  or  $I(P_{j}) \subset I(P_{k})$ .
+
+Intuitively, Definition 3.3 states that the relation between different ears forms a forest, in that each ear is nested on its parent. Moreover, the nested intervals either do not intersect or have inclusion relations for different children of the same parent ear. We give illustrations of NED in Figure 1.
+
+In this paper, we considerably extend the concept of NED to several variants defined below:
+
+- Endpoint-shared NED: a NED is called endpoint-shared if all ears with non-empty nested intervals share a common endpoint (see Figure 1(b,1)).  
+- **Strong NED:** a NED is called strong if for any two children  $P_j$ ,  $P_k$  ( $j < k$ ) nested on the same parent ear, we have  $I(P_j) \subset I(P_k)$  (see Figure 1(b,2)).  
+- Almost-strong NED: a NED is called almost-strong if for any children  $P_j$ ,  $P_k$  ( $j < k$ ) nested on the same parent ear and  $|I(P_j)| > 1$ , we have  $I(P_j) \subset I(P_k)$  (see Figure 1(b,3)).
+
+We are now ready to present our main results:
+
+Theorem 3.4. For all GNN models  $M$  defined in Section 2, the graph family  $\mathcal{F}^M$  satisfying Definition 3.1 exists (and is unique). Moreover, each  $\mathcal{F}^M$  can be separately described below:
+
+- MPNN:  $\mathcal{F}^{\mathrm{MP}} = \{F:F$  is a forest\};  
+- Subgraph GNN:  $\mathcal{F}^{\mathrm{Sub}} = \{F:F$  has an endpoint-shared NED\};  
+- Local 2-GNN:  $\mathcal{F}^{\mathsf{L}} = \{F:F$  has a strong NED\};  
+- Local 2-FGNN:  $\mathcal{F}^{\mathrm{LF}} = \{F:F$  has an almost-strong NED\};  
+- 2-FGNN:  $\mathcal{F}^{\mathrm{F}} = \{F:F$  has a NED\}.
+
+Theorem 3.4 gives a unified description of the homomorphism expressivity for all popular GNN models defined in Section 2. Despite the elegant conclusion, the proof process is actually involved and represents a major technical contribution, so we present a proof sketch below. Our proof is divided into three parts, presented in Appendices C.2 to C.4. First, we show the existence of  $\mathcal{F}^M$  for each model  $M$  based on a beautiful theory developed in Dell et al. (2018). Using the technique of unfolding tree, we prove that  $\mathcal{F}^M$  at least contains all graphs  $F$  that allow a specific type of tree decomposition (Diestel, 2017), and the homomorphism information of these graphs determines the representation of any graph  $G$  computed by  $M$  (i.e., Definition 3.1(a) holds). However, characterizing  $\mathcal{F}^M$  in terms of tree decomposition is sophisticated and not intuitive for most models  $M$ . In the next step, we give an equivalent description of  $\mathcal{F}^M$  based on novel extensions of NED proposed in Definition 3.3, which is simpler and more elegant. In the last step, we prove that  $\mathcal{F}^M$  does not contain other graphs. This is achieved by building non-trivial relations between three distinct theoretical tools: tree decomposition, pebble game (Cai et al., 1992), and Fürer graph (Fürer, 2001). Through a fine-grained analysis of the Fürer graphs expanded by  $F \notin \mathcal{F}^M$  (see Theorems C.47 and C.53), we show they are precisely a pair of graphs satisfying Definition 3.1(b), thus concluding the proof.
+
+Discussions with Dvořák (2010); Dell et al. (2018). Our work significantly extends a beautiful theory developed in Dvořák (2010); Dell et al. (2018), which showed that a pair of graphs  $G$ ,  $H$  are indistinguishable by 1-WL iff  $\operatorname{hom}(F, G) = \operatorname{hom}(F, H)$  for all trees  $F$ , and more generally, they are indistinguishable by  $k$ -FWL iff  $\operatorname{hom}(F, G) = \operatorname{hom}(F, H)$  for all graphs  $F$  of bounded treewidth  $k$ . In this paper, we successfully generalize these results to a broad range of practical GNN models. Moreover, two distinct contributions are worth discussing. First, we highlight a key insight that homomorphism can serve as a fundamental expressivity measure, which has far-reaching consequences as will be elaborated in Section 4. To show that  $\mathcal{F}^M$  is a valid expressivity measure, we prove an extra and non-trivial result that  $\mathcal{F}^M$  is maximal (Definition 3.1(b)). Without this crucial property,  $\mathcal{F}^{M_1} \supsetneq \mathcal{F}^{M_2}$  will not necessarily mean that model  $M_1$  is strictly more expressive than  $M_2$ , thus preventing any quantitative comparison between models. Second, Dell et al. (2018) leveraged treewidth to describe results, which, unfortunately, cannot be applied to most GNN models studied here. Instead, we resort to the novel concept of NED, by which we successfully derive unified and elegant descriptions for all models. Moreover, as will be shown later, NED is quite flexible and can be naturally generalized to node/edge-level expressivity, which is not studied in prior work.
+
+Finally, we remark that one can derive an equivalent (perhaps simpler) description of  $\mathcal{F}^{\mathrm{Sub}}$ , based on the fact that a graph  $F$  has an endpoint-shared NED iff  $F$  becomes a forest when deleting the shared endpoint. Formally, denoting by  $F \backslash \{u\}$  the induced subgraph of  $F$  over  $V_F \backslash \{u\}$ , we have
+
+Corollary 3.5.  $\mathcal{F}^{\mathrm{Sub}} = \{F:\exists u\in V_F$  s.t.  $F\backslash \{u\}$  is a forest}.
+
+# 3.3 EXTENDING TO NODE/EDGE-LEVEL EXPRESSIVITY
+
+So far, this paper mainly focuses on the graph-level expressivity, i.e., what information is encoded in the graph representation. In this subsection, we extend all results in Theorem 3.4 to the more fine-grained node/edge-level expressivity by answering what information is encoded in the node/edge features of a GNN (i.e.,  $\chi_G^M(u)$  or  $\chi_G^M(u,v)$  in Section 2). This yields the following definition:
+
+Definition 3.6. The node-level homomorphism expressivity of a GNN model  $M$ , denoted by  $\mathcal{F}_{\mathrm{n}}^{M}$ , is a family of connected rooted graphs satisfying the following conditions:
+
+a) For any connected graphs  $G,H$  and vertices  $u\in V_G,v\in V_H,\chi_G^M (u) = \chi_H^M (v)$  iff  $\operatorname {hom}(F^w,G^u) = \operatorname {hom}(F^w,H^v)$  for all  $F^w\in \mathcal{F}_{\mathfrak{n}}^{M}$  
+b) For any connected rooted graph  $F^w \notin \mathcal{F}_{\mathfrak{n}}^{M}$ , there exists a pair of connected graphs  $G, H$  and vertices  $u \in V_G, v \in V_H$  such that  $\chi_G^M(u) = \chi_H^M(v)$  and  $\operatorname{hom}(F^w, G^u) \neq \operatorname{hom}(F^w, H^v)$ .
+
+One can similarly define the edge-level homomorphism expressivity  $\mathcal{F}_{\mathrm{e}}^{M}$  to be a family of connected rooted graphs, each marking two special vertices (we omit the definition for clarity). The following result exactly characterizes  $\mathcal{F}_{\mathrm{n}}^{M}$  and  $\mathcal{F}_{\mathrm{e}}^{M}$  for all models  $M$  considered in this paper:
+
+Theorem 3.7. For all model  $M$  defined in Section 2,  $\mathcal{F}_{\mathfrak{n}}^{M}$  and  $\mathcal{F}_{\mathrm{e}}^{M}$  (except MPNN) exist. Moreover,
+
+- MPNN:  $\mathcal{F}_{\mathfrak{n}}^{\mathsf{MP}} = \{F^{w}: F \text{ is a tree}\}$ ;  
+- Subgraph GNN:
+
+$\mathcal{F}_{\mathtt{n}}^{\mathtt{Sub}} = \{F^{w}:F$  has a NED with shared endpoint  $w\} = \{F^{w}:F\backslash \{w\}$  is a forest},  $\mathcal{F}_{\mathtt{e}}^{\mathtt{Sub}} = \{F^{wx}:F$  has a NED with shared endpoint  $w\} = \{F^{wx}:F\backslash \{w\}$  is a forest};
+
+- 2-FGNN: \(\mathcal{F}_{\mathfrak{n}}^{\mathsf{F}} = \{F^{w}: F\) has a NED where \(w\) is an endpoint of the first ear}\(, \)\mathcal{F}_{\mathrm{e}}^{\mathrm{F}} = \{F^{wx}: F\) has a NED where \(w\) and \(x\) are endpoints of the first ear\}$.
+
+The cases of Local 2-GNN and Local 2-FGNN are similar to 2-FGNN by replacing "NED" with "strong NED" and "almost-strong NED", respectively.
+
+In summary, the node/edge-level homomorphism expressivity can be naturally described using NED by further specifying the endpoints of the first ear.
+
+# 3.4 EXTENDING TO HIGHER-ORDER GNNS
+
+Finally, we discuss how our results can be naturally extended to higher-order GNNs, thus providing a complete picture of the homomorphism expressivity hierarchy for infinitely many architectures. We focus on three representative examples: Subgraph  $k$ -GNN (Qian et al., 2022), Local  $k$ -GNN (Morris et al., 2020), and  $k$ -FGNN (Azizian & Lelarge, 2021). Subgraph  $k$ -GNN extracts a graph  $G^u$  for each vertex  $k$ -tuple  $u \in V_G^k$  and runs MPNNs independently, which recovers Subgraph GNN when  $k = 1$ . As the reader may have guessed, the following result exactly parallels Corollary 3.5:
+
+Theorem 3.8. The homomorphism expressivity of Subgraph  $k$ -GNN exists and can be described as  $\mathcal{F}^{\mathrm{Sub}(k)} = \{F : \exists U \subset V_F \text{ s.t. } |U| \leq k \text{ and } F \backslash U \text{ is a forest}\}$ .
+
+We next turn to Local  $k$ -GNN. To describe the result, we introduce a novel extension of Definition 3.3, called the  $k$ -order ear. Intuitively, it is formed by a graph of no more than  $k$  vertices, plus  $k$  paths each linking to a vertex in the graph (see Figure 2(a) for an illustration). Note that a 2-order ear is exactly a simple path. Then, we can naturally define the nested "interval" (see the solid orange lines in Figure 2(b) for an illustration) and thus define the concept of  $k$ -order strong NED. Due to space limit, a formal definition is deferred to Definition E.3. We have the following main result:
+
+![](images/963dfb2c3d1f660e84e4e3dfed0f1c537c67de224bfb1ed809069755040a7d30.jpg)  
+Figure 2: Illustration of higher-order ears. Each curve indicates a path (with possibly zero length) and each straight segment indicates an edge.
+
+Theorem 3.9. The homomorphism expressivity of Local  $k$ -GNN exists and can be described as  $\mathcal{F}^{\mathsf{L}(k)} = \{F : F \text{ has a } k \text{-order strong NED}\}$ .
+
+Finally, let us consider the standard  $k$ -FGNN (or equivalently, the  $k$ -FWL). Unfortunately, we cannot find a description of its homomorphism expressivity based on some form of higher-order NED; nevertheless, it is easy to describe the results using the notion of treewidth (see Definition C.2). Specifically, denoting  $\mathrm{tw}(F)$  to be the treewidth of graph  $F$ , we have the following result:
+
+Theorem 3.10. The homomorphism expressivity of  $k$ -FGNN exists and can be described as  $\mathcal{F}^{\mathsf{F}(k)} = \{F : \mathrm{tw}(F) \leq k\}$ .
+
+Interestingly, one can see that  $\mathcal{F}^{\mathrm{Sub}(0)}$ ,  $\mathcal{F}^{\mathrm{L}(1)}$ , and  $\mathcal{F}^{\mathrm{F}(1)}$  all degenerate to the family of forests, which coincides with the fact that all these higher-order GNNs reduces to MPNN for the base case.
+
+# 4 IMPLICATIONS
+
+The previous section has provided a complete description of the homomorphism expressivity for a variety of GNN models. In this section, we highlight the significance of these results through three different contexts. We will show how homomorphism expressivity can be used to link different GNN subareas, provide new insights into various known results, and answer a number of open problems.
+
+# 4.1 QUALITATIVE EXPRESSIVITY COMPARISON
+
+One direct corollary of Theorem 3.4 is that it readily enables expressivity comparison among all models in Section 2. This can be summarized below:
+
+Corollary 4.1. Under the notation of Theorem 3.4,  $\mathcal{F}^{\mathrm{MP}}\subsetneq \mathcal{F}^{\mathrm{Sub}}\subsetneq \mathcal{F}^{\mathrm{L}}\subsetneq \mathcal{F}^{\mathrm{LF}}\subsetneq \mathcal{F}^{\mathrm{F}}$ . Thus, the expressive power of the following GNN models strictly increases in order (in terms of distinguishing non-isomorphic graphs): MPNN, Subgraph GNN, Local 2-GNN, Local 2-FGNN, and 2-FGNN.
+
+Proof.  $\mathcal{F}^{\mathrm{MP}}\subset \mathcal{F}^{\mathrm{Sub}}$  follows from Corollary 3.5 and the fact that deleting any vertex of a forest yields a forest.  $\mathcal{F}^{\mathrm{Sub}}\subset \mathcal{F}^{\mathrm{L}}$  follows by the fact that any endpoint-shared NED is a strong NED.  $\mathcal{F}^{\mathrm{L}}\subset \mathcal{F}^{\mathrm{LF}}\subset \mathcal{F}^{\mathrm{F}}$  follows similarly since any strong NED is an almost-strong NED and any almost-strong NED is a NED. To prove strict separation results, one can check that the four graphs in Figure 1(b) precisely reveal the gap between each pair of graph families, thus concluding the proof.
+
+Corollary 4.1 recovers a series of results recently proved in Zhang et al. (2023a); Frasca et al. (2022). Compared to their results, our approach draws a much clearer picture of the expressivity gap between different architectures and essentially answers how large the gaps are. Moreover, we provide systematic guidance for finding counterexample graphs that unveil the expressivity gap: as shown in Corollary C.54, any graph  $F \in \mathcal{F}^{M_2} \backslash \mathcal{F}^{M_1}$  immediately gives a pair of non-isomorphic graphs that reveals the gap between models  $M_1$  and  $M_2$ . We note that this readily recovers the counterexamples constructed in Zhang et al. (2023a) and greatly simplifies their sophisticated case-by-case analysis.
+
+We next turn to three types of higher-order GNNs studied in Section 3.4, for which we can establish a complete expressiveness hierarchy, as presented in Corollary 4.2. A graphical illustration of these results is given in Figure 3.
+
+![](images/fca9ed91942914b08d04507aa0dd77121cccd68d26f7701733b23158a27efb31.jpg)  
+Figure 3: Expressiveness hierarchy of MPNN, Subgraph GNN, Local GNN, and FGNN.
+
+Corollary 4.2. Under the notations in Section 3.4, for any  $k > 0$ , the following hold:
+
+a)  $\mathcal{F}^{\mathrm{Sub}(k - 1)}\subsetneq \mathcal{F}^{\mathrm{Sub}(k)}$  . I.e., the expressive power of Subgraph  $k$  GNN strictly increases with  $k$  
+b)  $\mathcal{F}^{\mathsf{L}(k)}\subsetneq \mathcal{F}^{\mathsf{L}(k + 1)}$  . I.e., the expressive power of Local  $k$  -GNN strictly increases with  $k$  
+c)  $\mathcal{F}^{\mathrm{Sub}(k)}\subsetneq \mathcal{F}^{\mathrm{L}(k + 1)}$  . I.e., Local  $(k + 1)$  -GNN is strictly more expressive than Subgraph  $k$  GNN;  
+d)  $\mathcal{F}^{\mathsf{F}(k)}\subsetneq \mathcal{F}^{\mathsf{L}(k + 1)}\subsetneq \mathcal{F}^{\mathsf{F}(k + 1)}$  . I.e., the expressive power of Local  $(k + 1)$  -GNN lies strictly between  $k$  -FWL and  $(k + 1)$  -FWL;  
+e)  $\mathcal{F}^{\mathrm{Sub}(k)}\subsetneq \mathcal{F}^{\mathrm{F}(k + 1)}$ , and for all  $k > 1$ ,  $\mathcal{F}^{\mathrm{Sub}(k)}\backslash \mathcal{F}^{\mathrm{F}(k + 1)}\neq \emptyset$  and  $\mathcal{F}^{\mathrm{F}(k + 1)}\backslash \mathcal{F}^{\mathrm{Sub}(k)}\neq \emptyset$ . In other words, the expressive power of Subgraph  $k$ -GNN lies strictly within  $(k + 1)$ -FWL, but it is incomparable to  $k$ -FWL when  $k > 1$ .
+
+Corollary 4.2 recovers results in Morris et al. (2020); Qian et al. (2022) and further answers two open problems. First, Corollary 4.2(c) is a new result that bridges Morris et al. (2020) with Qian et al. (2022) and partially answers an open question in Zhang et al. (2023a, Appendix C). Another new result is Corollary 4.2(d), which essentially answers a fundamental open problem raised in Frasca et al. (2022, Appendix E), showing that their proposed  $\mathsf{RelGN}(k)$  model is bounded by  $k$ -FWL with an inherent expressivity gap (see Appendix E.4 for a detailed discussion). To sum up, all these challenging open problems become straightforward through the lens of homomorphism expressivity.
+
+# 4.2 SUBGRAPH COUNTING POWER
+
+The significance of homomorphism expressivity can go much beyond qualitative comparisons between models. As another implication, it provides a systematic way to study GNNs' ability to encode structural information such as subgraph count, which has been found crucial in numerous practical applications. Specifically, a well-known result in graph theory states that, for any graphs  $F, G$ , the subgraph count  $\operatorname{sub}(F, G)$  can be determined by  $\operatorname{hom}(\tilde{F}, G)$  where  $\tilde{F}$  ranges over all homomorphic images of  $F$  (i.e.,  $\operatorname{Spasm}(F)$ , see Section 2) (Lovász, 2012; Curticapean et al., 2017).
+
+Mathematically, given any graph  $F$ , let  $\operatorname{Spasm}^{\mathcal{Z}}(F)$  be any maximal set of pairwise non-isomorphic graphs chosen from  $\operatorname{Spasm}(F)$  (see Figure 4(a) for an illustration). Then, we have the following linear relation for all graph  $G$ :
+
+$$
+\operatorname {s u b} (F, G) = \sum_ {\tilde {F} \in \operatorname {S p a s m} ^ {\neq} (F)} \alpha (F, \tilde {F}) \cdot \operatorname {h o m} (\tilde {F}, G), \tag {6}
+$$
+
+where  $\alpha (F,\tilde{F})\neq 0$  is a constant scalar coefficient independent of  $G$ . Based on this formula, we can easily study the subgraph counting power of GNN models as shown in Proposition 4.4.
+
+Definition 4.3. Given a GNN model  $M$ , we say  $M$  can subgraph-count graph  $F$  at graph-level if  $\chi_G^M (G) = \chi_H^M (H)$  implies  $\operatorname {sub}(F,G) = \operatorname {sub}(F,H)$  for any graphs  $G,H$ . We say  $M$  can subgraphcount rooted graph  $F^{w}$  at node-level if  $\chi_G^M (u) = \chi_H^M (v)$  implies  $\operatorname {sub}(F^w,G^u) = \operatorname {sub}(F^w,H^v)$  for any graphs  $G,H$  and vertices  $u\in V_{G},v\in V_{H}$ . We can similarly define the edge-level subgraph counting ability for rooted graphs marking two special vertices.
+
+Proposition 4.4. For any GNN model  $M$  defined in Section 2, it can subgraph-count graph  $F$  (at graph-level) if  $\tilde{F} \in \mathcal{F}^M$  for all  $\tilde{F} \in \operatorname{Spasm}(F)$ . It can subgraph-count  $F^w$  (at node-level) if  $\tilde{F}^w \in \mathcal{F}_{\mathfrak{n}}^M$  for all  $\tilde{F}^w \in \operatorname{Spasm}(F^w)$ . A similar result holds for edge-level subgraph counting.
+
+The above proposition offers a simple way to affirm the ability of a GNN model  $M$  to subgraph-count any pattern at graph/node/edge-level. On the other hand, one may wonder whether the converse direction also holds, i.e.,  $M$  cannot subgraph-count  $F$  if there exists a homomorphic image  $\tilde{F} \in \mathsf{Spasm}(F)$  such that  $\tilde{F} \notin \mathcal{F}^M$ . We find that it is indeed the case. Specifically, if the set  $\mathsf{Spasm}(F) \backslash \mathcal{F}^M$  is not empty, then one can always find a pair of counterexample graphs  $G, H$  such that  $\chi_G^M(G) = \chi_H^M(H)$  but  $\mathsf{sub}(F, G) \neq \mathsf{sub}(F, H)$ . We eventually arrive at the following main theorem (see Appendix G.1 for a proof):
+
+Theorem 4.5. For any GNN model  $M$  such that their homomorphism expressivity  $\mathcal{F}^M$  exists,  $M$  can subgraph-count  $F$  iff  $\operatorname{Spasm}(F) \subset \mathcal{F}^M$ . Similar results hold for rooted graphs  $F^u / F^{uv}$  by replacing  $\mathcal{F}^M$  with node/edge-level homomorphism expressivity  $\mathcal{F}_{\mathfrak{n}}^M / \mathcal{F}_{\mathfrak{e}}^M$ .
+
+Example 4.6. As an example, we can readily characterize the cycle/path counting power of various GNNs. Denote by  $C_n / P_n$  the simple cycle/path of  $n$  vertices. Let  $\{u, v\} \in E_{C_n}$  be any edge in  $C_n$ , and  $\{w, x\} \in E_{P_n}$  be any edge in  $P_n$  where  $w$  is an endpoint of  $P_n$ . The following table lists exactly all cycles/paths each model can count at graph/node/edge-level.
+
+<table><tr><td rowspan="2">Structure Model</td><td colspan="3">Cycle</td><td colspan="3">Path</td></tr><tr><td>\( C_n \)</td><td>\( C_n^u \)</td><td>\( C_n^{uv} \)</td><td>\( P_n \)</td><td>\( P_n^w \)</td><td>\( P_n^{wx} \)</td></tr><tr><td>MPNN</td><td>None</td><td>None</td><td>None</td><td>n≤3</td><td>n≤3</td><td>n≤3</td></tr><tr><td>Subgraph GNN</td><td>n≤7</td><td>n≤4</td><td>n≤4</td><td>n≤7</td><td>n≤4</td><td>n≤4</td></tr><tr><td>Local 2-GNN</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Local 2-FGNN</td><td colspan="3">n≤7</td><td colspan="3">n≤7</td></tr><tr><td>2-FGNN</td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+![](images/0b8c8167c57ee57277750953d00c94ad88af1049db49016e3f2c9112fed7d752.jpg)  
+(a)  $\mathsf{Spasm}^{\neq}(C_6)$  has 10 graphs. (b) Rooted  $C_6$  Figure 4: Illustration of homomorphic images of the 6-cycle and rooted 6-cycle.
+
+![](images/b30b49dc435d40d1f52f70b72039407f218c2615658acf1500e994bd1a523e89.jpg)
+
+(b) Rooted  $C_6$
+
+**Discussions with prior work.** Our results significantly extend Huang et al. (2023) in several aspects. First, we show Subgraph GNN can count 6-cycle at graph-level by simply enumerating its spasm (see Figure 4(a)). However, it cannot count rooted 5/6-cycle at node-level because the homomorphic image can contain cycles that do not pass the marked vertex (see Figure 4(b)). This provides novel insights into Huang et al. (2023) and extends their results (albeit with a simpler analysis). Second, we reveal that Local 2-GNN can already count all cycles/pathways that 2-FWL can count (even at edge-level). This identifies a new architecture with both efficiency and strong expressiveness in subgraph counting, considerably extending the finding in the concurrent work of Zhou et al. (2023b).
+
+In Appendix G.2 (Tables 4 and 5), we summarize the statistics of all moderate-size patterns each model can count under homomorphisms/subgraphs, which enables quantitative expressivity comparisons of different models in a clear and exact manner. We also comprehensively list the counting ability of all moderate-size patterns in Table 6, which we believe can be helpful for future research.
+
+# 4.3 POLYNOMIAL EXPRESSIVITY
+
+As the third implication, homomorphism expressivity is closely related to the polynomial expressivity recently proposed in Puny et al. (2023). Concretely, given a model  $M$ , a graph  $F$  is in  $\mathcal{F}^M$  if  $M$  can express the invariant graph polynomial  $P_F$  (defined in Puny et al. (2023), Section 2.2), and a rooted graph  $F^{uv}$  is in  $\mathcal{F}_{\mathrm{e}}^{M}$  if  $M$  can express the equivariant graph polynomial  $P_{F^{uv}}$ . Based on this connection, our work introduces a novel toolbox for studying polynomial expressivity via the NED framework and offers new insights into which graph polynomials can be computed for a variety of practical GNNs. Moreover, we readily settle an open question in Puny et al. (2023), which upper bounds the polynomial expressivity for their proposed PPGN++.
+
+Corollary 4.7.  $PPGN++$  is bounded by (and thus as expressive as) the Prototypical edge-based model defined in Puny et al. (2023) for computing equivariant graph polynomials.
+
+Due to space limit, please refer to Appendix H for proof and more discussions.
+
+# 5 EXPERIMENTS
+
+This section aims to verify our theory through a comprehensive set of experiments. In each experiment, we implement four types of GNN models listed in Section 2, i.e., MPNN, Subgraph GNN, Local 2-GNN, and Local 2-FGNN. Note that all of these models are much more efficient than 2-FWL. Our primary objective here is not to produce SOTA results, but rather to provide a unified and equitable empirical comparison among these models. To ensure fairness, we employ the same GIN-based design (Xu et al., 2019) for all models and control their model sizes and training budgets to be roughly the same on each task. Details of model configurations are given in Appendix I. Our code is available at https://github.com/subgraph23/homomorphism-expressivity.
+
+Synthetic task. We first test whether these GNN models can easily learn homomorphism information from data as our theory predicts. We use the benchmark dataset from Zhao et al. (2022a) and comprehensively test the homomorphism expressivity at graph/node/edge-level by carefully selecting 8 substructures shown in Table 1. The reported performance is measured by the normalized
+
+Table 1: Experimental results on homomorphism counting. Red/blue nodes indicate marked vertices.  
+
+<table><tr><td rowspan="2">Task Model</td><td colspan="3">Graph-level</td><td colspan="2">Node-level</td><td colspan="3">Edge-level</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>MPNN</td><td>.300</td><td>.233</td><td>.254</td><td>.505</td><td>.478</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Subgraph GNN</td><td>.011</td><td>.015</td><td>.012</td><td>.004</td><td>.058</td><td>.003</td><td>.058</td><td>.048</td></tr><tr><td>Local 2-GNN</td><td>.008</td><td>.008</td><td>.010</td><td>.003</td><td>.004</td><td>.005</td><td>.006</td><td>.008</td></tr><tr><td>Local 2-FGNN</td><td>.003</td><td>.005</td><td>.004</td><td>.005</td><td>.005</td><td>.007</td><td>.007</td><td>.008</td></tr></table>
+
+Table 2: Experimental results on ZINC and Alchemy datasets. See Appendix I.4 for comparisons of more GNN models in literature.  
+
+<table><tr><td rowspan="2">Task 
+Model</td><td colspan="2">ZINC</td><td rowspan="2">Alchemy</td></tr><tr><td>Subset</td><td>Full</td></tr><tr><td>MPNN</td><td>.138 ± .006</td><td>.030 ± .002</td><td>.122 ± .002</td></tr><tr><td>Subgraph GNN</td><td>.110 ± .007</td><td>.028 ± .002</td><td>.116 ± .001</td></tr><tr><td>Local 2-GNN</td><td>.069 ± .001</td><td>.024 ± .002</td><td>.114 ± .001</td></tr><tr><td>Local 2-FGNN</td><td>.064 ± .002</td><td>.023 ± .001</td><td>.111 ± .001</td></tr></table>
+
+Table 3: Experimental results on the (Chordal) Cycle Counting task.  
+
+<table><tr><td rowspan="2">Task Model</td><td colspan="6">Graph-level</td><td colspan="6">Node-level</td><td colspan="6">Edge-level</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>MPNN</td><td>.358</td><td>.208</td><td>.188</td><td>.146</td><td>.261</td><td>.205</td><td>.600</td><td>.413</td><td>.300</td><td>.207</td><td>.318</td><td>.237</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Subgraph GNN</td><td>.010</td><td>.020</td><td>.024</td><td>.046</td><td>.007</td><td>.027</td><td>.003</td><td>.005</td><td>.092</td><td>.082</td><td>.050</td><td>.073</td><td>.001</td><td>.003</td><td>.090</td><td>.096</td><td>.038</td><td>.065</td></tr><tr><td>Local 2-GNN</td><td>.008</td><td>.011</td><td>.017</td><td>.034</td><td>.007</td><td>.016</td><td>.002</td><td>.005</td><td>.010</td><td>.023</td><td>.004</td><td>.015</td><td>.001</td><td>.005</td><td>.010</td><td>.019</td><td>.005</td><td>.014</td></tr><tr><td>Local 2-FGNN</td><td>.003</td><td>.004</td><td>.010</td><td>.020</td><td>.003</td><td>.010</td><td>.004</td><td>.006</td><td>.012</td><td>.021</td><td>.004</td><td>.014</td><td>.003</td><td>.006</td><td>.012</td><td>.022</td><td>.005</td><td>.012</td></tr></table>
+
+Mean Absolute Error (MAE) on the test dataset. It can be seen that the model performance indeed correlates to our theoretical predictions: (i) MPNN cannot encode any substructure under homomorphism; (ii) Subgraph GNN cannot encode the 2th, 3rd, 5th, 7th, 8th substructures; (iii) Local 2-GNN cannot encode the 3rd and 8th substructures; (iv) Local 2-FGNN can encode all substructures.
+
+Cycle counting power. Cycles are important structures in numerous graph learning tasks, yet encoding them is notoriously hard for GNNs. We next test the ability of different GNN models to subgraph-count (chordal) cycles at graph/node/edge-level. We follow the setting in Frasca et al. (2022); Zhang et al. (2023a); Huang et al. (2023) and present results in Table 3 (measured by the normalized test MAE). Remarkably, despite the same computational cost and model size, Local 2-(F)GNN performs significantly better than Subgraph GNN and achieves good performance for counting all  $3/4/5/6$ -cycles as well as chordal  $4/5$ -cycles (even at edge-level). These results match Example 4.6 and may suggest Local 2-(F)GNN as generic, efficient, yet powerful architectures in solving chemical and biological tasks where counting cycles is essential (e.g., benzene rings).
+
+Real-world tasks. We finally test these GNN models on three real-world benchmarks: ZINC-subset, ZINC-full (Dwivedi et al., 2020), and Alchemy (Chen et al., 2019a). Following the standard configuration, all models obey a 500K parameter budget. The results are shown in Table 2. It can be seen that the performance continues to improve when a more expressive model is used. In particular, Local 2-FGNN achieves the best performance on all tasks, suggesting that its theoretical expressivity guarantee can translate to practical performance in real-world settings.
+
+# 6 CONCLUSION
+
+In this paper, we present a new framework for systematically and quantitatively studying the expressive power of various GNN architectures. Through the lens of homomorphism expressivity, we give exact descriptions of the graph family each model can encode in terms of homomorphism counting. Our framework stands as a valuable toolbox to unify the landscape between different subareas in the GNN community, providing deep insights into a number of prior works and answering their open problems. In particular, one can establish a complete expressiveness hierarchy between models, determine the subgraph counting capabilities of GNNs at graph/node/edge-level, and understand their polynomial expressivity. On the theoretical side, our results establish deep connections with a series of fundamental topics in graph theory (see Appendix A.2); On the practical side, these results closely correlate with the empirical performance of GNN models, as demonstrated through extensive experiments. Finally, Appendix B outlines several open directions for further exploration, and we believe that the homomorphism expressivity framework paves a fresh way for future study of more expressive GNNs.
+
+Acknowledgement Bohang Zhang would like to thank Shengjie Luo for helpful discussions. We sincerely appreciate all reviewers for the valuable suggestions. This work is supported by National Science and Technology Major Project (2022ZD0114902) and National Science Foundation of China (NSFC62276005, NSFC62376007).
+
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+Jiaxuan You, Jonathan M Gomes-Selman, Rex Ying, and Jure Leskovec. Identity-aware graph neural networks. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pp. 10737-10745, 2021.  
+Bohang Zhang, Guhao Feng, Yiheng Du, Di He, and Liwei Wang. A complete expressiveness hierarchy for subgraph GNNs via subgraph weisfeiler-lehman tests. In International Conference on Machine Learning, volume 202, pp. 41019-41077. PMLR, 2023a.  
+Bohang Zhang, Shengjie Luo, Di He, and Liwei Wang. Rethinking the expressive power of gnu's via graph biconnectivity. In International Conference on Learning Representations, 2023b.  
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+
+# Appendix
+
+# Table of Contents
+
+A More Related Work 18
+
+A.1 Expressive Graph Neural Networks 18  
+A.2 Broader impacts and additional discussions 20
+
+B Limitations and Open Directions 21  
+C Proof of Theorem 3.4 21
+
+C.1 Preliminary 22  
+C.2 Part 1: tree decomposition 25  
+C.3 Part 2: nested ear decomposition 39  
+C.4 Part 3: pebble game 43
+
+D Node/edge-level Expressivity 50
+
+D.1 Related to tree decomposition and ear decomposition 51  
+D.2 Counterexamples 52
+
+E Higher-order GNNs 54
+
+E.1 Definition of higher-order GNNs 54  
+E.2 Higher-order strong NED 55  
+E.3 Proofs in Section 3.4 55  
+E.4 Expressivity gap between higher-order GNNs 56
+
+F Additional Discussions 57
+
+F.1 Regarding the definition of homomorphism expressivity 57  
+F.2 Regarding the definition of NED 57
+
+G Homomorphism and Subgraph Counting Power 58
+
+G.1 Proof of Theorem 4.5 58  
+G.2 Graph statistics and examples 59
+
+H Polynomial Expressivity 69
+
+I Experimental Details 70
+
+I.1 Datasets 70  
+I.2 Model details 70  
+I.3 Training details 72  
+I.4 Performance of baseline models in literature 72
+
+# A MORE RELATED WORK
+
+# A.1 EXPRESSIVE GRAPH NEURAL NETWORKS
+
+Since Morris et al. (2019); Xu et al. (2019) discovered the limited expressive power of MPNNs in distinguishing non-isomorphic graphs, a large amount of work has been devoted to developing GNNs with better expressiveness. Below, we briefly review representative approaches in this area. For a comprehensive survey on expressive GNNs, we refer readers to Morris et al. (2023).
+
+Higher-order GNNs. Inspired by the relation between MPNNs and the 1-WL test, a natural approach to designing provably more expressive GNNs is to mimic the higher-order WL tests. This gives rise to two fundamental types of higher-order GNNs. One type of GNNs mimics the  $k$ -WL test (Grohe, 2017), and representative architectures include  $k$ -GNN (Morris et al., 2019) and  $k$ -IGN (Maron et al., 2019b;c); the other type of GNNs mimics the  $k$ -FWL (Folklore WL) test (Cai et al., 1992) and is referred to as the the  $k$ -FGNN (Maron et al., 2019a). Azizian & Lelarge (2021); Geerts & Reutter (2022) proved that each of these architectures is exactly as expressive as the corresponding higher-order WL/FWL test. Therefore, the expressiveness grows strictly as the order  $k$  increases; when  $k$  approaches infinity, they can universally approximate any continuous graph functions (Chen et al., 2019b; Keriven & Peyré, 2019). However, due to the inherent computation/memory complexity, these architectures are generally not practical in real-world applications.
+
+Local GNNs. To improve computational efficiency, a subsequent line of work seeks to develop more scalable and practical GNN architectures by taking into account the local/sparse nature of graphs. Locality/sparsity is also an important inductive bias for graphs but is not well-exploited in higher-order GNNs, since their layer aggregation is inherently global and the graph adjacency information is only encoded in initial node features. To address these shortcomings, Morris et al. (2020) proposed the Local  $k$ -GNN (and several variants) as a replacement of  $k$ -GNN, which directly incorporates graph adjacency into network layers and only aggregates neighboring information instead of the global one. The authors further proved that Local  $k$ -GNN is strictly more expressive than  $k$ -GNN. Building upon Local  $k$ -GNN, Morris et al. (2022) proposed the  $(k,s)$ -SpeqNet that further reduces the computational cost by considering a subset of  $k$ -tuples whose vertices can be grouped into no more than  $s$  connected components. A similar idea appeared in Zhao et al. (2022b), in which the authors proposed the  $(k,s)$ -SetGNN by considering  $k$ -sets instead of  $k$ -tuples. Besides Local  $k$ -GNN, recent architectures proposed in Frasca et al. (2022) and Zhang et al. (2023a) can be analogously understood as Local  $k$ -IGN and Local  $k$ -FGNN, respectively. Very recently, Feng et al. (2023); Zhou et al. (2023b) generalized the Local  $k$ -FGNN to a broad class of Folklore-type GNNs and achieved good performance on several benchmark datasets.
+
+Subgraph GNNs. Graphs that are indistinguishable by WL tests typically possess a high degree of symmetry. In light of this observation, Subgraph GNNs have recently emerged as a compelling approach to designing expressive GNNs. The basic idea is to break symmetry by transforming the original graph into a collection of slightly modified subgraphs and feeding these subgraphs into a GNN model. The earliest forms of Subgraph GNNs may track back to Cotta et al. (2021); Papp et al. (2021) (albeit with a different motivation), where the authors proposed to feed node-deleted subgraphs into an MPNN. Papp & Wattenhofer (2022) later argued to use node marking instead of node deletion for better expressive power, resulting in the standard Subgraph GNN studied in this paper. Zhang & Li (2021); You et al. (2021) proposed the Nested GNN and Identity-aware GNN, both of which can be treated as variants of Subgraph GNNs that use ego networks as subgraphs. In particular, the heterogeneous message passing proposed in You et al. (2021) can also be seen as a form of node marking. We note that the model proposed in Vignac et al. (2020) can also be interpreted as a Subgraph GNN. Qian et al. (2022) proposed the higher-order Subgraph GNN by marking  $k$  nodes per subgraph, resulting in  $n^k$  different subgraphs when the original graph has  $n$  vertices. We call this architecture Subgraph  $k$ -GNN in this paper. The authors proved that Subgraph  $k$ -GNN is strictly bounded by  $(k + 1)$ -FWL and is incomparable to  $k$ -FWL when  $k > 1$ . Zhou et al. (2023a) further generalized Subgraph  $k$ -GNN to  $(l, k)$ -GNN by using  $l$ -GNN instead of MPNN to process each subgraph. It was proved that  $(l, k)$ -GNN is bounded by  $(k + l)$ -GNN for  $l \geq 2$ .
+
+Recently, Subgraph GNNs have been greatly extended by further allowing interactions between subgraphs. This is achieved by designing cross-subgraph aggregation layers (rather than feeding each subgraph independently into a GNN). Bevilacqua et al. (2022) developed the Equivariant Subgraph Aggregation Network that introduces a global aggregation between subgraphs. A similar design is
+
+proposed in the concurrent work of Zhao et al. (2022a). These architectures were later proved to strictly improve the expressivity of the original Subgraph GNNs (Zhang et al., 2023b;a). Frasca et al. (2022) built a general design space of Subgraph GNNs that unifies prior work and showed that all models in this space are bounded by a variant of 2-IGN (dubbed the Local 2-IGN in this paper), which is then bounded by 2-FWL. Zhang et al. (2023a) later proved that Local 2-IGN is as expressive as Local 2-GNN and strictly less expressive than 2-FWL (2-FGNN). In this paper, we still use the term "Subgraph GNN" to refer to the original architecture in the previous paragraph, while using "Local 2-GNN" to refer to the general architecture in Frasca et al. (2022); Zhang et al. (2023a).
+
+Substructure-based GNNs. Another line of work sought to develop expressive GNNs from practical considerations. In particular, Chen et al. (2020) pointed out that the ability of GNNs to detect/count graph substructures like path, cycle, and clique is crucial in numerous applications. Yet, MPNNs cannot subgraph-count any cycles/cliques. While higher-order WL tests can be more powerful in counting cycles (Fürer, 2017; Arvind et al., 2020), they suffer from substantial computational cost. As such, several works proposed to directly incorporate substructure/homomorphism counting into the node features as a preprocessing step to boost the expressiveness of MPNNs (Bouritsas et al., 2022; Barceló et al., 2021) or serve as an expressive graph kernel (Grohe, 2020; Nguyen & Maehara, 2020; Welke et al., 2023). Going beyond node features, Bodnar et al. (2021b;a); Giusti et al. (2023) further proposed a message-passing framework that enables interaction between nodes, edges, and higher-order substructures like cycles and cliques. We note that the Autobahn, TOGL, and Cy2C-GNN proposed in Thiede et al. (2021); Horn et al. (2022); Choi et al. (2022) can also be viewed as Substructure-based GNNs. However, most of the above approaches consider a fixed, predefined set of substructures rather than designing generic architectures that can learn substructures in an end-to-end fashion. Recently, Tahmasebi et al. (2023) proposed the RNP-GNN, an architecture that can count any substructure by recursively splitting a graph into a collection of vertex-marked subgraphs. We note that this design shares interesting similarities to higher-order subgraph GNNs and also the SpeqNet (Morris et al., 2022). Huang et al. (2023) proposed a generic model called  $\mathrm{I}^2$ -GNN based on a variant of Subgraph 2-GNN, which can count 6-cycle at node-level. In this paper, we show the Local 2-GNN can already count all cycles/paths within 7 nodes even at edge-level while being more efficient than  $\mathrm{I}^2$ -GNN (when using a similar ego network design). Finally, we remark that the polynomial expressivity proposed in Puny et al. (2023) can also be seen as a generalization of substructure counting, which further takes into account the real-valued node/edge features.
+
+Distance-based GNNs. Besides structural information, distance serves as another fundamental attribute of a graph, which, again, is not captured by MPNNs and the 1-WL test. Li et al. (2020) first proposed to improve the expressive power of GNNs by augmenting node features with Distance Encoding (DE). Related to DE, another approach to injecting distance information is the  $k$ -hop MPNN, which aggregates  $k$ -hop neighbors in a message-passing layer (Feng et al., 2022; Abboud et al., 2022; Wang et al., 2023). Feng et al. (2022); Zhang et al. (2023a) proved that the expressive power of  $k$ -hop MPNN is strictly bounded by 2-FWL. Distance can also be naturally incorporated in Graph Transformers through relative positional encoding, yielding the Graphormer architecture that has achieved remarkable performance across various benchmarks (Ying et al., 2021). Recently, Zhang et al. (2023b) built an interesting connection between distance and biconnectivity properties, showing that distance-enhanced GNNs can detect cut vertices and cut edges of a graph. This provides insights into the practical superiority of these models as biconnectivity is closely linked to real applications in chemistry and social network analysis. Zhang et al. (2023a) proved that Local 2-GNN can provably encode the distance (and thus biconnectivity) of a graph.
+
+Spectral-based GNNs. Graph spectra are also a class of fundamental properties and have long been used to design GNN models (Bruna et al., 2014; Defferrard et al., 2016). Balcilar et al. (2021b;a) showed that designing GNNs in the spectral domain can easily break the 1-WL expressivity. For Graph Transformers, Kreuzer et al. (2021); Dwivedi & Bresson (2020); Dwivedi et al. (2022) proposed to incorporate the spectra of the graph Laplacian matrix to boost the expressive power beyond the 1-WL test. Lim et al. (2023) further designed a principled equivariant architecture that takes the Laplacian eigenvalues and eigenvectors as inputs, which generalizes prior work.
+
+Other approaches. Murphy et al. (2019); Chen et al. (2020) proposed Relational Pooling as a general approach to designing expressive GNN architectures, whose basic idea is to implement a permutation-invariant GNN by symmetrizing permutation-sensitive base models. Wijesinghe & Wang (2022b) proposed the GraphSNN, which improves the expressive power of MPNNs by using more distinguishing edge features. Specifically, each edge feature encodes the structure of the over-
+
+lap subgraph of two 1-hop ego networks centered on the two endpoints of the edge. Very recently, Dimitrov et al. (2023) designed a GNN model that can distinguish all planar graphs, thus achieving a strong expressivity in chemical applications since molecular graphs are often planar.
+
+# A.2 BROADER IMPACTS AND ADDITIONAL DISCUSSIONS
+
+Broader impact in graph theory. Due to the fundamental nature of GNN architectures studied in this paper, our theoretical results may potentially have broader impacts on the graph theory community. Specifically, we study the color refinement (CR) algorithms corresponding to four types of (higher-order) GNNs: Subgraph  $(k - 1)$ -GNN, Local  $k$ -GNN, Local  $k$ -FGNN, and  $k$ -FGNN. All algorithms can be seen as natural extensions of the 1-WL test since they all reduce to 1-WL when  $k = 1$ . In the graph theory community, Subgraph GNN has another name known as the vertex-individualized CR algorithm, which appears widely in literature (Babai, 2016; Rattan & Seppelt, 2023; Neuen & Schweitzer, 2018) and has become part of the core algorithm for fast graph isomorphism testing software (e.g., McKay & Piperno, 2014). On the other hand, Local  $k$ -GNN and Local  $k$ -FGNN are surprisingly related to the guarded logic (Barceló et al., 2020; De Rijke, 2000; Baader, 2003), since the aggregations are purely local (guarded by the edge). From this perspective, these CR algorithms can be seen as natural extensions of guarded logic in higher-order scenarios.
+
+Besides these CR algorithms, our new extensions of NED may also have implications in graph theory. In particular, the strong NED (as well as its higher-order version) is elegant and may serve as a descriptive tool to characterize certain graph families. In addition, we establish intrinsic connections between NED and tree decomposition, which may have value in understanding other graph topics related to tree decomposition.
+
+Finally, to our knowledge, the node/edge-level homomorphism and the corresponding subgraph counting abilities of different CR algorithms do not seem to have been systematically investigated before. Whereas in this paper, all graph/node/edge-level expressivity is studied in a unified manner. To achieve this, we introduce additional proof techniques which we believe may facilitate future study in related areas. For example, the original technique for constructing counterexample graphs satisfying Definition 3.1(b) does not apply to node/edge-level settings, since they are no longer counterexample graphs satisfying Definition 3.6(b) (no matter which vertices  $u \in V_G, v \in V_H$  are marked). To address the problem, we propose clique-augmented Fürer graphs, a novel class of counterexample graphs that extend several prior works (e.g., Cai et al., 1992; Fürer, 2001), and conduct a fine-grained analysis of their automorphism property (see Appendix D.2). We believe this new technique can be used to generalize other results from graph-level to node/edge-level settings.
+
+Discussions with Barceló et al. (2021). In the GNN community, Barceló et al. (2021) first proposed to incorporate the homomorphism count of predefined substructures into node features as an approach to enhancing the expressivity of MPNNs. They systematically investigated the questions of what substructures are useful and how the homomorphism information of these substructures can boost the model expressivity to even break out  $k$ -FWL. Yet, they only gave a partial (incomplete) characterization of the substructures that can be counted by the specific  $\mathcal{F}$ -MPNN architecture and did not answer what substructures cannot be encoded, whereas our paper fully addresses both questions for a variety of popular GNN models. Note that these aspects are crucial to ensure that homomorphism expressivity is well-defined. As such, our paper first identifies that homomorphism expressivity is a complete, quantitative expressivity measure to compare different GNN models.
+
+Discussions with the concurrent work of Neuen (2023). After the initial submission, we became aware of a concurrent work (Neuen, 2023), which proved that  $k$ -FWL cannot count any graph with treewidth larger than  $k$  under homomorphism. In our context, this result exactly shows that Definition 3.1(b) is satisfied, and thus the homomorphism expressivity of  $k$ -FWL is well-defined. Notably, their construction of counterexample graphs is also based on Fürer graphs. Nevertheless, the proof technique between the two works is quite different: the proof in Neuen (2023) is built upon a key concept called oddomorphism, while our proof is based on the relation between tree decomposition and the simplified pebble game developed in Zhang et al. (2023a). It is essential to underscore that our results and proof technique are more general and go beyond the standard  $k$ -FWL, in that (i) it applies to a broad range of color refinement algorithms related to practical GNN architectures and (ii) it further extends to node/edge-level homomorphism expressivity. Our theoretical results thus strictly incorporate the results in Neuen (2023). The approach in Neuen (2023) (based on oddomorphism) cannot be easily generalized to these settings.
+
+# B LIMITATIONS AND OPEN DIRECTIONS
+
+There are still several open questions that are not fully explored in this paper. We list them below as promising directions for future study.
+
+Existence of homomorphism expressivity for refinement-based GNN architectures. In this paper, we prove that homomorphism expressivity exists for a wide range of architectures defined in Section 2. On the other hand, we also note that it may not be well-defined for certain pathological GNNs, as illustrated in Appendix F.1. Given this observation, a fundamental question is: what conditions can guarantee that the homomorphism expressivity of a GNN exists? Here, we hypothesize that a very mild condition can suffice. Specifically, we conjecture that as long as a GNN architecture is defined following a general form of color refinement procedure that outputs stable color mappings, its homomorphism expressivity always exists. We leave this conjecture as an important open problem for future study.
+
+Regarding higher-order Local FGNN. This paper characterizes the homomorphism expressivity for three classes of higher-order GNNs: Subgraph  $k$ -GNN, Local  $k$ -GNN, and  $k$ -FGNN. In particular, we introduce the  $k$ -order ear and  $k$ -order strong NED as a way to describe the homomorphism expressivity of Local  $k$ -GNN. However, it remains unclear how to give a simple description of the homomorphism expressivity for Local  $k$ -FGNN that can generalize the concept of almost-strong NED for  $k = 2$ . As such, our current expressiveness hierarchy (Figure 3) does not support Local  $k$ -FGNN yet. We leave this as an open problem and make the following conjecture below. We note that a similar open question has been informally raised in Zhang et al. (2023a).
+
+Conjecture B.1. For all  $k \geq 2$ , Local  $k$ -FGNN is strictly more expressive than Local  $k$ -GNN and strictly less expressive than  $k$ -FGNN.
+
+Expressivity gap between Local 2-(F)GNN and 2-FGNN in practical aspects. We have proved that 2-FGNN is strictly more expressive than Local 2-(F)GNN. However, from a practical perspective, we surprisingly find that the subgraph counting ability of Local 2-(F)GNN matches that of 2-FGNN for all structures within a moderate size (see Table 5), although the former is much more efficient. This leads to the intriguing question of what fundamental gaps exist between the two models in practical aspects, or is the efficiency gain free?
+
+Other architectures. In this paper, we comprehensively study a variety of popular GNN architectures ranging from Subgraph GNNs and Local GNNs to higher-order GNNs, and further link these architectures to Substructure-based GNNs (see Appendix A.1). Yet, we still do not cover all popular GNN architectures, such as the GSWL-based Subgraph GNN (Zhang et al., 2023a; Bevilacqua et al., 2022), SpeqNet (Morris et al., 2022), and  $\mathrm{I}^2$ -GNN (Huang et al., 2023). Moreover, the classes of Distance-based GNNs and Spectral-based GNNs (Appendix A.1) are also widely used in practice, which deserve future study. We would like to raise the question of characterizing the homomorphism expressivity of Distance-based GNNs and Spectral-based GNNs as an important open question. In this way, one can gain deep insights into these models' true expressivity and enable quantitative comparisons between all mainstream architectures. Moreover, it will become clear to what extent other GNN models can encode distance and spectral information about a graph.
+
+# C PROOF OF THEOREM 3.4
+
+This section gives the proof of the main theorem. For ease of reading, we first restate Theorem 3.4:
+
+Theorem 3.4. For all GNN models  $M$  defined in Section 2, the graph family  $\mathcal{F}^M$  satisfying Definition 3.1 exists (and is unique). Moreover, each  $\mathcal{F}^M$  can be separately described below:
+
+- Subgraph GNN:  $\mathcal{F}^{\mathrm{Sub}} = \{F:F$  has an endpoint-shared NED\};  
+- Local 2-GNN:  $\mathcal{F}^{\mathsf{L}} = \{F:F$  has a strong NED\};  
+- Local 2-FGNN:  $\mathcal{F}^{\mathrm{LF}} = \{F:F$  has an almost-strong NED\};  
+- 2-FGNN:  $\mathcal{F}^{\mathrm{F}} = \{F:F$  has a NED\}.
+
+For MPNN, since it is a special case of Subgraph  $k$ -GNN, the proof can be found in Appendix E.3.
+
+# C.1 PRELIMINARY
+
+Additional notations and concepts. Besides the notations defined in Section 2, we further define the following notations. We use the symbol  $\mathcal{G}$  to denote the set of all finite, simple, undirected, labeled graphs. Let  $G = (V_G,E_G,\ell_G)$  be a graph in  $\mathcal{G}$ . When the label is the same for all vertices, we can omit the term  $\ell_G$  and write  $G = (V_G,E_G)$ . Given vertex  $u\in V_{G}$ , denote the degree of  $u$  as  $\deg_G(u) = |N_G(u)|$ , and denote the closed neighborhood of  $u$  as  $N_G[u] = N_G(u)\cup \{u\}$ . The shortest path distance between vertices  $u$  and  $v$  is denoted by  $\mathrm{dis}_G(u,v)$ .
+
+Given a vertex set  $S \subset V_G$ , the induced subgraph of  $G$  over  $S$ , denoted as  $G[S]$ , is the subgraph of  $G$  with vertex set  $S$  and edge set  $\{\{u, v\} \in E_G : u, v \in S\}$ . Similarly, without abuse of notation, given an edge set  $R \subset E_G$ , the induced subgraph of  $G$  over  $R$ , denoted as  $G[R]$ , is the subgraph of  $G$  with vertex set  $\bigcup_{\{u, v\} \in R} \{u, v\}$  and edge set  $R$ . Given a vertex tuple  $\mathbf{u} = (u_1, \dots, u_k)$ , denote by  $G^{\mathbf{u}} = G^{u_1, \dots, u_k}$  the rooted graph obtained from  $G$  by marking vertices  $u_1, \dots, u_k$ . The atomic type of  $G$  over  $\mathbf{u}$ , denoted by  $\mathrm{atp}_G(\mathbf{u})$ , is a  $k \times k$  matrix where the element at position  $(i, j)$  is the tuple  $(\mathbb{I}[u_i = u_j], \mathbb{I}[\{u_i, u_j\} \in E_G])$ . Given two graphs  $G, H$ , the graph union  $G \cup H$  is defined to be the graph  $(V_G \cup V_H, E_G \cup E_H, \ell_{G \cup H})$ , where  $\ell_{G \cup H}(u) := \ell_G(u)$  for all  $u \in V_G$  and  $\ell_{G \cup H}(u) := \ell_H(u)$  for all  $u \in V_H$ . It is well-defined iff  $\ell_G(u) = \ell_H(u)$  for all  $u \in V_G \cap V_H$ . Finally, we use the notation  $G \simeq H$  to denote that  $G$  and  $H$  are isomorphic graphs.
+
+A graph  $T = (V_T, E_T, \ell_T)$  is called a tree if it does not contain cycles. Let  $T^r$  be a rooted tree where  $r$  is the root vertex. For each vertex  $t \in V_T$ , define its depth  $\mathsf{dep}_{T^r}(t) := \mathsf{dis}_T(t, r)$  to be the distance to the root, and denote by  $\mathsf{Desc}_{T^r}(t)$  the set of descendants of  $t$ , namely,  $s \in \mathsf{Desc}_{T^r}(t)$  iff  $\mathsf{dep}_{T^r}(s) = \mathsf{dep}_{T^r}(t) + \mathsf{dis}_T(t, s)$ . For each  $t \in V_T \setminus \{r\}$ , denote by  $\mathsf{pa}_{T^r}(t)$  the parent vertex of  $t$ , i.e., the unique vertex  $s \in N_T(t)$  satisfying  $\mathsf{dep}_{T^r}(t) = \mathsf{dep}_{T^r}(s) + 1$ . Define the subtree of  $T^r$  rooted at node  $t$  by  $T^r[t]$ , which is exactly the induced subgraph  $T[\mathsf{Desc}_{T^r}(t)]^t$  with root  $t$ .
+
+Tree decomposition. Our proof is based on a central concept in graph theory, called tree decomposition. It can be formally defined below:
+
+Definition C.1 (Tree decomposition). Given a graph  $G = (V_G, E_G, \ell_G)$ , its tree decomposition is a tree  $T = (V_T, E_T, \beta_T)$ , where the label function  $\beta_T : V_T \to 2^{V_G}$  satisfies the following conditions:
+
+a) Each tree node  $t \in V_T$  is associated to a non-empty subset of vertices  $\beta_T(t) \subset V_G$  in  $G$ , called a bag. We say tree node  $t$  contains vertex  $u$  if  $u \in \beta_T(t)$ ;  
+b) For each edge  $\{u,v\} \in V_G$  , there exists at least one tree node  $t\in V_T$  that contains the edge, i.e.,  $\{u,v\} \subset \beta_T(t)$  
+c) For each vertex  $u \in V_G$ , all tree nodes  $t$  containing  $u$  form a (non-empty) connected subtree. Formally, denoting  $B_T(u) = \{t \in V_T : u \in \beta_T(t)\}$ , then  $T[B_T(u)]$  is connected.
+
+If  $T$  is a tree decomposition of  $G$ , we call the pair  $(G, T)$  a tree-decomposed graph.
+
+We remark that given a graph  $G$ , there are multiple ways to decompose it and thus its tree decomposition is not unique. Several examples of tree decomposition are given in Figure 5.
+
+Definition C.2 (Treewidth). The width of a tree decomposition is defined as one less than the maximum bag size, i.e.,  $\max_{t\in T}|\beta_T(t)| - 1$ . The treewidth of a graph  $G$ , denoted as  $\mathrm{tw}(G)$ , is the minimum positive integer  $k$  such that there exists a tree decomposition of width  $k$ .
+
+Some important facts about treewidth are listed below:
+
+Fact C.3. For any graph  $G$ , the following hold:
+
+- The treewidth of  $G$  is at most  $|V_G| - 1$ , i.e., a trivial tree decomposition that only has one node  $t$  and  $\beta_T(t) = V_G$ .  
+-  $\operatorname{tw}(G) = |V_G| - 1$  iff  $G$  is a clique.  
+-  $\operatorname{tw}(G) = 1$  iff  $G$  is a forest.
+
+The above definition of tree decomposition is quite flexible without constraints on the structure of the tree or the size of each bag. Below, we define several restricted variants of tree decomposition, which (we will later see) are closely related to the GNN architectures studied in this paper. To begin with, we first define a general concept that slight modifies the original definition (Definition C.1) such that the tree becomes rooted and each bag is a multiset of vertices rather than a set.
+
+![](images/f880c48c3a49ad53d640c5d9cb66668fb8cf6308777fdb2eb36d2e6debcf0923.jpg)  
+Figure 5: Illustration of tree decomposition.
+
+Definition C.4 (Canonical tree decomposition). Given a graph  $G = (V_G, E_G, \ell_G)$ , a canonical tree decomposition of width  $k$  is a rooted tree  $T^r = (V_T, E_T, \beta_T)$  satisfying the following conditions:
+
+a) The depth of  $T$  is even, i.e.  $\max_{t\in V_T}\mathsf{dep}_{T^r}(t)$  is even;  
+b) Each tree node  $t \in V_T$  is associated to a multiset of vertices  $\beta_T(t) \subset V_G$ , called a bag. Moreover,  $|\beta_T(t)| = k$  if  $\mathsf{dep}_{T^r}(t)$  is even and  $|\beta_T(t)| = k + 1$  if  $\mathsf{dep}_{T^r}(t)$  is odd;  
+c) For all tree edges  $\{s,t\} \in E_T$  where  $\mathrm{dep}_{T^r}(s)$  is even and  $\mathrm{dep}_{T^r}(t)$  is odd,  $\beta_{T}(s)\subset \beta_{T}(t)$  (where “ $\subset$ ” denotes the multiset inclusion relation);  
+d) The conditions (b) and (c) in Definition C.1 are satisfied.
+
+As examples, one can check that the tree decomposition of all graphs in Figure 5(b) is canonical, but the tree decomposition in Figure 5(a) is not. An important observation about canonical tree decomposition is shown below:
+
+Proposition C.5. Let  $(F, T^r)$  be any tree-decomposed graph where  $T^r$  is a canonical tree decomposition of  $F$ . For any vertices  $u, v \in V_F$ , either of the following holds:
+
+-  $u$  and  $v$  are in the same bag of  $T^r$ , i.e., there is a node  $t \in V_T$  such that  $\{\{u, v\}\} \subset \beta_T(t)$ ;  
+- The induced subgraph  $T[B_T(u) \cup B_T(v)]$  is disconnected.
+
+Proof. Assume that  $u$  and  $v$  are not in the same bag, i.e.,  $B_{T}(u) \cap B_{T}(v) = \emptyset$ . Pick  $s \in B_{T}(u)$  and  $t \in B_{T}(v)$  such that  $\mathsf{dep}_{T^r}(s)$  and  $\mathsf{dep}_{T^r}(t)$  are minimal, respectively. Without loss of generality, assume that  $\mathsf{dep}_{T^r}(s) \leq \mathsf{dep}_{T^r}(t)$ . Then,  $t \neq r$  is not the root node and thus we can pick its parent  $\mathsf{pa}_{T^r}(t)$ . It follows that  $\mathsf{dep}_{T^r}(t)$  is odd and  $\beta_{T}(\mathsf{pa}_{T^{r}}(t)) \subset \beta_{T}(t)$  by definition of canonical tree decomposition. Therefore,  $u \notin \beta_{T}(\mathsf{pa}_{T^{r}}(t))$ . Moreover, by the assumption that  $\mathsf{dep}_{T^r}(s) \leq \mathsf{dep}_{T^r}(t)$ , any node in  $B_{T}(u)$  is not a descendant of  $t$ . We thus conclude that there does not exist a tree edge such that the two endpoints contain  $u$  and  $v$ , respectively.
+
+Now we are ready to define several restricted variants of canonical tree decomposition:
+
+Definition C.6. Define four families of tree-decomposed graphs  $S^{\mathrm{Sub}}$ ,  $S^{\mathrm{L}}$ ,  $S^{\mathrm{LF}}$ , and  $S^{\mathrm{F}}$  as follows:
+
+a)  $(F,T^{r})\in S^{\mathsf{F}}$  iff  $(F,T^{r})$  satisfies Definition C.4 with width  $k = 2$  
+b)  $(F,T^{r})\in S^{\mathrm{LF}}$  iff  $(F,T^{r})$  satisfies Definition C.4 with width  $k = 2$ , and for any tree node  $t$  of odd depth, it has only one child if  $w\notin \{v:v\in N_G[u],u\in \beta_T(s)\}$  where  $s$  is the parent node of  $t$  and  $w$  is the unique vertex in  $\beta_T(t)\backslash \beta_T(s)$ ;  
+c)  $(F,T^r)\in S^{\mathsf{L}}$  iff  $(F,T^{r})$  satisfies Definition C.4 with width  $k = 2$ , and any tree node  $t$  of odd depth has only one child;  
+d)  $(F, T^r) \in S^{\mathrm{Sub}}$  iff  $(F, T^r)$  satisfies Definition C.4 with width  $k = 2$ , and there exists a vertex  $u \in V_G$  such that  $u \in \beta_T(t)$  for all  $t \in V_T$ .
+
+Examples of tree-decomposed graphs in the four families are illustrated in Figure 5(b).
+
+Before closing this subsection, we define several notations for tree-decomposed graphs:
+
+Definition C.7. Given canonical tree-decomposed graph  $(G,T^r)$  and node  $t\in V_T$  , denote by  $G[T^{r}[t]]$  the subgraph of  $G$  induced by the vertex set  $\{u:u\in \beta_{T^r}(s),s\in \mathsf{Desc}_{T^r}(t)\}$
+
+Definition C.8. Given two canonical tree-decomposed graphs  $(G, T^r)$  and  $(\tilde{G}, \tilde{T}^s)$ , a pair of mappings  $(\rho, \tau)$  is called an isomorphism from  $(G, T^r)$  to  $(\tilde{G}, \tilde{T}^s)$ , denoted by  $(G, T^r) \simeq (\tilde{G}, \tilde{T}^s)$ , if the following hold:
+
+a)  $\rho$  is an isomorphism from  $G$  to  $\tilde{G}$ ;  
+b)  $\tau$  is an isomorphism from  $T^r$  to  $\tilde{T}^s$  (ignoring labels  $\beta$ );  
+c) For any  $t\in T^r$ $\rho (\beta_T(t)) = \beta_{\bar{T}}(\tau (t))$
+
+Equivalent formulation of GNN architectures. Below, we give equivalent definitions for several GNN architectures presented in Section 2, which will be used in subsequent analysis. Let  $G$  be a graph and  $u,v\in V_G$ . For all models  $M$  including Subgraph GNN, Local 2-GNN, Local 2-FGNN, and 2-FGNN, we define the initial color  $\tilde{\chi}_G^{M,(0)}(u,v)$  to be the isomorphism type of vertex pair  $(u,v)$  (i.e.,  $\mathsf{atp}_G(u,v)$  plus labels of each vertex). Note that this matches the original definition except for Subgraph GNN. Then in each iteration  $t$ , the color is updated according to the following formula. Here, for clarity, we denote  $\tilde{\chi}_G^{M,(t)}(u,S) = \{\{\tilde{\chi}_G^{M,(t)}(u,v):v\in S\}\}$  and  $\tilde{\chi}_G^{M,(t)}(S,v) = \{\{\tilde{\chi}_G^{M,(t)}(u,v):u\in S\}\}$  for any model  $M$  and set  $S$ .
+
+- Subgraph GNN:
+
+$$
+\tilde {\chi} _ {G} ^ {\text {S u b}, (t + 1)} (u, v) = \operatorname {h a s h} \left(\tilde {\chi} _ {G} ^ {\text {S u b}, (t)} (u, v), \tilde {\chi} _ {G} ^ {\text {S u b}, (t)} (u, N _ {G} (v)), \tilde {\chi} _ {G} ^ {\text {S u b}, (t)} (u, V _ {G})\right). \tag {7}
+$$
+
+- Local 2-GNN:
+
+$$
+\begin{array}{l} \tilde {\chi} _ {G} ^ {\mathsf {L}, (t + 1)} (u, v) = \operatorname {h a s h} \left(\tilde {\chi} _ {G} ^ {\mathsf {L}, (t)} (u, v), \tilde {\chi} _ {G} ^ {\mathsf {L}, (t)} (u, N _ {G} (v)), \tilde {\chi} _ {G} ^ {\mathsf {L}, (t)} \left(N _ {G} (u), v\right), \right. \tag {8} \\ \left. \tilde {\chi} _ {G} ^ {\mathsf {L}, (t)} (u, V _ {G}), \tilde {\chi} _ {G} ^ {\mathsf {L}, (t)} (V _ {G}, v)\right). \\ \end{array}
+$$
+
+- Local 2-FGNN:
+
+$$
+\begin{array}{l} \tilde {\chi} _ {G} ^ {\mathsf {L F}, (t + 1)} (u, v) = \mathsf {h a s h} \left(\tilde {\chi} _ {G} ^ {\mathsf {L F}, (t)} (u, v), \{\{\left(\tilde {\chi} _ {G} ^ {\mathsf {L F}, (t)} (w, v), \tilde {\chi} _ {G} ^ {\mathsf {L F}, (t)} (u, w)\right): w \in N _ {G} [ u ] \cup N _ {G} [ v ] \}, \right. \\ \left. \tilde {\chi} _ {G} ^ {\mathrm {L F}, (t)} \left(u, V _ {G}\right), \tilde {\chi} _ {G} ^ {\mathrm {L F}, (t)} \left(V _ {G}, v\right)\right). \tag {9} \\ \end{array}
+$$
+
+2-FGNN:
+
+$$
+\begin{array}{l} \tilde {\chi} _ {G} ^ {\mathsf {F}, (t + 1)} (u, v) = \operatorname {h a s h} \left(\tilde {\chi} _ {G} ^ {\mathsf {F}, (t)} (u, v), \{\{\left(\tilde {\chi} _ {G} ^ {\mathsf {F}, (t)} (w, v), \tilde {\chi} _ {G} ^ {\mathsf {F}, (t)} (u, w)\right): w \in V _ {G} \} \}, \right. \tag {10} \\ \left. \tilde {\chi} _ {G} ^ {\mathsf {F}, (t)} (u, V _ {G}), \tilde {\chi} _ {G} ^ {\mathsf {F}, (t)} (V _ {G}, v)\right). \\ \end{array}
+$$
+
+It can be seen that we additionally add global aggregations for these architectures. Moreover, in Local 2-FGNN we replace the neighbors by closed neighbors  $N_G[u] \cup N_G[v]$ . The stable color of  $(u,v)$  for different models is denoted by  $\tilde{\chi}_G^{\mathrm{Sub}}(u,v)$ ,  $\tilde{\chi}_G^{\mathrm{L}}(u,v)$ ,  $\tilde{\chi}_G^{\mathrm{LF}}(u,v)$ ,  $\tilde{\chi}_G^{\mathrm{F}}(u,v)$ , respectively. We have the following result:
+
+Proposition C.9. Let  $M \in \{\text{Sub}, \mathsf{L}, \mathsf{LF}, \mathsf{F}\}$  be any model. For any graphs  $G, H$ ,  $\tilde{\chi}_G^M(G) = \tilde{\chi}_H^M(H)$  iff  $\chi_G^M(G) = \chi_H^M(H)$ . Furthermore, if  $G$  and  $H$  are connected, then for any vertices  $u, v \in V_G$  and  $x, y \in V_H$ ,  $\tilde{\chi}_G^M(u, v) = \tilde{\chi}_H^M(x, y)$  iff  $\chi_G^M(u, v) = \chi_H^M(x, y)$ . In other words, the color mapping  $\tilde{\chi}^M$  is as fine as the original one  $\chi^M$ .
+
+Proof. The proof simply follows from Zhang et al. (2023a, Proposition 4.2 and Theorem 4.4), because (i) node marking in the initial color of Subgraph GNN is as expressive as using the isomorphism type, (ii) the global aggregation does not improve the expressivity when the corresponding local aggregation is presented, (iii) the single-point aggregation does not improve the expressivity in Local 2-FGNN.  $\square$
+
+# C.2 PART 1: TREE DECOMPOSITION
+
+We first define the unfolding tree of different CR algorithms, which is a standard tool in analyzing GNN expressivity.
+
+Definition C.10 (Unfolding tree of Subgraph GNN). Given a graph  $G$ , vertices  $u, v \in V_G$ , and a non-negative integer  $D$ , the depth-2D Subgraph GNN unfolding tree of graph  $G$  at  $(u, v)$ , denoted as  $\left(F_G^{\mathrm{Sub},(D)}(u, v), T_G^{\mathrm{Sub},(D)}(u, v)\right)$ , is a tree-decomposed graph  $(F, T^r) \in S^{\mathrm{Sub}}$  constructed as follows:
+
+1. Initialization. At the beginning,  $F = G[\{u,v\}]$  (if  $u = v$ ,  $F$  only has one vertex), and  $T$  only has a root node  $r$  with  $\beta_T(r) = \{\{u,v\}\}$ . Define a mapping  $\pi : V_F \to V_G$  as  $\pi(u) = u$  and  $\pi(v) = v$ .  
+2. Loop for  $D$  rounds. For each leaf node  $t$  in  $T^r$ , do the following procedure:
+
+Let  $\beta_T(t) = \{\{u,x\}\}$ . For each  $w \in V_G$ , add a fresh child node  $t_w$  to  $T^r$  and designate  $t$  as its parent. Then, consider the following three cases:
+
+a) If  $w \neq \pi(u)$  and  $w \neq \pi(x)$ , then add a fresh vertex  $z$  to  $F$  and extend  $\pi$  with  $\pi(z) = w$ . The label of  $z$  in  $F$  is set by  $\ell_F(z) = \ell_G(w)$ . Define  $\beta_T(t_w) = \beta_T(t) \cup \{\{z\}\}$ . Then, we add edges between  $z$  and  $\beta_T(t)$ , so that  $\pi$  is an isomorphism from  $F[\beta_T(t_w)]$  to  $G[\pi(\beta_T(t_w))]$ .  
+b) If  $w = \pi(u)$ , then we simply set  $\beta_T(t_w) = \beta_T(t) \cup \{\{u\}\}$  without modifying graph  $F$ .  
+c) If  $w = \pi(x)$ , then we simply set  $\beta_T(t_w) = \beta_T(t) \cup \{x\}$  without modifying graph  $F$ .
+
+Finally, add a fresh child node  $t_w'$  to  $T^r$ , designate  $t_w$  as its parent, and set  $\beta_T(t_w')$  based on the following three cases:
+
+a) If  $w \neq \pi(u)$  and  $w \neq \pi(x)$ , then  $\beta_T(t_w') = \{\{u, z\}\}$ .  
+b) If  $w = \pi(u)$ , then  $\beta_T(t_w') = \{\{u, u\}\}$ .  
+c) If  $w = \pi(x)$ , then  $\beta_T(t_w') = \{\{u, x\}\}$ .
+
+It is easy to see that the depth of tree  $T^r$  increases by 2 after each round,  $T^r$  is always a canonical tree decomposition of  $F$ , and  $(F, T^r) \in S^{\mathsf{Sub}}$ . An illustration of the construction of unfolding tree is given in Figure 6(a).
+
+We next define the unfolding tree of Local 2-GNN, which differs in the loop part such that the bags  $\beta_{T}(t_{w})$  and  $\beta_{T}(t_{w}^{\prime})$  now do not necessarily contain  $u$ .
+
+Definition C.11 (Unfolding tree of Local 2-GNN). Given a graph  $G$ , vertices  $u, v \in V_G$ , and a non-negative integer  $D$ , the depth-2D Local 2-GNN unfolding tree of graph  $G$  at  $(u, v)$ , denoted as  $\left(F_G^{\mathsf{L},(D)}(u, v), T_G^{\mathsf{L},(D)}(u, v)\right)$ , is a tree-decomposed graph  $(F, T^r) \in S^\mathsf{L}$  constructed as follows:
+
+1. Initialization. The procedure is exactly the same as Subgraph GNN (Definition C.10).  
+2. Loop for  $D$  rounds. For each leaf node  $t$  in  $T^r$ , do the following procedure:
+
+Let  $\beta_T(t) = \{\{x,y\}\}$ . For each  $w \in V_G$ , add a fresh child node  $t_w$  to  $T$  and designate  $t$  as its parent. Then, consider the following three cases:
+
+a) If  $w \neq \pi(x)$  and  $w \neq \pi(y)$ , then add a fresh vertex  $z$  to  $F$  and extend  $\pi$  with  $\pi(z) = w$ . The label of  $z$  in  $F$  is set by  $\ell_F(z) = \ell_G(w)$ . Define  $\beta_T(t_w) = \beta_T(t) \cup \{\{z\}\}$ . Then, we add edges between  $z$  and  $\beta_T(t)$ , so that  $\pi$  is an isomorphism from  $F[\beta_T(t_w)]$  to  $G[\pi(\beta_T(t_w))]$ .  
+b) If  $w = \pi(x)$ , then we simply set  $\beta_T(t_w) = \beta_T(t) \cup \{\{x\}\}$  without modifying graph  $F$ .  
+c) If  $w = \pi(y)$ , then we simply set  $\beta_T(t_w) = \beta_T(t) \cup \{\{y\}\}$  without modifying graph  $F$ .
+
+Next, add a fresh child node  $t_w'$  in  $T^r$ , designate  $t_w$  as its parent, and set  $\beta_T(t_w')$  based on the following three cases:
+
+a) If  $w \neq \pi(x)$  and  $w \neq \pi(y)$ , then  $\beta_T(t'_w) = \{\{x, z\}\}$ .  
+b) If  $w = \pi(x)$ , then  $\beta_T(t'_w) = \{ \{x, x\} \}$ .  
+c) If  $w = \pi(y)$ , then  $\beta_T(t'_w) = \{ \{x, y\} \}$ .
+
+Finally, we repeat the above procedure (point 2) once more, but this time the bag  $\beta_{T}(t_{w}^{\prime})$  is replaced by the following three cases (changing  $x$  to  $y$ ):
+
+![](images/f0970c4983fc1d23205d9e3b12f7c482f8c80d3c67a5fab62872798fcbbc3dbd.jpg)  
+(a)
+
+![](images/a29759fcfc14622ca3b6dfb31fdd215d97ea759c5e7316ee08dd80f7289f62eb.jpg)
+
+![](images/be2394636d3e86199f69686eb0f0270d9b6185180e6484739b5b7398b8625fcd.jpg)  
+G
+
+![](images/6631bd04768c7185219cc1ca7a5e3965c85649a0466c8f69e7b21241f8453157.jpg)  
+$F_{G}^{\mathrm{Sub},(1)}(1,2)$  
+$F_{G}^{\mathrm{L},(1)}(1,2)$
+
+![](images/71f5d05d3ed4a4d84d30274fa19b958612f767fdc93337ffd13051fa99926960.jpg)
+
+![](images/b28344026f6219b8397cd3fe316014f356eb016486e72d6a450eda55c0d121a1.jpg)  
+G
+
+![](images/3516d23cb44d351fed1e422bf9fed664343c07d6e0a360cebbb829994a4633d3.jpg)  
+$④^{\prime \prime}$
+
+![](images/a7ab05cc814a7595f7b27f26a603cbf6b6f76e8d7c57596895f586e833eb22cd.jpg)  
+$T_{G}^{\mathrm{Sub},(1)}(1,2)$  
+$T_{G}^{\mathbf{L},(1)}(1,2)$
+
+![](images/3b8bc1169f7a22dd55632cff04fc6152c8f9a737f66895f8a4cdde961909ff22.jpg)  
+(c)  
+G  
+(d)  
+G
+
+![](images/aab77d78b922d03191b4a92518164af91a056fbe1945b45b2bfb0e9dd74403ba.jpg)  
+$F_{G}^{\mathrm{LF},(1)}(1,2)$  
+$F_{G}^{\mathsf{F},(1)}(1,2)$
+
+![](images/c35c72d8b7d7993aec491b451ea27c376b5bc355a36234d3db5ea16a290f7eac.jpg)  
+$T_{G}^{\mathrm{LF},(1)}$  (1,2)
+
+![](images/50334674602320888695697b2c7dc06d34e49c04fca5e6372ad63b686fb99431.jpg)  
+$T_{G}^{\mathsf{F},(1)}(1,2)$  
+Figure 6: The depth-2 unfolding tree of graph  $G$  at vertex pair (1,2) for Subgraph GNN, Local 2-GNN, Local 2-FGNN, and 2-FGNN, respectively.
+
+a) If  $w \neq \pi(x)$  and  $w \neq \pi(y)$ , then  $\beta_T(t'_w) = \{\{y, z\}\}$ .  
+b) If  $w = \pi(y)$ , then  $\beta_T(t'_w) = \{\{y, y\}\}$ .  
+c) If  $w = \pi(x)$ , then  $\beta_T(t_w') = \{\{x, y\}\}$ .
+
+An illustration of the construction of unfolding tree is given in Figure 6(b).
+
+We next define the unfolding tree of Local 2-FGNN, which differs in the loop part such that the node  $t_w$  can have two children under certain conditions.
+
+Definition C.12 (Unfolding tree of Local 2-FGNN). Given a graph  $G$ , vertices  $u, v \in V_G$ , and a non-negative integer  $D$ , the depth-2D Local 2-FGNN unfolding tree of graph  $G$  at  $(u, v)$ , denoted as  $\left(F_G^{\mathsf{LF},(D)}(u, v), T_G^{\mathsf{LF},(D)}(u, v)\right)$ , is a tree-decomposed graph  $(F, T^r) \in S^{\mathsf{LF}}$  constructed as follows:
+
+1. Initialization. The procedure is exactly the same as Subgraph GNN (Definition C.10).  
+2. Loop for  $D$  rounds. For each leaf node  $t$  in  $T^r$ , do the following procedure:
+
+Let  $\beta_T(t) = \{\{x,y\}\}$ . For each  $w \in N_G[\pi(x)] \cup N_G[\pi(y)]$ , add a fresh child node  $t_w$  to  $T$  and designate  $t$  as its parent. Then, consider the following three cases:
+
+a) If  $w \neq \pi(x)$  and  $w \neq \pi(y)$ , then add a fresh vertex  $z$  to  $F$  and extend  $\pi$  with  $\pi(z) = w$ . The label of  $z$  in  $F$  is set by  $\ell_F(z) = \ell_G(w)$ . Define  $\beta_T(t_w) = \beta_T(t) \cup \{\{z\}\}$ . Then, we add edges between  $z$  and  $\beta_T(t)$ , so that  $\pi$  is an isomorphism from  $F[\beta_T(t_w)]$  to  $G[\pi(\beta_T(t_w))]$ .  
+b) If  $w = \pi(x)$ , then we simply set  $\beta_T(t_w) = \beta_T(t) \cup \{\{x\}\}$  without modifying graph  $F$ .  
+c) If  $w = \pi(y)$ , then we simply set  $\beta_T(t_w) = \beta_T(t) \cup \{\{y\}\}$  without modifying graph  $F$ .
+
+Next, add two fresh children  $t_w'$  and  $t_w''$  in  $T^r$ , designate  $t_w$  as their parent, and set  $\beta_T(t_w')$  and  $\beta_T(t_w'')$  based on the following three cases:
+
+a) If  $w \neq \pi(x)$  and  $w \neq \pi(y)$ , then  $\beta_T(t_w') = \{\{x, z\}\}$  and  $\beta_T(t_w'') = \{\{y, z\}\}$ .  
+b) If  $w = \pi(x)$ , then  $\beta_T(t_w') = \{ \{x, x\} \}$  and  $\beta_T(t_w'') = \{ \{x, y\} \}$ .  
+c) If  $w = \pi(y)$ , then  $\beta_T(t_w') = \{ \{x, y\} \}$  and  $\beta_T(t_w'') = \{ \{y, x\} \}$ .
+
+For each  $w \notin N_G[\pi(x)] \cup N_G[\pi(y)]$ , follow the same procedure as Local 2-GNN (Definition C.11).
+
+An illustration of the construction of unfolding tree is given in Figure 6(c).
+
+We finally define the unfolding tree of 2-FGNN, which differs in the loop part such that all nodes  $t_w$  have two children.
+
+Definition C.13 (Unfolding tree of 2-FGNN). Given a graph  $G$ , vertices  $u, v \in V_G$ , and a non-negative integer  $D$ , the depth-2D 2-FGNN unfolding tree of graph  $G$  at  $(u, v)$ , denoted as  $\left(F_G^{\mathrm{Sub},(D)}(u, v), T_G^{\mathrm{Sub},(D)}(u, v)\right)$ , is a tree-decomposed graph  $(F, T^r) \in S^{\mathsf{F}}$  constructed as follows:
+
+1. Initialization. The procedure is exactly the same as Local 2-FGNN (Definition C.12).  
+2. Loop for  $D$  rounds. The procedure is similar to Local 2-FGNN (Definition C.12) except that the condition  $w \in N_G[\pi(x)] \cup N_G[\pi(y)]$  is relaxed to all vertices.
+
+An illustration of the construction of unfolding tree is given in Figure 6(d).
+
+We are now ready to present the first core result:
+
+Lemma C.14. Let  $M \in \{\text{Sub}, \mathsf{L}, \mathsf{LF}, \mathsf{F}\}$  be any model. For any two graphs  $G, H$ , any vertices  $u, v \in V_G, x, y \in V_H$ , and any non-negative integer  $D$ ,  $\tilde{\chi}_G^{M,(D)}(u,v) = \tilde{\chi}_H^{M,(D)}(x,y)$  iff there exists an isomorphism  $(\rho, \tau)$  from  $\left(F_G^{M,(D)}(u,v), T_G^{M,(D)}(u,v)\right)$  to  $\left(F_H^{M,(D)}(x,y), T_H^{M,(D)}(x,y)\right)$  such that  $\rho(u) = x, \rho(v) = y$ .
+
+Proof. Here, we only give the proof for Local 2-GNN, and the proofs for Subgraph GNN, Local 2-FGNN and 2-FGNN are almost the same so we omit them for clarity.
+
+Proof for Local 2-GNN. The proof is based on induction over  $D$ . When  $D = 0$ , the theorem obviously holds. Now assume that the theorem holds for  $D \leq d$ , and consider  $D = d + 1$ . Below, we omit  $\mathsf{L}$  in the corner mark for clarity.
+
+1. We first prove that  $\tilde{\chi}_G^{(d + 1)}(u,v) = \tilde{\chi}_H^{(d + 1)}(x,y)$  implies that there exists an isomorphism  $(\rho ,\tau)$  from  $\left(F_G^{(d + 1)}(u,v),T_G^{(d + 1)}(u,v)\right)$  to  $\left(F_H^{(d + 1)}(x,y),T_H^{(d + 1)}(x,y)\right)$  such that  $\rho (u) = x,\rho (v) = y$  . If  $\tilde{\chi}_G^{(d + 1)}(u,v) = \tilde{\chi}_H^{(d + 1)}(x,y)$  , then
+
+$$
+\begin{array}{l} \left\{\left\{\left(\tilde {\chi} _ {G} ^ {(d)} (u, w), \mathsf {a t p} _ {G} (u, v, w)\right): w \in V _ {G} \right\} \right\} = \left\{\left\{\left(\tilde {\chi} _ {H} ^ {(d)} (x, z), \mathsf {a t p} _ {H} (x, y, z)\right): z \in V _ {H} \right\}, \right. (11) \\ \{\{(\tilde {\chi} _ {G} ^ {(d)} (w, v), \mathsf {a t p} _ {G} (u, v, w)): w \in V _ {G} \} \} = \{\{(\tilde {\chi} _ {H} ^ {(d)} (z, y), \mathsf {a t p} _ {H} (x, y, z)): z \in V _ {H} \}. (12) \\ \end{array}
+$$
+
+Let  $n = |V_G| = |V_H|$ . Thus, we can denote  $V_G = \{w_1, \dots, w_n\} = \{w_1', \dots, w_n'\}$  and  $V_H = \{z_1, \dots, z_n\} = \{z_1', \dots, z_n'\}$  such that
+
+-  $(\tilde{\chi}_G^{(d)}(u, w_i), \mathsf{atp}_G(u, v, w_i)) = (\tilde{\chi}_H^{(d)}(x, z_i), \mathsf{atp}_H(x, y, z_i))$  for all  $i \in [n]$ ;  
+-  $(\tilde{\chi}_G^{(d)}(w_i', v), \mathsf{atp}_G(u, v, w_i')) = (\tilde{\chi}_H^{(d)}(z_i', y), \mathsf{atp}_H(x, y, z_i'))$  for all  $i \in [n]$ .
+
+On the other hand, by definition of tree unfolding, we have
+
+$$
+F _ {G} ^ {(d + 1)} (u, v) = \left(\bigcup_ {w _ {i}} F _ {G} ^ {(d)} (u, w _ {i})\right) \cup \left(\bigcup_ {w _ {i} ^ {\prime}} F _ {G} ^ {(d)} (w _ {i} ^ {\prime}, v)\right) \cup F _ {G} ^ {(1)} (u, v),
+$$
+
+$$
+F _ {H} ^ {(d + 1)} (x, y) = \left(\bigcup_ {z _ {i}} F _ {H} ^ {(d)} (x, z _ {i})\right) \cup \left(\bigcup_ {z _ {i} ^ {\prime}} F _ {H} ^ {(d)} (z _ {i} ^ {\prime}, y)\right) \cup F _ {H} ^ {(1)} (x, y),
+$$
+
+![](images/674f09770f4f210781d42e760384377169b82fa8963ecb20080e13cd099ef355.jpg)  
+Figure 7: Illustration of the proof of Lemma C.14.
+
+![](images/bed23a54f04d7099280690fd9344766e4f3300f57d3451cb048c4c79d9187c73.jpg)
+
+where  $\cup$  represents the graph union. Here, all  $w_{i}, w_{j}^{\prime} \notin \{u, v\}$  in different graphs are treated as different vertices when taking the union, while  $u, v$  in different graphs are shared. See Figure 7 for an illustration of the above equations.
+
+By induction, there exists an isomorphism  $(\rho_i,\tau_i)$  from  $\left(F_G^{(d)}(u,w_i),T_G^{(d)}(u,w_i)\right)$  to  $\left(F_H^{(d)}(x,z_i),T_H^{(d)}(x,z_i)\right)$  such that  $\rho_{i}(u) = x$ ,  $\rho_{i}(w_{i}) = z_{i}$ $(i\in [n])$ , and there exist an isomorphism  $\rho_{i}^{\prime}$  from  $\left(F_G^{(d)}(w_i',v),T_G^{(d)}(w_i',v)\right)$  to  $\left(F_H^{(d)}(z_i',y),T_H^{(d)}(z_i',y)\right)$  such that  $\rho_{i}^{\prime}(v) = y$ ,  $\rho_{i}^{\prime}(w_{i}^{\prime}) = z_{i}^{\prime}$ $(i\in [n])$ . Moreover, we have  $\mathsf{atp}_G(u,v,w_i) = \mathsf{atp}_H(x,y,z_i)$  and  $\mathsf{atp}_G(u,v,w_i') = \mathsf{atp}_H(x,y,z_i')$  for  $i\in [n]$ , which implies that  $F_{G}^{(1)}(u,v)$  is isomorphic to  $F_H^{(1)}(x,y)$ . Therefore, if we construct  $\tilde{\rho}$  by merging all  $\rho_{i}$  and  $\rho_{i}'$ $(i\in [n])$ , and construct  $\tilde{\tau}$  by merging all  $\tau_{i}$  and  $\tau_{i}'$  and further specifying an appropriate mapping between tree nodes of depth no more than 1 in  $T_G^{(d + 1)}(u,v)$  and  $T_H^{(d + 1)}(x,y)$ , then it is straightforward to see that  $(\tilde{\rho},\tilde{\tau})$  is well-defined and is an isomorphism from  $\left(F_G^{(d + 1)}(u,v),T_G^{(d + 1)}(u,v)\right)$  to  $\left(F_H^{(d + 1)}(x,y),T_H^{(d + 1)}(x,y)\right)$  such that  $\tilde{\rho}(u) = x$ ,  $\tilde{\rho}(v) = y$ .
+
+2. We next prove that if there exists an isomorphism  $(\rho, \tau)$  from the tree-decomposed graph  $\left(F_G^{(d+1)}(u, v), T_G^{(d+1)}(u, v)\right)$  to  $\left(F_H^{(d+1)}(x, y), T_H^{(d+1)}(x, y)\right)$  such that  $\rho(u) = x$ ,  $\rho(v) = y$ , then  $\tilde{\chi}_G^{(d+1)}(u, v) = \tilde{\chi}_H^{(d+1)}(x, y)$ .
+
+Without loss of generality, assume  $u \neq v$  and  $x \neq y$ . Since  $\tau$  is an isomorphism from  $T_G^{(d+1)}(u,v)$  to  $T_H^{(d+1)}(x,y)$ ,  $\tau$  maps all tree nodes of depth 1 in  $T_G^{(d+1)}(u,v)$  to all tree nodes of depth 1 in  $T_H^{(d+1)}(x,y)$ . Let  $s_1, \dots, s_n$  be all nodes of depth 2 in  $T_G^{(d+1)}(u,v)$  such that  $u \in \beta_{T_G^{(d+1)}(u,v)}(s_i)$  (it follows that  $n = |V_G|$ ), and let  $s_i'$  be the parent of  $s_i$ . Similarly, let  $t_1, \dots, t_n$  be all nodes of depth 2 in  $T_H^{(d+1)}(x,y)$  such that  $x \in \beta_{T_H^{(d+1)}(x,y)}(t_i)$ , and let  $t_i'$  be the parent of  $t_i$ . Moreover, we can arrange the order so that the following are satisfied (for each  $i \in [n]$ ):
+
+a)  $\tau$  is an isomorphism from the subtree  $T_{G}^{(d + 1)}(u,v)[s_i]$  to the subtree  $T_{H}^{(d + 1)}(x,y)[t_i]$ .  
+b) For all  $s \in \mathsf{Desc}_{T_G^{(d + 1)}(u,v)}(s_i)$ ,  $\rho(\beta_{T_G^{(d + 1)}(u,v)}(s)) = \beta_{T_H^{(d + 1)}(x,y)}(\tau(s))$ .  
+c) By definition of the unfolding tree,  $\rho$  is an isomorphism from the induced subgraph  $F_{G}^{(d + 1)}(u,v)[T_{G}^{(d + 1)}(u,v)[s_{i}]]$  to the induced subgraph  $F_{H}^{(d + 1)}(x,y)[T_{H}^{(d + 1)}(x,y)[t_{i}]]$  (see Definition C.7).  
+d) Let  $\beta_{T_G^{(d + 1)}(u,v)}(s_i') = \{\{u,v,\tilde{w}_i\}\}$  and  $\beta_{T_H^{(d + 1)}(x,y)}(t_i') = \{\{x,y,\tilde{z}_i\}\}$ . Then,  $\rho (\tilde{w}_i) = \tilde{z}_i$ , and thus  $\{v,\tilde{w}_i\} \in E_{F_G^{(d + 1)}(u,v)}$  iff  $\{y,\tilde{z}_i\} \in E_{F_H^{(d + 1)}(x,y)}$ .
+
+By items (a) to (c),  $\left(F_{G}^{(d + 1)}(u,v)\left[T_{G}^{(d + 1)}(u,v)[s_{i}]\right],T_{G}^{(d + 1)}(u,v)[s_{i}]\right)$  is isomorphic to  $\left(F_H^{(d + 1)}(x,y)\left[T_H^{(d + 1)}(x,y)[t_i]\right],T_H^{(d + 1)}(x,y)[t_i]\right)$ . On the other hand, by definition of the unfolding tree,  $\left(F_G^{(d + 1)}(u,v)\left[T_G^{(d + 1)}(u,v)[s_i]\right],T_G^{(d + 1)}(u,v)[s_i]\right)$  is isomorphic to
+
+the depth-2d unfolding tree  $\left(F_{G}^{(d)}(u,w_{i}),T_{G}^{(d)}(u,w_{i})\right)$  for some  $w_{i}\in V_{G}$  satisfying that  $\{w_i,v\} \in E_G$  iff  $\{\tilde{w}_i,v\} \in E_{F_G^{(d + 1)}(u,v)}$
+
+Similarly,  $\left(F_H^{(d + 1)}(x,y)\left[T_H^{(d + 1)}(x,y)[t_i]\right],T_H^{(d + 1)}(x,y)[t_i]\right)$  is isomorphic to  $\left(F_{H}^{(d)}(x,z_{i}),T_{H}^{(d)}(x,z_{i})\right)$  for some  $z_{i}\in V_{H}$  satisfying that  $\{z_i,y\} \in E_H$  iff  $\{\tilde{z}_i,y\} \in E_{F_H^{(d + 1)}(x,y)}$ . Combining all the above equivalence yields that  $\left(F_G^{(d)}(u,w_i),T_G^{(d)}(u,w_i)\right)$  is isomorphic to  $\left(F_H^{(d)}(x,z_i),T_H^{(d)}(x,z_i)\right)$ , and  $\{w_i,v\} \in E_G$  iff  $\{z_i,y\} \in E_H$ .
+
+By induction, we have  $\tilde{\chi}_G^{(d)}(u,w_i) = \tilde{\chi}_H^{(d)}(x,z_i)$ . Moreover, we clearly have that  $\{u,w_i\} \in E_G$  iff  $\{x,z_i\} \in E_H$ , and  $\{u,v\} \in E_G$  iff  $\{x,y\} \in E_H$ . Therefore,
+
+$$
+\left(\tilde {\chi} _ {G} ^ {(d)} (u, w _ {i}), \mathsf {a t p} _ {G} (u, v, w _ {i})\right) = \left(\tilde {\chi} _ {H} ^ {(d)} (x, z _ {i}), \mathsf {a t p} _ {H} (x, y, z _ {i})\right). \tag {13}
+$$
+
+Next, note that  $\tilde{w}_i$  are different from each other for  $i\in [n]$  by definition of unfolding tree. Thus,  $w_{i}$  are also different from each other. It follows that
+
+$$
+\{\{(\tilde {\chi} _ {G} ^ {(d)} (u, w), \mathsf {a t p} _ {G} (u, v, w)): w \in V _ {G} \} \} = \{\{(\tilde {\chi} _ {H} ^ {(d)} (x, z), \mathsf {a t p} _ {H} (x, y, z)): z \in V _ {H} \} \}. \tag {14}
+$$
+
+Again using the same analysis as before, we obtain
+
+$$
+\{\{(\tilde {\chi} _ {G} ^ {(d)} (w, v), \mathsf {a t p} _ {G} (u, v, w)): w \in V _ {G} \} \} = \{\{(\tilde {\chi} _ {H} ^ {(d)} (z, y), \mathsf {a t p} _ {H} (x, y, z)): z \in V _ {H} \} \}. \tag {15}
+$$
+
+It remains to prove that  $\tilde{\chi}_G^{(d)}(u,v) = \tilde{\chi}_H^{(d)}(x,y)$ . To prove this, note that Equation (14) implies that
+
+$$
+\left\{\left\{\left(\tilde {\chi} _ {G} ^ {(d ^ {\prime})} (u, w), \mathsf {a t p} _ {G} (u, v, w)\right): w \in V _ {G} \right\} \right\} = \left\{\left\{\left(\tilde {\chi} _ {H} ^ {(d ^ {\prime})} (x, z), \mathsf {a t p} _ {H} (x, y, z)\right): z \in V _ {H} \right\} \right\} \tag {16}
+$$
+
+holds for all  $0 \leq d' \leq d$ , and Equation (15) implies that
+
+$$
+\left\{\left\{\left(\tilde {\chi} _ {G} ^ {(d ^ {\prime})} (w, v), \mathsf {a t p} _ {G} (u, v, w)\right): w \in V _ {G} \right\}\right\} = \left\{ \right.\left\{\left(\tilde {\chi} _ {H} ^ {(d)} (z, y), \mathsf {a t p} _ {H} (x, y, z)\right): z \in V _ {H} \right\}. \tag {17}
+$$
+
+holds for all  $0 \leq d' \leq d$ . Combined with Equations (16) and (17) and the fact that  $\tilde{\chi}_{G}^{(0)}(u,v) = \tilde{\chi}_{H}^{(0)}(x,y)$ , we can incrementally prove that  $\tilde{\chi}_{G}^{(d')}(u,v) = \tilde{\chi}_{H}^{(d')}(x,y)$  for all  $d' \leq d + 1$ .
+
+We have thus concluded the proof.
+
+Definition C.15. Let  $M \in \{\text{Sub}, \mathsf{L}, \mathsf{LF}, \mathsf{F}\}$  be any model. Given a graph  $G$  and a tree-decomposed graph  $(F, T^r)$ , define
+
+$$
+\mathsf {c n t} ^ {M} \left(\left(F, T ^ {r}\right), G\right) := \left| \left\{\left(u, v\right) \in V _ {G} ^ {2}: \exists D \in \mathbb {N} _ {+} \text {s . t .} \left(F _ {G} ^ {M, (D)} (u, v), T _ {G} ^ {M, (D)} (u, v)\right) \simeq \left(F, T ^ {r}\right) \right\} \right|,
+$$
+
+where  $\left(F_G^{M,(D)}(u,v),T_G^{M,(D)}(u,v)\right)$  is the depth-2D unfolding tree of  $G$  at  $(u,v)$  for model  $M$ .
+
+Corollary C.16. Let  $M \in \{\text{Sub}, \mathsf{L}, \mathsf{LF}, \mathsf{F}\}$  be any model. For any graphs  $G, H$ ,  $\chi_G^M(G) = \chi_H^M(H)$  iff  $\mathsf{cnt}^M((F, T^r), G) = \mathsf{cnt}^M((F, T^r), H)$  holds for all  $(F, T^r) \in S^M$ .
+
+Proof. “ $\Rightarrow$ ”. If  $\chi_G^M (G) = \chi_H^M (H)$ , then  $\{\{\chi_G^M (u,v):u,v\in V_G\} \} = \{\{\chi_H^M (x,y):x,y\in V_H\}\}$ . For each color  $c$  in the above multiset, pick  $u,v\in V_{G}$  with  $\chi_G^M (u,v) = c$ . It follows that if  $(F,T^{r})\simeq (F_{G}^{M,(D)}(u,v),T_{G}^{M,(D)}(u,v))\in S^{M}$  for some  $D$ , then  $\mathsf{cnt}^M ((F,T^r),G) = |\{(u,v)\in V_G^2:\chi_G^M (u,v) = c\} | = |\{(x,y)\in V_H^2:\chi_H^M (x,y) = c\} | = \mathsf{cnt}^M ((F,T^r),H)$  by Lemma C.14. On the other hand, if  $(F,T^{r})\neq (F_{G}^{M,(D)}(u,v),T_{G}^{M,(D)}(u,v))$  for all  $u,v\in V_{G}$  and all  $D$ , then clearly  $\mathsf{cnt}^M ((F,T^r),G) = \mathsf{cnt}^M ((F,T^r),H) = 0$ .
+
+$\Longleftarrow$  . If  $\mathsf{cnt}^M ((F,T^r),G) = \mathsf{cnt}^M ((F,T^r),H)$  holds for all  $(F,T^{r})\in S^{M}$  , it clearly holds for all  $(F_M^{(D)}(u,v),T_M^{(D)}(u,v))$  with  $u,v\in V_G$  and a sufficiently large  $D$  . This guarantees that for all color  $c,\left|\{(u,v)\in V_G^2:\chi_G^M (u,v) = c\} \right| = \left|\{(x,y)\in V_H^2:\chi_H^M (x,y) = c\} \right|$  by Lemma C.14. Therefore,  $\{\{\chi_G^M (u,v):u,v\in V_G\} \} = \{\{\chi_H^M (x,y):x,y\in V_H\} \}$  , concluding the proof.
+
+We next define an important concept called bag isomorphism (Dell et al., 2018).
+
+Definition C.17. Given a tree-decomposed graph  $(F, T^r)$  and a graph  $G$ , a bag isomorphism from  $(F, T^r)$  to  $G$  (abbreviated as "bIso") is a homomorphism  $f$  from  $F$  to  $G$  such that, for all  $t \in V_T$ ,  $f$  is an isomorphism from  $F[\beta_T(t)]$  to  $G[f(\beta_T(t))]$ . Denote  $\mathsf{BIso}((F, T^r), G)$  to be the set of all bag isomorphisms from  $(F, T^r)$  to  $G$ , and denote  $\mathsf{bIso}((F, T^r), G) = |\mathsf{BIso}((F, T^r), G)|$ .
+
+Remark C.18. To prove that a mapping  $f$  from  $(F, T^r)$  to  $G$  is a bIso, it suffices to prove the following conditions:
+
+1. For any two different vertices  $u, v \in V_G$  in the same bag,  $f(u) \neq f(v)$ ;  
+2. For any two vertices  $u, v \in V_G$  in the same bag,  $\{u, v\} \in E_F$  iff  $\{f(u), f(v)\} \in E_G$ ;  
+3. For any  $u\in V_G,\ell_F(u) = \ell_G(f(u))$
+
+The following fact is straightforward from the construction of the unfolding tree:
+
+Fact C.19. Let  $M \in \{\text{Sub}, \mathsf{L}, \mathsf{LF}, \mathsf{F}\}$  be any model considered above. For any graph  $G$ , any vertex pair  $(u, v) \in V_G^2$ , and any non-negative integer  $D$ , there is a bIso  $\pi$  from  $\left(F_G^{M,(D)}(u, v), T_G^{M,(D)}(u, v)\right)$  to  $G$ .
+
+Similarly, we need the following concept to describe the relation between two tree-decomposed graphs. We note that these technical concepts also appeared in Dell et al. (2018).
+
+Definition C.20. Given two tree-decomposed graphs  $(F, T^r)$  and  $(\tilde{F}, \tilde{T}^s)$ , a pair of mappings  $(\rho, \tau)$  is called homomorphism from  $(F, T^r)$  to  $(\tilde{F}, \tilde{T}^s)$  if it satisfies the following conditions:
+
+a)  $\tau$  is a homomorphism from  $T$  to  $\tilde{T}$  (ignoring labels  $\beta$ ) and is depth-preserving, i.e.,  $\mathsf{dep}_{T^r}(t) = \mathsf{dep}_{\tilde{T}^s}(\tau(t))$  for all  $t \in V_T$ ;  
+b) For all  $t \in V_T$ ,  $\rho$  is a homomorphism from  $F[\beta_T(t)]$  to  $\tilde{F}[\beta_{\tilde{T}}(\tau(t))]$ . Note that this implies that  $\rho$  is a homomorphism from  $F$  to  $\tilde{F}$ .  
+c) The depth of  $T^r$  is equal to the depth of  $\hat{T}^s$ .
+
+Definition C.21. Under Definition C.20,  $(\rho, \tau)$  is further called a bag-isomorphism homomorphism (abbreviated as "bIsoHom") from  $(F, T^r)$  to  $(\tilde{F}, \tilde{T}^s)$  if it is a homomorphism satisfying that, for all  $t \in V_T$ ,  $\rho$  is an isomorphism from  $F[\beta_T(t)]$  to  $\tilde{F}[\beta_{\tilde{T}}(\tau(t))]$ . Furthermore,  $(\rho, \tau)$  is called a bIsoSurj if  $\tau$  is surjective; and  $(\rho, \tau)$  is called a bIsoInj if  $\tau$  is injective. We use BIsoHom  $\left((F, T^r), (\tilde{F}, \tilde{T}^s)\right)$  to denote the set of bIsoHoms from  $(F, T^r)$  to  $(\tilde{F}, \tilde{T}^s)$ , and let bIsoHom  $\left((F, T^r), (\tilde{F}, \tilde{T}^s)\right) = \left|\text{BIsoHom}\left((F, T^r), (\tilde{F}, \tilde{T}^s)\right)\right|$ . The notations BIsoSurj  $\left((F, T^r), (\tilde{F}, \tilde{T}^s)\right)$ , bIsoSurj  $\left((F, T^r), (\tilde{F}, \tilde{T}^s)\right)$ , BIsolnj  $\left((F, T^r), (\tilde{F}, \tilde{T}^s)\right)$ , and bIsolnj  $\left((F, T^r), (\tilde{F}, \tilde{T}^s)\right)$  are defined accordingly.
+
+Remark C.22. In the above definition, the depth of a tree  $T^r$  is the maximal depth among all tree nodes in  $T^r$ . Note that we do not require that all leaf nodes have the same depth in  $T^r$ .
+
+We are now ready to present the second core result:
+
+Lemma C.23. Let  $M \in \{\text{Sub}, \mathsf{L}, \mathsf{LF}, \mathsf{F}\}$  be any model. For any graph  $G$  and tree-decomposed graph  $(F, T^r) \in S^M$ ,
+
+$$
+\operatorname {b l s o} \left(\left(F, T ^ {r}\right), G\right) = \sum_ {\left(\tilde {F}, \tilde {T} ^ {s}\right) \in \mathcal {S} ^ {M}} \operatorname {b l s o H o m} \left(\left(F, T ^ {r}\right), \left(\tilde {F}, \tilde {T} ^ {s}\right)\right) \cdot \operatorname {c n t} ^ {M} \left(\left(\tilde {F}, \tilde {T} ^ {s}\right), G\right). \tag {18}
+$$
+
+Here, the summation ranges over all non-isomorphic (tree-decomposed) graphs in  $S^M$  and is well-defined as there are only a finite number of graphs making the value in the summation non-zero.
+
+Proof. Here, we only give the proof for Local 2-GNN, and the proofs for Subgraph GNN, Local 2-FGNN and 2-FGNN are almost the same so we omit them for clarity.
+
+Proof for Local 2-GNN. We assume that the root bag of  $(F, T^r)$  is  $\{\{u, v\}\}$ , and the depth of  $(F, T^r)$  is  $2d$ . Let  $y, z \in V_G$  be any vertices in  $G$ , and denote  $(F_G^{(d)}(y, z), T_G^{(d)}(y, z))$  as the depth-2d Local 2-GNN unfolding tree at  $(y, z)$ . Define the following two sets:
+
+$$
+\begin{array}{l} S _ {1} (y, z) = \left\{g: g \in \mathsf {B l s o} \left((F, T ^ {r}), G\right), g (u) = y, g (v) = z \right\}, \\ S _ {2} (y, z) = \left\{(\rho , \tau): (\rho , \tau) \in \mathsf {B l s o H o m} \left((F, T ^ {r}), \left(F _ {G} ^ {(d)} (y, z), T _ {G} ^ {(d)} (y, z)\right)\right), \rho (u) = y, \rho (v) = z \right\}. \\ \end{array}
+$$
+
+Then, Lemma C.23 is equivalent to the following equation:
+
+$$
+\sum_ {y, z \in V _ {G}} | S _ {1} (y, z) | = \sum_ {y, z \in V _ {G}} | S _ {2} (y, z) |.
+$$
+
+We will prove that  $|S_{1}(y,z)| = |S_{2}(y,z)|$  for all  $y,z\in V_G$ .
+
+Given  $y, z \in V_G$ , according to Fact C.19, there exists a bIso  $\pi$  from  $(F_G^{(d)}(y,z), T_G^{(d)}(y,z))$  to graph  $G$ . Define a mapping  $\sigma$  such that  $\sigma(\rho, \tau) = \pi \circ \rho$  for all  $(\rho, \tau) \in S_2(y,z)$ . It suffices to prove that  $\sigma$  is a bijection from  $S_2(y,z)$  to  $S_1(y,z)$ .
+
+1. We first prove that  $\sigma$  is a mapping from  $S_{2}(y,z)$  to  $S_{1}(y,z)$ , i.e.,  $\pi \circ \rho \in S_{1}(y,z)$  for all  $(\rho, \tau) \in S_{2}(y,z)$ . First, we clearly have  $(\pi \circ \rho)(u) = \pi(y) = y$ ,  $(\pi \circ \rho)(v) = \pi(z) = z$ . We next prove that  $\pi \circ \rho \in \mathsf{BlsO}((F, T^{r}), G)$ . The proof is based on Remark C.18.
+
+a) Let  $w, x \in V_F$ ,  $w \neq x$  be any vertices in the same bag of  $T^r$ . Since  $(\rho, \tau)$  is a bIsoHom,  $\rho(w) \neq \rho(x)$  and  $\rho(w)$  and  $\rho(x)$  are in the same bag of  $T_G^{(d)}(y, z)$ . Again, since  $\pi$  is a bIso, we have  $\pi(\rho(w)) \neq \pi(\rho(x))$ .  
+b) Let  $w, x \in V_F$  be any vertices in the same bag of  $T^r$ . Since  $(\rho, \tau)$  is a bIso-Hom,  $\rho(w)$  and  $\rho(x)$  are in the same bag of  $T_G^{(d)}(y,z)$ , and  $\{w,x\} \in E_F$  iff  $\{\rho(w),\rho(x)\} \in E_{F_G^{(d)}(y,z)}$ . Again, since  $\pi$  is a bIso,  $\{\rho(w),\rho(x)\} \in E_{F_G^{(d)}(y,z)}$  iff  $\{\pi(\rho(w)),\pi(\rho(x))\} \in E_G$ . Therefore,  $\{w,x\} \in E_F$  iff  $\{\pi(\rho(w)),\pi(\rho(x))\} \in E_G$ .  
+c) We clearly have  $\ell_F(w) = \ell_{F_G^{(d)}(y,z)}(\rho (w)) = \ell_G(\pi (\rho (w)))$
+
+We have proved that  $\pi \circ \rho \in \mathrm{BlsO}((F,T^r),G)$
+
+2. We then prove that  $\sigma$  is a surjection. For all  $g\in S_1(y,z)$ , we define a mapping  $(\rho ,\tau)$  from  $(F,T^{r})$  to  $(F_{G}^{(d)}(y,z),T_{G}^{(d)}(y,z))$  as follows. First define  $\rho (u) = y$ ,  $\rho (v) = z$ , and set  $\tau (r)$  to be the root of  $(F_{G}^{(d)}(y,z),T_{G}^{(d)}(y,z))$ . Let  $w_{1},\dots ,w_{m}\in V_{F}$  and  $w_1^{\prime},\dots ,w_{m^{\prime}}^{\prime}\in V_{F}$  be vertices such that all  $\{u,w_i\}$  and  $\{w_i',v\}$  correspond to bags of  $T^{r}$  associated to all tree nodes of depth 2. Similarly, by definition of the Local 2-GNN unfolding tree, let  $x_{1},\dots ,x_{n}\in V_{F_{G}^{(d)}(y,z)}$  be different vertices and  $x_1^\prime ,\dots ,x_n^\prime \in V_{F_G^{(d)}(y,z)}$  be different vertices such that all  $\{y,x_i\}$  and  $\{x_i',z\}$  correspond to bags of  $T_{G}^{(d)}(y,z)$  associated to all tree nodes of depth 2. Since  $g$  and  $\pi$  are bIsos, we have:
+
+- For every  $w_{i}$  ( $i \in [m]$ ), there exists  $x_{j}$  ( $j \in [n]$ ), such that  $g(w_{i}) = \pi(x_{j}) = \tilde{x}_{j}$  for some  $\tilde{x}_{j} \in V_{G}$  and  $F[\{\{u, v, w_{i}\}\}] \simeq F_{G}^{(d)}(y, z)[\{\{y, z, x_{j}\}\}] \simeq G[\{\{y, z, \tilde{x}_{j}\}\}$ ;  
+- For every  $w_{i}^{\prime}$  ( $i \in [m']$ ), there exists  $x_{j}^{\prime}$  ( $j \in [n]$ ), such that  $g(w_{i}^{\prime}) = \pi(x_{j}^{\prime}) = \tilde{x}_{j}^{\prime}$  for some  $\tilde{x}_{j}^{\prime} \in V_{G}$  and  $F[\{\{u, v, w_{i}^{\prime}\}\}] \simeq F_{G}^{(d)}(y, z)[\{\{y, z, x_{j}^{\prime}\}\}] \simeq G[\{\{y, z, \tilde{x}_{j}^{\prime}\}]$ .
+
+We then define  $\rho(w_i) = x_j$  for each  $i \in [m]$  and  $\rho(w_i') = x_j'$  for each  $i \in [m']$ . Based on the above two items, one can easily define  $\tau$  such that each node  $s$  in  $T^r$  of depth 1 or 2 is mapped by  $\tau$  to a node  $t$  in  $T_G^{(d)}(y,z)$  of the same depth,  $\rho(\beta_T(s)) = \beta_{T_G^{(d)}}(t)$ , and  $\rho$  is an isomorphism from  $F[\beta_T(s)]$  to  $F_G^{(d)}(y,z)[\beta_{T_G^{(d)}(y,z)}(t)]$ .
+
+Next, we can recursively define  $\rho$ 's image on  $F[T^r[s]]$  for each tree node  $s$  of depth 2 following the same construction above. This is because  $g$  is still a bIso from  $(F[T^r[s]], T^r[s])$  to  $G$ ,  $\pi$  is still a bIso from  $\left(F_G^{(d)}(y,z)[T_G^{(d)}(y,z)[\tau(s)]], T_G^{(d)}(y,z)[\tau(s)]\right)$  to  $G$ , and  $g(\beta_T(s)) = \pi(\beta_{T_G^{(d)}}(\tau(s)))$ . Recursively applying this procedure, we can construct  $(\rho, \tau)$  such that it is a
+
+bIsoHom from  $(F, T^r)$  to  $\left(F_G^{(d)}(y, z), T_G^{(d)}(y, z)\right)$ . That is to say, we have proved that for all  $g \in S_1(u, v)$ , there is a preimage  $(\rho, \tau) \in S_2(y, z)$  such that  $\sigma(\rho, \tau) = g$ .
+
+3. We finally prove that  $\sigma$  is an injection. Let  $(\rho_1, \tau_1), (\rho_2, \tau_2) \in S_2(y, z)$  such that  $\pi \circ \rho_1 = \pi \circ \rho_2$ . Let  $w_1, \dots, w_m \in V_F$ ,  $w_1', \dots, w_{m'}' \in V_F$ ,  $x_1, \dots, x_n \in V_{F_G^{(d)}(y,z)}$ , and  $x_1', \dots, x_n' \in V_{F_G^{(d)}(y,z)}$  be defined as in the previous item. For each  $i \in [m]$ , let  $j_1(i)$  and  $j_2(i)$  be indices satisfying  $\rho_1(w_i) = x_{j_1(i)}$  and  $\rho_2(w_i) = x_{j_2(i)}$ . It follows that  $\pi(x_{j_1(i)}) = \pi(x_{j_2(i)})$ . By definition of the Local 2-GNN unfolding tree, we must have  $x_{j_1(i)} = x_{j_2(i)}$ , and thus  $\rho_1(w_i) = \rho_2(w_i)$ . Using a similar approach, we can prove that  $\rho_1(w_i') = \rho_2(w_i')$  for each  $i \in [m']$ . Next, we can recursively apply the above procedure to the subtree  $T^r[s]$  for each tree node of depth 2 following the previous item, and finally prove that  $\rho_1 = \rho_2$ . Therefore,  $\sigma$  is an injection.
+
+Combining the above three items completes the proof.
+
+Proposition C.24. Under Definition C.21, (i) if  $(\rho, \tau)$  is a bIsoSurj, then  $\rho$  is a surjection from  $F$  to  $\tilde{F}$  on both vertices and edges; (ii) if  $(\rho, \tau)$  is a bIsoInj, then  $\rho$  is an injection from  $F$  to  $\tilde{F}$  on both vertices and edges.
+
+Proof. We first prove that  $\rho$  is surjective if  $(\rho, \tau)$  is a bIsoSurj. We will only prove that  $\rho$  is surjective on edges, as proving that  $\rho$  is surjective on vertices is almost the same. For any  $\{x, y\} \in E_{\tilde{F}}$ , by Definition C.1(b) we can pick  $\tilde{t} \in V_{\tilde{T}}$  such that  $\{x, y\} \in \beta_{\tilde{T}}(\tilde{t})$ . Since  $\tau$  is surjective, there exists  $t \in V_T$  such that  $\tau(t) = \tilde{t}$ . By definition of bag isomorphism, there exists  $u, v \in \beta_T(t)$  such that  $\rho(u) = x$ ,  $\rho(v) = y$ , and  $F[\{u, v\}] \simeq \tilde{F}[\{x, y\}]$ . Therefore,  $\{u, v\} \in E_F$ .
+
+We next prove that  $\rho$  is injective if  $(\rho, \tau)$  is a bIsoInj. Pick any  $u \in V_F$ . It suffices to prove that  $\rho(u) = \rho(v)$  iff  $u = v$  for any  $v \in V_F$ . If the result does not hold, consider two cases:
+
+- There exists  $v \in V_F, v \neq u$  such that  $\rho(u) = \rho(v)$  and  $\{u, v\}$  are in the same bag of  $F$ . This contradicts the definition of bag isomorphism.  
+- For all  $v \in V_F$  such that  $v \neq u$  and  $\rho(u) = \rho(v)$ ,  $\{u, v\}$  are not in the same bag of  $F$ . By Proposition C.5,  $T\left[\bigcup_{v \in V_F: \rho(u) = \rho(v)} B_T(v)\right]$  is disconnected. We can thus pick a path  $P$  in  $T$  such that the endpoints  $t_1$  and  $t_2$  are in different connected components of  $T\left[\bigcup_{v \in V_F: \rho(u) = \rho(v)} B_T(v)\right]$ . This implies that there is a tree node  $t_3$  in  $P$  such that  $\beta_T(t_3) \cap \{v \in V_F: \rho(u) = \rho(v)\} = \emptyset$ . Consequently,  $\rho(u) \in \rho(\beta_T(t_1)) = \beta_{\tilde{T}}(\tau(t_1))$ ,  $\rho(u) \in \rho(\beta_T(t_2)) = \beta_{\tilde{T}}(\tau(t_2))$ , but  $\rho(u) \notin \beta_{\tilde{T}}(\tau(t_3))$ . On the other hand, since  $\tau$  is injective,  $\tilde{T}\left[\tau(V_P)\right]$  is also a path and  $\tau(t_3)$  is on the path between  $\tau(t_1)$  and  $\tau(t_2)$  in  $\tilde{T}$ . This contradicts the definition of tree decomposition (Definition C.1(c)).
+
+Combining the two cases concludes the proof.
+
+Lemma C.25. Let  $M \in \{\mathrm{Sub},\mathrm{L},\mathrm{LF},\mathrm{F}\}$  be any model. For any tree-decomposed graphs  $(F,T^r),(\tilde{F},\tilde{T}^s) \in S^M$ ,
+
+$$
+\mathsf {b l s o H o m} ((F, T ^ {r}), (\tilde {F}, \tilde {T} ^ {s})) = \sum_ {(\widehat {F}, \widehat {T} ^ {t}) \in \mathcal {S} ^ {M}} \frac {\mathsf {b l s o S u r j} \left((F , T ^ {r}) , (\widehat {F} , \widehat {T} ^ {t})\right) \cdot \mathsf {b l s o l n j} \left((\widehat {F} , \widehat {T} ^ {t}) , (\tilde {F} , \tilde {T} ^ {s})\right)}{\mathsf {a u t} (\widehat {F} , \widehat {T} ^ {t})},
+$$
+
+where  $\operatorname{aut}(\widehat{F},\widehat{T}^t)$  denotes the number of automorphisms of  $(\widehat{F},\widehat{T}^t)$ . Here, the summation ranges over all non-isomorphic (tree-decomposed) graphs in  $S^M$  and is well-defined as there are only a finite number of graphs making the value in the summation non-zero.
+
+Proof. We define the following set of three-tuples:
+
+$$
+\begin{array}{l} S = \left\{\left((\widehat {F}, \widehat {T} ^ {t}), (\rho^ {\mathsf {S}}, \tau^ {\mathsf {S}}), (\rho^ {\mathsf {I}}, \tau^ {\mathsf {I}})\right): (\widehat {F}, \widehat {T} ^ {t}) \in \mathcal {S} ^ {M}, \right. \\ \left. \left(\rho^ {\mathsf {S}}, \tau^ {\mathsf {S}}\right) \in \mathsf {B l s o S u r j} \left(\left(F, T ^ {r}\right), \left(\widehat {F}, \widehat {T} ^ {t}\right)\right), \left(\rho^ {\mathsf {I}}, \tau^ {\mathsf {I}}\right) \in \mathsf {B l s o l n j} \left(\left(\widehat {F}, \widehat {T} ^ {t}\right), \left(\tilde {F}, \tilde {T} ^ {s}\right)\right) \right\}. \\ \end{array}
+$$
+
+Define a mappings  $\sigma$  such that
+
+$$
+\sigma \left(\left(\widehat {F}, \widehat {T} ^ {t}\right), \left(\rho^ {\mathsf {S}}, \tau^ {\mathsf {S}}\right), \left(\rho^ {\mathsf {I}}, \tau^ {\mathsf {I}}\right)\right) = \left(\rho^ {\mathsf {I}} \circ \rho^ {\mathsf {S}}, \tau^ {\mathsf {I}} \circ \tau^ {\mathsf {S}}\right)
+$$
+
+for all  $\left((\widehat{F},\widehat{T}^t),(\rho^\mathsf{S},\tau^\mathsf{S}),(\rho^\mathsf{I},\tau^\mathsf{I})\right)\in S.$  It suffices to prove the following three statements:
+
+1.  $\sigma$  is a mapping from  $S$  to  $\mathsf{B}\mathsf{I}\mathsf{o}\mathsf{s}\mathsf{H}\mathsf{o}\mathsf{m}((F,T^r),(\tilde{F},\tilde{T}^s))$  
+2.  $\sigma$  is surjective;  
+3.  $\sigma\left((\widehat{F}_1, \widehat{T}_1^{t_1}), (\rho_1^{\mathsf{S}}, \tau_1^{\mathsf{S}}), (\rho_1^{\mathsf{I}}, \tau_1^{\mathsf{I}})\right) = \sigma\left((\widehat{F}_2, \widehat{T}_2^{t_2}), (\rho_2^{\mathsf{S}}, \tau_2^{\mathsf{S}}), (\rho_2^{\mathsf{I}}, \tau_2^{\mathsf{I}})\right)$  iff there exists an isomorphism  $(\widehat{\rho}, \widehat{\tau})$  from  $(\widehat{F}_1, \widehat{T}_1^{t_1})$  to  $(\widehat{F}_2, \widehat{T}_2^{t_2})$  such that  $\widehat{\rho} \circ \rho_1^{\mathsf{S}} = \rho_2^{\mathsf{S}}, \widehat{\tau} \circ \tau_1^{\mathsf{S}} = \tau_2^{\mathsf{S}}, \rho_1^{\mathsf{I}} = \rho_2^{\mathsf{I}} \circ \widehat{\rho}, \tau_1^{\mathsf{I}} = \tau_2^{\mathsf{I}} \circ \widehat{\tau}$ .
+
+We will prove these statements one by one.
+
+1. We first prove that  $\sigma$  is a mapping from  $S$  to  $\mathsf{BIsoHom}((F,T^r),(\tilde{F},\tilde{T}^s))$ . This simply follows from the fact that both bIsoSurj and bIsoInj are bIsoHom, and the composition of two bIsoHoms are still a bIsoHom.  
+2. We next prove that  $\sigma$  is surjective. Given  $(\rho^{\mathsf{H}},\tau^{\mathsf{H}})\in \mathrm{BlsoHom}((F,T^{r}),(\tilde{F},\tilde{T}^{s}))$ , we define  $(\widehat{F},\widehat{T}^{t})$ ,  $(\rho^{\mathsf{S}},\tau^{\mathsf{S}})$ , and  $(\rho^{\mathsf{I}},\tau^{\mathsf{I}})$  as follows:
+
+a) Let  $\widehat{F} = \tilde{F}[\rho^{\mathsf{H}}(V_F)]$  and  $\widehat{T}^t = \tilde{T}^s[\tau^{\mathsf{H}}(V_T)]$ . We clearly have  $(\widehat{F}, \widehat{T}^t) \in S^M$ .  
+b) Let  $\rho^{\mathsf{S}} = \rho^{\mathsf{H}}$  and  $\tau^{\mathsf{S}} = \tau^{\mathsf{H}}$ . Obviously,  $(\rho^{\mathsf{S}},\tau^{\mathsf{S}})$  is a bIsoSurj from  $(F,T^{r})$  to  $(\widehat{F},\widehat{T}^{t})$ .  
+c) Define identity mappings  $\rho^{\mathsf{I}}(u) = u$  for all  $u\in V_{\widehat{F}}$  and  $\tau^1 (t) = t$  for all  $t\in V_{\widehat{T}}$ . Obviously,  $(\rho^{\mathsf{I}},\tau^{\mathsf{I}})$  is a bIsoInj from  $(\widehat{F},\widehat{T}^{t})$  to  $(\tilde{F},\tilde{T}^{s})$ .
+
+We clearly have  $\rho^{\mathsf{H}} = \rho^{\mathsf{I}}\circ \rho^{\mathsf{S}}$  and  $\tau^{\mathrm{H}} = \tau^{\mathrm{I}}\circ \tau^{\mathrm{S}}$ . Thus,  $\sigma$  is a surjection.
+
+3. We finally prove the aforementioned item 3. It suffices to prove only one direction, namely,  $\sigma\left((\widehat{F}_1,\widehat{T}_1^{t_1}),(p_1^{\mathsf{S}},\tau_1^{\mathsf{S}}),(p_1^{\mathsf{l}},\tau_1^{\mathsf{l}})\right) = \sigma\left((\widehat{F}_2,\widehat{T}_2^{t_2}),(p_2^{\mathsf{S}},\tau_2^{\mathsf{S}}),(p_2^{\mathsf{l}},\tau_2^{\mathsf{l}})\right)$  implies that there exists an isomorphism  $(\widehat{\rho},\widehat{\tau})$  from  $(\widehat{F}_1,\widehat{T}_1^{t_1})$  to  $(\widehat{F}_2,\widehat{T}_2^{t_2})$  such that  $\widehat{\rho}\circ \rho_{1}^{\mathsf{S}} = \rho_{2}^{\mathsf{S}},\widehat{\tau}\circ \tau_{1}^{\mathsf{S}} = \tau_{2}^{\mathsf{S}},$ $\rho_1^{\mathrm{l}} = \rho_2^{\mathrm{l}}\circ \widehat{\rho},\tau_1^{\mathrm{l}} = \tau_2^{\mathrm{l}}\circ \widehat{\tau}$
+
+a) We first prove that  $\widehat{F}_1\simeq \widehat{F}_2$  and  $\widehat{T}_1^{t_1}\simeq \widehat{T}_2^{t_2}$
+
+For any  $u, v \in V_F$ , if  $\rho_1^{\mathsf{S}}(u) \neq \rho_1^{\mathsf{S}}(v)$ , then  $\rho_1^{\mathsf{I}}(\rho_1^{\mathsf{S}}(u)) \neq \rho_1^{\mathsf{I}}(\rho_1^{\mathsf{S}}(v))$  since  $\rho_1^{\mathsf{I}}$  is an injection (by Proposition C.24). Therefore,  $\rho_2^{\mathsf{I}}(\rho_2^{\mathsf{S}}(u)) \neq \rho_2^{\mathsf{I}}(\rho_2^{\mathsf{S}}(v))$ , and thus  $\rho_2^{\mathsf{S}}(u) \neq \rho_2^{\mathsf{S}}(v)$ . By symmetry, we also have that  $\rho_2^{\mathsf{S}}(u) \neq \rho_2^{\mathsf{S}}(v)$  implies  $\rho_1^{\mathsf{S}}(u) \neq \rho_1^{\mathsf{S}}(v)$ . This proves that  $\rho_1^{\mathsf{S}}(u) = \rho_1^{\mathsf{S}}(v)$  iff  $\rho_2^{\mathsf{S}}(u) = \rho_2^{\mathsf{S}}(v)$ .
+
+For any  $u, v \in V_F$ , if  $\{\rho_1^S(u), \rho_1^S(v)\} \in E_{\widehat{F}_1}$ , then  $\{\rho_1^1(\rho_1^S(u)), \rho_1^1(\rho_1^S(v))\} \in E_{\widehat{F}}$  since  $\rho_1^1$  is a homomorphism. Therefore,  $\{\rho_2^1(\rho_2^S(u)), \rho_2^1(\rho_2^S(v))\} \in E_{\widehat{F}}$ . This implies that  $\rho_2^1(\rho_2^S(u))$  and  $\rho_2^1(\rho_2^S(v))$  are in the same bag of  $\tilde{T}^s$ . Therefore, there are vertices  $x, y \in V_{\widehat{F}_2}$  in the same bag of  $\widehat{T}_2^{s_2}$  such that  $\rho_2^1(x) = \rho_2^1(\rho_2^S(u))$  and  $\rho_2^1(y) = \rho_2^1(\rho_2^S(v))$ , and by definition of bag isomorphism we have  $\{x, y\} \in E_{\widehat{F}_2}$ . Since  $\rho_2$  is injective,  $x = \rho_2^S(u)$  and  $y = \rho_2^S(v)$ , namely,  $\{\rho_2^S(u), \rho_2^S(v)\} \in E_{\widehat{F}_2}$ . By symmetry, we can prove that  $\{\rho_1^S(u), \rho_1^S(v)\} \in E_{\widehat{F}_1}$  iff  $\{\rho_2^S(u), \rho_2^S(v)\} \in E_{\widehat{F}_2}$ .
+
+Finally, noting that  $\rho_1^{\mathsf{S}}$  and  $\rho_2^{\mathsf{S}}$  are surjective (by Proposition C.24) and  $\ell_{\widehat{F}_1}(\rho_1^{\mathsf{S}}(u)) = \ell_F(u) = \ell_{\widehat{F}_2}(\rho_2^{\mathsf{S}}(u))$  for all  $u \in V_F$ , we obtain that  $\widehat{F}_1 \simeq \widehat{F}_2$ . Following the same procedure, we can prove that  $\widehat{T}_1^{t_1} \simeq \widehat{T}_2^{t_2}$ .
+
+b) Consequently, there exist isomorphisms  $\widehat{\rho}$  and  $\widehat{\tau}$  such that  $\widehat{\rho} \circ \rho_1^{\mathsf{S}} = \rho_2^{\mathsf{S}}, \widehat{\tau} \circ \tau_1^{\mathsf{S}} = \tau_2^{\mathsf{S}}$ . For any node  $q \in V_T$ ,
+
+$$
+\widehat {\rho} (\beta_ {\widehat {T} _ {1}} (\tau_ {1} ^ {\mathsf {S}} (q))) = \widehat {\rho} (\rho_ {1} ^ {\mathsf {S}} (\beta_ {T} (q))) = \rho_ {2} ^ {\mathsf {S}} (\beta_ {T} (q))) = \beta_ {\widehat {T} _ {2}} (\tau_ {2} ^ {\mathsf {S}} (q)) = \beta_ {\widehat {T} _ {2}} (\widehat {\tau} (\tau_ {1} ^ {\mathsf {S}} (q))).
+$$
+
+Since  $\tau_1^S$  is surjective,  $\tau_1^S (q)$  ranges over all nodes in  $\widehat{T}_1^{s_1}$  when  $q$  ranges over  $V_{T}$ . We thus conclude that  $(\rho ,\tau)$  is an isomorphism from  $(\widehat{F}_1,\widehat{T}_1^{t_1})$  to  $(\widehat{F}_2,\widehat{T}_2^{t_2})$  (see Definition C.8).
+
+c) We finally prove that  $\rho_1^{\mathsf{I}} = \rho_2^{\mathsf{I}}\circ \widehat{\rho}$  and  $\tau_1^{\mathsf{I}} = \tau_2^{\mathsf{I}}\circ \widehat{\tau}$ . Pick any  $u\in V_F$ , we have  $\rho_2^{\mathsf{I}}(\widehat{\rho} (\rho_1^{\mathsf{S}}(u))) = \rho_2^{\mathsf{I}}(\rho_2^{\mathsf{S}}(u)) = \rho_1^{\mathsf{I}}(\rho_1^{\mathsf{S}}(u))$ . Since  $\rho_{1}^{\mathsf{S}}$  is surjective,  $\rho_{1}^{\mathsf{S}}(u)$  ranges over all nodes in  $\widehat{F}_1$  when  $u$  ranges over  $V_{F}$ . This proves that  $\rho_1^{\mathsf{I}} = \rho_2^{\mathsf{I}}\circ \widehat{\rho}$ . Following the same procedure, we can prove that  $\tau_1^{\mathsf{I}} = \tau_2^{\mathsf{I}}\circ \widehat{\tau}$ .
+
+Combining the above three items concludes the proof.
+
+![](images/2c363b7f4c30baf190b496a4a88db627c65ea10677b711930d0c51fc336999c8.jpg)
+
+In the following, we further define two technical concepts that will be used to describe the next result. We note that these technical concepts also appeared in Dell et al. (2018).
+
+Definition C.26. A tree-decomposed graph  $(\tilde{F},\tilde{T}^{s})$  is called a bag extension of another tree-decomposed graph  $(F,T^r)$  if there is a graph  $H$  and a mapping  $(\rho ,\tau)$  such that  $F$  is a subgraph of  $H$  and  $(\rho ,\tau)$  is an isomorphism from the tree-decomposed graph  $(H,T^r)$  to  $(\tilde{F},\tilde{T}^{s})$ . Define  $\mathsf{BExt}\left((F,T^r),(\tilde{F},\tilde{T}^s)\right)$  to be the set of all mappings  $(\rho ,\tau)$  that satisfies the above conditions, and define  $\mathsf{bExt}\left((F,T^r),(\tilde{F},\tilde{T}^s)\right) = \left|\mathsf{BExt}\left((F,T^r),(\tilde{F},\tilde{T}^s)\right)\right|$ .
+
+Remark C.27. In other words, a bag extension of a tree-decomposed graph  $(F,T^r)$  can be obtained by adding an arbitrary number of edges to  $F$  while ensuring that each added edge is contained in a tree node in  $T^r$ . A immediate fact is that  $(\rho ,\tau)$  is a homomorphism from  $(F,T^{r})$  to  $(\tilde{F},\tilde{T}^{s})$ .
+
+Definition C.28. Given a tree-decomposed graph  $(F,T^r)$  and a graph  $G$ , a bag-strong homomorphism from  $(F,T^r)$  to  $G$  (abbreviated as "bStrHom") is a homomorphism  $f$  from  $F$  to  $G$  such that, for all  $t\in V_T$ ,  $f$  is a strong homomorphism from  $F[\beta_T(t)]$  to  $G[f(\beta_T(t))]$ , i.e.,  $\{u,v\} \in E_{F[\beta_T(t)]}$  iff  $\{f(u),f(v)\} \in E_{G[f(\beta_T(t))]}$ . Denote BStrHom  $((F,T^r),G)$  to be the set of all bStrHom from  $(F,T^r)$  to  $G$ , and denote bStrHom  $((F,T^r),G) = |\mathsf{BStrHom}((F,T^r),G)|$ .
+
+The following equation is straightforward:
+
+Lemma C.29. Let  $M \in \{\text{Sub}, \mathsf{L}, \mathsf{LF}, \mathsf{F}\}$  be any model. For any graph  $G$  and tree-decomposed graph  $(F, T^r) \in S^M$ ,
+
+$$
+\operatorname {h o m} (F, G) = \sum_ {(\tilde {F}, \tilde {T} ^ {s}) \in \mathcal {S} ^ {M}} \frac {\mathrm {b E x t} ((F , T ^ {r}) , (\tilde {F} , \tilde {T} ^ {s})) \cdot \mathrm {b S t r H o m} ((\tilde {F} , \tilde {T} ^ {s}) , G)}{\operatorname {a u t} (\tilde {F} , \tilde {T} ^ {s})},
+$$
+
+where  $\operatorname{aut}(\tilde{F},\tilde{T}^{s})$  denotes the number of automorphisms of  $(\tilde{F},\tilde{T}^{s})$ . Here, the summation ranges over all non-isomorphic (tree-decomposed) graphs in  $S^M$  and is well-defined as there are only a finite number of graphs making the value in the summation non-zero.
+
+Proof. The proof has a similar structure to the previous lemma. We define the following set of three-tuples:
+
+$$
+S = \left\{\left((\tilde {F}, \tilde {T} ^ {s}), (\rho , \tau), g\right): (\tilde {F}, \tilde {T} ^ {s}) \in \mathcal {S} ^ {M}, (\rho , \tau) \in \mathsf {B E x t} \Big ((F, T ^ {r}), (\tilde {F}, \tilde {T} ^ {s}) \Big), g \in \mathsf {B S t r H o m} ((\tilde {F}, \tilde {T} ^ {s}), G) \right\}.
+$$
+
+Define a mappings  $\sigma$  such that  $\sigma\left((\tilde{F},\tilde{T}^s),(\rho,\tau),g\right)=g\circ\rho$  for all  $\left((\tilde{F},\tilde{T}^s),(\rho,\tau),g\right)\in S$ . It is straightforward to see that  $\sigma$  is a mapping from  $S$  to  $\mathsf{BlsoHom}((F,T^r),(\tilde{F},\tilde{T}^s))$ , namely,  $g\circ\rho$  is a homomorphism from  $F$  to  $G$  for all  $\left((\tilde{F},\tilde{T}^s),(\rho,\tau),g\right)\in S$ .
+
+We then prove that  $\sigma$  is surjective. Given  $h\in \mathrm{Hom}(F,G)$ , define  $(\tilde{F},\tilde{T}^{s})$ ,  $(\rho ,\tau)$ , and  $g$  as follows:
+
+a) Define  $\tilde{F}$  be the graph obtained from  $F$  by adding edges  $\{\{u,v\} : \exists t \in V_T \text{s.t.} u,v \in \beta_T(t), \{h(u),h(v)\} \in E_G\}$ , and let  $\tilde{T}^s = T^r$ . Clearly,  $(\tilde{F},\tilde{T}^s)$  is a bag extension of  $(F,T^r)$ .  
+b) Define identity mappings  $\rho(u) = u$  for all  $u \in V_F$  and  $\tau(t) = t$  for all  $t \in V_T$ . Clearly,  $(\rho, \tau) \in \mathsf{BExt}((F, T^r), (\tilde{F}, \tilde{T}^s))$ .  
+c) Let  $g = h$ . It is easy to see that  $g$  is a strong homomorphism from  $\tilde{F}[\beta_{\tilde{T}}(t)]$  to  $G[g(\beta_{\tilde{T}}(t))]$  for each  $t \in V_{\tilde{T}}$ . Thus,  $g \in \mathsf{BStrHom}((\tilde{F}, \tilde{T}^s), G)$ .
+
+Noting that  $h = g = g \circ \rho$ , we have proved that  $\sigma$  is a surjection.
+
+We finally prove that  $\sigma\left((\tilde{F}_1, \tilde{T}_1^{s_1}), (\rho_1, \tau_1), g_1\right) = \sigma\left((\tilde{F}_2, \tilde{T}_2^{s_2}), (\rho_2, \tau_2), g_2\right)$  implies that there exists an isomorphism  $(\tilde{\rho}, \tilde{\tau})$  from  $(\tilde{F}_1, \tilde{T}_1^{s_1})$  to  $(\tilde{F}_2, \tilde{T}_2^{s_2})$  such that  $\tilde{\rho} \circ \rho_1 = \rho_2$ ,  $\tilde{\tau} \circ \tau_1 = \tau_2$ ,  $g_1 = g_2 \circ \tilde{\rho}$ . We first prove that  $\tilde{F}_1 \simeq \tilde{F}_2$  and  $\tilde{T}_1^{s_1} \simeq \tilde{T}_2^{s_2}$ . (i) For any  $u, v \in V_F$ , we obviously have  $\rho_1(u) = \rho_1(v)$  iff  $u = v$  iff  $\rho_2(u) = \rho_2(v)$ . (ii) Let  $i \in \{1, 2\}$ . For any  $u, v \in V_F$ ,  $\{\rho_i(u), \rho_i(v)\} \in E_{\tilde{F}_i}$  iff  $\{u, v\} \in E_F$  or  $u, v$  are in the same bag of  $T^r$  and  $\{g_i(\rho_i(u)), g(\rho_i(v))\} \in E_G$ . Since  $g_1 \circ \rho_1 = g_2 \circ \rho_2$ , we have  $\{\rho_1(u), \rho_1(v)\} \in E_{\tilde{F}_1}$  iff  $\{\rho_2(u), \rho_2(v)\} \in E_{\tilde{F}_2}$ . (iii) Finally, noting that  $\rho_1$  and  $\rho_2$  are bijective and  $\ell_{\tilde{F}_1}(\rho_1(u)) = \ell_F(u) = \ell_{\tilde{F}_2}(\rho_2(u))$  for all  $u \in V_F$ , we obtain that  $\tilde{F}_1 \simeq \tilde{F}_2$ . On the other hand,  $\tilde{T}_1^{s_1} \simeq \tilde{T}_2^{s_2}$  trivially holds. The remaining procedure is almost the same as in the previous lemma.
+
+We next show that the mapping bStrHom can be further decomposed as shown in Lemma C.31. We need an auxiliary concept:
+
+Definition C.30. Given two tree-decomposed graphs  $(F,T^r)$  and  $(\tilde{F},\tilde{T}^{s})$  , a homomorphism  $(\rho ,\tau)$  from  $(F,T^{r})$  to  $(\tilde{F},\tilde{T}^{s})$  is called bag-strong surjective (abbreviated as "bStrSurj") if  $\rho$  is a bagstrong homomorphism from  $(F,T^{r})$  to  $\tilde{F}$  and is surjective on both vertices and edges, and  $\tau$  is an isomorphism from  $T^r$  to  $\tilde{T}^s$  . Denote BStrSurj  $((F,T^r),(\tilde{F},\tilde{T}^{s}))$  to be the set of all bStrSurj from  $(F,T^{r})$  to  $(\tilde{F},\tilde{T}^{s})$  , and denote bStrSurj  $((F,T^r),(\tilde{F},\tilde{T}^{s})) = |\mathsf{BStrSurj}((F,T^r),(\tilde{F},\tilde{T}^{s}))|$
+
+Lemma C.31. Let  $M \in \{\text{Sub}, \mathsf{L}, \mathsf{LF}, \mathsf{F}\}$  be any model. For any graph  $G$  and tree-decomposed graph  $(F, T^r) \in S^M$ ,
+
+$$
+\mathsf{bStrHom}((F,T^{r}),G) = \sum_{(\tilde{F},\tilde{T}^{s})\in \mathcal{S}^{M}}\frac{\mathsf{bStrSurj}\left((F,T^{r}),( \tilde{F},\tilde{T}^{s})\right)\cdot\mathsf{blso}\left((\tilde{F},\tilde{T}^{s}),G\right)}{\mathsf{aut}(\tilde{F},\tilde{T}^{s})},
+$$
+
+where  $\operatorname{aut}(\tilde{F},\tilde{T}^s)$  denotes the number of automorphisms of  $(\tilde{F},\tilde{T}^{s})$  . Here, the summation ranges over all non-isomorphic (tree-decomposed) graphs in  $S^M$  and is well-defined as there are only a finite number of graphs making the value in the summation non-zero.
+
+Proof. The proof has a similar structure to the previous lemma. We define the following set of three-tuples:
+
+$$
+S = \left\{\left(\left(\tilde {F}, \tilde {T} ^ {s}\right), (\rho , \tau), g\right): \left(\tilde {F}, \tilde {T} ^ {s}\right) \in \mathcal {S} ^ {M}, (\rho , \tau) \in \mathrm {B S t r S u r j} \left(\left(F, T ^ {r}\right), \left(\tilde {F}, \tilde {T} ^ {s}\right)\right), g \in \mathrm {B l s o} \left(\left(\tilde {F}, \tilde {T} ^ {s}\right), G\right) \right\}.
+$$
+
+Define a mappings  $\sigma$  such that  $\sigma\left((\tilde{F},\tilde{T}^s),(\rho,\tau),g\right) = g\circ\rho$  for all  $\left((\tilde{F},\tilde{T}^s),(\rho,\tau),g\right)\in S$ . It suffices to prove the following three statements:
+
+1.  $\sigma$  is a mapping from  $S$  to  $\mathsf{BStrHom}((F,T^r),G)$  
+2.  $\sigma$  is surjective;  
+3.  $\sigma\left((\tilde{F}_1, \tilde{T}_1^{s_1}), (\rho_1, \tau_1), g_1\right) = \sigma\left((\tilde{F}_2, \tilde{T}_2^{s_2}), (\rho_2, \tau_2), g_2\right)$  iff there exists an isomorphism  $(\tilde{\rho}, \tilde{\tau})$  from  $(\tilde{F}_1, \tilde{T}_1^{s_1})$  to  $(\tilde{F}_2, \tilde{T}_2^{s_2})$  such that  $\tilde{\rho} \circ \rho_1 = \rho_2$ ,  $\tilde{\tau} \circ \tau_1 = \tau_2$ ,  $g_1 = g_2 \circ \tilde{\rho}$ .
+
+We will prove these statements one by one.
+
+1. We first prove that  $\sigma$  is a mapping from  $S$  to  $\mathsf{BStrHom}((F,T^r),G)$ . Pick any  $\left((\tilde{F},\tilde{T}^s),(\rho ,\tau),g\right)\in S$ . Pick any  $t\in V_T$  and  $u,v\in \beta_{T}(t)$ . Then,  $\{u,v\} \in E_F$  iff  $\{\rho (u),\rho (v)\} \in E_{\tilde{F}}$  (since  $\rho$  is a strong homomorphism from  $F[\beta_T(t)]$  to  $\tilde{F} [\rho (\beta_T(t))]$ ). Also,  $\rho (u),\rho (v)\in \beta_{\tilde{F}}(\tau (t))$  are in the same bag. Similarly,  $\{\rho (u),\rho (v)\} \in E_{\tilde{F}}$  iff  $\{g(\rho (u)),g(\rho (v))\} \in E_G$  (since  $g$  is a bIso). Thus,  $g\circ \rho$  is a bag-strong homomorphism.  
+2. We next prove that  $\sigma$  is surjective. Given  $h\in \mathsf{BStrHom}((F,T^r),G)$  , define  $(\tilde{F},\tilde{T}^{s}),(\rho ,\tau)$  and  $g$  as follows. First define a relation  $\sim$  on set  $V_{F}$  such that  $u\sim v$  iff the following hold:
+
+a)  $h(u) = h(v)$
+
+b) There exists a path  $P$  in  $T^r$  with endpoints  $t_1, t_2 \in V_T$  such that  $u \in \beta_T(t_1)$ ,  $v \in \beta_T(t_2)$ , and all node  $t$  on path  $P$  satisfies that  $h(u) \in h(\beta_T(t))$ .
+
+It is easy to see that  $\sim$  is an equivalence relation on  $V_{F}$ . We can thus define a mapping  $\rho$  that respects the relation, i.e.,  $\rho(u) = \rho(v)$  iff  $u \sim v$  for all  $u, v \in V_{F}$ . Moreover, for any edge  $\{u, v\} \in E_{F}$ ,  $\rho(u) \neq \rho(v)$  (since  $h$  is a homomorphism and  $h(u) \neq h(v)$ ). This implies that we can define  $\tilde{F}$  to be the homomorphic image of  $F$  such that  $\rho$  is the surjective homomorphism on both vertices and edges.
+
+We then define the mapping  $g: V_{\tilde{F}} \to V_G$  such that  $g(\rho(u)) = h(u)$  for all  $u \in V_F$ . Note that  $g$  is well-defined since  $\rho(u) = \rho(v)$  implies  $h(u) = h(v)$  for all  $u, v \in V_F$ , and  $\rho: V_F \to V_{\tilde{F}}$  is surjective. It follows that  $h = g \circ \rho$ . To prove that  $g$  is a homomorphism, note that for all  $\{x, y\} \in E_{\tilde{F}}$ , there exists an edge  $\{u, v\} \in E_F$  with  $\rho(u) = x$ ,  $\rho(v) = y$ , which implies that  $\{h(u), h(v)\} \in E_G$  (since  $h$  is a homomorphism), namely,  $\{g(x), g(y)\} \in E_G$ .
+
+We next define tree  $\tilde{T}^s = (V_T, E_T, \beta_{\tilde{T}})$ ,  $s = t$ , and identity mapping  $\tau$  so that  $\tau$  is an isomorphism from  $T_r$  to  $T_s$  (ignoring the labels). Set  $\beta_{\tilde{T}}(t) = \rho(\beta_T(t))$  for all  $t \in V_T$ . We will prove that  $(\tilde{F}, \tilde{T}^s) \in S^M$  is a valid tree decomposition. It suffices to prove that Definition C.1(c) holds. Pick any vertex  $x \in V_{\tilde{F}}$  and tree node  $t_1, t_2 \in B_{\tilde{T}}(x)$ . Then, there exists  $u \in \beta_T(t_1), v \in \beta_T(t_2)$  such that  $\rho(u) = x$ ,  $\rho(v) = x$ . Therefore,  $u \sim v$ . As such, there exists a path  $P$  in  $T^r$  such that all node  $t$  on  $P$  satisfies that there exists  $w \in \beta_T(t)$  with  $h(w) = h(u)$ , namely,  $w \sim u$ . Consequently,  $\rho(u) \in \beta_{\tilde{T}}(t)$ , implying that  $\tilde{T}^s[B_{\tilde{T}}(x)]$  is connected. This proves that  $(\tilde{F}, \tilde{T}^s) \in S^M$ . Also,  $(\rho, \tau)$  is clearly a homomorphism from  $(F, T^r)$  to  $(\tilde{F}, \tilde{T}^s)$  according to Definition C.20.
+
+It remains to prove that  $\rho$  is a bag-strong homomorphism and  $g$  is a bIso. Pick any  $t \in V_T$  and  $u, v \in \beta_T(t)$ . If  $\{u, v\} \notin E_F$ , then  $\{h(u), h(v)\} \notin E_G$  (since  $h$  is a bag-strong homomorphism). Therefore,  $\{\rho(u), \rho(v)\} \notin E_{\tilde{F}}$  (since  $g$  is a homomorphism), namely,  $\rho$  is a bag-strong homomorphism. Since  $\rho$  is surjective,  $\{\rho(u), \rho(v)\}$  ranges over all vertices in the same bag of  $\hat{T}^s$  when  $t \in V_T$  and  $u, v \in \beta_T(t)$  are arbitrary. Therefore,  $g$  is a bIso because  $\{\rho(u), \rho(v)\} \notin E_{\tilde{F}}$  iff  $\{h(u), h(v)\} \notin E_G$ .
+
+3. We finally prove that  $\sigma\left((\tilde{F}_1, \tilde{T}_1^{s_1}), (\rho_1, \tau_1), g_1\right) = \sigma\left((\tilde{F}_2, \tilde{T}_2^{s_2}), (\rho_2, \tau_2), g_2\right)$  implies there exists an isomorphism  $(\tilde{\rho}, \tilde{\tau})$  from  $(\tilde{F}_1, \tilde{T}_1^{s_1})$  to  $(\tilde{F}_2, \tilde{T}_2^{s_2})$  such that  $\tilde{\rho} \circ \rho_1 = \rho_2$ ,  $\tilde{\tau} \circ \tau_1 = \tau_2$ ,  $g_1 = g_2 \circ \tilde{\rho}$ . Let  $h = g_1 \circ \rho_1 = g_2 \circ \rho_2$ . Here, we will only prove that  $\tilde{F}_1 \simeq \tilde{F}_2$  since the remaining procedure is almost the same as previous proofs. Since both  $\tilde{F}_1$  and  $\tilde{F}_2$  are homomorphic images of  $F$ , it suffices to prove that, for all  $u, v \in V_F$ ,  $\rho_1(u) = \rho_1(v)$  iff the following hold:
+
+a)  $h(u) = h(v)$  
+b) There exists a path  $P$  in  $T^r$  with endpoints  $t_1, t_2 \in V_T$  such that  $u \in \beta_T(t_1)$ ,  $v \in \beta_T(t_2)$ , and all node  $t$  on path  $P$  satisfies that  $h(u) \in h(\beta_T(t))$ .
+
+On one hand, if  $\rho_{1}(u) = \rho_{1}(v)$ , we clearly have  $h(u) = h(v)$  and there exists  $t_1 \in B_T(u), t_2 \in B_T(v)$  such that  $t_1, t_2 \in B_{\tilde{T}_1}(\rho_1(u))$ . Since  $\tilde{T}_1[B_{\tilde{T}_1}(\rho_1(u))]$  is connected, there is a path  $P$  with endpoints  $t_1, t_2$  such that all nodes  $t$  on  $P$  satisfies  $\rho_{1}(u) \in \rho_{1}(\beta_{T}(t))$  and thus  $h(u) \in h(\beta_{T}(t))$ .
+
+On the other hand, if  $\rho_1(u) \neq \rho_1(v)$  but the above items (a) and (b) hold, consider two cases:
+
+-  $u$  and  $v$  are in the same bag of  $T$ . Then,  $\rho_1(u)$  and  $\rho_1(v)$  are in the same bag of  $\tilde{T}_1$ . Since  $g_1$  is a bIso,  $g_1(\rho_1(u)) \neq g_1(\rho_1(v))$ , which contradicts item (a).  
+-  $u$  and  $v$  are not in the same bag of  $T$ . Then, there exist two adjacent nodes  $t_1, t_2$  on  $P$  such that  $\rho_1(u) \in \beta_{\tilde{T}_1}(t_1), \rho_1(u) \notin \beta_{\tilde{T}_1}(t_2)$ . By Definition C.4(c),  $\beta_{\tilde{T}_1}(t_2) \subset \beta_{\tilde{T}_1}(t_1)$ . Also, item (b) implies that there exists  $w \in \beta_T(t_2)$  such that  $h(w) = h(u)$ . Therefore,  $\rho_1(u)$  and  $\rho_1(w)$  are two different nodes in  $\beta_{\tilde{T}_1}(t_1)$  with  $g_1(\rho_1(u)) = h(u) = h(w) = g_1(\rho_1(w))$ . This contradicts the condition that  $g_1$  is a bIso.
+
+This yields the desired result that  $\tilde{F}_1\simeq \tilde{F}_2$
+
+Combining the above three items concludes the proof.
+
+Let  $M \in \{\mathrm{Sub}, \mathrm{L}, \mathrm{LF}, \mathrm{F}\}$  be any model. We can list all non-isomorphic tree-decomposed (labeled) graphs in  $S^M$  into an infinite sequence  $(F_1, T_1^{r_1}), (F_2, T_2^{r_2}), \dots$ . Consider two types of ordering:
+
+- Ordered by the size of the graphs  $F_{i}$ . We denote this ordering as 1st. This ordering requires that for any  $i < j$ , either  $|V_{F_i}| < |V_{F_j}|$  or  $|V_{F_i}| = |V_{F_j}|$  and  $|E_{F_i}| \geq |E_{F_j}|$ . Note that when two graphs have an equal number of vertices, we place the graph with more edges to the front. When the number of edges is also the same, they can be arranged in any fixed order.  
+- Ordered by the size of the trees  $T_{i}$ . We denote this ordering as 2nd. This ordering requires that  $|V_{T_i}| \leq |V_{T_j}|$  for any  $i < j$ . When the number of tree nodes is the same, they can be arranged in any fixed order.
+
+Without loss of generality, we assume that the labels of each graph  $F_{i}$  are integers in the range of  $[1, |V_{F_i}|]$ . Note that using integer labels of a bounded range is already sufficient to represent all non-isomorphic labeled graphs up to a bijective label transformation. This ensures that the index  $i$  is countable.
+
+For the 1st ordering, define the following notations:
+
+a) Let  $f: \mathcal{S}^M \times \mathcal{S}^M \to \mathbb{N}$  be any mapping. Define the associated (infinite) matrix  $A^{f,M,1\mathrm{st}} \in \mathbb{N}^{\mathbb{N}_+ \times \mathbb{N}_+}$  such that  $A_{i,j}^{f,M,1\mathrm{st}} = f((F_i, T_i^{r_i}), (F_j, T_j^{r_j}))$ .  
+b) Let  $g: \mathcal{S}^M \times \mathcal{G} \to \mathbb{N}$  be any mapping. Given a graph  $G \in \mathcal{G}$ , define the (infinite) vector  $\pmb{p}_G^{g,M,1\mathrm{st}} \in \mathbb{N}^{\mathbb{N}_+}$  such that  $p_{G,i}^{f,M,1\mathrm{st}} = g((F_i,T_i^{r_i}),G)$ .  
+c) Let  $h: \mathcal{G} \times \mathcal{G} \to \mathbb{N}$  be any mapping. Given a graph  $G \in \mathcal{G}$ , define the (infinite) vector  $p_{G}^{h,M,1\mathrm{st}} \in \mathbb{N}^{\mathbb{N}_{+}}$  such that  $p_{G,i}^{h,M,1\mathrm{st}} = h(F_i, G)$ .
+
+We can similarly define  $\pmb{A}^{f,M,2\mathrm{nd}}$ ,  $\pmb{p}_G^{g,M,2\mathrm{nd}}$ ,  $\pmb{p}_G^{h,M,2\mathrm{nd}}$  for the ordering 2nd.
+
+Corollary C.32. Let  $M \in \{\text{Sub}, \mathsf{L}, \mathsf{LF}, \mathsf{F}\}$  be any model and  $G, H$  be two graphs. Then,  $\operatorname{hom}(F, G) = \operatorname{hom}(F, H)$  for all  $(F, T^r) \in S^M$  iff  $\text{blso}((F, T^r), G) = \text{blso}((F, T^r), H)$  for all  $(F, T^r) \in S^M$ .
+
+Proof. We separately consider each direction.
+
+1. We first prove that if  $\mathsf{blso}((F,T^r),G) = \mathsf{blso}((F,T^r),H)$  for all  $(F,T^{r})\in S^{M}$ , then  $\mathsf{hom}(F,G) = \mathsf{hom}(F,H)$  for all  $(F,T^{r})\in S^{M}$ . According to Lemmas C.29 and C.31, we can rewrite the corresponding equations into matrix forms for any  $F\in \mathcal{G}$ :
+
+$$
+\boldsymbol {p} _ {F} ^ {\text {h o m}, M, 1 \mathrm {s t}} = \boldsymbol {A} ^ {\text {b E x t}, M, 1 \mathrm {s t}} \left(\boldsymbol {A} ^ {\text {a u t}, M, 1 \mathrm {s t}}\right) ^ {- 1} \boldsymbol {p} _ {F} ^ {\text {b S t r H o m}, M, 1 \mathrm {s t}} \tag {19}
+$$
+
+$$
+\boldsymbol {p} _ {F} ^ {\mathrm {b S t r H o m}, M, 1 \mathrm {s t}} = \boldsymbol {A} ^ {\mathrm {b S t r S u r j}, M, 1 \mathrm {s t}} \left(\boldsymbol {A} ^ {\mathrm {a u t}, M, 1 \mathrm {s t}}\right) ^ {- 1} \boldsymbol {p} _ {F} ^ {\mathrm {b l s o}, M, 1 \mathrm {s t}}. \tag {20}
+$$
+
+From the above equations, we immediately obtain that  $p_{G}^{\mathrm{blso},M,1\mathrm{st}} = p_{H}^{\mathrm{blso},M,1\mathrm{st}}$  implies  $p_{G}^{\mathrm{hom},M,1\mathrm{st}} = p_{H}^{\mathrm{hom},M,1\mathrm{st}}$ .
+
+2. We next prove that if  $\mathsf{hom}(F,G) = \mathsf{hom}(F,H)$  for all  $(F,T^r)\in S^M$ , then  $\mathsf{blso}((F,T^r),G) = \mathsf{blso}((F,T^r),H)$  for all  $(F,T^{r})\in S^{M}$ . This can be seen from the following facts:
+
+a)  $A^{\mathrm{out},M,1\mathrm{st}}$  is a diagonal matrix and all diagonal elements are positive integers.  
+b)  $A^{\mathrm{bExt}, M, 1\mathrm{st}}$  is a lower triangular matrix and all diagonal elements are positive integers. This is because for any two tree-composed graphs  $(F_i, T_i^{r_i})$  and  $(F_j, T_j^{r_j})$ ,  $\mathsf{bExt}((F_i, T_i^{r_i}), (F_j, T_j^{r_j})) > 0$  only if  $|V_{F_i}| = |V_{F_j}|$  and  $|E_{F_i}| \leq |E_{F_j}|$ .  
+c)  $A^{\mathrm{bStrSurj}, M, 1\mathrm{st}}$  is also a lower triangular matrix and all diagonal elements are positive integers. This is because for any two tree-composed graphs  $(F_i, T_i^{r_i})$  and  $(F_j, T_j^{r_j})$ ,  $\mathsf{bStrSurj}((F_i, T_i^{r_i}), (F_j, T_j^{r_j})) > 0$  only if  $|V_{F_i}| > |V_{F_j}|$  or  $(|V_{F_i}| = |V_{F_j}|$  and  $|E_{F_i}| = |E_{F_j}|$ ).
+
+Therefore, the composition  $A^{\mathrm{bExt},M,1\mathrm{st}}(A^{\mathrm{aut},M,1\mathrm{st}})^{-1}A^{\mathrm{bStrSurj},M,1\mathrm{st}}(A^{\mathrm{aut},M,1\mathrm{st}})^{-1}$  is lower triangular and is invertible (although it is an infinite matrix). We thus arrive at the desired conclusion that  $p_{G}^{\mathrm{hom},M,1\mathrm{st}} = p_{H}^{\mathrm{hom},M,1\mathrm{st}}$  implies  $p_{G}^{\mathrm{blso},M,1\mathrm{st}} = p_{H}^{\mathrm{blso},M,1\mathrm{st}}$ .
+
+![](images/d51fa9c9fbdd38a72f32f1eb85b9a93490904c19b1a58b1587d2c50ae029aaec.jpg)
+
+Corollary C.33. Let  $M \in \{\text{Sub}, \mathsf{L}, \mathsf{LF}, \mathsf{F}\}$  be any model and  $G, H$  be two graphs. Then,  $\text{cnt}^M((F, T^r), G) = \text{cnt}^M((F, T^r), H)$  for all  $(F, T^r) \in \mathcal{S}^M$  iff  $\text{blso}((F, T^r), G) = \text{blso}((F, T^r), H)$  for all  $(F, T^r) \in \mathcal{S}^M$ .
+
+Proof. We separately consider each direction.
+
+1. We first prove that if  $\mathsf{cnt}^M((F, T^r), G) = \mathsf{cnt}^M((F, T^r), H)$  for all  $(F, T^r) \in S^M$ , then  $\mathsf{blso}((F, T^r), G) = \mathsf{blso}((F, T^r), H)$  for all  $(F, T^r) \in S^M$ . According to Lemma C.23, we can rewrite Equation (18) into the matrix form for all  $F \in \mathcal{G}$ :
+
+$$
+\boldsymbol {p} _ {F} ^ {\text {b l s o}, M, 2 \text {n d}} = \boldsymbol {A} ^ {\text {b l s o H o m}, M, 2 \text {n d}} \boldsymbol {p} _ {F} ^ {\text {c n t}, M, 2 \text {n d}}. \tag {21}
+$$
+
+This immediately obtains that  $\pmb{p}_G^{\mathrm{cnt},M,2\mathrm{nd}} = \pmb{p}_H^{\mathrm{cnt},M,2\mathrm{nd}}$  implies  $\pmb{p}_G^{\mathrm{blso},M,2\mathrm{nd}} = \pmb{p}_H^{\mathrm{blso},M,2\mathrm{nd}}$ .
+
+2. We next prove that  $\pmb{p}_G^{\mathrm{blo},M,2\mathrm{nd}} = \pmb{p}_H^{\mathrm{blo},M,2\mathrm{nd}}$  implies  $\pmb{p}_G^{\mathrm{cnt},M,2\mathrm{nd}} = \pmb{p}_H^{\mathrm{cnt},M,2\mathrm{nd}}$ . According to Lemma C.25, we have
+
+$$
+\boldsymbol {A} ^ {\text {b l s o H o m}, M, 2 \text {n d}} = \boldsymbol {A} ^ {\text {b l s o S u r j}, M, 2 \text {n d}} \left(\boldsymbol {A} ^ {\text {a u t}, M, 2 \text {n d}}\right) ^ {- 1} \boldsymbol {A} ^ {\text {b l s o l n j}, M, 2 \text {n d}}. \tag {22}
+$$
+
+Moreover, we have the following facts:
+
+a)  $A^{\mathrm{out},M,2\mathrm{nd}}$  is a diagonal matrix and all diagonal elements are positive integers.  
+b)  $A^{\mathrm{blsohnj},M,2\mathrm{nd}}$  is an upper triangular matrix and all diagonal elements are positive integers. This is because for any two tree-composed graphs  $(F_i,T_i^{r_i})$  and  $(F_j,T_j^{r_j})$ ,  $\mathsf{blsohnj}((F_i,T_i^{r_i}),(F_j,T_j^{r_j})) > 0$  only if  $|V_{T_i}| \leq |V_{T_j}|$ .  
+c) Similarly,  $\mathbf{A}^{\mathrm{bIsoSurj},M,2\mathrm{nd}}$  is a lower triangular matrix and all diagonal elements are positive integers.
+
+Unfortunately, since  $A^{\mathrm{blsolnj},M,2\mathrm{nd}}$  is an infinite upper triangular matrix, the inverse matrix is not well-defined. Nevertheless, we can use a special property of bIsoHom to complete our proof, namely,  $\mathrm{bIsOHom}((F_i,T_i^{r_i}),(F_j,T_j^{r_j})) > 0$  implies that the depth of  $T_{i}^{r_{i}}$  is equal to the depth of  $T_{j}^{r_{j}}$ . Therefore, denoting by  $A^{\mathrm{dep},d}$  the diagonal matrix where  $A_{ii}^{\mathrm{dep},d} = \mathbb{I}[\mathrm{the~depth~of~}T_i^{r_i}\mathrm{~is~}d]$ , we have
+
+$$
+\boldsymbol {A} ^ {\text {d e p}, d} \boldsymbol {p} _ {F} ^ {\text {b l s o}, M, 2 \text {n d}} = \boldsymbol {A} ^ {\text {b l s o S u r j}, M, 2 \text {n d}} \left(\boldsymbol {A} ^ {\text {a u t}, M, 2 \text {n d}}\right) ^ {- 1} \boldsymbol {A} ^ {\text {b l s o l n j}, M, 2 \text {n d}} \boldsymbol {A} ^ {\text {d e p}, d} \boldsymbol {p} _ {F} ^ {\text {c n t}, M, 2 \text {n d}} \tag {23}
+$$
+
+for all  $F \in \mathcal{G}$ . Fix any integer  $d \geq 0$ , and assume that  $A^{\mathrm{dep},d}p_{G}^{\mathrm{blso},M,2\mathrm{nd}} = A^{\mathrm{dep},d}p_{H}^{\mathrm{blso},M,2\mathrm{nd}}$ . We will prove that  $A^{\mathrm{dep},d}p_{G}^{\mathrm{cnt},M,2\mathrm{nd}} = A^{\mathrm{dep},d}p_{H}^{\mathrm{cnt},M,2\mathrm{nd}}$ . Since  $A^{\mathrm{blsoSurj},M,2\mathrm{nd}}$  is lower triangular with positive diagonal elements, it is invertible and thus
+
+$$
+\boldsymbol {A} ^ {\text {b l s o l n j}, M, 2 \text {n d}} \boldsymbol {A} ^ {\text {d e p}, d} \boldsymbol {p} _ {G} ^ {\text {c n t}, M, 2 \text {n d}} = \boldsymbol {A} ^ {\text {b l s o l n j}, M, 2 \text {n d}} \boldsymbol {A} ^ {\text {d e p}, d} \boldsymbol {p} _ {H} ^ {\text {c n t}, M, 2 \text {n d}}.
+$$
+
+Moreover, by definition of unfolding tree, there are only finite non-zero elements in both  $A^{\mathrm{dep},d}p_{G}^{\mathrm{cnt},M,2\mathrm{nd}}$  and  $A^{\mathrm{dep},d}p_{H}^{\mathrm{cnt},M,2\mathrm{nd}}$ , and the corresponding non-zero indices can only be in a fixed (finite) set. In this case, the upper triangular matrix  $A^{\mathrm{blosnj},M,2\mathrm{nd}}$  is reduced to a finite-dimensional matrix and thus  $A^{\mathrm{dep},d}p_{G}^{\mathrm{cnt},M,2\mathrm{nd}} = A^{\mathrm{dep},d}p_{H}^{\mathrm{cnt},M,2\mathrm{nd}}$ . By enumerating all  $d \geq 0$ , we obtain the desired result.
+
+![](images/3284e7d445ad611e884904d67281b951c3a22d1e19d1ad06f234e93520c4ec12.jpg)
+
+Combined with all previous results, we have arrived at the concluding corollary:
+
+Corollary C.34. Let  $M \in \{\mathrm{Sub},\mathsf{L},\mathsf{LF},\mathsf{F}\}$  be any model. For any two graphs  $G, H$ ,  $\chi_G^M (G) = \chi_H^M (H)$  iff  $\operatorname {hom}(F,G) = \operatorname {hom}(F,H)$  for all  $(F,T^{r})\in S^{M}$ .
+
+Proof. According to Corollary C.16, for any two graphs  $G,H$ , we have  $\chi_G^M (G) = \chi_H^M (H)$  iff  $\mathsf{cnt}^M ((F,T^r),G) = \mathsf{cnt}^M ((F,T^r),H)$  for all  $(F,T^{r})\in S^{M}$ . Then, Corollary C.33 implies that  $\mathsf{cnt}^M ((F,T^r),G) = \mathsf{cnt}^M ((F,T^r),H)$  for all  $(F,T^{r})\in S^{M}$  iff  $\mathsf{blos}((F,T^r),G) = \mathsf{blos}((F,T^r),H)$  for all  $(F,T^{r})\in S^{M}$ . Finally, Corollary C.32 implies that  $\mathsf{blos}((F,T^r),G) = \mathsf{blos}((F,T^r),H)$  for all  $(F,T^{r})\in S^{M}$  iff  $\mathsf{hom}(F,G) = \mathsf{hom}(F,H)$  for all  $(F,T^{r})\in S^{M}$ . We thus conclude the proof.
+
+# C.3 PART 2: NESTED EAR DECOMPOSITION
+
+In this part, we give equivalent formulations of the set  $S^M$  for any model  $M \in \{\mathrm{Sub}, \mathrm{L}, \mathrm{LF}, \mathrm{F}\}$  using the concept of NED defined in Definition 3.3.
+
+We first present some technical lemmas that will be used to deal with the tree decomposition of disconnected graphs.
+
+Lemma C.35. Let  $M \in \{\mathrm{Sub}, \mathrm{L}, \mathrm{LF}, \mathrm{F}\}$  be any model, and let  $(F, T^r) \in S^M$  be a tree-decomposed graph with  $\beta_T(t) = \{u, v\}$  for some  $u, v \in V_F$ . Assume that  $F$  has two connected components  $S_1, S_2 \subset V_F$  and  $u \in S_1$  and  $v \in S_2$  are in different components. Then, there exist tree decompositions  $(F[S_1], T_1^{r_1}), (F[S_2], T_2^{r_2}) \in S^M$  such that  $\beta_{T_1}(r_1) = \{u, u\}$  and  $\beta_{T_2}(r_2) = \{v, v\}$ .
+
+Proof. We can simply define  $T_{1} = (V_{T_{1}}, E_{T_{1}}, \beta_{T_{1}})$  with  $V_{T_{1}} = V_{T}$ ,  $E_{T_{1}} = E_{T}$ ,  $r_{1} = r$ , and  $\beta_{T_{1}}(t) = \{\{\phi_{1}(w): w \in \beta_{T}(t)\}\}$  with
+
+$$
+\phi_ {1} (w) = \left\{ \begin{array}{l l} w & \text {i f} w \in S _ {1}, \\ u & \text {i f} w \in S _ {2}. \end{array} \right.
+$$
+
+In other words,  $T_{1}$  has the same structure as  $T$  but with different bags such that all vertices in the same connected component as  $v$  are replaced by  $u$ . It is easy to see that  $T_{1}^{r_{1}}$  is a canonical tree decomposition of  $F[S_{1}]$  following Definition C.1. Moreover, we clearly have that for all  $M \in \{\mathrm{Sub}, \mathsf{L}, \mathsf{LF}, \mathsf{F}\}$ ,  $(F[S_{1}], T_{1}^{r_{1}}) \in S^{M}$ . We can construct  $T_{2} = (V_{T_{2}}, E_{T_{2}}, \beta_{T_{2}})$  by symmetry, which concludes the proof.
+
+The above lemma can be immediately generalized into the following one:
+
+Corollary C.36. Let  $M \in \{\text{Sub}, \mathsf{L}, \mathsf{LF}, \mathsf{F}\}$  be any model, and let  $(F, T^r) \in S^M$  be a tree-decomposed graph. For each connected component  $S \in V_F$  of  $F$ , pick any  $t \in V_T$  with the minimum depth such that  $S \cap \beta_T(t) \neq \emptyset$ . Then, there exists a tree decomposition  $\tilde{T}^s$  of  $F[S]$  satisfying that  $(F[S], \tilde{T}^s) \in S^M$  and  $S \cap \beta_T(t) \subset \beta_{\tilde{T}}(s)$ .
+
+Lemma C.37. Let  $M \in \{\text{Sub}, \mathsf{L}, \mathsf{LF}, \mathsf{F}\}$  be any model, and let  $(F, T^r) \in S^M$  be a tree-decomposed graph such that  $F$  is connected. Then, for each child node  $t$  of  $r$  in  $T^r$  and any vertex  $x$  in  $F[T^r[t]]$ , there is a path in  $F[T^r[t]]$  from  $x$  to some vertex in  $\beta_T(r)$ . Consequently, the number of connected components of  $F[T^r[t]]$  is bounded by the number of different elements in  $\beta_T(r)$ .
+
+Proof. Assume the above statement does not hold, and let  $S \subset V_{F[T^r[t]]}$  be the connected component of  $F[T^r[t]]$  such that  $S \cap \beta_T(r) = \emptyset$ . Note that  $F[S] = F[T^r[t]][S]$ . Since  $F$  is connected, there exists an edge  $\{v, w\} \in E_F$  such that  $v \in S$  and  $w \notin S$ . Since  $S$  is a connected component of  $F[T^r[t]]$ , we have  $w \notin V_{F[T^r[t]]}$ , and thus both  $v, w \notin \beta_T(r)$ . This yields a contradiction since  $\{v, w\}$  should be contained in a bag in  $T^r$  but  $B_T(v) \cap B_T(w) = \emptyset$ .
+
+To reduce duplication, below we only give proofs for Local 2-GNN and Local 2-FGNN. One can easily write a proof for Subgrapp HNN based on the proof of Local 2-GNN, and write a proof for 2-FGNN based on the proof of Local 2-FGNN.
+
+Lemma C.38. For any tree-decomposed graph  $(F, T^r) \in S^{\mathsf{L}}$ ,  $F$  has a strong NED.
+
+Proof. Based on Corollary C.36, we can assume that  $F$  is connected without loss of generality. We will prove the following stronger result: for any connected  $(F,T^r)\in S^{\mathsf{L}}$  with  $\beta_T(r) = \{\{u,v\}\}$ ,  $F$  has a strong NED where  $u,v$  are endpoints of the first ear. (For the case of  $u = v$ , the other endpoint can be arbitrary.)
+
+The proof is based on induction over the number the vertices in  $T^r$ . The above statement obviously holds for the base case of  $|V_T| = 1$ . Now assume that the statement holds when  $|V_T| \leq m$ , and consider the case of  $|V_T| = m + 1$ . Note that for any two different children  $t, t'$  of  $r$ ,  $F[T^r[t]]$  and  $F[T^r[t']]$  can only share vertices  $u, v$ . For each child node  $t$  of  $r$ , denote its unique child node as  $\tilde{t}$ . It is easy to see that  $T^r[\tilde{t}]$  is a canonical tree decomposition of  $F[T^r[\tilde{t}]]$  and  $(F[T^r[\tilde{t}]], T^r[\tilde{t}]) \in S^\perp$ . However, one needs to be cautious as  $F[T^r[\tilde{t}]]$  may not be connected (unlike the original graph  $F$ ). Below, we separately consider the following cases:
+
+1. First consider the case when  $u = v$ . In this case, Lemma C.37 implies that  $F[T^r[\tilde{t}]]$  is connected. According to the induction hypothesis,  $F[T^r[\tilde{t}]]$  has a strong NED (denoted as  $\mathcal{P}_{\tilde{t}}$ ) such that  $u$  is an endpoint of the first ear. We then merge the ear decompositions  $\mathcal{P}_{\tilde{t}}$  for all  $\tilde{t}$  into a whole  $\mathcal{P}$ , specify a root ear  $P_1$  in any  $\mathcal{P}_{\tilde{t}}$ , and let the first ear of other  $\mathcal{P}_{\tilde{t}}$  nest on  $P_1$  (with empty nested interval). It is easy to see that  $\mathcal{P}$  is a strong NED of  $F$  and  $u$  is an endpoint of the first ear.  
+2. Next consider the case when  $u \neq v$ . In this case, without loss of generality, assume  $\beta_T(\tilde{t}) = \{u, w\}$  for some  $w \in V_F$ .
+
+- Subcase 1:  $F[T^r[\tilde{t}]]$  is connected. Then,  $F[T^r[\tilde{t}]]$  has a strong NED  $\mathcal{P}_{\hat{t}}$  such that  $u$  and  $w$  are endpoints of the first ear.
+
+a) If  $\bar{w} = v$ , then  $F[T^r[t]] = F[T^r[\hat{t}]]$  and  $F[T^r[t]]$  clearly has a strong NED.  
+b) If  $w \notin N_F[v]$  or  $w = u$ , there are two additional subcases depending on whether  $\{u, v\} \in E_F$ . If  $\{u, v\} \in E_F$ , then  $F[T^r[t]]$  has a strong NED (which can be constructed from  $\mathcal{P}_{\tilde{t}}$  by adding an ear  $\{\{u, v\}\}$  and letting the first ear in  $\mathcal{P}_{\tilde{t}}$  nest on  $\{\{u, v\}\}$  (with empty nested interval). Otherwise,  $F[T^r[t]] \backslash \{v\}$  has a strong NED and  $v$  is an isolated vertex in  $F[T^r[t]]$ .  
+c) If  $w \in N_F(v)$  and  $w \neq u$ , then  $F[T^r[t]]$  also has a strong NED, which can be constructed from  $\mathcal{P}_{\tilde{t}}$  by extending the first ear to incorporate the edge  $\{w, v\}$  (we still need an additional ear  $\{\{u, v\}\}$  if  $\{u, v\} \in E_F$ ).
+
+- Subcase 2:  $F[T^r[\tilde{t}]]$  is disconnected. In this subcase, Lemma C.37 implies that  $F[T^r[\tilde{t}]]$  has exactly two connected components, and  $u$  and  $w$  are in different connected components (which can be easily proved by noting that  $v$  can only link to  $u$  or  $w$  in  $F[T^r[\tilde{t}]]$ ). We can thus invoke Lemma C.35, which shows that both connected components of  $F[T^r[\tilde{t}]]$ , denoted as  $\widehat{F}_1$  and  $\widehat{F}_2$ , admits tree decompositions  $\widehat{T}_1^{s_1}$  and  $\widehat{T}_2^{s_2}$  such that  $(\widehat{F}_1, \widehat{T}_1^{s_1}), (\widehat{F}_2, \widehat{T}_2^{s_2}) \in S^\mathsf{L}$ ,  $\beta_{\widehat{T}_1}(s_1) = \{\{u, u\}\}$  and  $\beta_{\widehat{T}_2}(s_2) = \{\{w, w\}\}$ . Moreover,  $|V_{\widehat{T}_1}| = |V_{\widehat{T}_2}| = |V_{T^r[\tilde{t}]}| < m + 1$ . Therefore, according to the induction hypothesis,  $\widehat{F}_1$  has a strong NED  $\mathcal{P}_{\tilde{t},1}$  with  $u$  as an endpoint of the first ear, and  $\widehat{F}_2$  has a strong NED  $\mathcal{P}_{\tilde{t},2}$  with  $w$  as an endpoint of the first ear. If  $\{w, v\} \in E_F$ , we can extend the first ear in  $\mathcal{P}_{\tilde{t},2}$  to include the edge  $\{w, v\}$ . Finally, if  $\{u, v\} \in E_F$ , by setting the first ear to be  $\{\{u, v\}\}$ , we can merge the two NEDs  $\mathcal{P}_{\tilde{t},1}$  and  $\mathcal{P}_{\tilde{t},2}$  to obtain the NED of  $F[T^r[t]]$  such that  $u, v$  are endpoints of the first ear.
+
+Overall, we always have that:
+
+a) If  $F[T^r[t]]$  is connected, then it admits a strong NED such that  $u, v$  are endpoints of the first ear.  
+b) If  $F[T^r[t]]$  is disconnected, then it has two connected components each admitting a strong NED such that  $u, v$  belong to an endpoint of the first ear for each of the two NEDs, respectively.
+
+Finally, noting that  $F = \bigcup_{t,\mathfrak{pa}_{Tr}(t) = r}F[T^r [t]]$ , we can merge all NEDs of  $F[T^{r}[t]]$  to obtain a strong NED of  $F$  satisfying that  $u,v$  are endpoints of the first ear.
+
+We thus conclude the proof of the induction step.
+
+Lemma C.39. For any graph  $F$ , if  $F$  admits a strong NED, then there exists a tree decomposition  $T^r$  of  $F$  such that  $(F,T^{r})\in S^{\mathsf{L}}$ .
+
+Proof. We can assume that  $F$  is connected without loss of generality, as it is easy to merge the tree decompositions of different connected components to form a single tree which is a tree decomposition of the whole graph. We will prove the following stronger result: for any connected graph  $F$ , if  $F$  admits a strong NED where  $u$  and  $v$  are two different endpoints of the first ear, then there exists a tree decomposition  $T^r$  of  $F$  such that  $(F, T^r) \in S^\mathbb{L}$  and  $\beta_T(r) = \{(u, v)\}$ .
+
+The proof is based on induction over the number the edges in  $F$ . The above statement obviously holds for the base case of  $|E_F| = 1$ . Now assume that the statement holds when  $|E_F| \leq m$ , and consider the case of  $|E_F| = m + 1$ . Denote the  $i$ -th ear as  $P_i$  and denote  $P_1 = \{\{w_0, w_1\}, \dots, \{w_{l-1}, w_l\}\}$  where  $w_0 = u$ ,  $w_l = v$ . Consider the following three cases:
+
+1. Either  $u$  or  $v$  is a cut vertex in  $F$ . Without loss of generality, assume that  $u$  is a cut vertex. In this case, there must exist a child ear  $P_i$  nested on  $P_1$  such that  $u$  is an endpoint of  $P_i$  and  $I(P_i) = \emptyset$ . We can thus split all ears into two parts  $\mathcal{P}_1$  and  $\mathcal{P}_2$ , where  $\mathcal{P}_1$  contains the ear  $P_i$  and its descendant ears in the NED tree, and  $\mathcal{P}_2$  contains other ears. Denote by  $F[\mathcal{P}_k]$  the connected graph induced by all edges in  $\mathcal{P}_k$  ( $k \in \{1, 2\}$ ). Then,  $\mathcal{P}_k$  is a strong NED of the graph  $F[\mathcal{P}_k]$ . Since both  $\mathcal{P}_1$  and  $\mathcal{P}_2$  contain no more than  $m$  edges, according to the induction hypothesis, there are tree decompositions  $T_k^{r_k}$  of  $F[\mathcal{P}_k]$  ( $k \in \{1, 2\}$ ) such that  $\beta_{T_1}(r_1) = \{\{u, w\}\}$  for some  $w \in V_F$ ,  $\beta_{T_2}(r_2) = \{\{u, v\}\}$ , and  $(F[\mathcal{P}_k], T_k^{r_k}) \in S^{\mathsf{L}}$  ( $k \in \{1, 2\}$ ). We can glue the tree  $T_1^{r_1}$  and  $T_2^{r_2}$  into a larger tree  $T^{r_2}$  by adding a new node  $t$  with  $\beta_T(t) = \{\{u, v, w\}\}$ , setting  $\mathsf{pa}_{T^r}(t) = r_2$  and  $\mathsf{pa}_{T^r}(r_1) = t$ . Clearly,  $T^{r_2}$  is a valid tree decomposition of  $F$ ,  $\beta_T(r_2) = \{\{u, v\}\}$ , and  $(F, T^{r_2}) \in S^{\mathsf{L}}$ .  
+2. There is an ear  $P_{i}$  nested on  $P_{1}$  with  $I(P_{i}) = P_{1}$ . In this case, we can split all ears into two parts  $\mathcal{P}_1$  and  $\mathcal{P}_2$ , where  $\mathcal{P}_1$  contains the ear  $P_{i}$  and its descendant ears in the NED tree, and  $\mathcal{P}_2$  contains other ears. Then,  $\mathcal{P}_k$  is a strong NED of the graph  $F[\mathcal{P}_k]$ . Since both  $\mathcal{P}_1$  and  $\mathcal{P}_2$  contain no more than  $m$  edges, according to the induction hypothesis, there is a tree decomposition  $T_k^{r_k}$  of  $F[\mathcal{P}_k]$  such that  $\beta_{T_k}(r_k) = \{\{u,v\}\}$  and  $(F[\mathcal{P}_k], T_k^{r_k}) \in S^{\mathsf{L}}$ . By merging the root node, we can glue the tree  $T_1^{r_1}$  and  $T_2^{r_2}$  into a larger tree  $T^r$ . Clearly,  $T^r$  is a valid tree decomposition of  $F$ ,  $\beta_T(r) = \{\{u,v\}\}$ , and  $(F,T^r) \in S^{\mathsf{L}}$ .  
+3. Neither  $u$  nor  $v$  is a cut vertex in  $F$  and all ears  $P_i$  nested on  $P_1$  satisfies  $I(P_i) \subsetneq P_1$ . In this case, we have either  $\{w_0, w_1\} \notin I(P_i)$  for all ear  $P_i$  nested on  $P_1$  or  $\{w_{l-1}, w_l\} \notin I(P_i)$  for all ear  $P_i$  nested on  $P_1$  (otherwise, it would contradict the definition of strong NED). Without loss of generality, assume  $\{w_0, w_1\} \notin I(P_i)$  for all ears  $P_i$  nested on  $P_1$ . Then, it is clear that  $l > 1$ , and the subgraph  $F \setminus \{u\}$  is connected and also admits a strong NED where  $w_1$  and  $v$  are two endpoints of the first ear. Therefore, according to the induction hypothesis, there is a tree decomposition  $\tilde{T}^s$  of  $F \setminus \{u\}$  satisfying  $(F \setminus \{u\}, \tilde{T}^s) \in S^\mathsf{L}$  and  $\beta_{\tilde{T}}(s) = \{w_1, v\}$ . We can then construct a tree  $T^r$  from  $\tilde{T}^s$  by adding two fresh nodes  $r$  and  $r'$  and setting  $\mathsf{pa}_{T^r}(s) = r'$  and  $\mathsf{pa}_{T^r}(r') = r$ . Set  $\beta_T(r) = \{u, v\}$ ,  $\beta_T(r') = \{u, v, w_1\}$ , and  $\beta_T(t) = \beta_{\tilde{T}}(t)$  for all  $t \in V_{\tilde{T}}$ . It is easy to see that the constructed  $T^r$  is a tree decomposition of  $F$  and  $(F, T^r) \in S^\mathsf{L}$ .
+
+Combining the above three cases concludes the induction step.
+
+![](images/3706a2b0f9b29c7dff4d686d5f1d771438cbdc11234be0e289fc186a18f43089.jpg)
+
+Theorem C.40. For any graph  $F$ ,  $F$  has a strong NED iff there is a tree decomposition  $T^r$  of  $F$  such that  $(F, T^r) \in S^\mathsf{L}$ .
+
+Proof. This is a direct consequence of Lemmas C.38 and C.39.
+
+![](images/7e164b3e67d390003e8e6b05ccb28558864c7e056100271d6404a714d427a1d9.jpg)
+
+We next turn to Local 2-FGNN, where the proof has a similar structure as that of Local 2-GNN.
+
+Lemma C.41. For any tree-decomposed graph  $(F, T^r) \in S^{\mathrm{LF}}$ ,  $F$  has an almost-strong NED.
+
+Proof. Based on Corollary C.36, we can assume that  $F$  is connected without loss of generality. We will prove the following stronger result: for any connected  $(F,T^r)\in S^{\mathrm{LF}}$  with  $\beta_T(r) = \{\{u,v\}\}$ ,  $F$  has an almost-strong NED where  $u,v$  are endpoints of the first ear. (For the case of  $u = v$ , the other endpoint can be arbitrary.)
+
+Similar to the proof of Lemma C.38, assume that the statement holds when  $|V_T| \leq m$ , and consider the case of  $|V_T| = m + 1$ . For each child node  $t$  of  $r$ , if  $t$  only has one child, the proof exactly follows the one in Lemma C.38. Therefore, it suffices to consider the case where  $t$  has two children  $q_1$  and  $q_2$ . Denote  $\beta_T(t) = \{\{u,v,w\}\}$ ,  $\beta_T(q_1) = \{\{u,w\}\}$ , and  $\beta_T(q_2) = \{\{v,w\}\}$ . Since  $w \in N_F[u] \cup N_F[v]$  (by definition of  $S^{\mathsf{LF}}$ ), we can assume that  $w \in N_F[v]$  without loss of generality. Recall that  $(F[T^r [q_1]], T^r [q_1])$ ,  $(F[T^r [q_2]], T^r [q_2]) \in S^{\mathsf{LF}}$ . Below, we separately consider the following cases:
+
+1. First consider the case when  $u = v$ . In this case, Lemma C.37 implies that both  $F[T^r[q_1]]$  and  $F[T^r[q_2]]$  are connected (since either  $u = w$  or  $\{u, w\} \in E_F$ ). According to the induction hypothesis, both  $F[T^r[q_1]]$  and  $F[T^r[q_2]]$  admit an almost-strong NED such that  $u$  and  $w$  are endpoints of the first ear. If  $w = u$ , we can merge the two almost-strong NEDs so that the first ear in one NED is nested on the first ear of the other (with an empty nested interval). Otherwise, the first ears in the two almost-strong NEDs share both endpoints and we can clearly merge them (in case of  $\{u, w\} \in E_F$ , there is a common ear  $\{\{u, w\}\}$  in both NEDs,
+
+which is taken only once). In both subcases, we obtain an almost-strong NED of  $F[T^r[t]]$  such that  $u$  is an endpoint of the first ear.
+
+2. Next consider the case when  $u \neq v$ . In this case,  $F[T^r[q_1]]$  and  $F[T^r[q_2]]$  share only one vertex  $w$ . Note that  $F[T^r[q_2]]$  is connected by Lemma C.37. We separately consider two subcases:
+
+- Subcase 1:  $F[T^r[q_1]]$  is connected. Then, according to the induction hypothesis,  $F[T^r[q_1]]$  admits an almost-strong NED  $\mathcal{P}_1$  such that  $u$  and  $w$  are endpoints of the first ear, and  $F[T^r[q_2]]$  admits an almost-strong NED  $\mathcal{P}_2$  such that  $w$  and  $v$  are endpoints of the first ear. (i) If  $w = v$ , we can merge the two almost-strong NEDs so that the first ear of  $\mathcal{P}_2$  is nested on the first ear of  $\mathcal{P}_1$  (with empty nested interval). (ii) If  $w = u$ , we can merge the two almost-strong NEDs so that the first ear of  $\mathcal{P}_1$  is nested on the first ear of  $\mathcal{P}_2$  (with empty nested interval). (iii) If  $w \in N_F(v)$  and  $w \neq u$ , the first ear of  $\mathcal{P}_2$  can be chosen as  $\{\{w, v\}\}$ . Then, we can merge the two almost-strong NEDs by gluing the first year in  $\mathcal{P}_1$  with the ear  $\{\{w, v\}\}$  in  $\mathcal{P}_2$ . One can see that the resulting NED is almost-strong. Overall, we always have that  $F[T^r[t]]$  admits an almost-strong NED such that  $u, v$  are endpoints of the first ear.  
+- Subcase 2:  $F[T^r[q_1]]$  is disconnected. In this subcase,  $w \neq u$ . Similar to the proof of Lemma C.38, we obtain that  $F[T^r[q_1]]$  has exactly two connected components, and  $u$  and  $w$  are in different connected components. We can thus invoke Lemma C.35, which shows that both connected components of  $F[T^r[q_1]]$ , denoted as  $\widehat{F}_1$  and  $\widehat{F}_2$ , admit almost-strong NEDs  $\mathcal{P}_1$  and  $\mathcal{P}_2$ , respectively. Moreover,  $u$  is an endpoint of the first ear in  $\mathcal{P}_1$ , and  $w$  is an endpoint of the first ear in  $\mathcal{P}_2$ . Besides,  $F[T^r[q_2]]$  admits an almost-strong NED  $\mathcal{P}_3$  with  $w, v$  as the endpoints of the first ear. By letting the first ear of  $\mathcal{P}_2$  nest on the first ear of  $\mathcal{P}_3$  (with an empty nested interval), we can merge  $\mathcal{P}_2$  and  $\mathcal{P}_3$ . Then, we can merge  $\mathcal{P}_1$  and  $\mathcal{P}_2 \cup \mathcal{P}_3$  following the same procedure as Subcase 2 in the proof of Lemma C.38.
+
+Overall, we always have that:
+
+a) If  $F[T^r[t]]$  is connected, then it admits an almost-strong NED such that  $u, v$  are endpoints of the first ear.  
+b) If  $F[T^r[t]]$  is disconnected, then it has two connected components each admitting an almost-strong NED such that  $u, v$  belong to an endpoint of the first ear for each of the two NEDs, respectively.
+
+In both subcases, it follows that  $F[T^{r}[t]]$  admits an almost-strong NED with  $u, v$  as two endpoints. Noting that  $F = \bigcup_{t, \mathfrak{pa}_{Tr}(t) = r} F[T^{r}[t]]$ , we can merge all NEDs of  $F[T^{r}[t]]$  to obtain an almost-strong NED of  $F$  with  $u, v$  as two endpoints.
+
+We thus conclude the proof of the induction step.
+
+Lemma C.42. For any graph  $F$ , if  $F$  admits an almost-strong NED, then there exists a tree decomposition  $T^r$  of  $F$  such that  $(F,T^{r})\in S^{\mathrm{LF}}$ .
+
+Proof. We can assume that  $F$  is connected without loss of generality. We will prove the following stronger result: for any graph  $F$ , if  $F$  admits an almost-strong NED where  $u$  and  $v$  are two different endpoints of the first ear, then there exists a tree decomposition  $T^r$  of  $F$  such that  $(F, T^r) \in S^{\mathsf{LF}}$  and  $\beta_T(r) = \{\{u, v\}\}$ .
+
+Similar to the proof of Lemma C.39, assume that the statement holds when  $|E_F| \leq m$ , and consider the case of  $|E_F| = m + 1$ . Denote the first ear as  $P_1 = \{\{w_0, w_1\}, \dots, \{w_{l-1}, w_l\}\}$  where  $w_0 = u$ ,  $w_l = v$ . Consider the following three cases:
+
+1. Either  $u$  or  $v$  is a cut vertex in  $F$ . This case is exactly the same as in Lemma C.39.  
+2. There is an ear  $P_{i}$  nested on  $P_{1}$  with  $I(P_{i}) = P_{1}$ . This case is also the same as in Lemma C.39.  
+3. Otherwise,  $l \geq 2$ .
+
+- If there is an ear  $P_i$  nested on  $P_1$  with  $\{w_0, w_1\}, \{w_1, w_2\} \in I(P_i)$ , then  $l \geq 3$ . By definition of almost-strong NED, there does not exist an ear  $P_j$  nested on  $P_1$  with  $\{w_{l-2}, w_{l-1}\}, \{w_{l-1}, w_l\} \in I(P_i)$ . In this case, we can split  $P_1$  into two parts:
+
+$P_{1,u} = P_1\backslash \{\{w_{l - 1},w_l\}\}$ , and  $P_{1,v} = \{\{w_{l - 1},w_l\}\}$ . Then, we can rearrange any ear nested on  $P_{1}$  so that it is nested on either  $P_{1,u}$  or  $P_{1,v}$ . In this way, we can split all ears into two sets  $\mathcal{P}_u$  and  $\mathcal{P}_v$ , one corresponding to  $P_{1,u}$  and its descendant ears, and the other corresponding to  $P_{1,v}$  and its descendant ears. Therefore, according to the induction hypothesis, there is a tree decomposition  $\tilde{T}_u^s$  of  $F[\mathcal{P}_u]$  satisfying  $(F[\mathcal{P}_u],\tilde{T}_u^s)\in S^{\mathrm{LF}}$  and  $\beta_{\tilde{T}_u}(s) = \{u,w_{l - 1}\}$ , and similarly, there is a tree decomposition  $\tilde{T}_v^{s'}$  of  $F[\mathcal{P}_v]$  satisfying  $(F[\mathcal{P}_v],\tilde{T}_v^{s'})\in S^{\mathrm{LF}}$  and  $\beta_{\tilde{T}_v}(s) = \{w_{l - 1},v\}$ . We can then construct a tree  $T^r$  by merging  $\tilde{T}_u^s$  and  $\tilde{T}_v^{s'}$  and adding two fresh nodes  $r,r'$ , where  $r$  is the root node,  $\mathsf{pa}_{T^r}(r') = r$  and  $\mathsf{pa}_{T^r}(s) = \mathsf{pa}_{T^r}(s') = r'$ . Set  $\beta_T(r) = \{u,v\}$  and  $\beta_T(r') = \{u,v,w_{l - 1}\}$ . It is easy to see that the constructed  $T^r$  is a tree decomposition of  $F$  and  $(F,T^r)\in S^{\mathrm{LF}}$ .
+
+- If there does not exist an ear  $P_{i}$  nested on  $P_{1}$  with  $\{w_{0}, w_{1}\}, \{w_{1}, w_{2}\} \in I(P_{i})$ , then we follow exactly the same analysis as in the previous item, except that now we split  $P_{1}$  into two parts:  $P_{1,u} = \{\{w_{0}, w_{1}\}\}$ , and  $P_{1,v} = P_{1} \backslash \{\{w_{0}, w_{1}\}\}$ . We can still construct a tree decomposition  $T^{r}$  of  $F$  such that  $(F, T^{r}) \in S^{\mathsf{LF}}$ .
+
+Combining the above three cases concludes the proof.
+
+![](images/f30cfda95c20f021405fe098e02b8544b38c7e149fb3a7dbe403bfc85e4ba0ae.jpg)
+
+Theorem C.43. For any graph  $F$ ,  $F$  has an almost-strong NED iff there is a tree decomposition  $T^r$  of  $F$  such that  $(F, T^r) \in S^{\mathsf{LF}}$ .
+
+Proof. This is a direct consequence of Lemmas C.41 and C.42.
+
+![](images/dc7856b177da299e7a9b26c6c4c4cfe5734d2fe1f098f50df076484b30cb27c0.jpg)
+
+# C.4 PART 3: PEBBLE GAME
+
+In this part, we prove that  $S^M$  is maximal (Definition 3.1(b)) for any  $M \in \{\mathrm{Sub}, \mathrm{L}, \mathrm{LF}, \mathrm{F}\}$ . To achieve this, we first introduce a general class of graphs which we call Fürer graphs (Fürer, 2001). Intuitions and illustrations of Fürer graphs can be found in Zhang et al. (2023a).
+
+Definition C.44 (Fürer graphs). Given any connected graph  $F = (V_F, E_F, \ell_F)$ , the Fürer graph  $G(F) = (V_{G(F)}, E_{G(F)}, \ell_{G(F)})$  is constructed as follows:
+
+$$
+V _ {G (F)} = \{(x, X): x \in V _ {F}, X \subset N _ {F} (x), | X | \bmod 2 = 0 \},
+$$
+
+$$
+E _ {G (F)} = \left\{\{(x, X), (y, Y) \} \subset V _ {G}: \{x, y \} \in E _ {F}, (x \in Y \leftrightarrow y \in X) \right\},
+$$
+
+$$
+\ell_ {G (F)} (x, X) = \ell_ {F} (x) \quad \forall (x, X) \in V _ {G (F)}.
+$$
+
+Here,  $x \in Y \leftrightarrow y \in X$  holds when either  $(x \in Y \text{ and } y \in X)$  or  $(x \notin Y \text{ and } y \notin X)$  holds. For each  $x \in V_F$ , denote the set
+
+$$
+\operatorname {M e t a} _ {F} (x) := \{(x, X): X \subset N _ {F} (x), | X | \bmod 2 = 0 \}, \tag {24}
+$$
+
+which is called the meta vertices of  $G(F)$  associated to  $x$ . Note that  $V_{G(F)} = \bigcup_{x\in V_F}\mathsf{Meta}_F(x)$ .
+
+We next define an operation called "twist":
+
+Definition C.45 (Twist). Let  $G(F) = (V_{G(F)}, E_{G(F)}, \ell_{G(F)})$  be the Fürer graph of  $F = (V_F, E_F, \ell_F)$ , and let  $\{x, y\} \in E_F$  be an edge of  $F$ . The twisted Fürer graph of  $G(F)$  for edge  $\{x, y\}$ , is constructed as follows:  $\text{twist}(G(F), \{x, y\}) := (V_{G(F)}, E_{\text{twist}(G(F), \{x, y\})}, \ell_{G(F)})$ , where
+
+$$
+E _ {\operatorname {w i s t} (G (F), \{x, y \})} := E _ {G (F)} \triangle \{\{\xi , \eta \}: \xi \in \operatorname {M e t a} _ {F} (x), \eta \in \operatorname {M e t a} _ {F} (y) \},
+$$
+
+and  $\triangle$  is the symmetric difference operator, i.e.,  $A\triangle B = (A\backslash B)\cup (B\backslash A)$ . For an edge set  $S = \{e_1,\dots ,e_k\} \subset E_F$ , we further define
+
+$$
+\operatorname {t w i s t} (G (F), S) := \operatorname {t w i s t} (\dots \operatorname {t w i s t} (G (F), e _ {1}) \dots , e _ {k}). \tag {25}
+$$
+
+Note that Equation (25) is well-defined as the resulting graph does not depend on the order of edges  $e_1, \dots, e_k$  for twisting.
+
+The following result is well-known (see e.g., Zhang et al., 2023a, Corollary I.5 and Lemma I.7)):
+
+Theorem C.46. For any connected graph  $F$  and any set  $S_1, S_2 \subset E_F$ ,  $\operatorname{twist}(G(F), S_1) \simeq \operatorname{twist}(G(F), S_2)$  iff  $|S_1| \mod 2 = |S_2| \mod 2$ .
+
+This result shows that the structure of twisted Fürer graph does not depend on which edge is twisted. Therefore, we can pick any edge  $\{u,v\} \in E_F$  and denote  $H(F) = \mathrm{twist}(G(F),\{u,v\})$ . We now show that  $G(F),H(F)$  can be distinguished via homomorphism information:
+
+Theorem C.47. For any graph  $F$ , let (twisted) Fürer graphs  $G(F)$ ,  $H(F)$  be defined above. Then,  $\operatorname{hom}(F, G(F)) \neq \operatorname{hom}(F, H(F))$ .
+
+Proof. In the proof below, we use  $g$  and  $h$  to denote mappings where  $g: V_F \to V_{G(F)}$  and  $h: V_F \to V_{H(F)}$ . The proof is divided into the following parts.
+
+1. We first prove that there is a homomorphism  $g \in \operatorname{Hom}(F, G(F))$  satisfying  $g(w) \in \operatorname{Meta}_F(w)$  for all  $w \in V_F$ , but there is no homomorphism  $h \in \operatorname{Hom}(F, H(F))$  satisfying  $h(w) \in \operatorname{Meta}_F(w)$  for all  $w \in V_F$ .
+
+a) Define a mapping  $g: V_F \to V_{G(F)}$  such that  $g(w) = (w, \emptyset)$  for all  $w \in V_F$ . We clearly have  $\{w, x\} \in E_F$  implies that  $\{g(w), g(x)\} \in E_{G(F)}$ . Moreover,  $\ell_{G(F)}(g(w)) = \ell_F(w)$ . Therefore,  $g \in \operatorname{Hom}(F, G(F))$  is indeed a homomorphism.  
+b) If we similarly define  $h: V_F \to V_{H(F)}$  such that  $h(w) = (w, \emptyset)$  for all  $w \in V_F$ , then for all  $\{w, x\} \in E_F \backslash \{\{u, v\}\}$  we have  $\{h(w), h(x)\} \in E_{H(F)}$ , but  $\{h(u), h(v)\} \notin E_{H(F)}$  since the edge  $\{u, v\}$  is twisted.
+
+It remains to prove that for all  $h: V_F \to V_{H(F)}$  of the form  $h(w) = (w, U_w)$  (for each  $w \in V_F$ ),  $h \notin \operatorname{Hom}(F, H(F))$ . It suffices to prove that there is an odd number of edges  $\{w, x\} \in E_F$  such that  $\{h(w), h(x)\} \notin E_{H(F)}$ . Let  $h, \tilde{h}$  be two such mappings that differ in only one vertex  $z$ , i.e.,  $h(w) = \tilde{h}(w)$  for all  $w \neq z$  but  $h(z) = (z, U_z) \neq (z, \tilde{U}_z) = \tilde{h}(z)$ . Denote  $D_z = \tilde{U}_z \triangle U_z$ . Based on the definition of Fürer graph, it follows that
+
+- for all  $\{w, x\} \in E_F$  with  $w \neq z, x \neq z$ , we have  $\{h(w), h(x)\} \in E_{H(F)}$  iff  $\{\tilde{h}(w), \tilde{h}(x)\} \in E_{H(F)}$ ;  
+- for all  $\{w, z\} \in E_F$  with  $w \notin D_z$ , we also have  $\{h(w), h(z)\} \in E_{H(F)}$  iff  $\{\tilde{h}(w), \tilde{h}(z)\} \in E_{H(F)}$ ;  
+- for all  $\{w, z\} \in E_F$  with  $w \in D_z$ , we have  $\{h(w), h(z)\} \in E_{H(F)}$  iff  $\{\tilde{h}(w), \tilde{h}(z)\} \notin E_{H(F)}$ .
+
+Since  $|U_z| \bmod 2 = 0$  and  $|\tilde{U}_z| \bmod 2 = 0$ , we have  $|D_z| \bmod 2 = 0$  and thus the number of edges  $\{w, x\} \in E_F$  such that  $\{h(w), h(x)\} \notin E_{H(F)}$  has the same parity as the number of edges such that  $\{\tilde{h}(w), \tilde{h}(x)\} \notin E_{H(F)}$ . Finally, noting that all mappings  $h$  can be obtained from the one in (b) by continually modifying  $U_w$  for each  $w \in V_F$  and the parity remains unchanged, we have concluded the proof of this part.
+
+2. We next prove that for any permutation  $\pi : V_F \to V_F$ , there does not exist a homomorphism  $h \in \operatorname{Hom}(F, H(F))$  satisfying  $h(w) \in \operatorname{Meta}_F(\pi(w))$  for all  $w \in V_F$ . Assume the conclusion does not hold and pick any  $h$  satisfying the above condition. Consider the following cases:
+
+a) Case 1: if  $\pi$  is an automorphism of  $F$ , then it is easy to see that  $h \circ \pi^{-1} \in \operatorname{Hom}(F, H(F))$ , because  $\{w, x\} \in E_F \Rightarrow \{\pi^{-1}(w), \pi^{-1}(x)\} \in E_F \Rightarrow \{h(\pi^{-1}(w)), h(\pi^{-1}(x))\} \in E_F$  for all  $w, x \in V_F$ , and  $\ell_F(w) = \ell_F(\pi^{-1}(w)) = \ell_{G(F)}(h(\pi^{-1}(w)))$  for all  $w \in V_F$ . Moreover,  $h \circ \pi^{-1}$  satisfies that  $h(\pi^{-1}(w)) \in \operatorname{Meta}_F(w)$ , yielding a contradiction to point 1.  
+b) Case 2: if  $\pi$  is not an automorphism of  $F$ , then there exists an edge  $\{w, x\} \in E_F$  such that  $\{\pi(w), \pi(x)\} \notin E_F$ . In this case, we must have  $\{h(w), h(x)\} \notin E_{H(F)}$  since by definition  $h(w) \in \mathsf{Meta}_F(\pi(w))$ ,  $h(x) \in \mathsf{Meta}_F(\pi(x))$ , and  $\{\pi(w), \pi(x)\}$  is not an edge of  $F$ .
+
+In both cases,  $h$  is invalid and thus there is no homomorphism  $h \in \operatorname{Hom}(F, H(F))$  satisfying  $h(w) \in \operatorname{Meta}_F(\pi(w))$  for all  $w \in V_F$ .
+
+3. We finally prove that the following two sets have equal size (i.e.,  $|S_G| = |S_H|$ ):
+
+$$
+S _ {G} := \left\{g \in \operatorname {H o m} (F, G (F)): \exists w, x, y \in V _ {F} \text {s . t .} x \neq y, g (x), g (y) \in \operatorname {M e t a} _ {F} (w) \right\},
+$$
+
+$$
+S _ {H} := \left\{h \in \operatorname {H o m} (F, H (F)): \exists w, x, y \in V _ {F} \text {s . t .} x \neq y, h (x), h (y) \in \operatorname {M e t a} _ {F} (w) \right\}.
+$$
+
+It suffices to prove that, for any proper subset  $U \subsetneq V_F$ , we have  $|S_G^U| = |S_H^U|$ , where  $S_G$  and  $S_H$  are defined as follows:
+
+$$
+S _ {G} ^ {U} := \left\{g \in \operatorname {H o m} (F, G (F)): g (x) \in \bigcup_ {w \in U} \operatorname {M e t a} _ {F} (w) \forall x \in V _ {F} \right\},
+$$
+
+$$
+S _ {H} ^ {U} := \left\{h \in \operatorname {H o m} (F, H (F)): h (x) \in \bigcup_ {w \in U} \operatorname {M e t a} _ {F} (w) \forall x \in V _ {F} \right\}.
+$$
+
+Fix  $U \subsetneq V_F$  and pick  $z \in V_F \backslash U$ . Let  $P$  be a simple path from  $u$  to  $z$  of the form  $P = \{\{w_0, w_1\}, \dots, \{w_{k-1}, w_k\}\} \subset E_F$  where  $\{w_0, w_1\} = \{u, v\}$ ,  $w_k = z$ . Define a mapping  $\sigma$  that takes  $g \in S_G^U$  as input and outputs a mapping  $h: V_F \to V_{H(F)}$ :
+
+$$
+h (x) = \left\{ \begin{array}{l l} g (x) & \text {i f} [ g (x) ] _ {0} \notin \{w _ {1}, \dots , w _ {k - 1} \}, \\ ([ g (x) ] _ {0}, [ g (x) ] _ {1} \bigtriangleup \{w _ {i - 1}, w _ {i + 1} \}) & \text {i f} [ g (x) ] _ {0} = w _ {i}, i \in [ k - 1 ], \end{array} \right.
+$$
+
+where we write  $g(x) = ([g(x)]_0, [g(x)]_1)$ . We will prove that  $h \in S_H^U$ . Since we clearly have  $h(w) \in U$  for all  $w \in V_F$ , it suffices to prove that  $h \in \mathsf{Hom}(F, H(F))$ . Let  $\{x, y\} \in E_F$  be any edge in  $F$ .
+
+- If  $[g(x)]_0, [g(y)]_0 \notin \{w_1, \dots, w_{k-1}\}$ , then  $\{h(x), h(y)\} = \{g(x), g(y)\} \in E_{G(F)}$ . Also, since  $\{[g(x)]_0, [g(y)]_0\} \neq \{u, v\}$ ,  $\{[g(x)]_0, [g(y)]_0\}$  is not twisted and thus  $\{h(x), h(y)\} \in E_{H(F)}$ .
+
+- If  $[g(x)]_0 \notin \{w_1, \dots, w_{k-1}\}$ ,  $[g(y)]_0 = w_i$  for some  $i \in [k-1]$ , and  $\{[g(x)]_0, [g(y)]_0\} \neq \{\{u, v\}, \{w_{k-1}, w_k\}\}$ , then
+
+$$
+\{h (x), h (y) \} = \left\{\left([ g (x) ] _ {0}, [ g (x) ] _ {1}\right), \left([ g (y) ] _ {0}, [ g (y) ] _ {1} \triangle \left\{w _ {i - 1}, w _ {i + 1} \right\}\right) \right\}.
+$$
+
+We have
+
+$$
+\begin{array}{l} \{g (x), g (y) \} \in E _ {G (F)} \iff ([ g (x) ] _ {0} \in [ g (y) ] _ {1}) \leftrightarrow ([ g (y) ] _ {0} \in [ g (x) ] _ {1}) \\ \Longleftrightarrow ([ g (x) ] _ {0} \in [ g (y) ] _ {1} \triangle \left\{w _ {i - 1}, w _ {i + 1} \right\}) \leftrightarrow ([ g (y) ] _ {0} \in [ g (x) ] _ {1}) \\ \iff \{h (x), h (y) \} \in E _ {G (F)}. \\ \end{array}
+$$
+
+Also, since  $\{[g(x)]_0, [g(y)]_0\}$  is not twisted,  $\{h(x), h(y)\} \in E_{H(F)}$ .
+
+- If  $[g(x)]_0, [g(y)]_0 \in \{w_1, \dots, w_{k-1}\}$ , the analysis is similar to the above one and we can still prove that  $\{h(x), h(y)\} \in E_{G(F)}$  and thus  $\{h(x), h(y)\} \in E_{G(F)}$ .  
+- If  $\{[g(x)]_0, [g(y)]_0\} = \{u, v\}$ , without loss of generality assume  $w_0 = u = [g(x)]_0$  and  $w_1 = v = [g(y)]_0$ . We have
+
+$$
+\begin{array}{l} \{g (x), g (y) \} \in E _ {G (F)} \iff (u \in [ g (y) ] _ {1}) \leftrightarrow (v \in [ g (x) ] _ {1}) \\ \Longleftrightarrow (u \notin [ g (y) ] _ {1} \triangle \{u, w _ {2} \}) \leftrightarrow (v \in [ g (x) ] _ {1}) \\ \Longleftrightarrow \{h (x), h (y) \} \notin E _ {G (F)}. \\ \end{array}
+$$
+
+However, since  $\{u,v\}$  is twisted, we still have  $\{h(x),h(y)\} \in E_{H(F)}$
+
+- Finally, we do not need to consider the case  $\{[g(x)]_0, [g(y)]_0\} = \{w_{k-1}, w_k\}$  since  $w_k = z \notin U$ .
+
+This proves that  $h \in S_H^U$ . Moreover, it is straightforward to see that the mapping  $\sigma(g) = h$  is a bijection from  $S_G^U$  to  $S_H^U$ . We have thus proved that  $|S_G^U| = |S_H^U|$ .
+
+Combining the above three items, we obtain  $\mathsf{Hom}(F,G(F)) > \mathsf{Hom}(F,H(F))$ , concluding the proof.
+
+In the subsequent analysis, we will prove that for all model  $M$  considered in Theorem 3.4 and any connected graph  $F$ ,  $\chi_{G(F)}^{M}(G(F)) = \chi_{H(F)}^{M}(H(F))$  if  $F \notin S^{M}$ . The proof is based on an important technique developed in Cai et al. (1992), called the pebble game. When restricting our analysis on Fürer graphs, the pebble game can be greatly simplified as shown in Fürer (2001); Zhang et al. (2023a). Below, we separately describe the corresponding pebble game for each model  $M$ .
+
+We first define a key concept called the connected component.
+
+Definition C.48 (Connected components). Let  $F = (V_F, E_F)$  be a connected graph and let  $U \subset V_F$  be a vertex set, called separation vertices. We say two edges  $\{u, v\}, \{x, y\} \in E_F$  are in the same connected component if there is a simple path  $\{\{y_0, y_1\}, \dots, \{y_{k-1}, y_k\}\}$  satisfying that  $\{y_0, y_1\} = \{u, v\}, \{y_{k-1}, y_k\} = \{x, y\}$  and  $y_i \notin U$  for all  $i \in [k-1]$ . It is easy to see that the above relation between edges forms an equivalence relation. Therefore, we can define a partition over the edge set, denoted by  $\mathbb{C}\mathbb{C}_F(U) = \{P_i : i \in [m]\}$  for some  $m$ , where each  $P_i \subset E_F$  is called a connected component.
+
+We are now ready to describe the game rule. There are two players (named Alice and Bob), a graph  $F$ , and several pebbles. At the beginning, all pebbles lie outside the graph. Through the game process, some pebbles will be placed on the vertices of  $F$  and thus separate the edges  $E_{F}$  into connected components according to Definition C.48. In each game round, Alice updates the location of pebbles, while Bob maintains a subset of connected components, ensuring that the number of selected components is odd. There are three major types of operations:
+
+1. Add a pebble  $\mathfrak{p}$ . Alice places a pebble  $\mathfrak{p}$  (previously outside the graph) on some vertex of  $F$ . If introducing this new pebble does not change the connected components, then Bob does nothing. Otherwise, there must be a connected component  $P$  separated by  $\mathfrak{p}$  into several components  $P = \bigcup_{i \in [m]} P_i$  for some  $m$ . Bob will update his selected components as follows: if  $P$  is selected, then remove  $P$  and add a subset of connected components in  $\{P_1, \dots, P_m\}$  while ensuring that the number of selected components in total is odd.  
+2. Remove a pebble  $\mathfrak{p}$ . Alice removes a pebble  $\mathfrak{p}$  (previously on some vertex) outside the graph. If introducing this new pebble does not change the connected components, then Bob does nothing. Otherwise, there are multiple connected components  $P_{1},\dots ,P_{m}$  getting merged into a whole  $P = \bigcup_{i\in [m]}P_{i}$ . Bob will update his selected components by removing all  $P_{i}$ ,  $i\in [m]$  (if selected) and optionally adding  $P$ , while ensuring that the number of selected components in total is odd.  
+3. Swap two pebbles  $\mathsf{p}$  and  $\mathsf{p}^{\prime}$ . Alice swaps the position of two pebbles  $\mathsf{p}$  and  $\mathsf{p}^{\prime}$ . This operation does not change the connected components and thus Bob does nothing.
+
+At any time, if there is an edge  $\{x,y\}$  such that both endpoints hold pebbles and the connected component  $\{\{x,y\}\}$  is selected by Bob, then Bob loses the game and Alice wins. If Alice cannot win through the game process, then Bob wins.
+
+We now define the concrete pebble game for each model  $M$  considered in this paper. In cases of Subgraph GNN, Local 2-GNN, Local 2-FGNN, and 2-FGNN, there are three pebbles  $\mathsf{p}_u$ ,  $\mathsf{p}_v$ ,  $\mathsf{p}_w$ . As described before, all pebbles lie outside the graph at the beginning. Alice first adds the pebble  $\mathsf{p}_u$  (operation 1) and then adds the pebble  $\mathsf{p}_v$  (operation 1). Next, the game cyclically executes the following process:
+
+- Subgraph GNN. Alice can choose either one of the following ways to play:
+
+- Remove the pebble  $\mathsf{p}_v$  (operation 2), and re-add the pebble  $\mathsf{p}_v$  (operation 1).  
+- Add the pebble  $\mathsf{p}_w$  (operation 1) adjacent to the pebble  $\mathsf{p}_v$ , swap pebble  $\mathsf{p}_v$  with  $\mathsf{p}_w$  (operation 3), and remove the pebble  $\mathsf{p}_w$  (operation 2).
+
+- Local 2-GNN. Alice can choose either one of the following ways to play:
+
+- Remove the pebble  $\mathsf{p}_u$  (operation 2), and re-add the pebble  $\mathsf{p}_u$  (operation 1).  
+- Remove the pebble  $\mathsf{p}_v$  (operation 2), and re-add the pebble  $\mathsf{p}_v$  (operation 1).  
+- Add the pebble  $\mathsf{p}_w$  (operation 1) adjacent to the pebble  $\mathsf{p}_u$ , swap pebble  $\mathsf{p}_u$  with  $\mathsf{p}_w$  (operation 3), and remove the pebble  $\mathsf{p}_w$  (operation 2).  
+- Add the pebble  $\mathsf{p}_w$  (operation 1) adjacent to the pebble  $\mathsf{p}_v$ , swap pebble  $\mathsf{p}_v$  with  $\mathsf{p}_w$  (operation 3), and remove the pebble  $\mathsf{p}_w$  (operation 2).
+
+- Local 2-FGNN. Alice can choose either one of the following ways to play:
+
+- Remove the pebble  $\mathsf{p}_u$  (operation 2), and re-add the pebble  $\mathsf{p}_u$  (operation 1).  
+- Remove the pebble  $\mathfrak{p}_v$  (operation 2), and re-add the pebble  $\mathfrak{p}_v$  (operation 1).  
+- Add the pebble  $\mathsf{p}_w$  (operation 1) adjacent to either pebble  $\mathsf{p}_u$  or pebble  $\mathsf{p}_v$ , swap pebble  $\mathsf{p}_w$  with any adjacent pebble (operation 3), and remove the pebble  $\mathsf{p}_w$  (operation 2).
+
+- 2-FGNN. Alice adds the pebble  $\mathsf{p}_w$  (operation 1), swap the pebbles  $\mathsf{p}_w$  with either  $\mathsf{p}_u$  or  $\mathsf{p}_v$  (operation 3), and remove the pebble  $\mathsf{p}_w$  (operation 2).
+
+Proposition C.49. For any model  $M \in \{\text{Sub}, \mathsf{L}, \mathsf{LF}, \mathsf{F}\}$  and any graph  $F$ , if Alice cannot win the pebble game associated to model  $M$  on graph  $F$ , then  $\chi_{G(F)}^{M}(G(F)) = \chi_{H(F)}^{M}(H(F))$ .
+
+Proof. Note that the pebble game does not depend on the vertex labels  $\ell_F$  in  $F$ . We first assume that the vertex labels are all different, i.e.,  $\ell_F(x) \neq \ell_F(y)$  for all  $x, y \in V_F$ . Then, the labels of vertices  $\xi \in V_{G(F)}, \eta \in V_{H(F)}$  are the same iff  $\xi$  and  $\eta$  belong to the same meta vertex, i.e.,  $\xi, \eta \in \mathsf{Meta}_F(x)$  for some  $x \in V_F$ . Based on Proposition C.9,  $\chi_{G(F)}^M(G(F)) = \chi_{H(F)}^M(H(F))$  iff  $\tilde{\chi}_{G(F)}^M(G(F)) = \tilde{\chi}_{H(F)}^M(H(F))$  where  $\tilde{\chi}^M$  is defined in Equations (7) to (10). In this case, the pebble games above exactly correspond to the aggregation formulas in Equations (7) to (10), and we can invoke the results in Zhang et al. (2023a, Theorem I.17) to show that  $\tilde{\chi}_{G(F)}^M(G(F)) = \tilde{\chi}_{H(F)}^M(H(F))$  (note that we do not need to consider the augmented Fürer graphs defined in their paper based on our label assignment). Therefore,  $\chi_{G(F)}^M(G(F)) = \chi_{H(F)}^M(H(F))$ .
+
+We next consider the general case when multiple vertices can have the same label in  $F$ . However, this can only make it harder to distinguish between  $G(F)$  and  $H(F)$ , and we clearly have  $\chi_{G(F)}^{M}(G(F)) = \chi_{H(F)}^{M}(H(F))$ .
+
+Based on the above result, in the subsequent proof we will analyze the players' strategy in the pebble game. Surprisingly, it turns out that given a graph  $F$ , if Alice can win the game, then her strategy can be described using the tree decomposition of  $F$  defined in Definition C.6.
+
+To illustrate this point, we need the concept of game state graph. Given a graph  $F$ , a game state is a three-tuple  $(u, v, Q)$  where  $u, v \in V_F \cup \{\emptyset\}$  and  $Q \subset \mathbb{C}\mathbb{C}_F(\{u, v\})$ , denoting the vertex that holds pebble  $\mathfrak{p}_u$ , the vertex that holds pebble  $\mathfrak{p}_v$ , and a subset of connected component selected by Bob, respectively. Here, the symbol  $\emptyset$  means that a pebble is left outside the graph. One can see that after any round, whether Alice can win the remaining game purely depends on this tuple. Now fix Alice's strategy. In each round, each game state will be transited to a finite number of states depending on how Bob plays, and all states and transitions form a directed graph, which we call the game state graph. The state  $(u, v, Q)$  is called a terminal state if  $\min_{P \in Q} |P| = 1$ . It is straightforward to see that Alice wins the game at any terminal state, as stated in the following result:
+
+Proposition C.50. Given graph  $F$  and model  $M \in \{\text{Sub}, \mathsf{L}, \mathsf{LF}, \mathsf{F}\}$ , let  $G^S$  be the game state graph defined above corresponding to an Alice's strategy. Then, Alice can win the game if there is an integer  $t$  such that any path in  $G^S$  of length  $t$  starting from the initial state  $(\emptyset, \emptyset, \{E_F\})$  goes through a terminal state.
+
+Proof. It suffices to prove that, at any terminal state, Alice can win the game. Let  $(u,v,Q)$  be a terminal state with  $\{x,y\} \in Q$ . If  $\{u,v\} = \{x,y\}$ , Alice already wins. If  $u \notin \{x,y\}$  and  $v \in \{x,y\}$ , Alice also wins in the next round since she can add the pebble  $\mathfrak{p}_w$  on some vertex  $w$  adjacent to  $\mathfrak{p}_v$  such that  $\{v,w\} = \{x,y\}$ . This is a valid game rule for all model  $M$ . Next, if  $u \in \{x,y\}$  and  $v \notin \{x,y\}$ , Alice also wins in the next round since she can first remove the pebble  $\mathfrak{p}_v$  and then place it on the unique vertex in  $\{x,y\} \backslash \{u\}$ . This is also a valid game rule for all model  $M$ . Note the removing  $\mathfrak{p}_v$  does not merge the connected component  $\{\{x,y\}\}$  since  $v \notin \{x,y\}$ . Finally, if both  $u \notin \{x,y\}$  and  $v \notin \{x,y\}$ ,  $F$  only has one edge and Alice can clearly win. Combining these cases, we conclude that Alice can always win.
+
+Based on the above proposition, a game state  $(u,v,Q)$  is called unreachable if any path starting from the initial state  $(\emptyset ,\emptyset ,\{E_F\})$  and ending at  $(u,v,Q)$  goes through some terminal state. We do not need to consider unreachable states since Alice always wins before reaching it. We next introduce an important technical concept:
+
+Definition C.51. Given a game state graph  $G^{\mathsf{S}}$ , a state  $(u, v, \{P\})$  is termed as "contracted" if for any transition  $((u, v, \{P\}), (u', v', \{P'\})) \in E_{G^{\mathsf{s}}}, P' \subset P$ . It is called strictly contracted if for any transition  $((u, v, \{P\}), (u', v', \{P'\})) \in E_{G^{\mathsf{s}}}, P' \subsetneq P$ .
+
+We have the following result:
+
+Lemma C.52. For any model  $M \in \{\text{Sub}, \text{L}, \text{LF}, \text{F}\}$  and any graph  $F$ , if Alice can win the pebble game associated to model  $M$  on graph  $F$ , then there exists a game state graph  $G^S$  corresponding to a winning strategy of Alice such that any reachable and non-terminal state is strictly contracted.
+
+Proof. We first prove that there is a strategy for Alice such that any reachable and non-terminal state is contracted. We will assume that for any reachable and non-terminal state  $(u, v, \{P\})$ , if there is only one transition of the form  $((u, v, \{P\}),(u', v', \{P'\}))$ , then  $P \notin P'$ , since making the connected component larger should be a bad strategy. (Otherwise, it is straightforward to construct a new strategy for Alice satisfying the assumption.) Now suppose that there exists a reachable state  $(u, v, \{P\})$  that is not contracted. Note that the game state graph induced by all reachable states is a Directed Acyclic Graph (DAG), so we can choose the state  $(u, v, \{P\})$  such that any path from the initial state  $(\emptyset, \emptyset, \{E_F\})$  to  $(u, v, \{P\})$  does not pass any intermediate state that is not contracted. Below, we will construct a new strategy that removes the state  $(u, v, \{P\})$  (making it unreachable).
+
+Note that we clearly have  $u \neq \emptyset$  and  $v \neq \emptyset$ . Without loss of generality, assume that  $((u,v,\{P\}),(u,v',\{P'\}))$  is a transition such that  $P' \notin P$  (the case of transition  $((u,v,\{P\}),(u',v,\{P'\}))$  is the same by symmetry). Moreover, we can assume that  $P \notin \mathbb{C}\mathbb{C}_F(\{u\})$  and  $P \notin \mathbb{C}\mathbb{C}_F(\{v\})$  (in other words, both  $u$  and  $v$  are at the boundary of the connected component  $P$ ). It would be easier to analyze the case where either  $u$  or  $v$  are not at the boundary. We separately consider the following cases:
+
+1. The transition  $((u,v,\{P\}),(u,v^{\prime},\{P^{\prime}\}))$  corresponds to Alice removing pebble  $\mathfrak{p}_v$  and placing it on  $v^{\prime}$ . Let  $(u_0,v_0,Q_0),\dots ,(u_T,v_T,Q_T)$  be any path from the initial state  $(\emptyset ,\emptyset ,\{E_F\})$  to  $(u,v,\{P\})$ , and let  $t\leq T$  be the maximal number such that  $v_{t}\neq v$ . We will construct a new strategy for Alice as follows.
+
+- At state  $(u_t, v_t, Q_t)$ , she removes pebble  $\mathfrak{p}_v$  and places it on  $v'$ , yielding (possibly multiple) states of the form  $(u_t, v', \tilde{Q}_t)$ .  
+- For all  $t < \tilde{t} < T$ , we have  $v_{t} = v$ . For each  $\tilde{t}$  and possible  $\tilde{Q}_{\tilde{t}}$ , she will apply the original strategy at  $(u_{\tilde{t}}, v_{\tilde{t}}, Q_{\tilde{t}})$  to the state  $(u_{\tilde{t}}, v', \tilde{Q}_{\tilde{t}})$ , namely, placing pebble  $\mathsf{p}_w$  on vertex  $u_{\tilde{t} + 1}$ , swapping pebble  $\mathsf{p}_u$  with  $\mathsf{p}_w$ , and leaving  $\mathsf{p}_w$  outside the graph. (Note that she cannot remove pebble  $\mathsf{p}_u$  first; otherwise the state  $(u_{\tilde{t}}, v_{\tilde{t}}, Q_{\tilde{t}})$  will not be contracted since  $u$  is at the boundary of  $Q_{\tilde{t}}$ .)
+
+It follows that  $((u_{\tilde{t}},v^{\prime},\tilde{Q}_{\tilde{t}}),(u_{\tilde{t} +1},v^{\prime},\tilde{Q}_{\tilde{t} +1}))$  is a transition of the new strategy,  $(u_{\tilde{t}},v^{\prime},\tilde{Q}_{\tilde{t}})$  is contracted, and all states of the form  $(u,v^{\prime},\tilde{Q}_{\tilde{T}})$  is reachable in the original strategy.
+
+We can repeat the above procedure for all paths from the initial state  $(\emptyset, \emptyset, \{E_F\})$  to state  $(u, v, \{P\})$ . Then, in the new strategy  $(u, v, \{P\})$  will be unreachable. However, the state  $(u_t, v_t, Q_t)$  now may violate the condition in Lemma C.52. Note that the assumption in the first paragraph is still satisfied for state  $(u_t, v_t, Q_t)$ . In this case, we can recursively apply the above procedure for state  $(u_t, v_t, Q_t)$ . Note that the procedure will only repeat a finite number of times, as the length of the path from the initial state to the state  $(u_t, v_t, Q_t)$  is strictly less than the length of the path from the initial state to the state  $(u, v, \{P\})$ .
+
+2. The transition  $((u,v,\{P\}),(u,v^{\prime},\{P^{\prime}\}))$  corresponds to Alice placing pebble  $\mathfrak{p}_w$  on  $v^{\prime}$ , swapping  $\mathfrak{p}_v$  and  $\mathfrak{p}_w$ , and removing pebble  $\mathfrak{p}_w$ . In this case, we must have  $P\notin \mathbb{C}\mathbb{C}_F(\{u,v,v'\})$  (i.e.,  $v^{\prime}$  further splits the component  $P$ ). Otherwise, the state  $(u,v,\{P\})$  must have only one transition and  $P\subset P^{\prime}$ , a contradiction to the first paragraph. Since  $P\notin \mathbb{C}\mathbb{C}_F(\{u,v,v'\})$ , it is thus easy to see that, if Alice changes her strategy by just removing pebble  $\mathfrak{p}_v$  and placing it on vertex  $v^{\prime}$ , all transitions starting from  $(u,v,\{P\})$  does not change. Therefore, we can just invoke the previous item to construct a desired strategy.
+
+Combining the two cases, we conclude that there is a strategy for Alice such that any reachable and non-terminal state is contracted. We next prove that any reachable and non-terminal state can be strictly contracted. Assume the result does not hold and the state  $(u,v,\{P\})$  is reachable, non-terminal, but not strictly contracted. Then, there is a transition  $((u,v,\{P\}),(u',v',\{P\})) \in E_{G^s}$ . Consider the following two cases:
+
+1.  $P \notin \mathbb{C}C_F(\{u\})$  and  $P \notin \mathbb{C}C_F(\{v\})$  (i.e., both  $u$  and  $v$  are at the boundary of the connected component  $P$ ). It follows that  $u' = u$  and  $v' = v$ . This implies that the game state graph is not acyclic, a contradiction.  
+2.  $P \in \mathbb{C}\mathbb{C}_F(\{u\})$  or  $P \in \mathbb{C}\mathbb{C}_F(\{v\})$ . Without loss of generality, assume that  $P \in \mathbb{C}\mathbb{C}_F(\{u\})$ . Since  $(u,v,\{P\})$  is non-terminal and not strictly contracted, there is a reachable and non-terminal state  $(\tilde{u},\tilde{v},\{P\})$  with either  $\tilde{u} = u$  or  $\tilde{v} = u$  such that there is a path from  $(u,v,\{P\})$
+
+to  $(\tilde{u},\tilde{v},\{P\})$  and  $(\tilde{u},\tilde{v},\{P\})$  is strictly contracted. We can then change the strategy at state  $(u,v,\{P\})$  to make it strictly contracted. Concretely, Alice can remove pebble  $\mathsf{p}_v$  (which does not merge connected components selected by Bob since  $P\in \mathbb{C}\mathbb{C}_F(\{u\}))$ , and place pebble  $\mathsf{p}_v$  on the vertex that corresponds to the strategy at state  $(\tilde{u},\tilde{v},\{P\})$  (possibly with a difference that  $u$  and  $v$  become exchanged). This makes the state  $(u,v,\{P\})$  strictly contracted.
+
+Combining the two cases concludes the proof.
+
+![](images/a72e207dcf82b05fef066d22c7af4a5edebfb8a1c197ac154832756dc7f72c6d.jpg)
+
+We are now ready to state the main theorem:
+
+Theorem C.53. Let  $M \in \{\text{Sub}, \mathsf{L}, \mathsf{LF}, \mathsf{F}\}$  be any model. Given any connected graph  $F$ , if Alice can win the pebble game associated with model  $M$  on graph  $F$ , then there is a tree decomposition  $T^r$  of  $F$  such that  $(F, T^r) \in S^M$ , where  $S^M$  is defined in Definition C.6.
+
+Proof. Let  $G^{\mathsf{S}}$  be the game state graph satisfying Lemma C.52. For each game state  $s$ , denote by  $\mathrm{next}_{G^{\mathsf{S}}}(\bar{s})$  the set of states  $s'$  such that  $(s, s')$  is a transition in  $G^{\mathsf{S}}$  and  $s'$  contains only a single connected component, i.e.,  $s'$  has the form  $(u, v, \{P\})$ . For a terminal state  $s$  of the form  $(u, v, \{\{x, y\}\})$ , define  $\mathrm{next}_{G^{\mathsf{S}}}(\bar{s}) = \{(x, y, \{\{x, y\}\})\}$ . By definition,  $\mathrm{next}_{G^{\mathsf{S}}}(\emptyset, \emptyset, \{E_F\}) = \{(u, \emptyset, Q_1), \dots, (u, \emptyset, Q_m)\}$  for some  $u \in V_F$  and  $Q_1, \dots, Q_m$  is the finest partition of  $\mathbb{C}\mathbb{C}_F(\{u\})$ .
+
+The tree  $T^r$  will be recursively constructed as follows. First create the tree root  $r$  with  $\beta_T(r) = \{u, u\}$ . As we will see later, the root node will be associated with the set of states  $S(r) := \mathrm{next}_{G^s}(\emptyset, \emptyset, \{E_F\})$ . We then do the following procedure:
+
+Let  $t$  be a leaf node in the current tree associated with a non-empty set of game states  $S(t)$  such that  $|\bigcup_{(u,v,\{P\}) \in S(t)} P| > 1$ . For each state  $(u,v,\{P\}) \in S(t)$ , create a new node  $\tilde{t}$  and set its parent to be  $t$ . Pick any state  $(u',v',\{P'\}) \in \mathrm{next}_{G^s}(u,v,\{P\})$ . Then, there must be a unique vertex  $w \in \{\{u',v'\}\} \setminus \{\{u,v\}\}$  and  $w$  does not depend on which  $(u',v',\{P'\})$  is picked (by definition of the game rule). Set  $\beta_T(\tilde{t}) = \beta_T(t) \cup \{\{w\}\}$ . Then, do the following constructions:
+
+- If there is a state of the form  $(u, w, \{P'\}) \in \mathrm{next}_{G^s}(u, v, \{P\})$ , then create a new node  $t'$  and connect it to the parent  $\tilde{t}$ . Set  $\beta_T(t') = \{(u, w)\}$ , and the node  $t'$  will be associated to the set of states  $S(t') = \{(u, w, \{P'\}): (u, w, \{P'\}) \in \mathrm{next}_{G^s}(u, v, \{P\})\}$ .  
+- If there is a state of the form  $(w, v, \{P''\}) \in \mathsf{next}_{G^{\mathsf{s}}}(u, v, \{P\})$ , then create a new node  $t''$  and connect it to the parent  $\tilde{t}$ . Set  $\beta_{T}(t'') = \{(w, v)\}$ , and the node  $t''$  will be associated to the set of states  $S(t'') = \{(w, v, \{P''\}): (w, v, \{P''\}) \in \mathsf{next}_{G^{\mathsf{s}}}(u, v, \{P\})\}$ .
+
+Note that (i) either  $w \neq u$  or  $w \neq v$  (since the game state graph is a DAG); (ii) both items can be used for one state  $(u, v, \{P\})$ , which happens for FWL-type GNNs. The construction of the tree completes when each leaf node is associated with only one game state of the form  $(x, y, \{\{x, y\}\})$ . It is easy to see that the above procedure terminates after adding a finite number of tree nodes.
+
+We now prove that  $T^r$  is a canonical tree decomposition of  $F$  and  $(F, T^r) \in S^M$ .
+
+1. We first prove that any edge in  $F$  is contained in some bag of  $T^r$ . Pick any non-leaf tree node  $t$  of even depth and any of its child  $\tilde{t}$ , denote  $(u, v, P) \in S(t)$  be the state associated with  $\tilde{t}$  in the above construction, and denote  $\beta_T(\tilde{t}) = \{\{u, v, w\}\}$ . We have
+
+$$
+P = \left(\bigcup_{\substack{t^{\prime},\mathsf{pa}_{T^{r}}(t^{\prime}) = \tilde{t},\\ (x,y,\{P^{\prime}\})\in S(t^{\prime})}}P^{\prime}\right)\cup \{\{u,w\} \in E_{F}\} \cup \{\{v,w\} \in E_{F}\} .
+$$
+
+Here,  $P' \subset P$  is a consequence of Lemma C.52, and the remaining edges  $\{u, w\}$  or  $\{v, w\}$  are in  $P$  because  $(u, v, P)$  is strictly contracted and  $w$  must be in the interior of  $P$ . Therefore,
+
+$$
+P\cup \{\{u,v\} \in E_{F}\} = \left(\bigcup_{\substack{t^{\prime},\mathsf{pa}_{Tr}(t^{\prime}) = \tilde{t},\\ (x,y,\{P^{\prime}\})\in S(t^{\prime})}}(P^{\prime}\cup \{\{x,y\} \in E_{F}\})\right)\cup \{\{x,y\} \in E_{F}:x,y\in \beta_{T}(\tilde{t})\} .
+$$
+
+Recursively applying the above equation yields the desired result
+
+$$
+E_{F} = \bigcup_{\substack{\tilde{t},\mathsf{dep}_{T}(\tilde{t})\text{is odd}}}\{\{x,y\} \in E_{F}:x,y\in \beta_{T}(\tilde{t})\} ,
+$$
+
+because (i)  $\bigcup_{(u,\emptyset,P)\in S(r)}P = E_F$ , and (ii) all leaf node  $t$  with  $S(t) = (x,y,\{\{x,y\}\})$  contains an edge  $\{x,y\}$  in its bag.
+
+2. We next prove that  $T^r$  satisfies the condition of Definition C.1(c). Fix any vertex  $w \in V_F$ , and let  $t$  be the tree node with minimal depth that contains  $w$ . Without loss of generality, assume that  $t$  is not the root. In this case, the depth of  $t$  is odd and  $\beta_T(t) = \{ \{u, v, w\} \}$  for some  $u \neq w, v \neq w$ . Let  $t'$  be a child node of  $t$  and we have  $w \in \beta_T(t')$ . It remains to prove that for any descendant  $\tilde{t} \in \mathsf{Desc}_T(t')$ , if  $w \in \beta_T(\tilde{t})$ , then  $w \in \beta_T(\hat{t})$  for all  $\hat{t}$  on the path between  $t'$  and  $\tilde{t}$ . This is actually a direct consequence of Lemma C.52, because when a pebble originally placed on  $w$  is removed, all edges linked to  $w$  will not be selected by Bob and thus any pebble can never be placed on  $w$  again.  
+3.  $T^r$  is canonical as Definition C.4 is clearly satisfied.  
+4. Finally, it is also easy to see that  $(F,T^{r})$  satisfies Definition C.6.
+
+We thus conclude the proof.
+
+![](images/4d53758aaeab6f723fd0adf83db534c477b1e660bc87bf25007e8c1d9d904a52.jpg)
+
+Corollary C.54. Let  $M \in \{\text{Sub}, \mathsf{L}, \mathsf{LF}, \mathsf{F}\}$  be any model. For any connected graph  $F \notin \mathcal{F}^M$ , let  $G(F)$  and  $H(F)$  be the Fürer graph and twisted Fürer graph with respect to  $F$ . Then,  $\text{hom}(F, G(F)) \neq \text{hom}(F, H(F))$  and  $\chi_{G(F)}^M(G(F)) = \chi_{H(F)}^M(H(F))$ .
+
+Proof. The proof directly follows from applying Theorems C.47 and C.53 and Proposition C.49.  $\square$
+
+![](images/efca4d81cb4eb459488c47a94aea190aa1e1b3a11b88ab86f90a692f33d4efc8.jpg)
+
+Finally, we remark that our construction can be easily generalized for disconnected graphs  $F \notin \mathcal{F}^M$ . Let  $F$  be the disjoint union of graphs  $\{F_i : i \in [m]\}$  where each  $F_i$  is the graph corresponding to a connected component of  $F$ . Assume that  $F_1$  is the connected component with the most number of edges (in case of a tie, pick the graph with the most number of vertices). Define  $\tilde{G}(F)$  be the disjoint union of  $G(F_1), F_2, \dots, F_m$  and  $\tilde{H}(F)$  to be the disjoint union of  $H(F_1), F_2, \dots, F_m$ . It follows that  $\chi_{\tilde{G}(F)}\tilde{G}(F) = \chi_{\tilde{H}(F)}\tilde{H}(F)$  and
+
+$$
+\begin{array}{l} \operatorname {h o m} (F, \tilde {G} (F)) = \prod_ {i \in [ m ]} \operatorname {h o m} (F _ {i}, \tilde {G} (F)) = \prod_ {i \in [ m ]} (\operatorname {h o m} (F _ {i}, G (F _ {1})) + \operatorname {h o m} (F _ {i}, F _ {2} \cup \dots \cup F _ {m})) \\ > \prod_ {i \in [ m ]} \left(\operatorname {h o m} \left(F _ {i}, H \left(F _ {1}\right)\right) + \operatorname {h o m} \left(F _ {i}, F _ {2} \cup \dots \cup F _ {m}\right)\right) = \operatorname {h o m} (F, \tilde {H} (F)), \\ \end{array}
+$$
+
+where we use the fact that  $\operatorname{hom}(F_i, G(F_1)) = \operatorname{hom}(F_i, H(F_1))$  when  $F_1 \notin \operatorname{Spasm}(F_i)$  (which can be easily proved following Theorem C.47). This concludes the proof of the general case.
+
+# D NODE/EDGE-LEVEL EXPRESSIVITY
+
+This section aims to prove Theorem 3.7. For the ease of reading, we first restate it below:
+
+Theorem 3.7 For all model  $M$  defined in Section 2,  $\mathcal{F}_{\mathfrak{n}}^{M}$  and  $\mathcal{F}_{\mathrm{e}}^{M}$  (except MPNN) exist. Moreover,
+
+- MPNN:  $\mathcal{F}_{\mathfrak{n}}^{\mathsf{MP}} = \{F^{w}: F \text{ is a tree}\}$ ;  
+- Subgraph GNN:
+
+$$
+\mathcal {F} _ {\mathfrak {n}} ^ {\mathrm {S u b}} = \left\{F ^ {w}: F \text {h a s a N E D w i t h s h a r d e n p o i n t} w \right\} = \left\{F ^ {w}: F \backslash \{w \} \text {i s a f o r e s t} \right\},
+$$
+
+$$
+\mathcal {F} _ {\mathrm {e}} ^ {\mathrm {S} _ {\mathrm {u b}}} = \left\{F ^ {w x}: F \text {h a s a N E D w i t h s h a r e d e n p o i n t} w \right\} = \left\{F ^ {w x}: F \backslash \{w \} \text {i s a f o r e s t} \right\};
+$$
+
+- 2-FGNN:  $\mathcal{F}_{\mathfrak{n}}^{\mathsf{F}} = \{F^{w}: F$  has a NED where  $w$  is an endpoint of the first ear},
+
+$$
+\mathcal {F} _ {\mathrm {e}} ^ {\mathrm {F}} = \left\{F ^ {w x}: F \text {h a s a N E D w h e r e} w \text {a n d} x \text {a r e e x i p o n t s o f t h e f i r s t e a r} \right\}.
+$$
+
+The cases of Local 2-GNN and Local 2-FGNN are similar to 2-FGNN by replacing "NED" with "strong NED" and "almost-strong NED", respectively.
+
+We remark that the definition of node/edge-level homomorphism expressivity involves only connected graphs for sake of simplicity. Similar to the proof of the graph-level expressivity, the proof of Theorem 3.7 consists of three parts. Among them, the proof related to tree decomposition and ear decomposition is quite similar to that of graph-level expressivity, so we only illustrate the proof sketch for clarity (Appendix D.1). However, the proof related to pebble game and counterexample graphs will require additional techniques, which is detailed in Appendix D.2.
+
+# D.1 RELATED TO TREE DECOMPOSITION AND EAR DECOMPOSITION
+
+We first extend several notations that are used in Appendix C.2.
+
+Definition D.1 (Tree decomposition for rooted graphs). Given a rooted graph  $G^u$  and tree  $T^r$ , we say  $T^r$  is a tree decomposition of  $G^u$  if  $T^r$  is a tree decomposition of  $G$  and all elements in  $u$  belong to the root bag, i.e.,  $u_i \in \beta_T(r)$  for all  $i$ .
+
+Based on this definition, given model  $M$ , we can define  $\mathcal{S}_{\mathfrak{n}}^{M}$  to be the family of tree-decomposed graphs  $(F^u,T^r)$  such that  $(F,T^r)\in S^M$ ; similarly, we define  $\mathcal{S}_{\mathrm{e}}^{M}$  to be the family of tree-decomposed graphs  $(F^{uv},T^{r})$  such that  $(F,T^{r})\in S^{M}$ .
+
+Definition D.2 (Bag isomorphism for rooted graphs). Given tree-decomposed graph  $(F^{\mathbf{u}}, T^{r})$  and rooted graph  $G^{\mathbf{v}}$  where  $\mathbf{u}$  and  $\mathbf{v}$  have equal length, a bag isomorphism from  $(F^{\mathbf{u}}, T^{r})$  to  $G^{\mathbf{v}}$  is a homomorphism  $f$  from  $F^{\mathbf{u}}$  to  $G^{\mathbf{v}}$  such that  $f$  is a bag isomorphism from  $(F, T^{r})$  to  $G$ .
+
+We can similarly define bag-isomorphism homomorphism (bIsoHom, bIsoSurj, bIsoInj), bag extension (bExt), and bag-strong surjective (bStrSurj) from tree-decomposed graph  $(F^u, T^r)$  to tree-decomposed graph  $(\tilde{F}^v, \tilde{T}^s)$ .
+
+Definition D.3 (Generalization of Definition C.15 for rooted graphs). Let  $M \in \{\mathrm{Sub}, \mathrm{L}, \mathrm{LF}, \mathrm{F}\}$  be any model. Given a rooted graph  $G^{u}$  and a tree-decomposed graph  $(F^w, T^r)$ , define
+
+$$
+\mathsf {c n t} ^ {M} \left(\left(F ^ {w}, T ^ {r}\right), G ^ {u}\right) := \left| \left\{v \in V _ {G}: \exists D \in \mathbb {N} _ {+} \text {s . t .} \left(\left[ F _ {G} ^ {M, (D)} (u, v) \right] ^ {u}, T _ {G} ^ {M, (D)} (u, v)\right) \simeq \left(F ^ {w}, T ^ {r}\right) \right\} \right|.
+$$
+
+Given a rooted graph  $G^{uv}$  and a tree-decomposed graph  $(F^{wx}, T^r)$ , define
+
+$$
+\mathsf {c n t} ^ {M} \left(\left(F ^ {w x}, T ^ {r}\right), G ^ {u v}\right) := \mathbb {I} \left[ \exists D \in \mathbb {N} _ {+} \text {s . t .} \left(\left[ F _ {G} ^ {M, (D)} (u, v) \right] ^ {u v}, T _ {G} ^ {M, (D)} (u, v)\right) \simeq \left(F ^ {w x}, T ^ {r}\right) \right].
+$$
+
+Here,  $\left(F_G^{M,(D)}(u,v),T_G^{M,(D)}(u,v)\right)$  is the depth-2D unfolding tree of  $G$  at  $(u,v)$  for model  $M$ .
+
+All the following lemmas are straightforward extensions of those in Appendix C.2.
+
+Lemma D.4. Let  $M \in \{\text{Sub}, \mathsf{L}, \mathsf{LF}, \mathsf{F}\}$  be any model. For any graph  $G^v$  and tree-decomposed graph  $(F^u, T^r) \in S_{\mathfrak{n}}^{M}$ , we have
+
+$$
+\operatorname {b l s o} \left(\left(F ^ {u}, T ^ {r}\right), G ^ {v}\right) = \sum_ {\left(\tilde {F} ^ {w}, \tilde {T} ^ {s}\right) \in \mathcal {S} _ {\mathfrak {n}} ^ {M}} \operatorname {b l s o H o m} \left(\left(F ^ {u}, T ^ {r}\right), \left(\tilde {F} ^ {w}, \tilde {T} ^ {s}\right)\right) \cdot \operatorname {c n t} ^ {M} \left(\left(\tilde {F} ^ {w}, \tilde {T} ^ {s}\right), G ^ {v}\right).
+$$
+
+The edge-level result is similar.
+
+Lemma D.5. Let  $M \in \{\mathrm{Sub},\mathrm{L},\mathrm{LF},\mathrm{F}\}$  be any model. For any tree-decomposed graphs  $(F^u,T^r),(\tilde{F}^v,\tilde{T}^s) \in S_{\mathfrak{n}}^{M}$ ,
+
+$$
+\mathsf{b l s o H o m}((F^{u},T^{r}),(\tilde{F}^{v},\tilde{T}^{s})) = \sum_{(\widehat{F}^{w},\widehat{T}^{t})\in \mathcal{S}_{n}^{M}}\frac{\mathsf{b l s o S u r j}\left((F^{u},T^{r}),( \widehat{F}^{w},\widehat{T}^{t})\right)\cdot\mathsf{b l s o I n j}\left((\widehat{F}^{w},\widehat{T}^{t}),( \tilde{F}^{v},\tilde{T}^{s})\right)}{\mathsf{a u t}(\widehat{F}^{w},\widehat{T}^{t})}.
+$$
+
+The edge-level result is similar.
+
+Lemma D.6. Let  $M \in \{\text{Sub}, \mathsf{L}, \mathsf{LF}, \mathsf{F}\}$  be any model. For any graph  $G^v$  and tree-decomposed graph  $(F^u, T^r) \in S_{\mathfrak{n}}^{M}$ ,
+
+$$
+\operatorname {h o m} (F ^ {u}, G ^ {v}) = \sum_ {(\tilde {F} ^ {w}, \tilde {T} ^ {s}) \in \mathcal {S} _ {n} ^ {M}} \frac {\mathrm {b E x t} ((F ^ {u} , T ^ {r}) , (\tilde {F} ^ {w} , \tilde {T} ^ {s})) \cdot \mathrm {b S t r H o m} ((\tilde {F} ^ {w} , \tilde {T} ^ {s}) , G ^ {v})}{\operatorname {a u t} (\tilde {F} ^ {w} , \tilde {T} ^ {s})}.
+$$
+
+The edge-level result is similar.
+
+Lemma D.7. Let  $M \in \{\text{Sub}, \mathsf{L}, \mathsf{LF}, \mathsf{F}\}$  be any model. For any graph  $G^v$  and tree-decomposed graph  $(F^u, T^r) \in S_{\mathfrak{n}}^{M}$ ,
+
+$$
+\mathsf{bStrHom}((F^{u},T^{r}),G^{v}) = \sum_{(\tilde{F}^{w},\tilde{T}^{s})\in \mathcal{S}_{n}^{M}}\frac{\mathsf{bStrSurj}\left((F^{u},T^{r}),( \tilde{F}^{w},\tilde{T}^{s})\right)\cdot\mathsf{blso}\left((\tilde{F}^{w},\tilde{T}^{s}),G^{v}\right)}{\mathsf{aut}(\tilde{F}^{w},\tilde{T}^{s})},
+$$
+
+The edge-level result is similar.
+
+Corollary D.8. Let  $M \in \{\text{Sub}, \mathsf{L}, \mathsf{LF}, \mathsf{F}\}$  be any model. For any two graphs  $G^u, H^v, \chi_G^M(u) = \chi_H^M(v)$  iff  $\operatorname{hom}(F^w, G^u) = \operatorname{hom}(F^w, H^v)$  for all  $(F^w, T^r) \in S_{\mathfrak{n}}^M$ . For any two graphs  $G^{uv}, H^{wx}, \chi_G^M(u, v) = \chi_H^M(w, x)$  iff  $\operatorname{hom}(F^{yz}, G^{uv}) = \operatorname{hom}(F^{yz}, H^{wx})$  for all  $(F^{yz}, T^r) \in S_{\mathfrak{e}}^M$ .
+
+Lemma D.9. Let  $M \in \{\mathrm{Sub}, \mathrm{L}, \mathrm{LF}, \mathrm{F}\}$  be any model.
+
+- For any rooted connected graph  $F^u$ ,  $F^u \in \mathcal{F}_{\mathfrak{n}}^{M}$  iff there is a tree decomposition  $T^{r}$  of  $F^u$  such that  $(F^u, T^r) \in S_{\mathfrak{n}}^{M}$ .  
+- For any rooted connected graph  $F^{uv}$ ,  $F^{uv} \in \mathcal{F}_{\mathrm{e}}^{M}$  iff there is a tree decomposition  $T^{r}$  of  $F^{uv}$  such that  $(F^{uv}, T^{r}) \in \mathcal{S}_{\mathrm{e}}^{M}$ .
+
+By combining the proof of previous lemmas and theorems, we can prove that both  $\mathcal{F}_{\mathrm{n}}^{M}$  and  $\mathcal{F}_{\mathrm{e}}^{M}$  satisfy Definition 3.6(a) for all  $M \in \{\mathrm{Sub}, \mathrm{L}, \mathrm{LF}, \mathrm{F}\}$ .
+
+# D.2 COUNTEREXAMPLES
+
+Let  $F^{\mathbf{w}}$  be a rooted graph that marks the special vertices  $w_{1},\dots ,w_{m}$ . Assume that  $F^{\mathbf{w}} \notin \mathcal{F}_{\mathfrak{n}}^{M}$  ( $m = 1$ ) or  $F^{\mathbf{w}} \notin \mathcal{F}_{\mathfrak{e}}^{M}$  ( $m = 2$ ). In this subsection, we will construct a pair of graphs  $G^{\mathbf{u}}$  and  $H^{\mathbf{v}}$  such that  $\chi_G^M (\pmb {u}) = \chi_H^M (\pmb {v})$  and  $\mathrm{hom}(F^{\mathbf{w}},G^{\mathbf{u}})\neq \mathrm{hom}(F^{\mathbf{w}},H^{\mathbf{v}})$ . However, it turns out that, if we naively follow the proof in Appendix C.4 by constructing the Fürer graph and twisted Fürer graph with respect to  $F$  without considering the marked vertices  $\pmb{w}$ , then the graphs may no longer be counterexamples here. For example, for the edge-level expressivity,  $F^{w_1,w_2}\notin \mathcal{F}_{\mathfrak{e}}^{M}$  does not imply that  $F\notin \mathcal{F}^{M}$  where  $M$  can be any model studied in this paper.
+
+To address the problem, we instead introduce a novel construction of counterexample graphs defined as follows:
+
+Definition D.10 (Clique-augmented Fürer graphs). Let  $F$  be any connected graph and  $w_{1}, \dots, w_{m} \in V_{F}$  be a sequence of vertices. Given an integer  $k \geq m$ , the  $k$ -clique-augmented Fürer graph with respect to  $F^{\mathbf{w}}$ , denoted by  $G_{k}(F^{\mathbf{w}})$ , is the Fürer graph of  $\tilde{F}$  where  $\tilde{F}$  is the union of graph  $F$  and a  $k$ -clique that contains  $w_{1}, \dots, w_{m}$  (and does not contain other vertices in  $F$ ). The twisted Fürer graph of  $G_{k}(F^{\mathbf{w}})$  is denoted as  $H_{k}(F^{\mathbf{w}})$ .
+
+We have the following main result:
+
+Theorem D.11. Let  $M \in \{\mathrm{Sub}, \mathrm{L}, \mathrm{LF}, \mathrm{F}\}$  be any model defined in Section 2.
+
+- For any rooted graph  $F^w$  marking vertex  $w$ , if  $F^w \notin \mathcal{F}_{\mathfrak{n}}^{M}$ , then there is a vertex  $(w, U) \in \operatorname{Meta}_{\tilde{F}}(w)$  such that  $U \subset V_F$  and  $\chi_{G_4(F^w)}^M((w, \emptyset)) = \chi_{H_4(F^w)}^M((w, U))$ ;  
+- For any rooted graph  $F^w$  marking two vertices  $w_1, w_2$ , if  $F^w \notin \mathcal{F}_{\mathrm{e}}^{M}$ , then there are two vertices  $(w_1, U_1) \in \operatorname{Meta}_{\tilde{F}}(w_1)$  and  $(w_2, U_2) \in \operatorname{Meta}_{\tilde{F}}(w_2)$  such that  $U_1, U_2 \subset V_F$  and  $\chi_{G_4(F^w)}^{M}((w_1, \emptyset), (w_2, \emptyset)) = \chi_{H_4(F^w)}^{M}((w_1, U_1), (w_2, U_2))$ .
+
+Proof. Below, we only give a proof of the second item. Let  $\tilde{F}$  be the union of  $F$  and the 4-clique according to Definition D.10. Similar to the graph-level expressivity, we will extend the pebble game defined in Appendix C.4 to edge-level. For the edge-level pebble game, the two pebbles  $\mathfrak{p}_u,\mathfrak{p}_v$  are initially placed on two predefined vertices  $w_{1},w_{2}$ , and Bob initially selects a predefined set of connected components  $Q\subset \mathbb{C}\mathbb{C}_{\tilde{F}}(\{w_1,w_2\})$  (of odd size). Then, the game executes the same process as in the text above Proposition C.49. The winning criterion for the two players is also the same. We denote the above edge-level pebble game as  $\mathsf{PG}_{\tilde{F}}(w_1,w_2,Q)$ .
+
+Based on the isomorphism property of Fürer graphs established in Zhang et al. (2023a, Theorem I.9 and Theorem I.17), it is straightforward to see that, for any vertices  $\xi, \xi' \in \mathsf{Meta}_{\tilde{F}}(w_1)$  and
+
+$\eta, \eta' \in \mathsf{Meta}_{\tilde{F}}(w_2), \chi_{G_4(F^w)}^M(\xi, \eta) = \chi_{H_4(F^w)}^M(\xi', \eta')$  iff Alice cannot win the edge-level pebble game  $\mathsf{PG}_{\tilde{F}}(w_1, w_2, Q)$  for some  $Q \subset \mathbb{C}\mathbb{C}_{\tilde{F}}(\{w_1, w_2\})$  (depending on  $\xi, \xi', \eta, \eta'$ ). Moreover, let  $\xi' = (w_1, \emptyset)$ ,  $\eta' = (w_2, \emptyset)$  and let  $\xi_2$  and  $\eta_2$  range over all even-size subsets of  $N_F(w_1)$  and  $N_F(w_2)$ , respectively, that is to say,  $\xi_2$  and  $\eta_2$  does not contain vertices that are only in the 4-clique. Then, the equivalent pebble game ranges over all  $Q$  of odd size such that every connected component in  $Q$  contains only vertices in  $F$ , i.e.,  $Q \subset \mathbb{C}\mathbb{C}_F(\{w_1, w_2\})$ . Therefore, if Alice cannot win the edge-level pebble game  $\mathsf{PG}_{\tilde{F}}(w_1, w_2, Q)$  for some  $Q \subset \mathbb{C}\mathbb{C}_F(\{w_1, w_2\})$ , then there are two vertices  $\xi \in \mathsf{Meta}_{\tilde{F}}(w_1)$  and  $\eta \in \mathsf{Meta}_{\tilde{F}}(w_2)$  such that  $\xi_2, \eta_2 \subset V_F$  and  $\chi_{G_4(F^w)}^M((w_1, \emptyset), (w_2, \emptyset)) = \chi_{H_4(F^w)}^M(\xi, \eta)$ .
+
+It thus remains to prove that, if Alice can win the edge-level pebble game  $\mathsf{PG}_{\tilde{F}}(w_1,w_2,Q)$  for all  $Q\subset \mathbb{C}C_F(\{w_1,w_2\})$ , then  $F^{\boldsymbol{w}}\in \mathcal{F}_{\mathrm{e}}^{M}$ . To prove this result, we similarly define the game state graph as in Appendix C.4, with the only difference that the initial state is now  $(w_{1},w_{2},Q)$ . The proof is divided into the following steps:
+
+1. Assume that  $Q$  contains a single connected component, i.e.,  $Q = \{P\}$ . We will prove that, if Alice can win the game  $\mathsf{PG}_{\tilde{F}}(w_1, w_2, \{P\})$ , then there is a game strategy for Alice such that for any reachable and non-terminal state  $(x, y, \{P'\})$ ,  $P' \subset P$ . If it is not the case, pick any state  $(x, y, \{P'\})$  such that  $P' \notin P$  and any path from  $(w_1, w_2, \{P\})$  to  $(x, y, \{P'\})$  does not pass any intermediate state that is not contracted. It follows that either  $w_1$  or  $w_2$  does not hold any pebble, and all edges connected to this vertex is in  $P'$ . Therefore,  $P'$  contains all vertices in the 4-clique. In this case, Alice cannot win as Bob can always keep the connected component containing all vertices in the 4-clique in subsequent rounds, i.e., Bob always selects a component containing a vertex in the 4-clique that does not hold a pebble.  
+2. Based on the above result, in the game  $\mathsf{PG}_{\tilde{F}}(w_1,w_2,\{P\})$ , the connected components selected by Bob are always in  $F$ , and thus the game process is the same as  $\mathsf{PG}_F(w_1,w_2,\{P\})$ . We can then prove the same result as Lemma C.52. Concretely, if Alice can win the edge-level pebble game  $\mathsf{PG}_{\tilde{F}}(w_1,w_2,\{P\})$ , then there exists a game state graph  $G^{\mathbb{S}}$  corresponding to a winning strategy, such that for any transition  $((x,y,\{P'\}), (x',y',\{P''\}))$  where  $(x,y,\{P'\})$  is a reachable and non-terminal state, we have  $P'' \subsetneq P'$ .  
+3. Next, we can follow the same procedure in the proof of Theorem C.53 to construct a tree decomposition  $T^r$  for the subgraph  $F[P]$  containing all edges in  $P$ , such that  $(F[P], T^r) \in S_{\mathbf{e}}^{M}$  and  $\beta_T(r) = \{w_1, w_2\}$ .  
+4. Finally, let  $P$  ranges over all connected components in  $\mathbb{C}C_F(\{w_1, w_2\})$ , for each  $P$  we have a tree-decomposed graph. We can glue these tree-decomposed graphs by merging the root to obtain the tree decomposition of  $F$ , because the root bags of all  $T^r$  are the same and any vertex  $x \notin \{w_1, w_2\}$  appears in the bag of only one  $T^r$ . Moreover, the glued tree  $\tilde{T}^s$  clearly satisfies that  $(F, \tilde{T}^s) \in S_{\mathrm{e}}^{M}$ .
+
+Combining these items shows that  $F^{\boldsymbol{w}} \in \mathcal{F}_{\mathrm{e}}^{M}$  and concludes the proof.
+
+Corollary D.12. Let  $F^w$  any rooted graph marking  $m$  vertices such that  $F^w \notin \mathcal{F}_n^M$  ( $m = 1$ ) or  $F^w \notin \mathcal{F}_{\mathrm{e}}^{M}$  ( $m = 2$ ), and let  $G_{k}(F^{w})$  and  $H_{k}(F^{w})$  be clique-augmented Fuirer graphs defined above. Denote by  $\xi, \eta \in V_{G_k(F^w)}^m$  two vertex tuples of length  $m$  where  $\xi_{i} = (w_{i}, \emptyset)$  and  $\eta_{i} \in \operatorname{Meta}_{\tilde{F}}(w_{i})$  with  $\eta_{i,2} \subset V_{F}$  for  $i \in [m]$ . Then,  $\operatorname{hom}(F^w, [G_k(F^w)]^\xi) \neq \operatorname{hom}(F^w, [H_k(F^w)]^\eta)$ .
+
+Proof. The proof exactly parallels that of Theorem C.47 by separately considering three cases. We omit the proof for clarity.
+
+Corollary D.13. Let  $M \in \{\mathrm{Sub}, \mathrm{L}, \mathrm{LF}, \mathrm{F}\}$  be any model.
+
+- For any graph  $F^w \notin \mathcal{F}_{\mathfrak{n}}^{M}$ , let  $G_4(F^w)$  and  $H_4(F^w)$  be the clique-augmented Fürer graph and the corresponding twisted Fürer graph. Then, there exists two vertices  $\xi, \xi'$  such that  $\operatorname{hom}(F^w, [G_4(F^w)]^\xi) \neq \operatorname{hom}(F^w, [H_4(F^w)]^{\xi'})$  and  $\chi_{G_4(F^w)}^M(\xi) = \chi_{H_4(F^w)}^M(\xi')$ .  
+- For any graph  $F^{wx} \notin \mathcal{F}_{\mathrm{e}}^{M}$ , let  $G_{4}(F^{wx})$  and  $H_{4}(F^{wx})$  be the clique-augmented Fürer graph and the corresponding twisted Fürer graph defined in Definition D.10. Then, there exists
+
+four vertices  $\xi, \eta, \xi', \eta'$  such that  $\operatorname{hom}(F^{wx}, [G_4(F^{wx})]^{\xi \eta}) \neq \operatorname{hom}(F^{wx}, [H_4(F^{wx})]^{\xi' \eta'})$  and  $\chi_{G_4(F^{wx})}^M(\xi, \eta) = \chi_{H_4(F^{wx})}^M(\xi', \eta')$ .
+
+Proof. The proof directly follows from Corollary D.12 and Theorem D.11.  $\square$
+
+![](images/9755f732146412f3b844f348a388921eb520c4d51b5e6eb103ddb710f7c75feb.jpg)
+
+# E HIGHER-ORDER GNNS
+
+# E.1 DEFINITION OF HIGHER-ORDER GNNS
+
+In this subsection, we give formal definitions of the CR algorithms for various higher-order GNNs, each of which generalizes a model in Section 2.
+
+- Subgraph  $k$ -GNN. In a Subgraph  $k$ -GNN, a graph  $G$  is treated as a set of subgraphs  $\{G^u : u \in V_G^k\}$ , where each subgraph  $G^u$  is obtained from  $G$  by marking  $k$  special vertices  $u \in V_G^k$ , and thus there are  $n^k$  subgraphs when  $G$  has  $n$  vertices. Subgraph GNN maintains a color  $\chi_G^{\mathrm{Sub}(k)}(\pmb{u}, v)$  for each vertex  $v$  in graph  $G^u$ . Initially,  $\chi_G^{\mathrm{Sub}(k), (0)}(\pmb{u}, v) = (\ell_G(v), \mathsf{atp}_G(\pmb{u}), \mathbb{I}[u_1 = v], \dots, \mathbb{I}[u_k = v])$ . It then runs MPNNs independently on each graph  $G^u$ :
+
+$$
+\chi_ {G} ^ {\operatorname {S u b} (k), (t + 1)} (\boldsymbol {u}, v) = \operatorname {h a s h} \left(\chi_ {G} ^ {\operatorname {S u b} (k), (t)} (\boldsymbol {u}, v), \{\left\{\chi_ {G} ^ {\operatorname {S u b} (k), (t)} (\boldsymbol {u}, w): w \in N _ {G} (v) \right\} \}\right). \tag {26}
+$$
+
+Denote the stable color as  $\chi_G^{\mathrm{Sub}(k)}(\pmb{u},\upsilon)$ . Define  $\chi_G^{\mathrm{Sub}(k)}(\pmb{u})\coloneqq \{\{\chi_G^{\mathrm{Sub}(k)}(\pmb{u},\upsilon):v\in V_G\}\}$ . Then, the graph representation is defined as  $\chi_G^{\mathrm{Sub}(k)}(G) = \{\{\chi_G^{\mathrm{Sub}(k)}(\pmb {u}):\pmb {u}\in V_G\} \}$ . We remark that Subgraph  $k$  -GNN is precisely the  $k$  -VSAN proposed in Qian et al. (2022).
+
+- Local  $k$ -GNN. Local  $k$ -GNN maintains a color  $\chi_G^{\mathsf{L}(k)}(\pmb{u})$  for each vertex  $k$ -tuple  $\pmb{u} \in V_G^k$ . Initially,  $\chi_G^{\mathsf{L}(k),(0)}(\pmb{u}) = (\ell_G(u_1), \dots, \ell_G(u_k), \mathsf{atp}_G(\pmb{u}))$ , called the isomorphism type of vertex  $k$ -tuple  $\pmb{u}$ , where  $\mathsf{atp}_G(\pmb{u})$  is the atomic type of  $\pmb{u}$ . Then, in each iteration  $t + 1$ ,
+
+$$
+\begin{array}{l} \chi_ {G} ^ {\mathsf {L} (k), (t + 1)} (\boldsymbol {u}) = \operatorname {h a s h} \left(\chi_ {G} ^ {\mathsf {L} (k), (t)} (\boldsymbol {u}), \{\{\chi_ {G} ^ {\mathsf {L} (k), (t)} (\boldsymbol {v}): \boldsymbol {v} \in N _ {G} ^ {(1)} (\boldsymbol {u}) \} \}, \dots , \right. \\ \left. \left\{\left\{\chi_ {G} ^ {\mathrm {L} (k), (t)} (\boldsymbol {v}): \boldsymbol {v} \in N _ {G} ^ {(k)} (\boldsymbol {u}) \right\} \right\}\right), \tag {27} \\ \end{array}
+$$
+
+where  $N_G^{(j)}(\pmb{u}) = \{(u_1, \dots, u_{j-1}, w, u_{j+1}, \dots, u_k) : w \in N_G(u_j)\}$ . Denote the stable color as  $\chi_G^{\mathsf{L}(k)}(\pmb{u})$ . The graph representation is defined as  $\chi_G^{\mathsf{L}(k)}(G) := \{\chi_G^{\mathsf{L}(k)}(\pmb{u}) : \pmb{u} \in V_G\}$ . We remark that Local  $k$ -GNN is precisely the  $\delta$ - $k$ -LWL proposed in Morris et al. (2020).
+
+- Local  $k$ -FGNN. Local  $k$ -FGNN is almost the same as Local  $k$ -GNN except that the update formula is replaced by the following one:
+
+$$
+\begin{array}{l} \chi_ {G} ^ {\mathsf {L F} (k), (t + 1)} (\boldsymbol {u}) = \operatorname {h a s h} \left(\chi_ {G} ^ {\mathsf {L F} (k), (t)} (\boldsymbol {u}), \{\{\left(\chi_ {G} ^ {\mathsf {L F} (k), (t)} (w, u _ {2}, \dots , u _ {k}), \chi_ {G} ^ {\mathsf {L F} (k), (t)} (u _ {1}, w, u _ {3}, \dots , u _ {k}), \right. \right. \\ \left. \dots , \chi_ {G} ^ {\mathsf {L F} (k), (t)} \left(u _ {1}, \dots , u _ {k - 1}, w\right): w \in N _ {G} \left(u _ {1}\right) \cup \dots \cup N _ {G} \left(u _ {k}\right) \right\} \Bigg). \tag {28} \\ \end{array}
+$$
+
+We remark that Local  $k$ -FGNN is precisely the  $\mathrm{SLFWL}(k)$  proposed in Zhang et al. (2023a).
+
+-  $k$ -FGNN. It is just the standard  $k$ -FWL (Cai et al., 1992). Compared with Local  $k$ -FGNN, the update formula is now global:
+
+$$
+\begin{array}{l} \chi_ {G} ^ {\mathsf {F} (k), (t + 1)} (\boldsymbol {u}) = \operatorname {h a s h} \left(\chi_ {G} ^ {\mathsf {F} (k), (t)} (\boldsymbol {u}), \{\{\left(\chi_ {G} ^ {\mathsf {F} (k), (t)} (w, u _ {2}, \dots , u _ {k}), \chi_ {G} ^ {\mathsf {F} (k), (t)} (u _ {1}, w, u _ {3}, \dots , u _ {k}), \right. \right. \\ \left. \dots , \chi_ {G} ^ {\mathsf {F} (k), (t)} \left(u _ {1}, \dots , u _ {k - 1}, w\right): w \in V _ {G} \right\}. \Bigg). \tag {29} \\ \end{array}
+$$
+
+We note that the computational complexity of Subgraph  $(k - 1)$ -GNN, Local  $k$ -GNN, and Local  $k$ -FGNN is the same, i.e.,  $\Theta(n^{k - 1}m)$  for a graph of  $n$  vertices and  $m$  edges. Moreover, it is easy to see that Subgraph 0-GNN, Local 1-GNN, and Local 1-FGNN all reduce to MPNN. For  $k$ -FGNN, the computational complexity is  $\Theta(n^{k + 1})$ , which is strictly higher than Subgraph  $(k - 1)$ -GNN, Local  $k$ -GNN, and Local  $k$ -FGNN.
+
+# E.2 HIGHER-ORDER STRONG NED
+
+In this subsection, we generalize the strong NED into higher-order versions. We first define the concept of higher-order ear:
+
+Definition E.1 ( $k$ -order ear). Given integer  $k \geq 1$ , a  $k$ -order ear is a graph  $G$  formed by the union of  $k$  paths  $P_{1}, \dots, P_{k}$  (possibly with zero length) plus an edge set  $Q$  satisfying the following conditions:
+
+- For each path  $P_{i}$ , denote its two endpoints as  $u_{i}, v_{i}$ , called the outer endpoint and inner endpoint, respectively. Then, all edges in  $Q$  are linked between inner endpoints, i.e.,  $Q \subset \{\{v_{i}, v_{j}\} : 1 \leq i, j \leq k, v_{i} \neq v_{j}\}$ .  
+- Any different paths  $P_{i}, P_{j}$  do not intersect except at the inner endpoint (when  $v_{i} = v_{j}$ ).  
+-  $G$  is a connected graph.
+
+The endpoints of the  $k$ -order ear is defined to be all outer endpoints  $u_{1}, \dots, u_{k}$ .
+
+It is easy to see that a 2-order ear is precisely a simple path, since linking two different paths (possibly with an additional edge) still yields a path. Below, we denote by inner  $(G)$  and outer  $(G)$  the set of inner endpoints and outer endpoints in ear  $G$ , respectively. We also denote by path  $(G)$  the set of paths in ear  $G$ . It follows that  $|\text{inner}(G)| = |\text{outer}(G)| = |\text{path}(G)| = k$ . Finally, given a path  $P$  and two vertices  $w_{1}, w_{2}$  in  $P$ , denote by subpath  $_{P}(w_{1}, w_{2})$  the subpath in  $P$  such that the two endpoints are  $w_{1}, w_{2}$ .
+
+Definition E.2 (Nested interval). Let  $G$  and  $H$  be two  $k$ -order ears with inner  $(G) = \{v_{1},\dots ,v_{k}\}$ , outer  $(G) = \{u_{1},\dots ,u_{k}\}$ , and outer  $(H) = \{w_{1},\dots ,w_{k}\}$ , where each  $\{u_i,v_i\}$  corresponds to the endpoints of a path  $P_{i}\in \mathrm{path}(G)$ . We say  $H$  is nested on  $G$  if one or more endpoint  $w_{i}$  of  $H$  ( $i\in [k]$ ) is in path  $P_{i}$ , and all other vertices in  $H$  are not in  $G$ . The nested interval is defined to be the union of subpaths subpath  $P_{i}(w_{i},v_{i})$  for all  $i\in [k]$  satisfying that  $w_{i}$  is in  $P_{i}$ .
+
+We give an illustration of the nested interval of two 3-order ears in Figure 2. Equipped with the above definition, we are ready to introduce the higher-order strong NED:
+
+Definition E.3 ( $k$ -order strong NED). Given a graph  $G$ , a  $k$ -order strong NED  $\mathcal{P}$  is a partition of the edge set  $E_G$  into a sequence of edge sets  $Q_1, \dots, Q_m$ , which satisfies the following conditions:
+
+Each  $Q_{i}$  is a  $k$  -order ear.  
+- Any two ears  $Q_{i}$  and  $Q_{j}$  with indices  $1 \leq i < j \leq c$  do not intersect, where  $c$  is the number of connected components of  $G$ .  
+- For each  $Q_{j}$  with index  $j > c$ , it is nested on some  $k$ -order ear  $Q_{i}$  with index  $1 \leq i < j$ . Moreover, except for the endpoints of  $Q_{j}$  on  $Q_{i}$ , no other vertices in  $Q_{j}$  are in any previous ear  $Q_{k}$  for  $1 \leq k < i$ .  
+- Denote by  $I(Q_{j}) \subset Q_{i}$  the nested interval of  $Q_{j}$  in  $Q_{i}$ . For all  $Q_{j}, Q_{k}$  with  $c < j < k \leq m$ , if  $Q_{j}$  and  $Q_{k}$  are nested on the same ear, then  $I(Q_{j}) \subset I(Q_{k})$ .
+
+# E.3 PROOFS IN SECTION 3.4
+
+We first generalize the tree-decomposed graphs in Definition C.6 to higher-order versions:
+
+Definition E.4. Define four families of tree-decomposed graphs  $\mathcal{S}^{\mathrm{Sub}(k)}, \mathcal{S}^{\mathrm{L}(k)}, \mathcal{S}^{\mathrm{LF}(k)}$ , and  $\mathcal{S}^{\mathrm{F}(k)}$  as follows:
+
+a)  $(F,T^{r})\in S^{\mathsf{F}(k)}$  iff  $(F,T^{r})$  satisfies Definition C.4 with width  $k$  
+b)  $(F,T^{r})\in S^{\mathsf{LF}(k)}$  iff  $(F,T^{r})$  satisfies Definition C.4 with width  $k$ , and for any tree node  $t$  of odd depth, it has only one child if  $w\notin \{v:v\in N_G(u),u\in \beta_T(s)\}$  where  $s$  is the parent node of  $t$  and  $w$  is the unique vertex in  $\beta_T(t)\backslash \beta_T(s)$ ;  
+c)  $(F,T^{r})\in S^{\mathbb{L}(k)}$  iff  $(F,T^{r})$  satisfies Definition C.4 with width  $k$ , and any tree node  $t$  of odd depth has only one child;  
+d)  $(F,T^{r})\in S^{\mathrm{Sub}(k)}$  iff  $(F,T^{r})$  satisfies Definition C.4 with width  $k$ , and there exists a multiset  $U\subset V_{G}$  of size  $|U| = k$  such that  $U\subset \beta_T(t)$  for all  $t\in V_T$ .
+
+Then, we can analogously prove the following theorems. The proofs are almost the same as in Appendix C, so we omit them for clarity.
+
+Theorem E.5. Let  $M \in \{\mathsf{Sub}(k), \mathsf{L}(k), \mathsf{LF}(k), \mathsf{F}(k)\}$ . Then, any graphs  $G$  and  $H$  have the same representation under model  $M$  (i.e.,  $\chi_G^M(G) = \chi_H^M(H)$ ) iff  $\mathsf{hom}(F, G) = \mathsf{hom}(F, H)$  for all  $(F, T^r) \in S^M$ .
+
+Theorem E.6. Let  $M \in \{\mathrm{Sub}(k), \mathsf{L}(k), \mathsf{LF}(k), \mathsf{F}(k)\}$  be any model, and let  $F$  be any graph such that no tree decomposition  $(F, T^r) \in S^M$ . Let  $G(F)$  and  $H(F)$  be the Fürer graph and twisted Fürer graph with respect to  $F$ . Then,  $\mathrm{hom}(F, G(F)) \neq \mathrm{hom}(F, H(F))$  and  $\chi_{G(F)}^{M}(G(F)) = \chi_{H(F)}^{M}(H(F))$ .
+
+Theorem E.7. For any graph  $F$ , there is a tree decomposition  $T^r$  of  $F$  such that  $(F, T^r) \in S^{\mathrm{Sub}(k)}$  iff there exists  $U \subset V_F$  such that  $|U| \leq k$  and  $F \backslash U$  is a forest.
+
+Theorem E.8. For any graph  $F$ , there is a tree decomposition  $T^r$  of  $F$  such that  $(F, T^r) \in S^{\mathsf{L}(k)}$  iff  $F$  has a  $k$ -order strong NED.
+
+Theorem E.9. For any graph  $F$ , there is a tree decomposition  $T^r$  of  $F$  such that  $(F, T^r) \in S^{\mathsf{F}(k)}$  iff  $\operatorname{tw}(F) \leq k$ .
+
+# E.4 EXPRESSIVITY GAP BETWEEN HIGHER-ORDER GNNS
+
+In this subsection, we show how homomorphism expressivity can be used to build a complete expressiveness hierarchy for higher-order GNNs as shown in Corollary 4.2.
+
+Gap between Subgraph  $k$ -GNN and Local  $(k + 1)$ -GNN  $(k \geq 1)$ . The counterexample graph is a  $(k + 1) \times (2k + 2)$  grid consisting  $(k + 1) \times (2k + 2)$  vertices. First, it is easy to see that the graph is not in  $\mathcal{F}^{\mathrm{Sub}(k)}$ , i.e., deleting  $k$  vertices of the graph cannot yield a forest. To see this, note that the graph consists of  $k \times (2k + 1)$  "squares", and each vertex is related to at most four squares. Therefore, deleting  $k$  vertices cannot eliminate all squares when  $k \geq 2$  (because  $4k < k \times (2k + 1)$ ). For the case of  $k = 1$ , we clearly have that deleting one vertex cannot eliminate all squares.
+
+We next show that the  $(k + 1)\times (2k + 2)$  grid is in  $\mathcal{F}^{\mathsf{L}(k + 1)}$ . This is also simple as shown in Figure 8, where each color represents a  $k$ -order ear. It can be seen that the  $(k + 1)\times (2k + 2)$  grid has a  $(k + 1)$ -order strong NED.
+
+![](images/830e1ed6f48f52caf086dee6661a9f5214573c1207c8b1c416a1eb46510085f3.jpg)  
+Figure 8: An 4-order strong NED of the  $4 \times 8$  grid graph.
+
+Relation between Subgraph  $k$ -GNN and  $k$ -FGNN ( $k \geq 2$ ). To show that  $\mathcal{F}^{\mathrm{Sub}(k)} \not\subsetneq \mathcal{F}^{\mathrm{F}(k)}$ , consider the  $(k + 2)$ -clique. Clearly, deleting  $k$  vertices from the  $(k + 2)$ -clique yields a graph consisting of two vertices linked by an edge, which is a tree. On the other hand, the
+
+treewidth of a  $(k + 2)$ -clique is  $k + 1$  (Fact C.3). So the  $(k + 2)$ -clique is in  $\mathcal{F}^{\mathrm{Sub}(k)}$  but not in  $\mathcal{F}^{\mathsf{F}(k)}$ .
+
+To show that  $\mathcal{F}^{\mathsf{F}(k)} \not\subset \mathcal{F}^{\mathrm{Sub}(k)}$ , we can again use the grid graph, but this time consider the  $k \times (2k + 2)$  grid. On the one hand, a standard result in graph theory shows that the treewidth of a  $a \times b$  grid graph is  $\min(a, b)$ . On the other hand, following the similar analysis above, we can prove that deleting  $k$  vertices from the  $k \times (2k + 2)$  grid cannot eliminate all squares.
+
+Gap between Local  $k$ -GNN and  $k$ -FGNN ( $k \geq 2$ ). The counterexample graph  $F$  is the union of the following graphs  $K_0 \cup K_1 \cup \dots \cup K_k$ , where  $K_0$  is a  $(k+1)$ -clique with vertex set  $\{u_1, \dots, u_{k+1}\}$  and for  $i \in [k+1]$ ,  $K_i$  is a  $(k+1)$ -clique with vertex set  $\{u_1, \dots, u_{i-1}, v_i, u_{i+1}, \dots, u_{k+1}\}$  where  $v_i$  is a new vertex (not in  $K_0$ ) and  $v_i \neq v_j$  for  $i \neq j$ . Namely, each  $K_i$  has  $k$  common vertices with  $K_0$ . It is easy to construct a tree decomposition  $T^r$  of  $F$  such that  $(F, T^r) \in S^{\mathsf{F}(k)}$ . On the other hand, it is easy to see that the graph does not have a  $k$ -order strong NED (equivalently, one can easily check that  $F$  does not admit a tree decomposition  $T^r$  satisfying  $(F, T^r) \in S^{\mathsf{L}(k)}$ ).
+
+Gap between  $k$ -FGNN and Local  $(k + 1)$ -GNN  $(k \geq 2)$ . The counterexample graph is again the  $(k + 1) \times (2k + 2)$  grid. We have proved that the graph is in  $\mathcal{F}^{\mathsf{L}(k + 1)}$  but not in  $\mathcal{F}^{\mathsf{L}(k)}$ .
+
+Regarding Local  $k$ -IGN and Frasca et al. (2022). Finally, we remark that based on the above results, we essentially proved an open question raised in Frasca et al. (2022) regarding the expressive power of Local  $k$ -IGN (i.e., the  $\mathsf{RelGN}(k)$  proposed in their paper). Following Zhang et al. (2023a),
+
+![](images/7a6402c5f447a7ef59e95c70fd19c4236ad3df702d0cbce15941509b93644af3.jpg)  
+(a)
+
+![](images/85b2010bdf07964a946f085449200e5651ba79ad301064f30d516e29ed43346b.jpg)  
+(b)  
+Figure 9: More illustration of NED. All NEDs in these example graphs are strong but not endpoint-shared. See Appendix F.2 for additional discussions.
+
+![](images/99fd93c219affb4cee6c3df9c6ccbd5677d623576126da9d89f8fdb09e6c3fd4.jpg)  
+(c)
+
+it is straightforward to see that Local  $k$ -IGN is as expressive as Local  $k$ -GNN. Therefore, Local  $k$ -IGN is strictly more expressive than  $(k - 1)$ -FGNN and strictly less expressive than  $k$ -FGNN.
+
+# F ADDITIONAL DISCUSSIONS
+
+# F.1 REGARDING THE DEFINITION OF HOMOMORPHISM EXPRESSIVITY
+
+In this paper, we have shown that homomorphism expressivity exists for a variety of popular GNNs. Unfortunately, due to the "iff" statement in Definition 3.1, homomorphism expressivity may not always be well-defined in the general case. We remark in this subsection that, there do exist pathological, intentionally designed GNNs such that the homomorphism expressivity is not well-defined.
+
+Consider a simple GNN  $M$  that outputs the representation of a graph  $G$  as follows. If  $G$  is a cycle of odd length, it outputs  $(1,L)$  where  $L$  is the length of the cycle. Otherwise, it outputs  $(0,\chi_{G}^{\mathrm{MP}}(G))$ , namely, running a MPNN on the graph. It follows that  $M$  is strictly more powerful than MPNN, e.g., it can distinguish between the 9-cycle and three triangles, which MPNN fails to distinguish. As a result,  $M$  can count all trees under homomorphism. Moreover, it cannot count other patterns under homomorphism, as the counterexample graphs for MPNN (i.e., Fürer graphs) are not cycles of odd length, so they are still counterexample graphs for model  $M$ . Therefore, the homomorphism expressivity of  $M$  should be exactly the family of forests if it exists. However, the homomorphism information of forests cannot determine the representation of model  $M$  since  $M$  is strictly more powerful than MPNN. So we conclude that  $\mathcal{F}^{M}$  does not exist.
+
+Note that the above GNN construction is inherently unnatural and hardly appears in practice. Actually, when a GNN is defined via the message-passing paradigm, we suspect that its homomorphism expressivity is always well-defined. As stated in Appendix B, we conjecture that, for any GNN characterized by a color refinement algorithm that outputs stable colors, the homomorphism expressivity always exists.
+
+# F.2 REGARDING THE DEFINITION OF NED
+
+In the main text, we have illustrated several types of NED with simple example graphs. To gain a deeper understanding of Definition 3.3, in this section we will present a few more complex examples (see Figure 9). Notably, unlike graphs in Figure 1(b), for all graphs in Figure 9 their NED contains ears such that only one endpoint is in its nested ear (e.g., ears 2 and 4 in Figure 9(a)). In other words, these ears have empty nested intervals.
+
+The presence of ears with empty nested interval stems from the fact that the corresponding graph is not biconnected (Zhang et al., 2023b). Indeed, one can check that all graphs in Definition 3.3(b) do not have cut vertices, while all graphs in Figure 9 have cut vertices. Moreover, in these examples, the number of ears with empty nested interval always equals to the number of biconnected components minus one. In particular, for biconnected graphs, it can be proved that any NED does not contain an ear with empty nested interval.
+
+# G HOMOMORPHISM AND SUBGRAPH COUNTING POWER
+
+# G.1 PROOF OF THEOREM 4.5
+
+Our proof draws inspiration from a recent technique developed in Seppelt (2023). We note that the original results in Seppelt (2023) are described for unlabeled graphs, but it is easy to extend these results to labeled graphs. To begin with, we define a concept called the graph categorical product.
+
+Definition G.1. The categorical product of two graphs  $G, H$ , denoted as  $G \times H$ , is a graph where  $V_{G \times H} = V_G \times V_H$ ,  $\ell_{G \times H}(g, h) = (\ell_G(g), \ell_H(h))$  for all  $g \in V_G$ ,  $h \in V_H$ , and  $E_{G \times H} = \{ (g, h), (g', h') \} : \{ g, g' \} \in E_G, \{ h, h' \} \in E_H$ .
+
+Lemma G.2 (Seppelt (2023)). For any graphs  $F, G,$  and  $H$ ,  $\operatorname{hom}(F,G\times H) = \operatorname{hom}(F,G)\cdot$ $\operatorname{hom}(F,H)$ .
+
+Proof. We define a mapping  $\tau$  from  $\operatorname{Hom}(F, G) \times \operatorname{Hom}(F, H)$  to  $\operatorname{Hom}(F, G \times H)$  as follows. for all  $\sigma_1 \in \operatorname{Hom}(F, G)$  and  $\sigma_2 \in \operatorname{Hom}(F, H)$ , define  $\rho = \tau(\sigma_1, \sigma_2)$  where  $\rho(f) = (\sigma_1(f), \sigma_2(f))$  for all  $f \in V_F$ . It is easy to see that  $\tau$  is a bijective mapping from  $\operatorname{Hom}(F, G) \times \operatorname{Hom}(F, H)$  to  $\operatorname{Hom}(F, G \times H)$ .
+
+Definition G.3. Given graphs  $F$  and  $G$ , denote by  $\operatorname{Surj}(F, G)$  the set of all homomorphisms from  $F$  to  $G$  that are surjective on both the vertices and edges of  $G$ , and define  $\operatorname{surj}(F, G) = |\operatorname{Surj}(F, G)|$ .
+
+The following proposition is straightforward (similar to Lemma C.25):
+
+Proposition G.4.  $\operatorname{hom}(G, H) = \sum_{F} \operatorname{surj}(G, F) \cdot \operatorname{sub}(F, H)$ , where  $F$  ranges over all non-isomorphic graphs.
+
+Lemma G.5. Let  $M$  be a GNN model such that its homomorphism expressivity  $\mathcal{F}^M$  exists. Given a finite set of graphs  $\mathcal{L}$  and a function  $\alpha : \mathcal{L} \to \mathbb{R} / \{0\}$ , if
+
+$$
+\chi_ {G} ^ {M} (G) = \chi_ {H} ^ {M} (H) \Longrightarrow \sum_ {L \in \mathcal {L}} \alpha (L) \operatorname {h o m} (L, G) = \sum_ {L \in \mathcal {L}} \alpha (L) \operatorname {h o m} (L, H),
+$$
+
+holds for all graphs  $G$  and  $H$ , then  $\mathcal{L} \subset \mathcal{F}^M$ .
+
+Proof. Let  $n$  be the largest number of vertices for all graphs in  $\mathcal{L}$ , and let  $\tilde{\mathcal{L}}$  be the set of all non-isomorphic graphs with no more than  $n$  vertices. We can arrange all graphs in  $\tilde{\mathcal{L}}$  into a sequence  $L_{1}, L_{2}, \ldots, L_{N}$  satisfying the following property:  $|V_{L_{i}}| \leq |V_{L_{i+1}}|$  or  $(|V_{L_{i}}| = |V_{L_{i+1}}|$  and  $|E_{L_{i}}| \leq |E_{L_{i+1}}|)$  for all  $i = 1, 2, \ldots, N-1$ . We then define the matrices  $A^{\text{hom}}$ ,  $A^{\text{surj}}$ , and  $A^{\text{sub}}$ , where the elements in the  $i^{th}$  row and  $j^{th}$  column are  $\text{hom}(L_{i}, L_{j})$ ,  $\text{surj}(L_{i}, L_{j})$ , and  $\text{sub}(L_{i}, L_{j})$ , respectively. Proposition G.4 implies that  $A^{\text{hom}} = A^{\text{surj}} \cdot A^{\text{sub}}$ . Since  $A^{\text{surj}}$  is a lower triangular matrix with nonzero diagonal elements and  $A^{\text{sub}}$  is an upper triangular matrix with non-zero diagonal elements, the matrix  $A^{\text{hom}}$  is invertible.
+
+We next extend  $\alpha$  to a function  $\tilde{\alpha}:\tilde{\mathcal{L}}\to \mathbb{R}$  by setting  $\tilde{\alpha} (L) = \alpha (L)$  for all  $L\in \mathcal{L}$  and  $\tilde{\alpha} (\tilde{L}) = 0$  for all  $\tilde{L}\in \tilde{\mathcal{L}}\backslash \mathcal{L}$ . Additionally, if  $\chi_G^M (G) = \chi_H^M (H)$ , then  $\mathrm{hom}(F,G) = \mathrm{hom}(F,H)$  for all  $F\in \mathcal{F}^{M}$  (by definition of homomorphism expressivity). Then, given any graph  $K\in \tilde{\mathcal{L}}$ , Lemma G.2 implies that  $\mathrm{hom}(F,G\times K) = \mathrm{hom}(F,H\times K)$  for all  $F\in \mathcal{F}^{M}$ . This further implies that  $\chi_{G\times K}^{M}(G\times K) = \chi_{H\times K}^{M}(H\times K)$  by definition of homomorphism expressivity. Therefore, Lemma G.2 implies that
+
+$$
+\sum_ {L \in \mathcal {L}} \alpha (L) \operatorname {h o m} (L, G) \cdot \operatorname {h o m} (L, K) = \sum_ {L \in \mathcal {L}} \alpha (L) \operatorname {h o m} (L, H) \cdot \operatorname {h o m} (L, K). \tag {30}
+$$
+
+Namely,
+
+$$
+\sum_ {L \in \tilde {\mathcal {L}}} \tilde {\alpha} (L) \operatorname {h o m} (L, G) \cdot \operatorname {h o m} (L, K) = \sum_ {L \in \tilde {\mathcal {L}}} \tilde {\alpha} (L) \operatorname {h o m} (L, H) \cdot \operatorname {h o m} (L, K). \tag {31}
+$$
+
+Now define vectors  $\pmb{p}_G$  and  $\pmb{p}_H$ , where the  $i^{th}$  element of vector  $\pmb{p}_G$  and  $\pmb{p}_H$  is  $\tilde{\alpha}(L_i) \mathrm{hom}(L_i, G)$  and  $\tilde{\alpha}(L_i) \mathrm{hom}(L_i, H)$ , respectively. We then have the following equation:
+
+$$
+\boldsymbol {A} ^ {\text {h o m}} \cdot \boldsymbol {p} _ {G} = \boldsymbol {A} ^ {\text {h o m}} \cdot \boldsymbol {p} _ {H}.
+$$
+
+Since  $\mathbf{A}^{\mathrm{hom}}$  is invertible,  $\pmb{p}_G^{\mathrm{hom}} = \pmb{p}_H^{\mathrm{hom}}$ . Therefore,  $\tilde{\alpha}(L)\mathrm{hom}(L,G) = \tilde{\alpha}(L)\mathrm{hom}(L,H)$  for all  $L \in \tilde{\mathcal{L}}$ , namely,  $\mathrm{hom}(L,G) = \mathrm{hom}(L,H)$  for all  $L \in \mathcal{L}$ . To sum up, we have proved that for all graph  $G,H$ ,  $\chi_G^M (G) = \chi_H^M (H)$  implies that  $\mathrm{hom}(L,G) = \mathrm{hom}(L,H)$  for all  $L \in \mathcal{L}$ . By definition of homomorphism expressivity, we conclude that  $\mathcal{L} \subset \mathcal{F}^M$ .
+
+Theorem G.6. Let  $M \in \{\mathrm{Sub}, \mathsf{L}, \mathsf{LF}, \mathsf{F}\}$  be any model. For any graph  $F$ , if  $\mathrm{Spasm}(F) \backslash \mathcal{F}^M \neq \emptyset$ , then there exists a pair of graphs  $G, H$  such that  $\chi_G^M(G) = \chi_H^M(H)$  and  $\mathrm{sub}(F, G) \neq \mathrm{sub}(F, H)$ .
+
+Proof. Recall that  $\operatorname{sub}(F, G) = \sum_{\tilde{F} \in \operatorname{Spasm}^{\neq}(F)} \alpha(F, \tilde{F}) \cdot \operatorname{hom}(\tilde{F}, G)$  where  $\alpha(F, \tilde{F}) \neq 0$  for all  $\tilde{F} \in \operatorname{Spasm}^{\neq}(F)$ . If the above theorem does not hold, then for all graphs  $G, H$ ,  $\chi_G^M(G) = \chi_H^M(H)$  implies that
+
+$$
+\sum_ {\tilde {F} \in \operatorname {S p a s m} ^ {\mathcal {Z}} (F)} \alpha (F, \tilde {F}) \cdot \operatorname {h o m} (\tilde {F}, G) = \sum_ {\tilde {F} \in \operatorname {S p a s m} ^ {\mathcal {Z}} (F)} \alpha (F, \tilde {F}) \cdot \operatorname {h o m} (\tilde {F}, G).
+$$
+
+Then, Lemma G.5 implies that  $\operatorname{Spasm}(F) \subset \mathcal{F}^M$ , yielding a contradiction.
+
+![](images/57ebffe1e90b2e72ff9a5e60fade4ea4cb2a805920423b6c74c3c01178cff917.jpg)
+
+We next extend our analysis to the node/edge-level subgraph counting. Since the proof techniques are almost the same as the graph-level setting, we only present key definitions and lemmas below while omitting the detailed proofs for clarity.
+
+Definition G.7. Given two rooted graphs  $G^{\mathbf{u}}$  and  $H^{\mathbf{v}}$  with  $\mathbf{u} \in V_G^m$  and  $\mathbf{v} \in V_H^m$  for some  $m \in \mathbb{N}_+$ , the categorical product of  $G^{\mathbf{u}}$  and  $H^{\mathbf{v}}$ , denoted as  $(G \times H)^{(u_1, v_1), \dots, (u_m, v_m)}$ , is a rooted graph obtained from  $G \times H$  by marking vertices  $(u_1, v_1), \dots, (u_m, v_m)$ .
+
+Lemma G.8 (Extension of Lemma G.2). For any rooted graphs  $F^{\mathbf{w}}$ ,  $G^{\mathbf{u}}$ ,  $H^{\mathbf{v}}$  where  $|\mathbf{u}| = |\mathbf{v}| = | \mathbf{w} | = m$ ,
+
+$$
+\operatorname {h o m} \left(F ^ {\boldsymbol {w}}, (G \times H) ^ {(u _ {1}, v _ {1}), \dots , (u _ {m}, v _ {m})}\right) = \operatorname {h o m} \left(F ^ {\boldsymbol {w}}, G ^ {\boldsymbol {u}}\right) \cdot \operatorname {h o m} \left(F ^ {\boldsymbol {w}}, H ^ {\boldsymbol {v}}\right).
+$$
+
+Proposition G.9 (Extension of Proposition G.4). For any rooted graphs  $G^u$ ,  $H^v$  where  $|\boldsymbol{u}| = |\boldsymbol{v}| = m$ ,  $\operatorname{hom}(G^u, H^v) = \sum_{F^w} \operatorname{surj}(G^u, F^w) \cdot \operatorname{sub}(F^w, H^v)$ , where  $F^w$  ranges over all non-isomorphic rooted graphs marking  $m$  vertices.
+
+We next present the main lemma for node-level subgraph counting. We omit the edge-level result for clarity.
+
+Lemma G.10 (Extension of Lemma G.5). Let  $M$  be a GNN model such that its node-level homomorphism expressivity  $\mathcal{F}_{\mathfrak{n}}^{M}$  exists. Given a finite set of rooted graphs  $\mathcal{L}_{\mathfrak{n}}$  and a function  $\alpha : \mathcal{L}_{\mathfrak{n}} \to \mathbb{R} / \{0\}$ , if
+
+$$
+\chi_ {G} ^ {M} (u) = \chi_ {H} ^ {M} (v) \Longrightarrow \sum_ {L ^ {w} \in \mathcal {L} _ {\mathrm {n}}} \alpha (L ^ {w}) \operatorname {h o m} (L ^ {w}, G ^ {u}) = \sum_ {L ^ {w} \in \mathcal {L} _ {\mathrm {n}}} \alpha (L ^ {w}) \operatorname {h o m} (L ^ {w}, H ^ {v}),
+$$
+
+holds for all rooted graphs  $G^{u}$  and  $H^{v}$ , then  $\mathcal{L}_{\mathfrak{n}} \subset \mathcal{F}_{\mathfrak{n}}^{M}$ .
+
+Theorem G.11. Let  $M \in \{\text{Sub}, \mathsf{L}, \mathsf{LF}, \mathsf{F}\}$  be any model. For any rooted graph  $F^w$ , if  $\text{Spasm}(F^w) \backslash \mathcal{F}_\mathfrak{n}^M \neq \emptyset$ , then there exist a pair of graphs  $G, H$  and vertices  $u \in V_G, v \in V_H$ , such that  $\chi_G^M(u) = \chi_H^M(v)$  and  $\text{sub}(F^w, G^u) \neq \text{sub}(F^w, H^v)$ .
+
+# G.2 GRAPH STATISTICS AND EXAMPLES
+
+In this section, we list the statistics of all moderate-size graphs that can/cannot be counted for each model in Section 2 at graph/node/edge-level. Table 4 presents the statistics under homomorphism count, while Table 5 presents the statistics under subgraph count. These tables offer a clear picture into how large the expressivity gaps are between different models. Several important findings are discussed below:
+
+- For homomorphism counting, graphs of 8 edges suffice to reveal the expressivity gaps between each pair of architectures (at edge-level).  
+- However, for subgraph counting, moderate-size graphs cannot reveal the gap between Local 2-GNN, Local 2-FGNN, and 2-FGNN. Moreover, even Subgraph GNN already matches the power of 2-FWL in counting small subgraphs at graph-level, but it is quite weak in counting subgraphs at node/edge-level.
+
+- Subgraph counting is much more challenging than homomorphism counting. Intuitively, this is because the homomorphism image of a graph usually has 4-cliques, and any graph that contains a 4 clique as subgraph cannot be counted under homomorphism.
+
+In Table 6, we list all subgraphs at moderate size (no more than 6 vertices or 8 edges) that can/cannot be counted by each GNN model. Here, we only list the graph-level expressivity as the node/edge-level expressivity involves too many non-isomorphic rooted graphs and cannot be fully presented (see Tables 4 and 5). We believe these results is comprehensive enough to cover most substructures of interest in the GNN community.
+
+Table 4: The number of (connected) graphs (or rooted graphs) of  $n$  vertices or  $m$  edges that can be counted under homomorphism by different models. These statistics can be viewed as a quantitative expressivity comparison between models.  
+
+<table><tr><td rowspan="2" colspan="2"></td><td colspan="5">Number of vertices n</td><td colspan="7">Number of edges m</td><td></td></tr><tr><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td></tr><tr><td rowspan="6">Graph level</td><td>MPNN</td><td>1</td><td>1</td><td>2</td><td>3</td><td>6</td><td>1</td><td>1</td><td>2</td><td>3</td><td>6</td><td>11</td><td>23</td><td>47</td></tr><tr><td>Subgraph GNN</td><td>1</td><td>2</td><td>5</td><td>15</td><td>51</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>29</td><td>76</td><td>210</td></tr><tr><td>Local 2-GNN</td><td>1</td><td>2</td><td>5</td><td>15</td><td>55</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>29</td><td>77</td><td>216</td></tr><tr><td>Local 2-FGNN</td><td>1</td><td>2</td><td>5</td><td>15</td><td>56</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>29</td><td>77</td><td>216</td></tr><tr><td>2-FGNN</td><td>1</td><td>2</td><td>5</td><td>15</td><td>56</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>29</td><td>77</td><td>216</td></tr><tr><td>All</td><td>1</td><td>2</td><td>6</td><td>21</td><td>112</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>30</td><td>79</td><td>227</td></tr><tr><td rowspan="6">Node level</td><td>MPNN</td><td>1</td><td>2</td><td>4</td><td>9</td><td>20</td><td>1</td><td>2</td><td>4</td><td>9</td><td>20</td><td>48</td><td>115</td><td>286</td></tr><tr><td>Subgraph GNN</td><td>1</td><td>3</td><td>8</td><td>27</td><td>88</td><td>1</td><td>2</td><td>5</td><td>12</td><td>31</td><td>83</td><td>228</td><td>640</td></tr><tr><td>Local 2-GNN</td><td>1</td><td>3</td><td>10</td><td>44</td><td>215</td><td>1</td><td>2</td><td>5</td><td>13</td><td>37</td><td>113</td><td>361</td><td>1210</td></tr><tr><td>Local 2-FGNN</td><td>1</td><td>3</td><td>10</td><td>44</td><td>217</td><td>1</td><td>2</td><td>5</td><td>13</td><td>37</td><td>113</td><td>361</td><td>1210</td></tr><tr><td>2-FGNN</td><td>1</td><td>3</td><td>10</td><td>44</td><td>217</td><td>1</td><td>2</td><td>5</td><td>13</td><td>37</td><td>113</td><td>361</td><td>1210</td></tr><tr><td>All</td><td>1</td><td>3</td><td>11</td><td>58</td><td>407</td><td>1</td><td>2</td><td>5</td><td>13</td><td>37</td><td>114</td><td>367</td><td>1248</td></tr><tr><td rowspan="5">Edge level</td><td>Subgraph GNN</td><td>1</td><td>4</td><td>18</td><td>77</td><td>340</td><td>1</td><td>3</td><td>10</td><td>33</td><td>107</td><td>347</td><td>1126</td><td>3664</td></tr><tr><td>Local 2-GNN</td><td>1</td><td>4</td><td>21</td><td>116</td><td>693</td><td>1</td><td>3</td><td>10</td><td>35</td><td>124</td><td>450</td><td>1665</td><td>6267</td></tr><tr><td>Local 2-FGNN</td><td>1</td><td>4</td><td>21</td><td>118</td><td>735</td><td>1</td><td>3</td><td>10</td><td>35</td><td>124</td><td>451</td><td>1678</td><td>6373</td></tr><tr><td>2-FGNN</td><td>1</td><td>4</td><td>21</td><td>118</td><td>735</td><td>1</td><td>3</td><td>10</td><td>35</td><td>124</td><td>451</td><td>1678</td><td>6374</td></tr><tr><td>All</td><td>1</td><td>4</td><td>23</td><td>162</td><td>1549</td><td>1</td><td>3</td><td>10</td><td>35</td><td>125</td><td>460</td><td>1747</td><td>6830</td></tr></table>
+
+Table 5: The number of (connected) graphs (or rooted graphs) of  $n$  vertices or  $m$  edges that can be subgraph-counted by different models. These statistics can be viewed as a quantitative expressivity comparison between models.  
+
+<table><tr><td rowspan="2" colspan="2"></td><td colspan="5">Number of vertices n</td><td colspan="7">Number of edges m</td><td></td></tr><tr><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td></tr><tr><td rowspan="6">Graph level</td><td>MPNN</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Subgraph GNN</td><td>1</td><td>2</td><td>5</td><td>13</td><td>24</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>20</td><td>21</td><td>22</td></tr><tr><td>Local 2-GNN</td><td>1</td><td>2</td><td>5</td><td>13</td><td>24</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>20</td><td>21</td><td>22</td></tr><tr><td>Local 2-FGNN</td><td>1</td><td>2</td><td>5</td><td>13</td><td>24</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>20</td><td>21</td><td>22</td></tr><tr><td>2-FGNN</td><td>1</td><td>2</td><td>5</td><td>13</td><td>24</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>20</td><td>21</td><td>22</td></tr><tr><td>All</td><td>1</td><td>2</td><td>6</td><td>21</td><td>112</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>30</td><td>79</td><td>227</td></tr><tr><td rowspan="6">Node level</td><td>MPNN</td><td>1</td><td>2</td><td>2</td><td>2</td><td>2</td><td>1</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td></tr><tr><td>Subgraph GNN</td><td>1</td><td>3</td><td>8</td><td>18</td><td>27</td><td>1</td><td>2</td><td>5</td><td>10</td><td>16</td><td>23</td><td>30</td><td>38</td></tr><tr><td>Local 2-GNN</td><td>1</td><td>3</td><td>10</td><td>37</td><td>84</td><td>1</td><td>2</td><td>5</td><td>13</td><td>37</td><td>72</td><td>75</td><td>86</td></tr><tr><td>Local 2-FGNN</td><td>1</td><td>3</td><td>10</td><td>37</td><td>84</td><td>1</td><td>2</td><td>5</td><td>13</td><td>37</td><td>72</td><td>75</td><td>86</td></tr><tr><td>2-FGNN</td><td>1</td><td>3</td><td>10</td><td>37</td><td>84</td><td>1</td><td>2</td><td>5</td><td>13</td><td>37</td><td>72</td><td>75</td><td>86</td></tr><tr><td>All</td><td>1</td><td>3</td><td>11</td><td>58</td><td>407</td><td>1</td><td>2</td><td>5</td><td>13</td><td>37</td><td>114</td><td>367</td><td>1248</td></tr><tr><td rowspan="5">Edge level</td><td>Subgraph GNN</td><td>1</td><td>4</td><td>18</td><td>47</td><td>81</td><td>1</td><td>3</td><td>10</td><td>25</td><td>46</td><td>69</td><td>95</td><td>124</td></tr><tr><td>Local 2-GNN</td><td>1</td><td>4</td><td>21</td><td>92</td><td>208</td><td>1</td><td>3</td><td>10</td><td>35</td><td>105</td><td>171</td><td>179</td><td>216</td></tr><tr><td>Local 2-FGNN</td><td>1</td><td>4</td><td>21</td><td>92</td><td>208</td><td>1</td><td>3</td><td>10</td><td>35</td><td>105</td><td>171</td><td>179</td><td>216</td></tr><tr><td>2-FGNN</td><td>1</td><td>4</td><td>21</td><td>92</td><td>208</td><td>1</td><td>3</td><td>10</td><td>35</td><td>105</td><td>171</td><td>179</td><td>216</td></tr><tr><td>All</td><td>1</td><td>4</td><td>23</td><td>162</td><td>1549</td><td>1</td><td>3</td><td>10</td><td>35</td><td>125</td><td>460</td><td>1747</td><td>6830</td></tr></table>
+
+Table 6: The ability of GNNs to homomorphism-count and subgraph-count different graphs  $F$  within a bounded size of  $n \leq  6$  vertices or  $m \leq  8$  edges. When one or more GNNs fail to subgraphcount  $F$  ,this table also gives a homomorphic image  $\widetilde{F} \in  \operatorname{Spasm}\left( F\right)$  that can be used to construct counterexample graphs (see Section 4.2).  
+
+<table><tr><td>nmF</td><td>Homomorphism MP Sub L LF F</td><td>Subgraph MP Sub L LF F</td><td>F</td><td>nmF</td><td>Homomorphism MP Sub L LF F</td><td>Subgraph MP Sub L LF F</td><td>F</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>2 1←</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>-</td><td>5 7</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>3 2←</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>-</td><td>5 8</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>3 3</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>4 3</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>4 3</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>-</td><td>5 10</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>4 4</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 5</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>4 4</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 5</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>4 5</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 5</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>4 6</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>6 5</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 4</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 5</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 4</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 5</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 4</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>-</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 5</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 5</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 5</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 5</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>√</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 5</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 5</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✗</td><td>x</td></tr><tr><td>5 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>5 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>6 6</td><td>x</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr></table>
+
+(see the next page)
+
+<table><tr><td>n m</td><td>F</td><td>Homomorphism</td><td>Subgraph</td><td>MP Sub L F</td><td>MP Sub L F</td><td>F</td><td>n m</td><td>F</td><td>Homomorphism</td><td>Subgraph</td><td>MP Sub L F</td><td>MP Sub L F</td><td>F</td><td>n m</td><td>F</td><td>Homomorphism</td><td>Subgraph</td><td>MP Sub L F</td><td>MP Sub L F</td><td>F</td></tr><tr><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>X</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>X</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>X</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td><td>69</td></tr><tr><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td></tr><tr><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td></tr><tr><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td></tr><tr><td>X</td><td>X</td><td>X</td><td>X</td><td>x</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td></tr><tr><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td></tr><tr><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>x</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td></tr></table>
+
+(see the next page)
+
+<table><tr><td>n m F</td><td>Homomorphism Subgraph MPSubL F F</td><td>F</td><td>m F</td><td>Homomorphism Subgraph MPSubL F F</td><td>F</td></tr><tr><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>611</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>611</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>611</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>611</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>y</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>y</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>69</td><td>69</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>77</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>77</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>77</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>77</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>77</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>77</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>77</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr></table>
+
+(see the next page)
+
+<table><tr><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td>77</td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>x</td><td>x</td><td>x</td><td>y</td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td></td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td></td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td></td><td></td><td></td></tr></table>
+
+(see the next page)
+
+<table><tr><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td>78</td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>x</td><td>x</td><td>x</td><td></td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td></td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td></td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>x</td><td></td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td></td><td></td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td></td><td></td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td></td><td></td><td></td></tr><tr><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>y</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td></td><td></td><td></td></tr></table>
+
+(see the next page)
+
+<table><tr><td>n m</td><td>F</td><td>Homomorphism Subgraph MP Sub L F</td><td>MP Sub L F</td><td>F</td><td>n m</td><td>F</td><td>Homomorphism Subgraph MP Sub L F</td><td>MP Sub L F</td><td>F</td><td>n m</td><td>F</td><td>Homomorphism Subgraph MP Sub L F</td><td>MP Sub L F</td><td>F</td><td>n m</td><td>F</td><td>Homomorphism Subgraph MP Sub L F</td><td>MP Sub L F</td><td>F</td></tr><tr><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td></tr><tr><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td><td></td></tr><tr><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td></td></tr><tr><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td></td></tr><tr><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td><td></td></tr><tr><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td><td></td></tr><tr><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td></td></tr><tr><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td></td></tr><tr><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td></td></tr></table>
+
+(see the next page)
+
+<table><tr><td>n m</td><td>F</td><td>Homomorphism</td><td>Subgraph</td><td>MP Sub L F F</td><td>MP Sub L F F</td></tr><tr><td>x</td><td>√</td><td>√</td><td>√</td><td>x</td><td>√</td></tr><tr><td>x</td><td>√</td><td>√</td><td>√</td><td>x</td><td>√</td></tr><tr><td>x</td><td>√</td><td>√</td><td>√</td><td>x</td><td>√</td></tr><tr><td>x</td><td>√</td><td>√</td><td>√</td><td>x</td><td>√</td></tr><tr><td>x</td><td>√</td><td>√</td><td>√</td><td>x</td><td>√</td></tr><tr><td>x</td><td>√</td><td>√</td><td>√</td><td>x</td><td>√</td></tr><tr><td>x</td><td>√</td><td>√</td><td>√</td><td>x</td><td>√</td></tr><tr><td>x</td><td>√</td><td>√</td><td>√</td><td>x</td><td>√</td></tr></table>
+
+(see the next page)
+
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m</td><td>F</td><td colspan="101">Homonorphism Subgraph MP Sub L F F</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+# H POLYNOMIAL EXPRESSIVITY
+
+Puny et al. (2023) proposed the equivariant graph polynomials, which are polynomials  $P$  that take real squared matrices  $\mathbf{X} \in \mathbb{R}^{n \times n}$  as input and outputs  $P(\mathbf{X}) \in \mathbb{R}^{n \times n}$ , such that  $P$  is equivariant under permutations, i.e.,  $P(\pi \cdot \mathbf{X}) = \pi \cdot P(\mathbf{X})$  for all permutation  $\pi \in S_n$ . The authors gave a concrete basis of equivariant polynomials, where each basis element  $P_{F^{uv}}$  corresponds to a rooted multi-graph  $F^{uv}$  marking two vertices  $u, v$ . Puny et al. (2023) showed that when  $\mathbf{X}$  is restricted to be the adjacency matrix of an undirected simple graph  $G$ , each  $F^{uv}$  will also reduce to an undirected simple graph, and  $P_{F^{uv}}(\mathbf{X})$  precisely computes the (unlabeled) homomorphism count  $\mathrm{hom}(F^{uv}, G^{wx})$  for all  $w, x \in V_G$ . Besides the original definition, Puny et al. (2023) also proposed the invariant graph polynomials and node-level equivariant graph polynomials, which are similarly related to the graph-level and node-level homomorphism count.
+
+As a direct consequence, if a GNN model  $M$  cannot count graph  $F^{uv} / F^u /F$  under homomorphism, it then cannot compute the equivariant/invariant graph polynomial  $P_{F^{uv}} / P_{F^u} / P_F$ . Based on these connections, our results can be directly used to provide insights into which equivariant graph polynomials cannot be computed by model  $M$ . This recovers several results in their paper and answers an open problem shown below.
+
+Corollary H.1. MPNN and  $PPGN++$  are bounded by the Prototypical node-based model and Prototypical edge-based model (defined in Puny et al. (2023)) for computing node-level and edge-level equivariant graph polynomials, respectively.
+
+Proof. Without loss of generality, we assume that the corresponding graphs of all equivariant polynomials are connected. According to Puny et al. (2023, Proposition H.2), the Prototypical node-based model can compute all  $P_{F^u}$  where  $F^u$  is a (rooted) tree and cannot compute other graph polynomials. If MPNN is not bounded by the Prototypical node-based model, then it can compute some  $P_{\tilde{F}^u}$  where  $\tilde{F}$  is not a tree. However, this is impossible since MPNN can only count forests under homomorphism according to Theorem 3.7. Note that the MPNN defined in their paper is equivalent to our definition when only considering connected graphs (the extra global aggregation  $11^\top X$  in their definition (9) does not improve the homomorphism expressivity).
+
+We next turn to  $\mathrm{PPGN}++$ , and the proof is similar (but more involved). We first show that the Prototypical edge-based model can compute any  $P_{F^{uv}}$  satisfying that the treewidth of the graph  $\tilde{F} := (V_F, E_F \cup \{\{u, v\}\}, \ell_F)$  is no more than 2. If  $\mathsf{tw}(\tilde{F}) \leq 2$ , it is a partial 2-tree. Thus, there is an ordering  $w_1, \dots, w_n$  of the vertex set  $V_F$  such that when deleting each vertex  $w_i$  and all incident edges in turn, we only ever delete vertices of degree at most 2. Now we claim that we can always order the two vertices  $u, v$  at the end, i.e.,  $w_{n-1} = u$  and  $w_n = v$ . Otherwise, there is a subset  $U \subset V_F$  such that all vertices in the induced subgraph  $\tilde{F}[U]$  are of degree at least 3 except  $u, v$ . It follows that the  $\mathsf{tw}(\tilde{F}[U]) \geq \mathsf{tw}(H) \geq 3$  where graph  $H$  is the graph obtained from  $\tilde{F}[U]$  by contracting  $u$  (or  $v$ ) if  $\deg_{\tilde{F}[U]}(u) \leq 2$  (or  $\deg_{\tilde{F}[U]}(v) \leq 2$ ). This yields a contradiction and verifies the claim that we can always set  $w_{n-1} = u$  and  $w_n = v$ .
+
+Now, following the proof in Puny et al. (2023, Proposition H.3), the Prototypical edge-based model can contract  $\tilde{F}^{uv}$  to a graph with only two vertices  $u, v$  and thus can compute the edge-level polynomial  $P_{\tilde{F}^{uv}}$ . If PPGN++ is not bounded by the Prototypical edge-based model, then it can compute some  $P_{\tilde{F}^{uv}}$  where  $\mathrm{tw}(\tilde{F}) \geq 3$ . Therefore, it can count the graph  $\tilde{F}$  under homomorphism at graph-level (since it can already count  $\tilde{F}$  at edge-level). This implies that PPGN++ is strictly more expressive than 2-FGNN (2-FWL) because we have proved that all graphs in  $\mathcal{F}^{\mathrm{F}}$  have a treewidth no more than 2. This yields a contradiction since PPGN++ is still bounded by 2-FWL in distinguishing non-isomorphic graphs.
+
+We also provide insights into the following results in their paper:
+
+Corollary H.2. The Prototypical node-based model is not 3-node-polynomial-exact. The Prototypical edge-based model is not 6-node-polynomial-exact and not 5-edge-polynomial-exact.
+
+This is simply because the triangle is not a tree and the 4-clique does not have a NED (or equivalently, the treewidth of a 4-clique is 3). It is also clear why the degree of the edge-based polynomial is 6, which is one less than that of the node-based polynomial using the concept of NED.
+
+# I EXPERIMENTAL DETAILS
+
+In this section, we provide all the experimental details in Section 5.
+
+# I.1 DATASETS
+
+We conduct experiments on tive tasks: (i) graph homomorphism counting, (ii) subgraph counting, (iii) ZINC-subset (Dwivedi et al., 2020), (iii) ZINC-full (Dwivedi et al., 2020), and (iv) Alchemy (Chen et al., 2019a).
+
+Homomorphism/Subgraph Counting. For both homomorphism and subgraph counting tasks, we use the standard synthetic graph dataset constructed in Zhao et al. (2022a) (which has been used in a number of papers, see e.g., Frasca et al., 2022; Huang et al., 2023; Zhang et al., 2023a). For homomorphism counting, we count the number of graph/node/edge-level homomorphisms for each pattern in Table 1 and normalize the value by the mean and variance across all graphs in the dataset. The evaluation metric of graph-level expressivity is chosen as the Mean Absolute Error (MAE). For node/edge-level expressivity, the error on each graph is defined to be the sum of absolute error over all vertices/edges. We then report the MAE across all graphs in the dataset. This ensures that graph/node/edge-level errors are roughly at the same scale (since  $\mathrm{hom}(F,G) = \sum_{w\in V_G}\mathrm{hom}(F^u,G^w) = \sum_{w,x\in V_G}\mathrm{hom}(F^{uv},G^{wx})$  for all  $u,v\in V_F$ ). For subgraph counting, the data processing and evaluation metric is similar to homomorphism counting, but there is a slight difference in the node/edge-level setting: there are no marked vertices in the pattern graph  $F$  (see Table 3). Instead, given a graph  $G$  in the dataset and a vertex  $w\in V_G$ , we count the number of subgraphs containing  $w$  that are isomorphic to  $F$  and  $w$  can be mapped to any vertex in  $F$ . Due to this difference, in the node/edge-level setting, the error on each graph is defined to be the average of absolute error over all vertices/edges.
+
+ZINC. ZINC (Dwivedi et al., 2020) is a standard real-world dataset for benchmarking molecular property prediction. The dataset consists of 250K molecular graphs, and the task is to predict the constrained solubility of the given molecule. In addition to the full dataset (denoted as ZINC-full), ZINC-subset is a sampled dataset with 12k molecules from the ZINC-full dataset. We train and test our models on both datasets following the standard protocol from Dwivedi et al. (2020).
+
+Alchemy. Alchemy (Chen et al., 2019a) is another real-world dataset with 12 graph-level quantum mechanical properties. We follow the sampling and training protocol from Lim et al. (2023); Puny et al. (2023), using 100K samples for training, 10K samples for testing, and 10K samples for validation.
+
+# I.2 MODEL DETAILS
+
+All models are implemented using the PyTorch (Paszke et al., 2019) framework and the PyTorch Geometric library (Fey & Lenssen, 2019). We consider four types of GNNs defined in Section 2: MPNN, Subgraph GNN, Local 2-GNN, and Local 2-FGNN. For each GNN model, the feature initialization, message-passing layers, and final pooling operation are separately defined below.
+
+Initialization. On both ZINC and Alchemy datasets, each graph node is an atom. We maintain a learnable atom embedding for each type of atom and use it to initialize features in GNN models. For MPNN, the initial feature  $h^{(0)}(u)$  of node  $u$  is simply the atom embedding, denoted as  $h^{(0)}(u) = e_{\mathrm{atom}(u)}^{\mathrm{N}}$ . For other models, the initial feature  $h^{(0)}(u,v)$  of node pair  $(u,v)$  consists of two parts: the first part is the node embedding of  $v$ , and the second part is a distance encoding that embeds the shortest path distance between  $u$  and  $v$ , as adopted in Zhang et al. (2023a). We note that while incorporating distance encoding does not increase the models' theoretical expressive power (see Zhang et al. (2023a)), it may add an inductive bias that can be helpful in real-world tasks. Formally, the initial feature can be written as  $h^{(0)}(u,v) = [e_{\mathrm{atom}(v)}^{\mathrm{N}}, e_{\mathrm{clip}(\mathrm{dis}(u,v))}^{\mathrm{D}}]$ . Here, we clip the distance to a predefined value max_dis so that there are a finite number of distance embeddings, and distances greater than the hyper-parameter max_dis (including the disconnected case) share the embedding.
+
+Propagation. On both ZINC and Alchemy datasets, each edge in a graph corresponds to a chemical bond and has a bond type. We maintain a learnable edge embedding for each type of edges in each layer and denote the embedding of edge  $\{u,v\}$  in layer  $l$  as  $g^{(l)}(u,v)$ . For MPNN, we use the
+
+standard GIN architecture proposed in Xu et al. (2019), which has the following form:
+
+$$
+h ^ {(l + 1)} (u) = \operatorname {R e L U} \left(\mathsf {B N} ^ {(l)} \left(f ^ {(l)} (u)\right)\right), \tag {32}
+$$
+
+$$
+f ^ {(l)} (u) = \mathsf {G I N} ^ {(l)} \left(h ^ {(l)} (u), \sum_ {v \in N _ {G} (u)} \operatorname {R e L U} \left(\mathsf {F C} ^ {(l)} \left(h ^ {(l)} (v)\right) + g ^ {(l)} (u, v)\right)\right), \tag {33}
+$$
+
+where
+
+$$
+\operatorname {G I N} ^ {(l)} (\boldsymbol {x}, \boldsymbol {y}) = \operatorname {M L P} ^ {(l)} \left((1 + \epsilon^ {(l)}) \boldsymbol {x} + \boldsymbol {y}\right). \tag {34}
+$$
+
+Here,  $\mathsf{FC}^{(l)}$  is a parameterized linear transformation,  $\epsilon^{(l)}$  is a learnable parameter,  $\mathsf{BN}^{(l)}$  is the batch normalization (Ioffe & Szegedy, 2015), and  $\mathsf{MLP}^{(l)}$  is a two-layer feed-forward network with another batch normalization in the hidden layer.
+
+For other architectures, the  $l$ -th GNN layer analogously has the following form:
+
+$$
+h ^ {(l + 1)} (u, v) = \operatorname {R e L U} \left(\mathsf {B N} ^ {(l)} \left(f ^ {(l)} (u, v)\right)\right), \tag {35}
+$$
+
+where the term  $f^{(l)}(u,v)$  is defined separately for each model:
+
+- Subgraph GNN:
+
+$$
+f ^ {(l)} (u, v) = \mathsf {G I N} ^ {(l)} \left(h ^ {(l)} (u, v), \sum_ {w \in N _ {G} (v)} \operatorname {R e L U} \left(\mathsf {F C} ^ {(l)} \left(h ^ {(l)} (u, w)\right) + g ^ {(l)} (w, v)\right)\right). \tag {36}
+$$
+
+- Local 2-GNN:
+
+$$
+\begin{array}{l} f ^ {(l)} (u, v) = \mathsf {G I N} ^ {(l, 1)} \left(h ^ {(l)} (u, v), \sum_ {w \in N _ {G} (u)} \operatorname {R e L U} \left(\mathsf {F C} ^ {(l, 1)} \left(h ^ {(l)} (w, v)\right) + g ^ {(l)} (u, w)\right)\right) \\ + \operatorname {G I N} ^ {(l, 2)} \left(h ^ {(l)} (u, v), \sum_ {w \in N _ {G} (v)} \operatorname {R e L U} \left(\mathsf {F C} ^ {(l, 2)} \left(h ^ {(l)} (u, w)\right) + g ^ {(l)} (w, v)\right)\right). \tag {37} \\ \end{array}
+$$
+
+- Local 2-FGNN:
+
+$$
+\begin{array}{l} f ^ {(l)} (u, v) \\ = \mathsf {G I N} ^ {(l, 1)} \left(h ^ {(l)} (u, v), \sum_ {w \in N _ {G} (u)} \mathsf {R e L U} \left(\mathsf {F C} ^ {(l, 1)} \left(h ^ {(l)} (u, w)\right) + \mathsf {F C} ^ {(l, 2)} \left(h ^ {(l)} (w, v)\right) + g ^ {(l)} (u, w)\right)\right) \\ + \mathsf {G I N} ^ {(l, 2)} \left(h ^ {(l)} (u, v), \sum_ {w \in N _ {G} (v)} \operatorname {R e L U} \left(\mathsf {F C} ^ {(l, 1)} \left(h ^ {(l)} (w, v)\right) + \mathsf {F C} ^ {(l, 2)} \left(h ^ {(l)} (u, w)\right) + g ^ {(l)} (w, v)\right)\right). \tag {38} \\ \end{array}
+$$
+
+For all the above GNN architectures, it can be seen that each layer only aggregates the local neighborhood of vertices or vertex pairs. This design will have shortcomings for disconnected graphs since a vertex cannot aggregate information from other connected components no matter how deep the model is (Barceló et al., 2020). Note that there do exist disconnected graphs in real-world datasets like ZINC. Therefore, on real-world datasets like ZINC and Alchemy, we also incorporate a global aggregation with the following form for each layer (similar to the global aggregation in Frasca et al. (2022); Zhang et al. (2023a)):
+
+$$
+\operatorname {G I N} ^ {(l, G)} \left(h ^ {(l)} (u, v), \sum_ {w \in V _ {G}} h ^ {(l)} (u, w)\right) \tag {39}
+$$
+
+Note that the global aggregation does not increase model's theoretical expressive power according to Zhang et al. (2023a).
+
+Pooling. Except for edge-level tasks, a final pooling layer is used to produce node-level features  $h(u)$  for all nodes  $u$ . It is implemented as follows:
+
+$$
+h (u) = \operatorname {M L P} \left(\sum_ {v \in \mathcal {V}} h ^ {(L)} (u, v)\right), \tag {40}
+$$
+
+where MLP is a 2-layer perceptron. For graph-level tasks, we further use a mean pooling layer to aggregate all  $h(u)$  and obtain the graph representation.
+
+# I.3 TRAINING DETAILS
+
+All experiments are run on a single NVIDIA Tesla V100 GPU. For all tasks, we use the distance encoding hyper-parameter max_dis = 5. To enable a fair comparison between models, for each task we keep the same depth for different models while varying the hidden dimension so that the number of model parameters is roughly the same. Note that the dimensions are chosen such that all models roughly obey the 500K parameter budget in ZINC and Alchemy. The hidden dimension size and the number of model parameters are listed as follows. All models are trained using the Adam optimizer.
+
+Table 7: Model size in different tasks.  
+
+<table><tr><td rowspan="2">Task</td><td colspan="3">Hidden dimension</td><td colspan="3"># Parameters</td></tr><tr><td>Counting</td><td>ZINC</td><td>Alchemy</td><td>Counting</td><td>ZINC</td><td>Alchemy</td></tr><tr><td>MPNN</td><td>128</td><td>150</td><td>150</td><td>314,119</td><td>510,158</td><td>509,719</td></tr><tr><td>Subgraph GNN</td><td>128</td><td>120</td><td>120</td><td>314,759</td><td>503,774</td><td>503,425</td></tr><tr><td>Local 2-GNN</td><td>96</td><td>96</td><td>96</td><td>317,388</td><td>495,188</td><td>494,911</td></tr><tr><td>Local 2-FGNN</td><td>96</td><td>96</td><td>96</td><td>317,388</td><td>495,188</td><td>494,911</td></tr></table>
+
+Homomorphism/Subgraph Counting. We use a model depth of  $L = 5$  in all experiments. Following prior work (Huang et al., 2023), we remove all BN layers in all models. The initial learning rate is chosen as 0.001 and is decayed by a factor of 0.9 once the MAE on the validation set plateaus for 10 epochs. Each model is trained for 1200 epochs with a batch size of 512. We ran each experiment 4 times independently with different seeds and reported the average performance at the last epoch. We found that the standard deviation among different seeds is negligible.
+
+ZINC. We use a model depth of  $L = 6$  in all experiments for both ZINC-subset and ZINC-full. Following prior work (Zhang et al., 2023a; Frasca et al., 2022), The initial learning rate is chosen as 0.001 and is decayed by a factor of 0.5 once the MAE on the validation set plateaus for 20 epochs. Each model is trained for 400 epochs on ZINC-subset and 500 epochs on ZINC-full, both with a batch size of 128. We report the MAE for the model checkpoint with the best validation performance. We ran each experiment 10 times independently with different seeds and reported the average performance as well as the standard deviation.
+
+Alchemy. We use a model depth of  $L = 6$  in all experiments. Following prior work (Lim et al., 2023; Puny et al., 2023), The initial learning rate is chosen as 0.002 and is decayed by a factor of 0.5 once the MAE on the validation set plateaus for 20 epochs. Each model is trained for 500 epochs with a batch size of 128. We report the MAE for the model checkpoint with the best validation performance. We ran each experiment 10 times independently with different seeds and reported the average performance as well as the standard deviation.
+
+# I.4 PERFORMANCE OF BASELINE MODELS IN LITERATURE
+
+For completeness, in this subsection we give a comprehensive list of the performance of GNN models in the literature on ZINC and Alchemy datasets. The numbers in each table below are directly taken from the original papers.
+
+Table 8: Performance of different GNN models on ZINC dataset reported in the literature.  
+
+<table><tr><td>Method</td><td>Model</td><td>Reference</td><td>Test MAE</td></tr><tr><td rowspan="8">MPNN</td><td>GIN</td><td>(Xu et al., 2019)</td><td>0.526±0.051</td></tr><tr><td>GraphSAGE</td><td>(Hamilton et al., 2017)</td><td>0.398±0.002</td></tr><tr><td>GAT</td><td>(Veličković et al., 2018)</td><td>0.384±0.007</td></tr><tr><td>GCN</td><td>(Kipf &amp; Welling, 2017)</td><td>0.367±0.011</td></tr><tr><td>MoNet</td><td>(Monti et al., 2017)</td><td>0.292±0.006</td></tr><tr><td>GatedGCN-PE</td><td>(Bresson &amp; Laurent, 2017)</td><td>0.214±0.006</td></tr><tr><td>MPNN(sum)</td><td>(Gilmer et al., 2017)</td><td>0.145±0.007</td></tr><tr><td>PNA</td><td>(Corso et al., 2020)</td><td>0.142±0.010</td></tr><tr><td rowspan="4">Higher-order GNN</td><td>RingGNN</td><td>(Chen et al., 2019b)</td><td>0.353±0.019</td></tr><tr><td>PPGN</td><td>(Maron et al., 2019a)</td><td>0.303±0.068</td></tr><tr><td>PPGN</td><td>(Puny et al., 2023)</td><td>0.079±0.005</td></tr><tr><td>PPGN++</td><td>(Puny et al., 2023)</td><td>0.076±0.003</td></tr><tr><td rowspan="7">Subgraph GNN</td><td>NGNN</td><td>(Zhang &amp; Li, 2021)</td><td>0.111±0.003</td></tr><tr><td>GNN-AK</td><td>(Zhao et al., 2022a)</td><td>0.105±0.010</td></tr><tr><td>GNN-AK+</td><td>(Zhao et al., 2022a)</td><td>0.091±0.002</td></tr><tr><td>ESAN</td><td>(Bevilacqua et al., 2022)</td><td>0.102±0.003</td></tr><tr><td>SUN</td><td>(Frasca et al., 2022)</td><td>0.083±0.003</td></tr><tr><td>I2-GNN</td><td>(Huang et al., 2023)</td><td>0.083±0.001</td></tr><tr><td>ID-MPNN</td><td>(Zhou et al., 2023a)</td><td>0.083±0.003</td></tr><tr><td rowspan="4">Local (F)GNN</td><td>SetGNN</td><td>(Zhao et al., 2022b)</td><td>0.075±0.003</td></tr><tr><td>GNN-SSWL</td><td>(Zhang et al., 2023a)</td><td>0.082±0.010</td></tr><tr><td>GNN-SSWL+</td><td>(Zhang et al., 2023a)</td><td>0.070±0.005</td></tr><tr><td>N2-GNN</td><td>(Feng et al., 2023)</td><td>0.059±0.002</td></tr><tr><td rowspan="4">Substructure-based GNN</td><td>GSN</td><td>(Bouritsas et al., 2022)</td><td>0.101±0.010</td></tr><tr><td>CIN (Small)</td><td>(Bodnar et al., 2021a)</td><td>0.094±0.004</td></tr><tr><td>CIN</td><td>(Bodnar et al., 2021a)</td><td>0.079±0.006</td></tr><tr><td>CIN++</td><td>(Giusti et al., 2023)</td><td>0.077±0.004</td></tr><tr><td rowspan="6">Graph Transformer</td><td>SAN</td><td>(Kreuzer et al., 2021)</td><td>0.139±0.006</td></tr><tr><td>K-Subgraph SAT</td><td>(Chen et al., 2022)</td><td>0.094±0.008</td></tr><tr><td>Graphormer</td><td>(Ying et al., 2021)</td><td>0.122±0.006</td></tr><tr><td>URPE</td><td>(Luo et al., 2022)</td><td>0.086±0.007</td></tr><tr><td>Graphormer-GD</td><td>(Zhang et al., 2023b)</td><td>0.081±0.009</td></tr><tr><td>GPS</td><td>(Rampasek et al., 2022)</td><td>0.070±0.004</td></tr><tr><td rowspan="6">Other</td><td>PF-GNN</td><td>(Dupty et al., 2021)</td><td>0.122±0.010</td></tr><tr><td>KP-GIN</td><td>(Feng et al., 2022)</td><td>0.093±0.007</td></tr><tr><td>SignNet</td><td>(Lim et al., 2023)</td><td>0.084±0.006</td></tr><tr><td>PathNN</td><td>(Michel et al., 2023)</td><td>0.090±0.004</td></tr><tr><td>PPGN++(6)</td><td>(Puny et al., 2023)</td><td>0.071±0.001</td></tr><tr><td>PlanE</td><td>(Dimitrov et al., 2023)</td><td>0.076±0.003</td></tr></table>
+
+Table 9: Performance of different GNN models on Alchemy dataset reported in the literature.  
+
+<table><tr><td>Model</td><td>Reference</td><td>Test MAE</td></tr><tr><td>GIN</td><td>(Xu et al., 2019)</td><td>0.180±0.006</td></tr><tr><td>PF-GNN</td><td>(Dupty et al., 2021)</td><td>0.111±0.010</td></tr><tr><td>δ-2-GNN</td><td>(Morris et al., 2020)</td><td>0.118±0.001</td></tr><tr><td>Recon-GNN</td><td>(Cotta et al., 2021)</td><td>0.125±0.001</td></tr><tr><td>SpeqNet</td><td>(Morris et al., 2022)</td><td>0.115±0.001</td></tr><tr><td>SignNet</td><td>(Lim et al., 2023)</td><td>0.113±0.002</td></tr><tr><td>PPGN</td><td>(Puny et al., 2023)</td><td>0.113±0.001</td></tr><tr><td>PPGN++</td><td>(Puny et al., 2023)</td><td>0.111±0.002</td></tr><tr><td>PPGN++(6)</td><td>(Puny et al., 2023)</td><td>0.109±0.001</td></tr></table>
\ No newline at end of file
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+                                    "content": "Designing expressive Graph Neural Networks (GNNs) is a fundamental topic in the graph learning community. So far, GNN expressiveness has been primarily assessed via the Weisfeiler-Lehman (WL) hierarchy. However, such an expressivity measure has notable limitations: it is inherently coarse, qualitative, and may not well reflect practical requirements (e.g., the ability to encode substructures). In this paper, we introduce a novel framework for quantitatively studying the expressiveness of GNN architectures, addressing all the above limitations. Specifically, we identify a fundamental expressivity measure termed homomorphism expressivity, which quantifies the ability of GNN models to count graphs under homomorphism. Homomorphism expressivity offers a complete and practical assessment tool: the completeness enables direct expressivity comparisons between GNN models, while the practicality allows for understanding concrete GNN abilities such as subgraph counting. By examining four classes of prominent GNNs as case studies, we derive simple, unified, and elegant descriptions of their homomorphism expressivity for both invariant and equivariant settings. Our results provide novel insights into a series of previous work, unify the landscape of different subareas in the community, and settle several open questions. Empirically, extensive experiments on both synthetic and real-world tasks verify our theory, showing that the practical performance of GNN models aligns well with the proposed metric."
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+                                    "content": "1 INTRODUCTION"
+                                }
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+                                    "content": "Owing to the ubiquity of graph-structured data in numerous applications, Graph Neural Networks (GNNs) have achieved enormous success in the field of machine learning over the past few years. However, one of the most prominent drawbacks of popular GNNs lies in the limited expressive power. In particular, Morris et al. (2019); Xu et al. (2019) showed that Message Passing GNNs (MPNNs) are intrinsically bounded by the 1-dimensional Weisfeiler-Lehman test (1-WL) in distinguishing non-isomorphic graphs (Weisfeiler & Lehman, 1968). Since then, the Weisfeiler-Lehman hierarchy has become a yardstick to measure the expressiveness and guide designing more powerful GNN architectures (see Appendix A.1 for an overview of representative approaches in this area)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
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+                                    "content": "However, as more and more architectures have been proposed, the limitations of the WL hierarchy are becoming increasingly evident. First, the WL hierarchy is arguably too coarse to evaluate the expressive power of practical GNN models (Morris et al., 2022; Puny et al., 2023). On one hand, architectures inspired by higher-order WL tests (Maron et al., 2019b;a; Morris et al., 2019) often suffer from substantial computation/memory costs. On the other hand, most practical and efficient GNNs are only proved to be strictly more expressive than 1-WL by leveraging toy example graphs (e.g., Zhang & Li, 2021; Bevilacqua et al., 2022; Wijesinghe & Wang, 2022a). Such a qualitative characterization may provide little insight into the models' true expressiveness. Besides, the expressive power brought from the WL hierarchy often does not align well with the one required in practice (Veličković, 2022). Hence, how to study the expressiveness of GNN models in a quantitative, systematic, and practical way remains a central research direction for the GNN community."
+                                }
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+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "type": "text",
+                                    "content": "*Equal technical contributions."
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+                                    "type": "text",
+                                    "content": " Project lead."
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+                                    "content": "To address the above limitations, this paper takes a different approach by studying GNN expressivity from the following practical angle: What structural information can a GNN model encode? Since the ability to detect/count graph substructures is crucial in various real-world applications (Chen et al., 2020; Huang et al., 2023; Tahmasebi et al., 2023), many expressive GNNs have been proposed based on preprocessing substructure information (Bouritsas et al., 2022; Barceló et al., 2021; Bodnar et al., 2021b;a). However, instead of augmenting GNNs by manually preprocessed (task-specific) substructures, it is nowadays more desirable to design generic, domain-agnostic GNNs that can end-to-end learn different structural information suitable for diverse applications. This naturally gives rise to the fundamental question of characterizing the complete set of substructures prevalent GNN models can encode. Unfortunately, this problem is widely recognized as challenging even when examining simple structures like cycles (Fürer, 2017; Arvind et al., 2020; Huang et al., 2023)."
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+                                    "type": "text",
+                                    "content": "Our contributions. Motivated by GNNs' ability to encode substructures, this paper presents a novel framework for quantitatively analyzing the expressive power of GNN models. Our approach is rooted in a critical discovery: given a GNN model "
+                                },
+                                {
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+                                    "content": ", the model's output representation for any graph "
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+                                    "type": "text",
+                                    "content": " can be fully determined by the structural information of "
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+                                    "type": "text",
+                                    "content": " over some pattern family "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ", where "
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+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " corresponds to precisely all (and only) those substructures that can be \"coded\" by model "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ". In this way, the set "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " can be naturally viewed as an expressivity description of "
+                                },
+                                {
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+                                    "type": "inline_equation",
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ": after identifying "
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+                                {
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+                                    "type": "inline_equation",
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " for each model "
+                                },
+                                {
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+                                    "type": "inline_equation",
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+                                    ],
+                                    "type": "text",
+                                    "content": ", the expressivity of different models can then be qualitatively/quantitatively compared by simply looking at their set inclusion relation and set difference."
+                                }
+                            ]
+                        }
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+                                    "type": "text",
+                                    "content": "The crux here is to define an appropriate notion of \"encodability\" so that "
+                                },
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+                                    "type": "text",
+                                    "content": " can admit a simple description. We identify that a good candidate is the homomorphism expressivity: i.e., "
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": " consists of all substructures that can be counted by model "
+                                },
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+                                    "type": "inline_equation",
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " under homomorphism (see Section 2 for a formal definition). Homomorphism is a foundational concept in graph theory (Lovasz, 2012) and is linked to many important topics such as graph coloring, graph matching, and subgraph counting. With this concept, we are able to give complete, unified, and surprisingly elegant descriptions of the pattern family "
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": " for a wide range of mainstream GNN architectures listed below:"
+                                }
+                            ]
+                        }
+                    ],
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+                {
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+                                            "type": "text",
+                                            "content": "- MPNN (e.g., Gilmer et al., 2017; Hamilton et al., 2017; Kipf & Welling, 2017; Xu et al., 2019);"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 4
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+                                            "type": "text",
+                                            "content": "- Subgraph GNN (You et al., 2021; Zhang & Li, 2021; Bevilacqua et al., 2022; Qian et al., 2022);"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 5
+                        },
+                        {
+                            "bbox": [
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+                                            "bbox": [
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+                                                414,
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+                                            "type": "text",
+                                            "content": "- Local GNN (Morris et al., 2020; 2022; Zhang et al., 2023a; Frasca et al., 2022);"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 6
+                        },
+                        {
+                            "bbox": [
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+                                            "type": "text",
+                                            "content": "- Folklore-type GNN (Maron et al., 2019a; Zhang et al., 2023a; Feng et al., 2023)."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 7
+                        }
+                    ],
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+                                    "type": "text",
+                                    "content": "Technically, the descriptions are based on a novel application and extension of the concept of nested ear decomposition (NED) in graph theory (Eppstein, 1992). We prove that: (i) (necessity) each model "
+                                },
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+                                    "type": "text",
+                                    "content": " above can count (under homomorphism) a specific family of patterns "
+                                },
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+                                    "type": "text",
+                                    "content": ", characterized by a specific type of NED; (ii) (sufficiency) any pattern "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "F \\notin \\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " cannot be counted under homomorphism by model "
+                                },
+                                {
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+                                    "type": "inline_equation",
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "; (iii) (completeness) for any graph, information collected from the homomorphism count in pattern family "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " determines its representation computed by model "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ". Therefore, homomorphism expressivity is well-defined and is a complete expressivity measure for GNN models."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
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+                                    "type": "text",
+                                    "content": "Our theory can be generalized in various aspects. One significant extension is the node-level and edge-level expressivity for equivariant GNNs (Azizian & Lelarge, 2021; Geerts & Reutter, 2022), which can be naturally tackled by a fine-grained analysis of NED. As another non-trivial generalization, we study higher-order GNN variants for several of the above architectures and derive results by defining higher-order NED. Both aspects demonstrate the flexibility of our proposed framework, suggesting it as a general recipe for analyzing future architectures."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
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+                                    "type": "text",
+                                    "content": "Implications. Homomorphism expressivity serves as a powerful toolbox for bridging different sub-areas in the GNN community, providing fresh understandings of a series of known results that were previously proved in complex ways, and answering a set of unresolved open problems. First, our results can readily establish a complete expressiveness hierarchy among all the aforementioned architectures and their higher-order extensions. This recovers and extends a number of results in Morris et al. (2020); Qian et al. (2022); Zhang et al. (2023a); Frasca et al. (2022) and answers their open problems (Section 4.1). In fact, our results go far beyond revealing the expressivity gap between models: we essentially answer how large the gap is and establish a systematic approach to constructing counterexample graphs. Second, based on the relation between homomorphism and subgraph count, we are able to characterize the subgraph counting power of GNN models for all patterns at graph, node, and edge levels, significantly advancing an open direction initiated in Fürer (2017); Arvind et al. (2020) (Section 4.2). As a special case, our results extend recent findings in"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                }
+            ],
+            "discarded_blocks": [
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "type": "text",
+                                    "content": "2"
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+                            ]
+                        }
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+                    "index": 12
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+                                    "type": "text",
+                                    "content": "Huang et al. (2023) about the cycle counting power of GNN models, highlighting that Local 2-GNN can already subgraph-count all cycles/paths within 7 nodes (even at edge-level). Third, our results provide a new toolbox for studying the polynomial expressivity proposed recently in Puny et al. (2023), extending it to various practical architectures and answering an open question (Section 4.3). Empirically, an extensive set of experiments verifies our theory, showing that the homomorphism expressivity of different models matches well with their practical performance in diverse tasks."
+                                }
+                            ]
+                        }
+                    ],
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+                                    "type": "text",
+                                    "content": "2 PRELIMINARY"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Notations. We use "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": " to denote sets and multisets, respectively. Given a (multi)set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": ", its cardinality is denoted as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "|S|"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": ". In this paper, we consider finite, undirected, vertex-labeled graphs with no self-loops or repeated edges. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G = (V_G, E_G, \\ell_G)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": " be a graph with vertex set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "V_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": ", edge set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "E_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": ", and label function "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": ", where each edge in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "E_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": " is a set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{u, v\\} \\subset V_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": " of cardinality two, and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell_G(u)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": " is the label of vertex "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": ". The rooted graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": " is a graph obtained from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": " by marking the special vertex "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\in V_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": "; we can similarly consider marking two special vertices "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v \\in V_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": " (denote by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^{uv}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": "). The neighbors of vertex "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": " is denoted as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "N_G(u) := \\{v \\in V_G : \\{u, v\\} \\in E_G\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": ". A graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F = (V_F, E_F, \\ell_F)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": " is a subgraph of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": " if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "V_F \\subset V_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "E_F \\subset E_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell_F(u) = \\ell_G(u)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": ". A simple path "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": " in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": " is an edge set of the form "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\{w_0, w_1\\}, \\dots, \\{w_{k-1}, w_k\\}\\} \\subset E_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_i \\neq w_j"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i \\neq j"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": ". Here, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": " are called endpoints of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        190,
+                                        506,
+                                        302
+                                    ],
+                                    "type": "text",
+                                    "content": " and other vertices are called internal points."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        306,
+                        505,
+                        406
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                306,
+                                505,
+                                406
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": "Homomorphism, isomorphism, and subgraph count. Given two graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": ", a homomorphism from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": " is a mapping "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f: V_F \\to V_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": " that preserves edges and labels, i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell_F(u) = \\ell_G(f(u))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{f(u), f(v)\\} \\in E_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{u, v\\} \\in E_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": ". When the mapping "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": " exists, we say "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": " is homomorphic to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": ". We denote by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{Hom}(F, G)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": " the set of all homomorphisms from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": " and define "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{hom}(F, G) = |\\operatorname{Hom}(F, G)|"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": ", which counts the number of homomorphisms for pattern "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": " in graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": ". If "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": " is further surjective on both vertices and edges, we call "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": " a homomorphic image of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": ". Denote by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{Spasm}(F)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": " the set of all homomorphic images of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": ", called the spasm of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": ". For rooted graphs, homomorphism should additionally preserve vertex marking: i.e., if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": " is a homomorphism from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F^{uv}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^{xy}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": ", then "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f(u) = x"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f(v) = y"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        505,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        410,
+                        505,
+                        456
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                410,
+                                505,
+                                456
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        410,
+                                        505,
+                                        456
+                                    ],
+                                    "type": "text",
+                                    "content": "A mapping "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        410,
+                                        505,
+                                        456
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f: V_F \\to V_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        410,
+                                        505,
+                                        456
+                                    ],
+                                    "type": "text",
+                                    "content": " is called an isomorphism if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        410,
+                                        505,
+                                        456
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        410,
+                                        505,
+                                        456
+                                    ],
+                                    "type": "text",
+                                    "content": " is a bijection and both "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        410,
+                                        505,
+                                        456
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        410,
+                                        505,
+                                        456
+                                    ],
+                                    "type": "text",
+                                    "content": " and its inverse "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        410,
+                                        505,
+                                        456
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f^{-1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        410,
+                                        505,
+                                        456
+                                    ],
+                                    "type": "text",
+                                    "content": " are homomorphisms. We denote by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        410,
+                                        505,
+                                        456
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{Sub}(F, G)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        410,
+                                        505,
+                                        456
+                                    ],
+                                    "type": "text",
+                                    "content": " the set of all subgraphs of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        410,
+                                        505,
+                                        456
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        410,
+                                        505,
+                                        456
+                                    ],
+                                    "type": "text",
+                                    "content": " isomorphic to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        410,
+                                        505,
+                                        456
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        410,
+                                        505,
+                                        456
+                                    ],
+                                    "type": "text",
+                                    "content": " and define "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        410,
+                                        505,
+                                        456
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{sub}(F, G) = |\\operatorname{Sub}(F, G)|"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        410,
+                                        505,
+                                        456
+                                    ],
+                                    "type": "text",
+                                    "content": ", which counts the number of patterns "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        410,
+                                        505,
+                                        456
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        410,
+                                        505,
+                                        456
+                                    ],
+                                    "type": "text",
+                                    "content": " occurred in graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        410,
+                                        505,
+                                        456
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        410,
+                                        505,
+                                        456
+                                    ],
+                                    "type": "text",
+                                    "content": " as a subgraph. We note that a similar definition holds for rooted graphs (e.g., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        410,
+                                        505,
+                                        456
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{sub}(F^{uv}, G^{xy})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        410,
+                                        505,
+                                        456
+                                    ],
+                                    "type": "text",
+                                    "content": ")."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        460,
+                        505,
+                        527
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                460,
+                                505,
+                                527
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        460,
+                                        505,
+                                        527
+                                    ],
+                                    "type": "text",
+                                    "content": "Graph neural networks. GNNs can be generally described as graph functions that are invariant under isomorphism. To achieve such invariance, most popular GNN models follow a color refinement (CR) paradigm: they maintain a feature representation (color) for each vertex or vertex tuples and iteratively refine these features through equivariant aggregation layers. Finally, there is a global pooling layer to merge all features and obtain the graph representation. Below, we separately define the corresponding CR algorithms for four mainstream classes of GNNs studied in this paper."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        120,
+                        529,
+                        504,
+                        567
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                120,
+                                529,
+                                504,
+                                567
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        120,
+                                        529,
+                                        504,
+                                        567
+                                    ],
+                                    "type": "text",
+                                    "content": "- MPNN. Given a graph "
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        529,
+                                        504,
+                                        567
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        529,
+                                        504,
+                                        567
+                                    ],
+                                    "type": "text",
+                                    "content": ", MPNN maintains a color "
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        529,
+                                        504,
+                                        567
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^{\\mathrm{MP}}(u)"
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        529,
+                                        504,
+                                        567
+                                    ],
+                                    "type": "text",
+                                    "content": " for each vertex "
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        529,
+                                        504,
+                                        567
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\in V_G"
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        529,
+                                        504,
+                                        567
+                                    ],
+                                    "type": "text",
+                                    "content": ". Initially, the color only depends on the vertex label, i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        529,
+                                        504,
+                                        567
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^{\\mathrm{MP},(0)}(u) = \\ell_G(u)"
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        529,
+                                        504,
+                                        567
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, in each iteration, the color is refined by the following update formula (where hash is a perfect hash function):"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        184,
+                        567,
+                        504,
+                        587
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                184,
+                                567,
+                                504,
+                                587
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        184,
+                                        567,
+                                        504,
+                                        587
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\chi_ {G} ^ {\\mathsf {M P}, (t + 1)} (u) = \\operatorname {h a s h} \\left(\\chi_ {G} ^ {\\mathsf {M P}, (t)} (u), \\{\\{\\chi_ {G} ^ {\\mathsf {M P}, (t)} (v): v \\in N _ {G} (u) \\} \\}\\right). \\tag {1}",
+                                    "image_path": "00953a677972111ecd2c6bec64ab21fc86e79c42e5903dfae0749c8d90cd9b8a.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        129,
+                        589,
+                        505,
+                        624
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                589,
+                                505,
+                                624
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        589,
+                                        505,
+                                        624
+                                    ],
+                                    "type": "text",
+                                    "content": "After a sufficient number of iterations, the colors become stable. We denote by "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        589,
+                                        505,
+                                        624
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^{\\mathrm{MP}}(u)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        589,
+                                        505,
+                                        624
+                                    ],
+                                    "type": "text",
+                                    "content": " the stable color of "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        589,
+                                        505,
+                                        624
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        589,
+                                        505,
+                                        624
+                                    ],
+                                    "type": "text",
+                                    "content": ", which is also the node feature of "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        589,
+                                        505,
+                                        624
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        589,
+                                        505,
+                                        624
+                                    ],
+                                    "type": "text",
+                                    "content": " computed by the MPNN. The graph representation is defined as the multiset of node colors, i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        589,
+                                        505,
+                                        624
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^{\\mathrm{MP}}(G) = \\{\\{\\chi_G^{\\mathrm{MP}}(u): u \\in V_G\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        589,
+                                        505,
+                                        624
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        122,
+                        624,
+                        504,
+                        672
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                122,
+                                624,
+                                504,
+                                672
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        122,
+                                        624,
+                                        504,
+                                        672
+                                    ],
+                                    "type": "text",
+                                    "content": "- Subgraph GNN. It treats a graph "
+                                },
+                                {
+                                    "bbox": [
+                                        122,
+                                        624,
+                                        504,
+                                        672
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        122,
+                                        624,
+                                        504,
+                                        672
+                                    ],
+                                    "type": "text",
+                                    "content": " as a set of subgraphs "
+                                },
+                                {
+                                    "bbox": [
+                                        122,
+                                        624,
+                                        504,
+                                        672
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\{G^u : u \\in V_G\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        122,
+                                        624,
+                                        504,
+                                        672
+                                    ],
+                                    "type": "text",
+                                    "content": ", each obtained from "
+                                },
+                                {
+                                    "bbox": [
+                                        122,
+                                        624,
+                                        504,
+                                        672
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        122,
+                                        624,
+                                        504,
+                                        672
+                                    ],
+                                    "type": "text",
+                                    "content": " by marking a special vertex "
+                                },
+                                {
+                                    "bbox": [
+                                        122,
+                                        624,
+                                        504,
+                                        672
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\in V_G"
+                                },
+                                {
+                                    "bbox": [
+                                        122,
+                                        624,
+                                        504,
+                                        672
+                                    ],
+                                    "type": "text",
+                                    "content": ". Subgraph GNN maintains a color "
+                                },
+                                {
+                                    "bbox": [
+                                        122,
+                                        624,
+                                        504,
+                                        672
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^{\\mathrm{Sub}}(u, v)"
+                                },
+                                {
+                                    "bbox": [
+                                        122,
+                                        624,
+                                        504,
+                                        672
+                                    ],
+                                    "type": "text",
+                                    "content": " for each vertex "
+                                },
+                                {
+                                    "bbox": [
+                                        122,
+                                        624,
+                                        504,
+                                        672
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v"
+                                },
+                                {
+                                    "bbox": [
+                                        122,
+                                        624,
+                                        504,
+                                        672
+                                    ],
+                                    "type": "text",
+                                    "content": " in graph "
+                                },
+                                {
+                                    "bbox": [
+                                        122,
+                                        624,
+                                        504,
+                                        672
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^u"
+                                },
+                                {
+                                    "bbox": [
+                                        122,
+                                        624,
+                                        504,
+                                        672
+                                    ],
+                                    "type": "text",
+                                    "content": ". Initially, "
+                                },
+                                {
+                                    "bbox": [
+                                        122,
+                                        624,
+                                        504,
+                                        672
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^{\\mathrm{Sub}, (0)}(u, v) = (\\ell_G(v), \\mathbb{I}[u = v])"
+                                },
+                                {
+                                    "bbox": [
+                                        122,
+                                        624,
+                                        504,
+                                        672
+                                    ],
+                                    "type": "text",
+                                    "content": ", where the latter term distinguishes the special mark. It then runs MPNNs independently on each graph "
+                                },
+                                {
+                                    "bbox": [
+                                        122,
+                                        624,
+                                        504,
+                                        672
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^u"
+                                },
+                                {
+                                    "bbox": [
+                                        122,
+                                        624,
+                                        504,
+                                        672
+                                    ],
+                                    "type": "text",
+                                    "content": ":"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        167,
+                        673,
+                        504,
+                        693
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                167,
+                                673,
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+                                693
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        167,
+                                        673,
+                                        504,
+                                        693
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\chi_ {G} ^ {\\text {S u b}, (t + 1)} (u, v) = \\operatorname {h a s h} \\left(\\chi_ {G} ^ {\\text {S u b}, (t)} (u, v), \\{\\{\\chi_ {G} ^ {\\text {S u b}, (t)} (u, w): w \\in N _ {G} (v) \\} \\}\\right). \\tag {2}",
+                                    "image_path": "4255a7e04aebc0264118f5001476017794249ca252626053218b5212bb1c3800.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        129,
+                        695,
+                        505,
+                        734
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                695,
+                                505,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        695,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": "Denote the stable color of "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        695,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u,v)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        695,
+                                        505,
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+                                    ],
+                                    "type": "text",
+                                    "content": " as "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        695,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^{\\mathrm{Sub}}(u,v)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        695,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": ". The node feature of "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        695,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        695,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": " computed by Subgraph GNN is defined by merging all colors in "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        695,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^{u}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        695,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": ", i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        695,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^{\\mathrm{Sub}}(u) \\coloneqq \\mathsf{hash}\\left(\\{\\chi_G^{\\mathrm{Sub}}(u,v):v\\in V_G\\}\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        695,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": ". Finally, the graph representation is defined as "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        695,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^{\\mathrm{Sub}}(G) = \\{\\{\\chi_G^{\\mathrm{Sub}}(u):u\\in V_G\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        695,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
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+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
+                                        105,
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+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
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+                    ],
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "3"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 2
+        },
+        {
+            "para_blocks": [
+                {
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+                    ],
+                    "type": "text",
+                    "angle": 0,
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+                        {
+                            "bbox": [
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+                                {
+                                    "bbox": [
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+                                        127
+                                    ],
+                                    "type": "text",
+                                    "content": "- Local GNN. Inspired by the "
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        82,
+                                        504,
+                                        127
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        82,
+                                        504,
+                                        127
+                                    ],
+                                    "type": "text",
+                                    "content": "-WL test (Grohe, 2017), Local "
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        82,
+                                        504,
+                                        127
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        82,
+                                        504,
+                                        127
+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN is defined by replacing all global aggregations in "
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        82,
+                                        504,
+                                        127
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        82,
+                                        504,
+                                        127
+                                    ],
+                                    "type": "text",
+                                    "content": "-WL by sparse ones that only aggregate local neighbors, yielding a much more efficient CR algorithm. As an example, the iteration of Local 2-GNN has the following form and enjoys the same computational complexity as a Subgraph GNN."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        130,
+                        128,
+                        504,
+                        158
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                130,
+                                128,
+                                504,
+                                158
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        130,
+                                        128,
+                                        504,
+                                        158
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\chi_ {G} ^ {\\mathrm {L}, (t + 1)} (u, v) = \\operatorname {h a s h} \\left(\\chi_ {G} ^ {\\mathrm {L}, (t)} (u, v), \\{\\{\\chi_ {G} ^ {\\mathrm {L}, (t)} (w, v): w \\in N _ {G} (u) \\} \\}, \\{\\{\\chi_ {G} ^ {\\mathrm {L}, (t)} (u, w): w \\in N _ {G} (v) \\} \\}\\right). \\tag {3}",
+                                    "image_path": "9bc4f5449f79d13d1e38c6b9a39f360dc75ac1113c288559c3c484d53d3b9053.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        127,
+                        158,
+                        504,
+                        196
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                127,
+                                158,
+                                504,
+                                196
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        127,
+                                        158,
+                                        504,
+                                        196
+                                    ],
+                                    "type": "text",
+                                    "content": "Initially, "
+                                },
+                                {
+                                    "bbox": [
+                                        127,
+                                        158,
+                                        504,
+                                        196
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^{\\mathsf{L},(0)}(u,v) = (\\ell_G(u),\\ell_G(v),\\mathbb{I}[u = v],\\mathbb{I}[\\{u,v\\} \\in E_G])"
+                                },
+                                {
+                                    "bbox": [
+                                        127,
+                                        158,
+                                        504,
+                                        196
+                                    ],
+                                    "type": "text",
+                                    "content": ", which is called the isomorphism type of vertex pair "
+                                },
+                                {
+                                    "bbox": [
+                                        127,
+                                        158,
+                                        504,
+                                        196
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u,v)"
+                                },
+                                {
+                                    "bbox": [
+                                        127,
+                                        158,
+                                        504,
+                                        196
+                                    ],
+                                    "type": "text",
+                                    "content": ". We similarly denote the stable color as "
+                                },
+                                {
+                                    "bbox": [
+                                        127,
+                                        158,
+                                        504,
+                                        196
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^{\\mathsf{L}}(u,v)"
+                                },
+                                {
+                                    "bbox": [
+                                        127,
+                                        158,
+                                        504,
+                                        196
+                                    ],
+                                    "type": "text",
+                                    "content": " and define the node feature "
+                                },
+                                {
+                                    "bbox": [
+                                        127,
+                                        158,
+                                        504,
+                                        196
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^{\\mathsf{L}}(u)"
+                                },
+                                {
+                                    "bbox": [
+                                        127,
+                                        158,
+                                        504,
+                                        196
+                                    ],
+                                    "type": "text",
+                                    "content": " and graph representation "
+                                },
+                                {
+                                    "bbox": [
+                                        127,
+                                        158,
+                                        504,
+                                        196
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^{\\mathsf{L}}(G)"
+                                },
+                                {
+                                    "bbox": [
+                                        127,
+                                        158,
+                                        504,
+                                        196
+                                    ],
+                                    "type": "text",
+                                    "content": " as in the Subgraph GNN."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        121,
+                        196,
+                        504,
+                        219
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                196,
+                                504,
+                                219
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        196,
+                                        504,
+                                        219
+                                    ],
+                                    "type": "text",
+                                    "content": "- Folklore-type GNN. The Folklore GNN (FGNN) is inspired by the standard "
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        196,
+                                        504,
+                                        219
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        196,
+                                        504,
+                                        219
+                                    ],
+                                    "type": "text",
+                                    "content": "-FWL test (Cai et al., 1992). As an example, the iteration formula of 2-FGNN is written as follows:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        156,
+                        220,
+                        504,
+                        239
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                156,
+                                220,
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+                                239
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        156,
+                                        220,
+                                        504,
+                                        239
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\chi_ {G} ^ {\\mathsf {F}, (t + 1)} (u, v) = \\operatorname {h a s h} \\left(\\chi_ {G} ^ {\\mathsf {F}, (t)} (u, v), \\{\\left(\\chi_ {G} ^ {\\mathsf {F}, (t)} (w, v), \\chi_ {G} ^ {\\mathsf {F}, (t)} (u, w)\\right): w \\in V _ {G} \\} \\}\\right). \\tag {4}",
+                                    "image_path": "313bc0650c0e5bef592495efff66bcf1023ca395cb3a7c501918d02cdfed33fe.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        129,
+                        240,
+                        504,
+                        262
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                240,
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+                                262
+                            ],
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+                                {
+                                    "bbox": [
+                                        129,
+                                        240,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "text",
+                                    "content": "One can similarly consider the more efficient Local 2-FGNN by only aggregating local neighbors, which has the same computational complexity as Local 2-GNN and Subgraph GNN:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\chi_ {G} ^ {\\mathrm {L F}, (t + 1)} (u, v) = \\operatorname {h a s h} \\left(\\chi_ {G} ^ {\\mathrm {L F}, (t)} (u, v), \\{\\{\\left(\\chi_ {G} ^ {\\mathrm {L F}, (t)} (w, v), \\chi_ {G} ^ {\\mathrm {L F}, (t)} (u, w)\\right): w \\in N _ {G} (u) \\cup N _ {G} (v) \\} \\}\\right). \\\\ \\text {T h e s t a b l e c o l o r , n o d e f e a t u r e , a n d g r a p h r e p r e s e n t a t i o n c a n b e s i m a r l y d e f i n e d .} \\end{array} \\tag {5}",
+                                    "image_path": "2f499e5afa110c8258fe19f7aa3d4a2f01d436c4e22c6b1619a202032145c0fa.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
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+                                    "type": "text",
+                                    "content": "Finally, we note that the latter three types of GNNs can be naturally generalized into higher-order variants. We give a general definition of all these architectures in Appendix E.1. For the base case of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    "type": "inline_equation",
+                                    "content": "k = 1"
+                                },
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+                                    ],
+                                    "type": "text",
+                                    "content": ", Subgraph "
+                                },
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(k - 1)"
+                                },
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+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN, Local "
+                                },
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": "-GNN, and Local "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "-FGNN all reduce to the MPNN."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
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+                    "type": "title",
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+                                    "type": "text",
+                                    "content": "3 HOMOMORPHISM EXPRESSIVITY OF GRAPH NEURAL NETWORKS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
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+                                    "type": "text",
+                                    "content": "3.1 HOMOMORPHISM EXPRESSIVITY"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Given a GNN model "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "M"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " and a substructure "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "F"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ", we say "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " can count graph "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " under homomorphism if, for any graph "
+                                },
+                                {
+                                    "bbox": [
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+                                        450
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": ", the graph representation "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        394,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^M (G)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        394,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "text",
+                                    "content": " determines the homomorphism count "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        394,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{hom}(F,G)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        394,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "text",
+                                    "content": ". In other words, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        394,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^M (G) = \\chi_H^M (H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        394,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "text",
+                                    "content": " implies "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                        450
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{hom}(F,G) = \\mathrm{hom}(F,H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        394,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "text",
+                                    "content": " for any graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        394,
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+                                        450
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G,H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        394,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "text",
+                                    "content": ". The central question studied in this paper is, what substructures "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        394,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        394,
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+                                        450
+                                    ],
+                                    "type": "text",
+                                    "content": " can a GNN model "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        394,
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+                                        450
+                                    ],
+                                    "type": "text",
+                                    "content": " count under homomorphism? This gives rise to the notion of homomorphism expressivity defined below:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Definition 3.1. The homomorphism expressivity of a GNN model "
+                                },
+                                {
+                                    "bbox": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ", denoted by "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": ", is a family of (labeled) graphs satisfying the following conditions<sup>1</sup>:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
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+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": "a) For any two graphs "
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+                                            "bbox": [
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+                                            "type": "inline_equation",
+                                            "content": "G,H,\\chi_G^M (G) = \\chi_H^M (H)"
+                                        },
+                                        {
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+                                            "type": "text",
+                                            "content": " iff "
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+                                            "type": "inline_equation",
+                                            "content": "\\operatorname {hom}(F,G) = \\operatorname {hom}(F,H)"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": " for all "
+                                        },
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+                                            "type": "inline_equation",
+                                            "content": "F\\in \\mathcal{F}^{M}"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 13
+                        },
+                        {
+                            "bbox": [
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+                                            "type": "text",
+                                            "content": "b) "
+                                        },
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+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{F}^M"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": " is maximal, i.e., for any graph "
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F \\notin \\mathcal{F}^M"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                491,
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+                                            ],
+                                            "type": "text",
+                                            "content": ", there exists a pair of graphs "
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "G, H"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
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+                                                515
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\chi_G^M(G) = \\chi_H^M(H)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                491,
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+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
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+                                                515
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\operatorname{hom}(F, G) \\neq \\operatorname{hom}(F, H)"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 14
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
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+                    ],
+                    "type": "text",
+                    "angle": 0,
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Example 3.2. As a simple example, consider a maximally expressive GNN "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                        550
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        516,
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+                                        550
+                                    ],
+                                    "type": "text",
+                                    "content": " that can solve the graph isomorphism problem, i.e., it computes the same representation for two graphs iff they are isomorphic. Then, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        516,
+                                        504,
+                                        550
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        516,
+                                        504,
+                                        550
+                                    ],
+                                    "type": "text",
+                                    "content": " contains all graphs. This is a classic result proved in Lovász (1967)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
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+                        666
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "The significance of homomorphism expressivity can be justified in the following aspects. First, it is a complete expressivity measure. Based on item (a), the homomorphism count within "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        555,
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+                                        666
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        555,
+                                        505,
+                                        666
+                                    ],
+                                    "type": "text",
+                                    "content": " essentially captures all information embedded in the graph representation computed by model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        555,
+                                        505,
+                                        666
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        555,
+                                        505,
+                                        666
+                                    ],
+                                    "type": "text",
+                                    "content": ". This contrasts with previously studied metrics such as the ability to compute biconnectivity properties (Zhang et al., 2023b) or count cycles (Huang et al., 2023), which only reflects restricted aspects of expressivity. Second, homomorphism expressivity is a quantitative measure and is much finer than qualitative expressivity results obtained from the graph isomorphism test. Specifically, by item (a), a GNN model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        555,
+                                        505,
+                                        666
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M_1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        555,
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+                                    ],
+                                    "type": "text",
+                                    "content": " is more expressive than another model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        555,
+                                        505,
+                                        666
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M_2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        555,
+                                        505,
+                                        666
+                                    ],
+                                    "type": "text",
+                                    "content": " in distinguishing non-isomorphic graphs iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        555,
+                                        505,
+                                        666
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^{M_2} \\subset \\mathcal{F}^{M_1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        555,
+                                        505,
+                                        666
+                                    ],
+                                    "type": "text",
+                                    "content": ". Furthermore, by item (b), "
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+                                    "content": ", homomorphism expressivity provides a novel way to analyze and compare the expressivity of GNN models. In the next subsection, we will give exact characterizations of "
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+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "content": "<sup>1</sup>While homomorphism expressivity exists for all common GNNs such as the ones in Section 2, we note that it may not be well-defined for certain pathological GNNs. See Appendix F.1 for a deep discussion on it."
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+                                            "content": "Figure 1: Illustration of NED and its variants. The number "
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+                                            "content": ". Different colors represent different ears. See Figure 9 for more examples."
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+                                    "content": "To derive our main results, we leverage a concept in graph theory known as nested ear decomposition (NED), which is originally introduced in Eppstein (1992). Here, we adapt the definition as follows:"
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+                                    "content": "Definition 3.3. Given a graph "
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+                                    "type": "text",
+                                    "content": ", a NED "
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+                                    "type": "inline_equation",
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+                                    "type": "text",
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+                                    "content": " into a sequence of simple paths "
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+                                    "type": "text",
+                                    "content": " (called ears), which satisfies the following conditions:"
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+                                            "content": " with indices "
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+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
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+                                            ],
+                                            "type": "text",
+                                            "content": "- For each ear "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
+                                                505,
+                                                312
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_{j}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
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+                                            ],
+                                            "type": "text",
+                                            "content": " with index "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "j > c"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
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+                                            ],
+                                            "type": "text",
+                                            "content": ", there is an ear "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_{i}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
+                                                505,
+                                                312
+                                            ],
+                                            "type": "text",
+                                            "content": " with index "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
+                                                505,
+                                                312
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "1 \\leq i < j"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
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+                                            ],
+                                            "type": "text",
+                                            "content": " such that one or two endpoints of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_{j}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
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+                                            ],
+                                            "type": "text",
+                                            "content": " lie in ear "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_{i}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
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+                                            ],
+                                            "type": "text",
+                                            "content": " (we say "
+                                        },
+                                        {
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_{j}"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": " is nested on "
+                                        },
+                                        {
+                                            "bbox": [
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+                                                255,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_{i}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
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+                                                312
+                                            ],
+                                            "type": "text",
+                                            "content": "). Moreover, except for the endpoints lying in ear "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
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+                                                312
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_{i}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
+                                                505,
+                                                312
+                                            ],
+                                            "type": "text",
+                                            "content": ", no other vertices in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
+                                                505,
+                                                312
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_{j}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
+                                                505,
+                                                312
+                                            ],
+                                            "type": "text",
+                                            "content": " are in any previous ear "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
+                                                505,
+                                                312
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_{k}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
+                                                505,
+                                                312
+                                            ],
+                                            "type": "text",
+                                            "content": " for "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
+                                                505,
+                                                312
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "1 \\leq k < j"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
+                                                505,
+                                                312
+                                            ],
+                                            "type": "text",
+                                            "content": ". If both endpoints of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
+                                                505,
+                                                312
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_{j}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
+                                                505,
+                                                312
+                                            ],
+                                            "type": "text",
+                                            "content": " lie in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
+                                                505,
+                                                312
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_{i}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
+                                                505,
+                                                312
+                                            ],
+                                            "type": "text",
+                                            "content": ", the subpath in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
+                                                505,
+                                                312
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_{i}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
+                                                505,
+                                                312
+                                            ],
+                                            "type": "text",
+                                            "content": " that shares the endpoints of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
+                                                505,
+                                                312
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_{j}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
+                                                505,
+                                                312
+                                            ],
+                                            "type": "text",
+                                            "content": " is called the nested interval of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
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+                                                312
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_{j}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
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+                                                312
+                                            ],
+                                            "type": "text",
+                                            "content": " in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_{i}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
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+                                            ],
+                                            "type": "text",
+                                            "content": ", denoted as "
+                                        },
+                                        {
+                                            "bbox": [
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+                                                312
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "I(P_{j}) \\subset P_{i}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
+                                                505,
+                                                312
+                                            ],
+                                            "type": "text",
+                                            "content": ". If only one endpoint lies in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
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+                                                312
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_{i}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
+                                                505,
+                                                312
+                                            ],
+                                            "type": "text",
+                                            "content": ", define "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
+                                                505,
+                                                312
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "I(P_{j}) = \\emptyset"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                255,
+                                                505,
+                                                312
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 14
+                        },
+                        {
+                            "bbox": [
+                                121,
+                                312,
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+                                327
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                    ],
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+                                        {
+                                            "bbox": [
+                                                121,
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+                                            ],
+                                            "type": "text",
+                                            "content": "- For all ears "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                312,
+                                                477,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_{j}, P_{k}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                312,
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+                                            ],
+                                            "type": "text",
+                                            "content": " with "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "c < j < k \\leq m"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                312,
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+                                            ],
+                                            "type": "text",
+                                            "content": ", either "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "I(P_{j}) \\cap I(P_{k}) = \\emptyset"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
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+                                            ],
+                                            "type": "text",
+                                            "content": " or "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
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+                                                327
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "I(P_{j}) \\subset I(P_{k})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                312,
+                                                477,
+                                                327
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 15
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        104,
+                        330,
+                        504,
+                        364
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                330,
+                                504,
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+                                {
+                                    "bbox": [
+                                        104,
+                                        330,
+                                        504,
+                                        364
+                                    ],
+                                    "type": "text",
+                                    "content": "Intuitively, Definition 3.3 states that the relation between different ears forms a forest, in that each ear is nested on its parent. Moreover, the nested intervals either do not intersect or have inclusion relations for different children of the same parent ear. We give illustrations of NED in Figure 1."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        104,
+                        369,
+                        473,
+                        380
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                369,
+                                473,
+                                380
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        369,
+                                        473,
+                                        380
+                                    ],
+                                    "type": "text",
+                                    "content": "In this paper, we considerably extend the concept of NED to several variants defined below:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
+                        121,
+                        385,
+                        504,
+                        456
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 22,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                121,
+                                385,
+                                504,
+                                409
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        385,
+                                        504,
+                                        409
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                121,
+                                                385,
+                                                504,
+                                                409
+                                            ],
+                                            "type": "text",
+                                            "content": "- Endpoint-shared NED: a NED is called endpoint-shared if all ears with non-empty nested intervals share a common endpoint (see Figure 1(b,1))."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 19
+                        },
+                        {
+                            "bbox": [
+                                121,
+                                409,
+                                504,
+                                433
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        121,
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+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                121,
+                                                409,
+                                                504,
+                                                433
+                                            ],
+                                            "type": "text",
+                                            "content": "- **Strong NED:** a NED is called strong if for any two children "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                409,
+                                                504,
+                                                433
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_j"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                409,
+                                                504,
+                                                433
+                                            ],
+                                            "type": "text",
+                                            "content": ", "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                409,
+                                                504,
+                                                433
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_k"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                409,
+                                                504,
+                                                433
+                                            ],
+                                            "type": "text",
+                                            "content": " ("
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                409,
+                                                504,
+                                                433
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "j < k"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                409,
+                                                504,
+                                                433
+                                            ],
+                                            "type": "text",
+                                            "content": ") nested on the same parent ear, we have "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                409,
+                                                504,
+                                                433
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "I(P_j) \\subset I(P_k)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                409,
+                                                504,
+                                                433
+                                            ],
+                                            "type": "text",
+                                            "content": " (see Figure 1(b,2))."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 20
+                        },
+                        {
+                            "bbox": [
+                                121,
+                                433,
+                                504,
+                                456
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        433,
+                                        504,
+                                        456
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                121,
+                                                433,
+                                                504,
+                                                456
+                                            ],
+                                            "type": "text",
+                                            "content": "- Almost-strong NED: a NED is called almost-strong if for any children "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                433,
+                                                504,
+                                                456
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_j"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                433,
+                                                504,
+                                                456
+                                            ],
+                                            "type": "text",
+                                            "content": ", "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                433,
+                                                504,
+                                                456
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_k"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                433,
+                                                504,
+                                                456
+                                            ],
+                                            "type": "text",
+                                            "content": " ("
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                433,
+                                                504,
+                                                456
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "j < k"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                433,
+                                                504,
+                                                456
+                                            ],
+                                            "type": "text",
+                                            "content": ") nested on the same parent ear and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                433,
+                                                504,
+                                                456
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "|I(P_j)| > 1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                433,
+                                                504,
+                                                456
+                                            ],
+                                            "type": "text",
+                                            "content": ", we have "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                433,
+                                                504,
+                                                456
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "I(P_j) \\subset I(P_k)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                433,
+                                                504,
+                                                456
+                                            ],
+                                            "type": "text",
+                                            "content": " (see Figure 1(b,3))."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 21
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        105,
+                        461,
+                        291,
+                        472
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                461,
+                                291,
+                                472
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        461,
+                                        291,
+                                        472
+                                    ],
+                                    "type": "text",
+                                    "content": "We are now ready to present our main results:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 23
+                },
+                {
+                    "bbox": [
+                        104,
+                        474,
+                        504,
+                        497
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                                504,
+                                497
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        474,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem 3.4. For all GNN models "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        474,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        474,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "text",
+                                    "content": " defined in Section 2, the graph family "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        474,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        474,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfying Definition 3.1 exists (and is unique). Moreover, each "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        474,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        474,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "text",
+                                    "content": " can be separately described below:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 24
+                },
+                {
+                    "bbox": [
+                        121,
+                        498,
+                        392,
+                        562
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 30,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                121,
+                                498,
+                                280,
+                                510
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        498,
+                                        280,
+                                        510
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                121,
+                                                498,
+                                                280,
+                                                510
+                                            ],
+                                            "type": "text",
+                                            "content": "- MPNN: "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                498,
+                                                280,
+                                                510
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{F}^{\\mathrm{MP}} = \\{F:F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                498,
+                                                280,
+                                                510
+                                            ],
+                                            "type": "text",
+                                            "content": " is a forest\\};"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 25
+                        },
+                        {
+                            "bbox": [
+                                121,
+                                510,
+                                392,
+                                523
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        510,
+                                        392,
+                                        523
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                121,
+                                                510,
+                                                392,
+                                                523
+                                            ],
+                                            "type": "text",
+                                            "content": "- Subgraph GNN: "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                510,
+                                                392,
+                                                523
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{F}^{\\mathrm{Sub}} = \\{F:F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                510,
+                                                392,
+                                                523
+                                            ],
+                                            "type": "text",
+                                            "content": " has an endpoint-shared NED\\};"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 26
+                        },
+                        {
+                            "bbox": [
+                                121,
+                                524,
+                                332,
+                                536
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        524,
+                                        332,
+                                        536
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                121,
+                                                524,
+                                                332,
+                                                536
+                                            ],
+                                            "type": "text",
+                                            "content": "- Local 2-GNN: "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                524,
+                                                332,
+                                                536
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{F}^{\\mathsf{L}} = \\{F:F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                524,
+                                                332,
+                                                536
+                                            ],
+                                            "type": "text",
+                                            "content": " has a strong NED\\};"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 27
+                        },
+                        {
+                            "bbox": [
+                                121,
+                                536,
+                                376,
+                                549
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        536,
+                                        376,
+                                        549
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                121,
+                                                536,
+                                                376,
+                                                549
+                                            ],
+                                            "type": "text",
+                                            "content": "- Local 2-FGNN: "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                536,
+                                                376,
+                                                549
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{F}^{\\mathrm{LF}} = \\{F:F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                536,
+                                                376,
+                                                549
+                                            ],
+                                            "type": "text",
+                                            "content": " has an almost-strong NED\\};"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 28
+                        },
+                        {
+                            "bbox": [
+                                121,
+                                550,
+                                283,
+                                562
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        550,
+                                        283,
+                                        562
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                121,
+                                                550,
+                                                283,
+                                                562
+                                            ],
+                                            "type": "text",
+                                            "content": "- 2-FGNN: "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                550,
+                                                283,
+                                                562
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{F}^{\\mathrm{F}} = \\{F:F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                550,
+                                                283,
+                                                562
+                                            ],
+                                            "type": "text",
+                                            "content": " has a NED\\}."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 29
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        104,
+                        567,
+                        506,
+                        733
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Theorem 3.4 gives a unified description of the homomorphism expressivity for all popular GNN models defined in Section 2. Despite the elegant conclusion, the proof process is actually involved and represents a major technical contribution, so we present a proof sketch below. Our proof is divided into three parts, presented in Appendices C.2 to C.4. First, we show the existence of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        506,
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+                                    ],
+                                    "type": "text",
+                                    "content": " for each model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
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+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " based on a beautiful theory developed in Dell et al. (2018). Using the technique of unfolding tree, we prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        506,
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+                                    ],
+                                    "type": "text",
+                                    "content": " at least contains all graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
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+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " that allow a specific type of tree decomposition (Diestel, 2017), and the homomorphism information of these graphs determines the representation of any graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " computed by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " (i.e., Definition 3.1(a) holds). However, characterizing "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " in terms of tree decomposition is sophisticated and not intuitive for most models "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ". In the next step, we give an equivalent description of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " based on novel extensions of NED proposed in Definition 3.3, which is simpler and more elegant. In the last step, we prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " does not contain other graphs. This is achieved by building non-trivial relations between three distinct theoretical tools: tree decomposition, pebble game (Cai et al., 1992), and Fürer graph (Fürer, 2001). Through a fine-grained analysis of the Fürer graphs expanded by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F \\notin \\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        567,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " (see Theorems C.47 and C.53), we show they are precisely a pair of graphs satisfying Definition 3.1(b), thus concluding the proof."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 31
+                }
+            ],
+            "discarded_blocks": [
+                {
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+                    ],
+                    "type": "header",
+                    "angle": 0,
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
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+                                    "type": "text",
+                                    "content": "5"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 32
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 4
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
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+                        82,
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Discussions with Dvořák (2010); Dell et al. (2018). Our work significantly extends a beautiful theory developed in Dvořák (2010); Dell et al. (2018), which showed that a pair of graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        248
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        248
+                                    ],
+                                    "type": "text",
+                                    "content": " are indistinguishable by 1-WL iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        248
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{hom}(F, G) = \\operatorname{hom}(F, H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        248
+                                    ],
+                                    "type": "text",
+                                    "content": " for all trees "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        248
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        248
+                                    ],
+                                    "type": "text",
+                                    "content": ", and more generally, they are indistinguishable by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        248
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        248
+                                    ],
+                                    "type": "text",
+                                    "content": "-FWL iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        248
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{hom}(F, G) = \\operatorname{hom}(F, H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        248
+                                    ],
+                                    "type": "text",
+                                    "content": " for all graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        248
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        248
+                                    ],
+                                    "type": "text",
+                                    "content": " of bounded treewidth "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        248
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        248
+                                    ],
+                                    "type": "text",
+                                    "content": ". In this paper, we successfully generalize these results to a broad range of practical GNN models. Moreover, two distinct contributions are worth discussing. First, we highlight a key insight that homomorphism can serve as a fundamental expressivity measure, which has far-reaching consequences as will be elaborated in Section 4. To show that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        248
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        248
+                                    ],
+                                    "type": "text",
+                                    "content": " is a valid expressivity measure, we prove an extra and non-trivial result that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        248
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        248
+                                    ],
+                                    "type": "text",
+                                    "content": " is maximal (Definition 3.1(b)). Without this crucial property, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        248
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^{M_1} \\supsetneq \\mathcal{F}^{M_2}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        248
+                                    ],
+                                    "type": "text",
+                                    "content": " will not necessarily mean that model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        248
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M_1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        248
+                                    ],
+                                    "type": "text",
+                                    "content": " is strictly more expressive than "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        248
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M_2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        248
+                                    ],
+                                    "type": "text",
+                                    "content": ", thus preventing any quantitative comparison between models. Second, Dell et al. (2018) leveraged treewidth to describe results, which, unfortunately, cannot be applied to most GNN models studied here. Instead, we resort to the novel concept of NED, by which we successfully derive unified and elegant descriptions for all models. Moreover, as will be shown later, NED is quite flexible and can be naturally generalized to node/edge-level expressivity, which is not studied in prior work."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
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+                        504,
+                        286
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Finally, we remark that one can derive an equivalent (perhaps simpler) description of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        252,
+                                        504,
+                                        286
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^{\\mathrm{Sub}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        252,
+                                        504,
+                                        286
+                                    ],
+                                    "type": "text",
+                                    "content": ", based on the fact that a graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        252,
+                                        504,
+                                        286
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        252,
+                                        504,
+                                        286
+                                    ],
+                                    "type": "text",
+                                    "content": " has an endpoint-shared NED iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        252,
+                                        504,
+                                        286
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        252,
+                                        504,
+                                        286
+                                    ],
+                                    "type": "text",
+                                    "content": " becomes a forest when deleting the shared endpoint. Formally, denoting by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        252,
+                                        504,
+                                        286
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F \\backslash \\{u\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        252,
+                                        504,
+                                        286
+                                    ],
+                                    "type": "text",
+                                    "content": " the induced subgraph of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        252,
+                                        504,
+                                        286
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        252,
+                                        504,
+                                        286
+                                    ],
+                                    "type": "text",
+                                    "content": " over "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        252,
+                                        504,
+                                        286
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "V_F \\backslash \\{u\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        252,
+                                        504,
+                                        286
+                                    ],
+                                    "type": "text",
+                                    "content": ", we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        105,
+                        288,
+                        358,
+                        301
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                288,
+                                358,
+                                301
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        288,
+                                        358,
+                                        301
+                                    ],
+                                    "type": "text",
+                                    "content": "Corollary 3.5. "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        288,
+                                        358,
+                                        301
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^{\\mathrm{Sub}} = \\{F:\\exists u\\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        288,
+                                        358,
+                                        301
+                                    ],
+                                    "type": "text",
+                                    "content": " s.t. "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        288,
+                                        358,
+                                        301
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F\\backslash \\{u\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        288,
+                                        358,
+                                        301
+                                    ],
+                                    "type": "text",
+                                    "content": " is a forest}."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        105,
+                        311,
+                        347,
+                        322
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                311,
+                                347,
+                                322
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        311,
+                                        347,
+                                        322
+                                    ],
+                                    "type": "text",
+                                    "content": "3.3 EXTENDING TO NODE/EDGE-LEVEL EXPRESSIVITY"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        331,
+                        504,
+                        376
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                331,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        376
+                                    ],
+                                    "type": "text",
+                                    "content": "So far, this paper mainly focuses on the graph-level expressivity, i.e., what information is encoded in the graph representation. In this subsection, we extend all results in Theorem 3.4 to the more fine-grained node/edge-level expressivity by answering what information is encoded in the node/edge features of a GNN (i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        376
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^M(u)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        376
+                                    ],
+                                    "type": "text",
+                                    "content": " or "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        376
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^M(u,v)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        376
+                                    ],
+                                    "type": "text",
+                                    "content": " in Section 2). This yields the following definition:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        379,
+                        504,
+                        402
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        379,
+                                        504,
+                                        402
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition 3.6. The node-level homomorphism expressivity of a GNN model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        379,
+                                        504,
+                                        402
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        379,
+                                        504,
+                                        402
+                                    ],
+                                    "type": "text",
+                                    "content": ", denoted by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        379,
+                                        504,
+                                        402
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}_{\\mathrm{n}}^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        379,
+                                        504,
+                                        402
+                                    ],
+                                    "type": "text",
+                                    "content": ", is a family of connected rooted graphs satisfying the following conditions:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        115,
+                        403,
+                        506,
+                        454
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 9,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                116,
+                                403,
+                                506,
+                                428
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                403,
+                                                506,
+                                                428
+                                            ],
+                                            "type": "text",
+                                            "content": "a) For any connected graphs "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                403,
+                                                506,
+                                                428
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "G,H"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                403,
+                                                506,
+                                                428
+                                            ],
+                                            "type": "text",
+                                            "content": " and vertices "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                403,
+                                                506,
+                                                428
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u\\in V_G,v\\in V_H,\\chi_G^M (u) = \\chi_H^M (v)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                403,
+                                                506,
+                                                428
+                                            ],
+                                            "type": "text",
+                                            "content": " iff "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                403,
+                                                506,
+                                                428
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\operatorname {hom}(F^w,G^u) = \\operatorname {hom}(F^w,H^v)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                403,
+                                                506,
+                                                428
+                                            ],
+                                            "type": "text",
+                                            "content": " for all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                403,
+                                                506,
+                                                428
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F^w\\in \\mathcal{F}_{\\mathfrak{n}}^{M}"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 7
+                        },
+                        {
+                            "bbox": [
+                                115,
+                                430,
+                                504,
+                                454
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        115,
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+                                        454
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                115,
+                                                430,
+                                                504,
+                                                454
+                                            ],
+                                            "type": "text",
+                                            "content": "b) For any connected rooted graph "
+                                        },
+                                        {
+                                            "bbox": [
+                                                115,
+                                                430,
+                                                504,
+                                                454
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F^w \\notin \\mathcal{F}_{\\mathfrak{n}}^{M}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                115,
+                                                430,
+                                                504,
+                                                454
+                                            ],
+                                            "type": "text",
+                                            "content": ", there exists a pair of connected graphs "
+                                        },
+                                        {
+                                            "bbox": [
+                                                115,
+                                                430,
+                                                504,
+                                                454
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "G, H"
+                                        },
+                                        {
+                                            "bbox": [
+                                                115,
+                                                430,
+                                                504,
+                                                454
+                                            ],
+                                            "type": "text",
+                                            "content": " and vertices "
+                                        },
+                                        {
+                                            "bbox": [
+                                                115,
+                                                430,
+                                                504,
+                                                454
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u \\in V_G, v \\in V_H"
+                                        },
+                                        {
+                                            "bbox": [
+                                                115,
+                                                430,
+                                                504,
+                                                454
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                115,
+                                                430,
+                                                504,
+                                                454
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\chi_G^M(u) = \\chi_H^M(v)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                115,
+                                                430,
+                                                504,
+                                                454
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                115,
+                                                430,
+                                                504,
+                                                454
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\operatorname{hom}(F^w, G^u) \\neq \\operatorname{hom}(F^w, H^v)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                115,
+                                                430,
+                                                504,
+                                                454
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 8
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        104,
+                        457,
+                        504,
+                        492
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                457,
+                                504,
+                                492
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        457,
+                                        504,
+                                        492
+                                    ],
+                                    "type": "text",
+                                    "content": "One can similarly define the edge-level homomorphism expressivity "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        457,
+                                        504,
+                                        492
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}_{\\mathrm{e}}^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        457,
+                                        504,
+                                        492
+                                    ],
+                                    "type": "text",
+                                    "content": " to be a family of connected rooted graphs, each marking two special vertices (we omit the definition for clarity). The following result exactly characterizes "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        457,
+                                        504,
+                                        492
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}_{\\mathrm{n}}^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        457,
+                                        504,
+                                        492
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        457,
+                                        504,
+                                        492
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}_{\\mathrm{e}}^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        457,
+                                        504,
+                                        492
+                                    ],
+                                    "type": "text",
+                                    "content": " for all models "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        457,
+                                        504,
+                                        492
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        457,
+                                        504,
+                                        492
+                                    ],
+                                    "type": "text",
+                                    "content": " considered in this paper:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        105,
+                        494,
+                        504,
+                        506
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                494,
+                                504,
+                                506
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        494,
+                                        504,
+                                        506
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem 3.7. For all model "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        494,
+                                        504,
+                                        506
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        494,
+                                        504,
+                                        506
+                                    ],
+                                    "type": "text",
+                                    "content": " defined in Section 2, "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        494,
+                                        504,
+                                        506
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}_{\\mathfrak{n}}^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        494,
+                                        504,
+                                        506
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        494,
+                                        504,
+                                        506
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}_{\\mathrm{e}}^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        494,
+                                        504,
+                                        506
+                                    ],
+                                    "type": "text",
+                                    "content": " (except MPNN) exist. Moreover,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        121,
+                        508,
+                        279,
+                        535
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 14,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                121,
+                                508,
+                                279,
+                                521
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        508,
+                                        279,
+                                        521
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                121,
+                                                508,
+                                                279,
+                                                521
+                                            ],
+                                            "type": "text",
+                                            "content": "- MPNN: "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                508,
+                                                279,
+                                                521
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{F}_{\\mathfrak{n}}^{\\mathsf{MP}} = \\{F^{w}: F \\text{ is a tree}\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                508,
+                                                279,
+                                                521
+                                            ],
+                                            "type": "text",
+                                            "content": ";"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 12
+                        },
+                        {
+                            "bbox": [
+                                121,
+                                524,
+                                201,
+                                535
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        524,
+                                        201,
+                                        535
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                121,
+                                                524,
+                                                201,
+                                                535
+                                            ],
+                                            "type": "text",
+                                            "content": "- Subgraph GNN:"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 13
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        129,
+                        534,
+                        471,
+                        559
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                534,
+                                471,
+                                559
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        534,
+                                        471,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}_{\\mathtt{n}}^{\\mathtt{Sub}} = \\{F^{w}:F"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        534,
+                                        471,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": " has a NED with shared endpoint "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        534,
+                                        471,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w\\} = \\{F^{w}:F\\backslash \\{w\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        534,
+                                        471,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": " is a forest}, "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        534,
+                                        471,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}_{\\mathtt{e}}^{\\mathtt{Sub}} = \\{F^{wx}:F"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        534,
+                                        471,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": " has a NED with shared endpoint "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        534,
+                                        471,
+                                        559
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w\\} = \\{F^{wx}:F\\backslash \\{w\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        534,
+                                        471,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": " is a forest};"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        121,
+                        560,
+                        444,
+                        584
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                121,
+                                560,
+                                444,
+                                584
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        560,
+                                        444,
+                                        584
+                                    ],
+                                    "type": "text",
+                                    "content": "- 2-FGNN: \\(\\mathcal{F}_{\\mathfrak{n}}^{\\mathsf{F}} = \\{F^{w}: F\\) has a NED where \\(w\\) is an endpoint of the first ear}\\(, \\)\\mathcal{F}_{\\mathrm{e}}^{\\mathrm{F}} = \\{F^{wx}: F\\) has a NED where \\(w\\) and \\(x\\) are endpoints of the first ear\\}$."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        105,
+                        586,
+                        504,
+                        609
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                586,
+                                504,
+                                609
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        586,
+                                        504,
+                                        609
+                                    ],
+                                    "type": "text",
+                                    "content": "The cases of Local 2-GNN and Local 2-FGNN are similar to 2-FGNN by replacing \"NED\" with \"strong NED\" and \"almost-strong NED\", respectively."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        104,
+                        613,
+                        504,
+                        636
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                613,
+                                504,
+                                636
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        613,
+                                        504,
+                                        636
+                                    ],
+                                    "type": "text",
+                                    "content": "In summary, the node/edge-level homomorphism expressivity can be naturally described using NED by further specifying the endpoints of the first ear."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
+                        105,
+                        647,
+                        298,
+                        658
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                        658
+                                    ],
+                                    "type": "text",
+                                    "content": "3.4 EXTENDING TO HIGHER-ORDER GNNS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                },
+                {
+                    "bbox": [
+                        104,
+                        665,
+                        504,
+                        733
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                665,
+                                504,
+                                733
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        665,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "Finally, we discuss how our results can be naturally extended to higher-order GNNs, thus providing a complete picture of the homomorphism expressivity hierarchy for infinitely many architectures. We focus on three representative examples: Subgraph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        665,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        665,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN (Qian et al., 2022), Local "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        665,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        665,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN (Morris et al., 2020), and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        665,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        665,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "-FGNN (Azizian & Lelarge, 2021). Subgraph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        665,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        665,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN extracts a graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        665,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        665,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " for each vertex "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        665,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        665,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "-tuple "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        665,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\in V_G^k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        665,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " and runs MPNNs independently, which recovers Subgraph GNN when "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        665,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k = 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        665,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ". As the reader may have guessed, the following result exactly parallels Corollary 3.5:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        302,
+                        751,
+                        309,
+                        760
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                302,
+                                751,
+                                309,
+                                760
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        302,
+                                        751,
+                                        309,
+                                        760
+                                    ],
+                                    "type": "text",
+                                    "content": "6"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 21
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 5
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        82,
+                        504,
+                        106
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                82,
+                                504,
+                                106
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem 3.8. The homomorphism expressivity of Subgraph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN exists and can be described as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^{\\mathrm{Sub}(k)} = \\{F : \\exists U \\subset V_F \\text{ s.t. } |U| \\leq k \\text{ and } F \\backslash U \\text{ is a forest}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        110,
+                        333,
+                        232
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                110,
+                                333,
+                                232
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        333,
+                                        232
+                                    ],
+                                    "type": "text",
+                                    "content": "We next turn to Local "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        333,
+                                        232
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        333,
+                                        232
+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN. To describe the result, we introduce a novel extension of Definition 3.3, called the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        333,
+                                        232
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        333,
+                                        232
+                                    ],
+                                    "type": "text",
+                                    "content": "-order ear. Intuitively, it is formed by a graph of no more than "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        333,
+                                        232
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        333,
+                                        232
+                                    ],
+                                    "type": "text",
+                                    "content": " vertices, plus "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        333,
+                                        232
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        333,
+                                        232
+                                    ],
+                                    "type": "text",
+                                    "content": " paths each linking to a vertex in the graph (see Figure 2(a) for an illustration). Note that a 2-order ear is exactly a simple path. Then, we can naturally define the nested \"interval\" (see the solid orange lines in Figure 2(b) for an illustration) and thus define the concept of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        333,
+                                        232
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        110,
+                                        333,
+                                        232
+                                    ],
+                                    "type": "text",
+                                    "content": "-order strong NED. Due to space limit, a formal definition is deferred to Definition E.3. We have the following main result:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "type": "image",
+                    "bbox": [
+                        342,
+                        110,
+                        503,
+                        174
+                    ],
+                    "blocks": [
+                        {
+                            "bbox": [
+                                342,
+                                110,
+                                503,
+                                174
+                            ],
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        342,
+                                        110,
+                                        503,
+                                        174
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                342,
+                                                110,
+                                                503,
+                                                174
+                                            ],
+                                            "type": "image",
+                                            "image_path": "963dfb2c3d1f660e84e4e3dfed0f1c537c67de224bfb1ed809069755040a7d30.jpg"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 3,
+                            "angle": 0,
+                            "type": "image_body"
+                        },
+                        {
+                            "bbox": [
+                                337,
+                                176,
+                                505,
+                                221
+                            ],
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        337,
+                                        176,
+                                        505,
+                                        221
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                337,
+                                                176,
+                                                505,
+                                                221
+                                            ],
+                                            "type": "text",
+                                            "content": "Figure 2: Illustration of higher-order ears. Each curve indicates a path (with possibly zero length) and each straight segment indicates an edge."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 4,
+                            "angle": 0,
+                            "type": "image_caption"
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        235,
+                        504,
+                        258
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                235,
+                                504,
+                                258
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        235,
+                                        504,
+                                        258
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem 3.9. The homomorphism expressivity of Local "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        235,
+                                        504,
+                                        258
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        235,
+                                        504,
+                                        258
+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN exists and can be described as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        235,
+                                        504,
+                                        258
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^{\\mathsf{L}(k)} = \\{F : F \\text{ has a } k \\text{-order strong NED}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        235,
+                                        504,
+                                        258
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        262,
+                        504,
+                        308
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                262,
+                                504,
+                                308
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        262,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": "Finally, let us consider the standard "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        262,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        262,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": "-FGNN (or equivalently, the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        262,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        262,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": "-FWL). Unfortunately, we cannot find a description of its homomorphism expressivity based on some form of higher-order NED; nevertheless, it is easy to describe the results using the notion of treewidth (see Definition C.2). Specifically, denoting "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        262,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{tw}(F)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        262,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": " to be the treewidth of graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        262,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        262,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": ", we have the following result:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        105,
+                        311,
+                        504,
+                        335
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                311,
+                                504,
+                                335
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        311,
+                                        504,
+                                        335
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem 3.10. The homomorphism expressivity of "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        311,
+                                        504,
+                                        335
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        311,
+                                        504,
+                                        335
+                                    ],
+                                    "type": "text",
+                                    "content": "-FGNN exists and can be described as "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        311,
+                                        504,
+                                        335
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^{\\mathsf{F}(k)} = \\{F : \\mathrm{tw}(F) \\leq k\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        311,
+                                        504,
+                                        335
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        104,
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+                                    "type": "text",
+                                    "content": "Interestingly, one can see that "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^{\\mathrm{Sub}(0)}"
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+                                {
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+                                    "type": "text",
+                                    "content": ", "
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+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^{\\mathrm{L}(1)}"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^{\\mathrm{F}(1)}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " all degenerate to the family of forests, which coincides with the fact that all these higher-order GNNs reduces to MPNN for the base case."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "title",
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+                                    "type": "text",
+                                    "content": "4 IMPLICATIONS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "The previous section has provided a complete description of the homomorphism expressivity for a variety of GNN models. In this section, we highlight the significance of these results through three different contexts. We will show how homomorphism expressivity can be used to link different GNN subareas, provide new insights into various known results, and answer a number of open problems."
+                                }
+                            ]
+                        }
+                    ],
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+                },
+                {
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+                                    "type": "text",
+                                    "content": "4.1 QUALITATIVE EXPRESSIVITY COMPARISON"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "One direct corollary of Theorem 3.4 is that it readily enables expressivity comparison among all models in Section 2. This can be summarized below:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Corollary 4.1. Under the notation of Theorem 3.4, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^{\\mathrm{MP}}\\subsetneq \\mathcal{F}^{\\mathrm{Sub}}\\subsetneq \\mathcal{F}^{\\mathrm{L}}\\subsetneq \\mathcal{F}^{\\mathrm{LF}}\\subsetneq \\mathcal{F}^{\\mathrm{F}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        513,
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+                                    ],
+                                    "type": "text",
+                                    "content": ". Thus, the expressive power of the following GNN models strictly increases in order (in terms of distinguishing non-isomorphic graphs): MPNN, Subgraph GNN, Local 2-GNN, Local 2-FGNN, and 2-FGNN."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
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+                    ],
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+                    "angle": 0,
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        553,
+                                        504,
+                                        609
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^{\\mathrm{MP}}\\subset \\mathcal{F}^{\\mathrm{Sub}}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " follows from Corollary 3.5 and the fact that deleting any vertex of a forest yields a forest. "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^{\\mathrm{Sub}}\\subset \\mathcal{F}^{\\mathrm{L}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        553,
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+                                    ],
+                                    "type": "text",
+                                    "content": " follows by the fact that any endpoint-shared NED is a strong NED. "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^{\\mathrm{L}}\\subset \\mathcal{F}^{\\mathrm{LF}}\\subset \\mathcal{F}^{\\mathrm{F}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " follows similarly since any strong NED is an almost-strong NED and any almost-strong NED is a NED. To prove strict separation results, one can check that the four graphs in Figure 1(b) precisely reveal the gap between each pair of graph families, thus concluding the proof."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
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+                    "angle": 0,
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Corollary 4.1 recovers a series of results recently proved in Zhang et al. (2023a); Frasca et al. (2022). Compared to their results, our approach draws a much clearer picture of the expressivity gap between different architectures and essentially answers how large the gaps are. Moreover, we provide systematic guidance for finding counterexample graphs that unveil the expressivity gap: as shown in Corollary C.54, any graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F \\in \\mathcal{F}^{M_2} \\backslash \\mathcal{F}^{M_1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        504,
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+                                    ],
+                                    "type": "text",
+                                    "content": " immediately gives a pair of non-isomorphic graphs that reveals the gap between models "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "M_1"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "M_2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": ". We note that this readily recovers the counterexamples constructed in Zhang et al. (2023a) and greatly simplifies their sophisticated case-by-case analysis."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
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+                                    "type": "text",
+                                    "content": "We next turn to three types of higher-order GNNs studied in Section 3.4, for which we can establish a complete expressiveness hierarchy, as presented in Corollary 4.2. A graphical illustration of these results is given in Figure 3."
+                                }
+                            ]
+                        }
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "content": "7"
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+                                            "type": "text",
+                                            "content": "Figure 3: Expressiveness hierarchy of MPNN, Subgraph GNN, Local GNN, and FGNN."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 2,
+                            "angle": 0,
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+                                    "type": "text",
+                                    "content": "Corollary 4.2. Under the notations in Section 3.4, for any "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "k > 0"
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+                                    "type": "text",
+                                    "content": ", the following hold:"
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+                                            "content": "a) "
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+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{F}^{\\mathrm{Sub}(k - 1)}\\subsetneq \\mathcal{F}^{\\mathrm{Sub}(k)}"
+                                        },
+                                        {
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+                                            "type": "text",
+                                            "content": " . I.e., the expressive power of Subgraph "
+                                        },
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+                                            "type": "inline_equation",
+                                            "content": "k"
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+                                            "type": "text",
+                                            "content": " GNN strictly increases with "
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+                                            "content": "b) "
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+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{F}^{\\mathsf{L}(k)}\\subsetneq \\mathcal{F}^{\\mathsf{L}(k + 1)}"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": " . I.e., the expressive power of Local "
+                                        },
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+                                            "content": "k"
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+                                            "type": "text",
+                                            "content": " -GNN strictly increases with "
+                                        },
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+                                            "content": "c) "
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+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{F}^{\\mathrm{Sub}(k)}\\subsetneq \\mathcal{F}^{\\mathrm{L}(k + 1)}"
+                                        },
+                                        {
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+                                            "type": "text",
+                                            "content": " . I.e., Local "
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+                                            "type": "inline_equation",
+                                            "content": "(k + 1)"
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+                                            "type": "text",
+                                            "content": " -GNN is strictly more expressive than Subgraph "
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+                                            "type": "text",
+                                            "content": " GNN;"
+                                        }
+                                    ]
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+                            "index": 6
+                        },
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+                                            "type": "text",
+                                            "content": "d) "
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+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{F}^{\\mathsf{F}(k)}\\subsetneq \\mathcal{F}^{\\mathsf{L}(k + 1)}\\subsetneq \\mathcal{F}^{\\mathsf{F}(k + 1)}"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": " . I.e., the expressive power of Local "
+                                        },
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+                                            "bbox": [
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(k + 1)"
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+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": " -GNN lies strictly between "
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "k"
+                                        },
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+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": " -FWL and "
+                                        },
+                                        {
+                                            "bbox": [
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+                                                207,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(k + 1)"
+                                        },
+                                        {
+                                            "bbox": [
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+                                                231
+                                            ],
+                                            "type": "text",
+                                            "content": " -FWL;"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 7
+                        },
+                        {
+                            "bbox": [
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+                                232,
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+                                266
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
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+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": "e) "
+                                        },
+                                        {
+                                            "bbox": [
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+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{F}^{\\mathrm{Sub}(k)}\\subsetneq \\mathcal{F}^{\\mathrm{F}(k + 1)}"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": ", and for all "
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+                                            "bbox": [
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+                                            "type": "inline_equation",
+                                            "content": "k > 1"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": ", "
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{F}^{\\mathrm{Sub}(k)}\\backslash \\mathcal{F}^{\\mathrm{F}(k + 1)}\\neq \\emptyset"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
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+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
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+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{F}^{\\mathrm{F}(k + 1)}\\backslash \\mathcal{F}^{\\mathrm{Sub}(k)}\\neq \\emptyset"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": ". In other words, the expressive power of Subgraph "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "k"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": "-GNN lies strictly within "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                232,
+                                                504,
+                                                266
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(k + 1)"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": "-FWL, but it is incomparable to "
+                                        },
+                                        {
+                                            "bbox": [
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+                                                266
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "k"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": "-FWL when "
+                                        },
+                                        {
+                                            "bbox": [
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+                                                232,
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+                                                266
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "k > 1"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 8
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
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+                    ],
+                    "type": "text",
+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "Corollary 4.2 recovers results in Morris et al. (2020); Qian et al. (2022) and further answers two open problems. First, Corollary 4.2(c) is a new result that bridges Morris et al. (2020) with Qian et al. (2022) and partially answers an open question in Zhang et al. (2023a, Appendix C). Another new result is Corollary 4.2(d), which essentially answers a fundamental open problem raised in Frasca et al. (2022, Appendix E), showing that their proposed "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{RelGN}(k)"
+                                },
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+                                    "type": "text",
+                                    "content": " model is bounded by "
+                                },
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+                                    "type": "inline_equation",
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "-FWL with an inherent expressivity gap (see Appendix E.4 for a detailed discussion). To sum up, all these challenging open problems become straightforward through the lens of homomorphism expressivity."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
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+                    ],
+                    "type": "title",
+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "4.2 SUBGRAPH COUNTING POWER"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
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+                    "angle": 0,
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "The significance of homomorphism expressivity can go much beyond qualitative comparisons between models. As another implication, it provides a systematic way to study GNNs' ability to encode structural information such as subgraph count, which has been found crucial in numerous practical applications. Specifically, a well-known result in graph theory states that, for any graphs "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "F, G"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ", the subgraph count "
+                                },
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+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{sub}(F, G)"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " can be determined by "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{hom}(\\tilde{F}, G)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        382,
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+                                        452
+                                    ],
+                                    "type": "text",
+                                    "content": " ranges over all homomorphic images of "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " (i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{Spasm}(F)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        382,
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+                                        452
+                                    ],
+                                    "type": "text",
+                                    "content": ", see Section 2) (Lovász, 2012; Curticapean et al., 2017)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Mathematically, given any graph "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ", let "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{Spasm}^{\\mathcal{Z}}(F)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        457,
+                                        504,
+                                        491
+                                    ],
+                                    "type": "text",
+                                    "content": " be any maximal set of pairwise non-isomorphic graphs chosen from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        457,
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+                                        491
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{Spasm}(F)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " (see Figure 4(a) for an illustration). Then, we have the following linear relation for all graph "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ":"
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+                            ]
+                        }
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\operatorname {s u b} (F, G) = \\sum_ {\\tilde {F} \\in \\operatorname {S p a s m} ^ {\\neq} (F)} \\alpha (F, \\tilde {F}) \\cdot \\operatorname {h o m} (\\tilde {F}, G), \\tag {6}",
+                                    "image_path": "f09cb0eac0734276c259f7a256ce039f9fccf8ef32f64bdf15747dc4ca3c1f77.jpg"
+                                }
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+                {
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+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\alpha (F,\\tilde{F})\\neq 0"
+                                },
+                                {
+                                    "bbox": [
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+                                        527,
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+                                    ],
+                                    "type": "text",
+                                    "content": " is a constant scalar coefficient independent of "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        527,
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+                                    ],
+                                    "type": "text",
+                                    "content": ". Based on this formula, we can easily study the subgraph counting power of GNN models as shown in Proposition 4.4."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
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+                    "type": "text",
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+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Definition 4.3. Given a GNN model "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        552,
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+                                    ],
+                                    "type": "text",
+                                    "content": ", we say "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
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+                                        609
+                                    ],
+                                    "type": "text",
+                                    "content": " can subgraph-count graph "
+                                },
+                                {
+                                    "bbox": [
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+                                        609
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        552,
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+                                        609
+                                    ],
+                                    "type": "text",
+                                    "content": " at graph-level if "
+                                },
+                                {
+                                    "bbox": [
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+                                        609
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^M (G) = \\chi_H^M (H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                        609
+                                    ],
+                                    "type": "text",
+                                    "content": " implies "
+                                },
+                                {
+                                    "bbox": [
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+                                        609
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname {sub}(F,G) = \\operatorname {sub}(F,H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        552,
+                                        504,
+                                        609
+                                    ],
+                                    "type": "text",
+                                    "content": " for any graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        552,
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+                                        609
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G,H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        552,
+                                        504,
+                                        609
+                                    ],
+                                    "type": "text",
+                                    "content": ". We say "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        552,
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+                                        609
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        552,
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+                                        609
+                                    ],
+                                    "type": "text",
+                                    "content": " can subgraphcount rooted graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                        609
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F^{w}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " at node-level if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        552,
+                                        504,
+                                        609
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^M (u) = \\chi_H^M (v)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        552,
+                                        504,
+                                        609
+                                    ],
+                                    "type": "text",
+                                    "content": " implies "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        552,
+                                        504,
+                                        609
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname {sub}(F^w,G^u) = \\operatorname {sub}(F^w,H^v)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        552,
+                                        504,
+                                        609
+                                    ],
+                                    "type": "text",
+                                    "content": " for any graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        552,
+                                        504,
+                                        609
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G,H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        552,
+                                        504,
+                                        609
+                                    ],
+                                    "type": "text",
+                                    "content": " and vertices "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        552,
+                                        504,
+                                        609
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u\\in V_{G},v\\in V_{H}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        552,
+                                        504,
+                                        609
+                                    ],
+                                    "type": "text",
+                                    "content": ". We can similarly define the edge-level subgraph counting ability for rooted graphs marking two special vertices."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        104,
+                        611,
+                        506,
+                        650
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                611,
+                                506,
+                                650
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
+                                        506,
+                                        650
+                                    ],
+                                    "type": "text",
+                                    "content": "Proposition 4.4. For any GNN model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
+                                        506,
+                                        650
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
+                                        506,
+                                        650
+                                    ],
+                                    "type": "text",
+                                    "content": " defined in Section 2, it can subgraph-count graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
+                                        506,
+                                        650
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
+                                        506,
+                                        650
+                                    ],
+                                    "type": "text",
+                                    "content": " (at graph-level) if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
+                                        506,
+                                        650
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F} \\in \\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
+                                        506,
+                                        650
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
+                                        506,
+                                        650
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F} \\in \\operatorname{Spasm}(F)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
+                                        506,
+                                        650
+                                    ],
+                                    "type": "text",
+                                    "content": ". It can subgraph-count "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
+                                        506,
+                                        650
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F^w"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
+                                        506,
+                                        650
+                                    ],
+                                    "type": "text",
+                                    "content": " (at node-level) if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
+                                        506,
+                                        650
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F}^w \\in \\mathcal{F}_{\\mathfrak{n}}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
+                                        506,
+                                        650
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
+                                        506,
+                                        650
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F}^w \\in \\operatorname{Spasm}(F^w)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        611,
+                                        506,
+                                        650
+                                    ],
+                                    "type": "text",
+                                    "content": ". A similar result holds for edge-level subgraph counting."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        104,
+                        653,
+                        505,
+                        733
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                653,
+                                505,
+                                733
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        653,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "The above proposition offers a simple way to affirm the ability of a GNN model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        653,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        653,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " to subgraph-count any pattern at graph/node/edge-level. On the other hand, one may wonder whether the converse direction also holds, i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        653,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        653,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " cannot subgraph-count "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        653,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        653,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " if there exists a homomorphic image "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        653,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F} \\in \\mathsf{Spasm}(F)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        653,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        653,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F} \\notin \\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        653,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ". We find that it is indeed the case. Specifically, if the set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        653,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{Spasm}(F) \\backslash \\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        653,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " is not empty, then one can always find a pair of counterexample graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        653,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G, H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        653,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        653,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^M(G) = \\chi_H^M(H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        653,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " but "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        653,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{sub}(F, G) \\neq \\mathsf{sub}(F, H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        653,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ". We eventually arrive at the following main theorem (see Appendix G.1 for a proof):"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        302,
+                        751,
+                        308,
+                        760
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                302,
+                                751,
+                                308,
+                                760
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        302,
+                                        751,
+                                        308,
+                                        760
+                                    ],
+                                    "type": "text",
+                                    "content": "8"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 7
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        82,
+                        504,
+                        118
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                82,
+                                504,
+                                118
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem 4.5. For any GNN model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "text",
+                                    "content": " such that their homomorphism expressivity "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "text",
+                                    "content": " exists, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "text",
+                                    "content": " can subgraph-count "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{Spasm}(F) \\subset \\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "text",
+                                    "content": ". Similar results hold for rooted graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F^u / F^{uv}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "text",
+                                    "content": " by replacing "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "text",
+                                    "content": " with node/edge-level homomorphism expressivity "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}_{\\mathfrak{n}}^M / \\mathcal{F}_{\\mathfrak{e}}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        118,
+                        504,
+                        163
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                118,
+                                504,
+                                163
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        118,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "text",
+                                    "content": "Example 4.6. As an example, we can readily characterize the cycle/path counting power of various GNNs. Denote by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        118,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_n / P_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        118,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "text",
+                                    "content": " the simple cycle/path of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        118,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        118,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "text",
+                                    "content": " vertices. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        118,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{u, v\\} \\in E_{C_n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        118,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "text",
+                                    "content": " be any edge in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        118,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "C_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        118,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        118,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{w, x\\} \\in E_{P_n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        118,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "text",
+                                    "content": " be any edge in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        118,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        118,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        118,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        118,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "text",
+                                    "content": " is an endpoint of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        118,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        118,
+                                        504,
+                                        163
+                                    ],
+                                    "type": "text",
+                                    "content": ". The following table lists exactly all cycles/paths each model can count at graph/node/edge-level."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "type": "table",
+                    "bbox": [
+                        106,
+                        167,
+                        324,
+                        243
+                    ],
+                    "blocks": [
+                        {
+                            "bbox": [
+                                106,
+                                167,
+                                324,
+                                243
+                            ],
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        106,
+                                        167,
+                                        324,
+                                        243
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                106,
+                                                167,
+                                                324,
+                                                243
+                                            ],
+                                            "type": "table",
+                                            "html": "<table><tr><td rowspan=\"2\">Structure Model</td><td colspan=\"3\">Cycle</td><td colspan=\"3\">Path</td></tr><tr><td>\\( C_n \\)</td><td>\\( C_n^u \\)</td><td>\\( C_n^{uv} \\)</td><td>\\( P_n \\)</td><td>\\( P_n^w \\)</td><td>\\( P_n^{wx} \\)</td></tr><tr><td>MPNN</td><td>None</td><td>None</td><td>None</td><td>n≤3</td><td>n≤3</td><td>n≤3</td></tr><tr><td>Subgraph GNN</td><td>n≤7</td><td>n≤4</td><td>n≤4</td><td>n≤7</td><td>n≤4</td><td>n≤4</td></tr><tr><td>Local 2-GNN</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Local 2-FGNN</td><td colspan=\"3\">n≤7</td><td colspan=\"3\">n≤7</td></tr><tr><td>2-FGNN</td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>",
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+                                            "content": " Figure 4: Illustration of homomorphic images of the 6-cycle and rooted 6-cycle."
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+                                    "type": "text",
+                                    "content": "**Discussions with prior work.** Our results significantly extend Huang et al. (2023) in several aspects. First, we show Subgraph GNN can count 6-cycle at graph-level by simply enumerating its spasm (see Figure 4(a)). However, it cannot count rooted 5/6-cycle at node-level because the homomorphic image can contain cycles that do not pass the marked vertex (see Figure 4(b)). This provides novel insights into Huang et al. (2023) and extends their results (albeit with a simpler analysis). Second, we reveal that Local 2-GNN can already count all cycles/pathways that 2-FWL can count (even at edge-level). This identifies a new architecture with both efficiency and strong expressiveness in subgraph counting, considerably extending the finding in the concurrent work of Zhou et al. (2023b)."
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+                                    "content": "In Appendix G.2 (Tables 4 and 5), we summarize the statistics of all moderate-size patterns each model can count under homomorphisms/subgraphs, which enables quantitative expressivity comparisons of different models in a clear and exact manner. We also comprehensively list the counting ability of all moderate-size patterns in Table 6, which we believe can be helpful for future research."
+                                }
+                            ]
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+                                    "content": "4.3 POLYNOMIAL EXPRESSIVITY"
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+                                    "content": "As the third implication, homomorphism expressivity is closely related to the polynomial expressivity recently proposed in Puny et al. (2023). Concretely, given a model "
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+                                    "content": " (defined in Puny et al. (2023), Section 2.2), and a rooted graph "
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+                                    "content": " can express the equivariant graph polynomial "
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+                                    "type": "text",
+                                    "content": ". Based on this connection, our work introduces a novel toolbox for studying polynomial expressivity via the NED framework and offers new insights into which graph polynomials can be computed for a variety of practical GNNs. Moreover, we readily settle an open question in Puny et al. (2023), which upper bounds the polynomial expressivity for their proposed PPGN++."
+                                }
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+                                    "type": "text",
+                                    "content": "Corollary 4.7. "
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+                                    "type": "text",
+                                    "content": " is bounded by (and thus as expressive as) the Prototypical edge-based model defined in Puny et al. (2023) for computing equivariant graph polynomials."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
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+                {
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+                                    "type": "text",
+                                    "content": "Due to space limit, please refer to Appendix H for proof and more discussions."
+                                }
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+                                    "content": "5 EXPERIMENTS"
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+                                    "content": "This section aims to verify our theory through a comprehensive set of experiments. In each experiment, we implement four types of GNN models listed in Section 2, i.e., MPNN, Subgraph GNN, Local 2-GNN, and Local 2-FGNN. Note that all of these models are much more efficient than 2-FWL. Our primary objective here is not to produce SOTA results, but rather to provide a unified and equitable empirical comparison among these models. To ensure fairness, we employ the same GIN-based design (Xu et al., 2019) for all models and control their model sizes and training budgets to be roughly the same on each task. Details of model configurations are given in Appendix I. Our code is available at https://github.com/subgraph23/homomorphism-expressivity."
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+                                    "content": "Synthetic task. We first test whether these GNN models can easily learn homomorphism information from data as our theory predicts. We use the benchmark dataset from Zhao et al. (2022a) and comprehensively test the homomorphism expressivity at graph/node/edge-level by carefully selecting 8 substructures shown in Table 1. The reported performance is measured by the normalized"
+                                }
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+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                            "content": "Table 1: Experimental results on homomorphism counting. Red/blue nodes indicate marked vertices."
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+                                            "html": "<table><tr><td rowspan=\"2\">Task Model</td><td colspan=\"3\">Graph-level</td><td colspan=\"2\">Node-level</td><td colspan=\"3\">Edge-level</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>MPNN</td><td>.300</td><td>.233</td><td>.254</td><td>.505</td><td>.478</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Subgraph GNN</td><td>.011</td><td>.015</td><td>.012</td><td>.004</td><td>.058</td><td>.003</td><td>.058</td><td>.048</td></tr><tr><td>Local 2-GNN</td><td>.008</td><td>.008</td><td>.010</td><td>.003</td><td>.004</td><td>.005</td><td>.006</td><td>.008</td></tr><tr><td>Local 2-FGNN</td><td>.003</td><td>.005</td><td>.004</td><td>.005</td><td>.005</td><td>.007</td><td>.007</td><td>.008</td></tr></table>",
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+                                            "content": "Table 2: Experimental results on ZINC and Alchemy datasets. See Appendix I.4 for comparisons of more GNN models in literature."
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+                                            "html": "<table><tr><td rowspan=\"2\">Task \nModel</td><td colspan=\"2\">ZINC</td><td rowspan=\"2\">Alchemy</td></tr><tr><td>Subset</td><td>Full</td></tr><tr><td>MPNN</td><td>.138 ± .006</td><td>.030 ± .002</td><td>.122 ± .002</td></tr><tr><td>Subgraph GNN</td><td>.110 ± .007</td><td>.028 ± .002</td><td>.116 ± .001</td></tr><tr><td>Local 2-GNN</td><td>.069 ± .001</td><td>.024 ± .002</td><td>.114 ± .001</td></tr><tr><td>Local 2-FGNN</td><td>.064 ± .002</td><td>.023 ± .001</td><td>.111 ± .001</td></tr></table>",
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+                                            "content": "Table 3: Experimental results on the (Chordal) Cycle Counting task."
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+                                            "html": "<table><tr><td rowspan=\"2\">Task Model</td><td colspan=\"6\">Graph-level</td><td colspan=\"6\">Node-level</td><td colspan=\"6\">Edge-level</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>MPNN</td><td>.358</td><td>.208</td><td>.188</td><td>.146</td><td>.261</td><td>.205</td><td>.600</td><td>.413</td><td>.300</td><td>.207</td><td>.318</td><td>.237</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Subgraph GNN</td><td>.010</td><td>.020</td><td>.024</td><td>.046</td><td>.007</td><td>.027</td><td>.003</td><td>.005</td><td>.092</td><td>.082</td><td>.050</td><td>.073</td><td>.001</td><td>.003</td><td>.090</td><td>.096</td><td>.038</td><td>.065</td></tr><tr><td>Local 2-GNN</td><td>.008</td><td>.011</td><td>.017</td><td>.034</td><td>.007</td><td>.016</td><td>.002</td><td>.005</td><td>.010</td><td>.023</td><td>.004</td><td>.015</td><td>.001</td><td>.005</td><td>.010</td><td>.019</td><td>.005</td><td>.014</td></tr><tr><td>Local 2-FGNN</td><td>.003</td><td>.004</td><td>.010</td><td>.020</td><td>.003</td><td>.010</td><td>.004</td><td>.006</td><td>.012</td><td>.021</td><td>.004</td><td>.014</td><td>.003</td><td>.006</td><td>.012</td><td>.022</td><td>.005</td><td>.012</td></tr></table>",
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+                                    "type": "text",
+                                    "content": "Mean Absolute Error (MAE) on the test dataset. It can be seen that the model performance indeed correlates to our theoretical predictions: (i) MPNN cannot encode any substructure under homomorphism; (ii) Subgraph GNN cannot encode the 2th, 3rd, 5th, 7th, 8th substructures; (iii) Local 2-GNN cannot encode the 3rd and 8th substructures; (iv) Local 2-FGNN can encode all substructures."
+                                }
+                            ]
+                        }
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+                                    "type": "text",
+                                    "content": "Cycle counting power. Cycles are important structures in numerous graph learning tasks, yet encoding them is notoriously hard for GNNs. We next test the ability of different GNN models to subgraph-count (chordal) cycles at graph/node/edge-level. We follow the setting in Frasca et al. (2022); Zhang et al. (2023a); Huang et al. (2023) and present results in Table 3 (measured by the normalized test MAE). Remarkably, despite the same computational cost and model size, Local 2-(F)GNN performs significantly better than Subgraph GNN and achieves good performance for counting all "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "3/4/5/6"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "-cycles as well as chordal "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "4/5"
+                                },
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+                                    "type": "text",
+                                    "content": "-cycles (even at edge-level). These results match Example 4.6 and may suggest Local 2-(F)GNN as generic, efficient, yet powerful architectures in solving chemical and biological tasks where counting cycles is essential (e.g., benzene rings)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
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+                                    "type": "text",
+                                    "content": "Real-world tasks. We finally test these GNN models on three real-world benchmarks: ZINC-subset, ZINC-full (Dwivedi et al., 2020), and Alchemy (Chen et al., 2019a). Following the standard configuration, all models obey a 500K parameter budget. The results are shown in Table 2. It can be seen that the performance continues to improve when a more expressive model is used. In particular, Local 2-FGNN achieves the best performance on all tasks, suggesting that its theoretical expressivity guarantee can translate to practical performance in real-world settings."
+                                }
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+                                    "type": "text",
+                                    "content": "6 CONCLUSION"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
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+                                    "type": "text",
+                                    "content": "In this paper, we present a new framework for systematically and quantitatively studying the expressive power of various GNN architectures. Through the lens of homomorphism expressivity, we give exact descriptions of the graph family each model can encode in terms of homomorphism counting. Our framework stands as a valuable toolbox to unify the landscape between different subareas in the GNN community, providing deep insights into a number of prior works and answering their open problems. In particular, one can establish a complete expressiveness hierarchy between models, determine the subgraph counting capabilities of GNNs at graph/node/edge-level, and understand their polynomial expressivity. On the theoretical side, our results establish deep connections with a series of fundamental topics in graph theory (see Appendix A.2); On the practical side, these results closely correlate with the empirical performance of GNN models, as demonstrated through extensive experiments. Finally, Appendix B outlines several open directions for further exploration, and we believe that the homomorphism expressivity framework paves a fresh way for future study of more expressive GNNs."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
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+                                    "type": "text",
+                                    "content": "Acknowledgement Bohang Zhang would like to thank Shengjie Luo for helpful discussions. We sincerely appreciate all reviewers for the valuable suggestions. This work is supported by National Science and Technology Major Project (2022ZD0114902) and National Science Foundation of China (NSFC62276005, NSFC62376007)."
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+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "content": "REFERENCES"
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+                                            "content": "C Proof of Theorem 3.4 21"
+                                        }
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+                            "index": 8
+                        }
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+                                            "content": "C.3 Part 2: nested ear decomposition 39"
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+                                        }
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+                        }
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+                                    "content": "D Node/edge-level Expressivity 50"
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+                                    "content": "E Higher-order GNNs 54"
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+                                            "content": "E.1 Definition of higher-order GNNs 54"
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+                                            "content": "E.3 Proofs in Section 3.4 55"
+                                        }
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+                                            "content": "E.4 Expressivity gap between higher-order GNNs 56"
+                                        }
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+                            ],
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+                        }
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+                                    "type": "text",
+                                    "content": "F Additional Discussions 57"
+                                }
+                            ]
+                        }
+                    ],
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+                },
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+                                        }
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+                                        }
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+                        },
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+                                            "content": "G.2 Graph statistics and examples 59"
+                                        }
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+                        }
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+                                    "content": "H Polynomial Expressivity 69"
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+                                    "content": "I Experimental Details 70"
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+                                            "content": "I.4 Performance of baseline models in literature 72"
+                                        }
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+                        }
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+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "content": "17"
+                                }
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+                                    "type": "text",
+                                    "content": "A MORE RELATED WORK"
+                                }
+                            ]
+                        }
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+                },
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+                                    "type": "text",
+                                    "content": "A.1 EXPRESSIVE GRAPH NEURAL NETWORKS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Since Morris et al. (2019); Xu et al. (2019) discovered the limited expressive power of MPNNs in distinguishing non-isomorphic graphs, a large amount of work has been devoted to developing GNNs with better expressiveness. Below, we briefly review representative approaches in this area. For a comprehensive survey on expressive GNNs, we refer readers to Morris et al. (2023)."
+                                }
+                            ]
+                        }
+                    ],
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+                                    "type": "text",
+                                    "content": "Higher-order GNNs. Inspired by the relation between MPNNs and the 1-WL test, a natural approach to designing provably more expressive GNNs is to mimic the higher-order WL tests. This gives rise to two fundamental types of higher-order GNNs. One type of GNNs mimics the "
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+                                    "content": "-WL test (Grohe, 2017), and representative architectures include "
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+                                    "type": "text",
+                                    "content": "-GNN (Morris et al., 2019) and "
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+                                    "type": "text",
+                                    "content": "-IGN (Maron et al., 2019b;c); the other type of GNNs mimics the "
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "-FWL (Folklore WL) test (Cai et al., 1992) and is referred to as the the "
+                                },
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+                                    "type": "text",
+                                    "content": "-FGNN (Maron et al., 2019a). Azizian & Lelarge (2021); Geerts & Reutter (2022) proved that each of these architectures is exactly as expressive as the corresponding higher-order WL/FWL test. Therefore, the expressiveness grows strictly as the order "
+                                },
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+                                    "type": "text",
+                                    "content": " increases; when "
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": " approaches infinity, they can universally approximate any continuous graph functions (Chen et al., 2019b; Keriven & Peyré, 2019). However, due to the inherent computation/memory complexity, these architectures are generally not practical in real-world applications."
+                                }
+                            ]
+                        }
+                    ],
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+                },
+                {
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+                                    "type": "text",
+                                    "content": "Local GNNs. To improve computational efficiency, a subsequent line of work seeks to develop more scalable and practical GNN architectures by taking into account the local/sparse nature of graphs. Locality/sparsity is also an important inductive bias for graphs but is not well-exploited in higher-order GNNs, since their layer aggregation is inherently global and the graph adjacency information is only encoded in initial node features. To address these shortcomings, Morris et al. (2020) proposed the Local "
+                                },
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+                                    "type": "text",
+                                    "content": "-GNN (and several variants) as a replacement of "
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": "-GNN, which directly incorporates graph adjacency into network layers and only aggregates neighboring information instead of the global one. The authors further proved that Local "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "-GNN is strictly more expressive than "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "k"
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "-GNN. Building upon Local "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "-GNN, Morris et al. (2022) proposed the "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(k,s)"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "-SpeqNet that further reduces the computational cost by considering a subset of "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "-tuples whose vertices can be grouped into no more than "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " connected components. A similar idea appeared in Zhao et al. (2022b), in which the authors proposed the "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "(k,s)"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "-SetGNN by considering "
+                                },
+                                {
+                                    "bbox": [
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+                                        303,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "-sets instead of "
+                                },
+                                {
+                                    "bbox": [
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+                                        303,
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+                                        479
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "-tuples. Besides Local "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        303,
+                                        506,
+                                        479
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN, recent architectures proposed in Frasca et al. (2022) and Zhang et al. (2023a) can be analogously understood as Local "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        303,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "-IGN and Local "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        303,
+                                        506,
+                                        479
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "-FGNN, respectively. Very recently, Feng et al. (2023); Zhou et al. (2023b) generalized the Local "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        303,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "-FGNN to a broad class of Folklore-type GNNs and achieved good performance on several benchmark datasets."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Subgraph GNNs. Graphs that are indistinguishable by WL tests typically possess a high degree of symmetry. In light of this observation, Subgraph GNNs have recently emerged as a compelling approach to designing expressive GNNs. The basic idea is to break symmetry by transforming the original graph into a collection of slightly modified subgraphs and feeding these subgraphs into a GNN model. The earliest forms of Subgraph GNNs may track back to Cotta et al. (2021); Papp et al. (2021) (albeit with a different motivation), where the authors proposed to feed node-deleted subgraphs into an MPNN. Papp & Wattenhofer (2022) later argued to use node marking instead of node deletion for better expressive power, resulting in the standard Subgraph GNN studied in this paper. Zhang & Li (2021); You et al. (2021) proposed the Nested GNN and Identity-aware GNN, both of which can be treated as variants of Subgraph GNNs that use ego networks as subgraphs. In particular, the heterogeneous message passing proposed in You et al. (2021) can also be seen as a form of node marking. We note that the model proposed in Vignac et al. (2020) can also be interpreted as a Subgraph GNN. Qian et al. (2022) proposed the higher-order Subgraph GNN by marking "
+                                },
+                                {
+                                    "bbox": [
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+                                        684
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " nodes per subgraph, resulting in "
+                                },
+                                {
+                                    "bbox": [
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+                                        484,
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+                                        684
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n^k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        484,
+                                        506,
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+                                    ],
+                                    "type": "text",
+                                    "content": " different subgraphs when the original graph has "
+                                },
+                                {
+                                    "bbox": [
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+                                        684
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " vertices. We call this architecture Subgraph "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN in this paper. The authors proved that Subgraph "
+                                },
+                                {
+                                    "bbox": [
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+                                        484,
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+                                        684
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN is strictly bounded by "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(k + 1)"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "-FWL and is incomparable to "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "-FWL when "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "k > 1"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ". Zhou et al. (2023a) further generalized Subgraph "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "k"
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN to "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "(l, k)"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "-GNN by using "
+                                },
+                                {
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": "-GNN instead of MPNN to process each subgraph. It was proved that "
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+                                    "type": "inline_equation",
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+                                    "content": "-GNN is bounded by "
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": "-GNN for "
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+                                    "type": "inline_equation",
+                                    "content": "l \\geq 2"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
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+                                    "type": "text",
+                                    "content": "Recently, Subgraph GNNs have been greatly extended by further allowing interactions between subgraphs. This is achieved by designing cross-subgraph aggregation layers (rather than feeding each subgraph independently into a GNN). Bevilacqua et al. (2022) developed the Equivariant Subgraph Aggregation Network that introduces a global aggregation between subgraphs. A similar design is"
+                                }
+                            ]
+                        }
+                    ],
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "type": "text",
+                                    "content": "proposed in the concurrent work of Zhao et al. (2022a). These architectures were later proved to strictly improve the expressivity of the original Subgraph GNNs (Zhang et al., 2023b;a). Frasca et al. (2022) built a general design space of Subgraph GNNs that unifies prior work and showed that all models in this space are bounded by a variant of 2-IGN (dubbed the Local 2-IGN in this paper), which is then bounded by 2-FWL. Zhang et al. (2023a) later proved that Local 2-IGN is as expressive as Local 2-GNN and strictly less expressive than 2-FWL (2-FGNN). In this paper, we still use the term \"Subgraph GNN\" to refer to the original architecture in the previous paragraph, while using \"Local 2-GNN\" to refer to the general architecture in Frasca et al. (2022); Zhang et al. (2023a)."
+                                }
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+                                    "content": "Substructure-based GNNs. Another line of work sought to develop expressive GNNs from practical considerations. In particular, Chen et al. (2020) pointed out that the ability of GNNs to detect/count graph substructures like path, cycle, and clique is crucial in numerous applications. Yet, MPNNs cannot subgraph-count any cycles/cliques. While higher-order WL tests can be more powerful in counting cycles (Fürer, 2017; Arvind et al., 2020), they suffer from substantial computational cost. As such, several works proposed to directly incorporate substructure/homomorphism counting into the node features as a preprocessing step to boost the expressiveness of MPNNs (Bouritsas et al., 2022; Barceló et al., 2021) or serve as an expressive graph kernel (Grohe, 2020; Nguyen & Maehara, 2020; Welke et al., 2023). Going beyond node features, Bodnar et al. (2021b;a); Giusti et al. (2023) further proposed a message-passing framework that enables interaction between nodes, edges, and higher-order substructures like cycles and cliques. We note that the Autobahn, TOGL, and Cy2C-GNN proposed in Thiede et al. (2021); Horn et al. (2022); Choi et al. (2022) can also be viewed as Substructure-based GNNs. However, most of the above approaches consider a fixed, predefined set of substructures rather than designing generic architectures that can learn substructures in an end-to-end fashion. Recently, Tahmasebi et al. (2023) proposed the RNP-GNN, an architecture that can count any substructure by recursively splitting a graph into a collection of vertex-marked subgraphs. We note that this design shares interesting similarities to higher-order subgraph GNNs and also the SpeqNet (Morris et al., 2022). Huang et al. (2023) proposed a generic model called "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{I}^2"
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "-GNN based on a variant of Subgraph 2-GNN, which can count 6-cycle at node-level. In this paper, we show the Local 2-GNN can already count all cycles/paths within 7 nodes even at edge-level while being more efficient than "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{I}^2"
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+                                {
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+                                    "type": "text",
+                                    "content": "-GNN (when using a similar ego network design). Finally, we remark that the polynomial expressivity proposed in Puny et al. (2023) can also be seen as a generalization of substructure counting, which further takes into account the real-valued node/edge features."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
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+                                    "content": "Distance-based GNNs. Besides structural information, distance serves as another fundamental attribute of a graph, which, again, is not captured by MPNNs and the 1-WL test. Li et al. (2020) first proposed to improve the expressive power of GNNs by augmenting node features with Distance Encoding (DE). Related to DE, another approach to injecting distance information is the "
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+                                    "type": "text",
+                                    "content": "-hop MPNN, which aggregates "
+                                },
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+                                    "type": "text",
+                                    "content": "-hop neighbors in a message-passing layer (Feng et al., 2022; Abboud et al., 2022; Wang et al., 2023). Feng et al. (2022); Zhang et al. (2023a) proved that the expressive power of "
+                                },
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+                                    "type": "text",
+                                    "content": "-hop MPNN is strictly bounded by 2-FWL. Distance can also be naturally incorporated in Graph Transformers through relative positional encoding, yielding the Graphormer architecture that has achieved remarkable performance across various benchmarks (Ying et al., 2021). Recently, Zhang et al. (2023b) built an interesting connection between distance and biconnectivity properties, showing that distance-enhanced GNNs can detect cut vertices and cut edges of a graph. This provides insights into the practical superiority of these models as biconnectivity is closely linked to real applications in chemistry and social network analysis. Zhang et al. (2023a) proved that Local 2-GNN can provably encode the distance (and thus biconnectivity) of a graph."
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+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
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+                                    "content": "Spectral-based GNNs. Graph spectra are also a class of fundamental properties and have long been used to design GNN models (Bruna et al., 2014; Defferrard et al., 2016). Balcilar et al. (2021b;a) showed that designing GNNs in the spectral domain can easily break the 1-WL expressivity. For Graph Transformers, Kreuzer et al. (2021); Dwivedi & Bresson (2020); Dwivedi et al. (2022) proposed to incorporate the spectra of the graph Laplacian matrix to boost the expressive power beyond the 1-WL test. Lim et al. (2023) further designed a principled equivariant architecture that takes the Laplacian eigenvalues and eigenvectors as inputs, which generalizes prior work."
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+                            ]
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+                    ],
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+                                    "content": "Other approaches. Murphy et al. (2019); Chen et al. (2020) proposed Relational Pooling as a general approach to designing expressive GNN architectures, whose basic idea is to implement a permutation-invariant GNN by symmetrizing permutation-sensitive base models. Wijesinghe & Wang (2022b) proposed the GraphSNN, which improves the expressive power of MPNNs by using more distinguishing edge features. Specifically, each edge feature encodes the structure of the over-"
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "content": "19"
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+                                    "type": "text",
+                                    "content": "lap subgraph of two 1-hop ego networks centered on the two endpoints of the edge. Very recently, Dimitrov et al. (2023) designed a GNN model that can distinguish all planar graphs, thus achieving a strong expressivity in chemical applications since molecular graphs are often planar."
+                                }
+                            ]
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+                    ],
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+                                    "type": "text",
+                                    "content": "A.2 BROADER IMPACTS AND ADDITIONAL DISCUSSIONS"
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+                        }
+                    ],
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+                {
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+                                    "type": "text",
+                                    "content": "Broader impact in graph theory. Due to the fundamental nature of GNN architectures studied in this paper, our theoretical results may potentially have broader impacts on the graph theory community. Specifically, we study the color refinement (CR) algorithms corresponding to four types of (higher-order) GNNs: Subgraph "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "(k - 1)"
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+                                    "type": "text",
+                                    "content": "-GNN, Local "
+                                },
+                                {
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": "-GNN, Local "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "k"
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "-FGNN, and "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "-FGNN. All algorithms can be seen as natural extensions of the 1-WL test since they all reduce to 1-WL when "
+                                },
+                                {
+                                    "bbox": [
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+                                        148,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k = 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": ". In the graph theory community, Subgraph GNN has another name known as the vertex-individualized CR algorithm, which appears widely in literature (Babai, 2016; Rattan & Seppelt, 2023; Neuen & Schweitzer, 2018) and has become part of the core algorithm for fast graph isomorphism testing software (e.g., McKay & Piperno, 2014). On the other hand, Local "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "k"
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN and Local "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "-FGNN are surprisingly related to the guarded logic (Barceló et al., 2020; De Rijke, 2000; Baader, 2003), since the aggregations are purely local (guarded by the edge). From this perspective, these CR algorithms can be seen as natural extensions of guarded logic in higher-order scenarios."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                    "type": "text",
+                                    "content": "Besides these CR algorithms, our new extensions of NED may also have implications in graph theory. In particular, the strong NED (as well as its higher-order version) is elegant and may serve as a descriptive tool to characterize certain graph families. In addition, we establish intrinsic connections between NED and tree decomposition, which may have value in understanding other graph topics related to tree decomposition."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
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+                                    "type": "text",
+                                    "content": "Finally, to our knowledge, the node/edge-level homomorphism and the corresponding subgraph counting abilities of different CR algorithms do not seem to have been systematically investigated before. Whereas in this paper, all graph/node/edge-level expressivity is studied in a unified manner. To achieve this, we introduce additional proof techniques which we believe may facilitate future study in related areas. For example, the original technique for constructing counterexample graphs satisfying Definition 3.1(b) does not apply to node/edge-level settings, since they are no longer counterexample graphs satisfying Definition 3.6(b) (no matter which vertices "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "u \\in V_G, v \\in V_H"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " are marked). To address the problem, we propose clique-augmented Fürer graphs, a novel class of counterexample graphs that extend several prior works (e.g., Cai et al., 1992; Fürer, 2001), and conduct a fine-grained analysis of their automorphism property (see Appendix D.2). We believe this new technique can be used to generalize other results from graph-level to node/edge-level settings."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
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+                                    "type": "text",
+                                    "content": "Discussions with Barceló et al. (2021). In the GNN community, Barceló et al. (2021) first proposed to incorporate the homomorphism count of predefined substructures into node features as an approach to enhancing the expressivity of MPNNs. They systematically investigated the questions of what substructures are useful and how the homomorphism information of these substructures can boost the model expressivity to even break out "
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": "-FWL. Yet, they only gave a partial (incomplete) characterization of the substructures that can be counted by the specific "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "-MPNN architecture and did not answer what substructures cannot be encoded, whereas our paper fully addresses both questions for a variety of popular GNN models. Note that these aspects are crucial to ensure that homomorphism expressivity is well-defined. As such, our paper first identifies that homomorphism expressivity is a complete, quantitative expressivity measure to compare different GNN models."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
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+                                    "type": "text",
+                                    "content": "Discussions with the concurrent work of Neuen (2023). After the initial submission, we became aware of a concurrent work (Neuen, 2023), which proved that "
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": "-FWL cannot count any graph with treewidth larger than "
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": " under homomorphism. In our context, this result exactly shows that Definition 3.1(b) is satisfied, and thus the homomorphism expressivity of "
+                                },
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+                                {
+                                    "bbox": [
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+                                        589,
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+                                    "type": "text",
+                                    "content": "-FWL is well-defined. Notably, their construction of counterexample graphs is also based on Fürer graphs. Nevertheless, the proof technique between the two works is quite different: the proof in Neuen (2023) is built upon a key concept called oddomorphism, while our proof is based on the relation between tree decomposition and the simplified pebble game developed in Zhang et al. (2023a). It is essential to underscore that our results and proof technique are more general and go beyond the standard "
+                                },
+                                {
+                                    "bbox": [
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+                                        589,
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+                                        732
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+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "-FWL, in that (i) it applies to a broad range of color refinement algorithms related to practical GNN architectures and (ii) it further extends to node/edge-level homomorphism expressivity. Our theoretical results thus strictly incorporate the results in Neuen (2023). The approach in Neuen (2023) (based on oddomorphism) cannot be easily generalized to these settings."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                }
+            ],
+            "discarded_blocks": [
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+                    "type": "header",
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
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+                                    "type": "text",
+                                    "content": "20"
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+                        }
+                    ],
+                    "index": 8
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+            ],
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+                                    "type": "text",
+                                    "content": "B LIMITATIONS AND OPEN DIRECTIONS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "There are still several open questions that are not fully explored in this paper. We list them below as promising directions for future study."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Existence of homomorphism expressivity for refinement-based GNN architectures. In this paper, we prove that homomorphism expressivity exists for a wide range of architectures defined in Section 2. On the other hand, we also note that it may not be well-defined for certain pathological GNNs, as illustrated in Appendix F.1. Given this observation, a fundamental question is: what conditions can guarantee that the homomorphism expressivity of a GNN exists? Here, we hypothesize that a very mild condition can suffice. Specifically, we conjecture that as long as a GNN architecture is defined following a general form of color refinement procedure that outputs stable color mappings, its homomorphism expressivity always exists. We leave this conjecture as an important open problem for future study."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
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+                    ],
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+                    "angle": 0,
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+                                {
+                                    "bbox": [
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+                                        329
+                                    ],
+                                    "type": "text",
+                                    "content": "Regarding higher-order Local FGNN. This paper characterizes the homomorphism expressivity for three classes of higher-order GNNs: Subgraph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        240,
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+                                    ],
+                                    "type": "inline_equation",
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+                                },
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN, Local "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        240,
+                                        504,
+                                        329
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        240,
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+                                        329
+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN, and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        240,
+                                        504,
+                                        329
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        240,
+                                        504,
+                                        329
+                                    ],
+                                    "type": "text",
+                                    "content": "-FGNN. In particular, we introduce the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        240,
+                                        504,
+                                        329
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        240,
+                                        504,
+                                        329
+                                    ],
+                                    "type": "text",
+                                    "content": "-order ear and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        240,
+                                        504,
+                                        329
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        240,
+                                        504,
+                                        329
+                                    ],
+                                    "type": "text",
+                                    "content": "-order strong NED as a way to describe the homomorphism expressivity of Local "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        240,
+                                        504,
+                                        329
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        240,
+                                        504,
+                                        329
+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN. However, it remains unclear how to give a simple description of the homomorphism expressivity for Local "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        240,
+                                        504,
+                                        329
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        240,
+                                        504,
+                                        329
+                                    ],
+                                    "type": "text",
+                                    "content": "-FGNN that can generalize the concept of almost-strong NED for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        240,
+                                        504,
+                                        329
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k = 2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        240,
+                                        504,
+                                        329
+                                    ],
+                                    "type": "text",
+                                    "content": ". As such, our current expressiveness hierarchy (Figure 3) does not support Local "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        240,
+                                        504,
+                                        329
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        240,
+                                        504,
+                                        329
+                                    ],
+                                    "type": "text",
+                                    "content": "-FGNN yet. We leave this as an open problem and make the following conjecture below. We note that a similar open question has been informally raised in Zhang et al. (2023a)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                            ],
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Conjecture B.1. For all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        333,
+                                        506,
+                                        357
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k \\geq 2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        333,
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+                                        357
+                                    ],
+                                    "type": "text",
+                                    "content": ", Local "
+                                },
+                                {
+                                    "bbox": [
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+                                        357
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        333,
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+                                        357
+                                    ],
+                                    "type": "text",
+                                    "content": "-FGNN is strictly more expressive than Local "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        333,
+                                        506,
+                                        357
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        333,
+                                        506,
+                                        357
+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN and strictly less expressive than "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        333,
+                                        506,
+                                        357
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        333,
+                                        506,
+                                        357
+                                    ],
+                                    "type": "text",
+                                    "content": "-FGNN."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
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+                        506,
+                        434
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
+                                        104,
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+                                        434
+                                    ],
+                                    "type": "text",
+                                    "content": "Expressivity gap between Local 2-(F)GNN and 2-FGNN in practical aspects. We have proved that 2-FGNN is strictly more expressive than Local 2-(F)GNN. However, from a practical perspective, we surprisingly find that the subgraph counting ability of Local 2-(F)GNN matches that of 2-FGNN for all structures within a moderate size (see Table 5), although the former is much more efficient. This leads to the intriguing question of what fundamental gaps exist between the two models in practical aspects, or is the efficiency gain free?"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
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+                        506,
+                        561
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
+                                        104,
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+                                        506,
+                                        561
+                                    ],
+                                    "type": "text",
+                                    "content": "Other architectures. In this paper, we comprehensively study a variety of popular GNN architectures ranging from Subgraph GNNs and Local GNNs to higher-order GNNs, and further link these architectures to Substructure-based GNNs (see Appendix A.1). Yet, we still do not cover all popular GNN architectures, such as the GSWL-based Subgraph GNN (Zhang et al., 2023a; Bevilacqua et al., 2022), SpeqNet (Morris et al., 2022), and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        438,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{I}^2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        438,
+                                        506,
+                                        561
+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN (Huang et al., 2023). Moreover, the classes of Distance-based GNNs and Spectral-based GNNs (Appendix A.1) are also widely used in practice, which deserve future study. We would like to raise the question of characterizing the homomorphism expressivity of Distance-based GNNs and Spectral-based GNNs as an important open question. In this way, one can gain deep insights into these models' true expressivity and enable quantitative comparisons between all mainstream architectures. Moreover, it will become clear to what extent other GNN models can encode distance and spectral information about a graph."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
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+                        255,
+                        591
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "C PROOF OF THEOREM 3.4"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
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+                    ],
+                    "type": "text",
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+                                    ],
+                                    "type": "text",
+                                    "content": "This section gives the proof of the main theorem. For ease of reading, we first restate Theorem 3.4:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
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+                        646
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        622,
+                                        504,
+                                        646
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem 3.4. For all GNN models "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        622,
+                                        504,
+                                        646
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        622,
+                                        504,
+                                        646
+                                    ],
+                                    "type": "text",
+                                    "content": " defined in Section 2, the graph family "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        622,
+                                        504,
+                                        646
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        622,
+                                        504,
+                                        646
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfying Definition 3.1 exists (and is unique). Moreover, each "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        622,
+                                        504,
+                                        646
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        622,
+                                        504,
+                                        646
+                                    ],
+                                    "type": "text",
+                                    "content": " can be separately described below:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
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+                        392,
+                        715
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 15,
+                    "blocks": [
+                        {
+                            "bbox": [
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+                                649,
+                                392,
+                                662
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                120,
+                                                649,
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+                                                662
+                                            ],
+                                            "type": "text",
+                                            "content": "- Subgraph GNN: "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                649,
+                                                392,
+                                                662
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{F}^{\\mathrm{Sub}} = \\{F:F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                649,
+                                                392,
+                                                662
+                                            ],
+                                            "type": "text",
+                                            "content": " has an endpoint-shared NED\\};"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 11
+                        },
+                        {
+                            "bbox": [
+                                121,
+                                667,
+                                332,
+                                680
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        667,
+                                        332,
+                                        680
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                121,
+                                                667,
+                                                332,
+                                                680
+                                            ],
+                                            "type": "text",
+                                            "content": "- Local 2-GNN: "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                667,
+                                                332,
+                                                680
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{F}^{\\mathsf{L}} = \\{F:F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                667,
+                                                332,
+                                                680
+                                            ],
+                                            "type": "text",
+                                            "content": " has a strong NED\\};"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 12
+                        },
+                        {
+                            "bbox": [
+                                121,
+                                685,
+                                376,
+                                697
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        685,
+                                        376,
+                                        697
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                121,
+                                                685,
+                                                376,
+                                                697
+                                            ],
+                                            "type": "text",
+                                            "content": "- Local 2-FGNN: "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                685,
+                                                376,
+                                                697
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{F}^{\\mathrm{LF}} = \\{F:F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                685,
+                                                376,
+                                                697
+                                            ],
+                                            "type": "text",
+                                            "content": " has an almost-strong NED\\};"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 13
+                        },
+                        {
+                            "bbox": [
+                                121,
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+                                715
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        702,
+                                        283,
+                                        715
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                121,
+                                                702,
+                                                283,
+                                                715
+                                            ],
+                                            "type": "text",
+                                            "content": "- 2-FGNN: "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                702,
+                                                283,
+                                                715
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{F}^{\\mathrm{F}} = \\{F:F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                702,
+                                                283,
+                                                715
+                                            ],
+                                            "type": "text",
+                                            "content": " has a NED\\}."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 14
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        104,
+                        720,
+                        503,
+                        733
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                720,
+                                503,
+                                733
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        720,
+                                        503,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "For MPNN, since it is a special case of Subgraph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        720,
+                                        503,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        720,
+                                        503,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN, the proof can be found in Appendix E.3."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
+                        750,
+                        310,
+                        760
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                299,
+                                750,
+                                310,
+                                760
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        299,
+                                        750,
+                                        310,
+                                        760
+                                    ],
+                                    "type": "text",
+                                    "content": "21"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 20
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        82,
+                        194,
+                        94
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                82,
+                                194,
+                                94
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        82,
+                                        194,
+                                        94
+                                    ],
+                                    "type": "text",
+                                    "content": "C.1 PRELIMINARY"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        102,
+                        504,
+                        170
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                102,
+                                504,
+                                170
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        170
+                                    ],
+                                    "type": "text",
+                                    "content": "Additional notations and concepts. Besides the notations defined in Section 2, we further define the following notations. We use the symbol "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        170
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{G}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        170
+                                    ],
+                                    "type": "text",
+                                    "content": " to denote the set of all finite, simple, undirected, labeled graphs. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        170
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G = (V_G,E_G,\\ell_G)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        170
+                                    ],
+                                    "type": "text",
+                                    "content": " be a graph in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        170
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{G}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        170
+                                    ],
+                                    "type": "text",
+                                    "content": ". When the label is the same for all vertices, we can omit the term "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        170
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        170
+                                    ],
+                                    "type": "text",
+                                    "content": " and write "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        170
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G = (V_G,E_G)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        170
+                                    ],
+                                    "type": "text",
+                                    "content": ". Given vertex "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        170
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u\\in V_{G}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        170
+                                    ],
+                                    "type": "text",
+                                    "content": ", denote the degree of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        170
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        170
+                                    ],
+                                    "type": "text",
+                                    "content": " as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        170
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\deg_G(u) = |N_G(u)|"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        170
+                                    ],
+                                    "type": "text",
+                                    "content": ", and denote the closed neighborhood of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        170
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        170
+                                    ],
+                                    "type": "text",
+                                    "content": " as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        170
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "N_G[u] = N_G(u)\\cup \\{u\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        170
+                                    ],
+                                    "type": "text",
+                                    "content": ". The shortest path distance between vertices "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        170
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        170
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        170
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        170
+                                    ],
+                                    "type": "text",
+                                    "content": " is denoted by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        170
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{dis}_G(u,v)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        170
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        174,
+                        506,
+                        287
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                174,
+                                506,
+                                287
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": "Given a vertex set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S \\subset V_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": ", the induced subgraph of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": " over "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": ", denoted as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G[S]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": ", is the subgraph of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": " with vertex set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": " and edge set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\{u, v\\} \\in E_G : u, v \\in S\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": ". Similarly, without abuse of notation, given an edge set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "R \\subset E_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": ", the induced subgraph of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": " over "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "R"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": ", denoted as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G[R]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": ", is the subgraph of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": " with vertex set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\bigcup_{\\{u, v\\} \\in R} \\{u, v\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": " and edge set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "R"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": ". Given a vertex tuple "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{u} = (u_1, \\dots, u_k)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": ", denote by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^{\\mathbf{u}} = G^{u_1, \\dots, u_k}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": " the rooted graph obtained from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": " by marking vertices "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u_1, \\dots, u_k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": ". The atomic type of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": " over "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{u}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": ", denoted by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{atp}_G(\\mathbf{u})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": ", is a "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k \\times k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": " matrix where the element at position "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(i, j)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": " is the tuple "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\mathbb{I}[u_i = u_j], \\mathbb{I}[\\{u_i, u_j\\} \\in E_G])"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": ". Given two graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G, H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": ", the graph union "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G \\cup H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": " is defined to be the graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(V_G \\cup V_H, E_G \\cup E_H, \\ell_{G \\cup H})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": ", where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell_{G \\cup H}(u) := \\ell_G(u)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\in V_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell_{G \\cup H}(u) := \\ell_H(u)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\in V_H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": ". It is well-defined iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell_G(u) = \\ell_H(u)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\in V_G \\cap V_H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": ". Finally, we use the notation "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G \\simeq H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": " to denote that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        174,
+                                        506,
+                                        287
+                                    ],
+                                    "type": "text",
+                                    "content": " are isomorphic graphs."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        292,
+                        504,
+                        360
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                292,
+                                504,
+                                360
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "text",
+                                    "content": "A graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T = (V_T, E_T, \\ell_T)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "text",
+                                    "content": " is called a tree if it does not contain cycles. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "text",
+                                    "content": " be a rooted tree where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "text",
+                                    "content": " is the root vertex. For each vertex "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t \\in V_T"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "text",
+                                    "content": ", define its depth "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{dep}_{T^r}(t) := \\mathsf{dis}_T(t, r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "text",
+                                    "content": " to be the distance to the root, and denote by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{Desc}_{T^r}(t)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "text",
+                                    "content": " the set of descendants of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "text",
+                                    "content": ", namely, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s \\in \\mathsf{Desc}_{T^r}(t)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{dep}_{T^r}(s) = \\mathsf{dep}_{T^r}(t) + \\mathsf{dis}_T(t, s)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "text",
+                                    "content": ". For each "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t \\in V_T \\setminus \\{r\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "text",
+                                    "content": ", denote by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{pa}_{T^r}(t)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "text",
+                                    "content": " the parent vertex of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "text",
+                                    "content": ", i.e., the unique vertex "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s \\in N_T(t)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfying "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{dep}_{T^r}(t) = \\mathsf{dep}_{T^r}(s) + 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "text",
+                                    "content": ". Define the subtree of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "text",
+                                    "content": " rooted at node "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "text",
+                                    "content": " by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r[t]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "text",
+                                    "content": ", which is exactly the induced subgraph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T[\\mathsf{Desc}_{T^r}(t)]^t"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "text",
+                                    "content": " with root "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        292,
+                                        504,
+                                        360
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        364,
+                        504,
+                        386
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                364,
+                                504,
+                                386
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        504,
+                                        386
+                                    ],
+                                    "type": "text",
+                                    "content": "Tree decomposition. Our proof is based on a central concept in graph theory, called tree decomposition. It can be formally defined below:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        388,
+                        504,
+                        412
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                388,
+                                504,
+                                412
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        504,
+                                        412
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition C.1 (Tree decomposition). Given a graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        504,
+                                        412
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G = (V_G, E_G, \\ell_G)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        504,
+                                        412
+                                    ],
+                                    "type": "text",
+                                    "content": ", its tree decomposition is a tree "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        504,
+                                        412
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T = (V_T, E_T, \\beta_T)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        504,
+                                        412
+                                    ],
+                                    "type": "text",
+                                    "content": ", where the label function "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        504,
+                                        412
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_T : V_T \\to 2^{V_G}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        504,
+                                        412
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfies the following conditions:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        116,
+                        416,
+                        503,
+                        491
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 10,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                116,
+                                416,
+                                503,
+                                437
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        416,
+                                        503,
+                                        437
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                416,
+                                                503,
+                                                437
+                                            ],
+                                            "type": "text",
+                                            "content": "a) Each tree node "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                416,
+                                                503,
+                                                437
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t \\in V_T"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                416,
+                                                503,
+                                                437
+                                            ],
+                                            "type": "text",
+                                            "content": " is associated to a non-empty subset of vertices "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                416,
+                                                503,
+                                                437
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(t) \\subset V_G"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                416,
+                                                503,
+                                                437
+                                            ],
+                                            "type": "text",
+                                            "content": " in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                416,
+                                                503,
+                                                437
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "G"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                416,
+                                                503,
+                                                437
+                                            ],
+                                            "type": "text",
+                                            "content": ", called a bag. We say tree node "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                416,
+                                                503,
+                                                437
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                416,
+                                                503,
+                                                437
+                                            ],
+                                            "type": "text",
+                                            "content": " contains vertex "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                416,
+                                                503,
+                                                437
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                416,
+                                                503,
+                                                437
+                                            ],
+                                            "type": "text",
+                                            "content": " if "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                416,
+                                                503,
+                                                437
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u \\in \\beta_T(t)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                416,
+                                                503,
+                                                437
+                                            ],
+                                            "type": "text",
+                                            "content": ";"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 7
+                        },
+                        {
+                            "bbox": [
+                                116,
+                                442,
+                                503,
+                                464
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        442,
+                                        503,
+                                        464
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                442,
+                                                503,
+                                                464
+                                            ],
+                                            "type": "text",
+                                            "content": "b) For each edge "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                442,
+                                                503,
+                                                464
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{u,v\\} \\in V_G"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                442,
+                                                503,
+                                                464
+                                            ],
+                                            "type": "text",
+                                            "content": " , there exists at least one tree node "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
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+                                            "type": "inline_equation",
+                                            "content": "t\\in V_T"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": " that contains the edge, i.e., "
+                                        },
+                                        {
+                                            "bbox": [
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+                                                442,
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+                                                464
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{u,v\\} \\subset \\beta_T(t)"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 8
+                        },
+                        {
+                            "bbox": [
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+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
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+                                    "bbox": [
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+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": "c) For each vertex "
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+                                            "bbox": [
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+                                            "type": "inline_equation",
+                                            "content": "u \\in V_G"
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+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": ", all tree nodes "
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+                                            "type": "inline_equation",
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+                                            "type": "text",
+                                            "content": " containing "
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+                                            "type": "inline_equation",
+                                            "content": "u"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": " form a (non-empty) connected subtree. Formally, denoting "
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+                                            "bbox": [
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+                                            "type": "inline_equation",
+                                            "content": "B_T(u) = \\{t \\in V_T : u \\in \\beta_T(t)\\}"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": ", then "
+                                        },
+                                        {
+                                            "bbox": [
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+                                            "type": "inline_equation",
+                                            "content": "T[B_T(u)]"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": " is connected."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 9
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+                    ],
+                    "sub_type": "text"
+                },
+                {
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+                    "type": "text",
+                    "angle": 0,
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "If "
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+                                    "bbox": [
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+                                    "type": "inline_equation",
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " is a tree decomposition of "
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+                                {
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+                                    "type": "inline_equation",
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ", we call the pair "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(G, T)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " a tree-decomposed graph."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
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+                        540
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
+                                        104,
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+                                        540
+                                    ],
+                                    "type": "text",
+                                    "content": "We remark that given a graph "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ", there are multiple ways to decompose it and thus its tree decomposition is not unique. Several examples of tree decomposition are given in Figure 5."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Definition C.2 (Treewidth). The width of a tree decomposition is defined as one less than the maximum bag size, i.e., "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\max_{t\\in T}|\\beta_T(t)| - 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": ". The treewidth of a graph "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ", denoted as "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{tw}(G)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": ", is the minimum positive integer "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " such that there exists a tree decomposition of width "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "type": "text",
+                                    "content": "Some important facts about treewidth are listed below:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
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+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "Fact C.3. For any graph "
+                                },
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+                                    "type": "text",
+                                    "content": ", the following hold:"
+                                }
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+                        }
+                    ],
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+                },
+                {
+                    "bbox": [
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+                                            "type": "text",
+                                            "content": "- The treewidth of "
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+                                            "content": " is at most "
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+                                            "type": "inline_equation",
+                                            "content": "|V_G| - 1"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": ", i.e., a trivial tree decomposition that only has one node "
+                                        },
+                                        {
+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": " and "
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+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(t) = V_G"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": "."
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+                                    ]
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+                                            "content": "\\operatorname{tw}(G) = |V_G| - 1"
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+                                            "type": "text",
+                                            "content": " is a clique."
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+                                            "type": "text",
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+                                    "content": "The above definition of tree decomposition is quite flexible without constraints on the structure of the tree or the size of each bag. Below, we define several restricted variants of tree decomposition, which (we will later see) are closely related to the GNN architectures studied in this paper. To begin with, we first define a general concept that slight modifies the original definition (Definition C.1) such that the tree becomes rooted and each bag is a multiset of vertices rather than a set."
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+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "content": "Definition C.4 (Canonical tree decomposition). Given a graph "
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G = (V_G, E_G, \\ell_G)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        284,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": ", a canonical tree decomposition of width "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        284,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        284,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": " is a rooted tree "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        284,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r = (V_T, E_T, \\beta_T)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        284,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfying the following conditions:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        116,
+                        312,
+                        504,
+                        390
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 8,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                116,
+                                312,
+                                353,
+                                324
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        312,
+                                        353,
+                                        324
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                312,
+                                                353,
+                                                324
+                                            ],
+                                            "type": "text",
+                                            "content": "a) The depth of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                312,
+                                                353,
+                                                324
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                312,
+                                                353,
+                                                324
+                                            ],
+                                            "type": "text",
+                                            "content": " is even, i.e. "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                312,
+                                                353,
+                                                324
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\max_{t\\in V_T}\\mathsf{dep}_{T^r}(t)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                312,
+                                                353,
+                                                324
+                                            ],
+                                            "type": "text",
+                                            "content": " is even;"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 4
+                        },
+                        {
+                            "bbox": [
+                                116,
+                                327,
+                                504,
+                                350
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        327,
+                                        504,
+                                        350
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                327,
+                                                504,
+                                                350
+                                            ],
+                                            "type": "text",
+                                            "content": "b) Each tree node "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                327,
+                                                504,
+                                                350
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t \\in V_T"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                327,
+                                                504,
+                                                350
+                                            ],
+                                            "type": "text",
+                                            "content": " is associated to a multiset of vertices "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                327,
+                                                504,
+                                                350
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(t) \\subset V_G"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                327,
+                                                504,
+                                                350
+                                            ],
+                                            "type": "text",
+                                            "content": ", called a bag. Moreover, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                327,
+                                                504,
+                                                350
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "|\\beta_T(t)| = k"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                327,
+                                                504,
+                                                350
+                                            ],
+                                            "type": "text",
+                                            "content": " if "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                327,
+                                                504,
+                                                350
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{dep}_{T^r}(t)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                327,
+                                                504,
+                                                350
+                                            ],
+                                            "type": "text",
+                                            "content": " is even and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                327,
+                                                504,
+                                                350
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "|\\beta_T(t)| = k + 1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                327,
+                                                504,
+                                                350
+                                            ],
+                                            "type": "text",
+                                            "content": " if "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                327,
+                                                504,
+                                                350
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{dep}_{T^r}(t)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                327,
+                                                504,
+                                                350
+                                            ],
+                                            "type": "text",
+                                            "content": " is odd;"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 5
+                        },
+                        {
+                            "bbox": [
+                                116,
+                                353,
+                                504,
+                                375
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        353,
+                                        504,
+                                        375
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                353,
+                                                504,
+                                                375
+                                            ],
+                                            "type": "text",
+                                            "content": "c) For all tree edges "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                353,
+                                                504,
+                                                375
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{s,t\\} \\in E_T"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                353,
+                                                504,
+                                                375
+                                            ],
+                                            "type": "text",
+                                            "content": " where "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                353,
+                                                504,
+                                                375
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathrm{dep}_{T^r}(s)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                353,
+                                                504,
+                                                375
+                                            ],
+                                            "type": "text",
+                                            "content": " is even and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                353,
+                                                504,
+                                                375
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathrm{dep}_{T^r}(t)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                353,
+                                                504,
+                                                375
+                                            ],
+                                            "type": "text",
+                                            "content": " is odd, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                353,
+                                                504,
+                                                375
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_{T}(s)\\subset \\beta_{T}(t)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                353,
+                                                504,
+                                                375
+                                            ],
+                                            "type": "text",
+                                            "content": " (where “"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                353,
+                                                504,
+                                                375
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\subset"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                353,
+                                                504,
+                                                375
+                                            ],
+                                            "type": "text",
+                                            "content": "” denotes the multiset inclusion relation);"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 6
+                        },
+                        {
+                            "bbox": [
+                                116,
+                                379,
+                                359,
+                                390
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        379,
+                                        359,
+                                        390
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                379,
+                                                359,
+                                                390
+                                            ],
+                                            "type": "text",
+                                            "content": "d) The conditions (b) and (c) in Definition C.1 are satisfied."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 7
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        104,
+                        399,
+                        504,
+                        433
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                399,
+                                504,
+                                433
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        399,
+                                        504,
+                                        433
+                                    ],
+                                    "type": "text",
+                                    "content": "As examples, one can check that the tree decomposition of all graphs in Figure 5(b) is canonical, but the tree decomposition in Figure 5(a) is not. An important observation about canonical tree decomposition is shown below:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
+                        436,
+                        504,
+                        460
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                436,
+                                504,
+                                460
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        436,
+                                        504,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": "Proposition C.5. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        436,
+                                        504,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        436,
+                                        504,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": " be any tree-decomposed graph where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        436,
+                                        504,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        436,
+                                        504,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": " is a canonical tree decomposition of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        436,
+                                        504,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        436,
+                                        504,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": ". For any vertices "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        436,
+                                        504,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        436,
+                                        504,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": ", either of the following holds:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        120,
+                        464,
+                        490,
+                        493
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 13,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                120,
+                                464,
+                                490,
+                                475
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        120,
+                                        464,
+                                        490,
+                                        475
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                120,
+                                                464,
+                                                490,
+                                                475
+                                            ],
+                                            "type": "text",
+                                            "content": "- "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                464,
+                                                490,
+                                                475
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                464,
+                                                490,
+                                                475
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                464,
+                                                490,
+                                                475
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                464,
+                                                490,
+                                                475
+                                            ],
+                                            "type": "text",
+                                            "content": " are in the same bag of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                464,
+                                                490,
+                                                475
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T^r"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                464,
+                                                490,
+                                                475
+                                            ],
+                                            "type": "text",
+                                            "content": ", i.e., there is a node "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                464,
+                                                490,
+                                                475
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t \\in V_T"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                464,
+                                                490,
+                                                475
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                464,
+                                                490,
+                                                475
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{\\{u, v\\}\\} \\subset \\beta_T(t)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                464,
+                                                490,
+                                                475
+                                            ],
+                                            "type": "text",
+                                            "content": ";"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 11
+                        },
+                        {
+                            "bbox": [
+                                121,
+                                480,
+                                368,
+                                493
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        480,
+                                        368,
+                                        493
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                121,
+                                                480,
+                                                368,
+                                                493
+                                            ],
+                                            "type": "text",
+                                            "content": "- The induced subgraph "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                480,
+                                                368,
+                                                493
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T[B_T(u) \\cup B_T(v)]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                480,
+                                                368,
+                                                493
+                                            ],
+                                            "type": "text",
+                                            "content": " is disconnected."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 12
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        104,
+                        504,
+                        506,
+                        583
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                504,
+                                506,
+                                583
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Assume that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": " are not in the same bag, i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "B_{T}(u) \\cap B_{T}(v) = \\emptyset"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": ". Pick "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s \\in B_{T}(u)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t \\in B_{T}(v)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{dep}_{T^r}(s)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{dep}_{T^r}(t)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": " are minimal, respectively. Without loss of generality, assume that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{dep}_{T^r}(s) \\leq \\mathsf{dep}_{T^r}(t)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t \\neq r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": " is not the root node and thus we can pick its parent "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{pa}_{T^r}(t)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": ". It follows that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{dep}_{T^r}(t)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": " is odd and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_{T}(\\mathsf{pa}_{T^{r}}(t)) \\subset \\beta_{T}(t)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": " by definition of canonical tree decomposition. Therefore, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\notin \\beta_{T}(\\mathsf{pa}_{T^{r}}(t))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": ". Moreover, by the assumption that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{dep}_{T^r}(s) \\leq \\mathsf{dep}_{T^r}(t)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": ", any node in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "B_{T}(u)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": " is not a descendant of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": ". We thus conclude that there does not exist a tree edge such that the two endpoints contain "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        504,
+                                        506,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": ", respectively."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        104,
+                        594,
+                        455,
+                        605
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Now we are ready to define several restricted variants of canonical tree decomposition:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Definition C.6. Define four families of tree-decomposed graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        608,
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+                                        620
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S^{\\mathrm{Sub}}"
+                                },
+                                {
+                                    "bbox": [
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+                                        608,
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+                                        620
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
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+                                        608,
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+                                        620
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S^{\\mathrm{L}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        608,
+                                        503,
+                                        620
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        608,
+                                        503,
+                                        620
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S^{\\mathrm{LF}}"
+                                },
+                                {
+                                    "bbox": [
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+                                        608,
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+                                        620
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        608,
+                                        503,
+                                        620
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S^{\\mathrm{F}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        608,
+                                        503,
+                                        620
+                                    ],
+                                    "type": "text",
+                                    "content": " as follows:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
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+                        505,
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+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 21,
+                    "blocks": [
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+                            "bbox": [
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+                                638
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+                            "type": "text",
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+                                            "type": "text",
+                                            "content": "a) "
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+                                            "type": "inline_equation",
+                                            "content": "(F,T^{r})\\in S^{\\mathsf{F}}"
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+                                            "type": "text",
+                                            "content": " iff "
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+                                            "type": "inline_equation",
+                                            "content": "(F,T^{r})"
+                                        },
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+                                            "bbox": [
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+                                                638
+                                            ],
+                                            "type": "text",
+                                            "content": " satisfies Definition C.4 with width "
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+                                            "bbox": [
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+                                                400,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "k = 2"
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+                                    ]
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+                            ],
+                            "index": 17
+                        },
+                        {
+                            "bbox": [
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+                            "type": "text",
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+                                            "type": "inline_equation",
+                                            "content": "(F,T^{r})\\in S^{\\mathrm{LF}}"
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+                                        {
+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": " iff "
+                                        },
+                                        {
+                                            "bbox": [
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+                                            "type": "inline_equation",
+                                            "content": "(F,T^{r})"
+                                        },
+                                        {
+                                            "bbox": [
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+                                                677
+                                            ],
+                                            "type": "text",
+                                            "content": " satisfies Definition C.4 with width "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "k = 2"
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+                                            "type": "text",
+                                            "content": ", and for any tree node "
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+                                            "type": "inline_equation",
+                                            "content": "t"
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+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": " of odd depth, it has only one child if "
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w\\notin \\{v:v\\in N_G[u],u\\in \\beta_T(s)\\}"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": " where "
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "s"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": " is the parent node of "
+                                        },
+                                        {
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t"
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+                                            "type": "text",
+                                            "content": " and "
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+                                            "bbox": [
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+                                            "type": "inline_equation",
+                                            "content": "w"
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+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": " is the unique vertex in "
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+                                            "bbox": [
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(t)\\backslash \\beta_T(s)"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": ";"
+                                        }
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+                            "bbox": [
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+                                            "type": "text",
+                                            "content": "c) "
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+                                            "type": "inline_equation",
+                                            "content": "(F,T^r)\\in S^{\\mathsf{L}}"
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+                                        {
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+                                            "type": "text",
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+                                            "type": "inline_equation",
+                                            "content": "(F,T^{r})"
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+                                            "type": "text",
+                                            "content": " satisfies Definition C.4 with width "
+                                        },
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+                                            "type": "inline_equation",
+                                            "content": "k = 2"
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+                                            "type": "text",
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+                                            "type": "text",
+                                            "content": " of odd depth has only one child;"
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+                                            "type": "inline_equation",
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+                                            "type": "inline_equation",
+                                            "content": "u \\in V_G"
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+                                            "type": "text",
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+                                            "type": "text",
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+                                    ],
+                                    "type": "text",
+                                    "content": "Examples of tree-decomposed graphs in the four families are illustrated in Figure 5(b)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        99,
+                        457,
+                        111
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                99,
+                                457,
+                                111
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        99,
+                                        457,
+                                        111
+                                    ],
+                                    "type": "text",
+                                    "content": "Before closing this subsection, we define several notations for tree-decomposed graphs:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        113,
+                        504,
+                        137
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                113,
+                                504,
+                                137
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        113,
+                                        504,
+                                        137
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition C.7. Given canonical tree-decomposed graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        113,
+                                        504,
+                                        137
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(G,T^r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        113,
+                                        504,
+                                        137
+                                    ],
+                                    "type": "text",
+                                    "content": " and node "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        113,
+                                        504,
+                                        137
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t\\in V_T"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        113,
+                                        504,
+                                        137
+                                    ],
+                                    "type": "text",
+                                    "content": " , denote by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        113,
+                                        504,
+                                        137
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G[T^{r}[t]]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        113,
+                                        504,
+                                        137
+                                    ],
+                                    "type": "text",
+                                    "content": " the subgraph of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        113,
+                                        504,
+                                        137
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        113,
+                                        504,
+                                        137
+                                    ],
+                                    "type": "text",
+                                    "content": " induced by the vertex set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        113,
+                                        504,
+                                        137
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{u:u\\in \\beta_{T^r}(s),s\\in \\mathsf{Desc}_{T^r}(t)\\}"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        140,
+                        506,
+                        176
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                140,
+                                506,
+                                176
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        140,
+                                        506,
+                                        176
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition C.8. Given two canonical tree-decomposed graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        140,
+                                        506,
+                                        176
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(G, T^r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        140,
+                                        506,
+                                        176
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        140,
+                                        506,
+                                        176
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{G}, \\tilde{T}^s)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        140,
+                                        506,
+                                        176
+                                    ],
+                                    "type": "text",
+                                    "content": ", a pair of mappings "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        140,
+                                        506,
+                                        176
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\rho, \\tau)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        140,
+                                        506,
+                                        176
+                                    ],
+                                    "type": "text",
+                                    "content": " is called an isomorphism from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        140,
+                                        506,
+                                        176
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(G, T^r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        140,
+                                        506,
+                                        176
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        140,
+                                        506,
+                                        176
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{G}, \\tilde{T}^s)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        140,
+                                        506,
+                                        176
+                                    ],
+                                    "type": "text",
+                                    "content": ", denoted by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        140,
+                                        506,
+                                        176
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(G, T^r) \\simeq (\\tilde{G}, \\tilde{T}^s)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        140,
+                                        506,
+                                        176
+                                    ],
+                                    "type": "text",
+                                    "content": ", if the following hold:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        116,
+                        182,
+                        355,
+                        230
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 8,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                116,
+                                182,
+                                270,
+                                194
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        182,
+                                        270,
+                                        194
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                182,
+                                                270,
+                                                194
+                                            ],
+                                            "type": "text",
+                                            "content": "a) "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                182,
+                                                270,
+                                                194
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                182,
+                                                270,
+                                                194
+                                            ],
+                                            "type": "text",
+                                            "content": " is an isomorphism from "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                182,
+                                                270,
+                                                194
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "G"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                182,
+                                                270,
+                                                194
+                                            ],
+                                            "type": "text",
+                                            "content": " to "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                182,
+                                                270,
+                                                194
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tilde{G}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                182,
+                                                270,
+                                                194
+                                            ],
+                                            "type": "text",
+                                            "content": ";"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 5
+                        },
+                        {
+                            "bbox": [
+                                116,
+                                200,
+                                355,
+                                213
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        200,
+                                        355,
+                                        213
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                200,
+                                                355,
+                                                213
+                                            ],
+                                            "type": "text",
+                                            "content": "b) "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                200,
+                                                355,
+                                                213
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tau"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                200,
+                                                355,
+                                                213
+                                            ],
+                                            "type": "text",
+                                            "content": " is an isomorphism from "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                200,
+                                                355,
+                                                213
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T^r"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                200,
+                                                355,
+                                                213
+                                            ],
+                                            "type": "text",
+                                            "content": " to "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                200,
+                                                355,
+                                                213
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tilde{T}^s"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                200,
+                                                355,
+                                                213
+                                            ],
+                                            "type": "text",
+                                            "content": " (ignoring labels "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                200,
+                                                355,
+                                                213
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                200,
+                                                355,
+                                                213
+                                            ],
+                                            "type": "text",
+                                            "content": ");"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 6
+                        },
+                        {
+                            "bbox": [
+                                116,
+                                217,
+                                286,
+                                230
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        217,
+                                        286,
+                                        230
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                217,
+                                                286,
+                                                230
+                                            ],
+                                            "type": "text",
+                                            "content": "c) For any "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                217,
+                                                286,
+                                                230
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t\\in T^r"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                217,
+                                                286,
+                                                230
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho (\\beta_T(t)) = \\beta_{\\bar{T}}(\\tau (t))"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 7
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        104,
+                        237,
+                        506,
+                        336
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                237,
+                                506,
+                                336
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        237,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "text",
+                                    "content": "Equivalent formulation of GNN architectures. Below, we give equivalent definitions for several GNN architectures presented in Section 2, which will be used in subsequent analysis. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        237,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        237,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "text",
+                                    "content": " be a graph and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        237,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u,v\\in V_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        237,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "text",
+                                    "content": ". For all models "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        237,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        237,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "text",
+                                    "content": " including Subgraph GNN, Local 2-GNN, Local 2-FGNN, and 2-FGNN, we define the initial color "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        237,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\chi}_G^{M,(0)}(u,v)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        237,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "text",
+                                    "content": " to be the isomorphism type of vertex pair "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        237,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u,v)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        237,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "text",
+                                    "content": " (i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        237,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{atp}_G(u,v)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        237,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "text",
+                                    "content": " plus labels of each vertex). Note that this matches the original definition except for Subgraph GNN. Then in each iteration "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        237,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        237,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "text",
+                                    "content": ", the color is updated according to the following formula. Here, for clarity, we denote "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        237,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\chi}_G^{M,(t)}(u,S) = \\{\\{\\tilde{\\chi}_G^{M,(t)}(u,v):v\\in S\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        237,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        237,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\chi}_G^{M,(t)}(S,v) = \\{\\{\\tilde{\\chi}_G^{M,(t)}(u,v):u\\in S\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        237,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "text",
+                                    "content": " for any model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        237,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        237,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "text",
+                                    "content": " and set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        237,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        237,
+                                        506,
+                                        336
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        121,
+                        340,
+                        198,
+                        352
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                121,
+                                340,
+                                198,
+                                352
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        340,
+                                        198,
+                                        352
+                                    ],
+                                    "type": "text",
+                                    "content": "- Subgraph GNN:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        159,
+                        355,
+                        505,
+                        376
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                159,
+                                355,
+                                505,
+                                376
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        159,
+                                        355,
+                                        505,
+                                        376
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\tilde {\\chi} _ {G} ^ {\\text {S u b}, (t + 1)} (u, v) = \\operatorname {h a s h} \\left(\\tilde {\\chi} _ {G} ^ {\\text {S u b}, (t)} (u, v), \\tilde {\\chi} _ {G} ^ {\\text {S u b}, (t)} (u, N _ {G} (v)), \\tilde {\\chi} _ {G} ^ {\\text {S u b}, (t)} (u, V _ {G})\\right). \\tag {7}",
+                                    "image_path": "6b8fdf24290badbb1bb3a070c665ab195237d9d390c40b2a975af68951f78a20.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        121,
+                        380,
+                        192,
+                        392
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                121,
+                                380,
+                                192,
+                                392
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        380,
+                                        192,
+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": "- Local 2-GNN:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        169,
+                        395,
+                        504,
+                        437
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                169,
+                                395,
+                                504,
+                                437
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        169,
+                                        395,
+                                        504,
+                                        437
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\tilde {\\chi} _ {G} ^ {\\mathsf {L}, (t + 1)} (u, v) = \\operatorname {h a s h} \\left(\\tilde {\\chi} _ {G} ^ {\\mathsf {L}, (t)} (u, v), \\tilde {\\chi} _ {G} ^ {\\mathsf {L}, (t)} (u, N _ {G} (v)), \\tilde {\\chi} _ {G} ^ {\\mathsf {L}, (t)} \\left(N _ {G} (u), v\\right), \\right. \\tag {8} \\\\ \\left. \\tilde {\\chi} _ {G} ^ {\\mathsf {L}, (t)} (u, V _ {G}), \\tilde {\\chi} _ {G} ^ {\\mathsf {L}, (t)} (V _ {G}, v)\\right). \\\\ \\end{array}",
+                                    "image_path": "4f00d76da0cf6c5aa62548534c44bc416330e1a40d43677bbf2853f03e85f832.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "- Local 2-FGNN:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\tilde {\\chi} _ {G} ^ {\\mathsf {L F}, (t + 1)} (u, v) = \\mathsf {h a s h} \\left(\\tilde {\\chi} _ {G} ^ {\\mathsf {L F}, (t)} (u, v), \\{\\{\\left(\\tilde {\\chi} _ {G} ^ {\\mathsf {L F}, (t)} (w, v), \\tilde {\\chi} _ {G} ^ {\\mathsf {L F}, (t)} (u, w)\\right): w \\in N _ {G} [ u ] \\cup N _ {G} [ v ] \\}, \\right. \\\\ \\left. \\tilde {\\chi} _ {G} ^ {\\mathrm {L F}, (t)} \\left(u, V _ {G}\\right), \\tilde {\\chi} _ {G} ^ {\\mathrm {L F}, (t)} \\left(V _ {G}, v\\right)\\right). \\tag {9} \\\\ \\end{array}",
+                                    "image_path": "b26def3f2133f655115d73c8f9266140a27f9952bbb710ac44c7e91cdcd9a3d9.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
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+                    "type": "text",
+                    "angle": 0,
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "2-FGNN:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
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+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        150,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\tilde {\\chi} _ {G} ^ {\\mathsf {F}, (t + 1)} (u, v) = \\operatorname {h a s h} \\left(\\tilde {\\chi} _ {G} ^ {\\mathsf {F}, (t)} (u, v), \\{\\{\\left(\\tilde {\\chi} _ {G} ^ {\\mathsf {F}, (t)} (w, v), \\tilde {\\chi} _ {G} ^ {\\mathsf {F}, (t)} (u, w)\\right): w \\in V _ {G} \\} \\}, \\right. \\tag {10} \\\\ \\left. \\tilde {\\chi} _ {G} ^ {\\mathsf {F}, (t)} (u, V _ {G}), \\tilde {\\chi} _ {G} ^ {\\mathsf {F}, (t)} (V _ {G}, v)\\right). \\\\ \\end{array}",
+                                    "image_path": "fc23dc0845831971a17643e662075d8ac2b4e68dd679bff6cb7d92e316377642.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
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+                        613
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                                506,
+                                613
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "It can be seen that we additionally add global aggregations for these architectures. Moreover, in Local 2-FGNN we replace the neighbors by closed neighbors "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "N_G[u] \\cup N_G[v]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": ". The stable color of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u,v)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": " for different models is denoted by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\chi}_G^{\\mathrm{Sub}}(u,v)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\chi}_G^{\\mathrm{L}}(u,v)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\chi}_G^{\\mathrm{LF}}(u,v)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\chi}_G^{\\mathrm{F}}(u,v)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        506,
+                                        613
+                                    ],
+                                    "type": "text",
+                                    "content": ", respectively. We have the following result:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
+                        104,
+                        616,
+                        506,
+                        665
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                616,
+                                506,
+                                665
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": "Proposition C.9. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": " be any model. For any graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G, H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\chi}_G^M(G) = \\tilde{\\chi}_H^M(H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^M(G) = \\chi_H^M(H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": ". Furthermore, if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": " are connected, then for any vertices "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v \\in V_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x, y \\in V_H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\chi}_G^M(u, v) = \\tilde{\\chi}_H^M(x, y)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^M(u, v) = \\chi_H^M(x, y)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": ". In other words, the color mapping "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\chi}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": " is as fine as the original one "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        616,
+                                        506,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                },
+                {
+                    "bbox": [
+                        104,
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+                        731
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                            ],
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+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. The proof simply follows from Zhang et al. (2023a, Proposition 4.2 and Theorem 4.4), because (i) node marking in the initial color of Subgraph GNN is as expressive as using the isomorphism type, (ii) the global aggregation does not improve the expressivity when the corresponding local aggregation is presented, (iii) the single-point aggregation does not improve the expressivity in Local 2-FGNN. "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\square"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                }
+            ],
+            "discarded_blocks": [
+                {
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+                    ],
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
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+                                    "type": "text",
+                                    "content": "24"
+                                }
+                            ]
+                        }
+                    ],
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+                }
+            ],
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+                        82,
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+                        94
+                    ],
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "C.2 PART 1: TREE DECOMPOSITION"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "We first define the unfolding tree of different CR algorithms, which is a standard tool in analyzing GNN expressivity."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
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+                    ],
+                    "type": "text",
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+                                {
+                                    "bbox": [
+                                        104,
+                                        126,
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+                                        178
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition C.10 (Unfolding tree of Subgraph GNN). Given a graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        126,
+                                        506,
+                                        178
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        126,
+                                        506,
+                                        178
+                                    ],
+                                    "type": "text",
+                                    "content": ", vertices "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        126,
+                                        506,
+                                        178
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v \\in V_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        126,
+                                        506,
+                                        178
+                                    ],
+                                    "type": "text",
+                                    "content": ", and a non-negative integer "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        126,
+                                        506,
+                                        178
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "D"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        126,
+                                        506,
+                                        178
+                                    ],
+                                    "type": "text",
+                                    "content": ", the depth-2D Subgraph GNN unfolding tree of graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        126,
+                                        506,
+                                        178
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        126,
+                                        506,
+                                        178
+                                    ],
+                                    "type": "text",
+                                    "content": " at "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        126,
+                                        506,
+                                        178
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u, v)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        126,
+                                        506,
+                                        178
+                                    ],
+                                    "type": "text",
+                                    "content": ", denoted as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        126,
+                                        506,
+                                        178
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(F_G^{\\mathrm{Sub},(D)}(u, v), T_G^{\\mathrm{Sub},(D)}(u, v)\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        126,
+                                        506,
+                                        178
+                                    ],
+                                    "type": "text",
+                                    "content": ", is a tree-decomposed graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        126,
+                                        506,
+                                        178
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^{\\mathrm{Sub}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        126,
+                                        506,
+                                        178
+                                    ],
+                                    "type": "text",
+                                    "content": " constructed as follows:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        116,
+                        183,
+                        505,
+                        234
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 6,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                117,
+                                183,
+                                505,
+                                218
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        183,
+                                        505,
+                                        218
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                117,
+                                                183,
+                                                505,
+                                                218
+                                            ],
+                                            "type": "text",
+                                            "content": "1. Initialization. At the beginning, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                183,
+                                                505,
+                                                218
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F = G[\\{u,v\\}]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                183,
+                                                505,
+                                                218
+                                            ],
+                                            "type": "text",
+                                            "content": " (if "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                183,
+                                                505,
+                                                218
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u = v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                183,
+                                                505,
+                                                218
+                                            ],
+                                            "type": "text",
+                                            "content": ", "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                183,
+                                                505,
+                                                218
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                183,
+                                                505,
+                                                218
+                                            ],
+                                            "type": "text",
+                                            "content": " only has one vertex), and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                183,
+                                                505,
+                                                218
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                183,
+                                                505,
+                                                218
+                                            ],
+                                            "type": "text",
+                                            "content": " only has a root node "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                183,
+                                                505,
+                                                218
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "r"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                183,
+                                                505,
+                                                218
+                                            ],
+                                            "type": "text",
+                                            "content": " with "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                183,
+                                                505,
+                                                218
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(r) = \\{\\{u,v\\}\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                183,
+                                                505,
+                                                218
+                                            ],
+                                            "type": "text",
+                                            "content": ". Define a mapping "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                183,
+                                                505,
+                                                218
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pi : V_F \\to V_G"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                183,
+                                                505,
+                                                218
+                                            ],
+                                            "type": "text",
+                                            "content": " as "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                183,
+                                                505,
+                                                218
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pi(u) = u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                183,
+                                                505,
+                                                218
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                183,
+                                                505,
+                                                218
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pi(v) = v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                183,
+                                                505,
+                                                218
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 4
+                        },
+                        {
+                            "bbox": [
+                                116,
+                                222,
+                                439,
+                                234
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        222,
+                                        439,
+                                        234
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                222,
+                                                439,
+                                                234
+                                            ],
+                                            "type": "text",
+                                            "content": "2. Loop for "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                222,
+                                                439,
+                                                234
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "D"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                222,
+                                                439,
+                                                234
+                                            ],
+                                            "type": "text",
+                                            "content": " rounds. For each leaf node "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                222,
+                                                439,
+                                                234
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                222,
+                                                439,
+                                                234
+                                            ],
+                                            "type": "text",
+                                            "content": " in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                222,
+                                                439,
+                                                234
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T^r"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                222,
+                                                439,
+                                                234
+                                            ],
+                                            "type": "text",
+                                            "content": ", do the following procedure:"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 5
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        129,
+                        234,
+                        504,
+                        257
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                234,
+                                504,
+                                257
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        234,
+                                        504,
+                                        257
+                                    ],
+                                    "type": "text",
+                                    "content": "Let "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        234,
+                                        504,
+                                        257
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_T(t) = \\{\\{u,x\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        234,
+                                        504,
+                                        257
+                                    ],
+                                    "type": "text",
+                                    "content": ". For each "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        234,
+                                        504,
+                                        257
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w \\in V_G"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        234,
+                                        504,
+                                        257
+                                    ],
+                                    "type": "text",
+                                    "content": ", add a fresh child node "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        234,
+                                        504,
+                                        257
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t_w"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        234,
+                                        504,
+                                        257
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        234,
+                                        504,
+                                        257
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        234,
+                                        504,
+                                        257
+                                    ],
+                                    "type": "text",
+                                    "content": " and designate "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        234,
+                                        504,
+                                        257
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        234,
+                                        504,
+                                        257
+                                    ],
+                                    "type": "text",
+                                    "content": " as its parent. Then, consider the following three cases:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        141,
+                        261,
+                        505,
+                        330
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 11,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                141,
+                                261,
+                                505,
+                                306
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        261,
+                                        505,
+                                        306
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "text",
+                                            "content": "a) If "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\neq \\pi(u)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\neq \\pi(x)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "text",
+                                            "content": ", then add a fresh vertex "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "z"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "text",
+                                            "content": " to "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "text",
+                                            "content": " and extend "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pi"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "text",
+                                            "content": " with "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pi(z) = w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "text",
+                                            "content": ". The label of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "z"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "text",
+                                            "content": " in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "text",
+                                            "content": " is set by "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\ell_F(z) = \\ell_G(w)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "text",
+                                            "content": ". Define "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(t_w) = \\beta_T(t) \\cup \\{\\{z\\}\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "text",
+                                            "content": ". Then, we add edges between "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "z"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(t)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "text",
+                                            "content": ", so that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pi"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "text",
+                                            "content": " is an isomorphism from "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[\\beta_T(t_w)]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "text",
+                                            "content": " to "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "G[\\pi(\\beta_T(t_w))]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                261,
+                                                505,
+                                                306
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 8
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                305,
+                                501,
+                                318
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        305,
+                                        501,
+                                        318
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                305,
+                                                501,
+                                                318
+                                            ],
+                                            "type": "text",
+                                            "content": "b) If "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                305,
+                                                501,
+                                                318
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w = \\pi(u)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                305,
+                                                501,
+                                                318
+                                            ],
+                                            "type": "text",
+                                            "content": ", then we simply set "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                305,
+                                                501,
+                                                318
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(t_w) = \\beta_T(t) \\cup \\{\\{u\\}\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                305,
+                                                501,
+                                                318
+                                            ],
+                                            "type": "text",
+                                            "content": " without modifying graph "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                305,
+                                                501,
+                                                318
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                305,
+                                                501,
+                                                318
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 9
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                318,
+                                501,
+                                330
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        318,
+                                        501,
+                                        330
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                318,
+                                                501,
+                                                330
+                                            ],
+                                            "type": "text",
+                                            "content": "c) If "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                318,
+                                                501,
+                                                330
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w = \\pi(x)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                318,
+                                                501,
+                                                330
+                                            ],
+                                            "type": "text",
+                                            "content": ", then we simply set "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                318,
+                                                501,
+                                                330
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(t_w) = \\beta_T(t) \\cup \\{x\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                318,
+                                                501,
+                                                330
+                                            ],
+                                            "type": "text",
+                                            "content": " without modifying graph "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                318,
+                                                501,
+                                                330
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                318,
+                                                501,
+                                                330
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 10
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        129,
+                        334,
+                        504,
+                        357
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                334,
+                                504,
+                                357
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        334,
+                                        504,
+                                        357
+                                    ],
+                                    "type": "text",
+                                    "content": "Finally, add a fresh child node "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "t_w'"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        334,
+                                        504,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        334,
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+                                    ],
+                                    "type": "text",
+                                    "content": ", designate "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        334,
+                                        504,
+                                        357
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t_w"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        334,
+                                        504,
+                                        357
+                                    ],
+                                    "type": "text",
+                                    "content": " as its parent, and set "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        334,
+                                        504,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_T(t_w')"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        334,
+                                        504,
+                                        357
+                                    ],
+                                    "type": "text",
+                                    "content": " based on the following three cases:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        141,
+                        361,
+                        366,
+                        397
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 16,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                141,
+                                361,
+                                366,
+                                374
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                        {
+                                            "bbox": [
+                                                141,
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+                                            ],
+                                            "type": "text",
+                                            "content": "a) If "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                361,
+                                                366,
+                                                374
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\neq \\pi(u)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                361,
+                                                366,
+                                                374
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                361,
+                                                366,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\neq \\pi(x)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                361,
+                                                366,
+                                                374
+                                            ],
+                                            "type": "text",
+                                            "content": ", then "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                361,
+                                                366,
+                                                374
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(t_w') = \\{\\{u, z\\}\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                361,
+                                                366,
+                                                374
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 13
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                374,
+                                307,
+                                386
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        374,
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+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                374,
+                                                307,
+                                                386
+                                            ],
+                                            "type": "text",
+                                            "content": "b) If "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                374,
+                                                307,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w = \\pi(u)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                374,
+                                                307,
+                                                386
+                                            ],
+                                            "type": "text",
+                                            "content": ", then "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                374,
+                                                307,
+                                                386
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(t_w') = \\{\\{u, u\\}\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                374,
+                                                307,
+                                                386
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 14
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                386,
+                                307,
+                                397
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
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+                                        307,
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+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                386,
+                                                307,
+                                                397
+                                            ],
+                                            "type": "text",
+                                            "content": "c) If "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                386,
+                                                307,
+                                                397
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w = \\pi(x)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                386,
+                                                307,
+                                                397
+                                            ],
+                                            "type": "text",
+                                            "content": ", then "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                386,
+                                                307,
+                                                397
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(t_w') = \\{\\{u, x\\}\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                386,
+                                                307,
+                                                397
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 15
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        104,
+                        402,
+                        505,
+                        437
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                402,
+                                505,
+                                437
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        402,
+                                        505,
+                                        437
+                                    ],
+                                    "type": "text",
+                                    "content": "It is easy to see that the depth of tree "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        402,
+                                        505,
+                                        437
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        402,
+                                        505,
+                                        437
+                                    ],
+                                    "type": "text",
+                                    "content": " increases by 2 after each round, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        402,
+                                        505,
+                                        437
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        402,
+                                        505,
+                                        437
+                                    ],
+                                    "type": "text",
+                                    "content": " is always a canonical tree decomposition of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        402,
+                                        505,
+                                        437
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        402,
+                                        505,
+                                        437
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        402,
+                                        505,
+                                        437
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^{\\mathsf{Sub}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        402,
+                                        505,
+                                        437
+                                    ],
+                                    "type": "text",
+                                    "content": ". An illustration of the construction of unfolding tree is given in Figure 6(a)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        104,
+                        443,
+                        504,
+                        466
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                443,
+                                504,
+                                466
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        443,
+                                        504,
+                                        466
+                                    ],
+                                    "type": "text",
+                                    "content": "We next define the unfolding tree of Local 2-GNN, which differs in the loop part such that the bags "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        443,
+                                        504,
+                                        466
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_{T}(t_{w})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        443,
+                                        504,
+                                        466
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        443,
+                                        504,
+                                        466
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_{T}(t_{w}^{\\prime})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        443,
+                                        504,
+                                        466
+                                    ],
+                                    "type": "text",
+                                    "content": " now do not necessarily contain "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        443,
+                                        504,
+                                        466
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        443,
+                                        504,
+                                        466
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
+                        104,
+                        467,
+                        505,
+                        509
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                467,
+                                505,
+                                509
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        467,
+                                        505,
+                                        509
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition C.11 (Unfolding tree of Local 2-GNN). Given a graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        467,
+                                        505,
+                                        509
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        467,
+                                        505,
+                                        509
+                                    ],
+                                    "type": "text",
+                                    "content": ", vertices "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        467,
+                                        505,
+                                        509
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v \\in V_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        467,
+                                        505,
+                                        509
+                                    ],
+                                    "type": "text",
+                                    "content": ", and a non-negative integer "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        467,
+                                        505,
+                                        509
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "D"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        467,
+                                        505,
+                                        509
+                                    ],
+                                    "type": "text",
+                                    "content": ", the depth-2D Local 2-GNN unfolding tree of graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        467,
+                                        505,
+                                        509
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        467,
+                                        505,
+                                        509
+                                    ],
+                                    "type": "text",
+                                    "content": " at "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        467,
+                                        505,
+                                        509
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u, v)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        467,
+                                        505,
+                                        509
+                                    ],
+                                    "type": "text",
+                                    "content": ", denoted as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        467,
+                                        505,
+                                        509
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(F_G^{\\mathsf{L},(D)}(u, v), T_G^{\\mathsf{L},(D)}(u, v)\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        467,
+                                        505,
+                                        509
+                                    ],
+                                    "type": "text",
+                                    "content": ", is a tree-decomposed graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        467,
+                                        505,
+                                        509
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^\\mathsf{L}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        467,
+                                        505,
+                                        509
+                                    ],
+                                    "type": "text",
+                                    "content": " constructed as follows:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                },
+                {
+                    "bbox": [
+                        117,
+                        513,
+                        479,
+                        541
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 22,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                117,
+                                513,
+                                479,
+                                525
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        513,
+                                        479,
+                                        525
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                117,
+                                                513,
+                                                479,
+                                                525
+                                            ],
+                                            "type": "text",
+                                            "content": "1. Initialization. The procedure is exactly the same as Subgraph GNN (Definition C.10)."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 20
+                        },
+                        {
+                            "bbox": [
+                                117,
+                                529,
+                                439,
+                                541
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        529,
+                                        439,
+                                        541
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                117,
+                                                529,
+                                                439,
+                                                541
+                                            ],
+                                            "type": "text",
+                                            "content": "2. Loop for "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                529,
+                                                439,
+                                                541
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "D"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                529,
+                                                439,
+                                                541
+                                            ],
+                                            "type": "text",
+                                            "content": " rounds. For each leaf node "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                529,
+                                                439,
+                                                541
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                529,
+                                                439,
+                                                541
+                                            ],
+                                            "type": "text",
+                                            "content": " in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                529,
+                                                439,
+                                                541
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T^r"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                529,
+                                                439,
+                                                541
+                                            ],
+                                            "type": "text",
+                                            "content": ", do the following procedure:"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 21
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        129,
+                        541,
+                        504,
+                        563
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                541,
+                                504,
+                                563
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        541,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "text",
+                                    "content": "Let "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        541,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_T(t) = \\{\\{x,y\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        541,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "text",
+                                    "content": ". For each "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        541,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w \\in V_G"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        541,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "text",
+                                    "content": ", add a fresh child node "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        541,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t_w"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        541,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        541,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        541,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "text",
+                                    "content": " and designate "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        541,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        541,
+                                        504,
+                                        563
+                                    ],
+                                    "type": "text",
+                                    "content": " as its parent. Then, consider the following three cases:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 23
+                },
+                {
+                    "bbox": [
+                        141,
+                        569,
+                        505,
+                        637
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 27,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                141,
+                                569,
+                                505,
+                                613
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
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+                                                505,
+                                                613
+                                            ],
+                                            "type": "text",
+                                            "content": "a) If "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\neq \\pi(x)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\neq \\pi(y)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "text",
+                                            "content": ", then add a fresh vertex "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "z"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "text",
+                                            "content": " to "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "text",
+                                            "content": " and extend "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pi"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "text",
+                                            "content": " with "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pi(z) = w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "text",
+                                            "content": ". The label of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "z"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "text",
+                                            "content": " in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "text",
+                                            "content": " is set by "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\ell_F(z) = \\ell_G(w)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "text",
+                                            "content": ". Define "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(t_w) = \\beta_T(t) \\cup \\{\\{z\\}\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "text",
+                                            "content": ". Then, we add edges between "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "z"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(t)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "text",
+                                            "content": ", so that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pi"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "text",
+                                            "content": " is an isomorphism from "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[\\beta_T(t_w)]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "text",
+                                            "content": " to "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "G[\\pi(\\beta_T(t_w))]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                569,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 24
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                613,
+                                501,
+                                625
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        613,
+                                        501,
+                                        625
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                613,
+                                                501,
+                                                625
+                                            ],
+                                            "type": "text",
+                                            "content": "b) If "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                613,
+                                                501,
+                                                625
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w = \\pi(x)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                613,
+                                                501,
+                                                625
+                                            ],
+                                            "type": "text",
+                                            "content": ", then we simply set "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                613,
+                                                501,
+                                                625
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(t_w) = \\beta_T(t) \\cup \\{\\{x\\}\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                613,
+                                                501,
+                                                625
+                                            ],
+                                            "type": "text",
+                                            "content": " without modifying graph "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                613,
+                                                501,
+                                                625
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                613,
+                                                501,
+                                                625
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 25
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                625,
+                                501,
+                                637
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        625,
+                                        501,
+                                        637
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                625,
+                                                501,
+                                                637
+                                            ],
+                                            "type": "text",
+                                            "content": "c) If "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                625,
+                                                501,
+                                                637
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w = \\pi(y)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                625,
+                                                501,
+                                                637
+                                            ],
+                                            "type": "text",
+                                            "content": ", then we simply set "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                625,
+                                                501,
+                                                637
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(t_w) = \\beta_T(t) \\cup \\{\\{y\\}\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                625,
+                                                501,
+                                                637
+                                            ],
+                                            "type": "text",
+                                            "content": " without modifying graph "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                625,
+                                                501,
+                                                637
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                625,
+                                                501,
+                                                637
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 26
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        129,
+                        641,
+                        504,
+                        665
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Next, add a fresh child node "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        641,
+                                        504,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t_w'"
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+                                    "bbox": [
+                                        129,
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+                                    "type": "text",
+                                    "content": " in "
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+                                {
+                                    "bbox": [
+                                        129,
+                                        641,
+                                        504,
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+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
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+                                    ],
+                                    "type": "text",
+                                    "content": ", designate "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        641,
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+                                    "type": "inline_equation",
+                                    "content": "t_w"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
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+                                    "type": "text",
+                                    "content": " as its parent, and set "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        641,
+                                        504,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_T(t_w')"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
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+                                    ],
+                                    "type": "text",
+                                    "content": " based on the following three cases:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 28
+                },
+                {
+                    "bbox": [
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+                        365,
+                        704
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 32,
+                    "blocks": [
+                        {
+                            "bbox": [
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+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
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+                                    "bbox": [
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+                                            "type": "text",
+                                            "content": "a) If "
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+                                        {
+                                            "bbox": [
+                                                141,
+                                                669,
+                                                365,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\neq \\pi(x)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
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+                                            ],
+                                            "type": "text",
+                                            "content": " and "
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+                                            "bbox": [
+                                                141,
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+                                            "type": "inline_equation",
+                                            "content": "w \\neq \\pi(y)"
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+                                        {
+                                            "bbox": [
+                                                141,
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+                                            ],
+                                            "type": "text",
+                                            "content": ", then "
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+                                        {
+                                            "bbox": [
+                                                141,
+                                                669,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(t'_w) = \\{\\{x, z\\}\\}"
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+                                        {
+                                            "bbox": [
+                                                141,
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+                                                365,
+                                                681
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 29
+                        },
+                        {
+                            "bbox": [
+                                141,
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+                                307,
+                                693
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
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+                                    "bbox": [
+                                        141,
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+                                            "bbox": [
+                                                141,
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+                                            "type": "text",
+                                            "content": "b) If "
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+                                        {
+                                            "bbox": [
+                                                141,
+                                                681,
+                                                307,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w = \\pi(x)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                681,
+                                                307,
+                                                693
+                                            ],
+                                            "type": "text",
+                                            "content": ", then "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                681,
+                                                307,
+                                                693
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(t'_w) = \\{ \\{x, x\\} \\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
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+                                                307,
+                                                693
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 30
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                693,
+                                307,
+                                704
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
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+                                    "bbox": [
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+                                    "spans": [
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+                                            "bbox": [
+                                                141,
+                                                693,
+                                                307,
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+                                            ],
+                                            "type": "text",
+                                            "content": "c) If "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                693,
+                                                307,
+                                                704
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w = \\pi(y)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                693,
+                                                307,
+                                                704
+                                            ],
+                                            "type": "text",
+                                            "content": ", then "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                693,
+                                                307,
+                                                704
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(t'_w) = \\{ \\{x, y\\} \\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                693,
+                                                307,
+                                                704
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 31
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
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+                            "bbox": [
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+                                    "type": "text",
+                                    "content": "Finally, we repeat the above procedure (point 2) once more, but this time the bag "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\beta_{T}(t_{w}^{\\prime})"
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " is replaced by the following three cases (changing "
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+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x"
+                                },
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+                                    "type": "text",
+                                    "content": " to "
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+                                    "type": "inline_equation",
+                                    "content": "y"
+                                },
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+                                    ],
+                                    "type": "text",
+                                    "content": "):"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 33
+                }
+            ],
+            "discarded_blocks": [
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+                    ],
+                    "type": "header",
+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
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+                                    "content": "25"
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+                                            "content": "T_{G}^{\\mathsf{F},(1)}(1,2)"
+                                        }
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+                                            "type": "text",
+                                            "content": "Figure 6: The depth-2 unfolding tree of graph "
+                                        },
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+                                            "type": "text",
+                                            "content": " at vertex pair (1,2) for Subgraph GNN, Local 2-GNN, Local 2-FGNN, and 2-FGNN, respectively."
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+                                            "type": "text",
+                                            "content": "a) If "
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+                                            "bbox": [
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+                                            "type": "inline_equation",
+                                            "content": "w \\neq \\pi(x)"
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+                                            "type": "text",
+                                            "content": " and "
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+                                            "type": "inline_equation",
+                                            "content": "w \\neq \\pi(y)"
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+                                            "type": "text",
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+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(t'_w) = \\{\\{y, z\\}\\}"
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+                                            "type": "text",
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+                        {
+                            "bbox": [
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+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
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+                                            "type": "text",
+                                            "content": "b) If "
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+                                        {
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+                                            "type": "inline_equation",
+                                            "content": "w = \\pi(y)"
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+                                            "type": "text",
+                                            "content": ", then "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
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+                                                497
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(t'_w) = \\{\\{y, y\\}\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
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+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 31
+                        },
+                        {
+                            "bbox": [
+                                141,
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+                                306,
+                                510
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                    "spans": [
+                                        {
+                                            "bbox": [
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+                                                497,
+                                                306,
+                                                510
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+                                            "type": "text",
+                                            "content": "c) If "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                497,
+                                                306,
+                                                510
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w = \\pi(x)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                497,
+                                                306,
+                                                510
+                                            ],
+                                            "type": "text",
+                                            "content": ", then "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                497,
+                                                306,
+                                                510
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(t_w') = \\{\\{x, y\\}\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                497,
+                                                306,
+                                                510
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 32
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        105,
+                        514,
+                        406,
+                        526
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "An illustration of the construction of unfolding tree is given in Figure 6(b)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 34
+                },
+                {
+                    "bbox": [
+                        104,
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        535,
+                                        504,
+                                        557
+                                    ],
+                                    "type": "text",
+                                    "content": "We next define the unfolding tree of Local 2-FGNN, which differs in the loop part such that the node "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        535,
+                                        504,
+                                        557
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t_w"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        535,
+                                        504,
+                                        557
+                                    ],
+                                    "type": "text",
+                                    "content": " can have two children under certain conditions."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 35
+                },
+                {
+                    "bbox": [
+                        105,
+                        561,
+                        504,
+                        603
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                561,
+                                504,
+                                603
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        561,
+                                        504,
+                                        603
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition C.12 (Unfolding tree of Local 2-FGNN). Given a graph "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        561,
+                                        504,
+                                        603
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        561,
+                                        504,
+                                        603
+                                    ],
+                                    "type": "text",
+                                    "content": ", vertices "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        561,
+                                        504,
+                                        603
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v \\in V_G"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        561,
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+                                        603
+                                    ],
+                                    "type": "text",
+                                    "content": ", and a non-negative integer "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        561,
+                                        504,
+                                        603
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "D"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        561,
+                                        504,
+                                        603
+                                    ],
+                                    "type": "text",
+                                    "content": ", the depth-2D Local 2-FGNN unfolding tree of graph "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        561,
+                                        504,
+                                        603
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        561,
+                                        504,
+                                        603
+                                    ],
+                                    "type": "text",
+                                    "content": " at "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        561,
+                                        504,
+                                        603
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u, v)"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        561,
+                                        504,
+                                        603
+                                    ],
+                                    "type": "text",
+                                    "content": ", denoted as "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        561,
+                                        504,
+                                        603
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(F_G^{\\mathsf{LF},(D)}(u, v), T_G^{\\mathsf{LF},(D)}(u, v)\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        561,
+                                        504,
+                                        603
+                                    ],
+                                    "type": "text",
+                                    "content": ", is a tree-decomposed graph "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        561,
+                                        504,
+                                        603
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^{\\mathsf{LF}}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        561,
+                                        504,
+                                        603
+                                    ],
+                                    "type": "text",
+                                    "content": " constructed as follows:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 36
+                },
+                {
+                    "bbox": [
+                        116,
+                        607,
+                        479,
+                        635
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 39,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                116,
+                                607,
+                                479,
+                                620
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        607,
+                                        479,
+                                        620
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                607,
+                                                479,
+                                                620
+                                            ],
+                                            "type": "text",
+                                            "content": "1. Initialization. The procedure is exactly the same as Subgraph GNN (Definition C.10)."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 37
+                        },
+                        {
+                            "bbox": [
+                                116,
+                                624,
+                                439,
+                                635
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        624,
+                                        439,
+                                        635
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
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+                                            ],
+                                            "type": "text",
+                                            "content": "2. Loop for "
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "D"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                624,
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+                                            ],
+                                            "type": "text",
+                                            "content": " rounds. For each leaf node "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                624,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                624,
+                                                439,
+                                                635
+                                            ],
+                                            "type": "text",
+                                            "content": " in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                624,
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+                                            "type": "inline_equation",
+                                            "content": "T^r"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                624,
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+                                            ],
+                                            "type": "text",
+                                            "content": ", do the following procedure:"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 38
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        129,
+                        635,
+                        504,
+                        658
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                635,
+                                504,
+                                658
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        635,
+                                        504,
+                                        658
+                                    ],
+                                    "type": "text",
+                                    "content": "Let "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        635,
+                                        504,
+                                        658
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_T(t) = \\{\\{x,y\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        635,
+                                        504,
+                                        658
+                                    ],
+                                    "type": "text",
+                                    "content": ". For each "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        635,
+                                        504,
+                                        658
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w \\in N_G[\\pi(x)] \\cup N_G[\\pi(y)]"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        635,
+                                        504,
+                                        658
+                                    ],
+                                    "type": "text",
+                                    "content": ", add a fresh child node "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        635,
+                                        504,
+                                        658
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t_w"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        635,
+                                        504,
+                                        658
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        635,
+                                        504,
+                                        658
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        635,
+                                        504,
+                                        658
+                                    ],
+                                    "type": "text",
+                                    "content": " and designate "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        635,
+                                        504,
+                                        658
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        635,
+                                        504,
+                                        658
+                                    ],
+                                    "type": "text",
+                                    "content": " as its parent. Then, consider the following three cases:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 40
+                },
+                {
+                    "bbox": [
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+                        663,
+                        505,
+                        733
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 44,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                141,
+                                663,
+                                505,
+                                708
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        663,
+                                        505,
+                                        708
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                663,
+                                                505,
+                                                708
+                                            ],
+                                            "type": "text",
+                                            "content": "a) If "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                663,
+                                                505,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\neq \\pi(x)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                663,
+                                                505,
+                                                708
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                663,
+                                                505,
+                                                708
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\neq \\pi(y)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                663,
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+                                                708
+                                            ],
+                                            "type": "text",
+                                            "content": ", then add a fresh vertex "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "z"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
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+                                            ],
+                                            "type": "text",
+                                            "content": " to "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                663,
+                                                505,
+                                                708
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                663,
+                                                505,
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+                                            ],
+                                            "type": "text",
+                                            "content": " and extend "
+                                        },
+                                        {
+                                            "bbox": [
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+                                            "type": "text",
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+                                            "type": "inline_equation",
+                                            "content": "\\pi(z) = w"
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+                                            "type": "text",
+                                            "content": ". The label of "
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+                                            "type": "inline_equation",
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+                                            "type": "inline_equation",
+                                            "content": "\\ell_F(z) = \\ell_G(w)"
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+                                            "type": "text",
+                                            "content": ". Define "
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+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(t_w) = \\beta_T(t) \\cup \\{\\{z\\}\\}"
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+                                            "type": "text",
+                                            "content": ". Then, we add edges between "
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+                                            "type": "inline_equation",
+                                            "content": "F[\\beta_T(t_w)]"
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+                                            "content": ", then we simply set "
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+                                            "type": "inline_equation",
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+                                            "content": "c) If "
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+                                    "content": "For each "
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+                                    "bbox": [
+                                        105,
+                                        178,
+                                        406,
+                                        190
+                                    ],
+                                    "type": "text",
+                                    "content": "An illustration of the construction of unfolding tree is given in Figure 6(c)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        104,
+                        199,
+                        504,
+                        220
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                199,
+                                504,
+                                220
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        199,
+                                        504,
+                                        220
+                                    ],
+                                    "type": "text",
+                                    "content": "We finally define the unfolding tree of 2-FGNN, which differs in the loop part such that all nodes "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        199,
+                                        504,
+                                        220
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t_w"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        199,
+                                        504,
+                                        220
+                                    ],
+                                    "type": "text",
+                                    "content": " have two children."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        105,
+                        224,
+                        505,
+                        266
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                224,
+                                505,
+                                266
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        224,
+                                        505,
+                                        266
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition C.13 (Unfolding tree of 2-FGNN). Given a graph "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        224,
+                                        505,
+                                        266
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        224,
+                                        505,
+                                        266
+                                    ],
+                                    "type": "text",
+                                    "content": ", vertices "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        224,
+                                        505,
+                                        266
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v \\in V_G"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        224,
+                                        505,
+                                        266
+                                    ],
+                                    "type": "text",
+                                    "content": ", and a non-negative integer "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        224,
+                                        505,
+                                        266
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "D"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        224,
+                                        505,
+                                        266
+                                    ],
+                                    "type": "text",
+                                    "content": ", the depth-2D 2-FGNN unfolding tree of graph "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        224,
+                                        505,
+                                        266
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        224,
+                                        505,
+                                        266
+                                    ],
+                                    "type": "text",
+                                    "content": " at "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        224,
+                                        505,
+                                        266
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u, v)"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        224,
+                                        505,
+                                        266
+                                    ],
+                                    "type": "text",
+                                    "content": ", denoted as "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        224,
+                                        505,
+                                        266
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(F_G^{\\mathrm{Sub},(D)}(u, v), T_G^{\\mathrm{Sub},(D)}(u, v)\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        224,
+                                        505,
+                                        266
+                                    ],
+                                    "type": "text",
+                                    "content": ", is a tree-decomposed graph "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        224,
+                                        505,
+                                        266
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^{\\mathsf{F}}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        224,
+                                        505,
+                                        266
+                                    ],
+                                    "type": "text",
+                                    "content": " constructed as follows:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        117,
+                        270,
+                        504,
+                        310
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 12,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                117,
+                                270,
+                                476,
+                                282
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        270,
+                                        476,
+                                        282
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                117,
+                                                270,
+                                                476,
+                                                282
+                                            ],
+                                            "type": "text",
+                                            "content": "1. Initialization. The procedure is exactly the same as Local 2-FGNN (Definition C.12)."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 10
+                        },
+                        {
+                            "bbox": [
+                                117,
+                                287,
+                                504,
+                                310
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        287,
+                                        504,
+                                        310
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                117,
+                                                287,
+                                                504,
+                                                310
+                                            ],
+                                            "type": "text",
+                                            "content": "2. Loop for "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                287,
+                                                504,
+                                                310
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "D"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                287,
+                                                504,
+                                                310
+                                            ],
+                                            "type": "text",
+                                            "content": " rounds. The procedure is similar to Local 2-FGNN (Definition C.12) except that the condition "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                287,
+                                                504,
+                                                310
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\in N_G[\\pi(x)] \\cup N_G[\\pi(y)]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                287,
+                                                504,
+                                                310
+                                            ],
+                                            "type": "text",
+                                            "content": " is relaxed to all vertices."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 11
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        105,
+                        315,
+                        406,
+                        327
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                315,
+                                406,
+                                327
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        315,
+                                        406,
+                                        327
+                                    ],
+                                    "type": "text",
+                                    "content": "An illustration of the construction of unfolding tree is given in Figure 6(d)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        105,
+                        335,
+                        301,
+                        346
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                335,
+                                301,
+                                346
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        335,
+                                        301,
+                                        346
+                                    ],
+                                    "type": "text",
+                                    "content": "We are now ready to present the first core result:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        104,
+                        349,
+                        504,
+                        406
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                349,
+                                504,
+                                406
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        349,
+                                        504,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma C.14. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        349,
+                                        504,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        349,
+                                        504,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": " be any model. For any two graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        349,
+                                        504,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G, H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        349,
+                                        504,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": ", any vertices "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        349,
+                                        504,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v \\in V_G, x, y \\in V_H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        349,
+                                        504,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": ", and any non-negative integer "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        349,
+                                        504,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "D"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        349,
+                                        504,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        349,
+                                        504,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\chi}_G^{M,(D)}(u,v) = \\tilde{\\chi}_H^{M,(D)}(x,y)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        349,
+                                        504,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": " iff there exists an isomorphism "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        349,
+                                        504,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\rho, \\tau)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        349,
+                                        504,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": " from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        349,
+                                        504,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(F_G^{M,(D)}(u,v), T_G^{M,(D)}(u,v)\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        349,
+                                        504,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        349,
+                                        504,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(F_H^{M,(D)}(x,y), T_H^{M,(D)}(x,y)\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        349,
+                                        504,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        349,
+                                        504,
+                                        406
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho(u) = x, \\rho(v) = y"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        349,
+                                        504,
+                                        406
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        104,
+                        417,
+                        504,
+                        441
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                417,
+                                504,
+                                441
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        417,
+                                        504,
+                                        441
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Here, we only give the proof for Local 2-GNN, and the proofs for Subgraph GNN, Local 2-FGNN and 2-FGNN are almost the same so we omit them for clarity."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        104,
+                        445,
+                        504,
+                        479
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                445,
+                                504,
+                                479
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        445,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof for Local 2-GNN. The proof is based on induction over "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        445,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "D"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        445,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "text",
+                                    "content": ". When "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        445,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "D = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        445,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "text",
+                                    "content": ", the theorem obviously holds. Now assume that the theorem holds for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        445,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "D \\leq d"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        445,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "text",
+                                    "content": ", and consider "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        445,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "D = d + 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        445,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "text",
+                                    "content": ". Below, we omit "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        445,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{L}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        445,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "text",
+                                    "content": " in the corner mark for clarity."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        117,
+                        484,
+                        504,
+                        533
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                117,
+                                484,
+                                504,
+                                533
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        484,
+                                        504,
+                                        533
+                                    ],
+                                    "type": "text",
+                                    "content": "1. We first prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        484,
+                                        504,
+                                        533
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\chi}_G^{(d + 1)}(u,v) = \\tilde{\\chi}_H^{(d + 1)}(x,y)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        484,
+                                        504,
+                                        533
+                                    ],
+                                    "type": "text",
+                                    "content": " implies that there exists an isomorphism "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        484,
+                                        504,
+                                        533
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\rho ,\\tau)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        484,
+                                        504,
+                                        533
+                                    ],
+                                    "type": "text",
+                                    "content": " from "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        484,
+                                        504,
+                                        533
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(F_G^{(d + 1)}(u,v),T_G^{(d + 1)}(u,v)\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        484,
+                                        504,
+                                        533
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        484,
+                                        504,
+                                        533
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(F_H^{(d + 1)}(x,y),T_H^{(d + 1)}(x,y)\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        484,
+                                        504,
+                                        533
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        484,
+                                        504,
+                                        533
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho (u) = x,\\rho (v) = y"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        484,
+                                        504,
+                                        533
+                                    ],
+                                    "type": "text",
+                                    "content": " . If "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        484,
+                                        504,
+                                        533
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\chi}_G^{(d + 1)}(u,v) = \\tilde{\\chi}_H^{(d + 1)}(x,y)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        484,
+                                        504,
+                                        533
+                                    ],
+                                    "type": "text",
+                                    "content": " , then"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
+                        138,
+                        536,
+                        504,
+                        571
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                138,
+                                536,
+                                504,
+                                571
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        138,
+                                        536,
+                                        504,
+                                        571
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\left\\{\\left\\{\\left(\\tilde {\\chi} _ {G} ^ {(d)} (u, w), \\mathsf {a t p} _ {G} (u, v, w)\\right): w \\in V _ {G} \\right\\} \\right\\} = \\left\\{\\left\\{\\left(\\tilde {\\chi} _ {H} ^ {(d)} (x, z), \\mathsf {a t p} _ {H} (x, y, z)\\right): z \\in V _ {H} \\right\\}, \\right. (11) \\\\ \\{\\{(\\tilde {\\chi} _ {G} ^ {(d)} (w, v), \\mathsf {a t p} _ {G} (u, v, w)): w \\in V _ {G} \\} \\} = \\{\\{(\\tilde {\\chi} _ {H} ^ {(d)} (z, y), \\mathsf {a t p} _ {H} (x, y, z)): z \\in V _ {H} \\}. (12) \\\\ \\end{array}",
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+                                    "type": "inline_equation",
+                                    "content": "n = |V_G| = |V_H|"
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+                                    "content": ". Thus, we can denote "
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+                                    "type": "inline_equation",
+                                    "content": "V_G = \\{w_1, \\dots, w_n\\} = \\{w_1', \\dots, w_n'\\}"
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+                                    "content": "V_H = \\{z_1, \\dots, z_n\\} = \\{z_1', \\dots, z_n'\\}"
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+                                            "content": "(\\tilde{\\chi}_G^{(d)}(w_i', v), \\mathsf{atp}_G(u, v, w_i')) = (\\tilde{\\chi}_H^{(d)}(z_i', y), \\mathsf{atp}_H(x, y, z_i'))"
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+                                            "type": "inline_equation",
+                                            "content": "i \\in [n]"
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+                                            "type": "text",
+                                            "content": "."
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+                                    "type": "text",
+                                    "content": "On the other hand, by definition of tree unfolding, we have"
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+                                    "content": "F _ {G} ^ {(d + 1)} (u, v) = \\left(\\bigcup_ {w _ {i}} F _ {G} ^ {(d)} (u, w _ {i})\\right) \\cup \\left(\\bigcup_ {w _ {i} ^ {\\prime}} F _ {G} ^ {(d)} (w _ {i} ^ {\\prime}, v)\\right) \\cup F _ {G} ^ {(1)} (u, v),",
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+                                    "type": "interline_equation",
+                                    "content": "F _ {H} ^ {(d + 1)} (x, y) = \\left(\\bigcup_ {z _ {i}} F _ {H} ^ {(d)} (x, z _ {i})\\right) \\cup \\left(\\bigcup_ {z _ {i} ^ {\\prime}} F _ {H} ^ {(d)} (z _ {i} ^ {\\prime}, y)\\right) \\cup F _ {H} ^ {(1)} (x, y),",
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+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "type": "text",
+                                    "content": " represents the graph union. Here, all "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        228,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_{i}, w_{j}^{\\prime} \\notin \\{u, v\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        228,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "text",
+                                    "content": " in different graphs are treated as different vertices when taking the union, while "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        228,
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+                                        262
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
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+                                    ],
+                                    "type": "text",
+                                    "content": " in different graphs are shared. See Figure 7 for an illustration of the above equations."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "By induction, there exists an isomorphism "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\rho_i,\\tau_i)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
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+                                    ],
+                                    "type": "text",
+                                    "content": " from "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(F_G^{(d)}(u,w_i),T_G^{(d)}(u,w_i)\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
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+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(F_H^{(d)}(x,z_i),T_H^{(d)}(x,z_i)\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
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+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
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+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_{i}(u) = x"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_{i}(w_{i}) = z_{i}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(i\\in [n])"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
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+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": ", and there exist an isomorphism "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_{i}^{\\prime}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": " from "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(F_G^{(d)}(w_i',v),T_G^{(d)}(w_i',v)\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(F_H^{(d)}(z_i',y),T_H^{(d)}(z_i',y)\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_{i}^{\\prime}(v) = y"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_{i}^{\\prime}(w_{i}^{\\prime}) = z_{i}^{\\prime}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(i\\in [n])"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": ". Moreover, we have "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{atp}_G(u,v,w_i) = \\mathsf{atp}_H(x,y,z_i)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{atp}_G(u,v,w_i') = \\mathsf{atp}_H(x,y,z_i')"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": " for "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i\\in [n]"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": ", which implies that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F_{G}^{(1)}(u,v)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": " is isomorphic to "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F_H^{(1)}(x,y)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": ". Therefore, if we construct "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\rho}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": " by merging all "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_{i}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_{i}'"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(i\\in [n])"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": ", and construct "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\tau}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": " by merging all "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau_{i}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau_{i}'"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": " and further specifying an appropriate mapping between tree nodes of depth no more than 1 in "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T_G^{(d + 1)}(u,v)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T_H^{(d + 1)}(x,y)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": ", then it is straightforward to see that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{\\rho},\\tilde{\\tau})"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": " is well-defined and is an isomorphism from "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(F_G^{(d + 1)}(u,v),T_G^{(d + 1)}(u,v)\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(F_H^{(d + 1)}(x,y),T_H^{(d + 1)}(x,y)\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\rho}(u) = x"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\rho}(v) = y"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        267,
+                                        506,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        117,
+                        431,
+                        504,
+                        477
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                117,
+                                431,
+                                504,
+                                477
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        431,
+                                        504,
+                                        477
+                                    ],
+                                    "type": "text",
+                                    "content": "2. We next prove that if there exists an isomorphism "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        431,
+                                        504,
+                                        477
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\rho, \\tau)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        431,
+                                        504,
+                                        477
+                                    ],
+                                    "type": "text",
+                                    "content": " from the tree-decomposed graph "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        431,
+                                        504,
+                                        477
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(F_G^{(d+1)}(u, v), T_G^{(d+1)}(u, v)\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        431,
+                                        504,
+                                        477
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        431,
+                                        504,
+                                        477
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(F_H^{(d+1)}(x, y), T_H^{(d+1)}(x, y)\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        431,
+                                        504,
+                                        477
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        431,
+                                        504,
+                                        477
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho(u) = x"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        431,
+                                        504,
+                                        477
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        431,
+                                        504,
+                                        477
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho(v) = y"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        431,
+                                        504,
+                                        477
+                                    ],
+                                    "type": "text",
+                                    "content": ", then "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        431,
+                                        504,
+                                        477
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\chi}_G^{(d+1)}(u, v) = \\tilde{\\chi}_H^{(d+1)}(x, y)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        431,
+                                        504,
+                                        477
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        129,
+                        476,
+                        505,
+                        570
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                476,
+                                505,
+                                570
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "text",
+                                    "content": "Without loss of generality, assume "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\neq v"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x \\neq y"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "text",
+                                    "content": ". Since "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "text",
+                                    "content": " is an isomorphism from "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T_G^{(d+1)}(u,v)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T_H^{(d+1)}(x,y)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "text",
+                                    "content": " maps all tree nodes of depth 1 in "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T_G^{(d+1)}(u,v)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "text",
+                                    "content": " to all tree nodes of depth 1 in "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T_H^{(d+1)}(x,y)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "text",
+                                    "content": ". Let "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s_1, \\dots, s_n"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "text",
+                                    "content": " be all nodes of depth 2 in "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T_G^{(d+1)}(u,v)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\in \\beta_{T_G^{(d+1)}(u,v)}(s_i)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "text",
+                                    "content": " (it follows that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n = |V_G|"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "text",
+                                    "content": "), and let "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s_i'"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "text",
+                                    "content": " be the parent of "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s_i"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "text",
+                                    "content": ". Similarly, let "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t_1, \\dots, t_n"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "text",
+                                    "content": " be all nodes of depth 2 in "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T_H^{(d+1)}(x,y)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x \\in \\beta_{T_H^{(d+1)}(x,y)}(t_i)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "text",
+                                    "content": ", and let "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t_i'"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "text",
+                                    "content": " be the parent of "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t_i"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "text",
+                                    "content": ". Moreover, we can arrange the order so that the following are satisfied (for each "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i \\in [n]"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        476,
+                                        505,
+                                        570
+                                    ],
+                                    "type": "text",
+                                    "content": "):"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        141,
+                        574,
+                        504,
+                        673
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 12,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                141,
+                                574,
+                                498,
+                                590
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        574,
+                                        498,
+                                        590
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                574,
+                                                498,
+                                                590
+                                            ],
+                                            "type": "text",
+                                            "content": "a) "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                574,
+                                                498,
+                                                590
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tau"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                574,
+                                                498,
+                                                590
+                                            ],
+                                            "type": "text",
+                                            "content": " is an isomorphism from the subtree "
+                                        },
+                                        {
+                                            "bbox": [
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+                                                574,
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+                                            "type": "inline_equation",
+                                            "content": "T_{G}^{(d + 1)}(u,v)[s_i]"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": " to the subtree "
+                                        },
+                                        {
+                                            "bbox": [
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+                                                574,
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+                                                590
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T_{H}^{(d + 1)}(x,y)[t_i]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                574,
+                                                498,
+                                                590
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 8
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                590,
+                                448,
+                                605
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
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+                                    "bbox": [
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+                                    "spans": [
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": "b) For all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                590,
+                                                448,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "s \\in \\mathsf{Desc}_{T_G^{(d + 1)}(u,v)}(s_i)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                590,
+                                                448,
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+                                            ],
+                                            "type": "text",
+                                            "content": ", "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                590,
+                                                448,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho(\\beta_{T_G^{(d + 1)}(u,v)}(s)) = \\beta_{T_H^{(d + 1)}(x,y)}(\\tau(s))"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                590,
+                                                448,
+                                                605
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 9
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                605,
+                                504,
+                                640
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
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+                                    "bbox": [
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+                                            "bbox": [
+                                                141,
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+                                            ],
+                                            "type": "text",
+                                            "content": "c) By definition of the unfolding tree, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                605,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                605,
+                                                504,
+                                                640
+                                            ],
+                                            "type": "text",
+                                            "content": " is an isomorphism from the induced subgraph "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                605,
+                                                504,
+                                                640
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F_{G}^{(d + 1)}(u,v)[T_{G}^{(d + 1)}(u,v)[s_{i}]]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                605,
+                                                504,
+                                                640
+                                            ],
+                                            "type": "text",
+                                            "content": " to the induced subgraph "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                605,
+                                                504,
+                                                640
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F_{H}^{(d + 1)}(x,y)[T_{H}^{(d + 1)}(x,y)[t_{i}]]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                605,
+                                                504,
+                                                640
+                                            ],
+                                            "type": "text",
+                                            "content": " (see Definition C.7)."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 10
+                        },
+                        {
+                            "bbox": [
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+                                504,
+                                673
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
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+                                    "bbox": [
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+                                        504,
+                                        673
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+                                        {
+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": "d) Let "
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_{T_G^{(d + 1)}(u,v)}(s_i') = \\{\\{u,v,\\tilde{w}_i\\}\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                641,
+                                                504,
+                                                673
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                641,
+                                                504,
+                                                673
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_{T_H^{(d + 1)}(x,y)}(t_i') = \\{\\{x,y,\\tilde{z}_i\\}\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                641,
+                                                504,
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+                                            ],
+                                            "type": "text",
+                                            "content": ". Then, "
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho (\\tilde{w}_i) = \\tilde{z}_i"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                641,
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+                                                673
+                                            ],
+                                            "type": "text",
+                                            "content": ", and thus "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                641,
+                                                504,
+                                                673
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{v,\\tilde{w}_i\\} \\in E_{F_G^{(d + 1)}(u,v)}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                641,
+                                                504,
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+                                            ],
+                                            "type": "text",
+                                            "content": " iff "
+                                        },
+                                        {
+                                            "bbox": [
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+                                                641,
+                                                504,
+                                                673
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{y,\\tilde{z}_i\\} \\in E_{F_H^{(d + 1)}(x,y)}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                641,
+                                                504,
+                                                673
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 11
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "By items (a) to (c), "
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+                                    "type": "inline_equation",
+                                    "content": "\\left(F_{G}^{(d + 1)}(u,v)\\left[T_{G}^{(d + 1)}(u,v)[s_{i}]\\right],T_{G}^{(d + 1)}(u,v)[s_{i}]\\right)"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " is isomorphic to "
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+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\left(F_H^{(d + 1)}(x,y)\\left[T_H^{(d + 1)}(x,y)[t_i]\\right],T_H^{(d + 1)}(x,y)[t_i]\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
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+                                    ],
+                                    "type": "text",
+                                    "content": ". On the other hand, by definition of the unfolding tree, "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(F_G^{(d + 1)}(u,v)\\left[T_G^{(d + 1)}(u,v)[s_i]\\right],T_G^{(d + 1)}(u,v)[s_i]\\right)"
+                                },
+                                {
+                                    "bbox": [
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+                                        677,
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+                                    ],
+                                    "type": "text",
+                                    "content": " is isomorphic to"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                }
+            ],
+            "discarded_blocks": [
+                {
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+                    ],
+                    "type": "header",
+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
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+                                    "type": "text",
+                                    "content": "28"
+                                }
+                            ]
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+                    ],
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+                }
+            ],
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+        },
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+            "para_blocks": [
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+                    ],
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+                    "angle": 0,
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+                                {
+                                    "bbox": [
+                                        129,
+                                        80,
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+                                    ],
+                                    "type": "text",
+                                    "content": "the depth-2d unfolding tree "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        80,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(F_{G}^{(d)}(u,w_{i}),T_{G}^{(d)}(u,w_{i})\\right)"
+                                },
+                                {
+                                    "bbox": [
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+                                        80,
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+                                    "type": "text",
+                                    "content": " for some "
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+                                        80,
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+                                        114
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_{i}\\in V_{G}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        80,
+                                        504,
+                                        114
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfying that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        80,
+                                        504,
+                                        114
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{w_i,v\\} \\in E_G"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        80,
+                                        504,
+                                        114
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        80,
+                                        504,
+                                        114
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\tilde{w}_i,v\\} \\in E_{F_G^{(d + 1)}(u,v)}"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                114,
+                                504,
+                                199
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        114,
+                                        504,
+                                        199
+                                    ],
+                                    "type": "text",
+                                    "content": "Similarly, "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        114,
+                                        504,
+                                        199
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(F_H^{(d + 1)}(x,y)\\left[T_H^{(d + 1)}(x,y)[t_i]\\right],T_H^{(d + 1)}(x,y)[t_i]\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        114,
+                                        504,
+                                        199
+                                    ],
+                                    "type": "text",
+                                    "content": " is isomorphic to "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        114,
+                                        504,
+                                        199
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(F_{H}^{(d)}(x,z_{i}),T_{H}^{(d)}(x,z_{i})\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        114,
+                                        504,
+                                        199
+                                    ],
+                                    "type": "text",
+                                    "content": " for some "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        114,
+                                        504,
+                                        199
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z_{i}\\in V_{H}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        114,
+                                        504,
+                                        199
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfying that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        114,
+                                        504,
+                                        199
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{z_i,y\\} \\in E_H"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        114,
+                                        504,
+                                        199
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        114,
+                                        504,
+                                        199
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\tilde{z}_i,y\\} \\in E_{F_H^{(d + 1)}(x,y)}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        114,
+                                        504,
+                                        199
+                                    ],
+                                    "type": "text",
+                                    "content": ". Combining all the above equivalence yields that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        114,
+                                        504,
+                                        199
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(F_G^{(d)}(u,w_i),T_G^{(d)}(u,w_i)\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        114,
+                                        504,
+                                        199
+                                    ],
+                                    "type": "text",
+                                    "content": " is isomorphic to "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        114,
+                                        504,
+                                        199
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(F_H^{(d)}(x,z_i),T_H^{(d)}(x,z_i)\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        114,
+                                        504,
+                                        199
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        114,
+                                        504,
+                                        199
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{w_i,v\\} \\in E_G"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        114,
+                                        504,
+                                        199
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        114,
+                                        504,
+                                        199
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{z_i,y\\} \\in E_H"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        114,
+                                        504,
+                                        199
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        129,
+                        202,
+                        504,
+                        228
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                202,
+                                504,
+                                228
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        202,
+                                        504,
+                                        228
+                                    ],
+                                    "type": "text",
+                                    "content": "By induction, we have "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        202,
+                                        504,
+                                        228
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\chi}_G^{(d)}(u,w_i) = \\tilde{\\chi}_H^{(d)}(x,z_i)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        202,
+                                        504,
+                                        228
+                                    ],
+                                    "type": "text",
+                                    "content": ". Moreover, we clearly have that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        202,
+                                        504,
+                                        228
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{u,w_i\\} \\in E_G"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        202,
+                                        504,
+                                        228
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        202,
+                                        504,
+                                        228
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{x,z_i\\} \\in E_H"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        202,
+                                        504,
+                                        228
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        202,
+                                        504,
+                                        228
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{u,v\\} \\in E_G"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        202,
+                                        504,
+                                        228
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        202,
+                                        504,
+                                        228
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{x,y\\} \\in E_H"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        202,
+                                        504,
+                                        228
+                                    ],
+                                    "type": "text",
+                                    "content": ". Therefore,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        194,
+                        232,
+                        504,
+                        251
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                194,
+                                232,
+                                504,
+                                251
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        194,
+                                        232,
+                                        504,
+                                        251
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left(\\tilde {\\chi} _ {G} ^ {(d)} (u, w _ {i}), \\mathsf {a t p} _ {G} (u, v, w _ {i})\\right) = \\left(\\tilde {\\chi} _ {H} ^ {(d)} (x, z _ {i}), \\mathsf {a t p} _ {H} (x, y, z _ {i})\\right). \\tag {13}",
+                                    "image_path": "fb930476c5a59f43f9d29c7e59dc2c88c1c4a59608895ac5ba7ef6dbd39818eb.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
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+                        277
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                253,
+                                504,
+                                277
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        253,
+                                        504,
+                                        277
+                                    ],
+                                    "type": "text",
+                                    "content": "Next, note that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        253,
+                                        504,
+                                        277
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{w}_i"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        253,
+                                        504,
+                                        277
+                                    ],
+                                    "type": "text",
+                                    "content": " are different from each other for "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        253,
+                                        504,
+                                        277
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i\\in [n]"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        253,
+                                        504,
+                                        277
+                                    ],
+                                    "type": "text",
+                                    "content": " by definition of unfolding tree. Thus, "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        253,
+                                        504,
+                                        277
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_{i}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        253,
+                                        504,
+                                        277
+                                    ],
+                                    "type": "text",
+                                    "content": " are also different from each other. It follows that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        137,
+                        281,
+                        504,
+                        299
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                137,
+                                281,
+                                504,
+                                299
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        137,
+                                        281,
+                                        504,
+                                        299
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\{\\{(\\tilde {\\chi} _ {G} ^ {(d)} (u, w), \\mathsf {a t p} _ {G} (u, v, w)): w \\in V _ {G} \\} \\} = \\{\\{(\\tilde {\\chi} _ {H} ^ {(d)} (x, z), \\mathsf {a t p} _ {H} (x, y, z)): z \\in V _ {H} \\} \\}. \\tag {14}",
+                                    "image_path": "8d3c359ee788d88caa669c6582ee56c33f17e06aa7cfc21b8d0db79fdda53afb.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
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+                        338,
+                        313
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                130,
+                                301,
+                                338,
+                                313
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        130,
+                                        301,
+                                        338,
+                                        313
+                                    ],
+                                    "type": "text",
+                                    "content": "Again using the same analysis as before, we obtain"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        138,
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                138,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        138,
+                                        318,
+                                        504,
+                                        335
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\{\\{(\\tilde {\\chi} _ {G} ^ {(d)} (w, v), \\mathsf {a t p} _ {G} (u, v, w)): w \\in V _ {G} \\} \\} = \\{\\{(\\tilde {\\chi} _ {H} ^ {(d)} (z, y), \\mathsf {a t p} _ {H} (x, y, z)): z \\in V _ {H} \\} \\}. \\tag {15}",
+                                    "image_path": "7e57ea7b7da5b7c69c5665c3f1477ff043bff25818f1db50af4a8d97cf24e7f1.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                339,
+                                504,
+                                364
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        339,
+                                        504,
+                                        364
+                                    ],
+                                    "type": "text",
+                                    "content": "It remains to prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        339,
+                                        504,
+                                        364
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\chi}_G^{(d)}(u,v) = \\tilde{\\chi}_H^{(d)}(x,y)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        339,
+                                        504,
+                                        364
+                                    ],
+                                    "type": "text",
+                                    "content": ". To prove this, note that Equation (14) implies that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        136,
+                        368,
+                        504,
+                        385
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                136,
+                                368,
+                                504,
+                                385
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        136,
+                                        368,
+                                        504,
+                                        385
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left\\{\\left\\{\\left(\\tilde {\\chi} _ {G} ^ {(d ^ {\\prime})} (u, w), \\mathsf {a t p} _ {G} (u, v, w)\\right): w \\in V _ {G} \\right\\} \\right\\} = \\left\\{\\left\\{\\left(\\tilde {\\chi} _ {H} ^ {(d ^ {\\prime})} (x, z), \\mathsf {a t p} _ {H} (x, y, z)\\right): z \\in V _ {H} \\right\\} \\right\\} \\tag {16}",
+                                    "image_path": "c6de31dcf8228af294af78a04de20bb96f4dceb6d8936fa73499cde02d911572.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
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+                        401
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                388,
+                                355,
+                                401
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        388,
+                                        355,
+                                        401
+                                    ],
+                                    "type": "text",
+                                    "content": "holds for all "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        388,
+                                        355,
+                                        401
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "0 \\leq d' \\leq d"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        388,
+                                        355,
+                                        401
+                                    ],
+                                    "type": "text",
+                                    "content": ", and Equation (15) implies that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        136,
+                        406,
+                        504,
+                        423
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                136,
+                                406,
+                                504,
+                                423
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        136,
+                                        406,
+                                        504,
+                                        423
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\left\\{\\left\\{\\left(\\tilde {\\chi} _ {G} ^ {(d ^ {\\prime})} (w, v), \\mathsf {a t p} _ {G} (u, v, w)\\right): w \\in V _ {G} \\right\\}\\right\\} = \\left\\{ \\right.\\left\\{\\left(\\tilde {\\chi} _ {H} ^ {(d)} (z, y), \\mathsf {a t p} _ {H} (x, y, z)\\right): z \\in V _ {H} \\right\\}. \\tag {17}",
+                                    "image_path": "3d2e67f4d2c0d8fdc9ea75c6a0c999b311403c24f5cecc46af676d734c95dc28.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        129,
+                        426,
+                        504,
+                        464
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                426,
+                                504,
+                                464
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        426,
+                                        504,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": "holds for all "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        426,
+                                        504,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "0 \\leq d' \\leq d"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        426,
+                                        504,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": ". Combined with Equations (16) and (17) and the fact that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        426,
+                                        504,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\chi}_{G}^{(0)}(u,v) = \\tilde{\\chi}_{H}^{(0)}(x,y)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        426,
+                                        504,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": ", we can incrementally prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        426,
+                                        504,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\chi}_{G}^{(d')}(u,v) = \\tilde{\\chi}_{H}^{(d')}(x,y)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        426,
+                                        504,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        426,
+                                        504,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d' \\leq d + 1"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        426,
+                                        504,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        105,
+                        468,
+                        246,
+                        480
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                468,
+                                246,
+                                480
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        468,
+                                        246,
+                                        480
+                                    ],
+                                    "type": "text",
+                                    "content": "We have thus concluded the proof."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        104,
+                        487,
+                        504,
+                        510
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                                504,
+                                510
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        510
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition C.15. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        510
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        510
+                                    ],
+                                    "type": "text",
+                                    "content": " be any model. Given a graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        510
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        510
+                                    ],
+                                    "type": "text",
+                                    "content": " and a tree-decomposed graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        510
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        510
+                                    ],
+                                    "type": "text",
+                                    "content": ", define"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        104,
+                        514,
+                        507,
+                        536
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                514,
+                                507,
+                                536
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        514,
+                                        507,
+                                        536
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathsf {c n t} ^ {M} \\left(\\left(F, T ^ {r}\\right), G\\right) := \\left| \\left\\{\\left(u, v\\right) \\in V _ {G} ^ {2}: \\exists D \\in \\mathbb {N} _ {+} \\text {s . t .} \\left(F _ {G} ^ {M, (D)} (u, v), T _ {G} ^ {M, (D)} (u, v)\\right) \\simeq \\left(F, T ^ {r}\\right) \\right\\} \\right|,",
+                                    "image_path": "95f1f5c386a3e2a980c6cbccace5c1247d3d7761118d2a541209a598e73a7a45.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        104,
+                        541,
+                        498,
+                        560
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                541,
+                                498,
+                                560
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        541,
+                                        498,
+                                        560
+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        541,
+                                        498,
+                                        560
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(F_G^{M,(D)}(u,v),T_G^{M,(D)}(u,v)\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        541,
+                                        498,
+                                        560
+                                    ],
+                                    "type": "text",
+                                    "content": " is the depth-2D unfolding tree of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        541,
+                                        498,
+                                        560
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        541,
+                                        498,
+                                        560
+                                    ],
+                                    "type": "text",
+                                    "content": " at "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        541,
+                                        498,
+                                        560
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u,v)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        541,
+                                        498,
+                                        560
+                                    ],
+                                    "type": "text",
+                                    "content": " for model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        541,
+                                        498,
+                                        560
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        541,
+                                        498,
+                                        560
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        104,
+                        563,
+                        504,
+                        589
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                563,
+                                504,
+                                589
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        504,
+                                        589
+                                    ],
+                                    "type": "text",
+                                    "content": "Corollary C.16. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        504,
+                                        589
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        504,
+                                        589
+                                    ],
+                                    "type": "text",
+                                    "content": " be any model. For any graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        504,
+                                        589
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G, H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        504,
+                                        589
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        504,
+                                        589
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^M(G) = \\chi_H^M(H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        504,
+                                        589
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        504,
+                                        589
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{cnt}^M((F, T^r), G) = \\mathsf{cnt}^M((F, T^r), H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        504,
+                                        589
+                                    ],
+                                    "type": "text",
+                                    "content": " holds for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        504,
+                                        589
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        504,
+                                        589
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
+                        104,
+                        598,
+                        505,
+                        677
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                598,
+                                505,
+                                677
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        505,
+                                        677
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. “"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        505,
+                                        677
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Rightarrow"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        505,
+                                        677
+                                    ],
+                                    "type": "text",
+                                    "content": "”. If "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        505,
+                                        677
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^M (G) = \\chi_H^M (H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        505,
+                                        677
+                                    ],
+                                    "type": "text",
+                                    "content": ", then "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        505,
+                                        677
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\{\\chi_G^M (u,v):u,v\\in V_G\\} \\} = \\{\\{\\chi_H^M (x,y):x,y\\in V_H\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        505,
+                                        677
+                                    ],
+                                    "type": "text",
+                                    "content": ". For each color "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        505,
+                                        677
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "c"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        505,
+                                        677
+                                    ],
+                                    "type": "text",
+                                    "content": " in the above multiset, pick "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        505,
+                                        677
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u,v\\in V_{G}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        505,
+                                        677
+                                    ],
+                                    "type": "text",
+                                    "content": " with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        505,
+                                        677
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^M (u,v) = c"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        505,
+                                        677
+                                    ],
+                                    "type": "text",
+                                    "content": ". It follows that if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        505,
+                                        677
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^{r})\\simeq (F_{G}^{M,(D)}(u,v),T_{G}^{M,(D)}(u,v))\\in S^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        505,
+                                        677
+                                    ],
+                                    "type": "text",
+                                    "content": " for some "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        505,
+                                        677
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "D"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        505,
+                                        677
+                                    ],
+                                    "type": "text",
+                                    "content": ", then "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        505,
+                                        677
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{cnt}^M ((F,T^r),G) = |\\{(u,v)\\in V_G^2:\\chi_G^M (u,v) = c\\} | = |\\{(x,y)\\in V_H^2:\\chi_H^M (x,y) = c\\} | = \\mathsf{cnt}^M ((F,T^r),H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        505,
+                                        677
+                                    ],
+                                    "type": "text",
+                                    "content": " by Lemma C.14. On the other hand, if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        505,
+                                        677
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^{r})\\neq (F_{G}^{M,(D)}(u,v),T_{G}^{M,(D)}(u,v))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        505,
+                                        677
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        505,
+                                        677
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u,v\\in V_{G}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        505,
+                                        677
+                                    ],
+                                    "type": "text",
+                                    "content": " and all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        505,
+                                        677
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "D"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        505,
+                                        677
+                                    ],
+                                    "type": "text",
+                                    "content": ", then clearly "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        505,
+                                        677
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{cnt}^M ((F,T^r),G) = \\mathsf{cnt}^M ((F,T^r),H) = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        505,
+                                        677
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                },
+                {
+                    "bbox": [
+                        104,
+                        681,
+                        505,
+                        734
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                681,
+                                505,
+                                734
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        681,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Longleftarrow"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        681,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": " . If "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        681,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{cnt}^M ((F,T^r),G) = \\mathsf{cnt}^M ((F,T^r),H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        681,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": " holds for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        681,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^{r})\\in S^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        681,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": " , it clearly holds for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        681,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F_M^{(D)}(u,v),T_M^{(D)}(u,v))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        681,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": " with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        681,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u,v\\in V_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        681,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": " and a sufficiently large "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        681,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "D"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        681,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": " . This guarantees that for all color "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        681,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "c,\\left|\\{(u,v)\\in V_G^2:\\chi_G^M (u,v) = c\\} \\right| = \\left|\\{(x,y)\\in V_H^2:\\chi_H^M (x,y) = c\\} \\right|"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        681,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": " by Lemma C.14. Therefore, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        681,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\{\\chi_G^M (u,v):u,v\\in V_G\\} \\} = \\{\\{\\chi_H^M (x,y):x,y\\in V_H\\} \\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        681,
+                                        505,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": " , concluding the proof."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
+                        750,
+                        311,
+                        761
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                299,
+                                750,
+                                311,
+                                761
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        299,
+                                        750,
+                                        311,
+                                        761
+                                    ],
+                                    "type": "text",
+                                    "content": "29"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 21
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 28
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        82,
+                        428,
+                        94
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                82,
+                                428,
+                                94
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        428,
+                                        94
+                                    ],
+                                    "type": "text",
+                                    "content": "We next define an important concept called bag isomorphism (Dell et al., 2018)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        97,
+                        506,
+                        143
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                97,
+                                506,
+                                143
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition C.17. Given a tree-decomposed graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": " and a graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": ", a bag isomorphism from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": " (abbreviated as \"bIso\") is a homomorphism "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": " from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": " such that, for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t \\in V_T"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": " is an isomorphism from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[\\beta_T(t)]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G[f(\\beta_T(t))]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": ". Denote "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{BIso}((F, T^r), G)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": " to be the set of all bag isomorphisms from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": ", and denote "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{bIso}((F, T^r), G) = |\\mathsf{BIso}((F, T^r), G)|"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        97,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        146,
+                        504,
+                        168
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                146,
+                                504,
+                                168
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        146,
+                                        504,
+                                        168
+                                    ],
+                                    "type": "text",
+                                    "content": "Remark C.18. To prove that a mapping "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        146,
+                                        504,
+                                        168
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        146,
+                                        504,
+                                        168
+                                    ],
+                                    "type": "text",
+                                    "content": " from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        146,
+                                        504,
+                                        168
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        146,
+                                        504,
+                                        168
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        146,
+                                        504,
+                                        168
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        146,
+                                        504,
+                                        168
+                                    ],
+                                    "type": "text",
+                                    "content": " is a bIso, it suffices to prove the following conditions:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        116,
+                        171,
+                        474,
+                        214
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 7,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                117,
+                                171,
+                                418,
+                                184
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        171,
+                                        418,
+                                        184
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                117,
+                                                171,
+                                                418,
+                                                184
+                                            ],
+                                            "type": "text",
+                                            "content": "1. For any two different vertices "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                171,
+                                                418,
+                                                184
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u, v \\in V_G"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                171,
+                                                418,
+                                                184
+                                            ],
+                                            "type": "text",
+                                            "content": " in the same bag, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                171,
+                                                418,
+                                                184
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "f(u) \\neq f(v)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                171,
+                                                418,
+                                                184
+                                            ],
+                                            "type": "text",
+                                            "content": ";"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 4
+                        },
+                        {
+                            "bbox": [
+                                116,
+                                186,
+                                474,
+                                199
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        186,
+                                        474,
+                                        199
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                186,
+                                                474,
+                                                199
+                                            ],
+                                            "type": "text",
+                                            "content": "2. For any two vertices "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                186,
+                                                474,
+                                                199
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u, v \\in V_G"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                186,
+                                                474,
+                                                199
+                                            ],
+                                            "type": "text",
+                                            "content": " in the same bag, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                186,
+                                                474,
+                                                199
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{u, v\\} \\in E_F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                186,
+                                                474,
+                                                199
+                                            ],
+                                            "type": "text",
+                                            "content": " iff "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                186,
+                                                474,
+                                                199
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{f(u), f(v)\\} \\in E_G"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                186,
+                                                474,
+                                                199
+                                            ],
+                                            "type": "text",
+                                            "content": ";"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 5
+                        },
+                        {
+                            "bbox": [
+                                117,
+                                201,
+                                279,
+                                214
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        201,
+                                        279,
+                                        214
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                117,
+                                                201,
+                                                279,
+                                                214
+                                            ],
+                                            "type": "text",
+                                            "content": "3. For any "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                201,
+                                                279,
+                                                214
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u\\in V_G,\\ell_F(u) = \\ell_G(f(u))"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 6
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        104,
+                        223,
+                        430,
+                        235
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                223,
+                                430,
+                                235
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        223,
+                                        430,
+                                        235
+                                    ],
+                                    "type": "text",
+                                    "content": "The following fact is straightforward from the construction of the unfolding tree:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        238,
+                        504,
+                        281
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                238,
+                                504,
+                                281
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        281
+                                    ],
+                                    "type": "text",
+                                    "content": "Fact C.19. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        281
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        281
+                                    ],
+                                    "type": "text",
+                                    "content": " be any model considered above. For any graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        281
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        281
+                                    ],
+                                    "type": "text",
+                                    "content": ", any vertex pair "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        281
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u, v) \\in V_G^2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        281
+                                    ],
+                                    "type": "text",
+                                    "content": ", and any non-negative integer "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        281
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "D"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        281
+                                    ],
+                                    "type": "text",
+                                    "content": ", there is a bIso "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        281
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        281
+                                    ],
+                                    "type": "text",
+                                    "content": " from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        281
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(F_G^{M,(D)}(u, v), T_G^{M,(D)}(u, v)\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        281
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        281
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        281
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
+                        289,
+                        504,
+                        312
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                289,
+                                504,
+                                312
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        289,
+                                        504,
+                                        312
+                                    ],
+                                    "type": "text",
+                                    "content": "Similarly, we need the following concept to describe the relation between two tree-decomposed graphs. We note that these technical concepts also appeared in Dell et al. (2018)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        316,
+                        504,
+                        342
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                316,
+                                504,
+                                342
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        316,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition C.20. Given two tree-decomposed graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        316,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        316,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        316,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{F}, \\tilde{T}^s)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        316,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "text",
+                                    "content": ", a pair of mappings "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        316,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\rho, \\tau)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        316,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "text",
+                                    "content": " is called homomorphism from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        316,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        316,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        316,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{F}, \\tilde{T}^s)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        316,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "text",
+                                    "content": " if it satisfies the following conditions:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        116,
+                        347,
+                        504,
+                        418
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 15,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                116,
+                                347,
+                                504,
+                                372
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        347,
+                                        504,
+                                        372
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                347,
+                                                504,
+                                                372
+                                            ],
+                                            "type": "text",
+                                            "content": "a) "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                347,
+                                                504,
+                                                372
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tau"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                347,
+                                                504,
+                                                372
+                                            ],
+                                            "type": "text",
+                                            "content": " is a homomorphism from "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                347,
+                                                504,
+                                                372
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                347,
+                                                504,
+                                                372
+                                            ],
+                                            "type": "text",
+                                            "content": " to "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                347,
+                                                504,
+                                                372
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tilde{T}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                347,
+                                                504,
+                                                372
+                                            ],
+                                            "type": "text",
+                                            "content": " (ignoring labels "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                347,
+                                                504,
+                                                372
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                347,
+                                                504,
+                                                372
+                                            ],
+                                            "type": "text",
+                                            "content": ") and is depth-preserving, i.e., "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                347,
+                                                504,
+                                                372
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{dep}_{T^r}(t) = \\mathsf{dep}_{\\tilde{T}^s}(\\tau(t))"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                347,
+                                                504,
+                                                372
+                                            ],
+                                            "type": "text",
+                                            "content": " for all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                347,
+                                                504,
+                                                372
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t \\in V_T"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                347,
+                                                504,
+                                                372
+                                            ],
+                                            "type": "text",
+                                            "content": ";"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 12
+                        },
+                        {
+                            "bbox": [
+                                116,
+                                375,
+                                504,
+                                401
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        375,
+                                        504,
+                                        401
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                375,
+                                                504,
+                                                401
+                                            ],
+                                            "type": "text",
+                                            "content": "b) For all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                375,
+                                                504,
+                                                401
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t \\in V_T"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                375,
+                                                504,
+                                                401
+                                            ],
+                                            "type": "text",
+                                            "content": ", "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                375,
+                                                504,
+                                                401
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                375,
+                                                504,
+                                                401
+                                            ],
+                                            "type": "text",
+                                            "content": " is a homomorphism from "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                375,
+                                                504,
+                                                401
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[\\beta_T(t)]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                375,
+                                                504,
+                                                401
+                                            ],
+                                            "type": "text",
+                                            "content": " to "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                375,
+                                                504,
+                                                401
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tilde{F}[\\beta_{\\tilde{T}}(\\tau(t))]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                375,
+                                                504,
+                                                401
+                                            ],
+                                            "type": "text",
+                                            "content": ". Note that this implies that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                375,
+                                                504,
+                                                401
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                375,
+                                                504,
+                                                401
+                                            ],
+                                            "type": "text",
+                                            "content": " is a homomorphism from "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                375,
+                                                504,
+                                                401
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                375,
+                                                504,
+                                                401
+                                            ],
+                                            "type": "text",
+                                            "content": " to "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                375,
+                                                504,
+                                                401
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tilde{F}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                375,
+                                                504,
+                                                401
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 13
+                        },
+                        {
+                            "bbox": [
+                                116,
+                                405,
+                                307,
+                                418
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        405,
+                                        307,
+                                        418
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                405,
+                                                307,
+                                                418
+                                            ],
+                                            "type": "text",
+                                            "content": "c) The depth of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                405,
+                                                307,
+                                                418
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T^r"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                405,
+                                                307,
+                                                418
+                                            ],
+                                            "type": "text",
+                                            "content": " is equal to the depth of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                405,
+                                                307,
+                                                418
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\hat{T}^s"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                405,
+                                                307,
+                                                418
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 14
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        104,
+                        429,
+                        506,
+                        553
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                429,
+                                506,
+                                553
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition C.21. Under Definition C.20, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\rho, \\tau)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "text",
+                                    "content": " is further called a bag-isomorphism homomorphism (abbreviated as \"bIsoHom\") from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{F}, \\tilde{T}^s)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "text",
+                                    "content": " if it is a homomorphism satisfying that, for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t \\in V_T"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "text",
+                                    "content": " is an isomorphism from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[\\beta_T(t)]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F}[\\beta_{\\tilde{T}}(\\tau(t))]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "text",
+                                    "content": ". Furthermore, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\rho, \\tau)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "text",
+                                    "content": " is called a bIsoSurj if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "text",
+                                    "content": " is surjective; and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\rho, \\tau)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "text",
+                                    "content": " is called a bIsoInj if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "text",
+                                    "content": " is injective. We use BIsoHom "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left((F, T^r), (\\tilde{F}, \\tilde{T}^s)\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "text",
+                                    "content": " to denote the set of bIsoHoms from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{F}, \\tilde{T}^s)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "text",
+                                    "content": ", and let bIsoHom "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left((F, T^r), (\\tilde{F}, \\tilde{T}^s)\\right) = \\left|\\text{BIsoHom}\\left((F, T^r), (\\tilde{F}, \\tilde{T}^s)\\right)\\right|"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "text",
+                                    "content": ". The notations BIsoSurj "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left((F, T^r), (\\tilde{F}, \\tilde{T}^s)\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "text",
+                                    "content": ", bIsoSurj "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left((F, T^r), (\\tilde{F}, \\tilde{T}^s)\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "text",
+                                    "content": ", BIsolnj "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left((F, T^r), (\\tilde{F}, \\tilde{T}^s)\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "text",
+                                    "content": ", and bIsolnj "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left((F, T^r), (\\tilde{F}, \\tilde{T}^s)\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        429,
+                                        506,
+                                        553
+                                    ],
+                                    "type": "text",
+                                    "content": " are defined accordingly."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        104,
+                        556,
+                        504,
+                        580
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                556,
+                                504,
+                                580
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        556,
+                                        504,
+                                        580
+                                    ],
+                                    "type": "text",
+                                    "content": "Remark C.22. In the above definition, the depth of a tree "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        556,
+                                        504,
+                                        580
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        556,
+                                        504,
+                                        580
+                                    ],
+                                    "type": "text",
+                                    "content": " is the maximal depth among all tree nodes in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        556,
+                                        504,
+                                        580
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        556,
+                                        504,
+                                        580
+                                    ],
+                                    "type": "text",
+                                    "content": ". Note that we do not require that all leaf nodes have the same depth in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        556,
+                                        504,
+                                        580
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        556,
+                                        504,
+                                        580
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        104,
+                        589,
+                        314,
+                        601
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                589,
+                                314,
+                                601
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        589,
+                                        314,
+                                        601
+                                    ],
+                                    "type": "text",
+                                    "content": "We are now ready to present the second core result:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
+                        104,
+                        604,
+                        506,
+                        628
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                604,
+                                506,
+                                628
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        604,
+                                        506,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma C.23. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        604,
+                                        506,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        604,
+                                        506,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": " be any model. For any graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        604,
+                                        506,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        604,
+                                        506,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": " and tree-decomposed graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        604,
+                                        506,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        604,
+                                        506,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": ","
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                },
+                {
+                    "bbox": [
+                        120,
+                        634,
+                        504,
+                        664
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                120,
+                                634,
+                                504,
+                                664
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        120,
+                                        634,
+                                        504,
+                                        664
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\operatorname {b l s o} \\left(\\left(F, T ^ {r}\\right), G\\right) = \\sum_ {\\left(\\tilde {F}, \\tilde {T} ^ {s}\\right) \\in \\mathcal {S} ^ {M}} \\operatorname {b l s o H o m} \\left(\\left(F, T ^ {r}\\right), \\left(\\tilde {F}, \\tilde {T} ^ {s}\\right)\\right) \\cdot \\operatorname {c n t} ^ {M} \\left(\\left(\\tilde {F}, \\tilde {T} ^ {s}\\right), G\\right). \\tag {18}",
+                                    "image_path": "08177fc7c835ef073a28a2a28d39b954fd7848c86b9d068db10dbb3337279a13.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                },
+                {
+                    "bbox": [
+                        104,
+                        671,
+                        504,
+                        695
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                671,
+                                504,
+                                695
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        671,
+                                        504,
+                                        695
+                                    ],
+                                    "type": "text",
+                                    "content": "Here, the summation ranges over all non-isomorphic (tree-decomposed) graphs in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        671,
+                                        504,
+                                        695
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        671,
+                                        504,
+                                        695
+                                    ],
+                                    "type": "text",
+                                    "content": " and is well-defined as there are only a finite number of graphs making the value in the summation non-zero."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 21
+                },
+                {
+                    "bbox": [
+                        104,
+                        709,
+                        504,
+                        733
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                709,
+                                504,
+                                733
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        709,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Here, we only give the proof for Local 2-GNN, and the proofs for Subgraph GNN, Local 2-FGNN and 2-FGNN are almost the same so we omit them for clarity."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 22
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                26,
+                                293,
+                                38
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
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+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        299,
+                                        750,
+                                        312,
+                                        760
+                                    ],
+                                    "type": "text",
+                                    "content": "30"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 23
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 29
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        82,
+                        504,
+                        120
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                82,
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        120
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof for Local 2-GNN. We assume that the root bag of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        120
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        120
+                                    ],
+                                    "type": "text",
+                                    "content": " is "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        120
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\{u, v\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        120
+                                    ],
+                                    "type": "text",
+                                    "content": ", and the depth of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        120
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        120
+                                    ],
+                                    "type": "text",
+                                    "content": " is "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        120
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "2d"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        120
+                                    ],
+                                    "type": "text",
+                                    "content": ". Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        120
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "y, z \\in V_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        120
+                                    ],
+                                    "type": "text",
+                                    "content": " be any vertices in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        120
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        120
+                                    ],
+                                    "type": "text",
+                                    "content": ", and denote "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        120
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F_G^{(d)}(y, z), T_G^{(d)}(y, z))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        120
+                                    ],
+                                    "type": "text",
+                                    "content": " as the depth-2d Local 2-GNN unfolding tree at "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        120
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(y, z)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        120
+                                    ],
+                                    "type": "text",
+                                    "content": ". Define the following two sets:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        105,
+                        126,
+                        503,
+                        163
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                126,
+                                503,
+                                163
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        126,
+                                        503,
+                                        163
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} S _ {1} (y, z) = \\left\\{g: g \\in \\mathsf {B l s o} \\left((F, T ^ {r}), G\\right), g (u) = y, g (v) = z \\right\\}, \\\\ S _ {2} (y, z) = \\left\\{(\\rho , \\tau): (\\rho , \\tau) \\in \\mathsf {B l s o H o m} \\left((F, T ^ {r}), \\left(F _ {G} ^ {(d)} (y, z), T _ {G} ^ {(d)} (y, z)\\right)\\right), \\rho (u) = y, \\rho (v) = z \\right\\}. \\\\ \\end{array}",
+                                    "image_path": "b59d9a279af979192f09a864da77fb0e5de85de5a62b70d2e236ddbc93cc63e2.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        105,
+                        169,
+                        345,
+                        182
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                169,
+                                345,
+                                182
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        169,
+                                        345,
+                                        182
+                                    ],
+                                    "type": "text",
+                                    "content": "Then, Lemma C.23 is equivalent to the following equation:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        227,
+                        189,
+                        381,
+                        215
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                227,
+                                189,
+                                381,
+                                215
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        227,
+                                        189,
+                                        381,
+                                        215
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\sum_ {y, z \\in V _ {G}} | S _ {1} (y, z) | = \\sum_ {y, z \\in V _ {G}} | S _ {2} (y, z) |.",
+                                    "image_path": "bbe481b1a1a1e27f31625e8e92ab29ccf2f5213fb3cb2b2ff3f4497e4a15efbc.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        105,
+                        223,
+                        346,
+                        236
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                223,
+                                346,
+                                236
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        223,
+                                        346,
+                                        236
+                                    ],
+                                    "type": "text",
+                                    "content": "We will prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        223,
+                                        346,
+                                        236
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "|S_{1}(y,z)| = |S_{2}(y,z)|"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        223,
+                                        346,
+                                        236
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        223,
+                                        346,
+                                        236
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "y,z\\in V_G"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        223,
+                                        346,
+                                        236
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        242,
+                        504,
+                        278
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                242,
+                                504,
+                                278
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
+                                        504,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": "Given "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
+                                        504,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "y, z \\in V_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
+                                        504,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": ", according to Fact C.19, there exists a bIso "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
+                                        504,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
+                                        504,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": " from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
+                                        504,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F_G^{(d)}(y,z), T_G^{(d)}(y,z))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
+                                        504,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": " to graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
+                                        504,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
+                                        504,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": ". Define a mapping "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
+                                        504,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
+                                        504,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
+                                        504,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma(\\rho, \\tau) = \\pi \\circ \\rho"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
+                                        504,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
+                                        504,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\rho, \\tau) \\in S_2(y,z)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
+                                        504,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": ". It suffices to prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
+                                        504,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
+                                        504,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": " is a bijection from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
+                                        504,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S_2(y,z)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
+                                        504,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
+                                        504,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S_1(y,z)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        242,
+                                        504,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        117,
+                        281,
+                        505,
+                        316
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                117,
+                                281,
+                                505,
+                                316
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        281,
+                                        505,
+                                        316
+                                    ],
+                                    "type": "text",
+                                    "content": "1. We first prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        281,
+                                        505,
+                                        316
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        281,
+                                        505,
+                                        316
+                                    ],
+                                    "type": "text",
+                                    "content": " is a mapping from "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        281,
+                                        505,
+                                        316
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S_{2}(y,z)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        281,
+                                        505,
+                                        316
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        281,
+                                        505,
+                                        316
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S_{1}(y,z)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        281,
+                                        505,
+                                        316
+                                    ],
+                                    "type": "text",
+                                    "content": ", i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        281,
+                                        505,
+                                        316
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi \\circ \\rho \\in S_{1}(y,z)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        281,
+                                        505,
+                                        316
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        281,
+                                        505,
+                                        316
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\rho, \\tau) \\in S_{2}(y,z)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        281,
+                                        505,
+                                        316
+                                    ],
+                                    "type": "text",
+                                    "content": ". First, we clearly have "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        281,
+                                        505,
+                                        316
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\pi \\circ \\rho)(u) = \\pi(y) = y"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        281,
+                                        505,
+                                        316
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        281,
+                                        505,
+                                        316
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\pi \\circ \\rho)(v) = \\pi(z) = z"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        281,
+                                        505,
+                                        316
+                                    ],
+                                    "type": "text",
+                                    "content": ". We next prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        281,
+                                        505,
+                                        316
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi \\circ \\rho \\in \\mathsf{BlsO}((F, T^{r}), G)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        281,
+                                        505,
+                                        316
+                                    ],
+                                    "type": "text",
+                                    "content": ". The proof is based on Remark C.18."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        141,
+                        320,
+                        505,
+                        426
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 11,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                141,
+                                320,
+                                505,
+                                357
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        320,
+                                        505,
+                                        357
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                320,
+                                                505,
+                                                357
+                                            ],
+                                            "type": "text",
+                                            "content": "a) Let "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                320,
+                                                505,
+                                                357
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w, x \\in V_F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                320,
+                                                505,
+                                                357
+                                            ],
+                                            "type": "text",
+                                            "content": ", "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                320,
+                                                505,
+                                                357
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\neq x"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                320,
+                                                505,
+                                                357
+                                            ],
+                                            "type": "text",
+                                            "content": " be any vertices in the same bag of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                320,
+                                                505,
+                                                357
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T^r"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                320,
+                                                505,
+                                                357
+                                            ],
+                                            "type": "text",
+                                            "content": ". Since "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                320,
+                                                505,
+                                                357
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(\\rho, \\tau)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                320,
+                                                505,
+                                                357
+                                            ],
+                                            "type": "text",
+                                            "content": " is a bIsoHom, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                320,
+                                                505,
+                                                357
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho(w) \\neq \\rho(x)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                320,
+                                                505,
+                                                357
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                320,
+                                                505,
+                                                357
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho(w)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                320,
+                                                505,
+                                                357
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                320,
+                                                505,
+                                                357
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho(x)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                320,
+                                                505,
+                                                357
+                                            ],
+                                            "type": "text",
+                                            "content": " are in the same bag of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                320,
+                                                505,
+                                                357
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T_G^{(d)}(y, z)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                320,
+                                                505,
+                                                357
+                                            ],
+                                            "type": "text",
+                                            "content": ". Again, since "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                320,
+                                                505,
+                                                357
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pi"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                320,
+                                                505,
+                                                357
+                                            ],
+                                            "type": "text",
+                                            "content": " is a bIso, we have "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                320,
+                                                505,
+                                                357
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pi(\\rho(w)) \\neq \\pi(\\rho(x))"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                320,
+                                                505,
+                                                357
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 8
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                358,
+                                505,
+                                409
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        358,
+                                        505,
+                                        409
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                358,
+                                                505,
+                                                409
+                                            ],
+                                            "type": "text",
+                                            "content": "b) Let "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                358,
+                                                505,
+                                                409
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w, x \\in V_F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                358,
+                                                505,
+                                                409
+                                            ],
+                                            "type": "text",
+                                            "content": " be any vertices in the same bag of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                358,
+                                                505,
+                                                409
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T^r"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                358,
+                                                505,
+                                                409
+                                            ],
+                                            "type": "text",
+                                            "content": ". Since "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                358,
+                                                505,
+                                                409
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(\\rho, \\tau)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                358,
+                                                505,
+                                                409
+                                            ],
+                                            "type": "text",
+                                            "content": " is a bIso-Hom, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                358,
+                                                505,
+                                                409
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho(w)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                358,
+                                                505,
+                                                409
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                358,
+                                                505,
+                                                409
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho(x)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                358,
+                                                505,
+                                                409
+                                            ],
+                                            "type": "text",
+                                            "content": " are in the same bag of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                358,
+                                                505,
+                                                409
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T_G^{(d)}(y,z)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                358,
+                                                505,
+                                                409
+                                            ],
+                                            "type": "text",
+                                            "content": ", and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                358,
+                                                505,
+                                                409
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{w,x\\} \\in E_F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                358,
+                                                505,
+                                                409
+                                            ],
+                                            "type": "text",
+                                            "content": " iff "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                358,
+                                                505,
+                                                409
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{\\rho(w),\\rho(x)\\} \\in E_{F_G^{(d)}(y,z)}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                358,
+                                                505,
+                                                409
+                                            ],
+                                            "type": "text",
+                                            "content": ". Again, since "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                358,
+                                                505,
+                                                409
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pi"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                358,
+                                                505,
+                                                409
+                                            ],
+                                            "type": "text",
+                                            "content": " is a bIso, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                358,
+                                                505,
+                                                409
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{\\rho(w),\\rho(x)\\} \\in E_{F_G^{(d)}(y,z)}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                358,
+                                                505,
+                                                409
+                                            ],
+                                            "type": "text",
+                                            "content": " iff "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                358,
+                                                505,
+                                                409
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{\\pi(\\rho(w)),\\pi(\\rho(x))\\} \\in E_G"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                358,
+                                                505,
+                                                409
+                                            ],
+                                            "type": "text",
+                                            "content": ". Therefore, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                358,
+                                                505,
+                                                409
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{w,x\\} \\in E_F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                358,
+                                                505,
+                                                409
+                                            ],
+                                            "type": "text",
+                                            "content": " iff "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                358,
+                                                505,
+                                                409
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{\\pi(\\rho(w)),\\pi(\\rho(x))\\} \\in E_G"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                358,
+                                                505,
+                                                409
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 9
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                410,
+                                397,
+                                426
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        410,
+                                        397,
+                                        426
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                410,
+                                                397,
+                                                426
+                                            ],
+                                            "type": "text",
+                                            "content": "c) We clearly have "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                410,
+                                                397,
+                                                426
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\ell_F(w) = \\ell_{F_G^{(d)}(y,z)}(\\rho (w)) = \\ell_G(\\pi (\\rho (w)))"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 10
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        129,
+                        430,
+                        324,
+                        443
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                430,
+                                324,
+                                443
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        430,
+                                        324,
+                                        443
+                                    ],
+                                    "type": "text",
+                                    "content": "We have proved that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        430,
+                                        324,
+                                        443
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi \\circ \\rho \\in \\mathrm{BlsO}((F,T^r),G)"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        117,
+                        448,
+                        505,
+                        549
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                117,
+                                448,
+                                505,
+                                549
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "text",
+                                    "content": "2. We then prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "text",
+                                    "content": " is a surjection. For all "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g\\in S_1(y,z)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "text",
+                                    "content": ", we define a mapping "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\rho ,\\tau)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "text",
+                                    "content": " from "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^{r})"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F_{G}^{(d)}(y,z),T_{G}^{(d)}(y,z))"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "text",
+                                    "content": " as follows. First define "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho (u) = y"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho (v) = z"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "text",
+                                    "content": ", and set "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau (r)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "text",
+                                    "content": " to be the root of "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F_{G}^{(d)}(y,z),T_{G}^{(d)}(y,z))"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "text",
+                                    "content": ". Let "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_{1},\\dots ,w_{m}\\in V_{F}"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_1^{\\prime},\\dots ,w_{m^{\\prime}}^{\\prime}\\in V_{F}"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "text",
+                                    "content": " be vertices such that all "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{u,w_i\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{w_i',v\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "text",
+                                    "content": " correspond to bags of "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^{r}"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "text",
+                                    "content": " associated to all tree nodes of depth 2. Similarly, by definition of the Local 2-GNN unfolding tree, let "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_{1},\\dots ,x_{n}\\in V_{F_{G}^{(d)}(y,z)}"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "text",
+                                    "content": " be different vertices and "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_1^\\prime ,\\dots ,x_n^\\prime \\in V_{F_G^{(d)}(y,z)}"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "text",
+                                    "content": " be different vertices such that all "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{y,x_i\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{x_i',z\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "text",
+                                    "content": " correspond to bags of "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T_{G}^{(d)}(y,z)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "text",
+                                    "content": " associated to all tree nodes of depth 2. Since "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        448,
+                                        505,
+                                        549
+                                    ],
+                                    "type": "text",
+                                    "content": " are bIsos, we have:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        145,
+                        554,
+                        504,
+                        610
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 16,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                145,
+                                554,
+                                504,
+                                581
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        145,
+                                        554,
+                                        504,
+                                        581
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                145,
+                                                554,
+                                                504,
+                                                581
+                                            ],
+                                            "type": "text",
+                                            "content": "- For every "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                554,
+                                                504,
+                                                581
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w_{i}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                554,
+                                                504,
+                                                581
+                                            ],
+                                            "type": "text",
+                                            "content": " ("
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                554,
+                                                504,
+                                                581
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "i \\in [m]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                554,
+                                                504,
+                                                581
+                                            ],
+                                            "type": "text",
+                                            "content": "), there exists "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                554,
+                                                504,
+                                                581
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "x_{j}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                554,
+                                                504,
+                                                581
+                                            ],
+                                            "type": "text",
+                                            "content": " ("
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                554,
+                                                504,
+                                                581
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "j \\in [n]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                554,
+                                                504,
+                                                581
+                                            ],
+                                            "type": "text",
+                                            "content": "), such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                554,
+                                                504,
+                                                581
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "g(w_{i}) = \\pi(x_{j}) = \\tilde{x}_{j}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                554,
+                                                504,
+                                                581
+                                            ],
+                                            "type": "text",
+                                            "content": " for some "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                554,
+                                                504,
+                                                581
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tilde{x}_{j} \\in V_{G}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                554,
+                                                504,
+                                                581
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                554,
+                                                504,
+                                                581
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[\\{\\{u, v, w_{i}\\}\\}] \\simeq F_{G}^{(d)}(y, z)[\\{\\{y, z, x_{j}\\}\\}] \\simeq G[\\{\\{y, z, \\tilde{x}_{j}\\}\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                554,
+                                                504,
+                                                581
+                                            ],
+                                            "type": "text",
+                                            "content": ";"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 14
+                        },
+                        {
+                            "bbox": [
+                                145,
+                                582,
+                                504,
+                                610
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        145,
+                                        582,
+                                        504,
+                                        610
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                145,
+                                                582,
+                                                504,
+                                                610
+                                            ],
+                                            "type": "text",
+                                            "content": "- For every "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                582,
+                                                504,
+                                                610
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w_{i}^{\\prime}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                582,
+                                                504,
+                                                610
+                                            ],
+                                            "type": "text",
+                                            "content": " ("
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                582,
+                                                504,
+                                                610
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "i \\in [m']"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                582,
+                                                504,
+                                                610
+                                            ],
+                                            "type": "text",
+                                            "content": "), there exists "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                582,
+                                                504,
+                                                610
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "x_{j}^{\\prime}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                582,
+                                                504,
+                                                610
+                                            ],
+                                            "type": "text",
+                                            "content": " ("
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                582,
+                                                504,
+                                                610
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "j \\in [n]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                582,
+                                                504,
+                                                610
+                                            ],
+                                            "type": "text",
+                                            "content": "), such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                582,
+                                                504,
+                                                610
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "g(w_{i}^{\\prime}) = \\pi(x_{j}^{\\prime}) = \\tilde{x}_{j}^{\\prime}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                582,
+                                                504,
+                                                610
+                                            ],
+                                            "type": "text",
+                                            "content": " for some "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                582,
+                                                504,
+                                                610
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tilde{x}_{j}^{\\prime} \\in V_{G}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                582,
+                                                504,
+                                                610
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                582,
+                                                504,
+                                                610
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[\\{\\{u, v, w_{i}^{\\prime}\\}\\}] \\simeq F_{G}^{(d)}(y, z)[\\{\\{y, z, x_{j}^{\\prime}\\}\\}] \\simeq G[\\{\\{y, z, \\tilde{x}_{j}^{\\prime}\\}]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                582,
+                                                504,
+                                                610
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 15
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        129,
+                        614,
+                        505,
+                        674
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                614,
+                                505,
+                                674
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "text",
+                                    "content": "We then define "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho(w_i) = x_j"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "text",
+                                    "content": " for each "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i \\in [m]"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho(w_i') = x_j'"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "text",
+                                    "content": " for each "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i \\in [m']"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "text",
+                                    "content": ". Based on the above two items, one can easily define "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "text",
+                                    "content": " such that each node "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "text",
+                                    "content": " in "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "text",
+                                    "content": " of depth 1 or 2 is mapped by "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "text",
+                                    "content": " to a node "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "text",
+                                    "content": " in "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T_G^{(d)}(y,z)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "text",
+                                    "content": " of the same depth, "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho(\\beta_T(s)) = \\beta_{T_G^{(d)}}(t)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "text",
+                                    "content": " is an isomorphism from "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[\\beta_T(s)]"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F_G^{(d)}(y,z)[\\beta_{T_G^{(d)}(y,z)}(t)]"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        614,
+                                        505,
+                                        674
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        129,
+                        677,
+                        506,
+                        735
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                677,
+                                506,
+                                735
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        677,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": "Next, we can recursively define "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        677,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        677,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": "'s image on "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        677,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[T^r[s]]"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        677,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": " for each tree node "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        677,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        677,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": " of depth 2 following the same construction above. This is because "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        677,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        677,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": " is still a bIso from "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        677,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F[T^r[s]], T^r[s])"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        677,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        677,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        677,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        677,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        677,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": " is still a bIso from "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        677,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(F_G^{(d)}(y,z)[T_G^{(d)}(y,z)[\\tau(s)]], T_G^{(d)}(y,z)[\\tau(s)]\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        677,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        677,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        677,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        677,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g(\\beta_T(s)) = \\pi(\\beta_{T_G^{(d)}}(\\tau(s)))"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        677,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": ". Recursively applying this procedure, we can construct "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        677,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\rho, \\tau)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        677,
+                                        506,
+                                        735
+                                    ],
+                                    "type": "text",
+                                    "content": " such that it is a"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
+                        750,
+                        310,
+                        760
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                299,
+                                750,
+                                310,
+                                760
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        299,
+                                        750,
+                                        310,
+                                        760
+                                    ],
+                                    "type": "text",
+                                    "content": "31"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 30
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        128,
+                        81,
+                        504,
+                        111
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                128,
+                                81,
+                                504,
+                                111
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        128,
+                                        81,
+                                        504,
+                                        111
+                                    ],
+                                    "type": "text",
+                                    "content": "bIsoHom from "
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        81,
+                                        504,
+                                        111
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r)"
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        81,
+                                        504,
+                                        111
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        81,
+                                        504,
+                                        111
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(F_G^{(d)}(y, z), T_G^{(d)}(y, z)\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        81,
+                                        504,
+                                        111
+                                    ],
+                                    "type": "text",
+                                    "content": ". That is to say, we have proved that for all "
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        81,
+                                        504,
+                                        111
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g \\in S_1(u, v)"
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        81,
+                                        504,
+                                        111
+                                    ],
+                                    "type": "text",
+                                    "content": ", there is a preimage "
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        81,
+                                        504,
+                                        111
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\rho, \\tau) \\in S_2(y, z)"
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        81,
+                                        504,
+                                        111
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        81,
+                                        504,
+                                        111
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma(\\rho, \\tau) = g"
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        81,
+                                        504,
+                                        111
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        117,
+                        115,
+                        506,
+                        227
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                117,
+                                115,
+                                506,
+                                227
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": "3. We finally prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": " is an injection. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\rho_1, \\tau_1), (\\rho_2, \\tau_2) \\in S_2(y, z)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi \\circ \\rho_1 = \\pi \\circ \\rho_2"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": ". Let "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_1, \\dots, w_m \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_1', \\dots, w_{m'}' \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_1, \\dots, x_n \\in V_{F_G^{(d)}(y,z)}"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_1', \\dots, x_n' \\in V_{F_G^{(d)}(y,z)}"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": " be defined as in the previous item. For each "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i \\in [m]"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": ", let "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "j_1(i)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "j_2(i)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": " be indices satisfying "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_1(w_i) = x_{j_1(i)}"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_2(w_i) = x_{j_2(i)}"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": ". It follows that "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi(x_{j_1(i)}) = \\pi(x_{j_2(i)})"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": ". By definition of the Local 2-GNN unfolding tree, we must have "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x_{j_1(i)} = x_{j_2(i)}"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": ", and thus "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_1(w_i) = \\rho_2(w_i)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": ". Using a similar approach, we can prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_1(w_i') = \\rho_2(w_i')"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": " for each "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i \\in [m']"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": ". Next, we can recursively apply the above procedure to the subtree "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r[s]"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": " for each tree node of depth 2 following the previous item, and finally prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_1 = \\rho_2"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": ". Therefore, "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        115,
+                                        506,
+                                        227
+                                    ],
+                                    "type": "text",
+                                    "content": " is an injection."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        231,
+                        324,
+                        243
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                231,
+                                324,
+                                243
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        231,
+                                        324,
+                                        243
+                                    ],
+                                    "type": "text",
+                                    "content": "Combining the above three items completes the proof."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        248,
+                        504,
+                        285
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                248,
+                                504,
+                                285
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        248,
+                                        504,
+                                        285
+                                    ],
+                                    "type": "text",
+                                    "content": "Proposition C.24. Under Definition C.21, (i) if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        248,
+                                        504,
+                                        285
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\rho, \\tau)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        248,
+                                        504,
+                                        285
+                                    ],
+                                    "type": "text",
+                                    "content": " is a bIsoSurj, then "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        248,
+                                        504,
+                                        285
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        248,
+                                        504,
+                                        285
+                                    ],
+                                    "type": "text",
+                                    "content": " is a surjection from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        248,
+                                        504,
+                                        285
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        248,
+                                        504,
+                                        285
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        248,
+                                        504,
+                                        285
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        248,
+                                        504,
+                                        285
+                                    ],
+                                    "type": "text",
+                                    "content": " on both vertices and edges; (ii) if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        248,
+                                        504,
+                                        285
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\rho, \\tau)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        248,
+                                        504,
+                                        285
+                                    ],
+                                    "type": "text",
+                                    "content": " is a bIsoInj, then "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        248,
+                                        504,
+                                        285
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        248,
+                                        504,
+                                        285
+                                    ],
+                                    "type": "text",
+                                    "content": " is an injection from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        248,
+                                        504,
+                                        285
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        248,
+                                        504,
+                                        285
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        248,
+                                        504,
+                                        285
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        248,
+                                        504,
+                                        285
+                                    ],
+                                    "type": "text",
+                                    "content": " on both vertices and edges."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        296,
+                        504,
+                        358
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                296,
+                                504,
+                                358
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. We first prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "text",
+                                    "content": " is surjective if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\rho, \\tau)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "text",
+                                    "content": " is a bIsoSurj. We will only prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "text",
+                                    "content": " is surjective on edges, as proving that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "text",
+                                    "content": " is surjective on vertices is almost the same. For any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{x, y\\} \\in E_{\\tilde{F}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "text",
+                                    "content": ", by Definition C.1(b) we can pick "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{t} \\in V_{\\tilde{T}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{x, y\\} \\in \\beta_{\\tilde{T}}(\\tilde{t})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "text",
+                                    "content": ". Since "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "text",
+                                    "content": " is surjective, there exists "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t \\in V_T"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau(t) = \\tilde{t}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "text",
+                                    "content": ". By definition of bag isomorphism, there exists "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v \\in \\beta_T(t)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho(u) = x"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho(v) = y"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[\\{u, v\\}] \\simeq \\tilde{F}[\\{x, y\\}]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "text",
+                                    "content": ". Therefore, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{u, v\\} \\in E_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        296,
+                                        504,
+                                        358
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        361,
+                        504,
+                        385
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                361,
+                                504,
+                                385
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        361,
+                                        504,
+                                        385
+                                    ],
+                                    "type": "text",
+                                    "content": "We next prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        361,
+                                        504,
+                                        385
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        361,
+                                        504,
+                                        385
+                                    ],
+                                    "type": "text",
+                                    "content": " is injective if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        361,
+                                        504,
+                                        385
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\rho, \\tau)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        361,
+                                        504,
+                                        385
+                                    ],
+                                    "type": "text",
+                                    "content": " is a bIsoInj. Pick any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        361,
+                                        504,
+                                        385
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        361,
+                                        504,
+                                        385
+                                    ],
+                                    "type": "text",
+                                    "content": ". It suffices to prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        361,
+                                        504,
+                                        385
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho(u) = \\rho(v)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        361,
+                                        504,
+                                        385
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        361,
+                                        504,
+                                        385
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u = v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        361,
+                                        504,
+                                        385
+                                    ],
+                                    "type": "text",
+                                    "content": " for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        361,
+                                        504,
+                                        385
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        361,
+                                        504,
+                                        385
+                                    ],
+                                    "type": "text",
+                                    "content": ". If the result does not hold, consider two cases:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        120,
+                        389,
+                        506,
+                        524
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 9,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                120,
+                                389,
+                                504,
+                                413
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        120,
+                                        389,
+                                        504,
+                                        413
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                120,
+                                                389,
+                                                504,
+                                                413
+                                            ],
+                                            "type": "text",
+                                            "content": "- There exists "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                389,
+                                                504,
+                                                413
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "v \\in V_F, v \\neq u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                389,
+                                                504,
+                                                413
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                389,
+                                                504,
+                                                413
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho(u) = \\rho(v)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                389,
+                                                504,
+                                                413
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                389,
+                                                504,
+                                                413
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{u, v\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                389,
+                                                504,
+                                                413
+                                            ],
+                                            "type": "text",
+                                            "content": " are in the same bag of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                389,
+                                                504,
+                                                413
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                389,
+                                                504,
+                                                413
+                                            ],
+                                            "type": "text",
+                                            "content": ". This contradicts the definition of bag isomorphism."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 7
+                        },
+                        {
+                            "bbox": [
+                                120,
+                                417,
+                                506,
+                                524
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        120,
+                                        417,
+                                        506,
+                                        524
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "text",
+                                            "content": "- For all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "v \\in V_F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "v \\neq u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho(u) = \\rho(v)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "text",
+                                            "content": ", "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{u, v\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "text",
+                                            "content": " are not in the same bag of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "text",
+                                            "content": ". By Proposition C.5, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T\\left[\\bigcup_{v \\in V_F: \\rho(u) = \\rho(v)} B_T(v)\\right]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "text",
+                                            "content": " is disconnected. We can thus pick a path "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "text",
+                                            "content": " in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "text",
+                                            "content": " such that the endpoints "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t_1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t_2"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "text",
+                                            "content": " are in different connected components of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T\\left[\\bigcup_{v \\in V_F: \\rho(u) = \\rho(v)} B_T(v)\\right]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "text",
+                                            "content": ". This implies that there is a tree node "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t_3"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "text",
+                                            "content": " in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(t_3) \\cap \\{v \\in V_F: \\rho(u) = \\rho(v)\\} = \\emptyset"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "text",
+                                            "content": ". Consequently, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho(u) \\in \\rho(\\beta_T(t_1)) = \\beta_{\\tilde{T}}(\\tau(t_1))"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "text",
+                                            "content": ", "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho(u) \\in \\rho(\\beta_T(t_2)) = \\beta_{\\tilde{T}}(\\tau(t_2))"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "text",
+                                            "content": ", but "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho(u) \\notin \\beta_{\\tilde{T}}(\\tau(t_3))"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "text",
+                                            "content": ". On the other hand, since "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tau"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "text",
+                                            "content": " is injective, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tilde{T}\\left[\\tau(V_P)\\right]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "text",
+                                            "content": " is also a path and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tau(t_3)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "text",
+                                            "content": " is on the path between "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tau(t_1)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tau(t_2)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "text",
+                                            "content": " in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tilde{T}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                417,
+                                                506,
+                                                524
+                                            ],
+                                            "type": "text",
+                                            "content": ". This contradicts the definition of tree decomposition (Definition C.1(c))."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 8
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        104,
+                        529,
+                        292,
+                        541
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                529,
+                                292,
+                                541
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        529,
+                                        292,
+                                        541
+                                    ],
+                                    "type": "text",
+                                    "content": "Combining the two cases concludes the proof."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        546,
+                        504,
+                        572
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                546,
+                                504,
+                                572
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        546,
+                                        504,
+                                        572
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma C.25. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        546,
+                                        504,
+                                        572
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\mathrm{Sub},\\mathrm{L},\\mathrm{LF},\\mathrm{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        546,
+                                        504,
+                                        572
+                                    ],
+                                    "type": "text",
+                                    "content": " be any model. For any tree-decomposed graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        546,
+                                        504,
+                                        572
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^r),(\\tilde{F},\\tilde{T}^s) \\in S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        546,
+                                        504,
+                                        572
+                                    ],
+                                    "type": "text",
+                                    "content": ","
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        111,
+                        576,
+                        498,
+                        617
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                111,
+                                576,
+                                498,
+                                617
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        111,
+                                        576,
+                                        498,
+                                        617
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathsf {b l s o H o m} ((F, T ^ {r}), (\\tilde {F}, \\tilde {T} ^ {s})) = \\sum_ {(\\widehat {F}, \\widehat {T} ^ {t}) \\in \\mathcal {S} ^ {M}} \\frac {\\mathsf {b l s o S u r j} \\left((F , T ^ {r}) , (\\widehat {F} , \\widehat {T} ^ {t})\\right) \\cdot \\mathsf {b l s o l n j} \\left((\\widehat {F} , \\widehat {T} ^ {t}) , (\\tilde {F} , \\tilde {T} ^ {s})\\right)}{\\mathsf {a u t} (\\widehat {F} , \\widehat {T} ^ {t})},",
+                                    "image_path": "a070a4414e3c982961d7c187cee823e1327409348ccdd0b588f8d6d9a5e5a384.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        103,
+                        624,
+                        504,
+                        659
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                103,
+                                624,
+                                504,
+                                659
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        103,
+                                        624,
+                                        504,
+                                        659
+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        624,
+                                        504,
+                                        659
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{aut}(\\widehat{F},\\widehat{T}^t)"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        624,
+                                        504,
+                                        659
+                                    ],
+                                    "type": "text",
+                                    "content": " denotes the number of automorphisms of "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        624,
+                                        504,
+                                        659
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\widehat{F},\\widehat{T}^t)"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        624,
+                                        504,
+                                        659
+                                    ],
+                                    "type": "text",
+                                    "content": ". Here, the summation ranges over all non-isomorphic (tree-decomposed) graphs in "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        624,
+                                        504,
+                                        659
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        624,
+                                        504,
+                                        659
+                                    ],
+                                    "type": "text",
+                                    "content": " and is well-defined as there are only a finite number of graphs making the value in the summation non-zero."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        104,
+                        670,
+                        310,
+                        683
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                670,
+                                310,
+                                683
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        670,
+                                        310,
+                                        683
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. We define the following set of three-tuples:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
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+                                    "bbox": [
+                                        132,
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+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} S = \\left\\{\\left((\\widehat {F}, \\widehat {T} ^ {t}), (\\rho^ {\\mathsf {S}}, \\tau^ {\\mathsf {S}}), (\\rho^ {\\mathsf {I}}, \\tau^ {\\mathsf {I}})\\right): (\\widehat {F}, \\widehat {T} ^ {t}) \\in \\mathcal {S} ^ {M}, \\right. \\\\ \\left. \\left(\\rho^ {\\mathsf {S}}, \\tau^ {\\mathsf {S}}\\right) \\in \\mathsf {B l s o S u r j} \\left(\\left(F, T ^ {r}\\right), \\left(\\widehat {F}, \\widehat {T} ^ {t}\\right)\\right), \\left(\\rho^ {\\mathsf {I}}, \\tau^ {\\mathsf {I}}\\right) \\in \\mathsf {B l s o l n j} \\left(\\left(\\widehat {F}, \\widehat {T} ^ {t}\\right), \\left(\\tilde {F}, \\tilde {T} ^ {s}\\right)\\right) \\right\\}. \\\\ \\end{array}",
+                                    "image_path": "5bd3ed701d5647c19409bf5e53dd841a80a2557b79c6d970baaec53615b2123a.jpg"
+                                }
+                            ]
+                        }
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+            "discarded_blocks": [
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
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+                                    "type": "text",
+                                    "content": "32"
+                                }
+                            ]
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+                    ],
+                    "index": 16
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+            ],
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Define a mappings "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        83,
+                                        230,
+                                        95
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        83,
+                                        230,
+                                        95
+                                    ],
+                                    "type": "text",
+                                    "content": " such that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        203,
+                        97,
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+                        118
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                203,
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+                                406,
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+                                {
+                                    "bbox": [
+                                        203,
+                                        97,
+                                        406,
+                                        118
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\sigma \\left(\\left(\\widehat {F}, \\widehat {T} ^ {t}\\right), \\left(\\rho^ {\\mathsf {S}}, \\tau^ {\\mathsf {S}}\\right), \\left(\\rho^ {\\mathsf {I}}, \\tau^ {\\mathsf {I}}\\right)\\right) = \\left(\\rho^ {\\mathsf {I}} \\circ \\rho^ {\\mathsf {S}}, \\tau^ {\\mathsf {I}} \\circ \\tau^ {\\mathsf {S}}\\right)",
+                                    "image_path": "1f3e4a4408b31ffd05f9532509202a90ed1c916453f384f7e1abbc5d5b5c4bf4.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        121,
+                        470,
+                        140
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                                470,
+                                140
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        121,
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+                                        140
+                                    ],
+                                    "type": "text",
+                                    "content": "for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        121,
+                                        470,
+                                        140
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left((\\widehat{F},\\widehat{T}^t),(\\rho^\\mathsf{S},\\tau^\\mathsf{S}),(\\rho^\\mathsf{I},\\tau^\\mathsf{I})\\right)\\in S."
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        121,
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+                                        140
+                                    ],
+                                    "type": "text",
+                                    "content": " It suffices to prove the following three statements:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        116,
+                        145,
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+                        225
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 7,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                117,
+                                145,
+                                352,
+                                159
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                        145,
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+                                        {
+                                            "bbox": [
+                                                117,
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+                                            ],
+                                            "type": "text",
+                                            "content": "1. "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                145,
+                                                352,
+                                                159
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\sigma"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                145,
+                                                352,
+                                                159
+                                            ],
+                                            "type": "text",
+                                            "content": " is a mapping from "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                145,
+                                                352,
+                                                159
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "S"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                145,
+                                                352,
+                                                159
+                                            ],
+                                            "type": "text",
+                                            "content": " to "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                145,
+                                                352,
+                                                159
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{B}\\mathsf{I}\\mathsf{o}\\mathsf{s}\\mathsf{H}\\mathsf{o}\\mathsf{m}((F,T^r),(\\tilde{F},\\tilde{T}^s))"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 4
+                        },
+                        {
+                            "bbox": [
+                                117,
+                                163,
+                                193,
+                                175
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        163,
+                                        193,
+                                        175
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                117,
+                                                163,
+                                                193,
+                                                175
+                                            ],
+                                            "type": "text",
+                                            "content": "2. "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                163,
+                                                193,
+                                                175
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\sigma"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                163,
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+                                                175
+                                            ],
+                                            "type": "text",
+                                            "content": " is surjective;"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 5
+                        },
+                        {
+                            "bbox": [
+                                116,
+                                180,
+                                506,
+                                225
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        180,
+                                        506,
+                                        225
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                180,
+                                                506,
+                                                225
+                                            ],
+                                            "type": "text",
+                                            "content": "3. "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                180,
+                                                506,
+                                                225
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\sigma\\left((\\widehat{F}_1, \\widehat{T}_1^{t_1}), (\\rho_1^{\\mathsf{S}}, \\tau_1^{\\mathsf{S}}), (\\rho_1^{\\mathsf{I}}, \\tau_1^{\\mathsf{I}})\\right) = \\sigma\\left((\\widehat{F}_2, \\widehat{T}_2^{t_2}), (\\rho_2^{\\mathsf{S}}, \\tau_2^{\\mathsf{S}}), (\\rho_2^{\\mathsf{I}}, \\tau_2^{\\mathsf{I}})\\right)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                180,
+                                                506,
+                                                225
+                                            ],
+                                            "type": "text",
+                                            "content": " iff there exists an isomorphism "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                180,
+                                                506,
+                                                225
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(\\widehat{\\rho}, \\widehat{\\tau})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                180,
+                                                506,
+                                                225
+                                            ],
+                                            "type": "text",
+                                            "content": " from "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                180,
+                                                506,
+                                                225
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(\\widehat{F}_1, \\widehat{T}_1^{t_1})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                180,
+                                                506,
+                                                225
+                                            ],
+                                            "type": "text",
+                                            "content": " to "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                180,
+                                                506,
+                                                225
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(\\widehat{F}_2, \\widehat{T}_2^{t_2})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                180,
+                                                506,
+                                                225
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                180,
+                                                506,
+                                                225
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\widehat{\\rho} \\circ \\rho_1^{\\mathsf{S}} = \\rho_2^{\\mathsf{S}}, \\widehat{\\tau} \\circ \\tau_1^{\\mathsf{S}} = \\tau_2^{\\mathsf{S}}, \\rho_1^{\\mathsf{I}} = \\rho_2^{\\mathsf{I}} \\circ \\widehat{\\rho}, \\tau_1^{\\mathsf{I}} = \\tau_2^{\\mathsf{I}} \\circ \\widehat{\\tau}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                180,
+                                                506,
+                                                225
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 6
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        105,
+                        230,
+                        280,
+                        243
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                230,
+                                280,
+                                243
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        230,
+                                        280,
+                                        243
+                                    ],
+                                    "type": "text",
+                                    "content": "We will prove these statements one by one."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        116,
+                        247,
+                        505,
+                        313
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 11,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                117,
+                                247,
+                                504,
+                                282
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                            "bbox": [
+                                                117,
+                                                247,
+                                                504,
+                                                282
+                                            ],
+                                            "type": "text",
+                                            "content": "1. We first prove that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                247,
+                                                504,
+                                                282
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\sigma"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                247,
+                                                504,
+                                                282
+                                            ],
+                                            "type": "text",
+                                            "content": " is a mapping from "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                247,
+                                                504,
+                                                282
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "S"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                247,
+                                                504,
+                                                282
+                                            ],
+                                            "type": "text",
+                                            "content": " to "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                247,
+                                                504,
+                                                282
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{BIsoHom}((F,T^r),(\\tilde{F},\\tilde{T}^s))"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                247,
+                                                504,
+                                                282
+                                            ],
+                                            "type": "text",
+                                            "content": ". This simply follows from the fact that both bIsoSurj and bIsoInj are bIsoHom, and the composition of two bIsoHoms are still a bIsoHom."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 9
+                        },
+                        {
+                            "bbox": [
+                                116,
+                                286,
+                                505,
+                                313
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        286,
+                                        505,
+                                        313
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                286,
+                                                505,
+                                                313
+                                            ],
+                                            "type": "text",
+                                            "content": "2. We next prove that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                286,
+                                                505,
+                                                313
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\sigma"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                286,
+                                                505,
+                                                313
+                                            ],
+                                            "type": "text",
+                                            "content": " is surjective. Given "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                286,
+                                                505,
+                                                313
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(\\rho^{\\mathsf{H}},\\tau^{\\mathsf{H}})\\in \\mathrm{BlsoHom}((F,T^{r}),(\\tilde{F},\\tilde{T}^{s}))"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                286,
+                                                505,
+                                                313
+                                            ],
+                                            "type": "text",
+                                            "content": ", we define "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                286,
+                                                505,
+                                                313
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(\\widehat{F},\\widehat{T}^{t})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                286,
+                                                505,
+                                                313
+                                            ],
+                                            "type": "text",
+                                            "content": ", "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                286,
+                                                505,
+                                                313
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(\\rho^{\\mathsf{S}},\\tau^{\\mathsf{S}})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                286,
+                                                505,
+                                                313
+                                            ],
+                                            "type": "text",
+                                            "content": ", and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                286,
+                                                505,
+                                                313
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(\\rho^{\\mathsf{I}},\\tau^{\\mathsf{I}})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                286,
+                                                505,
+                                                313
+                                            ],
+                                            "type": "text",
+                                            "content": " as follows:"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 10
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        141,
+                        317,
+                        503,
+                        372
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 15,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                141,
+                                317,
+                                463,
+                                331
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        317,
+                                        463,
+                                        331
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                317,
+                                                463,
+                                                331
+                                            ],
+                                            "type": "text",
+                                            "content": "a) Let "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                317,
+                                                463,
+                                                331
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\widehat{F} = \\tilde{F}[\\rho^{\\mathsf{H}}(V_F)]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                317,
+                                                463,
+                                                331
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                317,
+                                                463,
+                                                331
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\widehat{T}^t = \\tilde{T}^s[\\tau^{\\mathsf{H}}(V_T)]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                317,
+                                                463,
+                                                331
+                                            ],
+                                            "type": "text",
+                                            "content": ". We clearly have "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                317,
+                                                463,
+                                                331
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(\\widehat{F}, \\widehat{T}^t) \\in S^M"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                317,
+                                                463,
+                                                331
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 12
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                331,
+                                498,
+                                346
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        331,
+                                        498,
+                                        346
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                331,
+                                                498,
+                                                346
+                                            ],
+                                            "type": "text",
+                                            "content": "b) Let "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                331,
+                                                498,
+                                                346
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho^{\\mathsf{S}} = \\rho^{\\mathsf{H}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                331,
+                                                498,
+                                                346
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                331,
+                                                498,
+                                                346
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tau^{\\mathsf{S}} = \\tau^{\\mathsf{H}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                331,
+                                                498,
+                                                346
+                                            ],
+                                            "type": "text",
+                                            "content": ". Obviously, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                331,
+                                                498,
+                                                346
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(\\rho^{\\mathsf{S}},\\tau^{\\mathsf{S}})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                331,
+                                                498,
+                                                346
+                                            ],
+                                            "type": "text",
+                                            "content": " is a bIsoSurj from "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                331,
+                                                498,
+                                                346
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(F,T^{r})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                331,
+                                                498,
+                                                346
+                                            ],
+                                            "type": "text",
+                                            "content": " to "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                331,
+                                                498,
+                                                346
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(\\widehat{F},\\widehat{T}^{t})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                331,
+                                                498,
+                                                346
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 13
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                346,
+                                503,
+                                372
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        346,
+                                        503,
+                                        372
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                346,
+                                                503,
+                                                372
+                                            ],
+                                            "type": "text",
+                                            "content": "c) Define identity mappings "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                346,
+                                                503,
+                                                372
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho^{\\mathsf{I}}(u) = u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                346,
+                                                503,
+                                                372
+                                            ],
+                                            "type": "text",
+                                            "content": " for all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                346,
+                                                503,
+                                                372
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u\\in V_{\\widehat{F}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                346,
+                                                503,
+                                                372
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                346,
+                                                503,
+                                                372
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tau^1 (t) = t"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                346,
+                                                503,
+                                                372
+                                            ],
+                                            "type": "text",
+                                            "content": " for all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                346,
+                                                503,
+                                                372
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t\\in V_{\\widehat{T}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                346,
+                                                503,
+                                                372
+                                            ],
+                                            "type": "text",
+                                            "content": ". Obviously, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                346,
+                                                503,
+                                                372
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(\\rho^{\\mathsf{I}},\\tau^{\\mathsf{I}})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                346,
+                                                503,
+                                                372
+                                            ],
+                                            "type": "text",
+                                            "content": " is a bIsoInj from "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                346,
+                                                503,
+                                                372
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(\\widehat{F},\\widehat{T}^{t})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                346,
+                                                503,
+                                                372
+                                            ],
+                                            "type": "text",
+                                            "content": " to "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                346,
+                                                503,
+                                                372
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(\\tilde{F},\\tilde{T}^{s})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                346,
+                                                503,
+                                                372
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 14
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        129,
+                        375,
+                        418,
+                        388
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                375,
+                                418,
+                                388
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        375,
+                                        418,
+                                        388
+                                    ],
+                                    "type": "text",
+                                    "content": "We clearly have "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        375,
+                                        418,
+                                        388
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho^{\\mathsf{H}} = \\rho^{\\mathsf{I}}\\circ \\rho^{\\mathsf{S}}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        375,
+                                        418,
+                                        388
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        375,
+                                        418,
+                                        388
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau^{\\mathrm{H}} = \\tau^{\\mathrm{I}}\\circ \\tau^{\\mathrm{S}}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        375,
+                                        418,
+                                        388
+                                    ],
+                                    "type": "text",
+                                    "content": ". Thus, "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        375,
+                                        418,
+                                        388
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        375,
+                                        418,
+                                        388
+                                    ],
+                                    "type": "text",
+                                    "content": " is a surjection."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        117,
+                        393,
+                        504,
+                        450
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                117,
+                                393,
+                                504,
+                                450
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        393,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "text",
+                                    "content": "3. We finally prove the aforementioned item 3. It suffices to prove only one direction, namely, "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        393,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma\\left((\\widehat{F}_1,\\widehat{T}_1^{t_1}),(p_1^{\\mathsf{S}},\\tau_1^{\\mathsf{S}}),(p_1^{\\mathsf{l}},\\tau_1^{\\mathsf{l}})\\right) = \\sigma\\left((\\widehat{F}_2,\\widehat{T}_2^{t_2}),(p_2^{\\mathsf{S}},\\tau_2^{\\mathsf{S}}),(p_2^{\\mathsf{l}},\\tau_2^{\\mathsf{l}})\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        393,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "text",
+                                    "content": " implies that there exists an isomorphism "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        393,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\widehat{\\rho},\\widehat{\\tau})"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        393,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "text",
+                                    "content": " from "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        393,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\widehat{F}_1,\\widehat{T}_1^{t_1})"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        393,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        393,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\widehat{F}_2,\\widehat{T}_2^{t_2})"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        393,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        393,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\widehat{\\rho}\\circ \\rho_{1}^{\\mathsf{S}} = \\rho_{2}^{\\mathsf{S}},\\widehat{\\tau}\\circ \\tau_{1}^{\\mathsf{S}} = \\tau_{2}^{\\mathsf{S}},"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        393,
+                                        504,
+                                        450
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_1^{\\mathrm{l}} = \\rho_2^{\\mathrm{l}}\\circ \\widehat{\\rho},\\tau_1^{\\mathrm{l}} = \\tau_2^{\\mathrm{l}}\\circ \\widehat{\\tau}"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        141,
+                        454,
+                        333,
+                        468
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                141,
+                                454,
+                                333,
+                                468
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        454,
+                                        333,
+                                        468
+                                    ],
+                                    "type": "text",
+                                    "content": "a) We first prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        454,
+                                        333,
+                                        468
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\widehat{F}_1\\simeq \\widehat{F}_2"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        454,
+                                        333,
+                                        468
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        454,
+                                        333,
+                                        468
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\widehat{T}_1^{t_1}\\simeq \\widehat{T}_2^{t_2}"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
+                        153,
+                        468,
+                        504,
+                        514
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                153,
+                                468,
+                                504,
+                                514
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        153,
+                                        468,
+                                        504,
+                                        514
+                                    ],
+                                    "type": "text",
+                                    "content": "For any "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        468,
+                                        504,
+                                        514
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        468,
+                                        504,
+                                        514
+                                    ],
+                                    "type": "text",
+                                    "content": ", if "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        468,
+                                        504,
+                                        514
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_1^{\\mathsf{S}}(u) \\neq \\rho_1^{\\mathsf{S}}(v)"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        468,
+                                        504,
+                                        514
+                                    ],
+                                    "type": "text",
+                                    "content": ", then "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        468,
+                                        504,
+                                        514
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_1^{\\mathsf{I}}(\\rho_1^{\\mathsf{S}}(u)) \\neq \\rho_1^{\\mathsf{I}}(\\rho_1^{\\mathsf{S}}(v))"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        468,
+                                        504,
+                                        514
+                                    ],
+                                    "type": "text",
+                                    "content": " since "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        468,
+                                        504,
+                                        514
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_1^{\\mathsf{I}}"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        468,
+                                        504,
+                                        514
+                                    ],
+                                    "type": "text",
+                                    "content": " is an injection (by Proposition C.24). Therefore, "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        468,
+                                        504,
+                                        514
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_2^{\\mathsf{I}}(\\rho_2^{\\mathsf{S}}(u)) \\neq \\rho_2^{\\mathsf{I}}(\\rho_2^{\\mathsf{S}}(v))"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        468,
+                                        504,
+                                        514
+                                    ],
+                                    "type": "text",
+                                    "content": ", and thus "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        468,
+                                        504,
+                                        514
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_2^{\\mathsf{S}}(u) \\neq \\rho_2^{\\mathsf{S}}(v)"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        468,
+                                        504,
+                                        514
+                                    ],
+                                    "type": "text",
+                                    "content": ". By symmetry, we also have that "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        468,
+                                        504,
+                                        514
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_2^{\\mathsf{S}}(u) \\neq \\rho_2^{\\mathsf{S}}(v)"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        468,
+                                        504,
+                                        514
+                                    ],
+                                    "type": "text",
+                                    "content": " implies "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        468,
+                                        504,
+                                        514
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_1^{\\mathsf{S}}(u) \\neq \\rho_1^{\\mathsf{S}}(v)"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        468,
+                                        504,
+                                        514
+                                    ],
+                                    "type": "text",
+                                    "content": ". This proves that "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        468,
+                                        504,
+                                        514
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_1^{\\mathsf{S}}(u) = \\rho_1^{\\mathsf{S}}(v)"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        468,
+                                        504,
+                                        514
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        468,
+                                        504,
+                                        514
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_2^{\\mathsf{S}}(u) = \\rho_2^{\\mathsf{S}}(v)"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        468,
+                                        504,
+                                        514
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                },
+                {
+                    "bbox": [
+                        153,
+                        512,
+                        504,
+                        606
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                153,
+                                512,
+                                504,
+                                606
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "text",
+                                    "content": "For any "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "text",
+                                    "content": ", if "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\rho_1^S(u), \\rho_1^S(v)\\} \\in E_{\\widehat{F}_1}"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "text",
+                                    "content": ", then "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\rho_1^1(\\rho_1^S(u)), \\rho_1^1(\\rho_1^S(v))\\} \\in E_{\\widehat{F}}"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "text",
+                                    "content": " since "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_1^1"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "text",
+                                    "content": " is a homomorphism. Therefore, "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\rho_2^1(\\rho_2^S(u)), \\rho_2^1(\\rho_2^S(v))\\} \\in E_{\\widehat{F}}"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "text",
+                                    "content": ". This implies that "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_2^1(\\rho_2^S(u))"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_2^1(\\rho_2^S(v))"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "text",
+                                    "content": " are in the same bag of "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{T}^s"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "text",
+                                    "content": ". Therefore, there are vertices "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x, y \\in V_{\\widehat{F}_2}"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "text",
+                                    "content": " in the same bag of "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\widehat{T}_2^{s_2}"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_2^1(x) = \\rho_2^1(\\rho_2^S(u))"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_2^1(y) = \\rho_2^1(\\rho_2^S(v))"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "text",
+                                    "content": ", and by definition of bag isomorphism we have "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{x, y\\} \\in E_{\\widehat{F}_2}"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "text",
+                                    "content": ". Since "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_2"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "text",
+                                    "content": " is injective, "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x = \\rho_2^S(u)"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "y = \\rho_2^S(v)"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "text",
+                                    "content": ", namely, "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\rho_2^S(u), \\rho_2^S(v)\\} \\in E_{\\widehat{F}_2}"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "text",
+                                    "content": ". By symmetry, we can prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\rho_1^S(u), \\rho_1^S(v)\\} \\in E_{\\widehat{F}_1}"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\rho_2^S(u), \\rho_2^S(v)\\} \\in E_{\\widehat{F}_2}"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
+                                        512,
+                                        504,
+                                        606
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                },
+                {
+                    "bbox": [
+                        154,
+                        605,
+                        504,
+                        650
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                154,
+                                605,
+                                504,
+                                650
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        154,
+                                        605,
+                                        504,
+                                        650
+                                    ],
+                                    "type": "text",
+                                    "content": "Finally, noting that "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        605,
+                                        504,
+                                        650
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_1^{\\mathsf{S}}"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        605,
+                                        504,
+                                        650
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        605,
+                                        504,
+                                        650
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_2^{\\mathsf{S}}"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        605,
+                                        504,
+                                        650
+                                    ],
+                                    "type": "text",
+                                    "content": " are surjective (by Proposition C.24) and "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        605,
+                                        504,
+                                        650
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell_{\\widehat{F}_1}(\\rho_1^{\\mathsf{S}}(u)) = \\ell_F(u) = \\ell_{\\widehat{F}_2}(\\rho_2^{\\mathsf{S}}(u))"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        605,
+                                        504,
+                                        650
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        605,
+                                        504,
+                                        650
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        605,
+                                        504,
+                                        650
+                                    ],
+                                    "type": "text",
+                                    "content": ", we obtain that "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        605,
+                                        504,
+                                        650
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\widehat{F}_1 \\simeq \\widehat{F}_2"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        605,
+                                        504,
+                                        650
+                                    ],
+                                    "type": "text",
+                                    "content": ". Following the same procedure, we can prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        605,
+                                        504,
+                                        650
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\widehat{T}_1^{t_1} \\simeq \\widehat{T}_2^{t_2}"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        605,
+                                        504,
+                                        650
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 21
+                },
+                {
+                    "bbox": [
+                        141,
+                        648,
+                        504,
+                        673
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                141,
+                                648,
+                                504,
+                                673
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        648,
+                                        504,
+                                        673
+                                    ],
+                                    "type": "text",
+                                    "content": "b) Consequently, there exist isomorphisms "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        648,
+                                        504,
+                                        673
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\widehat{\\rho}"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        648,
+                                        504,
+                                        673
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        648,
+                                        504,
+                                        673
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\widehat{\\tau}"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        648,
+                                        504,
+                                        673
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        648,
+                                        504,
+                                        673
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\widehat{\\rho} \\circ \\rho_1^{\\mathsf{S}} = \\rho_2^{\\mathsf{S}}, \\widehat{\\tau} \\circ \\tau_1^{\\mathsf{S}} = \\tau_2^{\\mathsf{S}}"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        648,
+                                        504,
+                                        673
+                                    ],
+                                    "type": "text",
+                                    "content": ". For any node "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        648,
+                                        504,
+                                        673
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "q \\in V_T"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        648,
+                                        504,
+                                        673
+                                    ],
+                                    "type": "text",
+                                    "content": ","
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 22
+                },
+                {
+                    "bbox": [
+                        168,
+                        674,
+                        490,
+                        692
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                168,
+                                674,
+                                490,
+                                692
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        168,
+                                        674,
+                                        490,
+                                        692
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\widehat {\\rho} (\\beta_ {\\widehat {T} _ {1}} (\\tau_ {1} ^ {\\mathsf {S}} (q))) = \\widehat {\\rho} (\\rho_ {1} ^ {\\mathsf {S}} (\\beta_ {T} (q))) = \\rho_ {2} ^ {\\mathsf {S}} (\\beta_ {T} (q))) = \\beta_ {\\widehat {T} _ {2}} (\\tau_ {2} ^ {\\mathsf {S}} (q)) = \\beta_ {\\widehat {T} _ {2}} (\\widehat {\\tau} (\\tau_ {1} ^ {\\mathsf {S}} (q))).",
+                                    "image_path": "95650bfee2b8de7493f365b6beabfe15dc85c6c10a316e59feb80e05775470bf.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 23
+                },
+                {
+                    "bbox": [
+                        154,
+                        694,
+                        504,
+                        732
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                154,
+                                694,
+                                504,
+                                732
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        154,
+                                        694,
+                                        504,
+                                        732
+                                    ],
+                                    "type": "text",
+                                    "content": "Since "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        694,
+                                        504,
+                                        732
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau_1^S"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        694,
+                                        504,
+                                        732
+                                    ],
+                                    "type": "text",
+                                    "content": " is surjective, "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        694,
+                                        504,
+                                        732
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau_1^S (q)"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        694,
+                                        504,
+                                        732
+                                    ],
+                                    "type": "text",
+                                    "content": " ranges over all nodes in "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        694,
+                                        504,
+                                        732
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\widehat{T}_1^{s_1}"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        694,
+                                        504,
+                                        732
+                                    ],
+                                    "type": "text",
+                                    "content": " when "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        694,
+                                        504,
+                                        732
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "q"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        694,
+                                        504,
+                                        732
+                                    ],
+                                    "type": "text",
+                                    "content": " ranges over "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        694,
+                                        504,
+                                        732
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "V_{T}"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        694,
+                                        504,
+                                        732
+                                    ],
+                                    "type": "text",
+                                    "content": ". We thus conclude that "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        694,
+                                        504,
+                                        732
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\rho ,\\tau)"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        694,
+                                        504,
+                                        732
+                                    ],
+                                    "type": "text",
+                                    "content": " is an isomorphism from "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        694,
+                                        504,
+                                        732
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\widehat{F}_1,\\widehat{T}_1^{t_1})"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        694,
+                                        504,
+                                        732
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        694,
+                                        504,
+                                        732
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\widehat{F}_2,\\widehat{T}_2^{t_2})"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        694,
+                                        504,
+                                        732
+                                    ],
+                                    "type": "text",
+                                    "content": " (see Definition C.8)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 24
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
+                        750,
+                        310,
+                        760
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                299,
+                                750,
+                                310,
+                                760
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        299,
+                                        750,
+                                        310,
+                                        760
+                                    ],
+                                    "type": "text",
+                                    "content": "33"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 25
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 32
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        141,
+                        81,
+                        504,
+                        130
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                141,
+                                81,
+                                504,
+                                130
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": "c) We finally prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_1^{\\mathsf{I}} = \\rho_2^{\\mathsf{I}}\\circ \\widehat{\\rho}"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau_1^{\\mathsf{I}} = \\tau_2^{\\mathsf{I}}\\circ \\widehat{\\tau}"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": ". Pick any "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u\\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": ", we have "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_2^{\\mathsf{I}}(\\widehat{\\rho} (\\rho_1^{\\mathsf{S}}(u))) = \\rho_2^{\\mathsf{I}}(\\rho_2^{\\mathsf{S}}(u)) = \\rho_1^{\\mathsf{I}}(\\rho_1^{\\mathsf{S}}(u))"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": ". Since "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_{1}^{\\mathsf{S}}"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": " is surjective, "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_{1}^{\\mathsf{S}}(u)"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": " ranges over all nodes in "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\widehat{F}_1"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": " when "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": " ranges over "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "V_{F}"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": ". This proves that "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_1^{\\mathsf{I}} = \\rho_2^{\\mathsf{I}}\\circ \\widehat{\\rho}"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": ". Following the same procedure, we can prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau_1^{\\mathsf{I}} = \\tau_2^{\\mathsf{I}}\\circ \\widehat{\\tau}"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        105,
+                        134,
+                        324,
+                        147
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                134,
+                                324,
+                                147
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        134,
+                                        324,
+                                        147
+                                    ],
+                                    "type": "text",
+                                    "content": "Combining the above three items concludes the proof."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "type": "image",
+                    "bbox": [
+                        494,
+                        135,
+                        504,
+                        144
+                    ],
+                    "blocks": [
+                        {
+                            "bbox": [
+                                494,
+                                135,
+                                504,
+                                144
+                            ],
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        494,
+                                        135,
+                                        504,
+                                        144
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                494,
+                                                135,
+                                                504,
+                                                144
+                                            ],
+                                            "type": "image",
+                                            "image_path": "2c363b7f4c30baf190b496a4a88db627c65ea10677b711930d0c51fc336999c8.jpg"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 3,
+                            "angle": 0,
+                            "type": "image_body"
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        160,
+                        504,
+                        184
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                160,
+                                504,
+                                184
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        160,
+                                        504,
+                                        184
+                                    ],
+                                    "type": "text",
+                                    "content": "In the following, we further define two technical concepts that will be used to describe the next result. We note that these technical concepts also appeared in Dell et al. (2018)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        188,
+                        504,
+                        262
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                188,
+                                504,
+                                262
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        188,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition C.26. A tree-decomposed graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        188,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{F},\\tilde{T}^{s})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        188,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "text",
+                                    "content": " is called a bag extension of another tree-decomposed graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        188,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        188,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "text",
+                                    "content": " if there is a graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        188,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        188,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "text",
+                                    "content": " and a mapping "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        188,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\rho ,\\tau)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        188,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        188,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        188,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "text",
+                                    "content": " is a subgraph of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        188,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        188,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        188,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\rho ,\\tau)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        188,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "text",
+                                    "content": " is an isomorphism from the tree-decomposed graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        188,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(H,T^r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        188,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        188,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{F},\\tilde{T}^{s})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        188,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "text",
+                                    "content": ". Define "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        188,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{BExt}\\left((F,T^r),(\\tilde{F},\\tilde{T}^s)\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        188,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "text",
+                                    "content": " to be the set of all mappings "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        188,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\rho ,\\tau)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        188,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "text",
+                                    "content": " that satisfies the above conditions, and define "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        188,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{bExt}\\left((F,T^r),(\\tilde{F},\\tilde{T}^s)\\right) = \\left|\\mathsf{BExt}\\left((F,T^r),(\\tilde{F},\\tilde{T}^s)\\right)\\right|"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        188,
+                                        504,
+                                        262
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        265,
+                        504,
+                        301
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                265,
+                                504,
+                                301
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        265,
+                                        504,
+                                        301
+                                    ],
+                                    "type": "text",
+                                    "content": "Remark C.27. In other words, a bag extension of a tree-decomposed graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        265,
+                                        504,
+                                        301
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        265,
+                                        504,
+                                        301
+                                    ],
+                                    "type": "text",
+                                    "content": " can be obtained by adding an arbitrary number of edges to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        265,
+                                        504,
+                                        301
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        265,
+                                        504,
+                                        301
+                                    ],
+                                    "type": "text",
+                                    "content": " while ensuring that each added edge is contained in a tree node in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        265,
+                                        504,
+                                        301
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        265,
+                                        504,
+                                        301
+                                    ],
+                                    "type": "text",
+                                    "content": ". A immediate fact is that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        265,
+                                        504,
+                                        301
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\rho ,\\tau)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        265,
+                                        504,
+                                        301
+                                    ],
+                                    "type": "text",
+                                    "content": " is a homomorphism from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        265,
+                                        504,
+                                        301
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^{r})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        265,
+                                        504,
+                                        301
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        265,
+                                        504,
+                                        301
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{F},\\tilde{T}^{s})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        265,
+                                        504,
+                                        301
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        304,
+                        504,
+                        363
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                304,
+                                504,
+                                363
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition C.28. Given a tree-decomposed graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": " and a graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": ", a bag-strong homomorphism from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": " (abbreviated as \"bStrHom\") is a homomorphism "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": " from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": " such that, for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t\\in V_T"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": " is a strong homomorphism from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[\\beta_T(t)]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G[f(\\beta_T(t))]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": ", i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{u,v\\} \\in E_{F[\\beta_T(t)]}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{f(u),f(v)\\} \\in E_{G[f(\\beta_T(t))]}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": ". Denote BStrHom "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "((F,T^r),G)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": " to be the set of all bStrHom from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": ", and denote bStrHom "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "((F,T^r),G) = |\\mathsf{BStrHom}((F,T^r),G)|"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        304,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        105,
+                        371,
+                        278,
+                        384
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                371,
+                                278,
+                                384
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        371,
+                                        278,
+                                        384
+                                    ],
+                                    "type": "text",
+                                    "content": "The following equation is straightforward:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        386,
+                        506,
+                        410
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                386,
+                                506,
+                                410
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        386,
+                                        506,
+                                        410
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma C.29. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        386,
+                                        506,
+                                        410
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        386,
+                                        506,
+                                        410
+                                    ],
+                                    "type": "text",
+                                    "content": " be any model. For any graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        386,
+                                        506,
+                                        410
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        386,
+                                        506,
+                                        410
+                                    ],
+                                    "type": "text",
+                                    "content": " and tree-decomposed graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        386,
+                                        506,
+                                        410
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        386,
+                                        506,
+                                        410
+                                    ],
+                                    "type": "text",
+                                    "content": ","
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        148,
+                        415,
+                        461,
+                        456
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                148,
+                                415,
+                                461,
+                                456
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        148,
+                                        415,
+                                        461,
+                                        456
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\operatorname {h o m} (F, G) = \\sum_ {(\\tilde {F}, \\tilde {T} ^ {s}) \\in \\mathcal {S} ^ {M}} \\frac {\\mathrm {b E x t} ((F , T ^ {r}) , (\\tilde {F} , \\tilde {T} ^ {s})) \\cdot \\mathrm {b S t r H o m} ((\\tilde {F} , \\tilde {T} ^ {s}) , G)}{\\operatorname {a u t} (\\tilde {F} , \\tilde {T} ^ {s})},",
+                                    "image_path": "e41ecb29a2e9bd087d07ee410ddf4cdace908ce692c7d6a18ae2c8a3c51d3044.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        464,
+                        504,
+                        499
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                464,
+                                504,
+                                499
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        464,
+                                        504,
+                                        499
+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        464,
+                                        504,
+                                        499
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{aut}(\\tilde{F},\\tilde{T}^{s})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        464,
+                                        504,
+                                        499
+                                    ],
+                                    "type": "text",
+                                    "content": " denotes the number of automorphisms of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        464,
+                                        504,
+                                        499
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{F},\\tilde{T}^{s})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        464,
+                                        504,
+                                        499
+                                    ],
+                                    "type": "text",
+                                    "content": ". Here, the summation ranges over all non-isomorphic (tree-decomposed) graphs in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        464,
+                                        504,
+                                        499
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        464,
+                                        504,
+                                        499
+                                    ],
+                                    "type": "text",
+                                    "content": " and is well-defined as there are only a finite number of graphs making the value in the summation non-zero."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        104,
+                        513,
+                        506,
+                        536
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                513,
+                                506,
+                                536
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        513,
+                                        506,
+                                        536
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. The proof has a similar structure to the previous lemma. We define the following set of three-tuples:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        105,
+                        542,
+                        528,
+                        564
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                542,
+                                528,
+                                564
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        542,
+                                        528,
+                                        564
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "S = \\left\\{\\left((\\tilde {F}, \\tilde {T} ^ {s}), (\\rho , \\tau), g\\right): (\\tilde {F}, \\tilde {T} ^ {s}) \\in \\mathcal {S} ^ {M}, (\\rho , \\tau) \\in \\mathsf {B E x t} \\Big ((F, T ^ {r}), (\\tilde {F}, \\tilde {T} ^ {s}) \\Big), g \\in \\mathsf {B S t r H o m} ((\\tilde {F}, \\tilde {T} ^ {s}), G) \\right\\}.",
+                                    "image_path": "ed5dfe92593339543bfd21c9c7554fb1122fe7bf5970a5b1a8040c08793c4d95.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        104,
+                        570,
+                        504,
+                        622
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                570,
+                                504,
+                                622
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        504,
+                                        622
+                                    ],
+                                    "type": "text",
+                                    "content": "Define a mappings "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        504,
+                                        622
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        504,
+                                        622
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        504,
+                                        622
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma\\left((\\tilde{F},\\tilde{T}^s),(\\rho,\\tau),g\\right)=g\\circ\\rho"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        504,
+                                        622
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        504,
+                                        622
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left((\\tilde{F},\\tilde{T}^s),(\\rho,\\tau),g\\right)\\in S"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        504,
+                                        622
+                                    ],
+                                    "type": "text",
+                                    "content": ". It is straightforward to see that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        504,
+                                        622
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        504,
+                                        622
+                                    ],
+                                    "type": "text",
+                                    "content": " is a mapping from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        504,
+                                        622
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        504,
+                                        622
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        504,
+                                        622
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{BlsoHom}((F,T^r),(\\tilde{F},\\tilde{T}^s))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        504,
+                                        622
+                                    ],
+                                    "type": "text",
+                                    "content": ", namely, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        504,
+                                        622
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g\\circ\\rho"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        504,
+                                        622
+                                    ],
+                                    "type": "text",
+                                    "content": " is a homomorphism from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        504,
+                                        622
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        504,
+                                        622
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        504,
+                                        622
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        504,
+                                        622
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        504,
+                                        622
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left((\\tilde{F},\\tilde{T}^s),(\\rho,\\tau),g\\right)\\in S"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        570,
+                                        504,
+                                        622
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        105,
+                        626,
+                        503,
+                        641
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                626,
+                                503,
+                                641
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        626,
+                                        503,
+                                        641
+                                    ],
+                                    "type": "text",
+                                    "content": "We then prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        626,
+                                        503,
+                                        641
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        626,
+                                        503,
+                                        641
+                                    ],
+                                    "type": "text",
+                                    "content": " is surjective. Given "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        626,
+                                        503,
+                                        641
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h\\in \\mathrm{Hom}(F,G)"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        626,
+                                        503,
+                                        641
+                                    ],
+                                    "type": "text",
+                                    "content": ", define "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        626,
+                                        503,
+                                        641
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{F},\\tilde{T}^{s})"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        626,
+                                        503,
+                                        641
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        626,
+                                        503,
+                                        641
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\rho ,\\tau)"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        626,
+                                        503,
+                                        641
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        626,
+                                        503,
+                                        641
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        626,
+                                        503,
+                                        641
+                                    ],
+                                    "type": "text",
+                                    "content": " as follows:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        116,
+                        645,
+                        505,
+                        733
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 19,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                116,
+                                645,
+                                505,
+                                672
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        645,
+                                        505,
+                                        672
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                645,
+                                                505,
+                                                672
+                                            ],
+                                            "type": "text",
+                                            "content": "a) Define "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                645,
+                                                505,
+                                                672
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tilde{F}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                645,
+                                                505,
+                                                672
+                                            ],
+                                            "type": "text",
+                                            "content": " be the graph obtained from "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                645,
+                                                505,
+                                                672
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                645,
+                                                505,
+                                                672
+                                            ],
+                                            "type": "text",
+                                            "content": " by adding edges "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                645,
+                                                505,
+                                                672
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{\\{u,v\\} : \\exists t \\in V_T \\text{s.t.} u,v \\in \\beta_T(t), \\{h(u),h(v)\\} \\in E_G\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                645,
+                                                505,
+                                                672
+                                            ],
+                                            "type": "text",
+                                            "content": ", and let "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                645,
+                                                505,
+                                                672
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tilde{T}^s = T^r"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                645,
+                                                505,
+                                                672
+                                            ],
+                                            "type": "text",
+                                            "content": ". Clearly, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                645,
+                                                505,
+                                                672
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(\\tilde{F},\\tilde{T}^s)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                645,
+                                                505,
+                                                672
+                                            ],
+                                            "type": "text",
+                                            "content": " is a bag extension of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                645,
+                                                505,
+                                                672
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(F,T^r)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                645,
+                                                505,
+                                                672
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 16
+                        },
+                        {
+                            "bbox": [
+                                116,
+                                675,
+                                504,
+                                702
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        675,
+                                        504,
+                                        702
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                675,
+                                                504,
+                                                702
+                                            ],
+                                            "type": "text",
+                                            "content": "b) Define identity mappings "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                675,
+                                                504,
+                                                702
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho(u) = u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                675,
+                                                504,
+                                                702
+                                            ],
+                                            "type": "text",
+                                            "content": " for all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                675,
+                                                504,
+                                                702
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u \\in V_F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                675,
+                                                504,
+                                                702
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                675,
+                                                504,
+                                                702
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tau(t) = t"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                675,
+                                                504,
+                                                702
+                                            ],
+                                            "type": "text",
+                                            "content": " for all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                675,
+                                                504,
+                                                702
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t \\in V_T"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                675,
+                                                504,
+                                                702
+                                            ],
+                                            "type": "text",
+                                            "content": ". Clearly, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                675,
+                                                504,
+                                                702
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(\\rho, \\tau) \\in \\mathsf{BExt}((F, T^r), (\\tilde{F}, \\tilde{T}^s))"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                675,
+                                                504,
+                                                702
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 17
+                        },
+                        {
+                            "bbox": [
+                                116,
+                                706,
+                                504,
+                                733
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        706,
+                                        504,
+                                        733
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                706,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "text",
+                                            "content": "c) Let "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                706,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "g = h"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                706,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "text",
+                                            "content": ". It is easy to see that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                706,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "g"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                706,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "text",
+                                            "content": " is a strong homomorphism from "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                706,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tilde{F}[\\beta_{\\tilde{T}}(t)]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                706,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "text",
+                                            "content": " to "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                706,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "G[g(\\beta_{\\tilde{T}}(t))]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                706,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "text",
+                                            "content": " for each "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                706,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t \\in V_{\\tilde{T}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                706,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "text",
+                                            "content": ". Thus, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                706,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "g \\in \\mathsf{BStrHom}((\\tilde{F}, \\tilde{T}^s), G)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                706,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 18
+                        }
+                    ],
+                    "sub_type": "text"
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
+                        750,
+                        311,
+                        760
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                299,
+                                750,
+                                311,
+                                760
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        299,
+                                        750,
+                                        311,
+                                        760
+                                    ],
+                                    "type": "text",
+                                    "content": "34"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 33
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        82,
+                        367,
+                        95
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                82,
+                                367,
+                                95
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        367,
+                                        95
+                                    ],
+                                    "type": "text",
+                                    "content": "Noting that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        367,
+                                        95
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h = g = g \\circ \\rho"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        367,
+                                        95
+                                    ],
+                                    "type": "text",
+                                    "content": ", we have proved that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        367,
+                                        95
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        367,
+                                        95
+                                    ],
+                                    "type": "text",
+                                    "content": " is a surjection."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        100,
+                        506,
+                        216
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                100,
+                                506,
+                                216
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": "We finally prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma\\left((\\tilde{F}_1, \\tilde{T}_1^{s_1}), (\\rho_1, \\tau_1), g_1\\right) = \\sigma\\left((\\tilde{F}_2, \\tilde{T}_2^{s_2}), (\\rho_2, \\tau_2), g_2\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": " implies that there exists an isomorphism "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{\\rho}, \\tilde{\\tau})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": " from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{F}_1, \\tilde{T}_1^{s_1})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{F}_2, \\tilde{T}_2^{s_2})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\rho} \\circ \\rho_1 = \\rho_2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\tau} \\circ \\tau_1 = \\tau_2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g_1 = g_2 \\circ \\tilde{\\rho}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": ". We first prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F}_1 \\simeq \\tilde{F}_2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{T}_1^{s_1} \\simeq \\tilde{T}_2^{s_2}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": ". (i) For any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": ", we obviously have "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_1(u) = \\rho_1(v)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u = v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_2(u) = \\rho_2(v)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": ". (ii) Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i \\in \\{1, 2\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": ". For any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\rho_i(u), \\rho_i(v)\\} \\in E_{\\tilde{F}_i}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{u, v\\} \\in E_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": " or "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": " are in the same bag of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{g_i(\\rho_i(u)), g(\\rho_i(v))\\} \\in E_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": ". Since "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g_1 \\circ \\rho_1 = g_2 \\circ \\rho_2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": ", we have "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\rho_1(u), \\rho_1(v)\\} \\in E_{\\tilde{F}_1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\rho_2(u), \\rho_2(v)\\} \\in E_{\\tilde{F}_2}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": ". (iii) Finally, noting that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": " are bijective and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell_{\\tilde{F}_1}(\\rho_1(u)) = \\ell_F(u) = \\ell_{\\tilde{F}_2}(\\rho_2(u))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": ", we obtain that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F}_1 \\simeq \\tilde{F}_2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": ". On the other hand, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{T}_1^{s_1} \\simeq \\tilde{T}_2^{s_2}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        100,
+                                        506,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": " trivially holds. The remaining procedure is almost the same as in the previous lemma."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        228,
+                        504,
+                        251
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                228,
+                                504,
+                                251
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        228,
+                                        504,
+                                        251
+                                    ],
+                                    "type": "text",
+                                    "content": "We next show that the mapping bStrHom can be further decomposed as shown in Lemma C.31. We need an auxiliary concept:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        255,
+                        504,
+                        319
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                255,
+                                504,
+                                319
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition C.30. Given two tree-decomposed graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{F},\\tilde{T}^{s})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "text",
+                                    "content": " , a homomorphism "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\rho ,\\tau)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "text",
+                                    "content": " from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^{r})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{F},\\tilde{T}^{s})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "text",
+                                    "content": " is called bag-strong surjective (abbreviated as \"bStrSurj\") if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "text",
+                                    "content": " is a bagstrong homomorphism from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^{r})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "text",
+                                    "content": " and is surjective on both vertices and edges, and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "text",
+                                    "content": " is an isomorphism from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{T}^s"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "text",
+                                    "content": " . Denote BStrSurj "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "((F,T^r),(\\tilde{F},\\tilde{T}^{s}))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "text",
+                                    "content": " to be the set of all bStrSurj from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^{r})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{F},\\tilde{T}^{s})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "text",
+                                    "content": " , and denote bStrSurj "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        255,
+                                        504,
+                                        319
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "((F,T^r),(\\tilde{F},\\tilde{T}^{s})) = |\\mathsf{BStrSurj}((F,T^r),(\\tilde{F},\\tilde{T}^{s}))|"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        320,
+                        506,
+                        344
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                320,
+                                506,
+                                344
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        320,
+                                        506,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma C.31. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        320,
+                                        506,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        320,
+                                        506,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": " be any model. For any graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        320,
+                                        506,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        320,
+                                        506,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": " and tree-decomposed graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        320,
+                                        506,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        320,
+                                        506,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": ","
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        129,
+                        349,
+                        480,
+                        389
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                349,
+                                480,
+                                389
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        349,
+                                        480,
+                                        389
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathsf{bStrHom}((F,T^{r}),G) = \\sum_{(\\tilde{F},\\tilde{T}^{s})\\in \\mathcal{S}^{M}}\\frac{\\mathsf{bStrSurj}\\left((F,T^{r}),( \\tilde{F},\\tilde{T}^{s})\\right)\\cdot\\mathsf{blso}\\left((\\tilde{F},\\tilde{T}^{s}),G\\right)}{\\mathsf{aut}(\\tilde{F},\\tilde{T}^{s})},",
+                                    "image_path": "dbdbbf1005da5c46409f372de6d898bbaa398cc6893bd5413c0f9e637a02dfdb.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        396,
+                        504,
+                        431
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                396,
+                                504,
+                                431
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        431
+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        431
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{aut}(\\tilde{F},\\tilde{T}^s)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        431
+                                    ],
+                                    "type": "text",
+                                    "content": " denotes the number of automorphisms of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        431
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{F},\\tilde{T}^{s})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        431
+                                    ],
+                                    "type": "text",
+                                    "content": " . Here, the summation ranges over all non-isomorphic (tree-decomposed) graphs in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        431
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        396,
+                                        504,
+                                        431
+                                    ],
+                                    "type": "text",
+                                    "content": " and is well-defined as there are only a finite number of graphs making the value in the summation non-zero."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        104,
+                        443,
+                        506,
+                        466
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                443,
+                                506,
+                                466
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        443,
+                                        506,
+                                        466
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. The proof has a similar structure to the previous lemma. We define the following set of three-tuples:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        105,
+                        471,
+                        521,
+                        492
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                471,
+                                521,
+                                492
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        471,
+                                        521,
+                                        492
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "S = \\left\\{\\left(\\left(\\tilde {F}, \\tilde {T} ^ {s}\\right), (\\rho , \\tau), g\\right): \\left(\\tilde {F}, \\tilde {T} ^ {s}\\right) \\in \\mathcal {S} ^ {M}, (\\rho , \\tau) \\in \\mathrm {B S t r S u r j} \\left(\\left(F, T ^ {r}\\right), \\left(\\tilde {F}, \\tilde {T} ^ {s}\\right)\\right), g \\in \\mathrm {B l s o} \\left(\\left(\\tilde {F}, \\tilde {T} ^ {s}\\right), G\\right) \\right\\}.",
+                                    "image_path": "817f1ec57ab15c9693d600878f8c2d1ae53960265df312a324f61ad45b62bc2a.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
+                        498,
+                        506,
+                        527
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                498,
+                                506,
+                                527
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        498,
+                                        506,
+                                        527
+                                    ],
+                                    "type": "text",
+                                    "content": "Define a mappings "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        498,
+                                        506,
+                                        527
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        498,
+                                        506,
+                                        527
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        498,
+                                        506,
+                                        527
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma\\left((\\tilde{F},\\tilde{T}^s),(\\rho,\\tau),g\\right) = g\\circ\\rho"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        498,
+                                        506,
+                                        527
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        498,
+                                        506,
+                                        527
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left((\\tilde{F},\\tilde{T}^s),(\\rho,\\tau),g\\right)\\in S"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        498,
+                                        506,
+                                        527
+                                    ],
+                                    "type": "text",
+                                    "content": ". It suffices to prove the following three statements:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        116,
+                        532,
+                        504,
+                        599
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 14,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                117,
+                                532,
+                                329,
+                                544
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        532,
+                                        329,
+                                        544
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                117,
+                                                532,
+                                                329,
+                                                544
+                                            ],
+                                            "type": "text",
+                                            "content": "1. "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                532,
+                                                329,
+                                                544
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\sigma"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                532,
+                                                329,
+                                                544
+                                            ],
+                                            "type": "text",
+                                            "content": " is a mapping from "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                532,
+                                                329,
+                                                544
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "S"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                532,
+                                                329,
+                                                544
+                                            ],
+                                            "type": "text",
+                                            "content": " to "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                532,
+                                                329,
+                                                544
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{BStrHom}((F,T^r),G)"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 11
+                        },
+                        {
+                            "bbox": [
+                                117,
+                                549,
+                                192,
+                                561
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        549,
+                                        192,
+                                        561
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                117,
+                                                549,
+                                                192,
+                                                561
+                                            ],
+                                            "type": "text",
+                                            "content": "2. "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                549,
+                                                192,
+                                                561
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\sigma"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                549,
+                                                192,
+                                                561
+                                            ],
+                                            "type": "text",
+                                            "content": " is surjective;"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 12
+                        },
+                        {
+                            "bbox": [
+                                116,
+                                566,
+                                504,
+                                599
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        566,
+                                        504,
+                                        599
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                566,
+                                                504,
+                                                599
+                                            ],
+                                            "type": "text",
+                                            "content": "3. "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                566,
+                                                504,
+                                                599
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\sigma\\left((\\tilde{F}_1, \\tilde{T}_1^{s_1}), (\\rho_1, \\tau_1), g_1\\right) = \\sigma\\left((\\tilde{F}_2, \\tilde{T}_2^{s_2}), (\\rho_2, \\tau_2), g_2\\right)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                566,
+                                                504,
+                                                599
+                                            ],
+                                            "type": "text",
+                                            "content": " iff there exists an isomorphism "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                566,
+                                                504,
+                                                599
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(\\tilde{\\rho}, \\tilde{\\tau})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                566,
+                                                504,
+                                                599
+                                            ],
+                                            "type": "text",
+                                            "content": " from "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                566,
+                                                504,
+                                                599
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(\\tilde{F}_1, \\tilde{T}_1^{s_1})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                566,
+                                                504,
+                                                599
+                                            ],
+                                            "type": "text",
+                                            "content": " to "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                566,
+                                                504,
+                                                599
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(\\tilde{F}_2, \\tilde{T}_2^{s_2})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                566,
+                                                504,
+                                                599
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                566,
+                                                504,
+                                                599
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tilde{\\rho} \\circ \\rho_1 = \\rho_2"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                566,
+                                                504,
+                                                599
+                                            ],
+                                            "type": "text",
+                                            "content": ", "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                566,
+                                                504,
+                                                599
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tilde{\\tau} \\circ \\tau_1 = \\tau_2"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                566,
+                                                504,
+                                                599
+                                            ],
+                                            "type": "text",
+                                            "content": ", "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                566,
+                                                504,
+                                                599
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "g_1 = g_2 \\circ \\tilde{\\rho}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                566,
+                                                504,
+                                                599
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 13
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        105,
+                        603,
+                        280,
+                        615
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                603,
+                                280,
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+                                    "bbox": [
+                                        105,
+                                        603,
+                                        280,
+                                        615
+                                    ],
+                                    "type": "text",
+                                    "content": "We will prove these statements one by one."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        116,
+                        620,
+                        506,
+                        715
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 18,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                117,
+                                620,
+                                506,
+                                687
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        620,
+                                        506,
+                                        687
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "text",
+                                            "content": "1. We first prove that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\sigma"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "text",
+                                            "content": " is a mapping from "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "S"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "text",
+                                            "content": " to "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{BStrHom}((F,T^r),G)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "text",
+                                            "content": ". Pick any "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\left((\\tilde{F},\\tilde{T}^s),(\\rho ,\\tau),g\\right)\\in S"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "text",
+                                            "content": ". Pick any "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t\\in V_T"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u,v\\in \\beta_{T}(t)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "text",
+                                            "content": ". Then, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{u,v\\} \\in E_F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "text",
+                                            "content": " iff "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{\\rho (u),\\rho (v)\\} \\in E_{\\tilde{F}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "text",
+                                            "content": " (since "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "text",
+                                            "content": " is a strong homomorphism from "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[\\beta_T(t)]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "text",
+                                            "content": " to "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tilde{F} [\\rho (\\beta_T(t))]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "text",
+                                            "content": "). Also, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho (u),\\rho (v)\\in \\beta_{\\tilde{F}}(\\tau (t))"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "text",
+                                            "content": " are in the same bag. Similarly, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{\\rho (u),\\rho (v)\\} \\in E_{\\tilde{F}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "text",
+                                            "content": " iff "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{g(\\rho (u)),g(\\rho (v))\\} \\in E_G"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "text",
+                                            "content": " (since "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "g"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "text",
+                                            "content": " is a bIso). Thus, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "g\\circ \\rho"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                620,
+                                                506,
+                                                687
+                                            ],
+                                            "type": "text",
+                                            "content": " is a bag-strong homomorphism."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 16
+                        },
+                        {
+                            "bbox": [
+                                116,
+                                691,
+                                504,
+                                715
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        691,
+                                        504,
+                                        715
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                691,
+                                                504,
+                                                715
+                                            ],
+                                            "type": "text",
+                                            "content": "2. We next prove that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                691,
+                                                504,
+                                                715
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\sigma"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                691,
+                                                504,
+                                                715
+                                            ],
+                                            "type": "text",
+                                            "content": " is surjective. Given "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                691,
+                                                504,
+                                                715
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "h\\in \\mathsf{BStrHom}((F,T^r),G)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                691,
+                                                504,
+                                                715
+                                            ],
+                                            "type": "text",
+                                            "content": " , define "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                691,
+                                                504,
+                                                715
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(\\tilde{F},\\tilde{T}^{s}),(\\rho ,\\tau)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                691,
+                                                504,
+                                                715
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                691,
+                                                504,
+                                                715
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "g"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                691,
+                                                504,
+                                                715
+                                            ],
+                                            "type": "text",
+                                            "content": " as follows. First define a relation "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                691,
+                                                504,
+                                                715
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\sim"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                691,
+                                                504,
+                                                715
+                                            ],
+                                            "type": "text",
+                                            "content": " on set "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                691,
+                                                504,
+                                                715
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "V_{F}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                691,
+                                                504,
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+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                691,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u\\sim v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                691,
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+                                            ],
+                                            "type": "text",
+                                            "content": " iff the following hold:"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 17
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        141,
+                        720,
+                        212,
+                        732
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                141,
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+                            ],
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "a) "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h(u) = h(v)"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
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+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
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+                            ],
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+                                    "bbox": [
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+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
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+                    "type": "page_number",
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "35"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                }
+            ],
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+                612,
+                792
+            ],
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+        {
+            "para_blocks": [
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+                    ],
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+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "b) There exists a path "
+                                },
+                                {
+                                    "bbox": [
+                                        140,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P"
+                                },
+                                {
+                                    "bbox": [
+                                        140,
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+                                    ],
+                                    "type": "text",
+                                    "content": " in "
+                                },
+                                {
+                                    "bbox": [
+                                        140,
+                                        82,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        140,
+                                        82,
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+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": " with endpoints "
+                                },
+                                {
+                                    "bbox": [
+                                        140,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t_1, t_2 \\in V_T"
+                                },
+                                {
+                                    "bbox": [
+                                        140,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        140,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\in \\beta_T(t_1)"
+                                },
+                                {
+                                    "bbox": [
+                                        140,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        140,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v \\in \\beta_T(t_2)"
+                                },
+                                {
+                                    "bbox": [
+                                        140,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": ", and all node "
+                                },
+                                {
+                                    "bbox": [
+                                        140,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t"
+                                },
+                                {
+                                    "bbox": [
+                                        140,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": " on path "
+                                },
+                                {
+                                    "bbox": [
+                                        140,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P"
+                                },
+                                {
+                                    "bbox": [
+                                        140,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfies that "
+                                },
+                                {
+                                    "bbox": [
+                                        140,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h(u) \\in h(\\beta_T(t))"
+                                },
+                                {
+                                    "bbox": [
+                                        140,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        129,
+                        110,
+                        506,
+                        168
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                110,
+                                506,
+                                168
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        110,
+                                        506,
+                                        168
+                                    ],
+                                    "type": "text",
+                                    "content": "It is easy to see that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        110,
+                                        506,
+                                        168
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sim"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        110,
+                                        506,
+                                        168
+                                    ],
+                                    "type": "text",
+                                    "content": " is an equivalence relation on "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        110,
+                                        506,
+                                        168
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "V_{F}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        110,
+                                        506,
+                                        168
+                                    ],
+                                    "type": "text",
+                                    "content": ". We can thus define a mapping "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        110,
+                                        506,
+                                        168
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        110,
+                                        506,
+                                        168
+                                    ],
+                                    "type": "text",
+                                    "content": " that respects the relation, i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        110,
+                                        506,
+                                        168
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho(u) = \\rho(v)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        110,
+                                        506,
+                                        168
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        110,
+                                        506,
+                                        168
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\sim v"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        110,
+                                        506,
+                                        168
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        110,
+                                        506,
+                                        168
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v \\in V_{F}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        110,
+                                        506,
+                                        168
+                                    ],
+                                    "type": "text",
+                                    "content": ". Moreover, for any edge "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        110,
+                                        506,
+                                        168
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{u, v\\} \\in E_{F}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        110,
+                                        506,
+                                        168
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        110,
+                                        506,
+                                        168
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho(u) \\neq \\rho(v)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        110,
+                                        506,
+                                        168
+                                    ],
+                                    "type": "text",
+                                    "content": " (since "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        110,
+                                        506,
+                                        168
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        110,
+                                        506,
+                                        168
+                                    ],
+                                    "type": "text",
+                                    "content": " is a homomorphism and "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        110,
+                                        506,
+                                        168
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h(u) \\neq h(v)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        110,
+                                        506,
+                                        168
+                                    ],
+                                    "type": "text",
+                                    "content": "). This implies that we can define "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        110,
+                                        506,
+                                        168
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        110,
+                                        506,
+                                        168
+                                    ],
+                                    "type": "text",
+                                    "content": " to be the homomorphic image of "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        110,
+                                        506,
+                                        168
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        110,
+                                        506,
+                                        168
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        110,
+                                        506,
+                                        168
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        110,
+                                        506,
+                                        168
+                                    ],
+                                    "type": "text",
+                                    "content": " is the surjective homomorphism on both vertices and edges."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        129,
+                        172,
+                        504,
+                        229
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                172,
+                                504,
+                                229
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "text",
+                                    "content": "We then define the mapping "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g: V_{\\tilde{F}} \\to V_G"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g(\\rho(u)) = h(u)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "text",
+                                    "content": ". Note that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "text",
+                                    "content": " is well-defined since "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho(u) = \\rho(v)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "text",
+                                    "content": " implies "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h(u) = h(v)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho: V_F \\to V_{\\tilde{F}}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "text",
+                                    "content": " is surjective. It follows that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h = g \\circ \\rho"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "text",
+                                    "content": ". To prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "text",
+                                    "content": " is a homomorphism, note that for all "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{x, y\\} \\in E_{\\tilde{F}}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "text",
+                                    "content": ", there exists an edge "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{u, v\\} \\in E_F"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "text",
+                                    "content": " with "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho(u) = x"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho(v) = y"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "text",
+                                    "content": ", which implies that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{h(u), h(v)\\} \\in E_G"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "text",
+                                    "content": " (since "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "text",
+                                    "content": " is a homomorphism), namely, "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{g(x), g(y)\\} \\in E_G"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        172,
+                                        504,
+                                        229
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        129,
+                        235,
+                        504,
+                        344
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                235,
+                                504,
+                                344
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": "We next define tree "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{T}^s = (V_T, E_T, \\beta_{\\tilde{T}})"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s = t"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": ", and identity mapping "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": " so that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": " is an isomorphism from "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T_r"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T_s"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": " (ignoring the labels). Set "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_{\\tilde{T}}(t) = \\rho(\\beta_T(t))"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t \\in V_T"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": ". We will prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{F}, \\tilde{T}^s) \\in S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": " is a valid tree decomposition. It suffices to prove that Definition C.1(c) holds. Pick any vertex "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x \\in V_{\\tilde{F}}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": " and tree node "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t_1, t_2 \\in B_{\\tilde{T}}(x)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, there exists "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\in \\beta_T(t_1), v \\in \\beta_T(t_2)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho(u) = x"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho(v) = x"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": ". Therefore, "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\sim v"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": ". As such, there exists a path "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": " in "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": " such that all node "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": " on "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfies that there exists "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w \\in \\beta_T(t)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": " with "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h(w) = h(u)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": ", namely, "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w \\sim u"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": ". Consequently, "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho(u) \\in \\beta_{\\tilde{T}}(t)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": ", implying that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{T}^s[B_{\\tilde{T}}(x)]"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": " is connected. This proves that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{F}, \\tilde{T}^s) \\in S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": ". Also, "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\rho, \\tau)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": " is clearly a homomorphism from "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{F}, \\tilde{T}^s)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        235,
+                                        504,
+                                        344
+                                    ],
+                                    "type": "text",
+                                    "content": " according to Definition C.20."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        129,
+                        348,
+                        505,
+                        417
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                348,
+                                505,
+                                417
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "text",
+                                    "content": "It remains to prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "text",
+                                    "content": " is a bag-strong homomorphism and "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "text",
+                                    "content": " is a bIso. Pick any "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t \\in V_T"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v \\in \\beta_T(t)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "text",
+                                    "content": ". If "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{u, v\\} \\notin E_F"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "text",
+                                    "content": ", then "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{h(u), h(v)\\} \\notin E_G"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "text",
+                                    "content": " (since "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "text",
+                                    "content": " is a bag-strong homomorphism). Therefore, "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\rho(u), \\rho(v)\\} \\notin E_{\\tilde{F}}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "text",
+                                    "content": " (since "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "text",
+                                    "content": " is a homomorphism), namely, "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "text",
+                                    "content": " is a bag-strong homomorphism. Since "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "text",
+                                    "content": " is surjective, "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\rho(u), \\rho(v)\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "text",
+                                    "content": " ranges over all vertices in the same bag of "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\hat{T}^s"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "text",
+                                    "content": " when "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t \\in V_T"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v \\in \\beta_T(t)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "text",
+                                    "content": " are arbitrary. Therefore, "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "text",
+                                    "content": " is a bIso because "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\rho(u), \\rho(v)\\} \\notin E_{\\tilde{F}}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{h(u), h(v)\\} \\notin E_G"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        348,
+                                        505,
+                                        417
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        117,
+                        422,
+                        504,
+                        501
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                117,
+                                422,
+                                504,
+                                501
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "text",
+                                    "content": "3. We finally prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma\\left((\\tilde{F}_1, \\tilde{T}_1^{s_1}), (\\rho_1, \\tau_1), g_1\\right) = \\sigma\\left((\\tilde{F}_2, \\tilde{T}_2^{s_2}), (\\rho_2, \\tau_2), g_2\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "text",
+                                    "content": " implies there exists an isomorphism "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{\\rho}, \\tilde{\\tau})"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "text",
+                                    "content": " from "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{F}_1, \\tilde{T}_1^{s_1})"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{F}_2, \\tilde{T}_2^{s_2})"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\rho} \\circ \\rho_1 = \\rho_2"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\tau} \\circ \\tau_1 = \\tau_2"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g_1 = g_2 \\circ \\tilde{\\rho}"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "text",
+                                    "content": ". Let "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h = g_1 \\circ \\rho_1 = g_2 \\circ \\rho_2"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "text",
+                                    "content": ". Here, we will only prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F}_1 \\simeq \\tilde{F}_2"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "text",
+                                    "content": " since the remaining procedure is almost the same as previous proofs. Since both "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F}_1"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F}_2"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "text",
+                                    "content": " are homomorphic images of "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "text",
+                                    "content": ", it suffices to prove that, for all "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_1(u) = \\rho_1(v)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        422,
+                                        504,
+                                        501
+                                    ],
+                                    "type": "text",
+                                    "content": " iff the following hold:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        141,
+                        506,
+                        504,
+                        544
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 9,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                141,
+                                506,
+                                212,
+                                518
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
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+                                    "spans": [
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": "a) "
+                                        },
+                                        {
+                                            "bbox": [
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+                                                506,
+                                                212,
+                                                518
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "h(u) = h(v)"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 7
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                519,
+                                504,
+                                544
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        519,
+                                        504,
+                                        544
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                519,
+                                                504,
+                                                544
+                                            ],
+                                            "type": "text",
+                                            "content": "b) There exists a path "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                519,
+                                                504,
+                                                544
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                519,
+                                                504,
+                                                544
+                                            ],
+                                            "type": "text",
+                                            "content": " in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                519,
+                                                504,
+                                                544
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T^r"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                519,
+                                                504,
+                                                544
+                                            ],
+                                            "type": "text",
+                                            "content": " with endpoints "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                519,
+                                                504,
+                                                544
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t_1, t_2 \\in V_T"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                519,
+                                                504,
+                                                544
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                519,
+                                                504,
+                                                544
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u \\in \\beta_T(t_1)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                519,
+                                                504,
+                                                544
+                                            ],
+                                            "type": "text",
+                                            "content": ", "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                519,
+                                                504,
+                                                544
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "v \\in \\beta_T(t_2)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                519,
+                                                504,
+                                                544
+                                            ],
+                                            "type": "text",
+                                            "content": ", and all node "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                519,
+                                                504,
+                                                544
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                519,
+                                                504,
+                                                544
+                                            ],
+                                            "type": "text",
+                                            "content": " on path "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                519,
+                                                504,
+                                                544
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                519,
+                                                504,
+                                                544
+                                            ],
+                                            "type": "text",
+                                            "content": " satisfies that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                519,
+                                                504,
+                                                544
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "h(u) \\in h(\\beta_T(t))"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                519,
+                                                504,
+                                                544
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 8
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        129,
+                        548,
+                        504,
+                        595
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                548,
+                                504,
+                                595
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        548,
+                                        504,
+                                        595
+                                    ],
+                                    "type": "text",
+                                    "content": "On one hand, if "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        548,
+                                        504,
+                                        595
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_{1}(u) = \\rho_{1}(v)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        548,
+                                        504,
+                                        595
+                                    ],
+                                    "type": "text",
+                                    "content": ", we clearly have "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        548,
+                                        504,
+                                        595
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h(u) = h(v)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        548,
+                                        504,
+                                        595
+                                    ],
+                                    "type": "text",
+                                    "content": " and there exists "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        548,
+                                        504,
+                                        595
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t_1 \\in B_T(u), t_2 \\in B_T(v)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        548,
+                                        504,
+                                        595
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        548,
+                                        504,
+                                        595
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t_1, t_2 \\in B_{\\tilde{T}_1}(\\rho_1(u))"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        548,
+                                        504,
+                                        595
+                                    ],
+                                    "type": "text",
+                                    "content": ". Since "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        548,
+                                        504,
+                                        595
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{T}_1[B_{\\tilde{T}_1}(\\rho_1(u))]"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        548,
+                                        504,
+                                        595
+                                    ],
+                                    "type": "text",
+                                    "content": " is connected, there is a path "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        548,
+                                        504,
+                                        595
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        548,
+                                        504,
+                                        595
+                                    ],
+                                    "type": "text",
+                                    "content": " with endpoints "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        548,
+                                        504,
+                                        595
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t_1, t_2"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        548,
+                                        504,
+                                        595
+                                    ],
+                                    "type": "text",
+                                    "content": " such that all nodes "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        548,
+                                        504,
+                                        595
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        548,
+                                        504,
+                                        595
+                                    ],
+                                    "type": "text",
+                                    "content": " on "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        548,
+                                        504,
+                                        595
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        548,
+                                        504,
+                                        595
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfies "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        548,
+                                        504,
+                                        595
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_{1}(u) \\in \\rho_{1}(\\beta_{T}(t))"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        548,
+                                        504,
+                                        595
+                                    ],
+                                    "type": "text",
+                                    "content": " and thus "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        548,
+                                        504,
+                                        595
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h(u) \\in h(\\beta_{T}(t))"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        548,
+                                        504,
+                                        595
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        129,
+                        595,
+                        504,
+                        607
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                595,
+                                504,
+                                607
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        595,
+                                        504,
+                                        607
+                                    ],
+                                    "type": "text",
+                                    "content": "On the other hand, if "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        595,
+                                        504,
+                                        607
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho_1(u) \\neq \\rho_1(v)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        595,
+                                        504,
+                                        607
+                                    ],
+                                    "type": "text",
+                                    "content": " but the above items (a) and (b) hold, consider two cases:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        145,
+                        613,
+                        504,
+                        697
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 14,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                145,
+                                613,
+                                503,
+                                636
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        145,
+                                        613,
+                                        503,
+                                        636
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                145,
+                                                613,
+                                                503,
+                                                636
+                                            ],
+                                            "type": "text",
+                                            "content": "- "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                613,
+                                                503,
+                                                636
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                613,
+                                                503,
+                                                636
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                613,
+                                                503,
+                                                636
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                613,
+                                                503,
+                                                636
+                                            ],
+                                            "type": "text",
+                                            "content": " are in the same bag of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                613,
+                                                503,
+                                                636
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                613,
+                                                503,
+                                                636
+                                            ],
+                                            "type": "text",
+                                            "content": ". Then, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                613,
+                                                503,
+                                                636
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho_1(u)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                613,
+                                                503,
+                                                636
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                613,
+                                                503,
+                                                636
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho_1(v)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                613,
+                                                503,
+                                                636
+                                            ],
+                                            "type": "text",
+                                            "content": " are in the same bag of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                613,
+                                                503,
+                                                636
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tilde{T}_1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                613,
+                                                503,
+                                                636
+                                            ],
+                                            "type": "text",
+                                            "content": ". Since "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                613,
+                                                503,
+                                                636
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "g_1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                613,
+                                                503,
+                                                636
+                                            ],
+                                            "type": "text",
+                                            "content": " is a bIso, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                613,
+                                                503,
+                                                636
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "g_1(\\rho_1(u)) \\neq g_1(\\rho_1(v))"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                613,
+                                                503,
+                                                636
+                                            ],
+                                            "type": "text",
+                                            "content": ", which contradicts item (a)."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 12
+                        },
+                        {
+                            "bbox": [
+                                145,
+                                638,
+                                504,
+                                697
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        145,
+                                        638,
+                                        504,
+                                        697
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "text",
+                                            "content": "- "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "text",
+                                            "content": " are not in the same bag of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "text",
+                                            "content": ". Then, there exist two adjacent nodes "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t_1, t_2"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "text",
+                                            "content": " on "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho_1(u) \\in \\beta_{\\tilde{T}_1}(t_1), \\rho_1(u) \\notin \\beta_{\\tilde{T}_1}(t_2)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "text",
+                                            "content": ". By Definition C.4(c), "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_{\\tilde{T}_1}(t_2) \\subset \\beta_{\\tilde{T}_1}(t_1)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "text",
+                                            "content": ". Also, item (b) implies that there exists "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\in \\beta_T(t_2)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "h(w) = h(u)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "text",
+                                            "content": ". Therefore, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho_1(u)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\rho_1(w)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "text",
+                                            "content": " are two different nodes in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_{\\tilde{T}_1}(t_1)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "text",
+                                            "content": " with "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "g_1(\\rho_1(u)) = h(u) = h(w) = g_1(\\rho_1(w))"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "text",
+                                            "content": ". This contradicts the condition that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "g_1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                638,
+                                                504,
+                                                697
+                                            ],
+                                            "type": "text",
+                                            "content": " is a bIso."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 13
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        129,
+                        702,
+                        305,
+                        715
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                702,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        702,
+                                        305,
+                                        715
+                                    ],
+                                    "type": "text",
+                                    "content": "This yields the desired result that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        702,
+                                        305,
+                                        715
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F}_1\\simeq \\tilde{F}_2"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        105,
+                        720,
+                        324,
+                        732
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
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+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
+                                        720,
+                                        324,
+                                        732
+                                    ],
+                                    "type": "text",
+                                    "content": "Combining the above three items concludes the proof."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
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+                    ],
+                    "type": "header",
+                    "angle": 0,
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+                                    "bbox": [
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+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
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+                    ],
+                    "type": "page_number",
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "36"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 35
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        82,
+                        504,
+                        106
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                82,
+                                504,
+                                106
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": "Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\mathrm{Sub}, \\mathrm{L}, \\mathrm{LF}, \\mathrm{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": " be any model. We can list all non-isomorphic tree-decomposed (labeled) graphs in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": " into an infinite sequence "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F_1, T_1^{r_1}), (F_2, T_2^{r_2}), \\dots"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": ". Consider two types of ordering:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        121,
+                        110,
+                        505,
+                        190
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 4,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                121,
+                                110,
+                                504,
+                                155
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        110,
+                                        504,
+                                        155
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                121,
+                                                110,
+                                                504,
+                                                155
+                                            ],
+                                            "type": "text",
+                                            "content": "- Ordered by the size of the graphs "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                110,
+                                                504,
+                                                155
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F_{i}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                110,
+                                                504,
+                                                155
+                                            ],
+                                            "type": "text",
+                                            "content": ". We denote this ordering as 1st. This ordering requires that for any "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                110,
+                                                504,
+                                                155
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "i < j"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                110,
+                                                504,
+                                                155
+                                            ],
+                                            "type": "text",
+                                            "content": ", either "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                110,
+                                                504,
+                                                155
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "|V_{F_i}| < |V_{F_j}|"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                110,
+                                                504,
+                                                155
+                                            ],
+                                            "type": "text",
+                                            "content": " or "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                110,
+                                                504,
+                                                155
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "|V_{F_i}| = |V_{F_j}|"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                110,
+                                                504,
+                                                155
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                110,
+                                                504,
+                                                155
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "|E_{F_i}| \\geq |E_{F_j}|"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                110,
+                                                504,
+                                                155
+                                            ],
+                                            "type": "text",
+                                            "content": ". Note that when two graphs have an equal number of vertices, we place the graph with more edges to the front. When the number of edges is also the same, they can be arranged in any fixed order."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 2
+                        },
+                        {
+                            "bbox": [
+                                121,
+                                156,
+                                505,
+                                190
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        156,
+                                        505,
+                                        190
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                121,
+                                                156,
+                                                505,
+                                                190
+                                            ],
+                                            "type": "text",
+                                            "content": "- Ordered by the size of the trees "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                156,
+                                                505,
+                                                190
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T_{i}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                156,
+                                                505,
+                                                190
+                                            ],
+                                            "type": "text",
+                                            "content": ". We denote this ordering as 2nd. This ordering requires that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                156,
+                                                505,
+                                                190
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "|V_{T_i}| \\leq |V_{T_j}|"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                156,
+                                                505,
+                                                190
+                                            ],
+                                            "type": "text",
+                                            "content": " for any "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                156,
+                                                505,
+                                                190
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "i < j"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                156,
+                                                505,
+                                                190
+                                            ],
+                                            "type": "text",
+                                            "content": ". When the number of tree nodes is the same, they can be arranged in any fixed order."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 3
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        104,
+                        194,
+                        506,
+                        239
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                194,
+                                506,
+                                239
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        194,
+                                        506,
+                                        239
+                                    ],
+                                    "type": "text",
+                                    "content": "Without loss of generality, we assume that the labels of each graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        194,
+                                        506,
+                                        239
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F_{i}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        194,
+                                        506,
+                                        239
+                                    ],
+                                    "type": "text",
+                                    "content": " are integers in the range of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        194,
+                                        506,
+                                        239
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "[1, |V_{F_i}|]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        194,
+                                        506,
+                                        239
+                                    ],
+                                    "type": "text",
+                                    "content": ". Note that using integer labels of a bounded range is already sufficient to represent all non-isomorphic labeled graphs up to a bijective label transformation. This ensures that the index "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        194,
+                                        506,
+                                        239
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        194,
+                                        506,
+                                        239
+                                    ],
+                                    "type": "text",
+                                    "content": " is countable."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        105,
+                        244,
+                        315,
+                        257
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                244,
+                                315,
+                                257
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        244,
+                                        315,
+                                        257
+                                    ],
+                                    "type": "text",
+                                    "content": "For the 1st ordering, define the following notations:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        116,
+                        260,
+                        505,
+                        343
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 10,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                116,
+                                260,
+                                504,
+                                289
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        260,
+                                        504,
+                                        289
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                260,
+                                                504,
+                                                289
+                                            ],
+                                            "type": "text",
+                                            "content": "a) Let "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                260,
+                                                504,
+                                                289
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "f: \\mathcal{S}^M \\times \\mathcal{S}^M \\to \\mathbb{N}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                260,
+                                                504,
+                                                289
+                                            ],
+                                            "type": "text",
+                                            "content": " be any mapping. Define the associated (infinite) matrix "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                260,
+                                                504,
+                                                289
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "A^{f,M,1\\mathrm{st}} \\in \\mathbb{N}^{\\mathbb{N}_+ \\times \\mathbb{N}_+}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                260,
+                                                504,
+                                                289
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                260,
+                                                504,
+                                                289
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "A_{i,j}^{f,M,1\\mathrm{st}} = f((F_i, T_i^{r_i}), (F_j, T_j^{r_j}))"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                260,
+                                                504,
+                                                289
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 7
+                        },
+                        {
+                            "bbox": [
+                                116,
+                                289,
+                                505,
+                                316
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        289,
+                                        505,
+                                        316
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                289,
+                                                505,
+                                                316
+                                            ],
+                                            "type": "text",
+                                            "content": "b) Let "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                289,
+                                                505,
+                                                316
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "g: \\mathcal{S}^M \\times \\mathcal{G} \\to \\mathbb{N}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                289,
+                                                505,
+                                                316
+                                            ],
+                                            "type": "text",
+                                            "content": " be any mapping. Given a graph "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                289,
+                                                505,
+                                                316
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "G \\in \\mathcal{G}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                289,
+                                                505,
+                                                316
+                                            ],
+                                            "type": "text",
+                                            "content": ", define the (infinite) vector "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                289,
+                                                505,
+                                                316
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pmb{p}_G^{g,M,1\\mathrm{st}} \\in \\mathbb{N}^{\\mathbb{N}_+}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                289,
+                                                505,
+                                                316
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                289,
+                                                505,
+                                                316
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "p_{G,i}^{f,M,1\\mathrm{st}} = g((F_i,T_i^{r_i}),G)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                289,
+                                                505,
+                                                316
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 8
+                        },
+                        {
+                            "bbox": [
+                                116,
+                                316,
+                                504,
+                                343
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        316,
+                                        504,
+                                        343
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                316,
+                                                504,
+                                                343
+                                            ],
+                                            "type": "text",
+                                            "content": "c) Let "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                316,
+                                                504,
+                                                343
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "h: \\mathcal{G} \\times \\mathcal{G} \\to \\mathbb{N}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                316,
+                                                504,
+                                                343
+                                            ],
+                                            "type": "text",
+                                            "content": " be any mapping. Given a graph "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                316,
+                                                504,
+                                                343
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "G \\in \\mathcal{G}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                316,
+                                                504,
+                                                343
+                                            ],
+                                            "type": "text",
+                                            "content": ", define the (infinite) vector "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                316,
+                                                504,
+                                                343
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "p_{G}^{h,M,1\\mathrm{st}} \\in \\mathbb{N}^{\\mathbb{N}_{+}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                316,
+                                                504,
+                                                343
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                316,
+                                                504,
+                                                343
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "p_{G,i}^{h,M,1\\mathrm{st}} = h(F_i, G)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                316,
+                                                504,
+                                                343
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 9
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
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+                        363
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        347,
+                                        408,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": "We can similarly define "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        347,
+                                        408,
+                                        363
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{A}^{f,M,2\\mathrm{nd}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        347,
+                                        408,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        347,
+                                        408,
+                                        363
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{p}_G^{g,M,2\\mathrm{nd}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        347,
+                                        408,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        347,
+                                        408,
+                                        363
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{p}_G^{h,M,2\\mathrm{nd}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        347,
+                                        408,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": " for the ordering 2nd."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        104,
+                        364,
+                        506,
+                        398
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                364,
+                                506,
+                                398
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        506,
+                                        398
+                                    ],
+                                    "type": "text",
+                                    "content": "Corollary C.32. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        506,
+                                        398
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        506,
+                                        398
+                                    ],
+                                    "type": "text",
+                                    "content": " be any model and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        506,
+                                        398
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G, H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        506,
+                                        398
+                                    ],
+                                    "type": "text",
+                                    "content": " be two graphs. Then, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        506,
+                                        398
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{hom}(F, G) = \\operatorname{hom}(F, H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        506,
+                                        398
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        506,
+                                        398
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        506,
+                                        398
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        506,
+                                        398
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\text{blso}((F, T^r), G) = \\text{blso}((F, T^r), H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        506,
+                                        398
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        506,
+                                        398
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        506,
+                                        398
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        105,
+                        410,
+                        292,
+                        422
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                410,
+                                292,
+                                422
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        410,
+                                        292,
+                                        422
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. We separately consider each direction."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        117,
+                        426,
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+                        460
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
+                                        117,
+                                        426,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": "1. We first prove that if "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        426,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{blso}((F,T^r),G) = \\mathsf{blso}((F,T^r),H)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        426,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        426,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^{r})\\in S^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        426,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": ", then "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        426,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{hom}(F,G) = \\mathsf{hom}(F,H)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        426,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        426,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^{r})\\in S^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        426,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": ". According to Lemmas C.29 and C.31, we can rewrite the corresponding equations into matrix forms for any "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        426,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F\\in \\mathcal{G}"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        426,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": ":"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        212,
+                        464,
+                        504,
+                        479
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                212,
+                                464,
+                                504,
+                                479
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        212,
+                                        464,
+                                        504,
+                                        479
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\boldsymbol {p} _ {F} ^ {\\text {h o m}, M, 1 \\mathrm {s t}} = \\boldsymbol {A} ^ {\\text {b E x t}, M, 1 \\mathrm {s t}} \\left(\\boldsymbol {A} ^ {\\text {a u t}, M, 1 \\mathrm {s t}}\\right) ^ {- 1} \\boldsymbol {p} _ {F} ^ {\\text {b S t r H o m}, M, 1 \\mathrm {s t}} \\tag {19}",
+                                    "image_path": "7d45357b26f568ea1da11722301b6784044254482bf8065f844518d329e31b82.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        200,
+                        480,
+                        504,
+                        497
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                200,
+                                480,
+                                504,
+                                497
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        200,
+                                        480,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\boldsymbol {p} _ {F} ^ {\\mathrm {b S t r H o m}, M, 1 \\mathrm {s t}} = \\boldsymbol {A} ^ {\\mathrm {b S t r S u r j}, M, 1 \\mathrm {s t}} \\left(\\boldsymbol {A} ^ {\\mathrm {a u t}, M, 1 \\mathrm {s t}}\\right) ^ {- 1} \\boldsymbol {p} _ {F} ^ {\\mathrm {b l s o}, M, 1 \\mathrm {s t}}. \\tag {20}",
+                                    "image_path": "aa6028533b51aa109d2f083f563573ad5552cc7374c0266bcb774b91c0ecbf81.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        129,
+                        500,
+                        504,
+                        529
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                500,
+                                504,
+                                529
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        500,
+                                        504,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": "From the above equations, we immediately obtain that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        500,
+                                        504,
+                                        529
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p_{G}^{\\mathrm{blso},M,1\\mathrm{st}} = p_{H}^{\\mathrm{blso},M,1\\mathrm{st}}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        500,
+                                        504,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": " implies "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        500,
+                                        504,
+                                        529
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p_{G}^{\\mathrm{hom},M,1\\mathrm{st}} = p_{H}^{\\mathrm{hom},M,1\\mathrm{st}}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        500,
+                                        504,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        116,
+                        533,
+                        505,
+                        567
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                116,
+                                533,
+                                505,
+                                567
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        533,
+                                        505,
+                                        567
+                                    ],
+                                    "type": "text",
+                                    "content": "2. We next prove that if "
+                                },
+                                {
+                                    "bbox": [
+                                        116,
+                                        533,
+                                        505,
+                                        567
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{hom}(F,G) = \\mathsf{hom}(F,H)"
+                                },
+                                {
+                                    "bbox": [
+                                        116,
+                                        533,
+                                        505,
+                                        567
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        116,
+                                        533,
+                                        505,
+                                        567
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^r)\\in S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        116,
+                                        533,
+                                        505,
+                                        567
+                                    ],
+                                    "type": "text",
+                                    "content": ", then "
+                                },
+                                {
+                                    "bbox": [
+                                        116,
+                                        533,
+                                        505,
+                                        567
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{blso}((F,T^r),G) = \\mathsf{blso}((F,T^r),H)"
+                                },
+                                {
+                                    "bbox": [
+                                        116,
+                                        533,
+                                        505,
+                                        567
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        116,
+                                        533,
+                                        505,
+                                        567
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^{r})\\in S^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        116,
+                                        533,
+                                        505,
+                                        567
+                                    ],
+                                    "type": "text",
+                                    "content": ". This can be seen from the following facts:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
+                        141,
+                        572,
+                        505,
+                        673
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 22,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                141,
+                                572,
+                                468,
+                                584
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        572,
+                                        468,
+                                        584
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                572,
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+                                                584
+                                            ],
+                                            "type": "text",
+                                            "content": "a) "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                572,
+                                                468,
+                                                584
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "A^{\\mathrm{out},M,1\\mathrm{st}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                572,
+                                                468,
+                                                584
+                                            ],
+                                            "type": "text",
+                                            "content": " is a diagonal matrix and all diagonal elements are positive integers."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 19
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                584,
+                                504,
+                                622
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
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+                                        622
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                584,
+                                                504,
+                                                622
+                                            ],
+                                            "type": "text",
+                                            "content": "b) "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                584,
+                                                504,
+                                                622
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "A^{\\mathrm{bExt}, M, 1\\mathrm{st}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                584,
+                                                504,
+                                                622
+                                            ],
+                                            "type": "text",
+                                            "content": " is a lower triangular matrix and all diagonal elements are positive integers. This is because for any two tree-composed graphs "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                584,
+                                                504,
+                                                622
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(F_i, T_i^{r_i})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                584,
+                                                504,
+                                                622
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                584,
+                                                504,
+                                                622
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(F_j, T_j^{r_j})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                584,
+                                                504,
+                                                622
+                                            ],
+                                            "type": "text",
+                                            "content": ", "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                584,
+                                                504,
+                                                622
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{bExt}((F_i, T_i^{r_i}), (F_j, T_j^{r_j})) > 0"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                584,
+                                                504,
+                                                622
+                                            ],
+                                            "type": "text",
+                                            "content": " only if "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                584,
+                                                504,
+                                                622
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "|V_{F_i}| = |V_{F_j}|"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                584,
+                                                504,
+                                                622
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                584,
+                                                504,
+                                                622
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "|E_{F_i}| \\leq |E_{F_j}|"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                584,
+                                                504,
+                                                622
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 20
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                623,
+                                505,
+                                673
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        623,
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+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                623,
+                                                505,
+                                                673
+                                            ],
+                                            "type": "text",
+                                            "content": "c) "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                623,
+                                                505,
+                                                673
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "A^{\\mathrm{bStrSurj}, M, 1\\mathrm{st}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                623,
+                                                505,
+                                                673
+                                            ],
+                                            "type": "text",
+                                            "content": " is also a lower triangular matrix and all diagonal elements are positive integers. This is because for any two tree-composed graphs "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                623,
+                                                505,
+                                                673
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(F_i, T_i^{r_i})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                623,
+                                                505,
+                                                673
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
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+                                            "type": "inline_equation",
+                                            "content": "(F_j, T_j^{r_j})"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": ", "
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{bStrSurj}((F_i, T_i^{r_i}), (F_j, T_j^{r_j})) > 0"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
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+                                            ],
+                                            "type": "text",
+                                            "content": " only if "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                623,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "|V_{F_i}| > |V_{F_j}|"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                623,
+                                                505,
+                                                673
+                                            ],
+                                            "type": "text",
+                                            "content": " or "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                623,
+                                                505,
+                                                673
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(|V_{F_i}| = |V_{F_j}|"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                623,
+                                                505,
+                                                673
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                623,
+                                                505,
+                                                673
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "|E_{F_i}| = |E_{F_j}|"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                623,
+                                                505,
+                                                673
+                                            ],
+                                            "type": "text",
+                                            "content": ")."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 21
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        129,
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+                        505,
+                        716
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
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+                                    ],
+                                    "type": "text",
+                                    "content": "Therefore, the composition "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        677,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A^{\\mathrm{bExt},M,1\\mathrm{st}}(A^{\\mathrm{aut},M,1\\mathrm{st}})^{-1}A^{\\mathrm{bStrSurj},M,1\\mathrm{st}}(A^{\\mathrm{aut},M,1\\mathrm{st}})^{-1}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " is lower triangular and is invertible (although it is an infinite matrix). We thus arrive at the desired conclusion that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        677,
+                                        505,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p_{G}^{\\mathrm{hom},M,1\\mathrm{st}} = p_{H}^{\\mathrm{hom},M,1\\mathrm{st}}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
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+                                    ],
+                                    "type": "text",
+                                    "content": " implies "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        677,
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+                                        716
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "p_{G}^{\\mathrm{blso},M,1\\mathrm{st}} = p_{H}^{\\mathrm{blso},M,1\\mathrm{st}}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
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+                                        716
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 23
+                },
+                {
+                    "type": "image",
+                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
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+                    ],
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "37"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 25
+                }
+            ],
+            "page_size": [
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+            ],
+            "page_idx": 36
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        82,
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+                        118
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "text",
+                                    "content": "Corollary C.33. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "text",
+                                    "content": " be any model and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G, H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "text",
+                                    "content": " be two graphs. Then, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\text{cnt}^M((F, T^r), G) = \\text{cnt}^M((F, T^r), H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in \\mathcal{S}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\text{blso}((F, T^r), G) = \\text{blso}((F, T^r), H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in \\mathcal{S}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        118
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        105,
+                        128,
+                        291,
+                        140
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                128,
+                                291,
+                                140
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        128,
+                                        291,
+                                        140
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. We separately consider each direction."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        117,
+                        144,
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+                        179
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                117,
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+                                504,
+                                179
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        144,
+                                        504,
+                                        179
+                                    ],
+                                    "type": "text",
+                                    "content": "1. We first prove that if "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        144,
+                                        504,
+                                        179
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{cnt}^M((F, T^r), G) = \\mathsf{cnt}^M((F, T^r), H)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        144,
+                                        504,
+                                        179
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        144,
+                                        504,
+                                        179
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        144,
+                                        504,
+                                        179
+                                    ],
+                                    "type": "text",
+                                    "content": ", then "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        144,
+                                        504,
+                                        179
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{blso}((F, T^r), G) = \\mathsf{blso}((F, T^r), H)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        144,
+                                        504,
+                                        179
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        144,
+                                        504,
+                                        179
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        144,
+                                        504,
+                                        179
+                                    ],
+                                    "type": "text",
+                                    "content": ". According to Lemma C.23, we can rewrite Equation (18) into the matrix form for all "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        144,
+                                        504,
+                                        179
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F \\in \\mathcal{G}"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        144,
+                                        504,
+                                        179
+                                    ],
+                                    "type": "text",
+                                    "content": ":"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        235,
+                        184,
+                        504,
+                        200
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                235,
+                                184,
+                                504,
+                                200
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        235,
+                                        184,
+                                        504,
+                                        200
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\boldsymbol {p} _ {F} ^ {\\text {b l s o}, M, 2 \\text {n d}} = \\boldsymbol {A} ^ {\\text {b l s o H o m}, M, 2 \\text {n d}} \\boldsymbol {p} _ {F} ^ {\\text {c n t}, M, 2 \\text {n d}}. \\tag {21}",
+                                    "image_path": "2fa4eaef4a8ff316ddebef80dc838a58e9c6fb99609251b074bd961661feb35c.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        129,
+                        205,
+                        485,
+                        222
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                205,
+                                485,
+                                222
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        205,
+                                        485,
+                                        222
+                                    ],
+                                    "type": "text",
+                                    "content": "This immediately obtains that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        205,
+                                        485,
+                                        222
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{p}_G^{\\mathrm{cnt},M,2\\mathrm{nd}} = \\pmb{p}_H^{\\mathrm{cnt},M,2\\mathrm{nd}}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        205,
+                                        485,
+                                        222
+                                    ],
+                                    "type": "text",
+                                    "content": " implies "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        205,
+                                        485,
+                                        222
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{p}_G^{\\mathrm{blso},M,2\\mathrm{nd}} = \\pmb{p}_H^{\\mathrm{blso},M,2\\mathrm{nd}}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        205,
+                                        485,
+                                        222
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        117,
+                        224,
+                        504,
+                        250
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                117,
+                                224,
+                                504,
+                                250
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        224,
+                                        504,
+                                        250
+                                    ],
+                                    "type": "text",
+                                    "content": "2. We next prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        224,
+                                        504,
+                                        250
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{p}_G^{\\mathrm{blo},M,2\\mathrm{nd}} = \\pmb{p}_H^{\\mathrm{blo},M,2\\mathrm{nd}}"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        224,
+                                        504,
+                                        250
+                                    ],
+                                    "type": "text",
+                                    "content": " implies "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        224,
+                                        504,
+                                        250
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{p}_G^{\\mathrm{cnt},M,2\\mathrm{nd}} = \\pmb{p}_H^{\\mathrm{cnt},M,2\\mathrm{nd}}"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        224,
+                                        504,
+                                        250
+                                    ],
+                                    "type": "text",
+                                    "content": ". According to Lemma C.25, we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
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+                        253,
+                        504,
+                        269
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                190,
+                                253,
+                                504,
+                                269
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        190,
+                                        253,
+                                        504,
+                                        269
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\boldsymbol {A} ^ {\\text {b l s o H o m}, M, 2 \\text {n d}} = \\boldsymbol {A} ^ {\\text {b l s o S u r j}, M, 2 \\text {n d}} \\left(\\boldsymbol {A} ^ {\\text {a u t}, M, 2 \\text {n d}}\\right) ^ {- 1} \\boldsymbol {A} ^ {\\text {b l s o l n j}, M, 2 \\text {n d}}. \\tag {22}",
+                                    "image_path": "388b81ce503226b49878a4d979898642a3d7ef941b7a8c42bf439a203a4fb1c9.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
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+                        289,
+                        285
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                    "bbox": [
+                                        129,
+                                        274,
+                                        289,
+                                        285
+                                    ],
+                                    "type": "text",
+                                    "content": "Moreover, we have the following facts:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
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+                    ],
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+                            "type": "text",
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+                            "lines": [
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+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": "a) "
+                                        },
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+                                            "bbox": [
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+                                            "type": "inline_equation",
+                                            "content": "A^{\\mathrm{out},M,2\\mathrm{nd}}"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": " is a diagonal matrix and all diagonal elements are positive integers."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 9
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                301,
+                                504,
+                                341
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        301,
+                                        504,
+                                        341
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                301,
+                                                504,
+                                                341
+                                            ],
+                                            "type": "text",
+                                            "content": "b) "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                301,
+                                                504,
+                                                341
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "A^{\\mathrm{blsohnj},M,2\\mathrm{nd}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                301,
+                                                504,
+                                                341
+                                            ],
+                                            "type": "text",
+                                            "content": " is an upper triangular matrix and all diagonal elements are positive integers. This is because for any two tree-composed graphs "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                301,
+                                                504,
+                                                341
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(F_i,T_i^{r_i})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                301,
+                                                504,
+                                                341
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                301,
+                                                504,
+                                                341
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(F_j,T_j^{r_j})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                301,
+                                                504,
+                                                341
+                                            ],
+                                            "type": "text",
+                                            "content": ", "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                301,
+                                                504,
+                                                341
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{blsohnj}((F_i,T_i^{r_i}),(F_j,T_j^{r_j})) > 0"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                301,
+                                                504,
+                                                341
+                                            ],
+                                            "type": "text",
+                                            "content": " only if "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                301,
+                                                504,
+                                                341
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "|V_{T_i}| \\leq |V_{T_j}|"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                301,
+                                                504,
+                                                341
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 10
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                341,
+                                504,
+                                365
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        341,
+                                        504,
+                                        365
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                341,
+                                                504,
+                                                365
+                                            ],
+                                            "type": "text",
+                                            "content": "c) Similarly, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                341,
+                                                504,
+                                                365
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathbf{A}^{\\mathrm{bIsoSurj},M,2\\mathrm{nd}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                341,
+                                                504,
+                                                365
+                                            ],
+                                            "type": "text",
+                                            "content": " is a lower triangular matrix and all diagonal elements are positive integers."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 11
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        129,
+                        369,
+                        505,
+                        431
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                369,
+                                505,
+                                431
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        369,
+                                        505,
+                                        431
+                                    ],
+                                    "type": "text",
+                                    "content": "Unfortunately, since "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        369,
+                                        505,
+                                        431
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A^{\\mathrm{blsolnj},M,2\\mathrm{nd}}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        369,
+                                        505,
+                                        431
+                                    ],
+                                    "type": "text",
+                                    "content": " is an infinite upper triangular matrix, the inverse matrix is not well-defined. Nevertheless, we can use a special property of bIsoHom to complete our proof, namely, "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        369,
+                                        505,
+                                        431
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{bIsOHom}((F_i,T_i^{r_i}),(F_j,T_j^{r_j})) > 0"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        369,
+                                        505,
+                                        431
+                                    ],
+                                    "type": "text",
+                                    "content": " implies that the depth of "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        369,
+                                        505,
+                                        431
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T_{i}^{r_{i}}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        369,
+                                        505,
+                                        431
+                                    ],
+                                    "type": "text",
+                                    "content": " is equal to the depth of "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        369,
+                                        505,
+                                        431
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T_{j}^{r_{j}}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        369,
+                                        505,
+                                        431
+                                    ],
+                                    "type": "text",
+                                    "content": ". Therefore, denoting by "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        369,
+                                        505,
+                                        431
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A^{\\mathrm{dep},d}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        369,
+                                        505,
+                                        431
+                                    ],
+                                    "type": "text",
+                                    "content": " the diagonal matrix where "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        369,
+                                        505,
+                                        431
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A_{ii}^{\\mathrm{dep},d} = \\mathbb{I}[\\mathrm{the~depth~of~}T_i^{r_i}\\mathrm{~is~}d]"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        369,
+                                        505,
+                                        431
+                                    ],
+                                    "type": "text",
+                                    "content": ", we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        146,
+                        437,
+                        504,
+                        453
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                146,
+                                437,
+                                504,
+                                453
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        146,
+                                        437,
+                                        504,
+                                        453
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\boldsymbol {A} ^ {\\text {d e p}, d} \\boldsymbol {p} _ {F} ^ {\\text {b l s o}, M, 2 \\text {n d}} = \\boldsymbol {A} ^ {\\text {b l s o S u r j}, M, 2 \\text {n d}} \\left(\\boldsymbol {A} ^ {\\text {a u t}, M, 2 \\text {n d}}\\right) ^ {- 1} \\boldsymbol {A} ^ {\\text {b l s o l n j}, M, 2 \\text {n d}} \\boldsymbol {A} ^ {\\text {d e p}, d} \\boldsymbol {p} _ {F} ^ {\\text {c n t}, M, 2 \\text {n d}} \\tag {23}",
+                                    "image_path": "89fa92648c26bb52e9bf6c2d66a01cb55dc8c6dedc3d2317586a5e054ef2c7ca.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        129,
+                        458,
+                        504,
+                        497
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                458,
+                                504,
+                                497
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        458,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "text",
+                                    "content": "for all "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        458,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F \\in \\mathcal{G}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        458,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "text",
+                                    "content": ". Fix any integer "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        458,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d \\geq 0"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        458,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "text",
+                                    "content": ", and assume that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        458,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A^{\\mathrm{dep},d}p_{G}^{\\mathrm{blso},M,2\\mathrm{nd}} = A^{\\mathrm{dep},d}p_{H}^{\\mathrm{blso},M,2\\mathrm{nd}}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        458,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "text",
+                                    "content": ". We will prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        458,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A^{\\mathrm{dep},d}p_{G}^{\\mathrm{cnt},M,2\\mathrm{nd}} = A^{\\mathrm{dep},d}p_{H}^{\\mathrm{cnt},M,2\\mathrm{nd}}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        458,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "text",
+                                    "content": ". Since "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        458,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A^{\\mathrm{blsoSurj},M,2\\mathrm{nd}}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        458,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "text",
+                                    "content": " is lower triangular with positive diagonal elements, it is invertible and thus"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        187,
+                        502,
+                        447,
+                        518
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                187,
+                                502,
+                                447,
+                                518
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        187,
+                                        502,
+                                        447,
+                                        518
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\boldsymbol {A} ^ {\\text {b l s o l n j}, M, 2 \\text {n d}} \\boldsymbol {A} ^ {\\text {d e p}, d} \\boldsymbol {p} _ {G} ^ {\\text {c n t}, M, 2 \\text {n d}} = \\boldsymbol {A} ^ {\\text {b l s o l n j}, M, 2 \\text {n d}} \\boldsymbol {A} ^ {\\text {d e p}, d} \\boldsymbol {p} _ {H} ^ {\\text {c n t}, M, 2 \\text {n d}}.",
+                                    "image_path": "bc8e73c59026cd3c68780a80eb7a6e94391a9e0f9d454de73fefc5cd5f2a3a30.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        129,
+                        522,
+                        504,
+                        583
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                522,
+                                504,
+                                583
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        522,
+                                        504,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": "Moreover, by definition of unfolding tree, there are only finite non-zero elements in both "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        522,
+                                        504,
+                                        583
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A^{\\mathrm{dep},d}p_{G}^{\\mathrm{cnt},M,2\\mathrm{nd}}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        522,
+                                        504,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        522,
+                                        504,
+                                        583
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A^{\\mathrm{dep},d}p_{H}^{\\mathrm{cnt},M,2\\mathrm{nd}}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        522,
+                                        504,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": ", and the corresponding non-zero indices can only be in a fixed (finite) set. In this case, the upper triangular matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        522,
+                                        504,
+                                        583
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A^{\\mathrm{blosnj},M,2\\mathrm{nd}}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        522,
+                                        504,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": " is reduced to a finite-dimensional matrix and thus "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        522,
+                                        504,
+                                        583
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A^{\\mathrm{dep},d}p_{G}^{\\mathrm{cnt},M,2\\mathrm{nd}} = A^{\\mathrm{dep},d}p_{H}^{\\mathrm{cnt},M,2\\mathrm{nd}}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        522,
+                                        504,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": ". By enumerating all "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        522,
+                                        504,
+                                        583
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "d \\geq 0"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        522,
+                                        504,
+                                        583
+                                    ],
+                                    "type": "text",
+                                    "content": ", we obtain the desired result."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "type": "image",
+                    "bbox": [
+                        494,
+                        589,
+                        505,
+                        598
+                    ],
+                    "blocks": [
+                        {
+                            "bbox": [
+                                494,
+                                589,
+                                505,
+                                598
+                            ],
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        494,
+                                        589,
+                                        505,
+                                        598
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                494,
+                                                589,
+                                                505,
+                                                598
+                                            ],
+                                            "type": "image",
+                                            "image_path": "3284e7d445ad611e884904d67281b951c3a22d1e19d1ad06f234e93520c4ec12.jpg"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 18,
+                            "angle": 0,
+                            "type": "image_body"
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
+                        104,
+                        612,
+                        429,
+                        624
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                612,
+                                429,
+                                624
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        612,
+                                        429,
+                                        624
+                                    ],
+                                    "type": "text",
+                                    "content": "Combined with all previous results, we have arrived at the concluding corollary:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                },
+                {
+                    "bbox": [
+                        104,
+                        626,
+                        506,
+                        653
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                626,
+                                506,
+                                653
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        506,
+                                        653
+                                    ],
+                                    "type": "text",
+                                    "content": "Corollary C.34. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        506,
+                                        653
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\mathrm{Sub},\\mathsf{L},\\mathsf{LF},\\mathsf{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        506,
+                                        653
+                                    ],
+                                    "type": "text",
+                                    "content": " be any model. For any two graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        506,
+                                        653
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G, H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        506,
+                                        653
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        506,
+                                        653
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^M (G) = \\chi_H^M (H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        506,
+                                        653
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        506,
+                                        653
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname {hom}(F,G) = \\operatorname {hom}(F,H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        506,
+                                        653
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        506,
+                                        653
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^{r})\\in S^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        506,
+                                        653
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                },
+                {
+                    "bbox": [
+                        104,
+                        663,
+                        506,
+                        733
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                663,
+                                506,
+                                733
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. According to Corollary C.16, for any two graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G,H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ", we have "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^M (G) = \\chi_H^M (H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{cnt}^M ((F,T^r),G) = \\mathsf{cnt}^M ((F,T^r),H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^{r})\\in S^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, Corollary C.33 implies that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{cnt}^M ((F,T^r),G) = \\mathsf{cnt}^M ((F,T^r),H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^{r})\\in S^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{blos}((F,T^r),G) = \\mathsf{blos}((F,T^r),H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^{r})\\in S^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ". Finally, Corollary C.32 implies that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{blos}((F,T^r),G) = \\mathsf{blos}((F,T^r),H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^{r})\\in S^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{hom}(F,G) = \\mathsf{hom}(F,H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^{r})\\in S^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        506,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ". We thus conclude the proof."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 21
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
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+                        311,
+                        761
+                    ],
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+                    "angle": 0,
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+                            ],
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+                                {
+                                    "bbox": [
+                                        299,
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+                                        311,
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+                                    ],
+                                    "type": "text",
+                                    "content": "38"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 22
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 37
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        82,
+                        298,
+                        94
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                82,
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+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
+                                        82,
+                                        298,
+                                        94
+                                    ],
+                                    "type": "text",
+                                    "content": "C.3 PART 2: NESTED EAR DECOMPOSITION"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        102,
+                        504,
+                        126
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                102,
+                                504,
+                                126
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        126
+                                    ],
+                                    "type": "text",
+                                    "content": "In this part, we give equivalent formulations of the set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        126
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        126
+                                    ],
+                                    "type": "text",
+                                    "content": " for any model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        126
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\mathrm{Sub}, \\mathrm{L}, \\mathrm{LF}, \\mathrm{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        102,
+                                        504,
+                                        126
+                                    ],
+                                    "type": "text",
+                                    "content": " using the concept of NED defined in Definition 3.3."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        130,
+                        506,
+                        154
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                130,
+                                506,
+                                154
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        130,
+                                        506,
+                                        154
+                                    ],
+                                    "type": "text",
+                                    "content": "We first present some technical lemmas that will be used to deal with the tree decomposition of disconnected graphs."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        156,
+                        505,
+                        204
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                156,
+                                505,
+                                204
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        156,
+                                        505,
+                                        204
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma C.35. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        156,
+                                        505,
+                                        204
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\mathrm{Sub}, \\mathrm{L}, \\mathrm{LF}, \\mathrm{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        156,
+                                        505,
+                                        204
+                                    ],
+                                    "type": "text",
+                                    "content": " be any model, and let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        156,
+                                        505,
+                                        204
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        156,
+                                        505,
+                                        204
+                                    ],
+                                    "type": "text",
+                                    "content": " be a tree-decomposed graph with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        156,
+                                        505,
+                                        204
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_T(t) = \\{u, v\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        156,
+                                        505,
+                                        204
+                                    ],
+                                    "type": "text",
+                                    "content": " for some "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        156,
+                                        505,
+                                        204
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        156,
+                                        505,
+                                        204
+                                    ],
+                                    "type": "text",
+                                    "content": ". Assume that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        156,
+                                        505,
+                                        204
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        156,
+                                        505,
+                                        204
+                                    ],
+                                    "type": "text",
+                                    "content": " has two connected components "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        156,
+                                        505,
+                                        204
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S_1, S_2 \\subset V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        156,
+                                        505,
+                                        204
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        156,
+                                        505,
+                                        204
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\in S_1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        156,
+                                        505,
+                                        204
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        156,
+                                        505,
+                                        204
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v \\in S_2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        156,
+                                        505,
+                                        204
+                                    ],
+                                    "type": "text",
+                                    "content": " are in different components. Then, there exist tree decompositions "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        156,
+                                        505,
+                                        204
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F[S_1], T_1^{r_1}), (F[S_2], T_2^{r_2}) \\in S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        156,
+                                        505,
+                                        204
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        156,
+                                        505,
+                                        204
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_{T_1}(r_1) = \\{u, u\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        156,
+                                        505,
+                                        204
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        156,
+                                        505,
+                                        204
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_{T_2}(r_2) = \\{v, v\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        156,
+                                        505,
+                                        204
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        214,
+                        504,
+                        239
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                214,
+                                504,
+                                239
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        214,
+                                        504,
+                                        239
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. We can simply define "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        214,
+                                        504,
+                                        239
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T_{1} = (V_{T_{1}}, E_{T_{1}}, \\beta_{T_{1}})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        214,
+                                        504,
+                                        239
+                                    ],
+                                    "type": "text",
+                                    "content": " with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        214,
+                                        504,
+                                        239
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "V_{T_{1}} = V_{T}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        214,
+                                        504,
+                                        239
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        214,
+                                        504,
+                                        239
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "E_{T_{1}} = E_{T}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        214,
+                                        504,
+                                        239
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        214,
+                                        504,
+                                        239
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r_{1} = r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        214,
+                                        504,
+                                        239
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        214,
+                                        504,
+                                        239
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_{T_{1}}(t) = \\{\\{\\phi_{1}(w): w \\in \\beta_{T}(t)\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        214,
+                                        504,
+                                        239
+                                    ],
+                                    "type": "text",
+                                    "content": " with"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        244,
+                        243,
+                        359,
+                        270
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                244,
+                                243,
+                                359,
+                                270
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        244,
+                                        243,
+                                        359,
+                                        270
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\phi_ {1} (w) = \\left\\{ \\begin{array}{l l} w & \\text {i f} w \\in S _ {1}, \\\\ u & \\text {i f} w \\in S _ {2}. \\end{array} \\right.",
+                                    "image_path": "2aac957784a98599f9c65e6169603cebe4622d817114559b282e3281b66cf1b6.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        275,
+                        505,
+                        331
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                275,
+                                505,
+                                331
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        275,
+                                        505,
+                                        331
+                                    ],
+                                    "type": "text",
+                                    "content": "In other words, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        275,
+                                        505,
+                                        331
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T_{1}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        275,
+                                        505,
+                                        331
+                                    ],
+                                    "type": "text",
+                                    "content": " has the same structure as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        275,
+                                        505,
+                                        331
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        275,
+                                        505,
+                                        331
+                                    ],
+                                    "type": "text",
+                                    "content": " but with different bags such that all vertices in the same connected component as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        275,
+                                        505,
+                                        331
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        275,
+                                        505,
+                                        331
+                                    ],
+                                    "type": "text",
+                                    "content": " are replaced by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        275,
+                                        505,
+                                        331
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        275,
+                                        505,
+                                        331
+                                    ],
+                                    "type": "text",
+                                    "content": ". It is easy to see that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        275,
+                                        505,
+                                        331
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T_{1}^{r_{1}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        275,
+                                        505,
+                                        331
+                                    ],
+                                    "type": "text",
+                                    "content": " is a canonical tree decomposition of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        275,
+                                        505,
+                                        331
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[S_{1}]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        275,
+                                        505,
+                                        331
+                                    ],
+                                    "type": "text",
+                                    "content": " following Definition C.1. Moreover, we clearly have that for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        275,
+                                        505,
+                                        331
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\mathrm{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        275,
+                                        505,
+                                        331
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        275,
+                                        505,
+                                        331
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F[S_{1}], T_{1}^{r_{1}}) \\in S^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        275,
+                                        505,
+                                        331
+                                    ],
+                                    "type": "text",
+                                    "content": ". We can construct "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        275,
+                                        505,
+                                        331
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T_{2} = (V_{T_{2}}, E_{T_{2}}, \\beta_{T_{2}})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        275,
+                                        505,
+                                        331
+                                    ],
+                                    "type": "text",
+                                    "content": " by symmetry, which concludes the proof."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        104,
+                        343,
+                        402,
+                        357
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                343,
+                                402,
+                                357
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        343,
+                                        402,
+                                        357
+                                    ],
+                                    "type": "text",
+                                    "content": "The above lemma can be immediately generalized into the following one:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        358,
+                        504,
+                        409
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                358,
+                                504,
+                                409
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        358,
+                                        504,
+                                        409
+                                    ],
+                                    "type": "text",
+                                    "content": "Corollary C.36. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        358,
+                                        504,
+                                        409
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        358,
+                                        504,
+                                        409
+                                    ],
+                                    "type": "text",
+                                    "content": " be any model, and let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        358,
+                                        504,
+                                        409
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        358,
+                                        504,
+                                        409
+                                    ],
+                                    "type": "text",
+                                    "content": " be a tree-decomposed graph. For each connected component "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        358,
+                                        504,
+                                        409
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        358,
+                                        504,
+                                        409
+                                    ],
+                                    "type": "text",
+                                    "content": " of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        358,
+                                        504,
+                                        409
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        358,
+                                        504,
+                                        409
+                                    ],
+                                    "type": "text",
+                                    "content": ", pick any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        358,
+                                        504,
+                                        409
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t \\in V_T"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        358,
+                                        504,
+                                        409
+                                    ],
+                                    "type": "text",
+                                    "content": " with the minimum depth such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        358,
+                                        504,
+                                        409
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S \\cap \\beta_T(t) \\neq \\emptyset"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        358,
+                                        504,
+                                        409
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, there exists a tree decomposition "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        358,
+                                        504,
+                                        409
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{T}^s"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        358,
+                                        504,
+                                        409
+                                    ],
+                                    "type": "text",
+                                    "content": " of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        358,
+                                        504,
+                                        409
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[S]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        358,
+                                        504,
+                                        409
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfying that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        358,
+                                        504,
+                                        409
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F[S], \\tilde{T}^s) \\in S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        358,
+                                        504,
+                                        409
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        358,
+                                        504,
+                                        409
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S \\cap \\beta_T(t) \\subset \\beta_{\\tilde{T}}(s)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        358,
+                                        504,
+                                        409
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
+                        411,
+                        505,
+                        458
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                411,
+                                505,
+                                458
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        505,
+                                        458
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma C.37. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        505,
+                                        458
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        505,
+                                        458
+                                    ],
+                                    "type": "text",
+                                    "content": " be any model, and let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        505,
+                                        458
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        505,
+                                        458
+                                    ],
+                                    "type": "text",
+                                    "content": " be a tree-decomposed graph such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        505,
+                                        458
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        505,
+                                        458
+                                    ],
+                                    "type": "text",
+                                    "content": " is connected. Then, for each child node "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        505,
+                                        458
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        505,
+                                        458
+                                    ],
+                                    "type": "text",
+                                    "content": " of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        505,
+                                        458
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        505,
+                                        458
+                                    ],
+                                    "type": "text",
+                                    "content": " in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        505,
+                                        458
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        505,
+                                        458
+                                    ],
+                                    "type": "text",
+                                    "content": " and any vertex "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        505,
+                                        458
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        505,
+                                        458
+                                    ],
+                                    "type": "text",
+                                    "content": " in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        505,
+                                        458
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[T^r[t]]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        505,
+                                        458
+                                    ],
+                                    "type": "text",
+                                    "content": ", there is a path in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        505,
+                                        458
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[T^r[t]]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        505,
+                                        458
+                                    ],
+                                    "type": "text",
+                                    "content": " from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        505,
+                                        458
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        505,
+                                        458
+                                    ],
+                                    "type": "text",
+                                    "content": " to some vertex in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        505,
+                                        458
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_T(r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        505,
+                                        458
+                                    ],
+                                    "type": "text",
+                                    "content": ". Consequently, the number of connected components of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        505,
+                                        458
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[T^r[t]]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        505,
+                                        458
+                                    ],
+                                    "type": "text",
+                                    "content": " is bounded by the number of different elements in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        505,
+                                        458
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_T(r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        411,
+                                        505,
+                                        458
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        469,
+                        505,
+                        529
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                469,
+                                505,
+                                529
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Assume the above statement does not hold, and let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S \\subset V_{F[T^r[t]]}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": " be the connected component of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[T^r[t]]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S \\cap \\beta_T(r) = \\emptyset"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": ". Note that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[S] = F[T^r[t]][S]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": ". Since "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": " is connected, there exists an edge "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{v, w\\} \\in E_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v \\in S"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w \\notin S"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": ". Since "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": " is a connected component of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[T^r[t]]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": ", we have "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w \\notin V_{F[T^r[t]]}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": ", and thus both "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v, w \\notin \\beta_T(r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": ". This yields a contradiction since "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{v, w\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": " should be contained in a bag in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": " but "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "B_T(v) \\cap B_T(w) = \\emptyset"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        469,
+                                        505,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        104,
+                        540,
+                        504,
+                        574
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                540,
+                                504,
+                                574
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        504,
+                                        574
+                                    ],
+                                    "type": "text",
+                                    "content": "To reduce duplication, below we only give proofs for Local 2-GNN and Local 2-FGNN. One can easily write a proof for Subgrapp HNN based on the proof of Local 2-GNN, and write a proof for 2-FGNN based on the proof of Local 2-FGNN."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        104,
+                        576,
+                        443,
+                        590
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                576,
+                                443,
+                                590
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        576,
+                                        443,
+                                        590
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma C.38. For any tree-decomposed graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        576,
+                                        443,
+                                        590
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^{\\mathsf{L}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        576,
+                                        443,
+                                        590
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        576,
+                                        443,
+                                        590
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        576,
+                                        443,
+                                        590
+                                    ],
+                                    "type": "text",
+                                    "content": " has a strong NED."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        104,
+                        601,
+                        505,
+                        647
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                601,
+                                505,
+                                647
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        601,
+                                        505,
+                                        647
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Based on Corollary C.36, we can assume that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        601,
+                                        505,
+                                        647
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        601,
+                                        505,
+                                        647
+                                    ],
+                                    "type": "text",
+                                    "content": " is connected without loss of generality. We will prove the following stronger result: for any connected "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        601,
+                                        505,
+                                        647
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^r)\\in S^{\\mathsf{L}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        601,
+                                        505,
+                                        647
+                                    ],
+                                    "type": "text",
+                                    "content": " with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        601,
+                                        505,
+                                        647
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_T(r) = \\{\\{u,v\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        601,
+                                        505,
+                                        647
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        601,
+                                        505,
+                                        647
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        601,
+                                        505,
+                                        647
+                                    ],
+                                    "type": "text",
+                                    "content": " has a strong NED where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        601,
+                                        505,
+                                        647
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u,v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        601,
+                                        505,
+                                        647
+                                    ],
+                                    "type": "text",
+                                    "content": " are endpoints of the first ear. (For the case of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        601,
+                                        505,
+                                        647
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u = v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        601,
+                                        505,
+                                        647
+                                    ],
+                                    "type": "text",
+                                    "content": ", the other endpoint can be arbitrary.)"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        104,
+                        651,
+                        505,
+                        733
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                651,
+                                505,
+                                733
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "The proof is based on induction over the number the vertices in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ". The above statement obviously holds for the base case of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "|V_T| = 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ". Now assume that the statement holds when "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "|V_T| \\leq m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ", and consider the case of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "|V_T| = m + 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ". Note that for any two different children "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t, t'"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[T^r[t]]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[T^r[t']]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " can only share vertices "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ". For each child node "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ", denote its unique child node as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{t}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ". It is easy to see that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r[\\tilde{t}]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " is a canonical tree decomposition of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[T^r[\\tilde{t}]]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F[T^r[\\tilde{t}]], T^r[\\tilde{t}]) \\in S^\\perp"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ". However, one needs to be cautious as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[T^r[\\tilde{t}]]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " may not be connected (unlike the original graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        651,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "). Below, we separately consider the following cases:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
+                        750,
+                        311,
+                        761
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                299,
+                                750,
+                                311,
+                                761
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        299,
+                                        750,
+                                        311,
+                                        761
+                                    ],
+                                    "type": "text",
+                                    "content": "39"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 38
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        117,
+                        82,
+                        504,
+                        180
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 3,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                118,
+                                82,
+                                504,
+                                150
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        118,
+                                        82,
+                                        504,
+                                        150
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "text",
+                                            "content": "1. First consider the case when "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u = v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "text",
+                                            "content": ". In this case, Lemma C.37 implies that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[T^r[\\tilde{t}]]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "text",
+                                            "content": " is connected. According to the induction hypothesis, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[T^r[\\tilde{t}]]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "text",
+                                            "content": " has a strong NED (denoted as "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_{\\tilde{t}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "text",
+                                            "content": ") such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "text",
+                                            "content": " is an endpoint of the first ear. We then merge the ear decompositions "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_{\\tilde{t}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "text",
+                                            "content": " for all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tilde{t}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "text",
+                                            "content": " into a whole "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "text",
+                                            "content": ", specify a root ear "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "text",
+                                            "content": " in any "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_{\\tilde{t}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "text",
+                                            "content": ", and let the first ear of other "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_{\\tilde{t}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "text",
+                                            "content": " nest on "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "text",
+                                            "content": " (with empty nested interval). It is easy to see that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "text",
+                                            "content": " is a strong NED of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                150
+                                            ],
+                                            "type": "text",
+                                            "content": " is an endpoint of the first ear."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 1
+                        },
+                        {
+                            "bbox": [
+                                117,
+                                156,
+                                504,
+                                180
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        156,
+                                        504,
+                                        180
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                117,
+                                                156,
+                                                504,
+                                                180
+                                            ],
+                                            "type": "text",
+                                            "content": "2. Next consider the case when "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                156,
+                                                504,
+                                                180
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u \\neq v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                156,
+                                                504,
+                                                180
+                                            ],
+                                            "type": "text",
+                                            "content": ". In this case, without loss of generality, assume "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                156,
+                                                504,
+                                                180
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(\\tilde{t}) = \\{u, w\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                156,
+                                                504,
+                                                180
+                                            ],
+                                            "type": "text",
+                                            "content": " for some "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                156,
+                                                504,
+                                                180
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\in V_F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                156,
+                                                504,
+                                                180
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 2
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        141,
+                        186,
+                        504,
+                        209
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                141,
+                                186,
+                                504,
+                                209
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        186,
+                                        504,
+                                        209
+                                    ],
+                                    "type": "text",
+                                    "content": "- Subcase 1: "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        186,
+                                        504,
+                                        209
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[T^r[\\tilde{t}]]"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        186,
+                                        504,
+                                        209
+                                    ],
+                                    "type": "text",
+                                    "content": " is connected. Then, "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        186,
+                                        504,
+                                        209
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[T^r[\\tilde{t}]]"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        186,
+                                        504,
+                                        209
+                                    ],
+                                    "type": "text",
+                                    "content": " has a strong NED "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        186,
+                                        504,
+                                        209
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{P}_{\\hat{t}}"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        186,
+                                        504,
+                                        209
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        186,
+                                        504,
+                                        209
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        186,
+                                        504,
+                                        209
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        186,
+                                        504,
+                                        209
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        186,
+                                        504,
+                                        209
+                                    ],
+                                    "type": "text",
+                                    "content": " are endpoints of the first ear."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        161,
+                        209,
+                        504,
+                        309
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 8,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                161,
+                                209,
+                                477,
+                                221
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        161,
+                                        209,
+                                        477,
+                                        221
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                161,
+                                                209,
+                                                477,
+                                                221
+                                            ],
+                                            "type": "text",
+                                            "content": "a) If "
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                209,
+                                                477,
+                                                221
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\bar{w} = v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                209,
+                                                477,
+                                                221
+                                            ],
+                                            "type": "text",
+                                            "content": ", then "
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                209,
+                                                477,
+                                                221
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[T^r[t]] = F[T^r[\\hat{t}]]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                209,
+                                                477,
+                                                221
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                209,
+                                                477,
+                                                221
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[T^r[t]]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                209,
+                                                477,
+                                                221
+                                            ],
+                                            "type": "text",
+                                            "content": " clearly has a strong NED."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 5
+                        },
+                        {
+                            "bbox": [
+                                161,
+                                221,
+                                504,
+                                276
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        161,
+                                        221,
+                                        504,
+                                        276
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                161,
+                                                221,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "text",
+                                            "content": "b) If "
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                221,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\notin N_F[v]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                221,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "text",
+                                            "content": " or "
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                221,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w = u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                221,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "text",
+                                            "content": ", there are two additional subcases depending on whether "
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                221,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{u, v\\} \\in E_F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                221,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "text",
+                                            "content": ". If "
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                221,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{u, v\\} \\in E_F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                221,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "text",
+                                            "content": ", then "
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                221,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[T^r[t]]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                221,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "text",
+                                            "content": " has a strong NED (which can be constructed from "
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                221,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_{\\tilde{t}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                221,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "text",
+                                            "content": " by adding an ear "
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                221,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{\\{u, v\\}\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                221,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "text",
+                                            "content": " and letting the first ear in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                221,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_{\\tilde{t}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                221,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "text",
+                                            "content": " nest on "
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                221,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{\\{u, v\\}\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                221,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "text",
+                                            "content": " (with empty nested interval). Otherwise, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                221,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[T^r[t]] \\backslash \\{v\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                221,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "text",
+                                            "content": " has a strong NED and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                221,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                221,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "text",
+                                            "content": " is an isolated vertex in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                221,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[T^r[t]]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                221,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 6
+                        },
+                        {
+                            "bbox": [
+                                161,
+                                275,
+                                504,
+                                309
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        161,
+                                        275,
+                                        504,
+                                        309
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                161,
+                                                275,
+                                                504,
+                                                309
+                                            ],
+                                            "type": "text",
+                                            "content": "c) If "
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                275,
+                                                504,
+                                                309
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\in N_F(v)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                275,
+                                                504,
+                                                309
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                275,
+                                                504,
+                                                309
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\neq u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                275,
+                                                504,
+                                                309
+                                            ],
+                                            "type": "text",
+                                            "content": ", then "
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                275,
+                                                504,
+                                                309
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[T^r[t]]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                275,
+                                                504,
+                                                309
+                                            ],
+                                            "type": "text",
+                                            "content": " also has a strong NED, which can be constructed from "
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                275,
+                                                504,
+                                                309
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_{\\tilde{t}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                275,
+                                                504,
+                                                309
+                                            ],
+                                            "type": "text",
+                                            "content": " by extending the first ear to incorporate the edge "
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                275,
+                                                504,
+                                                309
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{w, v\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                275,
+                                                504,
+                                                309
+                                            ],
+                                            "type": "text",
+                                            "content": " (we still need an additional ear "
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                275,
+                                                504,
+                                                309
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{\\{u, v\\}\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                275,
+                                                504,
+                                                309
+                                            ],
+                                            "type": "text",
+                                            "content": " if "
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                275,
+                                                504,
+                                                309
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{u, v\\} \\in E_F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                161,
+                                                275,
+                                                504,
+                                                309
+                                            ],
+                                            "type": "text",
+                                            "content": ")."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 7
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        141,
+                        310,
+                        505,
+                        460
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                141,
+                                310,
+                                505,
+                                460
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": "- Subcase 2: "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[T^r[\\tilde{t}]]"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": " is disconnected. In this subcase, Lemma C.37 implies that "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[T^r[\\tilde{t}]]"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": " has exactly two connected components, and "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": " are in different connected components (which can be easily proved by noting that "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": " can only link to "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": " or "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": " in "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[T^r[\\tilde{t}]]"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": "). We can thus invoke Lemma C.35, which shows that both connected components of "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[T^r[\\tilde{t}]]"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": ", denoted as "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\widehat{F}_1"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\widehat{F}_2"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": ", admits tree decompositions "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\widehat{T}_1^{s_1}"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\widehat{T}_2^{s_2}"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\widehat{F}_1, \\widehat{T}_1^{s_1}), (\\widehat{F}_2, \\widehat{T}_2^{s_2}) \\in S^\\mathsf{L}"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_{\\widehat{T}_1}(s_1) = \\{\\{u, u\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_{\\widehat{T}_2}(s_2) = \\{\\{w, w\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": ". Moreover, "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "|V_{\\widehat{T}_1}| = |V_{\\widehat{T}_2}| = |V_{T^r[\\tilde{t}]}| < m + 1"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": ". Therefore, according to the induction hypothesis, "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\widehat{F}_1"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": " has a strong NED "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{P}_{\\tilde{t},1}"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": " with "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": " as an endpoint of the first ear, and "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\widehat{F}_2"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": " has a strong NED "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{P}_{\\tilde{t},2}"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": " with "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": " as an endpoint of the first ear. If "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{w, v\\} \\in E_F"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": ", we can extend the first ear in "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{P}_{\\tilde{t},2}"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": " to include the edge "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{w, v\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": ". Finally, if "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{u, v\\} \\in E_F"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": ", by setting the first ear to be "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\{u, v\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": ", we can merge the two NEDs "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{P}_{\\tilde{t},1}"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{P}_{\\tilde{t},2}"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": " to obtain the NED of "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[T^r[t]]"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v"
+                                },
+                                {
+                                    "bbox": [
+                                        141,
+                                        310,
+                                        505,
+                                        460
+                                    ],
+                                    "type": "text",
+                                    "content": " are endpoints of the first ear."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        130,
+                        467,
+                        251,
+                        478
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                130,
+                                467,
+                                251,
+                                478
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        130,
+                                        467,
+                                        251,
+                                        478
+                                    ],
+                                    "type": "text",
+                                    "content": "Overall, we always have that:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        141,
+                        483,
+                        504,
+                        540
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 13,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                141,
+                                483,
+                                504,
+                                505
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        483,
+                                        504,
+                                        505
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                483,
+                                                504,
+                                                505
+                                            ],
+                                            "type": "text",
+                                            "content": "a) If "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                483,
+                                                504,
+                                                505
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[T^r[t]]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                483,
+                                                504,
+                                                505
+                                            ],
+                                            "type": "text",
+                                            "content": " is connected, then it admits a strong NED such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                483,
+                                                504,
+                                                505
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u, v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                483,
+                                                504,
+                                                505
+                                            ],
+                                            "type": "text",
+                                            "content": " are endpoints of the first ear."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 11
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                506,
+                                504,
+                                540
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        506,
+                                        504,
+                                        540
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                506,
+                                                504,
+                                                540
+                                            ],
+                                            "type": "text",
+                                            "content": "b) If "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                506,
+                                                504,
+                                                540
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[T^r[t]]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                506,
+                                                504,
+                                                540
+                                            ],
+                                            "type": "text",
+                                            "content": " is disconnected, then it has two connected components each admitting a strong NED such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                506,
+                                                504,
+                                                540
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u, v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                506,
+                                                504,
+                                                540
+                                            ],
+                                            "type": "text",
+                                            "content": " belong to an endpoint of the first ear for each of the two NEDs, respectively."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 12
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        129,
+                        544,
+                        504,
+                        570
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                544,
+                                504,
+                                570
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        544,
+                                        504,
+                                        570
+                                    ],
+                                    "type": "text",
+                                    "content": "Finally, noting that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        544,
+                                        504,
+                                        570
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F = \\bigcup_{t,\\mathfrak{pa}_{Tr}(t) = r}F[T^r [t]]"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        544,
+                                        504,
+                                        570
+                                    ],
+                                    "type": "text",
+                                    "content": ", we can merge all NEDs of "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        544,
+                                        504,
+                                        570
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[T^{r}[t]]"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        544,
+                                        504,
+                                        570
+                                    ],
+                                    "type": "text",
+                                    "content": " to obtain a strong NED of "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        544,
+                                        504,
+                                        570
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        544,
+                                        504,
+                                        570
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfying that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        544,
+                                        504,
+                                        570
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u,v"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        544,
+                                        504,
+                                        570
+                                    ],
+                                    "type": "text",
+                                    "content": " are endpoints of the first ear."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        105,
+                        574,
+                        304,
+                        586
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                574,
+                                304,
+                                586
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        574,
+                                        304,
+                                        586
+                                    ],
+                                    "type": "text",
+                                    "content": "We thus conclude the proof of the induction step."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        104,
+                        592,
+                        504,
+                        615
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                592,
+                                504,
+                                615
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        592,
+                                        504,
+                                        615
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma C.39. For any graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        592,
+                                        504,
+                                        615
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        592,
+                                        504,
+                                        615
+                                    ],
+                                    "type": "text",
+                                    "content": ", if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        592,
+                                        504,
+                                        615
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        592,
+                                        504,
+                                        615
+                                    ],
+                                    "type": "text",
+                                    "content": " admits a strong NED, then there exists a tree decomposition "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        592,
+                                        504,
+                                        615
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        592,
+                                        504,
+                                        615
+                                    ],
+                                    "type": "text",
+                                    "content": " of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        592,
+                                        504,
+                                        615
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        592,
+                                        504,
+                                        615
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        592,
+                                        504,
+                                        615
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^{r})\\in S^{\\mathsf{L}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        592,
+                                        504,
+                                        615
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        104,
+                        626,
+                        504,
+                        683
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                626,
+                                504,
+                                683
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        683
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. We can assume that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        683
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        683
+                                    ],
+                                    "type": "text",
+                                    "content": " is connected without loss of generality, as it is easy to merge the tree decompositions of different connected components to form a single tree which is a tree decomposition of the whole graph. We will prove the following stronger result: for any connected graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        683
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        683
+                                    ],
+                                    "type": "text",
+                                    "content": ", if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        683
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        683
+                                    ],
+                                    "type": "text",
+                                    "content": " admits a strong NED where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        683
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        683
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        683
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        683
+                                    ],
+                                    "type": "text",
+                                    "content": " are two different endpoints of the first ear, then there exists a tree decomposition "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        683
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        683
+                                    ],
+                                    "type": "text",
+                                    "content": " of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        683
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        683
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        683
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^\\mathbb{L}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        683
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        683
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_T(r) = \\{(u, v)\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        683
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        104,
+                        687,
+                        504,
+                        733
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                687,
+                                504,
+                                733
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "The proof is based on induction over the number the edges in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ". The above statement obviously holds for the base case of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "|E_F| = 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ". Now assume that the statement holds when "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "|E_F| \\leq m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ", and consider the case of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "|E_F| = m + 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ". Denote the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "-th ear as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " and denote "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_1 = \\{\\{w_0, w_1\\}, \\dots, \\{w_{l-1}, w_l\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_0 = u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_l = v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ". Consider the following three cases:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
+                        751,
+                        312,
+                        760
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                299,
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+                            ],
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+                                {
+                                    "bbox": [
+                                        299,
+                                        751,
+                                        312,
+                                        760
+                                    ],
+                                    "type": "text",
+                                    "content": "40"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 39
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        117,
+                        82,
+                        504,
+                        411
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 4,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                118,
+                                82,
+                                504,
+                                205
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        118,
+                                        82,
+                                        504,
+                                        205
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": "1. Either "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": " or "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": " is a cut vertex in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": ". Without loss of generality, assume that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": " is a cut vertex. In this case, there must exist a child ear "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_i"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": " nested on "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": " is an endpoint of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_i"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "I(P_i) = \\emptyset"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": ". We can thus split all ears into two parts "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_2"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": ", where "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": " contains the ear "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_i"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": " and its descendant ears in the NED tree, and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_2"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": " contains other ears. Denote by "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[\\mathcal{P}_k]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": " the connected graph induced by all edges in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_k"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": " ("
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "k \\in \\{1, 2\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": "). Then, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_k"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": " is a strong NED of the graph "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[\\mathcal{P}_k]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": ". Since both "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_2"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": " contain no more than "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "m"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": " edges, according to the induction hypothesis, there are tree decompositions "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T_k^{r_k}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": " of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[\\mathcal{P}_k]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": " ("
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "k \\in \\{1, 2\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": ") such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_{T_1}(r_1) = \\{\\{u, w\\}\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": " for some "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\in V_F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": ", "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_{T_2}(r_2) = \\{\\{u, v\\}\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": ", and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(F[\\mathcal{P}_k], T_k^{r_k}) \\in S^{\\mathsf{L}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": " ("
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "k \\in \\{1, 2\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": "). We can glue the tree "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T_1^{r_1}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T_2^{r_2}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": " into a larger tree "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T^{r_2}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": " by adding a new node "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": " with "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(t) = \\{\\{u, v, w\\}\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": ", setting "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{pa}_{T^r}(t) = r_2"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{pa}_{T^r}(r_1) = t"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": ". Clearly, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T^{r_2}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": " is a valid tree decomposition of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": ", "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(r_2) = \\{\\{u, v\\}\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": ", and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(F, T^{r_2}) \\in S^{\\mathsf{L}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                82,
+                                                504,
+                                                205
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 1
+                        },
+                        {
+                            "bbox": [
+                                117,
+                                210,
+                                504,
+                                290
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        210,
+                                        504,
+                                        290
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "text",
+                                            "content": "2. There is an ear "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_{i}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "text",
+                                            "content": " nested on "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_{1}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "text",
+                                            "content": " with "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "I(P_{i}) = P_{1}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "text",
+                                            "content": ". In this case, we can split all ears into two parts "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_2"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "text",
+                                            "content": ", where "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "text",
+                                            "content": " contains the ear "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_{i}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "text",
+                                            "content": " and its descendant ears in the NED tree, and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_2"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "text",
+                                            "content": " contains other ears. Then, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_k"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "text",
+                                            "content": " is a strong NED of the graph "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[\\mathcal{P}_k]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "text",
+                                            "content": ". Since both "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_2"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "text",
+                                            "content": " contain no more than "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "m"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "text",
+                                            "content": " edges, according to the induction hypothesis, there is a tree decomposition "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T_k^{r_k}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "text",
+                                            "content": " of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[\\mathcal{P}_k]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_{T_k}(r_k) = \\{\\{u,v\\}\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(F[\\mathcal{P}_k], T_k^{r_k}) \\in S^{\\mathsf{L}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "text",
+                                            "content": ". By merging the root node, we can glue the tree "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T_1^{r_1}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T_2^{r_2}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "text",
+                                            "content": " into a larger tree "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T^r"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "text",
+                                            "content": ". Clearly, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T^r"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "text",
+                                            "content": " is a valid tree decomposition of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "text",
+                                            "content": ", "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(r) = \\{\\{u,v\\}\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "text",
+                                            "content": ", and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(F,T^r) \\in S^{\\mathsf{L}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                210,
+                                                504,
+                                                290
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 2
+                        },
+                        {
+                            "bbox": [
+                                117,
+                                294,
+                                504,
+                                411
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        294,
+                                        504,
+                                        411
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": "3. Neither "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": " nor "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": " is a cut vertex in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": " and all ears "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_i"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": " nested on "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": " satisfies "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "I(P_i) \\subsetneq P_1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": ". In this case, we have either "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{w_0, w_1\\} \\notin I(P_i)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": " for all ear "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_i"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": " nested on "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": " or "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{w_{l-1}, w_l\\} \\notin I(P_i)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": " for all ear "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_i"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": " nested on "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": " (otherwise, it would contradict the definition of strong NED). Without loss of generality, assume "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{w_0, w_1\\} \\notin I(P_i)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": " for all ears "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_i"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": " nested on "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": ". Then, it is clear that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "l > 1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": ", and the subgraph "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F \\setminus \\{u\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": " is connected and also admits a strong NED where "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w_1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": " are two endpoints of the first ear. Therefore, according to the induction hypothesis, there is a tree decomposition "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tilde{T}^s"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": " of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F \\setminus \\{u\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": " satisfying "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(F \\setminus \\{u\\}, \\tilde{T}^s) \\in S^\\mathsf{L}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_{\\tilde{T}}(s) = \\{w_1, v\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": ". We can then construct a tree "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T^r"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": " from "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tilde{T}^s"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": " by adding two fresh nodes "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "r"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "r'"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": " and setting "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{pa}_{T^r}(s) = r'"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{pa}_{T^r}(r') = r"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": ". Set "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(r) = \\{u, v\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": ", "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(r') = \\{u, v, w_1\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": ", and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(t) = \\beta_{\\tilde{T}}(t)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": " for all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t \\in V_{\\tilde{T}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": ". It is easy to see that the constructed "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T^r"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": " is a tree decomposition of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(F, T^r) \\in S^\\mathsf{L}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                294,
+                                                504,
+                                                411
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 3
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        105,
+                        415,
+                        358,
+                        426
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                415,
+                                358,
+                                426
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        415,
+                                        358,
+                                        426
+                                    ],
+                                    "type": "text",
+                                    "content": "Combining the above three cases concludes the induction step."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "type": "image",
+                    "bbox": [
+                        494,
+                        415,
+                        504,
+                        423
+                    ],
+                    "blocks": [
+                        {
+                            "bbox": [
+                                494,
+                                415,
+                                504,
+                                423
+                            ],
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        494,
+                                        415,
+                                        504,
+                                        423
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                494,
+                                                415,
+                                                504,
+                                                423
+                                            ],
+                                            "type": "image",
+                                            "image_path": "3706a2b0f9b29c7dff4d686d5f1d771438cbdc11234be0e289fc186a18f43089.jpg"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 6,
+                            "angle": 0,
+                            "type": "image_body"
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        433,
+                        504,
+                        456
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                433,
+                                504,
+                                456
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        433,
+                                        504,
+                                        456
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem C.40. For any graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        433,
+                                        504,
+                                        456
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        433,
+                                        504,
+                                        456
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        433,
+                                        504,
+                                        456
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        433,
+                                        504,
+                                        456
+                                    ],
+                                    "type": "text",
+                                    "content": " has a strong NED iff there is a tree decomposition "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        433,
+                                        504,
+                                        456
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        433,
+                                        504,
+                                        456
+                                    ],
+                                    "type": "text",
+                                    "content": " of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        433,
+                                        504,
+                                        456
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        433,
+                                        504,
+                                        456
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        433,
+                                        504,
+                                        456
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^\\mathsf{L}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        433,
+                                        504,
+                                        456
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        105,
+                        468,
+                        361,
+                        480
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                468,
+                                361,
+                                480
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        468,
+                                        361,
+                                        480
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. This is a direct consequence of Lemmas C.38 and C.39."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "type": "image",
+                    "bbox": [
+                        494,
+                        469,
+                        504,
+                        478
+                    ],
+                    "blocks": [
+                        {
+                            "bbox": [
+                                494,
+                                469,
+                                504,
+                                478
+                            ],
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        494,
+                                        469,
+                                        504,
+                                        478
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                494,
+                                                469,
+                                                504,
+                                                478
+                                            ],
+                                            "type": "image",
+                                            "image_path": "7e164b3e67d390003e8e6b05ccb28558864c7e056100271d6404a714d427a1d9.jpg"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 9,
+                            "angle": 0,
+                            "type": "image_body"
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        105,
+                        493,
+                        490,
+                        506
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                493,
+                                490,
+                                506
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        493,
+                                        490,
+                                        506
+                                    ],
+                                    "type": "text",
+                                    "content": "We next turn to Local 2-FGNN, where the proof has a similar structure as that of Local 2-GNN."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        105,
+                        508,
+                        482,
+                        521
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                508,
+                                482,
+                                521
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        508,
+                                        482,
+                                        521
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma C.41. For any tree-decomposed graph "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        508,
+                                        482,
+                                        521
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^{\\mathrm{LF}}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        508,
+                                        482,
+                                        521
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        508,
+                                        482,
+                                        521
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        508,
+                                        482,
+                                        521
+                                    ],
+                                    "type": "text",
+                                    "content": " has an almost-strong NED."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        104,
+                        533,
+                        504,
+                        578
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                533,
+                                504,
+                                578
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        504,
+                                        578
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Based on Corollary C.36, we can assume that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        504,
+                                        578
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        504,
+                                        578
+                                    ],
+                                    "type": "text",
+                                    "content": " is connected without loss of generality. We will prove the following stronger result: for any connected "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        504,
+                                        578
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^r)\\in S^{\\mathrm{LF}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        504,
+                                        578
+                                    ],
+                                    "type": "text",
+                                    "content": " with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        504,
+                                        578
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_T(r) = \\{\\{u,v\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        504,
+                                        578
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        504,
+                                        578
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        504,
+                                        578
+                                    ],
+                                    "type": "text",
+                                    "content": " has an almost-strong NED where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        504,
+                                        578
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u,v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        504,
+                                        578
+                                    ],
+                                    "type": "text",
+                                    "content": " are endpoints of the first ear. (For the case of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        504,
+                                        578
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u = v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        504,
+                                        578
+                                    ],
+                                    "type": "text",
+                                    "content": ", the other endpoint can be arbitrary.)"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        104,
+                        582,
+                        504,
+                        651
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                582,
+                                504,
+                                651
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": "Similar to the proof of Lemma C.38, assume that the statement holds when "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "|V_T| \\leq m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": ", and consider the case of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "|V_T| = m + 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": ". For each child node "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": " of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": ", if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": " only has one child, the proof exactly follows the one in Lemma C.38. Therefore, it suffices to consider the case where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": " has two children "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "q_1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "q_2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": ". Denote "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_T(t) = \\{\\{u,v,w\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_T(q_1) = \\{\\{u,w\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_T(q_2) = \\{\\{v,w\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": ". Since "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w \\in N_F[u] \\cup N_F[v]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": " (by definition of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S^{\\mathsf{LF}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": "), we can assume that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w \\in N_F[v]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": " without loss of generality. Recall that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F[T^r [q_1]], T^r [q_1])"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F[T^r [q_2]], T^r [q_2]) \\in S^{\\mathsf{LF}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        582,
+                                        504,
+                                        651
+                                    ],
+                                    "type": "text",
+                                    "content": ". Below, we separately consider the following cases:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        118,
+                        654,
+                        505,
+                        733
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                118,
+                                654,
+                                505,
+                                733
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        118,
+                                        654,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "1. First consider the case when "
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        654,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u = v"
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        654,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ". In this case, Lemma C.37 implies that both "
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        654,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[T^r[q_1]]"
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        654,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        654,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[T^r[q_2]]"
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        654,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " are connected (since either "
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        654,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u = w"
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        654,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " or "
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        654,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{u, w\\} \\in E_F"
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        654,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "). According to the induction hypothesis, both "
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        654,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[T^r[q_1]]"
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        654,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        654,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[T^r[q_2]]"
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        654,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " admit an almost-strong NED such that "
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        654,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        654,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        654,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w"
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        654,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " are endpoints of the first ear. If "
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        654,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w = u"
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        654,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ", we can merge the two almost-strong NEDs so that the first ear in one NED is nested on the first ear of the other (with an empty nested interval). Otherwise, the first ears in the two almost-strong NEDs share both endpoints and we can clearly merge them (in case of "
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        654,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{u, w\\} \\in E_F"
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        654,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ", there is a common ear "
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        654,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\{u, w\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        654,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " in both NEDs,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
+                        751,
+                        310,
+                        760
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                299,
+                                751,
+                                310,
+                                760
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        299,
+                                        751,
+                                        310,
+                                        760
+                                    ],
+                                    "type": "text",
+                                    "content": "41"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 40
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        129,
+                        82,
+                        504,
+                        105
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                82,
+                                504,
+                                105
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        82,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "text",
+                                    "content": "which is taken only once). In both subcases, we obtain an almost-strong NED of "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        82,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[T^r[t]]"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        82,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        82,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        82,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "text",
+                                    "content": " is an endpoint of the first ear."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        117,
+                        110,
+                        506,
+                        143
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                117,
+                                110,
+                                506,
+                                143
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        110,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": "2. Next consider the case when "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        110,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\neq v"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        110,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": ". In this case, "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        110,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[T^r[q_1]]"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        110,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        110,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[T^r[q_2]]"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        110,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": " share only one vertex "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        110,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        110,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": ". Note that "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        110,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[T^r[q_2]]"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        110,
+                                        506,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": " is connected by Lemma C.37. We separately consider two subcases:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        146,
+                        148,
+                        506,
+                        383
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 5,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                146,
+                                148,
+                                506,
+                                270
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        146,
+                                        148,
+                                        506,
+                                        270
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "text",
+                                            "content": "- Subcase 1: "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[T^r[q_1]]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "text",
+                                            "content": " is connected. Then, according to the induction hypothesis, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[T^r[q_1]]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "text",
+                                            "content": " admits an almost-strong NED "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "text",
+                                            "content": " are endpoints of the first ear, and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[T^r[q_2]]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "text",
+                                            "content": " admits an almost-strong NED "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_2"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "text",
+                                            "content": " are endpoints of the first ear. (i) If "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w = v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "text",
+                                            "content": ", we can merge the two almost-strong NEDs so that the first ear of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_2"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "text",
+                                            "content": " is nested on the first ear of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "text",
+                                            "content": " (with empty nested interval). (ii) If "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w = u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "text",
+                                            "content": ", we can merge the two almost-strong NEDs so that the first ear of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "text",
+                                            "content": " is nested on the first ear of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_2"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "text",
+                                            "content": " (with empty nested interval). (iii) If "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\in N_F(v)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\neq u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "text",
+                                            "content": ", the first ear of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_2"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "text",
+                                            "content": " can be chosen as "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{\\{w, v\\}\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "text",
+                                            "content": ". Then, we can merge the two almost-strong NEDs by gluing the first year in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "text",
+                                            "content": " with the ear "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{\\{w, v\\}\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "text",
+                                            "content": " in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_2"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "text",
+                                            "content": ". One can see that the resulting NED is almost-strong. Overall, we always have that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[T^r[t]]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "text",
+                                            "content": " admits an almost-strong NED such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u, v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                148,
+                                                506,
+                                                270
+                                            ],
+                                            "type": "text",
+                                            "content": " are endpoints of the first ear."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 3
+                        },
+                        {
+                            "bbox": [
+                                146,
+                                270,
+                                506,
+                                383
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        146,
+                                        270,
+                                        506,
+                                        383
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "text",
+                                            "content": "- Subcase 2: "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[T^r[q_1]]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "text",
+                                            "content": " is disconnected. In this subcase, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\neq u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "text",
+                                            "content": ". Similar to the proof of Lemma C.38, we obtain that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[T^r[q_1]]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "text",
+                                            "content": " has exactly two connected components, and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "text",
+                                            "content": " are in different connected components. We can thus invoke Lemma C.35, which shows that both connected components of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[T^r[q_1]]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "text",
+                                            "content": ", denoted as "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\widehat{F}_1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\widehat{F}_2"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "text",
+                                            "content": ", admit almost-strong NEDs "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_2"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "text",
+                                            "content": ", respectively. Moreover, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "text",
+                                            "content": " is an endpoint of the first ear in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "text",
+                                            "content": ", and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "text",
+                                            "content": " is an endpoint of the first ear in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_2"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "text",
+                                            "content": ". Besides, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[T^r[q_2]]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "text",
+                                            "content": " admits an almost-strong NED "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_3"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "text",
+                                            "content": " with "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w, v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "text",
+                                            "content": " as the endpoints of the first ear. By letting the first ear of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_2"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "text",
+                                            "content": " nest on the first ear of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_3"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "text",
+                                            "content": " (with an empty nested interval), we can merge "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_2"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_3"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "text",
+                                            "content": ". Then, we can merge "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{P}_2 \\cup \\mathcal{P}_3"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                270,
+                                                506,
+                                                383
+                                            ],
+                                            "type": "text",
+                                            "content": " following the same procedure as Subcase 2 in the proof of Lemma C.38."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 4
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        129,
+                        388,
+                        251,
+                        399
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                388,
+                                251,
+                                399
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        388,
+                                        251,
+                                        399
+                                    ],
+                                    "type": "text",
+                                    "content": "Overall, we always have that:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        141,
+                        404,
+                        505,
+                        460
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 9,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                141,
+                                404,
+                                504,
+                                426
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        404,
+                                        504,
+                                        426
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                404,
+                                                504,
+                                                426
+                                            ],
+                                            "type": "text",
+                                            "content": "a) If "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                404,
+                                                504,
+                                                426
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[T^r[t]]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                404,
+                                                504,
+                                                426
+                                            ],
+                                            "type": "text",
+                                            "content": " is connected, then it admits an almost-strong NED such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                404,
+                                                504,
+                                                426
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u, v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                404,
+                                                504,
+                                                426
+                                            ],
+                                            "type": "text",
+                                            "content": " are endpoints of the first ear."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 7
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                427,
+                                505,
+                                460
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        427,
+                                        505,
+                                        460
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                427,
+                                                505,
+                                                460
+                                            ],
+                                            "type": "text",
+                                            "content": "b) If "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                427,
+                                                505,
+                                                460
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[T^r[t]]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                427,
+                                                505,
+                                                460
+                                            ],
+                                            "type": "text",
+                                            "content": " is disconnected, then it has two connected components each admitting an almost-strong NED such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                427,
+                                                505,
+                                                460
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u, v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                427,
+                                                505,
+                                                460
+                                            ],
+                                            "type": "text",
+                                            "content": " belong to an endpoint of the first ear for each of the two NEDs, respectively."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 8
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        129,
+                        466,
+                        505,
+                        502
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                466,
+                                505,
+                                502
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        466,
+                                        505,
+                                        502
+                                    ],
+                                    "type": "text",
+                                    "content": "In both subcases, it follows that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        466,
+                                        505,
+                                        502
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[T^{r}[t]]"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        466,
+                                        505,
+                                        502
+                                    ],
+                                    "type": "text",
+                                    "content": " admits an almost-strong NED with "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        466,
+                                        505,
+                                        502
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        466,
+                                        505,
+                                        502
+                                    ],
+                                    "type": "text",
+                                    "content": " as two endpoints. Noting that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        466,
+                                        505,
+                                        502
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F = \\bigcup_{t, \\mathfrak{pa}_{Tr}(t) = r} F[T^{r}[t]]"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        466,
+                                        505,
+                                        502
+                                    ],
+                                    "type": "text",
+                                    "content": ", we can merge all NEDs of "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        466,
+                                        505,
+                                        502
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[T^{r}[t]]"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        466,
+                                        505,
+                                        502
+                                    ],
+                                    "type": "text",
+                                    "content": " to obtain an almost-strong NED of "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        466,
+                                        505,
+                                        502
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        466,
+                                        505,
+                                        502
+                                    ],
+                                    "type": "text",
+                                    "content": " with "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        466,
+                                        505,
+                                        502
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        466,
+                                        505,
+                                        502
+                                    ],
+                                    "type": "text",
+                                    "content": " as two endpoints."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        105,
+                        506,
+                        304,
+                        518
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                506,
+                                304,
+                                518
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        506,
+                                        304,
+                                        518
+                                    ],
+                                    "type": "text",
+                                    "content": "We thus conclude the proof of the induction step."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        104,
+                        524,
+                        504,
+                        548
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                524,
+                                504,
+                                548
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        524,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma C.42. For any graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        524,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        524,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "text",
+                                    "content": ", if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        524,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        524,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "text",
+                                    "content": " admits an almost-strong NED, then there exists a tree decomposition "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        524,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        524,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "text",
+                                    "content": " of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        524,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        524,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        524,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^{r})\\in S^{\\mathrm{LF}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        524,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        104,
+                        559,
+                        504,
+                        605
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                559,
+                                504,
+                                605
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        504,
+                                        605
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. We can assume that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        504,
+                                        605
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        504,
+                                        605
+                                    ],
+                                    "type": "text",
+                                    "content": " is connected without loss of generality. We will prove the following stronger result: for any graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        504,
+                                        605
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        504,
+                                        605
+                                    ],
+                                    "type": "text",
+                                    "content": ", if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        504,
+                                        605
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        504,
+                                        605
+                                    ],
+                                    "type": "text",
+                                    "content": " admits an almost-strong NED where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        504,
+                                        605
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        504,
+                                        605
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        504,
+                                        605
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        504,
+                                        605
+                                    ],
+                                    "type": "text",
+                                    "content": " are two different endpoints of the first ear, then there exists a tree decomposition "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        504,
+                                        605
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        504,
+                                        605
+                                    ],
+                                    "type": "text",
+                                    "content": " of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        504,
+                                        605
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        504,
+                                        605
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        504,
+                                        605
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^{\\mathsf{LF}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        504,
+                                        605
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        504,
+                                        605
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_T(r) = \\{\\{u, v\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        559,
+                                        504,
+                                        605
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        104,
+                        609,
+                        505,
+                        643
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                609,
+                                505,
+                                643
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        609,
+                                        505,
+                                        643
+                                    ],
+                                    "type": "text",
+                                    "content": "Similar to the proof of Lemma C.39, assume that the statement holds when "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        609,
+                                        505,
+                                        643
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "|E_F| \\leq m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        609,
+                                        505,
+                                        643
+                                    ],
+                                    "type": "text",
+                                    "content": ", and consider the case of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        609,
+                                        505,
+                                        643
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "|E_F| = m + 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        609,
+                                        505,
+                                        643
+                                    ],
+                                    "type": "text",
+                                    "content": ". Denote the first ear as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        609,
+                                        505,
+                                        643
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_1 = \\{\\{w_0, w_1\\}, \\dots, \\{w_{l-1}, w_l\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        609,
+                                        505,
+                                        643
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        609,
+                                        505,
+                                        643
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_0 = u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        609,
+                                        505,
+                                        643
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        609,
+                                        505,
+                                        643
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_l = v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        609,
+                                        505,
+                                        643
+                                    ],
+                                    "type": "text",
+                                    "content": ". Consider the following three cases:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        116,
+                        647,
+                        504,
+                        693
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 18,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                117,
+                                647,
+                                462,
+                                658
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        647,
+                                        462,
+                                        658
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                117,
+                                                647,
+                                                462,
+                                                658
+                                            ],
+                                            "type": "text",
+                                            "content": "1. Either "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                647,
+                                                462,
+                                                658
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                647,
+                                                462,
+                                                658
+                                            ],
+                                            "type": "text",
+                                            "content": " or "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                647,
+                                                462,
+                                                658
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                647,
+                                                462,
+                                                658
+                                            ],
+                                            "type": "text",
+                                            "content": " is a cut vertex in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                647,
+                                                462,
+                                                658
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                647,
+                                                462,
+                                                658
+                                            ],
+                                            "type": "text",
+                                            "content": ". This case is exactly the same as in Lemma C.39."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 15
+                        },
+                        {
+                            "bbox": [
+                                116,
+                                664,
+                                504,
+                                677
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        664,
+                                        504,
+                                        677
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                664,
+                                                504,
+                                                677
+                                            ],
+                                            "type": "text",
+                                            "content": "2. There is an ear "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                664,
+                                                504,
+                                                677
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_{i}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                664,
+                                                504,
+                                                677
+                                            ],
+                                            "type": "text",
+                                            "content": " nested on "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                664,
+                                                504,
+                                                677
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_{1}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                664,
+                                                504,
+                                                677
+                                            ],
+                                            "type": "text",
+                                            "content": " with "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                664,
+                                                504,
+                                                677
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "I(P_{i}) = P_{1}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                664,
+                                                504,
+                                                677
+                                            ],
+                                            "type": "text",
+                                            "content": ". This case is also the same as in Lemma C.39."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 16
+                        },
+                        {
+                            "bbox": [
+                                117,
+                                681,
+                                203,
+                                693
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        681,
+                                        203,
+                                        693
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                117,
+                                                681,
+                                                203,
+                                                693
+                                            ],
+                                            "type": "text",
+                                            "content": "3. Otherwise, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                681,
+                                                203,
+                                                693
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "l \\geq 2"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                681,
+                                                203,
+                                                693
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 17
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        146,
+                        698,
+                        506,
+                        734
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                146,
+                                698,
+                                506,
+                                734
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        146,
+                                        698,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": "- If there is an ear "
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        698,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_i"
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        698,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": " nested on "
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        698,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_1"
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        698,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": " with "
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        698,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{w_0, w_1\\}, \\{w_1, w_2\\} \\in I(P_i)"
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        698,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": ", then "
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        698,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "l \\geq 3"
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        698,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": ". By definition of almost-strong NED, there does not exist an ear "
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        698,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_j"
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        698,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": " nested on "
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        698,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_1"
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        698,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": " with "
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        698,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{w_{l-2}, w_{l-1}\\}, \\{w_{l-1}, w_l\\} \\in I(P_i)"
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        698,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": ". In this case, we can split "
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        698,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_1"
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        698,
+                                        506,
+                                        734
+                                    ],
+                                    "type": "text",
+                                    "content": " into two parts:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
+                        751,
+                        312,
+                        760
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                299,
+                                751,
+                                312,
+                                760
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        299,
+                                        751,
+                                        312,
+                                        760
+                                    ],
+                                    "type": "text",
+                                    "content": "42"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 41
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        154,
+                        81,
+                        504,
+                        213
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                154,
+                                81,
+                                504,
+                                213
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{1,u} = P_1\\backslash \\{\\{w_{l - 1},w_l\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{1,v} = \\{\\{w_{l - 1},w_l\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, we can rearrange any ear nested on "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{1}"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": " so that it is nested on either "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{1,u}"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": " or "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{1,v}"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": ". In this way, we can split all ears into two sets "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{P}_u"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{P}_v"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": ", one corresponding to "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{1,u}"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": " and its descendant ears, and the other corresponding to "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{1,v}"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": " and its descendant ears. Therefore, according to the induction hypothesis, there is a tree decomposition "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{T}_u^s"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": " of "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[\\mathcal{P}_u]"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfying "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F[\\mathcal{P}_u],\\tilde{T}_u^s)\\in S^{\\mathrm{LF}}"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_{\\tilde{T}_u}(s) = \\{u,w_{l - 1}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": ", and similarly, there is a tree decomposition "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{T}_v^{s'}"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": " of "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F[\\mathcal{P}_v]"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfying "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F[\\mathcal{P}_v],\\tilde{T}_v^{s'})\\in S^{\\mathrm{LF}}"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_{\\tilde{T}_v}(s) = \\{w_{l - 1},v\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": ". We can then construct a tree "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": " by merging "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{T}_u^s"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{T}_v^{s'}"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": " and adding two fresh nodes "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r,r'"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": ", where "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": " is the root node, "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{pa}_{T^r}(r') = r"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{pa}_{T^r}(s) = \\mathsf{pa}_{T^r}(s') = r'"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": ". Set "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_T(r) = \\{u,v\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_T(r') = \\{u,v,w_{l - 1}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": ". It is easy to see that the constructed "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": " is a tree decomposition of "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^r)\\in S^{\\mathrm{LF}}"
+                                },
+                                {
+                                    "bbox": [
+                                        154,
+                                        81,
+                                        504,
+                                        213
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        146,
+                        214,
+                        504,
+                        261
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                146,
+                                214,
+                                504,
+                                261
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        146,
+                                        214,
+                                        504,
+                                        261
+                                    ],
+                                    "type": "text",
+                                    "content": "- If there does not exist an ear "
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        214,
+                                        504,
+                                        261
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{i}"
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        214,
+                                        504,
+                                        261
+                                    ],
+                                    "type": "text",
+                                    "content": " nested on "
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        214,
+                                        504,
+                                        261
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{1}"
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        214,
+                                        504,
+                                        261
+                                    ],
+                                    "type": "text",
+                                    "content": " with "
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        214,
+                                        504,
+                                        261
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{w_{0}, w_{1}\\}, \\{w_{1}, w_{2}\\} \\in I(P_{i})"
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        214,
+                                        504,
+                                        261
+                                    ],
+                                    "type": "text",
+                                    "content": ", then we follow exactly the same analysis as in the previous item, except that now we split "
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        214,
+                                        504,
+                                        261
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{1}"
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        214,
+                                        504,
+                                        261
+                                    ],
+                                    "type": "text",
+                                    "content": " into two parts: "
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        214,
+                                        504,
+                                        261
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{1,u} = \\{\\{w_{0}, w_{1}\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        214,
+                                        504,
+                                        261
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        214,
+                                        504,
+                                        261
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{1,v} = P_{1} \\backslash \\{\\{w_{0}, w_{1}\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        214,
+                                        504,
+                                        261
+                                    ],
+                                    "type": "text",
+                                    "content": ". We can still construct a tree decomposition "
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        214,
+                                        504,
+                                        261
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^{r}"
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        214,
+                                        504,
+                                        261
+                                    ],
+                                    "type": "text",
+                                    "content": " of "
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        214,
+                                        504,
+                                        261
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        214,
+                                        504,
+                                        261
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        214,
+                                        504,
+                                        261
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^{r}) \\in S^{\\mathsf{LF}}"
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        214,
+                                        504,
+                                        261
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        265,
+                        323,
+                        277
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                265,
+                                323,
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+                            "spans": [
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+                                    "bbox": [
+                                        104,
+                                        265,
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+                                        277
+                                    ],
+                                    "type": "text",
+                                    "content": "Combining the above three cases concludes the proof."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "type": "image",
+                    "bbox": [
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+                    ],
+                    "blocks": [
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+                            "bbox": [
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+                            ],
+                            "lines": [
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+                                    "bbox": [
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+                                    "spans": [
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+                                            "bbox": [
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+                                            ],
+                                            "type": "image",
+                                            "image_path": "f30cfda95c20f021405fe098e02b8544b38c7e149fb3a7dbe403bfc85e4ba0ae.jpg"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 4,
+                            "angle": 0,
+                            "type": "image_body"
+                        }
+                    ],
+                    "index": 4
+                },
+                {
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        306
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem C.43. For any graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        306
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        306
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        306
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        306
+                                    ],
+                                    "type": "text",
+                                    "content": " has an almost-strong NED iff there is a tree decomposition "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        306
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        306
+                                    ],
+                                    "type": "text",
+                                    "content": " of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        306
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        306
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        306
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^{\\mathsf{LF}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        283,
+                                        504,
+                                        306
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        105,
+                        317,
+                        361,
+                        330
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                317,
+                                361,
+                                330
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        317,
+                                        361,
+                                        330
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. This is a direct consequence of Lemmas C.41 and C.42."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "type": "image",
+                    "bbox": [
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+                    ],
+                    "blocks": [
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+                                            "bbox": [
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+                                                327
+                                            ],
+                                            "type": "image",
+                                            "image_path": "dc7856b177da299e7a9b26c6c4c4cfe5734d2fe1f098f50df076484b30cb27c0.jpg"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 7,
+                            "angle": 0,
+                            "type": "image_body"
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        105,
+                        342,
+                        231,
+                        354
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
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+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "type": "text",
+                                    "content": "C.4 PART 3: PEBBLE GAME"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                        504,
+                                        397
+                                    ],
+                                    "type": "text",
+                                    "content": "In this part, we prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        362,
+                                        504,
+                                        397
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        362,
+                                        504,
+                                        397
+                                    ],
+                                    "type": "text",
+                                    "content": " is maximal (Definition 3.1(b)) for any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        362,
+                                        504,
+                                        397
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\mathrm{Sub}, \\mathrm{L}, \\mathrm{LF}, \\mathrm{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        362,
+                                        504,
+                                        397
+                                    ],
+                                    "type": "text",
+                                    "content": ". To achieve this, we first introduce a general class of graphs which we call Fürer graphs (Fürer, 2001). Intuitions and illustrations of Fürer graphs can be found in Zhang et al. (2023a)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        399,
+                                        504,
+                                        424
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition C.44 (Fürer graphs). Given any connected graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        399,
+                                        504,
+                                        424
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F = (V_F, E_F, \\ell_F)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        399,
+                                        504,
+                                        424
+                                    ],
+                                    "type": "text",
+                                    "content": ", the Fürer graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        399,
+                                        504,
+                                        424
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G(F) = (V_{G(F)}, E_{G(F)}, \\ell_{G(F)})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        399,
+                                        504,
+                                        424
+                                    ],
+                                    "type": "text",
+                                    "content": " is constructed as follows:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        160,
+                        428,
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        160,
+                                        428,
+                                        402,
+                                        441
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "V _ {G (F)} = \\{(x, X): x \\in V _ {F}, X \\subset N _ {F} (x), | X | \\bmod 2 = 0 \\},",
+                                    "image_path": "305c21404e4402e729a86a6cdc3e8d9726e0c2305eaececb8f6eaa9ba084b10a.jpg"
+                                }
+                            ]
+                        }
+                    ],
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+                    ],
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+                                {
+                                    "bbox": [
+                                        160,
+                                        443,
+                                        449,
+                                        457
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "E _ {G (F)} = \\left\\{\\{(x, X), (y, Y) \\} \\subset V _ {G}: \\{x, y \\} \\in E _ {F}, (x \\in Y \\leftrightarrow y \\in X) \\right\\},",
+                                    "image_path": "9196ef2ab8933c65a3ca6190ad24a2d258c31a0dc2fe728d2bfbda8df06941c8.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
+                                        161,
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\ell_ {G (F)} (x, X) = \\ell_ {F} (x) \\quad \\forall (x, X) \\in V _ {G (F)}.",
+                                    "image_path": "6d0b3ac60d81d017f5faf34482e473a9115af245f33bbf696e771fd2d5afd90a.jpg"
+                                }
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+                },
+                {
+                    "bbox": [
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+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                        500
+                                    ],
+                                    "type": "text",
+                                    "content": "Here, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        477,
+                                        504,
+                                        500
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x \\in Y \\leftrightarrow y \\in X"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        477,
+                                        504,
+                                        500
+                                    ],
+                                    "type": "text",
+                                    "content": " holds when either "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        477,
+                                        504,
+                                        500
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(x \\in Y \\text{ and } y \\in X)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        477,
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+                                        500
+                                    ],
+                                    "type": "text",
+                                    "content": " or "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        477,
+                                        504,
+                                        500
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(x \\notin Y \\text{ and } y \\notin X)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        477,
+                                        504,
+                                        500
+                                    ],
+                                    "type": "text",
+                                    "content": " holds. For each "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        477,
+                                        504,
+                                        500
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        477,
+                                        504,
+                                        500
+                                    ],
+                                    "type": "text",
+                                    "content": ", denote the set"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        191,
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\operatorname {M e t a} _ {F} (x) := \\{(x, X): X \\subset N _ {F} (x), | X | \\bmod 2 = 0 \\}, \\tag {24}",
+                                    "image_path": "4309076ded912c1a2e4115595af6048a54bab998fca43ac1c7b64aeec5b2850a.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        104,
+                        522,
+                        493,
+                        537
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                522,
+                                493,
+                                537
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        522,
+                                        493,
+                                        537
+                                    ],
+                                    "type": "text",
+                                    "content": "which is called the meta vertices of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        522,
+                                        493,
+                                        537
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G(F)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        522,
+                                        493,
+                                        537
+                                    ],
+                                    "type": "text",
+                                    "content": " associated to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        522,
+                                        493,
+                                        537
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        522,
+                                        493,
+                                        537
+                                    ],
+                                    "type": "text",
+                                    "content": ". Note that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        522,
+                                        493,
+                                        537
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "V_{G(F)} = \\bigcup_{x\\in V_F}\\mathsf{Meta}_F(x)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        522,
+                                        493,
+                                        537
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        105,
+                        544,
+                        279,
+                        555
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                544,
+                                279,
+                                555
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        544,
+                                        279,
+                                        555
+                                    ],
+                                    "type": "text",
+                                    "content": "We next define an operation called \"twist\":"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        104,
+                        558,
+                        504,
+                        594
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                558,
+                                504,
+                                594
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        558,
+                                        504,
+                                        594
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition C.45 (Twist). Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        558,
+                                        504,
+                                        594
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G(F) = (V_{G(F)}, E_{G(F)}, \\ell_{G(F)})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        558,
+                                        504,
+                                        594
+                                    ],
+                                    "type": "text",
+                                    "content": " be the Fürer graph of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        558,
+                                        504,
+                                        594
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F = (V_F, E_F, \\ell_F)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        558,
+                                        504,
+                                        594
+                                    ],
+                                    "type": "text",
+                                    "content": ", and let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        558,
+                                        504,
+                                        594
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{x, y\\} \\in E_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        558,
+                                        504,
+                                        594
+                                    ],
+                                    "type": "text",
+                                    "content": " be an edge of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        558,
+                                        504,
+                                        594
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        558,
+                                        504,
+                                        594
+                                    ],
+                                    "type": "text",
+                                    "content": ". The twisted Fürer graph of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        558,
+                                        504,
+                                        594
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G(F)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        558,
+                                        504,
+                                        594
+                                    ],
+                                    "type": "text",
+                                    "content": " for edge "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        558,
+                                        504,
+                                        594
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{x, y\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        558,
+                                        504,
+                                        594
+                                    ],
+                                    "type": "text",
+                                    "content": ", is constructed as follows: "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        558,
+                                        504,
+                                        594
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\text{twist}(G(F), \\{x, y\\}) := (V_{G(F)}, E_{\\text{twist}(G(F), \\{x, y\\})}, \\ell_{G(F)})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        558,
+                                        504,
+                                        594
+                                    ],
+                                    "type": "text",
+                                    "content": ", where"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
+                        159,
+                        599,
+                        451,
+                        613
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                159,
+                                599,
+                                451,
+                                613
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        159,
+                                        599,
+                                        451,
+                                        613
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "E _ {\\operatorname {w i s t} (G (F), \\{x, y \\})} := E _ {G (F)} \\triangle \\{\\{\\xi , \\eta \\}: \\xi \\in \\operatorname {M e t a} _ {F} (x), \\eta \\in \\operatorname {M e t a} _ {F} (y) \\},",
+                                    "image_path": "706b30d3fa29f2dc48a9d444585333aea1532da9fa54da0f8730841ed236938a.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                },
+                {
+                    "bbox": [
+                        104,
+                        617,
+                        504,
+                        641
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                617,
+                                504,
+                                641
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        617,
+                                        504,
+                                        641
+                                    ],
+                                    "type": "text",
+                                    "content": "and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        617,
+                                        504,
+                                        641
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\triangle"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        617,
+                                        504,
+                                        641
+                                    ],
+                                    "type": "text",
+                                    "content": " is the symmetric difference operator, i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        617,
+                                        504,
+                                        641
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A\\triangle B = (A\\backslash B)\\cup (B\\backslash A)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        617,
+                                        504,
+                                        641
+                                    ],
+                                    "type": "text",
+                                    "content": ". For an edge set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        617,
+                                        504,
+                                        641
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S = \\{e_1,\\dots ,e_k\\} \\subset E_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        617,
+                                        504,
+                                        641
+                                    ],
+                                    "type": "text",
+                                    "content": ", we further define"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                },
+                {
+                    "bbox": [
+                        194,
+                        646,
+                        504,
+                        658
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                194,
+                                646,
+                                504,
+                                658
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        194,
+                                        646,
+                                        504,
+                                        658
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\operatorname {t w i s t} (G (F), S) := \\operatorname {t w i s t} (\\dots \\operatorname {t w i s t} (G (F), e _ {1}) \\dots , e _ {k}). \\tag {25}",
+                                    "image_path": "2489b8c8683ab8349b7c4436469ed121cfb4a7e7b9252fd83e533002e6f6913b.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 21
+                },
+                {
+                    "bbox": [
+                        104,
+                        663,
+                        504,
+                        686
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                663,
+                                504,
+                                686
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        504,
+                                        686
+                                    ],
+                                    "type": "text",
+                                    "content": "Note that Equation (25) is well-defined as the resulting graph does not depend on the order of edges "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        504,
+                                        686
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "e_1, \\dots, e_k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        663,
+                                        504,
+                                        686
+                                    ],
+                                    "type": "text",
+                                    "content": " for twisting."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 22
+                },
+                {
+                    "bbox": [
+                        104,
+                        694,
+                        496,
+                        707
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                694,
+                                496,
+                                707
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        694,
+                                        496,
+                                        707
+                                    ],
+                                    "type": "text",
+                                    "content": "The following result is well-known (see e.g., Zhang et al., 2023a, Corollary I.5 and Lemma I.7)):"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 23
+                },
+                {
+                    "bbox": [
+                        104,
+                        709,
+                        504,
+                        733
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                709,
+                                504,
+                                733
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        709,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem C.46. For any connected graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        709,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        709,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " and any set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        709,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S_1, S_2 \\subset E_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        709,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        709,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{twist}(G(F), S_1) \\simeq \\operatorname{twist}(G(F), S_2)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        709,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        709,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "|S_1| \\mod 2 = |S_2| \\mod 2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        709,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 24
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
+                        751,
+                        310,
+                        760
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                299,
+                                751,
+                                310,
+                                760
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        299,
+                                        751,
+                                        310,
+                                        760
+                                    ],
+                                    "type": "text",
+                                    "content": "43"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 25
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 42
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        82,
+                        504,
+                        117
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                82,
+                                504,
+                                117
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        117
+                                    ],
+                                    "type": "text",
+                                    "content": "This result shows that the structure of twisted Fürer graph does not depend on which edge is twisted. Therefore, we can pick any edge "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        117
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{u,v\\} \\in E_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        117
+                                    ],
+                                    "type": "text",
+                                    "content": " and denote "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        117
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H(F) = \\mathrm{twist}(G(F),\\{u,v\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        117
+                                    ],
+                                    "type": "text",
+                                    "content": ". We now show that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        117
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G(F),H(F)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        117
+                                    ],
+                                    "type": "text",
+                                    "content": " can be distinguished via homomorphism information:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        118,
+                        504,
+                        143
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                118,
+                                504,
+                                143
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        118,
+                                        504,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem C.47. For any graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        118,
+                                        504,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        118,
+                                        504,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": ", let (twisted) Fürer graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        118,
+                                        504,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G(F)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        118,
+                                        504,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        118,
+                                        504,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H(F)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        118,
+                                        504,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": " be defined above. Then, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        118,
+                                        504,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{hom}(F, G(F)) \\neq \\operatorname{hom}(F, H(F))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        118,
+                                        504,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        152,
+                        505,
+                        176
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                152,
+                                505,
+                                176
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        152,
+                                        505,
+                                        176
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. In the proof below, we use "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        152,
+                                        505,
+                                        176
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        152,
+                                        505,
+                                        176
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        152,
+                                        505,
+                                        176
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        152,
+                                        505,
+                                        176
+                                    ],
+                                    "type": "text",
+                                    "content": " to denote mappings where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        152,
+                                        505,
+                                        176
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g: V_F \\to V_{G(F)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        152,
+                                        505,
+                                        176
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        152,
+                                        505,
+                                        176
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h: V_F \\to V_{H(F)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        152,
+                                        505,
+                                        176
+                                    ],
+                                    "type": "text",
+                                    "content": ". The proof is divided into the following parts."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        117,
+                        180,
+                        505,
+                        216
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                117,
+                                180,
+                                505,
+                                216
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        180,
+                                        505,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": "1. We first prove that there is a homomorphism "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        180,
+                                        505,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g \\in \\operatorname{Hom}(F, G(F))"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        180,
+                                        505,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfying "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        180,
+                                        505,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g(w) \\in \\operatorname{Meta}_F(w)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        180,
+                                        505,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        180,
+                                        505,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        180,
+                                        505,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": ", but there is no homomorphism "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        180,
+                                        505,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h \\in \\operatorname{Hom}(F, H(F))"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        180,
+                                        505,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfying "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        180,
+                                        505,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h(w) \\in \\operatorname{Meta}_F(w)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        180,
+                                        505,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        180,
+                                        505,
+                                        216
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        180,
+                                        505,
+                                        216
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        141,
+                        220,
+                        504,
+                        292
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 7,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                141,
+                                220,
+                                504,
+                                256
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        220,
+                                        504,
+                                        256
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                220,
+                                                504,
+                                                256
+                                            ],
+                                            "type": "text",
+                                            "content": "a) Define a mapping "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                220,
+                                                504,
+                                                256
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "g: V_F \\to V_{G(F)}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                220,
+                                                504,
+                                                256
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                220,
+                                                504,
+                                                256
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "g(w) = (w, \\emptyset)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                220,
+                                                504,
+                                                256
+                                            ],
+                                            "type": "text",
+                                            "content": " for all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                220,
+                                                504,
+                                                256
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\in V_F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                220,
+                                                504,
+                                                256
+                                            ],
+                                            "type": "text",
+                                            "content": ". We clearly have "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                220,
+                                                504,
+                                                256
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{w, x\\} \\in E_F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                220,
+                                                504,
+                                                256
+                                            ],
+                                            "type": "text",
+                                            "content": " implies that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                220,
+                                                504,
+                                                256
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{g(w), g(x)\\} \\in E_{G(F)}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                220,
+                                                504,
+                                                256
+                                            ],
+                                            "type": "text",
+                                            "content": ". Moreover, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                220,
+                                                504,
+                                                256
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\ell_{G(F)}(g(w)) = \\ell_F(w)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                220,
+                                                504,
+                                                256
+                                            ],
+                                            "type": "text",
+                                            "content": ". Therefore, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                220,
+                                                504,
+                                                256
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "g \\in \\operatorname{Hom}(F, G(F))"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                220,
+                                                504,
+                                                256
+                                            ],
+                                            "type": "text",
+                                            "content": " is indeed a homomorphism."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 5
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                256,
+                                504,
+                                292
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        256,
+                                        504,
+                                        292
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                256,
+                                                504,
+                                                292
+                                            ],
+                                            "type": "text",
+                                            "content": "b) If we similarly define "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                256,
+                                                504,
+                                                292
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "h: V_F \\to V_{H(F)}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                256,
+                                                504,
+                                                292
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                256,
+                                                504,
+                                                292
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "h(w) = (w, \\emptyset)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                256,
+                                                504,
+                                                292
+                                            ],
+                                            "type": "text",
+                                            "content": " for all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                256,
+                                                504,
+                                                292
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\in V_F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                256,
+                                                504,
+                                                292
+                                            ],
+                                            "type": "text",
+                                            "content": ", then for all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                256,
+                                                504,
+                                                292
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{w, x\\} \\in E_F \\backslash \\{\\{u, v\\}\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                256,
+                                                504,
+                                                292
+                                            ],
+                                            "type": "text",
+                                            "content": " we have "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                256,
+                                                504,
+                                                292
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{h(w), h(x)\\} \\in E_{H(F)}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                256,
+                                                504,
+                                                292
+                                            ],
+                                            "type": "text",
+                                            "content": ", but "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                256,
+                                                504,
+                                                292
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{h(u), h(v)\\} \\notin E_{H(F)}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                256,
+                                                504,
+                                                292
+                                            ],
+                                            "type": "text",
+                                            "content": " since the edge "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                256,
+                                                504,
+                                                292
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{u, v\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                256,
+                                                504,
+                                                292
+                                            ],
+                                            "type": "text",
+                                            "content": " is twisted."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 6
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        129,
+                        297,
+                        504,
+                        361
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                297,
+                                504,
+                                361
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        297,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "text",
+                                    "content": "It remains to prove that for all "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        297,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h: V_F \\to V_{H(F)}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        297,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "text",
+                                    "content": " of the form "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        297,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h(w) = (w, U_w)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        297,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "text",
+                                    "content": " (for each "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        297,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        297,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "text",
+                                    "content": "), "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        297,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h \\notin \\operatorname{Hom}(F, H(F))"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        297,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "text",
+                                    "content": ". It suffices to prove that there is an odd number of edges "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        297,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{w, x\\} \\in E_F"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        297,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        297,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{h(w), h(x)\\} \\notin E_{H(F)}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        297,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "text",
+                                    "content": ". Let "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        297,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h, \\tilde{h}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        297,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "text",
+                                    "content": " be two such mappings that differ in only one vertex "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        297,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        297,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "text",
+                                    "content": ", i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        297,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h(w) = \\tilde{h}(w)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        297,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        297,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w \\neq z"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        297,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "text",
+                                    "content": " but "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        297,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h(z) = (z, U_z) \\neq (z, \\tilde{U}_z) = \\tilde{h}(z)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        297,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "text",
+                                    "content": ". Denote "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        297,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "D_z = \\tilde{U}_z \\triangle U_z"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        297,
+                                        504,
+                                        361
+                                    ],
+                                    "type": "text",
+                                    "content": ". Based on the definition of Fürer graph, it follows that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        145,
+                        365,
+                        505,
+                        444
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 12,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                145,
+                                365,
+                                505,
+                                392
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        145,
+                                        365,
+                                        505,
+                                        392
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                145,
+                                                365,
+                                                505,
+                                                392
+                                            ],
+                                            "type": "text",
+                                            "content": "- for all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                365,
+                                                505,
+                                                392
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{w, x\\} \\in E_F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                365,
+                                                505,
+                                                392
+                                            ],
+                                            "type": "text",
+                                            "content": " with "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                365,
+                                                505,
+                                                392
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\neq z, x \\neq z"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                365,
+                                                505,
+                                                392
+                                            ],
+                                            "type": "text",
+                                            "content": ", we have "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                365,
+                                                505,
+                                                392
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{h(w), h(x)\\} \\in E_{H(F)}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                365,
+                                                505,
+                                                392
+                                            ],
+                                            "type": "text",
+                                            "content": " iff "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                365,
+                                                505,
+                                                392
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{\\tilde{h}(w), \\tilde{h}(x)\\} \\in E_{H(F)}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                365,
+                                                505,
+                                                392
+                                            ],
+                                            "type": "text",
+                                            "content": ";"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 9
+                        },
+                        {
+                            "bbox": [
+                                145,
+                                391,
+                                505,
+                                418
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        145,
+                                        391,
+                                        505,
+                                        418
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                145,
+                                                391,
+                                                505,
+                                                418
+                                            ],
+                                            "type": "text",
+                                            "content": "- for all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                391,
+                                                505,
+                                                418
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{w, z\\} \\in E_F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                391,
+                                                505,
+                                                418
+                                            ],
+                                            "type": "text",
+                                            "content": " with "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                391,
+                                                505,
+                                                418
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\notin D_z"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                391,
+                                                505,
+                                                418
+                                            ],
+                                            "type": "text",
+                                            "content": ", we also have "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                391,
+                                                505,
+                                                418
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{h(w), h(z)\\} \\in E_{H(F)}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                391,
+                                                505,
+                                                418
+                                            ],
+                                            "type": "text",
+                                            "content": " iff "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                391,
+                                                505,
+                                                418
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{\\tilde{h}(w), \\tilde{h}(z)\\} \\in E_{H(F)}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                391,
+                                                505,
+                                                418
+                                            ],
+                                            "type": "text",
+                                            "content": ";"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 10
+                        },
+                        {
+                            "bbox": [
+                                145,
+                                418,
+                                504,
+                                444
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        145,
+                                        418,
+                                        504,
+                                        444
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                145,
+                                                418,
+                                                504,
+                                                444
+                                            ],
+                                            "type": "text",
+                                            "content": "- for all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                418,
+                                                504,
+                                                444
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{w, z\\} \\in E_F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                418,
+                                                504,
+                                                444
+                                            ],
+                                            "type": "text",
+                                            "content": " with "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                418,
+                                                504,
+                                                444
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\in D_z"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                418,
+                                                504,
+                                                444
+                                            ],
+                                            "type": "text",
+                                            "content": ", we have "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                418,
+                                                504,
+                                                444
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{h(w), h(z)\\} \\in E_{H(F)}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                418,
+                                                504,
+                                                444
+                                            ],
+                                            "type": "text",
+                                            "content": " iff "
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                418,
+                                                504,
+                                                444
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{\\tilde{h}(w), \\tilde{h}(z)\\} \\notin E_{H(F)}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                145,
+                                                418,
+                                                504,
+                                                444
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 11
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        129,
+                        450,
+                        505,
+                        510
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                450,
+                                505,
+                                510
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        450,
+                                        505,
+                                        510
+                                    ],
+                                    "type": "text",
+                                    "content": "Since "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        450,
+                                        505,
+                                        510
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "|U_z| \\bmod 2 = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        450,
+                                        505,
+                                        510
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        450,
+                                        505,
+                                        510
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "|\\tilde{U}_z| \\bmod 2 = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        450,
+                                        505,
+                                        510
+                                    ],
+                                    "type": "text",
+                                    "content": ", we have "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        450,
+                                        505,
+                                        510
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "|D_z| \\bmod 2 = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        450,
+                                        505,
+                                        510
+                                    ],
+                                    "type": "text",
+                                    "content": " and thus the number of edges "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        450,
+                                        505,
+                                        510
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{w, x\\} \\in E_F"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        450,
+                                        505,
+                                        510
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        450,
+                                        505,
+                                        510
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{h(w), h(x)\\} \\notin E_{H(F)}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        450,
+                                        505,
+                                        510
+                                    ],
+                                    "type": "text",
+                                    "content": " has the same parity as the number of edges such that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        450,
+                                        505,
+                                        510
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\tilde{h}(w), \\tilde{h}(x)\\} \\notin E_{H(F)}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        450,
+                                        505,
+                                        510
+                                    ],
+                                    "type": "text",
+                                    "content": ". Finally, noting that all mappings "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        450,
+                                        505,
+                                        510
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        450,
+                                        505,
+                                        510
+                                    ],
+                                    "type": "text",
+                                    "content": " can be obtained from the one in (b) by continually modifying "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        450,
+                                        505,
+                                        510
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "U_w"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        450,
+                                        505,
+                                        510
+                                    ],
+                                    "type": "text",
+                                    "content": " for each "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        450,
+                                        505,
+                                        510
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        450,
+                                        505,
+                                        510
+                                    ],
+                                    "type": "text",
+                                    "content": " and the parity remains unchanged, we have concluded the proof of this part."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        117,
+                        514,
+                        504,
+                        548
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                117,
+                                514,
+                                504,
+                                548
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        514,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "text",
+                                    "content": "2. We next prove that for any permutation "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        514,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi : V_F \\to V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        514,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "text",
+                                    "content": ", there does not exist a homomorphism "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        514,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h \\in \\operatorname{Hom}(F, H(F))"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        514,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfying "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        514,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h(w) \\in \\operatorname{Meta}_F(\\pi(w))"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        514,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        514,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        514,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "text",
+                                    "content": ". Assume the conclusion does not hold and pick any "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        514,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        514,
+                                        504,
+                                        548
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfying the above condition. Consider the following cases:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        141,
+                        553,
+                        504,
+                        656
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 17,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                141,
+                                553,
+                                504,
+                                609
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        553,
+                                        504,
+                                        609
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                553,
+                                                504,
+                                                609
+                                            ],
+                                            "type": "text",
+                                            "content": "a) Case 1: if "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                553,
+                                                504,
+                                                609
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pi"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                553,
+                                                504,
+                                                609
+                                            ],
+                                            "type": "text",
+                                            "content": " is an automorphism of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                553,
+                                                504,
+                                                609
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                553,
+                                                504,
+                                                609
+                                            ],
+                                            "type": "text",
+                                            "content": ", then it is easy to see that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                553,
+                                                504,
+                                                609
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "h \\circ \\pi^{-1} \\in \\operatorname{Hom}(F, H(F))"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                553,
+                                                504,
+                                                609
+                                            ],
+                                            "type": "text",
+                                            "content": ", because "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                553,
+                                                504,
+                                                609
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{w, x\\} \\in E_F \\Rightarrow \\{\\pi^{-1}(w), \\pi^{-1}(x)\\} \\in E_F \\Rightarrow \\{h(\\pi^{-1}(w)), h(\\pi^{-1}(x))\\} \\in E_F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                553,
+                                                504,
+                                                609
+                                            ],
+                                            "type": "text",
+                                            "content": " for all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                553,
+                                                504,
+                                                609
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w, x \\in V_F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                553,
+                                                504,
+                                                609
+                                            ],
+                                            "type": "text",
+                                            "content": ", and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                553,
+                                                504,
+                                                609
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\ell_F(w) = \\ell_F(\\pi^{-1}(w)) = \\ell_{G(F)}(h(\\pi^{-1}(w)))"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                553,
+                                                504,
+                                                609
+                                            ],
+                                            "type": "text",
+                                            "content": " for all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                553,
+                                                504,
+                                                609
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\in V_F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                553,
+                                                504,
+                                                609
+                                            ],
+                                            "type": "text",
+                                            "content": ". Moreover, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                553,
+                                                504,
+                                                609
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "h \\circ \\pi^{-1}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                553,
+                                                504,
+                                                609
+                                            ],
+                                            "type": "text",
+                                            "content": " satisfies that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                553,
+                                                504,
+                                                609
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "h(\\pi^{-1}(w)) \\in \\operatorname{Meta}_F(w)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                553,
+                                                504,
+                                                609
+                                            ],
+                                            "type": "text",
+                                            "content": ", yielding a contradiction to point 1."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 15
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                609,
+                                504,
+                                656
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        609,
+                                        504,
+                                        656
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                141,
+                                                609,
+                                                504,
+                                                656
+                                            ],
+                                            "type": "text",
+                                            "content": "b) Case 2: if "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                609,
+                                                504,
+                                                656
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\pi"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                609,
+                                                504,
+                                                656
+                                            ],
+                                            "type": "text",
+                                            "content": " is not an automorphism of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                609,
+                                                504,
+                                                656
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                609,
+                                                504,
+                                                656
+                                            ],
+                                            "type": "text",
+                                            "content": ", then there exists an edge "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                609,
+                                                504,
+                                                656
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{w, x\\} \\in E_F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                609,
+                                                504,
+                                                656
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                609,
+                                                504,
+                                                656
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{\\pi(w), \\pi(x)\\} \\notin E_F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                609,
+                                                504,
+                                                656
+                                            ],
+                                            "type": "text",
+                                            "content": ". In this case, we must have "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                609,
+                                                504,
+                                                656
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{h(w), h(x)\\} \\notin E_{H(F)}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                609,
+                                                504,
+                                                656
+                                            ],
+                                            "type": "text",
+                                            "content": " since by definition "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                609,
+                                                504,
+                                                656
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "h(w) \\in \\mathsf{Meta}_F(\\pi(w))"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                609,
+                                                504,
+                                                656
+                                            ],
+                                            "type": "text",
+                                            "content": ", "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                609,
+                                                504,
+                                                656
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "h(x) \\in \\mathsf{Meta}_F(\\pi(x))"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                609,
+                                                504,
+                                                656
+                                            ],
+                                            "type": "text",
+                                            "content": ", and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                609,
+                                                504,
+                                                656
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{\\pi(w), \\pi(x)\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                141,
+                                                609,
+                                                504,
+                                                656
+                                            ],
+                                            "type": "text",
+                                            "content": " is not an edge of "
+                                        },
+                                        {
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+                                    "content": "In both cases, "
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+                                    "type": "text",
+                                    "content": " is invalid and thus there is no homomorphism "
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+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "h \\in \\operatorname{Hom}(F, H(F))"
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": " satisfying "
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+                                    "type": "inline_equation",
+                                    "content": "h(w) \\in \\operatorname{Meta}_F(\\pi(w))"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " for all "
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+                                    "type": "inline_equation",
+                                    "content": "w \\in V_F"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
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+                                    "type": "text",
+                                    "content": "3. We finally prove that the following two sets have equal size (i.e., "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "|S_G| = |S_H|"
+                                },
+                                {
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+                                    "type": "text",
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+                                    "type": "interline_equation",
+                                    "content": "S _ {G} := \\left\\{g \\in \\operatorname {H o m} (F, G (F)): \\exists w, x, y \\in V _ {F} \\text {s . t .} x \\neq y, g (x), g (y) \\in \\operatorname {M e t a} _ {F} (w) \\right\\},",
+                                    "image_path": "083b6762d08ca64d6f28305efda27237c5c17e77fbc19258de9b41b3bdd8e4b3.jpg"
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+                                    "type": "interline_equation",
+                                    "content": "S _ {H} := \\left\\{h \\in \\operatorname {H o m} (F, H (F)): \\exists w, x, y \\in V _ {F} \\text {s . t .} x \\neq y, h (x), h (y) \\in \\operatorname {M e t a} _ {F} (w) \\right\\}.",
+                                    "image_path": "1141a7e4478c5e4c410755eb95c86eacba302aa322053c229601c47499f6b1cc.jpg"
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
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+                                    "content": "44"
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+                                    "type": "text",
+                                    "content": "It suffices to prove that, for any proper subset "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "U \\subsetneq V_F"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ", we have "
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+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "|S_G^U| = |S_H^U|"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        81,
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+                                        105
+                                    ],
+                                    "type": "text",
+                                    "content": ", where "
+                                },
+                                {
+                                    "bbox": [
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+                                        81,
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+                                        105
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S_G"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        81,
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+                                        105
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        81,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S_H"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        81,
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+                                        105
+                                    ],
+                                    "type": "text",
+                                    "content": " are defined as follows:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
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+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
+                                        179,
+                                        109,
+                                        453,
+                                        142
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "S _ {G} ^ {U} := \\left\\{g \\in \\operatorname {H o m} (F, G (F)): g (x) \\in \\bigcup_ {w \\in U} \\operatorname {M e t a} _ {F} (w) \\forall x \\in V _ {F} \\right\\},",
+                                    "image_path": "72886dc4e7e92c4eac27aa13971802ce22a21c658ad6daf7cef00c4bc81829ba.jpg"
+                                }
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+                            "bbox": [
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+                                    "bbox": [
+                                        180,
+                                        144,
+                                        454,
+                                        177
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "S _ {H} ^ {U} := \\left\\{h \\in \\operatorname {H o m} (F, H (F)): h (x) \\in \\bigcup_ {w \\in U} \\operatorname {M e t a} _ {F} (w) \\forall x \\in V _ {F} \\right\\}.",
+                                    "image_path": "b31cbf7b8427b636e2b90921a5f2f2aa6fa040a847e83b7c06cb228bc83a84e7.jpg"
+                                }
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+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
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+                                504,
+                                217
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        182,
+                                        504,
+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": "Fix "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        182,
+                                        504,
+                                        217
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "U \\subsetneq V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        182,
+                                        504,
+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": " and pick "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        182,
+                                        504,
+                                        217
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z \\in V_F \\backslash U"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        182,
+                                        504,
+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": ". Let "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        182,
+                                        504,
+                                        217
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        182,
+                                        504,
+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": " be a simple path from "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        182,
+                                        504,
+                                        217
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        182,
+                                        504,
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+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        182,
+                                        504,
+                                        217
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "z"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        182,
+                                        504,
+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": " of the form "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        182,
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+                                        217
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P = \\{\\{w_0, w_1\\}, \\dots, \\{w_{k-1}, w_k\\}\\} \\subset E_F"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        182,
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+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        182,
+                                        504,
+                                        217
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{w_0, w_1\\} = \\{u, v\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        182,
+                                        504,
+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        182,
+                                        504,
+                                        217
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_k = z"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        182,
+                                        504,
+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": ". Define a mapping "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        182,
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+                                        217
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        182,
+                                        504,
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+                                    ],
+                                    "type": "text",
+                                    "content": " that takes "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        182,
+                                        504,
+                                        217
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g \\in S_G^U"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        182,
+                                        504,
+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": " as input and outputs a mapping "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h: V_F \\to V_{H(F)}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        182,
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+                                    ],
+                                    "type": "text",
+                                    "content": ":"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        157,
+                                        223,
+                                        470,
+                                        249
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "h (x) = \\left\\{ \\begin{array}{l l} g (x) & \\text {i f} [ g (x) ] _ {0} \\notin \\{w _ {1}, \\dots , w _ {k - 1} \\}, \\\\ ([ g (x) ] _ {0}, [ g (x) ] _ {1} \\bigtriangleup \\{w _ {i - 1}, w _ {i + 1} \\}) & \\text {i f} [ g (x) ] _ {0} = w _ {i}, i \\in [ k - 1 ], \\end{array} \\right.",
+                                    "image_path": "31c7ae4b37ed7a1029d64bdb60aec4a4b6e02d1e510b61d72817d1ef9a252898.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
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+                        255,
+                        504,
+                        289
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                255,
+                                504,
+                                289
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        255,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": "where we write "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        255,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g(x) = ([g(x)]_0, [g(x)]_1)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        255,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": ". We will prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        255,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h \\in S_H^U"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        255,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": ". Since we clearly have "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        255,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h(w) \\in U"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        255,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        255,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        255,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": ", it suffices to prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        255,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h \\in \\mathsf{Hom}(F, H(F))"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        255,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": ". Let "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        255,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{x, y\\} \\in E_F"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        255,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": " be any edge in "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "\\{[g(x)]_0, [g(y)]_0\\}"
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+                                    "type": "inline_equation",
+                                    "content": "\\{[g(x)]_0, [g(y)]_0\\} \\neq \\{\\{u, v\\}, \\{w_{k-1}, w_k\\}\\}"
+                                },
+                                {
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+                                    "type": "interline_equation",
+                                    "content": "\\{h (x), h (y) \\} = \\left\\{\\left([ g (x) ] _ {0}, [ g (x) ] _ {1}\\right), \\left([ g (y) ] _ {0}, [ g (y) ] _ {1} \\triangle \\left\\{w _ {i - 1}, w _ {i + 1} \\right\\}\\right) \\right\\}.",
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+                                    "type": "inline_equation",
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+                                    "content": "\\{h(x), h(y)\\} \\in E_{H(F)}"
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+                                    "type": "text",
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+                                            "content": "[g(x)]_0, [g(y)]_0 \\in \\{w_1, \\dots, w_{k-1}\\}"
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+                                            "content": ", the analysis is similar to the above one and we can still prove that "
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+                                            "type": "inline_equation",
+                                            "content": "\\{h(x), h(y)\\} \\in E_{G(F)}"
+                                        },
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+                                            "type": "text",
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+                                        },
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+                                            "type": "text",
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+                            "type": "text",
+                            "angle": 0,
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+                                            "type": "text",
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+                                        },
+                                        {
+                                            "bbox": [
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+                                            "type": "inline_equation",
+                                            "content": "\\{[g(x)]_0, [g(y)]_0\\} = \\{u, v\\}"
+                                        },
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+                                            "type": "text",
+                                            "content": ", without loss of generality assume "
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+                                            "bbox": [
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+                                            "type": "inline_equation",
+                                            "content": "w_0 = u = [g(x)]_0"
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+                                            "type": "text",
+                                            "content": " and "
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+                                            "bbox": [
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+                                            "type": "inline_equation",
+                                            "content": "w_1 = v = [g(y)]_0"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": ". We have"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 14
+                        }
+                    ],
+                    "sub_type": "text"
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+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\{g (x), g (y) \\} \\in E _ {G (F)} \\iff (u \\in [ g (y) ] _ {1}) \\leftrightarrow (v \\in [ g (x) ] _ {1}) \\\\ \\Longleftrightarrow (u \\notin [ g (y) ] _ {1} \\triangle \\{u, w _ {2} \\}) \\leftrightarrow (v \\in [ g (x) ] _ {1}) \\\\ \\Longleftrightarrow \\{h (x), h (y) \\} \\notin E _ {G (F)}. \\\\ \\end{array}",
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+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                153,
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "However, since "
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+                                        153,
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+                                    "type": "inline_equation",
+                                    "content": "\\{u,v\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        153,
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+                                    ],
+                                    "type": "text",
+                                    "content": " is twisted, we still have "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{h(x),h(y)\\} \\in E_{H(F)}"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                            ],
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "- Finally, we do not need to consider the case "
+                                },
+                                {
+                                    "bbox": [
+                                        146,
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+                                        588
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{[g(x)]_0, [g(y)]_0\\} = \\{w_{k-1}, w_k\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        146,
+                                        566,
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+                                    ],
+                                    "type": "text",
+                                    "content": " since "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_k = z \\notin U"
+                                },
+                                {
+                                    "bbox": [
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+                                        588
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
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+                        594,
+                        504,
+                        619
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        594,
+                                        504,
+                                        619
+                                    ],
+                                    "type": "text",
+                                    "content": "This proves that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        594,
+                                        504,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h \\in S_H^U"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        594,
+                                        504,
+                                        619
+                                    ],
+                                    "type": "text",
+                                    "content": ". Moreover, it is straightforward to see that the mapping "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        594,
+                                        504,
+                                        619
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma(g) = h"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        594,
+                                        504,
+                                        619
+                                    ],
+                                    "type": "text",
+                                    "content": " is a bijection from "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        594,
+                                        504,
+                                        619
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S_G^U"
+                                },
+                                {
+                                    "bbox": [
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+                                        619
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        594,
+                                        504,
+                                        619
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S_H^U"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        594,
+                                        504,
+                                        619
+                                    ],
+                                    "type": "text",
+                                    "content": ". We have thus proved that "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "|S_G^U| = |S_H^U|"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        594,
+                                        504,
+                                        619
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        623,
+                                        505,
+                                        646
+                                    ],
+                                    "type": "text",
+                                    "content": "Combining the above three items, we obtain "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        623,
+                                        505,
+                                        646
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{Hom}(F,G(F)) > \\mathsf{Hom}(F,H(F))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        623,
+                                        505,
+                                        646
+                                    ],
+                                    "type": "text",
+                                    "content": ", concluding the proof."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                },
+                {
+                    "bbox": [
+                        104,
+                        658,
+                        504,
+                        715
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                658,
+                                504,
+                                715
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        658,
+                                        504,
+                                        715
+                                    ],
+                                    "type": "text",
+                                    "content": "In the subsequent analysis, we will prove that for all model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        658,
+                                        504,
+                                        715
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        658,
+                                        504,
+                                        715
+                                    ],
+                                    "type": "text",
+                                    "content": " considered in Theorem 3.4 and any connected graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        658,
+                                        504,
+                                        715
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        658,
+                                        504,
+                                        715
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        658,
+                                        504,
+                                        715
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_{G(F)}^{M}(G(F)) = \\chi_{H(F)}^{M}(H(F))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        658,
+                                        504,
+                                        715
+                                    ],
+                                    "type": "text",
+                                    "content": " if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        658,
+                                        504,
+                                        715
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F \\notin S^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        658,
+                                        504,
+                                        715
+                                    ],
+                                    "type": "text",
+                                    "content": ". The proof is based on an important technique developed in Cai et al. (1992), called the pebble game. When restricting our analysis on Fürer graphs, the pebble game can be greatly simplified as shown in Fürer (2001); Zhang et al. (2023a). Below, we separately describe the corresponding pebble game for each model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        658,
+                                        504,
+                                        715
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        658,
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+                                        715
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 21
+                },
+                {
+                    "bbox": [
+                        105,
+                        720,
+                        356,
+                        732
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
+                                        720,
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+                                        732
+                                    ],
+                                    "type": "text",
+                                    "content": "We first define a key concept called the connected component."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 22
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
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+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
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+                    "angle": 0,
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "45"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 23
+                }
+            ],
+            "page_size": [
+                612,
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+            ],
+            "page_idx": 44
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        82,
+                        506,
+                        160
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                                160
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition C.48 (Connected components). Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F = (V_F, E_F)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": " be a connected graph and let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "U \\subset V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": " be a vertex set, called separation vertices. We say two edges "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{u, v\\}, \\{x, y\\} \\in E_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": " are in the same connected component if there is a simple path "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\{y_0, y_1\\}, \\dots, \\{y_{k-1}, y_k\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfying that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{y_0, y_1\\} = \\{u, v\\}, \\{y_{k-1}, y_k\\} = \\{x, y\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "y_i \\notin U"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i \\in [k-1]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": ". It is easy to see that the above relation between edges forms an equivalence relation. Therefore, we can define a partition over the edge set, denoted by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{C}\\mathbb{C}_F(U) = \\{P_i : i \\in [m]\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": " for some "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": ", where each "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_i \\subset E_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": " is called a connected component."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        169,
+                        506,
+                        237
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                169,
+                                506,
+                                237
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        169,
+                                        506,
+                                        237
+                                    ],
+                                    "type": "text",
+                                    "content": "We are now ready to describe the game rule. There are two players (named Alice and Bob), a graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        169,
+                                        506,
+                                        237
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        169,
+                                        506,
+                                        237
+                                    ],
+                                    "type": "text",
+                                    "content": ", and several pebbles. At the beginning, all pebbles lie outside the graph. Through the game process, some pebbles will be placed on the vertices of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        169,
+                                        506,
+                                        237
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        169,
+                                        506,
+                                        237
+                                    ],
+                                    "type": "text",
+                                    "content": " and thus separate the edges "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        169,
+                                        506,
+                                        237
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "E_{F}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        169,
+                                        506,
+                                        237
+                                    ],
+                                    "type": "text",
+                                    "content": " into connected components according to Definition C.48. In each game round, Alice updates the location of pebbles, while Bob maintains a subset of connected components, ensuring that the number of selected components is odd. There are three major types of operations:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        117,
+                        241,
+                        505,
+                        404
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 6,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                118,
+                                241,
+                                505,
+                                310
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        118,
+                                        241,
+                                        505,
+                                        310
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                118,
+                                                241,
+                                                505,
+                                                310
+                                            ],
+                                            "type": "text",
+                                            "content": "1. Add a pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                241,
+                                                505,
+                                                310
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathfrak{p}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                241,
+                                                505,
+                                                310
+                                            ],
+                                            "type": "text",
+                                            "content": ". Alice places a pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                241,
+                                                505,
+                                                310
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathfrak{p}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                241,
+                                                505,
+                                                310
+                                            ],
+                                            "type": "text",
+                                            "content": " (previously outside the graph) on some vertex of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                241,
+                                                505,
+                                                310
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                241,
+                                                505,
+                                                310
+                                            ],
+                                            "type": "text",
+                                            "content": ". If introducing this new pebble does not change the connected components, then Bob does nothing. Otherwise, there must be a connected component "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                241,
+                                                505,
+                                                310
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                241,
+                                                505,
+                                                310
+                                            ],
+                                            "type": "text",
+                                            "content": " separated by "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                241,
+                                                505,
+                                                310
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathfrak{p}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                241,
+                                                505,
+                                                310
+                                            ],
+                                            "type": "text",
+                                            "content": " into several components "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                241,
+                                                505,
+                                                310
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P = \\bigcup_{i \\in [m]} P_i"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                241,
+                                                505,
+                                                310
+                                            ],
+                                            "type": "text",
+                                            "content": " for some "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                241,
+                                                505,
+                                                310
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "m"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                241,
+                                                505,
+                                                310
+                                            ],
+                                            "type": "text",
+                                            "content": ". Bob will update his selected components as follows: if "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                241,
+                                                505,
+                                                310
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                241,
+                                                505,
+                                                310
+                                            ],
+                                            "type": "text",
+                                            "content": " is selected, then remove "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                241,
+                                                505,
+                                                310
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                241,
+                                                505,
+                                                310
+                                            ],
+                                            "type": "text",
+                                            "content": " and add a subset of connected components in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                241,
+                                                505,
+                                                310
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\{P_1, \\dots, P_m\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                241,
+                                                505,
+                                                310
+                                            ],
+                                            "type": "text",
+                                            "content": " while ensuring that the number of selected components in total is odd."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 3
+                        },
+                        {
+                            "bbox": [
+                                117,
+                                312,
+                                504,
+                                379
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        312,
+                                        504,
+                                        379
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                117,
+                                                312,
+                                                504,
+                                                379
+                                            ],
+                                            "type": "text",
+                                            "content": "2. Remove a pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                312,
+                                                504,
+                                                379
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathfrak{p}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                312,
+                                                504,
+                                                379
+                                            ],
+                                            "type": "text",
+                                            "content": ". Alice removes a pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                312,
+                                                504,
+                                                379
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathfrak{p}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                312,
+                                                504,
+                                                379
+                                            ],
+                                            "type": "text",
+                                            "content": " (previously on some vertex) outside the graph. If introducing this new pebble does not change the connected components, then Bob does nothing. Otherwise, there are multiple connected components "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                312,
+                                                504,
+                                                379
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_{1},\\dots ,P_{m}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                312,
+                                                504,
+                                                379
+                                            ],
+                                            "type": "text",
+                                            "content": " getting merged into a whole "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                312,
+                                                504,
+                                                379
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P = \\bigcup_{i\\in [m]}P_{i}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                312,
+                                                504,
+                                                379
+                                            ],
+                                            "type": "text",
+                                            "content": ". Bob will update his selected components by removing all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                312,
+                                                504,
+                                                379
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_{i}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                312,
+                                                504,
+                                                379
+                                            ],
+                                            "type": "text",
+                                            "content": ", "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                312,
+                                                504,
+                                                379
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "i\\in [m]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                312,
+                                                504,
+                                                379
+                                            ],
+                                            "type": "text",
+                                            "content": " (if selected) and optionally adding "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                312,
+                                                504,
+                                                379
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                312,
+                                                504,
+                                                379
+                                            ],
+                                            "type": "text",
+                                            "content": ", while ensuring that the number of selected components in total is odd."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 4
+                        },
+                        {
+                            "bbox": [
+                                117,
+                                381,
+                                504,
+                                404
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        381,
+                                        504,
+                                        404
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                117,
+                                                381,
+                                                504,
+                                                404
+                                            ],
+                                            "type": "text",
+                                            "content": "3. Swap two pebbles "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                381,
+                                                504,
+                                                404
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                381,
+                                                504,
+                                                404
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                381,
+                                                504,
+                                                404
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}^{\\prime}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                381,
+                                                504,
+                                                404
+                                            ],
+                                            "type": "text",
+                                            "content": ". Alice swaps the position of two pebbles "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                381,
+                                                504,
+                                                404
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                381,
+                                                504,
+                                                404
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                381,
+                                                504,
+                                                404
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}^{\\prime}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                381,
+                                                504,
+                                                404
+                                            ],
+                                            "type": "text",
+                                            "content": ". This operation does not change the connected components and thus Bob does nothing."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 5
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        104,
+                        409,
+                        505,
+                        443
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                409,
+                                505,
+                                443
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        409,
+                                        505,
+                                        443
+                                    ],
+                                    "type": "text",
+                                    "content": "At any time, if there is an edge "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        409,
+                                        505,
+                                        443
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{x,y\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        409,
+                                        505,
+                                        443
+                                    ],
+                                    "type": "text",
+                                    "content": " such that both endpoints hold pebbles and the connected component "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        409,
+                                        505,
+                                        443
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\{x,y\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        409,
+                                        505,
+                                        443
+                                    ],
+                                    "type": "text",
+                                    "content": " is selected by Bob, then Bob loses the game and Alice wins. If Alice cannot win through the game process, then Bob wins."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        104,
+                        449,
+                        506,
+                        505
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                449,
+                                506,
+                                505
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        506,
+                                        505
+                                    ],
+                                    "type": "text",
+                                    "content": "We now define the concrete pebble game for each model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        506,
+                                        505
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        506,
+                                        505
+                                    ],
+                                    "type": "text",
+                                    "content": " considered in this paper. In cases of Subgraph GNN, Local 2-GNN, Local 2-FGNN, and 2-FGNN, there are three pebbles "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        506,
+                                        505
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{p}_u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        506,
+                                        505
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        506,
+                                        505
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{p}_v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        506,
+                                        505
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        506,
+                                        505
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{p}_w"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        506,
+                                        505
+                                    ],
+                                    "type": "text",
+                                    "content": ". As described before, all pebbles lie outside the graph at the beginning. Alice first adds the pebble "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        506,
+                                        505
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{p}_u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        506,
+                                        505
+                                    ],
+                                    "type": "text",
+                                    "content": " (operation 1) and then adds the pebble "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        506,
+                                        505
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{p}_v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        506,
+                                        505
+                                    ],
+                                    "type": "text",
+                                    "content": " (operation 1). Next, the game cyclically executes the following process:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        121,
+                        509,
+                        436,
+                        521
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                121,
+                                509,
+                                436,
+                                521
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        509,
+                                        436,
+                                        521
+                                    ],
+                                    "type": "text",
+                                    "content": "- Subgraph GNN. Alice can choose either one of the following ways to play:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        144,
+                        523,
+                        503,
+                        557
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 12,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                144,
+                                523,
+                                460,
+                                534
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        144,
+                                        523,
+                                        460,
+                                        534
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                144,
+                                                523,
+                                                460,
+                                                534
+                                            ],
+                                            "type": "text",
+                                            "content": "- Remove the pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                523,
+                                                460,
+                                                534
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                523,
+                                                460,
+                                                534
+                                            ],
+                                            "type": "text",
+                                            "content": " (operation 2), and re-add the pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                523,
+                                                460,
+                                                534
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                523,
+                                                460,
+                                                534
+                                            ],
+                                            "type": "text",
+                                            "content": " (operation 1)."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 10
+                        },
+                        {
+                            "bbox": [
+                                144,
+                                535,
+                                503,
+                                557
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        144,
+                                        535,
+                                        503,
+                                        557
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                144,
+                                                535,
+                                                503,
+                                                557
+                                            ],
+                                            "type": "text",
+                                            "content": "- Add the pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                535,
+                                                503,
+                                                557
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                535,
+                                                503,
+                                                557
+                                            ],
+                                            "type": "text",
+                                            "content": " (operation 1) adjacent to the pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                535,
+                                                503,
+                                                557
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                535,
+                                                503,
+                                                557
+                                            ],
+                                            "type": "text",
+                                            "content": ", swap pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                535,
+                                                503,
+                                                557
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                535,
+                                                503,
+                                                557
+                                            ],
+                                            "type": "text",
+                                            "content": " with "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                535,
+                                                503,
+                                                557
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                535,
+                                                503,
+                                                557
+                                            ],
+                                            "type": "text",
+                                            "content": " (operation 3), and remove the pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                535,
+                                                503,
+                                                557
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                535,
+                                                503,
+                                                557
+                                            ],
+                                            "type": "text",
+                                            "content": " (operation 2)."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 11
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        121,
+                        559,
+                        427,
+                        571
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                121,
+                                559,
+                                427,
+                                571
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        559,
+                                        427,
+                                        571
+                                    ],
+                                    "type": "text",
+                                    "content": "- Local 2-GNN. Alice can choose either one of the following ways to play:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        144,
+                        573,
+                        503,
+                        643
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 18,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                144,
+                                573,
+                                461,
+                                585
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        144,
+                                        573,
+                                        461,
+                                        585
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                144,
+                                                573,
+                                                461,
+                                                585
+                                            ],
+                                            "type": "text",
+                                            "content": "- Remove the pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                573,
+                                                461,
+                                                585
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                573,
+                                                461,
+                                                585
+                                            ],
+                                            "type": "text",
+                                            "content": " (operation 2), and re-add the pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                573,
+                                                461,
+                                                585
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                573,
+                                                461,
+                                                585
+                                            ],
+                                            "type": "text",
+                                            "content": " (operation 1)."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 14
+                        },
+                        {
+                            "bbox": [
+                                144,
+                                586,
+                                460,
+                                597
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        144,
+                                        586,
+                                        460,
+                                        597
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                144,
+                                                586,
+                                                460,
+                                                597
+                                            ],
+                                            "type": "text",
+                                            "content": "- Remove the pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                586,
+                                                460,
+                                                597
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                586,
+                                                460,
+                                                597
+                                            ],
+                                            "type": "text",
+                                            "content": " (operation 2), and re-add the pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                586,
+                                                460,
+                                                597
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                586,
+                                                460,
+                                                597
+                                            ],
+                                            "type": "text",
+                                            "content": " (operation 1)."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 15
+                        },
+                        {
+                            "bbox": [
+                                144,
+                                597,
+                                503,
+                                620
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        144,
+                                        597,
+                                        503,
+                                        620
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                144,
+                                                597,
+                                                503,
+                                                620
+                                            ],
+                                            "type": "text",
+                                            "content": "- Add the pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                597,
+                                                503,
+                                                620
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                597,
+                                                503,
+                                                620
+                                            ],
+                                            "type": "text",
+                                            "content": " (operation 1) adjacent to the pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                597,
+                                                503,
+                                                620
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                597,
+                                                503,
+                                                620
+                                            ],
+                                            "type": "text",
+                                            "content": ", swap pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                597,
+                                                503,
+                                                620
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                597,
+                                                503,
+                                                620
+                                            ],
+                                            "type": "text",
+                                            "content": " with "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                597,
+                                                503,
+                                                620
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                597,
+                                                503,
+                                                620
+                                            ],
+                                            "type": "text",
+                                            "content": " (operation 3), and remove the pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                597,
+                                                503,
+                                                620
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                597,
+                                                503,
+                                                620
+                                            ],
+                                            "type": "text",
+                                            "content": " (operation 2)."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 16
+                        },
+                        {
+                            "bbox": [
+                                144,
+                                621,
+                                503,
+                                643
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        144,
+                                        621,
+                                        503,
+                                        643
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                144,
+                                                621,
+                                                503,
+                                                643
+                                            ],
+                                            "type": "text",
+                                            "content": "- Add the pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                621,
+                                                503,
+                                                643
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                621,
+                                                503,
+                                                643
+                                            ],
+                                            "type": "text",
+                                            "content": " (operation 1) adjacent to the pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                621,
+                                                503,
+                                                643
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                621,
+                                                503,
+                                                643
+                                            ],
+                                            "type": "text",
+                                            "content": ", swap pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                621,
+                                                503,
+                                                643
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                621,
+                                                503,
+                                                643
+                                            ],
+                                            "type": "text",
+                                            "content": " with "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                621,
+                                                503,
+                                                643
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                621,
+                                                503,
+                                                643
+                                            ],
+                                            "type": "text",
+                                            "content": " (operation 3), and remove the pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                621,
+                                                503,
+                                                643
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                621,
+                                                503,
+                                                643
+                                            ],
+                                            "type": "text",
+                                            "content": " (operation 2)."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 17
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        121,
+                        645,
+                        433,
+                        657
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                121,
+                                645,
+                                433,
+                                657
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        645,
+                                        433,
+                                        657
+                                    ],
+                                    "type": "text",
+                                    "content": "- Local 2-FGNN. Alice can choose either one of the following ways to play:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                },
+                {
+                    "bbox": [
+                        144,
+                        658,
+                        503,
+                        705
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 23,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                144,
+                                658,
+                                461,
+                                670
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        144,
+                                        658,
+                                        461,
+                                        670
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                144,
+                                                658,
+                                                461,
+                                                670
+                                            ],
+                                            "type": "text",
+                                            "content": "- Remove the pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                658,
+                                                461,
+                                                670
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                658,
+                                                461,
+                                                670
+                                            ],
+                                            "type": "text",
+                                            "content": " (operation 2), and re-add the pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                658,
+                                                461,
+                                                670
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                658,
+                                                461,
+                                                670
+                                            ],
+                                            "type": "text",
+                                            "content": " (operation 1)."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 20
+                        },
+                        {
+                            "bbox": [
+                                144,
+                                671,
+                                460,
+                                681
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        144,
+                                        671,
+                                        460,
+                                        681
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                144,
+                                                671,
+                                                460,
+                                                681
+                                            ],
+                                            "type": "text",
+                                            "content": "- Remove the pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                671,
+                                                460,
+                                                681
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathfrak{p}_v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                671,
+                                                460,
+                                                681
+                                            ],
+                                            "type": "text",
+                                            "content": " (operation 2), and re-add the pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                671,
+                                                460,
+                                                681
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathfrak{p}_v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                671,
+                                                460,
+                                                681
+                                            ],
+                                            "type": "text",
+                                            "content": " (operation 1)."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 21
+                        },
+                        {
+                            "bbox": [
+                                144,
+                                683,
+                                503,
+                                705
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        144,
+                                        683,
+                                        503,
+                                        705
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                144,
+                                                683,
+                                                503,
+                                                705
+                                            ],
+                                            "type": "text",
+                                            "content": "- Add the pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                683,
+                                                503,
+                                                705
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                683,
+                                                503,
+                                                705
+                                            ],
+                                            "type": "text",
+                                            "content": " (operation 1) adjacent to either pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                683,
+                                                503,
+                                                705
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                683,
+                                                503,
+                                                705
+                                            ],
+                                            "type": "text",
+                                            "content": " or pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                683,
+                                                503,
+                                                705
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                683,
+                                                503,
+                                                705
+                                            ],
+                                            "type": "text",
+                                            "content": ", swap pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                683,
+                                                503,
+                                                705
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                683,
+                                                503,
+                                                705
+                                            ],
+                                            "type": "text",
+                                            "content": " with any adjacent pebble (operation 3), and remove the pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                683,
+                                                503,
+                                                705
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                144,
+                                                683,
+                                                503,
+                                                705
+                                            ],
+                                            "type": "text",
+                                            "content": " (operation 2)."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 22
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        121,
+                        708,
+                        503,
+                        730
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                121,
+                                708,
+                                503,
+                                730
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        708,
+                                        503,
+                                        730
+                                    ],
+                                    "type": "text",
+                                    "content": "- 2-FGNN. Alice adds the pebble "
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        708,
+                                        503,
+                                        730
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{p}_w"
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        708,
+                                        503,
+                                        730
+                                    ],
+                                    "type": "text",
+                                    "content": " (operation 1), swap the pebbles "
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        708,
+                                        503,
+                                        730
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{p}_w"
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        708,
+                                        503,
+                                        730
+                                    ],
+                                    "type": "text",
+                                    "content": " with either "
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        708,
+                                        503,
+                                        730
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{p}_u"
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        708,
+                                        503,
+                                        730
+                                    ],
+                                    "type": "text",
+                                    "content": " or "
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        708,
+                                        503,
+                                        730
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{p}_v"
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        708,
+                                        503,
+                                        730
+                                    ],
+                                    "type": "text",
+                                    "content": " (operation 3), and remove the pebble "
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        708,
+                                        503,
+                                        730
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{p}_w"
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        708,
+                                        503,
+                                        730
+                                    ],
+                                    "type": "text",
+                                    "content": " (operation 2)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 24
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
+                        751,
+                        312,
+                        760
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                299,
+                                751,
+                                312,
+                                760
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        299,
+                                        751,
+                                        312,
+                                        760
+                                    ],
+                                    "type": "text",
+                                    "content": "46"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 25
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 45
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        82,
+                        504,
+                        108
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                82,
+                                504,
+                                108
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        108
+                                    ],
+                                    "type": "text",
+                                    "content": "Proposition C.49. For any model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        108
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        108
+                                    ],
+                                    "type": "text",
+                                    "content": " and any graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        108
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        108
+                                    ],
+                                    "type": "text",
+                                    "content": ", if Alice cannot win the pebble game associated to model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        108
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        108
+                                    ],
+                                    "type": "text",
+                                    "content": " on graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        108
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        108
+                                    ],
+                                    "type": "text",
+                                    "content": ", then "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        108
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_{G(F)}^{M}(G(F)) = \\chi_{H(F)}^{M}(H(F))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        108
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        119,
+                        506,
+                        231
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                119,
+                                506,
+                                231
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Note that the pebble game does not depend on the vertex labels "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "text",
+                                    "content": " in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "text",
+                                    "content": ". We first assume that the vertex labels are all different, i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell_F(x) \\neq \\ell_F(y)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x, y \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, the labels of vertices "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\xi \\in V_{G(F)}, \\eta \\in V_{H(F)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "text",
+                                    "content": " are the same iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\xi"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\eta"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "text",
+                                    "content": " belong to the same meta vertex, i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\xi, \\eta \\in \\mathsf{Meta}_F(x)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "text",
+                                    "content": " for some "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "x \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "text",
+                                    "content": ". Based on Proposition C.9, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_{G(F)}^M(G(F)) = \\chi_{H(F)}^M(H(F))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\chi}_{G(F)}^M(G(F)) = \\tilde{\\chi}_{H(F)}^M(H(F))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\chi}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "text",
+                                    "content": " is defined in Equations (7) to (10). In this case, the pebble games above exactly correspond to the aggregation formulas in Equations (7) to (10), and we can invoke the results in Zhang et al. (2023a, Theorem I.17) to show that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\chi}_{G(F)}^M(G(F)) = \\tilde{\\chi}_{H(F)}^M(H(F))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "text",
+                                    "content": " (note that we do not need to consider the augmented Fürer graphs defined in their paper based on our label assignment). Therefore, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_{G(F)}^M(G(F)) = \\chi_{H(F)}^M(H(F))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        119,
+                                        506,
+                                        231
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        235,
+                        505,
+                        271
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                235,
+                                505,
+                                271
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        235,
+                                        505,
+                                        271
+                                    ],
+                                    "type": "text",
+                                    "content": "We next consider the general case when multiple vertices can have the same label in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        235,
+                                        505,
+                                        271
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        235,
+                                        505,
+                                        271
+                                    ],
+                                    "type": "text",
+                                    "content": ". However, this can only make it harder to distinguish between "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        235,
+                                        505,
+                                        271
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G(F)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        235,
+                                        505,
+                                        271
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        235,
+                                        505,
+                                        271
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H(F)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        235,
+                                        505,
+                                        271
+                                    ],
+                                    "type": "text",
+                                    "content": ", and we clearly have "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        235,
+                                        505,
+                                        271
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_{G(F)}^{M}(G(F)) = \\chi_{H(F)}^{M}(H(F))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        235,
+                                        505,
+                                        271
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        282,
+                        504,
+                        316
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                282,
+                                504,
+                                316
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        282,
+                                        504,
+                                        316
+                                    ],
+                                    "type": "text",
+                                    "content": "Based on the above result, in the subsequent proof we will analyze the players' strategy in the pebble game. Surprisingly, it turns out that given a graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        282,
+                                        504,
+                                        316
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        282,
+                                        504,
+                                        316
+                                    ],
+                                    "type": "text",
+                                    "content": ", if Alice can win the game, then her strategy can be described using the tree decomposition of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        282,
+                                        504,
+                                        316
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        282,
+                                        504,
+                                        316
+                                    ],
+                                    "type": "text",
+                                    "content": " defined in Definition C.6."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        321,
+                        505,
+                        421
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                321,
+                                505,
+                                421
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        321,
+                                        505,
+                                        421
+                                    ],
+                                    "type": "text",
+                                    "content": "To illustrate this point, we need the concept of game state graph. Given a graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        321,
+                                        505,
+                                        421
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        321,
+                                        505,
+                                        421
+                                    ],
+                                    "type": "text",
+                                    "content": ", a game state is a three-tuple "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        321,
+                                        505,
+                                        421
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u, v, Q)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        321,
+                                        505,
+                                        421
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        321,
+                                        505,
+                                        421
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v \\in V_F \\cup \\{\\emptyset\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        321,
+                                        505,
+                                        421
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        321,
+                                        505,
+                                        421
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Q \\subset \\mathbb{C}\\mathbb{C}_F(\\{u, v\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        321,
+                                        505,
+                                        421
+                                    ],
+                                    "type": "text",
+                                    "content": ", denoting the vertex that holds pebble "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        321,
+                                        505,
+                                        421
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathfrak{p}_u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        321,
+                                        505,
+                                        421
+                                    ],
+                                    "type": "text",
+                                    "content": ", the vertex that holds pebble "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        321,
+                                        505,
+                                        421
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathfrak{p}_v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        321,
+                                        505,
+                                        421
+                                    ],
+                                    "type": "text",
+                                    "content": ", and a subset of connected component selected by Bob, respectively. Here, the symbol "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        321,
+                                        505,
+                                        421
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\emptyset"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        321,
+                                        505,
+                                        421
+                                    ],
+                                    "type": "text",
+                                    "content": " means that a pebble is left outside the graph. One can see that after any round, whether Alice can win the remaining game purely depends on this tuple. Now fix Alice's strategy. In each round, each game state will be transited to a finite number of states depending on how Bob plays, and all states and transitions form a directed graph, which we call the game state graph. The state "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        321,
+                                        505,
+                                        421
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u, v, Q)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        321,
+                                        505,
+                                        421
+                                    ],
+                                    "type": "text",
+                                    "content": " is called a terminal state if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        321,
+                                        505,
+                                        421
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\min_{P \\in Q} |P| = 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        321,
+                                        505,
+                                        421
+                                    ],
+                                    "type": "text",
+                                    "content": ". It is straightforward to see that Alice wins the game at any terminal state, as stated in the following result:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        423,
+                        505,
+                        468
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                423,
+                                505,
+                                468
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        423,
+                                        505,
+                                        468
+                                    ],
+                                    "type": "text",
+                                    "content": "Proposition C.50. Given graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        423,
+                                        505,
+                                        468
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        423,
+                                        505,
+                                        468
+                                    ],
+                                    "type": "text",
+                                    "content": " and model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        423,
+                                        505,
+                                        468
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        423,
+                                        505,
+                                        468
+                                    ],
+                                    "type": "text",
+                                    "content": ", let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        423,
+                                        505,
+                                        468
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^S"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        423,
+                                        505,
+                                        468
+                                    ],
+                                    "type": "text",
+                                    "content": " be the game state graph defined above corresponding to an Alice's strategy. Then, Alice can win the game if there is an integer "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        423,
+                                        505,
+                                        468
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        423,
+                                        505,
+                                        468
+                                    ],
+                                    "type": "text",
+                                    "content": " such that any path in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        423,
+                                        505,
+                                        468
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^S"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        423,
+                                        505,
+                                        468
+                                    ],
+                                    "type": "text",
+                                    "content": " of length "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        423,
+                                        505,
+                                        468
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        423,
+                                        505,
+                                        468
+                                    ],
+                                    "type": "text",
+                                    "content": " starting from the initial state "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        423,
+                                        505,
+                                        468
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\emptyset, \\emptyset, \\{E_F\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        423,
+                                        505,
+                                        468
+                                    ],
+                                    "type": "text",
+                                    "content": " goes through a terminal state."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        480,
+                        506,
+                        581
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                480,
+                                506,
+                                581
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. It suffices to prove that, at any terminal state, Alice can win the game. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u,v,Q)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": " be a terminal state with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{x,y\\} \\in Q"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": ". If "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{u,v\\} = \\{x,y\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": ", Alice already wins. If "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\notin \\{x,y\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v \\in \\{x,y\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": ", Alice also wins in the next round since she can add the pebble "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathfrak{p}_w"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": " on some vertex "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": " adjacent to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathfrak{p}_v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{v,w\\} = \\{x,y\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": ". This is a valid game rule for all model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": ". Next, if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\in \\{x,y\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v \\notin \\{x,y\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": ", Alice also wins in the next round since she can first remove the pebble "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathfrak{p}_v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": " and then place it on the unique vertex in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{x,y\\} \\backslash \\{u\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": ". This is also a valid game rule for all model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": ". Note the removing "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathfrak{p}_v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": " does not merge the connected component "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{\\{x,y\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": " since "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v \\notin \\{x,y\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": ". Finally, if both "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\notin \\{x,y\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v \\notin \\{x,y\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        480,
+                                        506,
+                                        581
+                                    ],
+                                    "type": "text",
+                                    "content": " only has one edge and Alice can clearly win. Combining these cases, we conclude that Alice can always win."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        104,
+                        593,
+                        504,
+                        638
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                593,
+                                504,
+                                638
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        504,
+                                        638
+                                    ],
+                                    "type": "text",
+                                    "content": "Based on the above proposition, a game state "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        504,
+                                        638
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u,v,Q)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        504,
+                                        638
+                                    ],
+                                    "type": "text",
+                                    "content": " is called unreachable if any path starting from the initial state "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        504,
+                                        638
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\emptyset ,\\emptyset ,\\{E_F\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        504,
+                                        638
+                                    ],
+                                    "type": "text",
+                                    "content": " and ending at "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        504,
+                                        638
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u,v,Q)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        593,
+                                        504,
+                                        638
+                                    ],
+                                    "type": "text",
+                                    "content": " goes through some terminal state. We do not need to consider unreachable states since Alice always wins before reaching it. We next introduce an important technical concept:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        640,
+                        504,
+                        676
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                640,
+                                504,
+                                676
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        640,
+                                        504,
+                                        676
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition C.51. Given a game state graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        640,
+                                        504,
+                                        676
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^{\\mathsf{S}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        640,
+                                        504,
+                                        676
+                                    ],
+                                    "type": "text",
+                                    "content": ", a state "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        640,
+                                        504,
+                                        676
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u, v, \\{P\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        640,
+                                        504,
+                                        676
+                                    ],
+                                    "type": "text",
+                                    "content": " is termed as \"contracted\" if for any transition "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        640,
+                                        504,
+                                        676
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "((u, v, \\{P\\}), (u', v', \\{P'\\})) \\in E_{G^{\\mathsf{s}}}, P' \\subset P"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        640,
+                                        504,
+                                        676
+                                    ],
+                                    "type": "text",
+                                    "content": ". It is called strictly contracted if for any transition "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        640,
+                                        504,
+                                        676
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "((u, v, \\{P\\}), (u', v', \\{P'\\})) \\in E_{G^{\\mathsf{s}}}, P' \\subsetneq P"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        640,
+                                        504,
+                                        676
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        105,
+                        683,
+                        226,
+                        696
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                683,
+                                226,
+                                696
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        683,
+                                        226,
+                                        696
+                                    ],
+                                    "type": "text",
+                                    "content": "We have the following result:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        698,
+                        504,
+                        733
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                698,
+                                504,
+                                733
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        698,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma C.52. For any model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        698,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\text{Sub}, \\text{L}, \\text{LF}, \\text{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        698,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " and any graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        698,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        698,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ", if Alice can win the pebble game associated to model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        698,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        698,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " on graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        698,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        698,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": ", then there exists a game state graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        698,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^S"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        698,
+                                        504,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": " corresponding to a winning strategy of Alice such that any reachable and non-terminal state is strictly contracted."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
+                        750,
+                        311,
+                        760
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                299,
+                                750,
+                                311,
+                                760
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        299,
+                                        750,
+                                        311,
+                                        760
+                                    ],
+                                    "type": "text",
+                                    "content": "47"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 46
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        82,
+                        506,
+                        183
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                82,
+                                506,
+                                183
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        183
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. We first prove that there is a strategy for Alice such that any reachable and non-terminal state is contracted. We will assume that for any reachable and non-terminal state "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        183
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u, v, \\{P\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        183
+                                    ],
+                                    "type": "text",
+                                    "content": ", if there is only one transition of the form "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        183
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "((u, v, \\{P\\}),(u', v', \\{P'\\}))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        183
+                                    ],
+                                    "type": "text",
+                                    "content": ", then "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        183
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P \\notin P'"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        183
+                                    ],
+                                    "type": "text",
+                                    "content": ", since making the connected component larger should be a bad strategy. (Otherwise, it is straightforward to construct a new strategy for Alice satisfying the assumption.) Now suppose that there exists a reachable state "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        183
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u, v, \\{P\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        183
+                                    ],
+                                    "type": "text",
+                                    "content": " that is not contracted. Note that the game state graph induced by all reachable states is a Directed Acyclic Graph (DAG), so we can choose the state "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        183
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u, v, \\{P\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        183
+                                    ],
+                                    "type": "text",
+                                    "content": " such that any path from the initial state "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        183
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\emptyset, \\emptyset, \\{E_F\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        183
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        183
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u, v, \\{P\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        183
+                                    ],
+                                    "type": "text",
+                                    "content": " does not pass any intermediate state that is not contracted. Below, we will construct a new strategy that removes the state "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        183
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u, v, \\{P\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        506,
+                                        183
+                                    ],
+                                    "type": "text",
+                                    "content": " (making it unreachable)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        186,
+                        506,
+                        255
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                186,
+                                506,
+                                255
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        255
+                                    ],
+                                    "type": "text",
+                                    "content": "Note that we clearly have "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        255
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\neq \\emptyset"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        255
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        255
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v \\neq \\emptyset"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        255
+                                    ],
+                                    "type": "text",
+                                    "content": ". Without loss of generality, assume that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        255
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "((u,v,\\{P\\}),(u,v',\\{P'\\}))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        255
+                                    ],
+                                    "type": "text",
+                                    "content": " is a transition such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        255
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P' \\notin P"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        255
+                                    ],
+                                    "type": "text",
+                                    "content": " (the case of transition "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        255
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "((u,v,\\{P\\}),(u',v,\\{P'\\}))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        255
+                                    ],
+                                    "type": "text",
+                                    "content": " is the same by symmetry). Moreover, we can assume that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        255
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P \\notin \\mathbb{C}\\mathbb{C}_F(\\{u\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        255
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        255
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P \\notin \\mathbb{C}\\mathbb{C}_F(\\{v\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        255
+                                    ],
+                                    "type": "text",
+                                    "content": " (in other words, both "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        255
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        255
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        255
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        255
+                                    ],
+                                    "type": "text",
+                                    "content": " are at the boundary of the connected component "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        255
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        255
+                                    ],
+                                    "type": "text",
+                                    "content": "). It would be easier to analyze the case where either "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        255
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        255
+                                    ],
+                                    "type": "text",
+                                    "content": " or "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        255
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        186,
+                                        506,
+                                        255
+                                    ],
+                                    "type": "text",
+                                    "content": " are not at the boundary. We separately consider the following cases:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        118,
+                        258,
+                        505,
+                        304
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                118,
+                                258,
+                                505,
+                                304
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        118,
+                                        258,
+                                        505,
+                                        304
+                                    ],
+                                    "type": "text",
+                                    "content": "1. The transition "
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        258,
+                                        505,
+                                        304
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "((u,v,\\{P\\}),(u,v^{\\prime},\\{P^{\\prime}\\}))"
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        258,
+                                        505,
+                                        304
+                                    ],
+                                    "type": "text",
+                                    "content": " corresponds to Alice removing pebble "
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        258,
+                                        505,
+                                        304
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathfrak{p}_v"
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        258,
+                                        505,
+                                        304
+                                    ],
+                                    "type": "text",
+                                    "content": " and placing it on "
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        258,
+                                        505,
+                                        304
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v^{\\prime}"
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        258,
+                                        505,
+                                        304
+                                    ],
+                                    "type": "text",
+                                    "content": ". Let "
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        258,
+                                        505,
+                                        304
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u_0,v_0,Q_0),\\dots ,(u_T,v_T,Q_T)"
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        258,
+                                        505,
+                                        304
+                                    ],
+                                    "type": "text",
+                                    "content": " be any path from the initial state "
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        258,
+                                        505,
+                                        304
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\emptyset ,\\emptyset ,\\{E_F\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        258,
+                                        505,
+                                        304
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        258,
+                                        505,
+                                        304
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u,v,\\{P\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        258,
+                                        505,
+                                        304
+                                    ],
+                                    "type": "text",
+                                    "content": ", and let "
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        258,
+                                        505,
+                                        304
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t\\leq T"
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        258,
+                                        505,
+                                        304
+                                    ],
+                                    "type": "text",
+                                    "content": " be the maximal number such that "
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        258,
+                                        505,
+                                        304
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v_{t}\\neq v"
+                                },
+                                {
+                                    "bbox": [
+                                        118,
+                                        258,
+                                        505,
+                                        304
+                                    ],
+                                    "type": "text",
+                                    "content": ". We will construct a new strategy for Alice as follows."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        146,
+                        308,
+                        506,
+                        397
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 6,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                146,
+                                308,
+                                504,
+                                334
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        146,
+                                        308,
+                                        504,
+                                        334
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                146,
+                                                308,
+                                                504,
+                                                334
+                                            ],
+                                            "type": "text",
+                                            "content": "- At state "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                308,
+                                                504,
+                                                334
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(u_t, v_t, Q_t)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                308,
+                                                504,
+                                                334
+                                            ],
+                                            "type": "text",
+                                            "content": ", she removes pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                308,
+                                                504,
+                                                334
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathfrak{p}_v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                308,
+                                                504,
+                                                334
+                                            ],
+                                            "type": "text",
+                                            "content": " and places it on "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                308,
+                                                504,
+                                                334
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "v'"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                308,
+                                                504,
+                                                334
+                                            ],
+                                            "type": "text",
+                                            "content": ", yielding (possibly multiple) states of the form "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                308,
+                                                504,
+                                                334
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(u_t, v', \\tilde{Q}_t)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                308,
+                                                504,
+                                                334
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 4
+                        },
+                        {
+                            "bbox": [
+                                146,
+                                338,
+                                506,
+                                397
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        146,
+                                        338,
+                                        506,
+                                        397
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "text",
+                                            "content": "- For all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t < \\tilde{t} < T"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "text",
+                                            "content": ", we have "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "v_{t} = v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "text",
+                                            "content": ". For each "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tilde{t}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "text",
+                                            "content": " and possible "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tilde{Q}_{\\tilde{t}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "text",
+                                            "content": ", she will apply the original strategy at "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(u_{\\tilde{t}}, v_{\\tilde{t}}, Q_{\\tilde{t}})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "text",
+                                            "content": " to the state "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(u_{\\tilde{t}}, v', \\tilde{Q}_{\\tilde{t}})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "text",
+                                            "content": ", namely, placing pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "text",
+                                            "content": " on vertex "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u_{\\tilde{t} + 1}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "text",
+                                            "content": ", swapping pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "text",
+                                            "content": " with "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "text",
+                                            "content": ", and leaving "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "text",
+                                            "content": " outside the graph. (Note that she cannot remove pebble "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{p}_u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "text",
+                                            "content": " first; otherwise the state "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(u_{\\tilde{t}}, v_{\\tilde{t}}, Q_{\\tilde{t}})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "text",
+                                            "content": " will not be contracted since "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "text",
+                                            "content": " is at the boundary of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "Q_{\\tilde{t}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                146,
+                                                338,
+                                                506,
+                                                397
+                                            ],
+                                            "type": "text",
+                                            "content": ".)"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 5
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        129,
+                        401,
+                        504,
+                        430
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                401,
+                                504,
+                                430
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        401,
+                                        504,
+                                        430
+                                    ],
+                                    "type": "text",
+                                    "content": "It follows that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        401,
+                                        504,
+                                        430
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "((u_{\\tilde{t}},v^{\\prime},\\tilde{Q}_{\\tilde{t}}),(u_{\\tilde{t} +1},v^{\\prime},\\tilde{Q}_{\\tilde{t} +1}))"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        401,
+                                        504,
+                                        430
+                                    ],
+                                    "type": "text",
+                                    "content": " is a transition of the new strategy, "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        401,
+                                        504,
+                                        430
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u_{\\tilde{t}},v^{\\prime},\\tilde{Q}_{\\tilde{t}})"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        401,
+                                        504,
+                                        430
+                                    ],
+                                    "type": "text",
+                                    "content": " is contracted, and all states of the form "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        401,
+                                        504,
+                                        430
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u,v^{\\prime},\\tilde{Q}_{\\tilde{T}})"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        401,
+                                        504,
+                                        430
+                                    ],
+                                    "type": "text",
+                                    "content": " is reachable in the original strategy."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        129,
+                        433,
+                        505,
+                        513
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                433,
+                                505,
+                                513
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        433,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "text",
+                                    "content": "We can repeat the above procedure for all paths from the initial state "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        433,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\emptyset, \\emptyset, \\{E_F\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        433,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "text",
+                                    "content": " to state "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        433,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u, v, \\{P\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        433,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, in the new strategy "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        433,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u, v, \\{P\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        433,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "text",
+                                    "content": " will be unreachable. However, the state "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        433,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u_t, v_t, Q_t)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        433,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "text",
+                                    "content": " now may violate the condition in Lemma C.52. Note that the assumption in the first paragraph is still satisfied for state "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        433,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u_t, v_t, Q_t)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        433,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "text",
+                                    "content": ". In this case, we can recursively apply the above procedure for state "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        433,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u_t, v_t, Q_t)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        433,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "text",
+                                    "content": ". Note that the procedure will only repeat a finite number of times, as the length of the path from the initial state to the state "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        433,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u_t, v_t, Q_t)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        433,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "text",
+                                    "content": " is strictly less than the length of the path from the initial state to the state "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        433,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u, v, \\{P\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        433,
+                                        505,
+                                        513
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        117,
+                        515,
+                        505,
+                        595
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                117,
+                                515,
+                                505,
+                                595
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "text",
+                                    "content": "2. The transition "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "((u,v,\\{P\\}),(u,v^{\\prime},\\{P^{\\prime}\\}))"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "text",
+                                    "content": " corresponds to Alice placing pebble "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathfrak{p}_w"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "text",
+                                    "content": " on "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v^{\\prime}"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "text",
+                                    "content": ", swapping "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathfrak{p}_v"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathfrak{p}_w"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "text",
+                                    "content": ", and removing pebble "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathfrak{p}_w"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "text",
+                                    "content": ". In this case, we must have "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P\\notin \\mathbb{C}\\mathbb{C}_F(\\{u,v,v'\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "text",
+                                    "content": " (i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v^{\\prime}"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "text",
+                                    "content": " further splits the component "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "text",
+                                    "content": "). Otherwise, the state "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u,v,\\{P\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "text",
+                                    "content": " must have only one transition and "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P\\subset P^{\\prime}"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "text",
+                                    "content": ", a contradiction to the first paragraph. Since "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P\\notin \\mathbb{C}\\mathbb{C}_F(\\{u,v,v'\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "text",
+                                    "content": ", it is thus easy to see that, if Alice changes her strategy by just removing pebble "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathfrak{p}_v"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "text",
+                                    "content": " and placing it on vertex "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v^{\\prime}"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "text",
+                                    "content": ", all transitions starting from "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u,v,\\{P\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        515,
+                                        505,
+                                        595
+                                    ],
+                                    "type": "text",
+                                    "content": " does not change. Therefore, we can just invoke the previous item to construct a desired strategy."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
+                        598,
+                        506,
+                        656
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                598,
+                                506,
+                                656
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        506,
+                                        656
+                                    ],
+                                    "type": "text",
+                                    "content": "Combining the two cases, we conclude that there is a strategy for Alice such that any reachable and non-terminal state is contracted. We next prove that any reachable and non-terminal state can be strictly contracted. Assume the result does not hold and the state "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        506,
+                                        656
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u,v,\\{P\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        506,
+                                        656
+                                    ],
+                                    "type": "text",
+                                    "content": " is reachable, non-terminal, but not strictly contracted. Then, there is a transition "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        506,
+                                        656
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "((u,v,\\{P\\}),(u',v',\\{P\\})) \\in E_{G^s}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        598,
+                                        506,
+                                        656
+                                    ],
+                                    "type": "text",
+                                    "content": ". Consider the following two cases:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        117,
+                        659,
+                        505,
+                        734
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 13,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                117,
+                                659,
+                                505,
+                                693
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        659,
+                                        505,
+                                        693
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                117,
+                                                659,
+                                                505,
+                                                693
+                                            ],
+                                            "type": "text",
+                                            "content": "1. "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                659,
+                                                505,
+                                                693
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P \\notin \\mathbb{C}C_F(\\{u\\})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                659,
+                                                505,
+                                                693
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                659,
+                                                505,
+                                                693
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P \\notin \\mathbb{C}C_F(\\{v\\})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                659,
+                                                505,
+                                                693
+                                            ],
+                                            "type": "text",
+                                            "content": " (i.e., both "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                659,
+                                                505,
+                                                693
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                659,
+                                                505,
+                                                693
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                659,
+                                                505,
+                                                693
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                659,
+                                                505,
+                                                693
+                                            ],
+                                            "type": "text",
+                                            "content": " are at the boundary of the connected component "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                659,
+                                                505,
+                                                693
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                659,
+                                                505,
+                                                693
+                                            ],
+                                            "type": "text",
+                                            "content": "). It follows that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                659,
+                                                505,
+                                                693
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u' = u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                659,
+                                                505,
+                                                693
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                659,
+                                                505,
+                                                693
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "v' = v"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                659,
+                                                505,
+                                                693
+                                            ],
+                                            "type": "text",
+                                            "content": ". This implies that the game state graph is not acyclic, a contradiction."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 11
+                        },
+                        {
+                            "bbox": [
+                                117,
+                                697,
+                                505,
+                                734
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        697,
+                                        505,
+                                        734
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                117,
+                                                697,
+                                                505,
+                                                734
+                                            ],
+                                            "type": "text",
+                                            "content": "2. "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                697,
+                                                505,
+                                                734
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P \\in \\mathbb{C}\\mathbb{C}_F(\\{u\\})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                697,
+                                                505,
+                                                734
+                                            ],
+                                            "type": "text",
+                                            "content": " or "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                697,
+                                                505,
+                                                734
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P \\in \\mathbb{C}\\mathbb{C}_F(\\{v\\})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                697,
+                                                505,
+                                                734
+                                            ],
+                                            "type": "text",
+                                            "content": ". Without loss of generality, assume that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                697,
+                                                505,
+                                                734
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P \\in \\mathbb{C}\\mathbb{C}_F(\\{u\\})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                697,
+                                                505,
+                                                734
+                                            ],
+                                            "type": "text",
+                                            "content": ". Since "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                697,
+                                                505,
+                                                734
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(u,v,\\{P\\})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                697,
+                                                505,
+                                                734
+                                            ],
+                                            "type": "text",
+                                            "content": " is non-terminal and not strictly contracted, there is a reachable and non-terminal state "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                697,
+                                                505,
+                                                734
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(\\tilde{u},\\tilde{v},\\{P\\})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                697,
+                                                505,
+                                                734
+                                            ],
+                                            "type": "text",
+                                            "content": " with either "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                697,
+                                                505,
+                                                734
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tilde{u} = u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                697,
+                                                505,
+                                                734
+                                            ],
+                                            "type": "text",
+                                            "content": " or "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                697,
+                                                505,
+                                                734
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tilde{v} = u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                697,
+                                                505,
+                                                734
+                                            ],
+                                            "type": "text",
+                                            "content": " such that there is a path from "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                697,
+                                                505,
+                                                734
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(u,v,\\{P\\})"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 12
+                        }
+                    ],
+                    "sub_type": "text"
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
+                        750,
+                        311,
+                        761
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                299,
+                                750,
+                                311,
+                                761
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        299,
+                                        750,
+                                        311,
+                                        761
+                                    ],
+                                    "type": "text",
+                                    "content": "48"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 47
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        129,
+                        82,
+                        504,
+                        139
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                82,
+                                504,
+                                139
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        82,
+                                        504,
+                                        139
+                                    ],
+                                    "type": "text",
+                                    "content": "to "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        82,
+                                        504,
+                                        139
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{u},\\tilde{v},\\{P\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        82,
+                                        504,
+                                        139
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        82,
+                                        504,
+                                        139
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{u},\\tilde{v},\\{P\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        82,
+                                        504,
+                                        139
+                                    ],
+                                    "type": "text",
+                                    "content": " is strictly contracted. We can then change the strategy at state "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        82,
+                                        504,
+                                        139
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u,v,\\{P\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        82,
+                                        504,
+                                        139
+                                    ],
+                                    "type": "text",
+                                    "content": " to make it strictly contracted. Concretely, Alice can remove pebble "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        82,
+                                        504,
+                                        139
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{p}_v"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        82,
+                                        504,
+                                        139
+                                    ],
+                                    "type": "text",
+                                    "content": " (which does not merge connected components selected by Bob since "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        82,
+                                        504,
+                                        139
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P\\in \\mathbb{C}\\mathbb{C}_F(\\{u\\}))"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        82,
+                                        504,
+                                        139
+                                    ],
+                                    "type": "text",
+                                    "content": ", and place pebble "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        82,
+                                        504,
+                                        139
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{p}_v"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        82,
+                                        504,
+                                        139
+                                    ],
+                                    "type": "text",
+                                    "content": " on the vertex that corresponds to the strategy at state "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        82,
+                                        504,
+                                        139
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{u},\\tilde{v},\\{P\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        82,
+                                        504,
+                                        139
+                                    ],
+                                    "type": "text",
+                                    "content": " (possibly with a difference that "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        82,
+                                        504,
+                                        139
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        82,
+                                        504,
+                                        139
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        82,
+                                        504,
+                                        139
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        82,
+                                        504,
+                                        139
+                                    ],
+                                    "type": "text",
+                                    "content": " become exchanged). This makes the state "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        82,
+                                        504,
+                                        139
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u,v,\\{P\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        82,
+                                        504,
+                                        139
+                                    ],
+                                    "type": "text",
+                                    "content": " strictly contracted."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        105,
+                        143,
+                        293,
+                        156
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                143,
+                                293,
+                                156
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        143,
+                                        293,
+                                        156
+                                    ],
+                                    "type": "text",
+                                    "content": "Combining the two cases concludes the proof."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "type": "image",
+                    "bbox": [
+                        494,
+                        144,
+                        504,
+                        153
+                    ],
+                    "blocks": [
+                        {
+                            "bbox": [
+                                494,
+                                144,
+                                504,
+                                153
+                            ],
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        494,
+                                        144,
+                                        504,
+                                        153
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                494,
+                                                144,
+                                                504,
+                                                153
+                                            ],
+                                            "type": "image",
+                                            "image_path": "a72e207dcf82b05fef066d22c7af4a5edebfb8a1c197ac154832756dc7f72c6d.jpg"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 3,
+                            "angle": 0,
+                            "type": "image_body"
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        105,
+                        169,
+                        286,
+                        180
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                169,
+                                286,
+                                180
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        169,
+                                        286,
+                                        180
+                                    ],
+                                    "type": "text",
+                                    "content": "We are now ready to state the main theorem:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        184,
+                        504,
+                        218
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                184,
+                                504,
+                                218
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        184,
+                                        504,
+                                        218
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem C.53. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        184,
+                                        504,
+                                        218
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        184,
+                                        504,
+                                        218
+                                    ],
+                                    "type": "text",
+                                    "content": " be any model. Given any connected graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        184,
+                                        504,
+                                        218
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        184,
+                                        504,
+                                        218
+                                    ],
+                                    "type": "text",
+                                    "content": ", if Alice can win the pebble game associated with model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        184,
+                                        504,
+                                        218
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        184,
+                                        504,
+                                        218
+                                    ],
+                                    "type": "text",
+                                    "content": " on graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        184,
+                                        504,
+                                        218
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        184,
+                                        504,
+                                        218
+                                    ],
+                                    "type": "text",
+                                    "content": ", then there is a tree decomposition "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        184,
+                                        504,
+                                        218
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        184,
+                                        504,
+                                        218
+                                    ],
+                                    "type": "text",
+                                    "content": " of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        184,
+                                        504,
+                                        218
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        184,
+                                        504,
+                                        218
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        184,
+                                        504,
+                                        218
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        184,
+                                        504,
+                                        218
+                                    ],
+                                    "type": "text",
+                                    "content": ", where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        184,
+                                        504,
+                                        218
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        184,
+                                        504,
+                                        218
+                                    ],
+                                    "type": "text",
+                                    "content": " is defined in Definition C.6."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        232,
+                        504,
+                        289
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                232,
+                                504,
+                                289
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^{\\mathsf{S}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": " be the game state graph satisfying Lemma C.52. For each game state "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": ", denote by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{next}_{G^{\\mathsf{S}}}(\\bar{s})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": " the set of states "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s'"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(s, s')"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": " is a transition in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^{\\mathsf{S}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s'"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": " contains only a single connected component, i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s'"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": " has the form "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u, v, \\{P\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": ". For a terminal state "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "s"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": " of the form "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u, v, \\{\\{x, y\\}\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": ", define "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{next}_{G^{\\mathsf{S}}}(\\bar{s}) = \\{(x, y, \\{\\{x, y\\}\\})\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": ". By definition, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{next}_{G^{\\mathsf{S}}}(\\emptyset, \\emptyset, \\{E_F\\}) = \\{(u, \\emptyset, Q_1), \\dots, (u, \\emptyset, Q_m)\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": " for some "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Q_1, \\dots, Q_m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": " is the finest partition of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbb{C}\\mathbb{C}_F(\\{u\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        232,
+                                        504,
+                                        289
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        293,
+                        504,
+                        327
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                293,
+                                504,
+                                327
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        293,
+                                        504,
+                                        327
+                                    ],
+                                    "type": "text",
+                                    "content": "The tree "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        293,
+                                        504,
+                                        327
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        293,
+                                        504,
+                                        327
+                                    ],
+                                    "type": "text",
+                                    "content": " will be recursively constructed as follows. First create the tree root "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        293,
+                                        504,
+                                        327
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        293,
+                                        504,
+                                        327
+                                    ],
+                                    "type": "text",
+                                    "content": " with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        293,
+                                        504,
+                                        327
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_T(r) = \\{u, u\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        293,
+                                        504,
+                                        327
+                                    ],
+                                    "type": "text",
+                                    "content": ". As we will see later, the root node will be associated with the set of states "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        293,
+                                        504,
+                                        327
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S(r) := \\mathrm{next}_{G^s}(\\emptyset, \\emptyset, \\{E_F\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        293,
+                                        504,
+                                        327
+                                    ],
+                                    "type": "text",
+                                    "content": ". We then do the following procedure:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        104,
+                        332,
+                        504,
+                        392
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                332,
+                                504,
+                                392
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": "Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": " be a leaf node in the current tree associated with a non-empty set of game states "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S(t)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "|\\bigcup_{(u,v,\\{P\\}) \\in S(t)} P| > 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": ". For each state "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u,v,\\{P\\}) \\in S(t)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": ", create a new node "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{t}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": " and set its parent to be "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": ". Pick any state "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u',v',\\{P'\\}) \\in \\mathrm{next}_{G^s}(u,v,\\{P\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, there must be a unique vertex "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w \\in \\{\\{u',v'\\}\\} \\setminus \\{\\{u,v\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": " does not depend on which "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u',v',\\{P'\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": " is picked (by definition of the game rule). Set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_T(\\tilde{t}) = \\beta_T(t) \\cup \\{\\{w\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        332,
+                                        504,
+                                        392
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, do the following constructions:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        120,
+                        396,
+                        504,
+                        473
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 11,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                120,
+                                396,
+                                504,
+                                433
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        120,
+                                        396,
+                                        504,
+                                        433
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                120,
+                                                396,
+                                                504,
+                                                433
+                                            ],
+                                            "type": "text",
+                                            "content": "- If there is a state of the form "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                396,
+                                                504,
+                                                433
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(u, w, \\{P'\\}) \\in \\mathrm{next}_{G^s}(u, v, \\{P\\})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                396,
+                                                504,
+                                                433
+                                            ],
+                                            "type": "text",
+                                            "content": ", then create a new node "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                396,
+                                                504,
+                                                433
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t'"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                396,
+                                                504,
+                                                433
+                                            ],
+                                            "type": "text",
+                                            "content": " and connect it to the parent "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                396,
+                                                504,
+                                                433
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tilde{t}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                396,
+                                                504,
+                                                433
+                                            ],
+                                            "type": "text",
+                                            "content": ". Set "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                396,
+                                                504,
+                                                433
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(t') = \\{(u, w)\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                396,
+                                                504,
+                                                433
+                                            ],
+                                            "type": "text",
+                                            "content": ", and the node "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                396,
+                                                504,
+                                                433
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t'"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                396,
+                                                504,
+                                                433
+                                            ],
+                                            "type": "text",
+                                            "content": " will be associated to the set of states "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                396,
+                                                504,
+                                                433
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "S(t') = \\{(u, w, \\{P'\\}): (u, w, \\{P'\\}) \\in \\mathrm{next}_{G^s}(u, v, \\{P\\})\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                396,
+                                                504,
+                                                433
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 9
+                        },
+                        {
+                            "bbox": [
+                                120,
+                                436,
+                                504,
+                                473
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        120,
+                                        436,
+                                        504,
+                                        473
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                120,
+                                                436,
+                                                504,
+                                                473
+                                            ],
+                                            "type": "text",
+                                            "content": "- If there is a state of the form "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                436,
+                                                504,
+                                                473
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(w, v, \\{P''\\}) \\in \\mathsf{next}_{G^{\\mathsf{s}}}(u, v, \\{P\\})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                436,
+                                                504,
+                                                473
+                                            ],
+                                            "type": "text",
+                                            "content": ", then create a new node "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                436,
+                                                504,
+                                                473
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t''"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                436,
+                                                504,
+                                                473
+                                            ],
+                                            "type": "text",
+                                            "content": " and connect it to the parent "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                436,
+                                                504,
+                                                473
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tilde{t}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                436,
+                                                504,
+                                                473
+                                            ],
+                                            "type": "text",
+                                            "content": ". Set "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                436,
+                                                504,
+                                                473
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_{T}(t'') = \\{(w, v)\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                436,
+                                                504,
+                                                473
+                                            ],
+                                            "type": "text",
+                                            "content": ", and the node "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                436,
+                                                504,
+                                                473
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t''"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                436,
+                                                504,
+                                                473
+                                            ],
+                                            "type": "text",
+                                            "content": " will be associated to the set of states "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                436,
+                                                504,
+                                                473
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "S(t'') = \\{(w, v, \\{P''\\}): (w, v, \\{P''\\}) \\in \\mathsf{next}_{G^{\\mathsf{s}}}(u, v, \\{P\\})\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                436,
+                                                504,
+                                                473
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 10
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        104,
+                        476,
+                        504,
+                        521
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                476,
+                                504,
+                                521
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        476,
+                                        504,
+                                        521
+                                    ],
+                                    "type": "text",
+                                    "content": "Note that (i) either "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        476,
+                                        504,
+                                        521
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w \\neq u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        476,
+                                        504,
+                                        521
+                                    ],
+                                    "type": "text",
+                                    "content": " or "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        476,
+                                        504,
+                                        521
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w \\neq v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        476,
+                                        504,
+                                        521
+                                    ],
+                                    "type": "text",
+                                    "content": " (since the game state graph is a DAG); (ii) both items can be used for one state "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        476,
+                                        504,
+                                        521
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u, v, \\{P\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        476,
+                                        504,
+                                        521
+                                    ],
+                                    "type": "text",
+                                    "content": ", which happens for FWL-type GNNs. The construction of the tree completes when each leaf node is associated with only one game state of the form "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        476,
+                                        504,
+                                        521
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(x, y, \\{\\{x, y\\}\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        476,
+                                        504,
+                                        521
+                                    ],
+                                    "type": "text",
+                                    "content": ". It is easy to see that the above procedure terminates after adding a finite number of tree nodes."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        105,
+                        526,
+                        434,
+                        539
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                526,
+                                434,
+                                539
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        526,
+                                        434,
+                                        539
+                                    ],
+                                    "type": "text",
+                                    "content": "We now prove that "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        526,
+                                        434,
+                                        539
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        526,
+                                        434,
+                                        539
+                                    ],
+                                    "type": "text",
+                                    "content": " is a canonical tree decomposition of "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        526,
+                                        434,
+                                        539
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        526,
+                                        434,
+                                        539
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        526,
+                                        434,
+                                        539
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        526,
+                                        434,
+                                        539
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        117,
+                        544,
+                        504,
+                        579
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                117,
+                                544,
+                                504,
+                                579
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        544,
+                                        504,
+                                        579
+                                    ],
+                                    "type": "text",
+                                    "content": "1. We first prove that any edge in "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        544,
+                                        504,
+                                        579
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        544,
+                                        504,
+                                        579
+                                    ],
+                                    "type": "text",
+                                    "content": " is contained in some bag of "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        544,
+                                        504,
+                                        579
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        544,
+                                        504,
+                                        579
+                                    ],
+                                    "type": "text",
+                                    "content": ". Pick any non-leaf tree node "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        544,
+                                        504,
+                                        579
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        544,
+                                        504,
+                                        579
+                                    ],
+                                    "type": "text",
+                                    "content": " of even depth and any of its child "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        544,
+                                        504,
+                                        579
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{t}"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        544,
+                                        504,
+                                        579
+                                    ],
+                                    "type": "text",
+                                    "content": ", denote "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        544,
+                                        504,
+                                        579
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u, v, P) \\in S(t)"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        544,
+                                        504,
+                                        579
+                                    ],
+                                    "type": "text",
+                                    "content": " be the state associated with "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        544,
+                                        504,
+                                        579
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{t}"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        544,
+                                        504,
+                                        579
+                                    ],
+                                    "type": "text",
+                                    "content": " in the above construction, and denote "
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        544,
+                                        504,
+                                        579
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\beta_T(\\tilde{t}) = \\{\\{u, v, w\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        117,
+                                        544,
+                                        504,
+                                        579
+                                    ],
+                                    "type": "text",
+                                    "content": ". We have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        184,
+                        585,
+                        449,
+                        642
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                184,
+                                585,
+                                449,
+                                642
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        184,
+                                        585,
+                                        449,
+                                        642
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "P = \\left(\\bigcup_{\\substack{t^{\\prime},\\mathsf{pa}_{T^{r}}(t^{\\prime}) = \\tilde{t},\\\\ (x,y,\\{P^{\\prime}\\})\\in S(t^{\\prime})}}P^{\\prime}\\right)\\cup \\{\\{u,w\\} \\in E_{F}\\} \\cup \\{\\{v,w\\} \\in E_{F}\\} .",
+                                    "image_path": "7364150863edfc512f18da95c33622784b8e622dbb273a25216dbf0e61fb4f55.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        129,
+                        647,
+                        504,
+                        671
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                647,
+                                504,
+                                671
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        647,
+                                        504,
+                                        671
+                                    ],
+                                    "type": "text",
+                                    "content": "Here, "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        647,
+                                        504,
+                                        671
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P' \\subset P"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        647,
+                                        504,
+                                        671
+                                    ],
+                                    "type": "text",
+                                    "content": " is a consequence of Lemma C.52, and the remaining edges "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        647,
+                                        504,
+                                        671
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{u, w\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        647,
+                                        504,
+                                        671
+                                    ],
+                                    "type": "text",
+                                    "content": " or "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        647,
+                                        504,
+                                        671
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{v, w\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        647,
+                                        504,
+                                        671
+                                    ],
+                                    "type": "text",
+                                    "content": " are in "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        647,
+                                        504,
+                                        671
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        647,
+                                        504,
+                                        671
+                                    ],
+                                    "type": "text",
+                                    "content": " because "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        647,
+                                        504,
+                                        671
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u, v, P)"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " is strictly contracted and "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        647,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        647,
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+                                    ],
+                                    "type": "text",
+                                    "content": " must be in the interior of "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        647,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        647,
+                                        504,
+                                        671
+                                    ],
+                                    "type": "text",
+                                    "content": ". Therefore,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        129,
+                        677,
+                        529,
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+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
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+                                    "type": "interline_equation",
+                                    "content": "P\\cup \\{\\{u,v\\} \\in E_{F}\\} = \\left(\\bigcup_{\\substack{t^{\\prime},\\mathsf{pa}_{Tr}(t^{\\prime}) = \\tilde{t},\\\\ (x,y,\\{P^{\\prime}\\})\\in S(t^{\\prime})}}(P^{\\prime}\\cup \\{\\{x,y\\} \\in E_{F}\\})\\right)\\cup \\{\\{x,y\\} \\in E_{F}:x,y\\in \\beta_{T}(\\tilde{t})\\} .",
+                                    "image_path": "96f13fb88db220af923f6261333bb35209007fb2bb1a63743d271ef0cd61f62f.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                }
+            ],
+            "discarded_blocks": [
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+                    "type": "header",
+                    "angle": 0,
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
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+                                    "type": "text",
+                                    "content": "49"
+                                }
+                            ]
+                        }
+                    ],
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+                }
+            ],
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+                                    "type": "text",
+                                    "content": "Recursively applying the above equation yields the desired result"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "E_{F} = \\bigcup_{\\substack{\\tilde{t},\\mathsf{dep}_{T}(\\tilde{t})\\text{is odd}}}\\{\\{x,y\\} \\in E_{F}:x,y\\in \\beta_{T}(\\tilde{t})\\} ,",
+                                    "image_path": "73cd5a7956f4b8841c607dc11e4a539f0fd58c69e2794e85ec36ace441504a38.jpg"
+                                }
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+                    "index": 2
+                },
+                {
+                    "bbox": [
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+                        134,
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+                        161
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                128,
+                                134,
+                                504,
+                                161
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        128,
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+                                    ],
+                                    "type": "text",
+                                    "content": "because (i) "
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        134,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\bigcup_{(u,\\emptyset,P)\\in S(r)}P = E_F"
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        134,
+                                        504,
+                                        161
+                                    ],
+                                    "type": "text",
+                                    "content": ", and (ii) all leaf node "
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        134,
+                                        504,
+                                        161
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t"
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        134,
+                                        504,
+                                        161
+                                    ],
+                                    "type": "text",
+                                    "content": " with "
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        134,
+                                        504,
+                                        161
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "S(t) = (x,y,\\{\\{x,y\\}\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        134,
+                                        504,
+                                        161
+                                    ],
+                                    "type": "text",
+                                    "content": " contains an edge "
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        134,
+                                        504,
+                                        161
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{x,y\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        134,
+                                        504,
+                                        161
+                                    ],
+                                    "type": "text",
+                                    "content": " in its bag."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
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+                        165,
+                        506,
+                        292
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 7,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                117,
+                                165,
+                                506,
+                                257
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                        165,
+                                        506,
+                                        257
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                117,
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+                                            ],
+                                            "type": "text",
+                                            "content": "2. We next prove that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T^r"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "text",
+                                            "content": " satisfies the condition of Definition C.1(c). Fix any vertex "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\in V_F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "text",
+                                            "content": ", and let "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "text",
+                                            "content": " be the tree node with minimal depth that contains "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "text",
+                                            "content": ". Without loss of generality, assume that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "text",
+                                            "content": " is not the root. In this case, the depth of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "text",
+                                            "content": " is odd and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(t) = \\{ \\{u, v, w\\} \\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "text",
+                                            "content": " for some "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u \\neq w, v \\neq w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "text",
+                                            "content": ". Let "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t'"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "text",
+                                            "content": " be a child node of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "text",
+                                            "content": " and we have "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\in \\beta_T(t')"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "text",
+                                            "content": ". It remains to prove that for any descendant "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tilde{t} \\in \\mathsf{Desc}_T(t')"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "text",
+                                            "content": ", if "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\in \\beta_T(\\tilde{t})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "text",
+                                            "content": ", then "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w \\in \\beta_T(\\hat{t})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "text",
+                                            "content": " for all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\hat{t}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "text",
+                                            "content": " on the path between "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t'"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tilde{t}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "text",
+                                            "content": ". This is actually a direct consequence of Lemma C.52, because when a pebble originally placed on "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
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+                                            ],
+                                            "type": "text",
+                                            "content": " is removed, all edges linked to "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "text",
+                                            "content": " will not be selected by Bob and thus any pebble can never be placed on "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
+                                                506,
+                                                257
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                165,
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+                                                257
+                                            ],
+                                            "type": "text",
+                                            "content": " again."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 4
+                        },
+                        {
+                            "bbox": [
+                                117,
+                                261,
+                                343,
+                                274
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                    ],
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+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": "3. "
+                                        },
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+                                            "bbox": [
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T^r"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": " is canonical as Definition C.4 is clearly satisfied."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 5
+                        },
+                        {
+                            "bbox": [
+                                117,
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+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": "4. Finally, it is also easy to see that "
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(F,T^{r})"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": " satisfies Definition C.6."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 6
+                        }
+                    ],
+                    "sub_type": "text"
+                },
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "We thus conclude the proof."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
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+                            "type": "image_body"
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        314,
+                                        504,
+                                        352
+                                    ],
+                                    "type": "text",
+                                    "content": "Corollary C.54. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        314,
+                                        504,
+                                        352
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        314,
+                                        504,
+                                        352
+                                    ],
+                                    "type": "text",
+                                    "content": " be any model. For any connected graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        314,
+                                        504,
+                                        352
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F \\notin \\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        314,
+                                        504,
+                                        352
+                                    ],
+                                    "type": "text",
+                                    "content": ", let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        314,
+                                        504,
+                                        352
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G(F)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        314,
+                                        504,
+                                        352
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        314,
+                                        504,
+                                        352
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H(F)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        314,
+                                        504,
+                                        352
+                                    ],
+                                    "type": "text",
+                                    "content": " be the Fürer graph and twisted Fürer graph with respect to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        314,
+                                        504,
+                                        352
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        314,
+                                        504,
+                                        352
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        314,
+                                        504,
+                                        352
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\text{hom}(F, G(F)) \\neq \\text{hom}(F, H(F))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        314,
+                                        504,
+                                        352
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        314,
+                                        504,
+                                        352
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_{G(F)}^M(G(F)) = \\chi_{H(F)}^M(H(F))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        314,
+                                        504,
+                                        352
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        364,
+                        487,
+                        376
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                364,
+                                487,
+                                376
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        487,
+                                        376
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. The proof directly follows from applying Theorems C.47 and C.53 and Proposition C.49. "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        364,
+                                        487,
+                                        376
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\square"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "type": "image",
+                    "bbox": [
+                        494,
+                        365,
+                        504,
+                        374
+                    ],
+                    "blocks": [
+                        {
+                            "bbox": [
+                                494,
+                                365,
+                                504,
+                                374
+                            ],
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        494,
+                                        365,
+                                        504,
+                                        374
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                494,
+                                                365,
+                                                504,
+                                                374
+                                            ],
+                                            "type": "image",
+                                            "image_path": "efca4d81cb4eb459488c47a94aea190aa1e1b3a11b88ab86f90a692f33d4efc8.jpg"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 12,
+                            "angle": 0,
+                            "type": "image_body"
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        104,
+                        388,
+                        506,
+                        464
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                388,
+                                506,
+                                464
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": "Finally, we remark that our construction can be easily generalized for disconnected graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F \\notin \\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": ". Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": " be the disjoint union of graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{F_i : i \\in [m]\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": " where each "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F_i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": " is the graph corresponding to a connected component of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": ". Assume that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F_1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": " is the connected component with the most number of edges (in case of a tie, pick the graph with the most number of vertices). Define "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{G}(F)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": " be the disjoint union of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G(F_1), F_2, \\dots, F_m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{H}(F)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": " to be the disjoint union of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H(F_1), F_2, \\dots, F_m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": ". It follows that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_{\\tilde{G}(F)}\\tilde{G}(F) = \\chi_{\\tilde{H}(F)}\\tilde{H}(F)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        506,
+                                        464
+                                    ],
+                                    "type": "text",
+                                    "content": " and"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        113,
+                        469,
+                        495,
+                        527
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                113,
+                                469,
+                                495,
+                                527
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        113,
+                                        469,
+                                        495,
+                                        527
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\operatorname {h o m} (F, \\tilde {G} (F)) = \\prod_ {i \\in [ m ]} \\operatorname {h o m} (F _ {i}, \\tilde {G} (F)) = \\prod_ {i \\in [ m ]} (\\operatorname {h o m} (F _ {i}, G (F _ {1})) + \\operatorname {h o m} (F _ {i}, F _ {2} \\cup \\dots \\cup F _ {m})) \\\\ > \\prod_ {i \\in [ m ]} \\left(\\operatorname {h o m} \\left(F _ {i}, H \\left(F _ {1}\\right)\\right) + \\operatorname {h o m} \\left(F _ {i}, F _ {2} \\cup \\dots \\cup F _ {m}\\right)\\right) = \\operatorname {h o m} (F, \\tilde {H} (F)), \\\\ \\end{array}",
+                                    "image_path": "5f817fdc7597ecc7b554cd95ca23ecac2eda45bb1d68915d0ceb8ecdb8e5e923.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        104,
+                        533,
+                        504,
+                        557
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                533,
+                                504,
+                                557
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        504,
+                                        557
+                                    ],
+                                    "type": "text",
+                                    "content": "where we use the fact that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        504,
+                                        557
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{hom}(F_i, G(F_1)) = \\operatorname{hom}(F_i, H(F_1))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        504,
+                                        557
+                                    ],
+                                    "type": "text",
+                                    "content": " when "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        504,
+                                        557
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F_1 \\notin \\operatorname{Spasm}(F_i)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        504,
+                                        557
+                                    ],
+                                    "type": "text",
+                                    "content": " (which can be easily proved following Theorem C.47). This concludes the proof of the general case."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        105,
+                        572,
+                        308,
+                        585
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                572,
+                                308,
+                                585
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        572,
+                                        308,
+                                        585
+                                    ],
+                                    "type": "text",
+                                    "content": "D NODE/EDGE-LEVEL EXPRESSIVITY"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        104,
+                        597,
+                        464,
+                        610
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                597,
+                                464,
+                                610
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        597,
+                                        464,
+                                        610
+                                    ],
+                                    "type": "text",
+                                    "content": "This section aims to prove Theorem 3.7. For the ease of reading, we first restate it below:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        104,
+                        614,
+                        504,
+                        628
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                614,
+                                504,
+                                628
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        614,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem 3.7 For all model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        614,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        614,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": " defined in Section 2, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        614,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}_{\\mathfrak{n}}^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        614,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        614,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}_{\\mathrm{e}}^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        614,
+                                        504,
+                                        628
+                                    ],
+                                    "type": "text",
+                                    "content": " (except MPNN) exist. Moreover,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
+                        121,
+                        630,
+                        279,
+                        658
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 21,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                121,
+                                630,
+                                279,
+                                644
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        630,
+                                        279,
+                                        644
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                121,
+                                                630,
+                                                279,
+                                                644
+                                            ],
+                                            "type": "text",
+                                            "content": "- MPNN: "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                630,
+                                                279,
+                                                644
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathcal{F}_{\\mathfrak{n}}^{\\mathsf{MP}} = \\{F^{w}: F \\text{ is a tree}\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                630,
+                                                279,
+                                                644
+                                            ],
+                                            "type": "text",
+                                            "content": ";"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 19
+                        },
+                        {
+                            "bbox": [
+                                121,
+                                647,
+                                201,
+                                658
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        647,
+                                        201,
+                                        658
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                121,
+                                                647,
+                                                201,
+                                                658
+                                            ],
+                                            "type": "text",
+                                            "content": "- Subgraph GNN:"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 20
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        129,
+                        657,
+                        471,
+                        670
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        657,
+                                        471,
+                                        670
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathcal {F} _ {\\mathfrak {n}} ^ {\\mathrm {S u b}} = \\left\\{F ^ {w}: F \\text {h a s a N E D w i t h s h a r d e n p o i n t} w \\right\\} = \\left\\{F ^ {w}: F \\backslash \\{w \\} \\text {i s a f o r e s t} \\right\\},",
+                                    "image_path": "37adc68ee41d23bb307d3bca3b8512e00460a4997036f76eeee7e62b8c1bb3ad.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 22
+                },
+                {
+                    "bbox": [
+                        129,
+                        669,
+                        471,
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+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
+                                        129,
+                                        669,
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+                                        681
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathcal {F} _ {\\mathrm {e}} ^ {\\mathrm {S} _ {\\mathrm {u b}}} = \\left\\{F ^ {w x}: F \\text {h a s a N E D w i t h s h a r e d e n p o i n t} w \\right\\} = \\left\\{F ^ {w x}: F \\backslash \\{w \\} \\text {i s a f o r e s t} \\right\\};",
+                                    "image_path": "b639d8b3d71685b6c170707a01fd92aafa496ef78d1dc028b806560aecf70d8b.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 23
+                },
+                {
+                    "bbox": [
+                        121,
+                        684,
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+                        696
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "type": "text",
+                                    "content": "- 2-FGNN: "
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+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}_{\\mathfrak{n}}^{\\mathsf{F}} = \\{F^{w}: F"
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+                                    "type": "text",
+                                    "content": " has a NED where "
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+                                    "type": "text",
+                                    "content": " is an endpoint of the first ear},"
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+                    "index": 24
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+                                    "type": "interline_equation",
+                                    "content": "\\mathcal {F} _ {\\mathrm {e}} ^ {\\mathrm {F}} = \\left\\{F ^ {w x}: F \\text {h a s a N E D w h e r e} w \\text {a n d} x \\text {a r e e x i p o n t s o f t h e f i r s t e a r} \\right\\}.",
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+                                    "type": "text",
+                                    "content": "The cases of Local 2-GNN and Local 2-FGNN are similar to 2-FGNN by replacing \"NED\" with \"strong NED\" and \"almost-strong NED\", respectively."
+                                }
+                            ]
+                        }
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "type": "text",
+                                    "content": "We remark that the definition of node/edge-level homomorphism expressivity involves only connected graphs for sake of simplicity. Similar to the proof of the graph-level expressivity, the proof of Theorem 3.7 consists of three parts. Among them, the proof related to tree decomposition and ear decomposition is quite similar to that of graph-level expressivity, so we only illustrate the proof sketch for clarity (Appendix D.1). However, the proof related to pebble game and counterexample graphs will require additional techniques, which is detailed in Appendix D.2."
+                                }
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+                                    "type": "text",
+                                    "content": "D.1 RELATED TO TREE DECOMPOSITION AND EAR DECOMPOSITION"
+                                }
+                            ]
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+                                    "type": "text",
+                                    "content": "We first extend several notations that are used in Appendix C.2."
+                                }
+                            ]
+                        }
+                    ],
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+                },
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+                                {
+                                    "bbox": [
+                                        104,
+                                        196,
+                                        504,
+                                        231
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition D.1 (Tree decomposition for rooted graphs). Given a rooted graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        196,
+                                        504,
+                                        231
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        196,
+                                        504,
+                                        231
+                                    ],
+                                    "type": "text",
+                                    "content": " and tree "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        196,
+                                        504,
+                                        231
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        196,
+                                        504,
+                                        231
+                                    ],
+                                    "type": "text",
+                                    "content": ", we say "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        196,
+                                        504,
+                                        231
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        196,
+                                        504,
+                                        231
+                                    ],
+                                    "type": "text",
+                                    "content": " is a tree decomposition of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        196,
+                                        504,
+                                        231
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        196,
+                                        504,
+                                        231
+                                    ],
+                                    "type": "text",
+                                    "content": " if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        196,
+                                        504,
+                                        231
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        196,
+                                        504,
+                                        231
+                                    ],
+                                    "type": "text",
+                                    "content": " is a tree decomposition of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        196,
+                                        504,
+                                        231
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        196,
+                                        504,
+                                        231
+                                    ],
+                                    "type": "text",
+                                    "content": " and all elements in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        196,
+                                        504,
+                                        231
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        196,
+                                        504,
+                                        231
+                                    ],
+                                    "type": "text",
+                                    "content": " belong to the root bag, i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        196,
+                                        504,
+                                        231
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u_i \\in \\beta_T(r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        196,
+                                        504,
+                                        231
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        196,
+                                        504,
+                                        231
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        196,
+                                        504,
+                                        231
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        238,
+                        504,
+                        273
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                238,
+                                504,
+                                273
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        273
+                                    ],
+                                    "type": "text",
+                                    "content": "Based on this definition, given model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        273
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        273
+                                    ],
+                                    "type": "text",
+                                    "content": ", we can define "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        273
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{S}_{\\mathfrak{n}}^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        273
+                                    ],
+                                    "type": "text",
+                                    "content": " to be the family of tree-decomposed graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        273
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F^u,T^r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        273
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        273
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^r)\\in S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        273
+                                    ],
+                                    "type": "text",
+                                    "content": "; similarly, we define "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        273
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{S}_{\\mathrm{e}}^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        273
+                                    ],
+                                    "type": "text",
+                                    "content": " to be the family of tree-decomposed graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        273
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F^{uv},T^{r})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        273
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        273
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F,T^{r})\\in S^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        238,
+                                        504,
+                                        273
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        274,
+                        505,
+                        309
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                274,
+                                505,
+                                309
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        505,
+                                        309
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition D.2 (Bag isomorphism for rooted graphs). Given tree-decomposed graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        505,
+                                        309
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F^{\\mathbf{u}}, T^{r})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        505,
+                                        309
+                                    ],
+                                    "type": "text",
+                                    "content": " and rooted graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        505,
+                                        309
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^{\\mathbf{v}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        505,
+                                        309
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        505,
+                                        309
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{u}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        505,
+                                        309
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        505,
+                                        309
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{v}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        505,
+                                        309
+                                    ],
+                                    "type": "text",
+                                    "content": " have equal length, a bag isomorphism from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        505,
+                                        309
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F^{\\mathbf{u}}, T^{r})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        505,
+                                        309
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        505,
+                                        309
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^{\\mathbf{v}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        505,
+                                        309
+                                    ],
+                                    "type": "text",
+                                    "content": " is a homomorphism "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        505,
+                                        309
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        505,
+                                        309
+                                    ],
+                                    "type": "text",
+                                    "content": " from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        505,
+                                        309
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F^{\\mathbf{u}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        505,
+                                        309
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        505,
+                                        309
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^{\\mathbf{v}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        505,
+                                        309
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        505,
+                                        309
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        505,
+                                        309
+                                    ],
+                                    "type": "text",
+                                    "content": " is a bag isomorphism from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        505,
+                                        309
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^{r})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        505,
+                                        309
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        505,
+                                        309
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        274,
+                                        505,
+                                        309
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        316,
+                        504,
+                        353
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                316,
+                                504,
+                                353
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        316,
+                                        504,
+                                        353
+                                    ],
+                                    "type": "text",
+                                    "content": "We can similarly define bag-isomorphism homomorphism (bIsoHom, bIsoSurj, bIsoInj), bag extension (bExt), and bag-strong surjective (bStrSurj) from tree-decomposed graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        316,
+                                        504,
+                                        353
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F^u, T^r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        316,
+                                        504,
+                                        353
+                                    ],
+                                    "type": "text",
+                                    "content": " to tree-decomposed graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        316,
+                                        504,
+                                        353
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(\\tilde{F}^v, \\tilde{T}^s)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        316,
+                                        504,
+                                        353
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        104,
+                        354,
+                        504,
+                        378
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                354,
+                                504,
+                                378
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        504,
+                                        378
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition D.3 (Generalization of Definition C.15 for rooted graphs). Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        504,
+                                        378
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\mathrm{Sub}, \\mathrm{L}, \\mathrm{LF}, \\mathrm{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        504,
+                                        378
+                                    ],
+                                    "type": "text",
+                                    "content": " be any model. Given a rooted graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        504,
+                                        378
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^{u}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        504,
+                                        378
+                                    ],
+                                    "type": "text",
+                                    "content": " and a tree-decomposed graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        504,
+                                        378
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F^w, T^r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        354,
+                                        504,
+                                        378
+                                    ],
+                                    "type": "text",
+                                    "content": ", define"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        381,
+                        526,
+                        403
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                381,
+                                526,
+                                403
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        381,
+                                        526,
+                                        403
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathsf {c n t} ^ {M} \\left(\\left(F ^ {w}, T ^ {r}\\right), G ^ {u}\\right) := \\left| \\left\\{v \\in V _ {G}: \\exists D \\in \\mathbb {N} _ {+} \\text {s . t .} \\left(\\left[ F _ {G} ^ {M, (D)} (u, v) \\right] ^ {u}, T _ {G} ^ {M, (D)} (u, v)\\right) \\simeq \\left(F ^ {w}, T ^ {r}\\right) \\right\\} \\right|.",
+                                    "image_path": "c78236545f1e7303d67da5398560e968a0458f7dce71505aaec552c316097915.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
+                        406,
+                        408,
+                        418
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                406,
+                                408,
+                                418
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        406,
+                                        408,
+                                        418
+                                    ],
+                                    "type": "text",
+                                    "content": "Given a rooted graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        406,
+                                        408,
+                                        418
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^{uv}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        406,
+                                        408,
+                                        418
+                                    ],
+                                    "type": "text",
+                                    "content": " and a tree-decomposed graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        406,
+                                        408,
+                                        418
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F^{wx}, T^r)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        406,
+                                        408,
+                                        418
+                                    ],
+                                    "type": "text",
+                                    "content": ", define"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        105,
+                        422,
+                        501,
+                        444
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                422,
+                                501,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        422,
+                                        501,
+                                        444
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathsf {c n t} ^ {M} \\left(\\left(F ^ {w x}, T ^ {r}\\right), G ^ {u v}\\right) := \\mathbb {I} \\left[ \\exists D \\in \\mathbb {N} _ {+} \\text {s . t .} \\left(\\left[ F _ {G} ^ {M, (D)} (u, v) \\right] ^ {u v}, T _ {G} ^ {M, (D)} (u, v)\\right) \\simeq \\left(F ^ {w x}, T ^ {r}\\right) \\right].",
+                                    "image_path": "4c6c22135758d8fcfecf0cb9731a99a7c3b23fcc76dacdefd1247cdfe4fb9cad.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        104,
+                        446,
+                        495,
+                        467
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        446,
+                                        495,
+                                        467
+                                    ],
+                                    "type": "text",
+                                    "content": "Here, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        446,
+                                        495,
+                                        467
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\left(F_G^{M,(D)}(u,v),T_G^{M,(D)}(u,v)\\right)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        446,
+                                        495,
+                                        467
+                                    ],
+                                    "type": "text",
+                                    "content": " is the depth-2D unfolding tree of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        446,
+                                        495,
+                                        467
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        446,
+                                        495,
+                                        467
+                                    ],
+                                    "type": "text",
+                                    "content": " at "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        446,
+                                        495,
+                                        467
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u,v)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        446,
+                                        495,
+                                        467
+                                    ],
+                                    "type": "text",
+                                    "content": " for model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        446,
+                                        495,
+                                        467
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        446,
+                                        495,
+                                        467
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        104,
+                        474,
+                        438,
+                        486
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                474,
+                                438,
+                                486
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        474,
+                                        438,
+                                        486
+                                    ],
+                                    "type": "text",
+                                    "content": "All the following lemmas are straightforward extensions of those in Appendix C.2."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        104,
+                        487,
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+                        512
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        512
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma D.4. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        512
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        512
+                                    ],
+                                    "type": "text",
+                                    "content": " be any model. For any graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        512
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        512
+                                    ],
+                                    "type": "text",
+                                    "content": " and tree-decomposed graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        512
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F^u, T^r) \\in S_{\\mathfrak{n}}^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        487,
+                                        504,
+                                        512
+                                    ],
+                                    "type": "text",
+                                    "content": ", we have"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        108,
+                        514,
+                        501,
+                        545
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                108,
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+                                501,
+                                545
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        108,
+                                        514,
+                                        501,
+                                        545
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\operatorname {b l s o} \\left(\\left(F ^ {u}, T ^ {r}\\right), G ^ {v}\\right) = \\sum_ {\\left(\\tilde {F} ^ {w}, \\tilde {T} ^ {s}\\right) \\in \\mathcal {S} _ {\\mathfrak {n}} ^ {M}} \\operatorname {b l s o H o m} \\left(\\left(F ^ {u}, T ^ {r}\\right), \\left(\\tilde {F} ^ {w}, \\tilde {T} ^ {s}\\right)\\right) \\cdot \\operatorname {c n t} ^ {M} \\left(\\left(\\tilde {F} ^ {w}, \\tilde {T} ^ {s}\\right), G ^ {v}\\right).",
+                                    "image_path": "1d9f1364ab67b3666842a9ef98f476dae58d91c707bcf95c1b0836da7dc5fbf9.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        104,
+                        548,
+                        233,
+                        559
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                                233,
+                                559
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        548,
+                                        233,
+                                        559
+                                    ],
+                                    "type": "text",
+                                    "content": "The edge-level result is similar."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        104,
+                        562,
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+                        588
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        562,
+                                        504,
+                                        588
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma D.5. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        562,
+                                        504,
+                                        588
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\mathrm{Sub},\\mathrm{L},\\mathrm{LF},\\mathrm{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        562,
+                                        504,
+                                        588
+                                    ],
+                                    "type": "text",
+                                    "content": " be any model. For any tree-decomposed graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        562,
+                                        504,
+                                        588
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F^u,T^r),(\\tilde{F}^v,\\tilde{T}^s) \\in S_{\\mathfrak{n}}^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        562,
+                                        504,
+                                        588
+                                    ],
+                                    "type": "text",
+                                    "content": ","
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        104,
+                        591,
+                        523,
+                        632
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathsf{b l s o H o m}((F^{u},T^{r}),(\\tilde{F}^{v},\\tilde{T}^{s})) = \\sum_{(\\widehat{F}^{w},\\widehat{T}^{t})\\in \\mathcal{S}_{n}^{M}}\\frac{\\mathsf{b l s o S u r j}\\left((F^{u},T^{r}),( \\widehat{F}^{w},\\widehat{T}^{t})\\right)\\cdot\\mathsf{b l s o I n j}\\left((\\widehat{F}^{w},\\widehat{T}^{t}),( \\tilde{F}^{v},\\tilde{T}^{s})\\right)}{\\mathsf{a u t}(\\widehat{F}^{w},\\widehat{T}^{t})}.",
+                                    "image_path": "3c9669650f0963c693f3765d4699d3cd9cf5eeb8c8db222ca3348fc16e2b75f5.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
+                        104,
+                        635,
+                        233,
+                        647
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
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+                                {
+                                    "bbox": [
+                                        104,
+                                        635,
+                                        233,
+                                        647
+                                    ],
+                                    "type": "text",
+                                    "content": "The edge-level result is similar."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                },
+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Lemma D.6. Let "
+                                },
+                                {
+                                    "bbox": [
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+                                        649,
+                                        505,
+                                        673
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        649,
+                                        505,
+                                        673
+                                    ],
+                                    "type": "text",
+                                    "content": " be any model. For any graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        649,
+                                        505,
+                                        673
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        649,
+                                        505,
+                                        673
+                                    ],
+                                    "type": "text",
+                                    "content": " and tree-decomposed graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        649,
+                                        505,
+                                        673
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F^u, T^r) \\in S_{\\mathfrak{n}}^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        649,
+                                        505,
+                                        673
+                                    ],
+                                    "type": "text",
+                                    "content": ","
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                },
+                {
+                    "bbox": [
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+                    ],
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+                                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "\\operatorname {h o m} (F ^ {u}, G ^ {v}) = \\sum_ {(\\tilde {F} ^ {w}, \\tilde {T} ^ {s}) \\in \\mathcal {S} _ {n} ^ {M}} \\frac {\\mathrm {b E x t} ((F ^ {u} , T ^ {r}) , (\\tilde {F} ^ {w} , \\tilde {T} ^ {s})) \\cdot \\mathrm {b S t r H o m} ((\\tilde {F} ^ {w} , \\tilde {T} ^ {s}) , G ^ {v})}{\\operatorname {a u t} (\\tilde {F} ^ {w} , \\tilde {T} ^ {s})}.",
+                                    "image_path": "f4b58983318a57eaf04a6edb7389b31707b8c01e7c4a839f1657e4485bfbf1e0.jpg"
+                                }
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+                        }
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "The edge-level result is similar."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 22
+                }
+            ],
+            "discarded_blocks": [
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
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+                                310,
+                                761
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        299,
+                                        750,
+                                        310,
+                                        761
+                                    ],
+                                    "type": "text",
+                                    "content": "51"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 23
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 50
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        82,
+                        504,
+                        106
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                504,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma D.7. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": " be any model. For any graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": " and tree-decomposed graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F^u, T^r) \\in S_{\\mathfrak{n}}^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        106
+                                    ],
+                                    "type": "text",
+                                    "content": ","
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        109,
+                        110,
+                        501,
+                        150
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                109,
+                                110,
+                                501,
+                                150
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        109,
+                                        110,
+                                        501,
+                                        150
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\mathsf{bStrHom}((F^{u},T^{r}),G^{v}) = \\sum_{(\\tilde{F}^{w},\\tilde{T}^{s})\\in \\mathcal{S}_{n}^{M}}\\frac{\\mathsf{bStrSurj}\\left((F^{u},T^{r}),( \\tilde{F}^{w},\\tilde{T}^{s})\\right)\\cdot\\mathsf{blso}\\left((\\tilde{F}^{w},\\tilde{T}^{s}),G^{v}\\right)}{\\mathsf{aut}(\\tilde{F}^{w},\\tilde{T}^{s})},",
+                                    "image_path": "4662faaf051d63180c8421e6f79a78a79412378cefc110c442200d9005c19e7d.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        105,
+                        155,
+                        233,
+                        167
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                155,
+                                233,
+                                167
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        155,
+                                        233,
+                                        167
+                                    ],
+                                    "type": "text",
+                                    "content": "The edge-level result is similar."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        168,
+                        504,
+                        206
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                168,
+                                504,
+                                206
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        168,
+                                        504,
+                                        206
+                                    ],
+                                    "type": "text",
+                                    "content": "Corollary D.8. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        168,
+                                        504,
+                                        206
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        168,
+                                        504,
+                                        206
+                                    ],
+                                    "type": "text",
+                                    "content": " be any model. For any two graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        168,
+                                        504,
+                                        206
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^u, H^v, \\chi_G^M(u) = \\chi_H^M(v)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        168,
+                                        504,
+                                        206
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        168,
+                                        504,
+                                        206
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{hom}(F^w, G^u) = \\operatorname{hom}(F^w, H^v)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        168,
+                                        504,
+                                        206
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        168,
+                                        504,
+                                        206
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F^w, T^r) \\in S_{\\mathfrak{n}}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        168,
+                                        504,
+                                        206
+                                    ],
+                                    "type": "text",
+                                    "content": ". For any two graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        168,
+                                        504,
+                                        206
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^{uv}, H^{wx}, \\chi_G^M(u, v) = \\chi_H^M(w, x)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        168,
+                                        504,
+                                        206
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        168,
+                                        504,
+                                        206
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{hom}(F^{yz}, G^{uv}) = \\operatorname{hom}(F^{yz}, H^{wx})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        168,
+                                        504,
+                                        206
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        168,
+                                        504,
+                                        206
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F^{yz}, T^r) \\in S_{\\mathfrak{e}}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        168,
+                                        504,
+                                        206
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        105,
+                        208,
+                        323,
+                        220
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                208,
+                                323,
+                                220
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        208,
+                                        323,
+                                        220
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma D.9. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        208,
+                                        323,
+                                        220
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\mathrm{Sub}, \\mathrm{L}, \\mathrm{LF}, \\mathrm{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        208,
+                                        323,
+                                        220
+                                    ],
+                                    "type": "text",
+                                    "content": " be any model."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        120,
+                        224,
+                        504,
+                        276
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 8,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                120,
+                                224,
+                                504,
+                                248
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        120,
+                                        224,
+                                        504,
+                                        248
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                120,
+                                                224,
+                                                504,
+                                                248
+                                            ],
+                                            "type": "text",
+                                            "content": "- For any rooted connected graph "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                224,
+                                                504,
+                                                248
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F^u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                224,
+                                                504,
+                                                248
+                                            ],
+                                            "type": "text",
+                                            "content": ", "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                224,
+                                                504,
+                                                248
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F^u \\in \\mathcal{F}_{\\mathfrak{n}}^{M}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                224,
+                                                504,
+                                                248
+                                            ],
+                                            "type": "text",
+                                            "content": " iff there is a tree decomposition "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                224,
+                                                504,
+                                                248
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T^{r}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                224,
+                                                504,
+                                                248
+                                            ],
+                                            "type": "text",
+                                            "content": " of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                224,
+                                                504,
+                                                248
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F^u"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                224,
+                                                504,
+                                                248
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                224,
+                                                504,
+                                                248
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(F^u, T^r) \\in S_{\\mathfrak{n}}^{M}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                224,
+                                                504,
+                                                248
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 6
+                        },
+                        {
+                            "bbox": [
+                                121,
+                                252,
+                                504,
+                                276
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        252,
+                                        504,
+                                        276
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                121,
+                                                252,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "text",
+                                            "content": "- For any rooted connected graph "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                252,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F^{uv}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                252,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "text",
+                                            "content": ", "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                252,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F^{uv} \\in \\mathcal{F}_{\\mathrm{e}}^{M}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                252,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "text",
+                                            "content": " iff there is a tree decomposition "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                252,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T^{r}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                252,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "text",
+                                            "content": " of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                252,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F^{uv}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                252,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                252,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(F^{uv}, T^{r}) \\in \\mathcal{S}_{\\mathrm{e}}^{M}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                252,
+                                                504,
+                                                276
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 7
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        104,
+                        284,
+                        504,
+                        308
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                284,
+                                504,
+                                308
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        284,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": "By combining the proof of previous lemmas and theorems, we can prove that both "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        284,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}_{\\mathrm{n}}^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        284,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        284,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}_{\\mathrm{e}}^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        284,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfy Definition 3.6(a) for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        284,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\mathrm{Sub}, \\mathrm{L}, \\mathrm{LF}, \\mathrm{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        284,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        105,
+                        320,
+                        223,
+                        331
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                320,
+                                223,
+                                331
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        320,
+                                        223,
+                                        331
+                                    ],
+                                    "type": "text",
+                                    "content": "D.2 COUNTEREXAMPLES"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        340,
+                        504,
+                        419
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                340,
+                                504,
+                                419
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "text",
+                                    "content": "Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F^{\\mathbf{w}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "text",
+                                    "content": " be a rooted graph that marks the special vertices "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_{1},\\dots ,w_{m}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "text",
+                                    "content": ". Assume that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F^{\\mathbf{w}} \\notin \\mathcal{F}_{\\mathfrak{n}}^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "text",
+                                    "content": " ("
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m = 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "text",
+                                    "content": ") or "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F^{\\mathbf{w}} \\notin \\mathcal{F}_{\\mathfrak{e}}^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "text",
+                                    "content": " ("
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m = 2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "text",
+                                    "content": "). In this subsection, we will construct a pair of graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^{\\mathbf{u}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H^{\\mathbf{v}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^M (\\pmb {u}) = \\chi_H^M (\\pmb {v})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{hom}(F^{\\mathbf{w}},G^{\\mathbf{u}})\\neq \\mathrm{hom}(F^{\\mathbf{w}},H^{\\mathbf{v}})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "text",
+                                    "content": ". However, it turns out that, if we naively follow the proof in Appendix C.4 by constructing the Fürer graph and twisted Fürer graph with respect to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "text",
+                                    "content": " without considering the marked vertices "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{w}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "text",
+                                    "content": ", then the graphs may no longer be counterexamples here. For example, for the edge-level expressivity, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F^{w_1,w_2}\\notin \\mathcal{F}_{\\mathfrak{e}}^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "text",
+                                    "content": " does not imply that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F\\notin \\mathcal{F}^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        340,
+                                        504,
+                                        419
+                                    ],
+                                    "type": "text",
+                                    "content": " can be any model studied in this paper."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        104,
+                        423,
+                        504,
+                        446
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                423,
+                                504,
+                                446
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        423,
+                                        504,
+                                        446
+                                    ],
+                                    "type": "text",
+                                    "content": "To address the problem, we instead introduce a novel construction of counterexample graphs defined as follows:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        104,
+                        449,
+                        505,
+                        507
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                449,
+                                505,
+                                507
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition D.10 (Clique-augmented Fürer graphs). Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": " be any connected graph and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_{1}, \\dots, w_{m} \\in V_{F}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": " be a sequence of vertices. Given an integer "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k \\geq m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": ", the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": "-clique-augmented Fürer graph with respect to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F^{\\mathbf{w}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": ", denoted by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G_{k}(F^{\\mathbf{w}})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": ", is the Fürer graph of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": " is the union of graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": " and a "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": "-clique that contains "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_{1}, \\dots, w_{m}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": " (and does not contain other vertices in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": "). The twisted Fürer graph of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G_{k}(F^{\\mathbf{w}})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": " is denoted as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H_{k}(F^{\\mathbf{w}})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        449,
+                                        505,
+                                        507
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        105,
+                        514,
+                        248,
+                        526
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                514,
+                                248,
+                                526
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        514,
+                                        248,
+                                        526
+                                    ],
+                                    "type": "text",
+                                    "content": "We have the following main result:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        105,
+                        529,
+                        414,
+                        541
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                529,
+                                414,
+                                541
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        529,
+                                        414,
+                                        541
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem D.11. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        529,
+                                        414,
+                                        541
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\mathrm{Sub}, \\mathrm{L}, \\mathrm{LF}, \\mathrm{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        529,
+                                        414,
+                                        541
+                                    ],
+                                    "type": "text",
+                                    "content": " be any model defined in Section 2."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        120,
+                        545,
+                        505,
+                        613
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 18,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                120,
+                                545,
+                                504,
+                                571
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        120,
+                                        545,
+                                        504,
+                                        571
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                120,
+                                                545,
+                                                504,
+                                                571
+                                            ],
+                                            "type": "text",
+                                            "content": "- For any rooted graph "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                545,
+                                                504,
+                                                571
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F^w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                545,
+                                                504,
+                                                571
+                                            ],
+                                            "type": "text",
+                                            "content": " marking vertex "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                545,
+                                                504,
+                                                571
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                545,
+                                                504,
+                                                571
+                                            ],
+                                            "type": "text",
+                                            "content": ", if "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                545,
+                                                504,
+                                                571
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F^w \\notin \\mathcal{F}_{\\mathfrak{n}}^{M}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                545,
+                                                504,
+                                                571
+                                            ],
+                                            "type": "text",
+                                            "content": ", then there is a vertex "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                545,
+                                                504,
+                                                571
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(w, U) \\in \\operatorname{Meta}_{\\tilde{F}}(w)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                545,
+                                                504,
+                                                571
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                545,
+                                                504,
+                                                571
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "U \\subset V_F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                545,
+                                                504,
+                                                571
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                545,
+                                                504,
+                                                571
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\chi_{G_4(F^w)}^M((w, \\emptyset)) = \\chi_{H_4(F^w)}^M((w, U))"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                545,
+                                                504,
+                                                571
+                                            ],
+                                            "type": "text",
+                                            "content": ";"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 16
+                        },
+                        {
+                            "bbox": [
+                                121,
+                                575,
+                                505,
+                                613
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        575,
+                                        505,
+                                        613
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                121,
+                                                575,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "text",
+                                            "content": "- For any rooted graph "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                575,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F^w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                575,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "text",
+                                            "content": " marking two vertices "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                575,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w_1, w_2"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                575,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "text",
+                                            "content": ", if "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                575,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F^w \\notin \\mathcal{F}_{\\mathrm{e}}^{M}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                575,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "text",
+                                            "content": ", then there are two vertices "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                575,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(w_1, U_1) \\in \\operatorname{Meta}_{\\tilde{F}}(w_1)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                575,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                575,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(w_2, U_2) \\in \\operatorname{Meta}_{\\tilde{F}}(w_2)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                575,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                575,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "U_1, U_2 \\subset V_F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                575,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                575,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\chi_{G_4(F^w)}^{M}((w_1, \\emptyset), (w_2, \\emptyset)) = \\chi_{H_4(F^w)}^{M}((w_1, U_1), (w_2, U_2))"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                575,
+                                                505,
+                                                613
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 17
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        104,
+                        626,
+                        504,
+                        705
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                626,
+                                504,
+                                705
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        705
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Below, we only give a proof of the second item. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        705
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        705
+                                    ],
+                                    "type": "text",
+                                    "content": " be the union of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        705
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        705
+                                    ],
+                                    "type": "text",
+                                    "content": " and the 4-clique according to Definition D.10. Similar to the graph-level expressivity, we will extend the pebble game defined in Appendix C.4 to edge-level. For the edge-level pebble game, the two pebbles "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        705
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathfrak{p}_u,\\mathfrak{p}_v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        705
+                                    ],
+                                    "type": "text",
+                                    "content": " are initially placed on two predefined vertices "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        705
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_{1},w_{2}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        705
+                                    ],
+                                    "type": "text",
+                                    "content": ", and Bob initially selects a predefined set of connected components "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        705
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Q\\subset \\mathbb{C}\\mathbb{C}_{\\tilde{F}}(\\{w_1,w_2\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        705
+                                    ],
+                                    "type": "text",
+                                    "content": " (of odd size). Then, the game executes the same process as in the text above Proposition C.49. The winning criterion for the two players is also the same. We denote the above edge-level pebble game as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        705
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{PG}_{\\tilde{F}}(w_1,w_2,Q)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        626,
+                                        504,
+                                        705
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                },
+                {
+                    "bbox": [
+                        104,
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+                    ],
+                    "type": "text",
+                    "angle": 0,
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Based on the isomorphism property of Fürer graphs established in Zhang et al. (2023a, Theorem I.9 and Theorem I.17), it is straightforward to see that, for any vertices "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        709,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\xi, \\xi' \\in \\mathsf{Meta}_{\\tilde{F}}(w_1)"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " and"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                }
+            ],
+            "discarded_blocks": [
+                {
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+                    ],
+                    "type": "header",
+                    "angle": 0,
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
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+                                    "type": "text",
+                                    "content": "52"
+                                }
+                            ]
+                        }
+                    ],
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+                }
+            ],
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+            "para_blocks": [
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+                    ],
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+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                        506,
+                                        192
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\eta, \\eta' \\in \\mathsf{Meta}_{\\tilde{F}}(w_2), \\chi_{G_4(F^w)}^M(\\xi, \\eta) = \\chi_{H_4(F^w)}^M(\\xi', \\eta')"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "text",
+                                    "content": " iff Alice cannot win the edge-level pebble game "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{PG}_{\\tilde{F}}(w_1, w_2, Q)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "text",
+                                    "content": " for some "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Q \\subset \\mathbb{C}\\mathbb{C}_{\\tilde{F}}(\\{w_1, w_2\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "text",
+                                    "content": " (depending on "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\xi, \\xi', \\eta, \\eta'"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "text",
+                                    "content": "). Moreover, let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\xi' = (w_1, \\emptyset)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\eta' = (w_2, \\emptyset)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "text",
+                                    "content": " and let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\xi_2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\eta_2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "text",
+                                    "content": " range over all even-size subsets of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "N_F(w_1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "N_F(w_2)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "text",
+                                    "content": ", respectively, that is to say, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\xi_2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\eta_2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "text",
+                                    "content": " does not contain vertices that are only in the 4-clique. Then, the equivalent pebble game ranges over all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Q"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "text",
+                                    "content": " of odd size such that every connected component in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Q"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "text",
+                                    "content": " contains only vertices in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "text",
+                                    "content": ", i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Q \\subset \\mathbb{C}\\mathbb{C}_F(\\{w_1, w_2\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "text",
+                                    "content": ". Therefore, if Alice cannot win the edge-level pebble game "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{PG}_{\\tilde{F}}(w_1, w_2, Q)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "text",
+                                    "content": " for some "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Q \\subset \\mathbb{C}\\mathbb{C}_F(\\{w_1, w_2\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "text",
+                                    "content": ", then there are two vertices "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\xi \\in \\mathsf{Meta}_{\\tilde{F}}(w_1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\eta \\in \\mathsf{Meta}_{\\tilde{F}}(w_2)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\xi_2, \\eta_2 \\subset V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_{G_4(F^w)}^M((w_1, \\emptyset), (w_2, \\emptyset)) = \\chi_{H_4(F^w)}^M(\\xi, \\eta)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        506,
+                                        192
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        197,
+                        504,
+                        243
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                197,
+                                504,
+                                243
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        197,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "text",
+                                    "content": "It thus remains to prove that, if Alice can win the edge-level pebble game "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        197,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{PG}_{\\tilde{F}}(w_1,w_2,Q)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        197,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        197,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Q\\subset \\mathbb{C}C_F(\\{w_1,w_2\\})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        197,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "text",
+                                    "content": ", then "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        197,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F^{\\boldsymbol{w}}\\in \\mathcal{F}_{\\mathrm{e}}^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        197,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "text",
+                                    "content": ". To prove this result, we similarly define the game state graph as in Appendix C.4, with the only difference that the initial state is now "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        197,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(w_{1},w_{2},Q)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        197,
+                                        504,
+                                        243
+                                    ],
+                                    "type": "text",
+                                    "content": ". The proof is divided into the following steps:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        117,
+                        247,
+                        505,
+                        523
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 7,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                118,
+                                247,
+                                504,
+                                348
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        118,
+                                        247,
+                                        504,
+                                        348
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                118,
+                                                247,
+                                                504,
+                                                348
+                                            ],
+                                            "type": "text",
+                                            "content": "1. Assume that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                247,
+                                                504,
+                                                348
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "Q"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                247,
+                                                504,
+                                                348
+                                            ],
+                                            "type": "text",
+                                            "content": " contains a single connected component, i.e., "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                247,
+                                                504,
+                                                348
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "Q = \\{P\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                247,
+                                                504,
+                                                348
+                                            ],
+                                            "type": "text",
+                                            "content": ". We will prove that, if Alice can win the game "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                247,
+                                                504,
+                                                348
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{PG}_{\\tilde{F}}(w_1, w_2, \\{P\\})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                247,
+                                                504,
+                                                348
+                                            ],
+                                            "type": "text",
+                                            "content": ", then there is a game strategy for Alice such that for any reachable and non-terminal state "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                247,
+                                                504,
+                                                348
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(x, y, \\{P'\\})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                247,
+                                                504,
+                                                348
+                                            ],
+                                            "type": "text",
+                                            "content": ", "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                247,
+                                                504,
+                                                348
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P' \\subset P"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                247,
+                                                504,
+                                                348
+                                            ],
+                                            "type": "text",
+                                            "content": ". If it is not the case, pick any state "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                247,
+                                                504,
+                                                348
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(x, y, \\{P'\\})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                247,
+                                                504,
+                                                348
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                247,
+                                                504,
+                                                348
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P' \\notin P"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                247,
+                                                504,
+                                                348
+                                            ],
+                                            "type": "text",
+                                            "content": " and any path from "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                247,
+                                                504,
+                                                348
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(w_1, w_2, \\{P\\})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                247,
+                                                504,
+                                                348
+                                            ],
+                                            "type": "text",
+                                            "content": " to "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                247,
+                                                504,
+                                                348
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(x, y, \\{P'\\})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                247,
+                                                504,
+                                                348
+                                            ],
+                                            "type": "text",
+                                            "content": " does not pass any intermediate state that is not contracted. It follows that either "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                247,
+                                                504,
+                                                348
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w_1"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                247,
+                                                504,
+                                                348
+                                            ],
+                                            "type": "text",
+                                            "content": " or "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                247,
+                                                504,
+                                                348
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w_2"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                247,
+                                                504,
+                                                348
+                                            ],
+                                            "type": "text",
+                                            "content": " does not hold any pebble, and all edges connected to this vertex is in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                247,
+                                                504,
+                                                348
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P'"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                247,
+                                                504,
+                                                348
+                                            ],
+                                            "type": "text",
+                                            "content": ". Therefore, "
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                247,
+                                                504,
+                                                348
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P'"
+                                        },
+                                        {
+                                            "bbox": [
+                                                118,
+                                                247,
+                                                504,
+                                                348
+                                            ],
+                                            "type": "text",
+                                            "content": " contains all vertices in the 4-clique. In this case, Alice cannot win as Bob can always keep the connected component containing all vertices in the 4-clique in subsequent rounds, i.e., Bob always selects a component containing a vertex in the 4-clique that does not hold a pebble."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 3
+                        },
+                        {
+                            "bbox": [
+                                117,
+                                353,
+                                505,
+                                419
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        353,
+                                        505,
+                                        419
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                117,
+                                                353,
+                                                505,
+                                                419
+                                            ],
+                                            "type": "text",
+                                            "content": "2. Based on the above result, in the game "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                353,
+                                                505,
+                                                419
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{PG}_{\\tilde{F}}(w_1,w_2,\\{P\\})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                353,
+                                                505,
+                                                419
+                                            ],
+                                            "type": "text",
+                                            "content": ", the connected components selected by Bob are always in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                353,
+                                                505,
+                                                419
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                353,
+                                                505,
+                                                419
+                                            ],
+                                            "type": "text",
+                                            "content": ", and thus the game process is the same as "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                353,
+                                                505,
+                                                419
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{PG}_F(w_1,w_2,\\{P\\})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                353,
+                                                505,
+                                                419
+                                            ],
+                                            "type": "text",
+                                            "content": ". We can then prove the same result as Lemma C.52. Concretely, if Alice can win the edge-level pebble game "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                353,
+                                                505,
+                                                419
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathsf{PG}_{\\tilde{F}}(w_1,w_2,\\{P\\})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                353,
+                                                505,
+                                                419
+                                            ],
+                                            "type": "text",
+                                            "content": ", then there exists a game state graph "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                353,
+                                                505,
+                                                419
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "G^{\\mathbb{S}}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                353,
+                                                505,
+                                                419
+                                            ],
+                                            "type": "text",
+                                            "content": " corresponding to a winning strategy, such that for any transition "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                353,
+                                                505,
+                                                419
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "((x,y,\\{P'\\}), (x',y',\\{P''\\}))"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                353,
+                                                505,
+                                                419
+                                            ],
+                                            "type": "text",
+                                            "content": " where "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                353,
+                                                505,
+                                                419
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(x,y,\\{P'\\})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                353,
+                                                505,
+                                                419
+                                            ],
+                                            "type": "text",
+                                            "content": " is a reachable and non-terminal state, we have "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                353,
+                                                505,
+                                                419
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P'' \\subsetneq P'"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                353,
+                                                505,
+                                                419
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 4
+                        },
+                        {
+                            "bbox": [
+                                117,
+                                424,
+                                504,
+                                459
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        424,
+                                        504,
+                                        459
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                117,
+                                                424,
+                                                504,
+                                                459
+                                            ],
+                                            "type": "text",
+                                            "content": "3. Next, we can follow the same procedure in the proof of Theorem C.53 to construct a tree decomposition "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                424,
+                                                504,
+                                                459
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T^r"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                424,
+                                                504,
+                                                459
+                                            ],
+                                            "type": "text",
+                                            "content": " for the subgraph "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                424,
+                                                504,
+                                                459
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F[P]"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                424,
+                                                504,
+                                                459
+                                            ],
+                                            "type": "text",
+                                            "content": " containing all edges in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                424,
+                                                504,
+                                                459
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                424,
+                                                504,
+                                                459
+                                            ],
+                                            "type": "text",
+                                            "content": ", such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                424,
+                                                504,
+                                                459
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(F[P], T^r) \\in S_{\\mathbf{e}}^{M}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                424,
+                                                504,
+                                                459
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                424,
+                                                504,
+                                                459
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(r) = \\{w_1, w_2\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                424,
+                                                504,
+                                                459
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 5
+                        },
+                        {
+                            "bbox": [
+                                117,
+                                463,
+                                504,
+                                523
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        117,
+                                        463,
+                                        504,
+                                        523
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                117,
+                                                463,
+                                                504,
+                                                523
+                                            ],
+                                            "type": "text",
+                                            "content": "4. Finally, let "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                463,
+                                                504,
+                                                523
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                463,
+                                                504,
+                                                523
+                                            ],
+                                            "type": "text",
+                                            "content": " ranges over all connected components in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                463,
+                                                504,
+                                                523
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\mathbb{C}C_F(\\{w_1, w_2\\})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                463,
+                                                504,
+                                                523
+                                            ],
+                                            "type": "text",
+                                            "content": ", for each "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                463,
+                                                504,
+                                                523
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                463,
+                                                504,
+                                                523
+                                            ],
+                                            "type": "text",
+                                            "content": " we have a tree-decomposed graph. We can glue these tree-decomposed graphs by merging the root to obtain the tree decomposition of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                463,
+                                                504,
+                                                523
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                463,
+                                                504,
+                                                523
+                                            ],
+                                            "type": "text",
+                                            "content": ", because the root bags of all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                463,
+                                                504,
+                                                523
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T^r"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                463,
+                                                504,
+                                                523
+                                            ],
+                                            "type": "text",
+                                            "content": " are the same and any vertex "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                463,
+                                                504,
+                                                523
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "x \\notin \\{w_1, w_2\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                463,
+                                                504,
+                                                523
+                                            ],
+                                            "type": "text",
+                                            "content": " appears in the bag of only one "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                463,
+                                                504,
+                                                523
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "T^r"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                463,
+                                                504,
+                                                523
+                                            ],
+                                            "type": "text",
+                                            "content": ". Moreover, the glued tree "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                463,
+                                                504,
+                                                523
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\tilde{T}^s"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                463,
+                                                504,
+                                                523
+                                            ],
+                                            "type": "text",
+                                            "content": " clearly satisfies that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                463,
+                                                504,
+                                                523
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(F, \\tilde{T}^s) \\in S_{\\mathrm{e}}^{M}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                117,
+                                                463,
+                                                504,
+                                                523
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 6
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        104,
+                        526,
+                        392,
+                        539
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                526,
+                                392,
+                                539
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        526,
+                                        392,
+                                        539
+                                    ],
+                                    "type": "text",
+                                    "content": "Combining these items shows that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        526,
+                                        392,
+                                        539
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F^{\\boldsymbol{w}} \\in \\mathcal{F}_{\\mathrm{e}}^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        526,
+                                        392,
+                                        539
+                                    ],
+                                    "type": "text",
+                                    "content": " and concludes the proof."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        549,
+                        504,
+                        599
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                549,
+                                504,
+                                599
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": "Corollary D.12. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F^w"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": " any rooted graph marking "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": " vertices such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F^w \\notin \\mathcal{F}_n^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": " ("
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m = 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": ") or "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F^w \\notin \\mathcal{F}_{\\mathrm{e}}^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": " ("
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m = 2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": "), and let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G_{k}(F^{w})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H_{k}(F^{w})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": " be clique-augmented Fuirer graphs defined above. Denote by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\xi, \\eta \\in V_{G_k(F^w)}^m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": " two vertex tuples of length "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\xi_{i} = (w_{i}, \\emptyset)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\eta_{i} \\in \\operatorname{Meta}_{\\tilde{F}}(w_{i})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": " with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\eta_{i,2} \\subset V_{F}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": " for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i \\in [m]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{hom}(F^w, [G_k(F^w)]^\\xi) \\neq \\operatorname{hom}(F^w, [H_k(F^w)]^\\eta)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        599
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
+                        615,
+                        504,
+                        638
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                615,
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+                                638
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        615,
+                                        504,
+                                        638
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. The proof exactly parallels that of Theorem C.47 by separately considering three cases. We omit the proof for clarity."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Corollary D.13. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\mathrm{Sub}, \\mathrm{L}, \\mathrm{LF}, \\mathrm{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        648,
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+                                    ],
+                                    "type": "text",
+                                    "content": " be any model."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
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+                    "type": "list",
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+                            "bbox": [
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+                                            "type": "text",
+                                            "content": "- For any graph "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                665,
+                                                504,
+                                                703
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F^w \\notin \\mathcal{F}_{\\mathfrak{n}}^{M}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
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+                                            ],
+                                            "type": "text",
+                                            "content": ", let "
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "G_4(F^w)"
+                                        },
+                                        {
+                                            "bbox": [
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+                                                665,
+                                                504,
+                                                703
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
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+                                                120,
+                                                665,
+                                                504,
+                                                703
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "H_4(F^w)"
+                                        },
+                                        {
+                                            "bbox": [
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+                                                504,
+                                                703
+                                            ],
+                                            "type": "text",
+                                            "content": " be the clique-augmented Fürer graph and the corresponding twisted Fürer graph. Then, there exists two vertices "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                665,
+                                                504,
+                                                703
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\xi, \\xi'"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
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+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
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+                                            "bbox": [
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\operatorname{hom}(F^w, [G_4(F^w)]^\\xi) \\neq \\operatorname{hom}(F^w, [H_4(F^w)]^{\\xi'})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
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+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
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+                                            "type": "inline_equation",
+                                            "content": "\\chi_{G_4(F^w)}^M(\\xi) = \\chi_{H_4(F^w)}^M(\\xi')"
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                665,
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+                                                703
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 12
+                        },
+                        {
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+                            "type": "text",
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+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": "- For any graph "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                708,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "F^{wx} \\notin \\mathcal{F}_{\\mathrm{e}}^{M}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                708,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "text",
+                                            "content": ", let "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                708,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "G_{4}(F^{wx})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                708,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
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+                                                504,
+                                                733
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "H_{4}(F^{wx})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                708,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "text",
+                                            "content": " be the clique-augmented Fürer graph and the corresponding twisted Fürer graph defined in Definition D.10. Then, there exists"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 13
+                        }
+                    ],
+                    "sub_type": "text"
+                }
+            ],
+            "discarded_blocks": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
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+                },
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+                        299,
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+                                    "type": "text",
+                                    "content": "53"
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+                                    "bbox": [
+                                        127,
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+                                        109
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+                                    "type": "text",
+                                    "content": "four vertices "
+                                },
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+                                    "bbox": [
+                                        127,
+                                        80,
+                                        506,
+                                        109
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\xi, \\eta, \\xi', \\eta'"
+                                },
+                                {
+                                    "bbox": [
+                                        127,
+                                        80,
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+                                        109
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        127,
+                                        80,
+                                        506,
+                                        109
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{hom}(F^{wx}, [G_4(F^{wx})]^{\\xi \\eta}) \\neq \\operatorname{hom}(F^{wx}, [H_4(F^{wx})]^{\\xi' \\eta'})"
+                                },
+                                {
+                                    "bbox": [
+                                        127,
+                                        80,
+                                        506,
+                                        109
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
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+                                    "bbox": [
+                                        127,
+                                        80,
+                                        506,
+                                        109
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_{G_4(F^{wx})}^M(\\xi, \\eta) = \\chi_{H_4(F^{wx})}^M(\\xi', \\eta')"
+                                },
+                                {
+                                    "bbox": [
+                                        127,
+                                        80,
+                                        506,
+                                        109
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
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+                    "index": 1
+                },
+                {
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+                    "angle": 0,
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. The proof directly follows from Corollary D.12 and Theorem D.11. "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\square"
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+                                    "type": "text",
+                                    "content": "E HIGHER-ORDER GNNS"
+                                }
+                            ]
+                        }
+                    ],
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+                },
+                {
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+                    ],
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+                                    ],
+                                    "type": "text",
+                                    "content": "E.1 DEFINITION OF HIGHER-ORDER GNNS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
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+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "In this subsection, we give formal definitions of the CR algorithms for various higher-order GNNs, each of which generalizes a model in Section 2."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
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+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "- Subgraph "
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
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+                                        292
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        120,
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+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN. In a Subgraph "
+                                },
+                                {
+                                    "bbox": [
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+                                        220,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
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+                                        220,
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+                                        292
+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN, a graph "
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
+                                        505,
+                                        292
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
+                                        505,
+                                        292
+                                    ],
+                                    "type": "text",
+                                    "content": " is treated as a set of subgraphs "
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
+                                        505,
+                                        292
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{G^u : u \\in V_G^k\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
+                                        505,
+                                        292
+                                    ],
+                                    "type": "text",
+                                    "content": ", where each subgraph "
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
+                                        505,
+                                        292
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^u"
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
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+                                        292
+                                    ],
+                                    "type": "text",
+                                    "content": " is obtained from "
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
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+                                        292
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
+                                        505,
+                                        292
+                                    ],
+                                    "type": "text",
+                                    "content": " by marking "
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
+                                        505,
+                                        292
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
+                                        505,
+                                        292
+                                    ],
+                                    "type": "text",
+                                    "content": " special vertices "
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
+                                        505,
+                                        292
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\in V_G^k"
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
+                                        505,
+                                        292
+                                    ],
+                                    "type": "text",
+                                    "content": ", and thus there are "
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
+                                        505,
+                                        292
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n^k"
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
+                                        505,
+                                        292
+                                    ],
+                                    "type": "text",
+                                    "content": " subgraphs when "
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
+                                        505,
+                                        292
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
+                                        505,
+                                        292
+                                    ],
+                                    "type": "text",
+                                    "content": " has "
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
+                                        505,
+                                        292
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n"
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
+                                        505,
+                                        292
+                                    ],
+                                    "type": "text",
+                                    "content": " vertices. Subgraph GNN maintains a color "
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
+                                        505,
+                                        292
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^{\\mathrm{Sub}(k)}(\\pmb{u}, v)"
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
+                                        505,
+                                        292
+                                    ],
+                                    "type": "text",
+                                    "content": " for each vertex "
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
+                                        505,
+                                        292
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v"
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
+                                        505,
+                                        292
+                                    ],
+                                    "type": "text",
+                                    "content": " in graph "
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
+                                        505,
+                                        292
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^u"
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
+                                        505,
+                                        292
+                                    ],
+                                    "type": "text",
+                                    "content": ". Initially, "
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
+                                        505,
+                                        292
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^{\\mathrm{Sub}(k), (0)}(\\pmb{u}, v) = (\\ell_G(v), \\mathsf{atp}_G(\\pmb{u}), \\mathbb{I}[u_1 = v], \\dots, \\mathbb{I}[u_k = v])"
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
+                                        505,
+                                        292
+                                    ],
+                                    "type": "text",
+                                    "content": ". It then runs MPNNs independently on each graph "
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
+                                        505,
+                                        292
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^u"
+                                },
+                                {
+                                    "bbox": [
+                                        120,
+                                        220,
+                                        505,
+                                        292
+                                    ],
+                                    "type": "text",
+                                    "content": ":"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        140,
+                        297,
+                        505,
+                        317
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                140,
+                                297,
+                                505,
+                                317
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        140,
+                                        297,
+                                        505,
+                                        317
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\chi_ {G} ^ {\\operatorname {S u b} (k), (t + 1)} (\\boldsymbol {u}, v) = \\operatorname {h a s h} \\left(\\chi_ {G} ^ {\\operatorname {S u b} (k), (t)} (\\boldsymbol {u}, v), \\{\\left\\{\\chi_ {G} ^ {\\operatorname {S u b} (k), (t)} (\\boldsymbol {u}, w): w \\in N _ {G} (v) \\right\\} \\}\\right). \\tag {26}",
+                                    "image_path": "35f150bb6e070dce50086ef966b85576d1adc35ce9c4f258f37c87f7d64bd3a4.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        129,
+                        323,
+                        504,
+                        363
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                323,
+                                504,
+                                363
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        323,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": "Denote the stable color as "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        323,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^{\\mathrm{Sub}(k)}(\\pmb{u},\\upsilon)"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        323,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": ". Define "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        323,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^{\\mathrm{Sub}(k)}(\\pmb{u})\\coloneqq \\{\\{\\chi_G^{\\mathrm{Sub}(k)}(\\pmb{u},\\upsilon):v\\in V_G\\}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        323,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, the graph representation is defined as "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        323,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^{\\mathrm{Sub}(k)}(G) = \\{\\{\\chi_G^{\\mathrm{Sub}(k)}(\\pmb {u}):\\pmb {u}\\in V_G\\} \\}"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        323,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": ". We remark that Subgraph "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        323,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        323,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": " -GNN is precisely the "
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        323,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        129,
+                                        323,
+                                        504,
+                                        363
+                                    ],
+                                    "type": "text",
+                                    "content": " -VSAN proposed in Qian et al. (2022)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        121,
+                        365,
+                        505,
+                        404
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                121,
+                                365,
+                                505,
+                                404
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        365,
+                                        505,
+                                        404
+                                    ],
+                                    "type": "text",
+                                    "content": "- Local "
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        365,
+                                        505,
+                                        404
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        365,
+                                        505,
+                                        404
+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN. Local "
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        365,
+                                        505,
+                                        404
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        365,
+                                        505,
+                                        404
+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN maintains a color "
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        365,
+                                        505,
+                                        404
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^{\\mathsf{L}(k)}(\\pmb{u})"
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        365,
+                                        505,
+                                        404
+                                    ],
+                                    "type": "text",
+                                    "content": " for each vertex "
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        365,
+                                        505,
+                                        404
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        365,
+                                        505,
+                                        404
+                                    ],
+                                    "type": "text",
+                                    "content": "-tuple "
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        365,
+                                        505,
+                                        404
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{u} \\in V_G^k"
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        365,
+                                        505,
+                                        404
+                                    ],
+                                    "type": "text",
+                                    "content": ". Initially, "
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        365,
+                                        505,
+                                        404
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^{\\mathsf{L}(k),(0)}(\\pmb{u}) = (\\ell_G(u_1), \\dots, \\ell_G(u_k), \\mathsf{atp}_G(\\pmb{u}))"
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        365,
+                                        505,
+                                        404
+                                    ],
+                                    "type": "text",
+                                    "content": ", called the isomorphism type of vertex "
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        365,
+                                        505,
+                                        404
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        365,
+                                        505,
+                                        404
+                                    ],
+                                    "type": "text",
+                                    "content": "-tuple "
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        365,
+                                        505,
+                                        404
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{u}"
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        365,
+                                        505,
+                                        404
+                                    ],
+                                    "type": "text",
+                                    "content": ", where "
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        365,
+                                        505,
+                                        404
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{atp}_G(\\pmb{u})"
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        365,
+                                        505,
+                                        404
+                                    ],
+                                    "type": "text",
+                                    "content": " is the atomic type of "
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        365,
+                                        505,
+                                        404
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{u}"
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        365,
+                                        505,
+                                        404
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, in each iteration "
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        365,
+                                        505,
+                                        404
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "t + 1"
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        365,
+                                        505,
+                                        404
+                                    ],
+                                    "type": "text",
+                                    "content": ","
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        167,
+                        409,
+                        504,
+                        452
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                167,
+                                409,
+                                504,
+                                452
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        167,
+                                        409,
+                                        504,
+                                        452
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\chi_ {G} ^ {\\mathsf {L} (k), (t + 1)} (\\boldsymbol {u}) = \\operatorname {h a s h} \\left(\\chi_ {G} ^ {\\mathsf {L} (k), (t)} (\\boldsymbol {u}), \\{\\{\\chi_ {G} ^ {\\mathsf {L} (k), (t)} (\\boldsymbol {v}): \\boldsymbol {v} \\in N _ {G} ^ {(1)} (\\boldsymbol {u}) \\} \\}, \\dots , \\right. \\\\ \\left. \\left\\{\\left\\{\\chi_ {G} ^ {\\mathrm {L} (k), (t)} (\\boldsymbol {v}): \\boldsymbol {v} \\in N _ {G} ^ {(k)} (\\boldsymbol {u}) \\right\\} \\right\\}\\right), \\tag {27} \\\\ \\end{array}",
+                                    "image_path": "a6077a6f564f9ab12c21a25b6dfaaa0b1982d29689b42cfa0de45ba4d558b2c3.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        128,
+                        458,
+                        504,
+                        497
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                128,
+                                458,
+                                504,
+                                497
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        128,
+                                        458,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "text",
+                                    "content": "where "
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        458,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "N_G^{(j)}(\\pmb{u}) = \\{(u_1, \\dots, u_{j-1}, w, u_{j+1}, \\dots, u_k) : w \\in N_G(u_j)\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        458,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "text",
+                                    "content": ". Denote the stable color as "
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        458,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^{\\mathsf{L}(k)}(\\pmb{u})"
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        458,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "text",
+                                    "content": ". The graph representation is defined as "
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        458,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^{\\mathsf{L}(k)}(G) := \\{\\chi_G^{\\mathsf{L}(k)}(\\pmb{u}) : \\pmb{u} \\in V_G\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        458,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "text",
+                                    "content": ". We remark that Local "
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        458,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        458,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN is precisely the "
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        458,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\delta"
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        458,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "text",
+                                    "content": "-"
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        458,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        128,
+                                        458,
+                                        504,
+                                        497
+                                    ],
+                                    "type": "text",
+                                    "content": "-LWL proposed in Morris et al. (2020)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        121,
+                        498,
+                        505,
+                        521
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                121,
+                                498,
+                                505,
+                                521
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        498,
+                                        505,
+                                        521
+                                    ],
+                                    "type": "text",
+                                    "content": "- Local "
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        498,
+                                        505,
+                                        521
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        498,
+                                        505,
+                                        521
+                                    ],
+                                    "type": "text",
+                                    "content": "-FGNN. Local "
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        498,
+                                        505,
+                                        521
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        498,
+                                        505,
+                                        521
+                                    ],
+                                    "type": "text",
+                                    "content": "-FGNN is almost the same as Local "
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        498,
+                                        505,
+                                        521
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        121,
+                                        498,
+                                        505,
+                                        521
+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN except that the update formula is replaced by the following one:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        129,
+                        526,
+                        507,
+                        577
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                526,
+                                507,
+                                577
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        526,
+                                        507,
+                                        577
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\chi_ {G} ^ {\\mathsf {L F} (k), (t + 1)} (\\boldsymbol {u}) = \\operatorname {h a s h} \\left(\\chi_ {G} ^ {\\mathsf {L F} (k), (t)} (\\boldsymbol {u}), \\{\\{\\left(\\chi_ {G} ^ {\\mathsf {L F} (k), (t)} (w, u _ {2}, \\dots , u _ {k}), \\chi_ {G} ^ {\\mathsf {L F} (k), (t)} (u _ {1}, w, u _ {3}, \\dots , u _ {k}), \\right. \\right. \\\\ \\left. \\dots , \\chi_ {G} ^ {\\mathsf {L F} (k), (t)} \\left(u _ {1}, \\dots , u _ {k - 1}, w\\right): w \\in N _ {G} \\left(u _ {1}\\right) \\cup \\dots \\cup N _ {G} \\left(u _ {k}\\right) \\right\\} \\Bigg). \\tag {28} \\\\ \\end{array}",
+                                    "image_path": "fa94ad97039df85c0bd8c24a36b2869271b15af71214c4d8d275d05d1869e43b.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        129,
+                        578,
+                        502,
+                        589
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                129,
+                                578,
+                                502,
+                                589
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        129,
+                                        578,
+                                        502,
+                                        589
+                                    ],
+                                    "type": "text",
+                                    "content": "We remark that Local "
+                                },
+                                {
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+                                    "type": "inline_equation",
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+                                },
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+                                    "type": "text",
+                                    "content": "-FGNN is precisely the "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{SLFWL}(k)"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " proposed in Zhang et al. (2023a)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
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+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "- "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "-FGNN. It is just the standard "
+                                },
+                                {
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+                                    "type": "inline_equation",
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "-FWL (Cai et al., 1992). Compared with Local "
+                                },
+                                {
+                                    "bbox": [
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+                                        613
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "-FGNN, the update formula is now global:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
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+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} \\chi_ {G} ^ {\\mathsf {F} (k), (t + 1)} (\\boldsymbol {u}) = \\operatorname {h a s h} \\left(\\chi_ {G} ^ {\\mathsf {F} (k), (t)} (\\boldsymbol {u}), \\{\\{\\left(\\chi_ {G} ^ {\\mathsf {F} (k), (t)} (w, u _ {2}, \\dots , u _ {k}), \\chi_ {G} ^ {\\mathsf {F} (k), (t)} (u _ {1}, w, u _ {3}, \\dots , u _ {k}), \\right. \\right. \\\\ \\left. \\dots , \\chi_ {G} ^ {\\mathsf {F} (k), (t)} \\left(u _ {1}, \\dots , u _ {k - 1}, w\\right): w \\in V _ {G} \\right\\}. \\Bigg). \\tag {29} \\\\ \\end{array}",
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+                                }
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+                {
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+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "We note that the computational complexity of Subgraph "
+                                },
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": "-GNN, Local "
+                                },
+                                {
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+                                    "type": "inline_equation",
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "-GNN, and Local "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        677,
+                                        505,
+                                        732
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "-FGNN is the same, i.e., "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Theta(n^{k - 1}m)"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " for a graph of "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " vertices and "
+                                },
+                                {
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " edges. Moreover, it is easy to see that Subgraph 0-GNN, Local 1-GNN, and Local 1-FGNN all reduce to MPNN. For "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "-FGNN, the computational complexity is "
+                                },
+                                {
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\Theta(n^{k + 1})"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ", which is strictly higher than Subgraph "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(k - 1)"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN, Local "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN, and Local "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
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+                                },
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+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "-FGNN."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                }
+            ],
+            "discarded_blocks": [
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
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+                                    "content": "54"
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+                                    "type": "text",
+                                    "content": "E.2 HIGHER-ORDER STRONG NED"
+                                }
+                            ]
+                        }
+                    ],
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+                },
+                {
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+                                    "type": "text",
+                                    "content": "In this subsection, we generalize the strong NED into higher-order versions. We first define the concept of higher-order ear:"
+                                }
+                            ]
+                        }
+                    ],
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+                },
+                {
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+                                    "type": "text",
+                                    "content": "Definition E.1 ("
+                                },
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+                                    "type": "inline_equation",
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "-order ear). Given integer "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "k \\geq 1"
+                                },
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+                                    "type": "text",
+                                    "content": ", a "
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+                                    "type": "inline_equation",
+                                    "content": "k"
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+                                {
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+                                    "type": "text",
+                                    "content": "-order ear is a graph "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " formed by the union of "
+                                },
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+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " paths "
+                                },
+                                {
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{1}, \\dots, P_{k}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " (possibly with zero length) plus an edge set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        127,
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+                                        159
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Q"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " satisfying the following conditions:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
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+                    "type": "list",
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+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
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+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": "- For each path "
+                                        },
+                                        {
+                                            "bbox": [
+                                                120,
+                                                166,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_{i}"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": ", denote its two endpoints as "
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "u_{i}, v_{i}"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": ", called the outer endpoint and inner endpoint, respectively. Then, all edges in "
+                                        },
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+                                            "bbox": [
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+                                            "type": "inline_equation",
+                                            "content": "Q"
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+                                            "type": "text",
+                                            "content": " are linked between inner endpoints, i.e., "
+                                        },
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "Q \\subset \\{\\{v_{i}, v_{j}\\} : 1 \\leq i, j \\leq k, v_{i} \\neq v_{j}\\}"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
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+                            "index": 4
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "type": "text",
+                            "angle": 0,
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+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": "- Any different paths "
+                                        },
+                                        {
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "P_{i}, P_{j}"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": " do not intersect except at the inner endpoint (when "
+                                        },
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "v_{i} = v_{j}"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": ")."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 5
+                        },
+                        {
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+                            "type": "text",
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+                                            "type": "text",
+                                            "content": "- "
+                                        },
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+                                            "type": "inline_equation",
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+                                        },
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+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": " is a connected graph."
+                                        }
+                                    ]
+                                }
+                            ],
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+                        }
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+                        251
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                239,
+                                426,
+                                251
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        239,
+                                        426,
+                                        251
+                                    ],
+                                    "type": "text",
+                                    "content": "The endpoints of the "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        239,
+                                        426,
+                                        251
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        239,
+                                        426,
+                                        251
+                                    ],
+                                    "type": "text",
+                                    "content": "-order ear is defined to be all outer endpoints "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        239,
+                                        426,
+                                        251
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u_{1}, \\dots, u_{k}"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        239,
+                                        426,
+                                        251
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        258,
+                        504,
+                        323
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                258,
+                                504,
+                                323
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        258,
+                                        504,
+                                        323
+                                    ],
+                                    "type": "text",
+                                    "content": "It is easy to see that a 2-order ear is precisely a simple path, since linking two different paths (possibly with an additional edge) still yields a path. Below, we denote by inner "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        258,
+                                        504,
+                                        323
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(G)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        258,
+                                        504,
+                                        323
+                                    ],
+                                    "type": "text",
+                                    "content": " and outer "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        258,
+                                        504,
+                                        323
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(G)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        258,
+                                        504,
+                                        323
+                                    ],
+                                    "type": "text",
+                                    "content": " the set of inner endpoints and outer endpoints in ear "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        258,
+                                        504,
+                                        323
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        258,
+                                        504,
+                                        323
+                                    ],
+                                    "type": "text",
+                                    "content": ", respectively. We also denote by path "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        258,
+                                        504,
+                                        323
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(G)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        258,
+                                        504,
+                                        323
+                                    ],
+                                    "type": "text",
+                                    "content": " the set of paths in ear "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        258,
+                                        504,
+                                        323
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        258,
+                                        504,
+                                        323
+                                    ],
+                                    "type": "text",
+                                    "content": ". It follows that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        258,
+                                        504,
+                                        323
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "|\\text{inner}(G)| = |\\text{outer}(G)| = |\\text{path}(G)| = k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        258,
+                                        504,
+                                        323
+                                    ],
+                                    "type": "text",
+                                    "content": ". Finally, given a path "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        258,
+                                        504,
+                                        323
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        258,
+                                        504,
+                                        323
+                                    ],
+                                    "type": "text",
+                                    "content": " and two vertices "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        258,
+                                        504,
+                                        323
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_{1}, w_{2}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        258,
+                                        504,
+                                        323
+                                    ],
+                                    "type": "text",
+                                    "content": " in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        258,
+                                        504,
+                                        323
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        258,
+                                        504,
+                                        323
+                                    ],
+                                    "type": "text",
+                                    "content": ", denote by subpath "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        258,
+                                        504,
+                                        323
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "_{P}(w_{1}, w_{2})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        258,
+                                        504,
+                                        323
+                                    ],
+                                    "type": "text",
+                                    "content": " the subpath in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        258,
+                                        504,
+                                        323
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        258,
+                                        504,
+                                        323
+                                    ],
+                                    "type": "text",
+                                    "content": " such that the two endpoints are "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        258,
+                                        504,
+                                        323
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_{1}, w_{2}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        258,
+                                        504,
+                                        323
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
+                        325,
+                        504,
+                        382
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                325,
+                                504,
+                                382
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition E.2 (Nested interval). Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "text",
+                                    "content": " be two "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "text",
+                                    "content": "-order ears with inner "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(G) = \\{v_{1},\\dots ,v_{k}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "text",
+                                    "content": ", outer "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(G) = \\{u_{1},\\dots ,u_{k}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "text",
+                                    "content": ", and outer "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(H) = \\{w_{1},\\dots ,w_{k}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "text",
+                                    "content": ", where each "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{u_i,v_i\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "text",
+                                    "content": " corresponds to the endpoints of a path "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{i}\\in \\mathrm{path}(G)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "text",
+                                    "content": ". We say "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "text",
+                                    "content": " is nested on "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "text",
+                                    "content": " if one or more endpoint "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_{i}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "text",
+                                    "content": " of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "text",
+                                    "content": " ("
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i\\in [k]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "text",
+                                    "content": ") is in path "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{i}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "text",
+                                    "content": ", and all other vertices in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "text",
+                                    "content": " are not in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "text",
+                                    "content": ". The nested interval is defined to be the union of subpaths subpath "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{i}(w_{i},v_{i})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i\\in [k]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfying that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_{i}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "text",
+                                    "content": " is in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{i}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        325,
+                                        504,
+                                        382
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        388,
+                        504,
+                        411
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                388,
+                                504,
+                                411
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        388,
+                                        504,
+                                        411
+                                    ],
+                                    "type": "text",
+                                    "content": "We give an illustration of the nested interval of two 3-order ears in Figure 2. Equipped with the above definition, we are ready to introduce the higher-order strong NED:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        104,
+                        412,
+                        505,
+                        436
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                412,
+                                505,
+                                436
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        412,
+                                        505,
+                                        436
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition E.3 ("
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        412,
+                                        505,
+                                        436
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        412,
+                                        505,
+                                        436
+                                    ],
+                                    "type": "text",
+                                    "content": "-order strong NED). Given a graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        412,
+                                        505,
+                                        436
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        412,
+                                        505,
+                                        436
+                                    ],
+                                    "type": "text",
+                                    "content": ", a "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        412,
+                                        505,
+                                        436
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        412,
+                                        505,
+                                        436
+                                    ],
+                                    "type": "text",
+                                    "content": "-order strong NED "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        412,
+                                        505,
+                                        436
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{P}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        412,
+                                        505,
+                                        436
+                                    ],
+                                    "type": "text",
+                                    "content": " is a partition of the edge set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        412,
+                                        505,
+                                        436
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "E_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        412,
+                                        505,
+                                        436
+                                    ],
+                                    "type": "text",
+                                    "content": " into a sequence of edge sets "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        412,
+                                        505,
+                                        436
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "Q_1, \\dots, Q_m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        412,
+                                        505,
+                                        436
+                                    ],
+                                    "type": "text",
+                                    "content": ", which satisfies the following conditions:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        121,
+                        441,
+                        504,
+                        548
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 17,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                121,
+                                441,
+                                233,
+                                452
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        441,
+                                        233,
+                                        452
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                121,
+                                                441,
+                                                233,
+                                                452
+                                            ],
+                                            "type": "text",
+                                            "content": "Each "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                441,
+                                                233,
+                                                452
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "Q_{i}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                441,
+                                                233,
+                                                452
+                                            ],
+                                            "type": "text",
+                                            "content": " is a "
+                                        },
+                                        {
+                                            "bbox": [
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+                                                441,
+                                                233,
+                                                452
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "k"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                441,
+                                                233,
+                                                452
+                                            ],
+                                            "type": "text",
+                                            "content": " -order ear."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 13
+                        },
+                        {
+                            "bbox": [
+                                121,
+                                457,
+                                504,
+                                479
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                        457,
+                                        504,
+                                        479
+                                    ],
+                                    "spans": [
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+                                            "bbox": [
+                                                121,
+                                                457,
+                                                504,
+                                                479
+                                            ],
+                                            "type": "text",
+                                            "content": "- Any two ears "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                457,
+                                                504,
+                                                479
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "Q_{i}"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "Q_{j}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                457,
+                                                504,
+                                                479
+                                            ],
+                                            "type": "text",
+                                            "content": " with indices "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                457,
+                                                504,
+                                                479
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "1 \\leq i < j \\leq c"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                457,
+                                                504,
+                                                479
+                                            ],
+                                            "type": "text",
+                                            "content": " do not intersect, where "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                457,
+                                                504,
+                                                479
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "c"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                457,
+                                                504,
+                                                479
+                                            ],
+                                            "type": "text",
+                                            "content": " is the number of connected components of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                457,
+                                                504,
+                                                479
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "G"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                457,
+                                                504,
+                                                479
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 14
+                        },
+                        {
+                            "bbox": [
+                                121,
+                                485,
+                                504,
+                                518
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        485,
+                                        504,
+                                        518
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                121,
+                                                485,
+                                                504,
+                                                518
+                                            ],
+                                            "type": "text",
+                                            "content": "- For each "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                485,
+                                                504,
+                                                518
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "Q_{j}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                485,
+                                                504,
+                                                518
+                                            ],
+                                            "type": "text",
+                                            "content": " with index "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                485,
+                                                504,
+                                                518
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "j > c"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                485,
+                                                504,
+                                                518
+                                            ],
+                                            "type": "text",
+                                            "content": ", it is nested on some "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
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+                                                518
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "k"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                485,
+                                                504,
+                                                518
+                                            ],
+                                            "type": "text",
+                                            "content": "-order ear "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                485,
+                                                504,
+                                                518
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "Q_{i}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                485,
+                                                504,
+                                                518
+                                            ],
+                                            "type": "text",
+                                            "content": " with index "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                485,
+                                                504,
+                                                518
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "1 \\leq i < j"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                485,
+                                                504,
+                                                518
+                                            ],
+                                            "type": "text",
+                                            "content": ". Moreover, except for the endpoints of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
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+                                                504,
+                                                518
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "Q_{j}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                485,
+                                                504,
+                                                518
+                                            ],
+                                            "type": "text",
+                                            "content": " on "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                485,
+                                                504,
+                                                518
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "Q_{i}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                485,
+                                                504,
+                                                518
+                                            ],
+                                            "type": "text",
+                                            "content": ", no other vertices in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                485,
+                                                504,
+                                                518
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "Q_{j}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                485,
+                                                504,
+                                                518
+                                            ],
+                                            "type": "text",
+                                            "content": " are in any previous ear "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                485,
+                                                504,
+                                                518
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "Q_{k}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                485,
+                                                504,
+                                                518
+                                            ],
+                                            "type": "text",
+                                            "content": " for "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                485,
+                                                504,
+                                                518
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "1 \\leq k < i"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                485,
+                                                504,
+                                                518
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 15
+                        },
+                        {
+                            "bbox": [
+                                121,
+                                523,
+                                504,
+                                548
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        121,
+                                        523,
+                                        504,
+                                        548
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                121,
+                                                523,
+                                                504,
+                                                548
+                                            ],
+                                            "type": "text",
+                                            "content": "- Denote by "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                523,
+                                                504,
+                                                548
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "I(Q_{j}) \\subset Q_{i}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                523,
+                                                504,
+                                                548
+                                            ],
+                                            "type": "text",
+                                            "content": " the nested interval of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                523,
+                                                504,
+                                                548
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "Q_{j}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                523,
+                                                504,
+                                                548
+                                            ],
+                                            "type": "text",
+                                            "content": " in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                523,
+                                                504,
+                                                548
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "Q_{i}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                523,
+                                                504,
+                                                548
+                                            ],
+                                            "type": "text",
+                                            "content": ". For all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                523,
+                                                504,
+                                                548
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "Q_{j}, Q_{k}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                523,
+                                                504,
+                                                548
+                                            ],
+                                            "type": "text",
+                                            "content": " with "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                523,
+                                                504,
+                                                548
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "c < j < k \\leq m"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                523,
+                                                504,
+                                                548
+                                            ],
+                                            "type": "text",
+                                            "content": ", if "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                523,
+                                                504,
+                                                548
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "Q_{j}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                523,
+                                                504,
+                                                548
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                523,
+                                                504,
+                                                548
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "Q_{k}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                523,
+                                                504,
+                                                548
+                                            ],
+                                            "type": "text",
+                                            "content": " are nested on the same ear, then "
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                523,
+                                                504,
+                                                548
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "I(Q_{j}) \\subset I(Q_{k})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                121,
+                                                523,
+                                                504,
+                                                548
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 16
+                        }
+                    ],
+                    "sub_type": "text"
+                },
+                {
+                    "bbox": [
+                        105,
+                        559,
+                        236,
+                        571
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                559,
+                                236,
+                                571
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                        236,
+                                        571
+                                    ],
+                                    "type": "text",
+                                    "content": "E.3 PROOFS IN SECTION 3.4"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
+                        104,
+                        579,
+                        467,
+                        591
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                579,
+                                467,
+                                591
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        579,
+                                        467,
+                                        591
+                                    ],
+                                    "type": "text",
+                                    "content": "We first generalize the tree-decomposed graphs in Definition C.6 to higher-order versions:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                },
+                {
+                    "bbox": [
+                        104,
+                        592,
+                        504,
+                        616
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                592,
+                                504,
+                                616
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        592,
+                                        504,
+                                        616
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition E.4. Define four families of tree-decomposed graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        592,
+                                        504,
+                                        616
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{S}^{\\mathrm{Sub}(k)}, \\mathcal{S}^{\\mathrm{L}(k)}, \\mathcal{S}^{\\mathrm{LF}(k)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        592,
+                                        504,
+                                        616
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        592,
+                                        504,
+                                        616
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{S}^{\\mathrm{F}(k)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        592,
+                                        504,
+                                        616
+                                    ],
+                                    "type": "text",
+                                    "content": " as follows:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                },
+                {
+                    "bbox": [
+                        116,
+                        620,
+                        506,
+                        733
+                    ],
+                    "type": "list",
+                    "angle": 0,
+                    "index": 25,
+                    "blocks": [
+                        {
+                            "bbox": [
+                                116,
+                                620,
+                                392,
+                                634
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
+                                        620,
+                                        392,
+                                        634
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                116,
+                                                620,
+                                                392,
+                                                634
+                                            ],
+                                            "type": "text",
+                                            "content": "a) "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                620,
+                                                392,
+                                                634
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(F,T^{r})\\in S^{\\mathsf{F}(k)}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                620,
+                                                392,
+                                                634
+                                            ],
+                                            "type": "text",
+                                            "content": " iff "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                620,
+                                                392,
+                                                634
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(F,T^{r})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                620,
+                                                392,
+                                                634
+                                            ],
+                                            "type": "text",
+                                            "content": " satisfies Definition C.4 with width "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                620,
+                                                392,
+                                                634
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "k"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 21
+                        },
+                        {
+                            "bbox": [
+                                116,
+                                639,
+                                506,
+                                674
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
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+                                                506,
+                                                674
+                                            ],
+                                            "type": "text",
+                                            "content": "b) "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                639,
+                                                506,
+                                                674
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(F,T^{r})\\in S^{\\mathsf{LF}(k)}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                639,
+                                                506,
+                                                674
+                                            ],
+                                            "type": "text",
+                                            "content": " iff "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                639,
+                                                506,
+                                                674
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(F,T^{r})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                639,
+                                                506,
+                                                674
+                                            ],
+                                            "type": "text",
+                                            "content": " satisfies Definition C.4 with width "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                639,
+                                                506,
+                                                674
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "k"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                639,
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+                                                674
+                                            ],
+                                            "type": "text",
+                                            "content": ", and for any tree node "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                639,
+                                                506,
+                                                674
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                639,
+                                                506,
+                                                674
+                                            ],
+                                            "type": "text",
+                                            "content": " of odd depth, it has only one child if "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                639,
+                                                506,
+                                                674
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w\\notin \\{v:v\\in N_G(u),u\\in \\beta_T(s)\\}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                639,
+                                                506,
+                                                674
+                                            ],
+                                            "type": "text",
+                                            "content": " where "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                639,
+                                                506,
+                                                674
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "s"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                639,
+                                                506,
+                                                674
+                                            ],
+                                            "type": "text",
+                                            "content": " is the parent node of "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                639,
+                                                506,
+                                                674
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                639,
+                                                506,
+                                                674
+                                            ],
+                                            "type": "text",
+                                            "content": " and "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                639,
+                                                506,
+                                                674
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "w"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                639,
+                                                506,
+                                                674
+                                            ],
+                                            "type": "text",
+                                            "content": " is the unique vertex in "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                639,
+                                                506,
+                                                674
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "\\beta_T(t)\\backslash \\beta_T(s)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                639,
+                                                506,
+                                                674
+                                            ],
+                                            "type": "text",
+                                            "content": ";"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 22
+                        },
+                        {
+                            "bbox": [
+                                116,
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+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
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+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": "c) "
+                                        },
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+                                            "bbox": [
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(F,T^{r})\\in S^{\\mathbb{L}(k)}"
+                                        },
+                                        {
+                                            "bbox": [
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+                                            "type": "text",
+                                            "content": " iff "
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+                                            "bbox": [
+                                                116,
+                                                679,
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+                                                704
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(F,T^{r})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                679,
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+                                                704
+                                            ],
+                                            "type": "text",
+                                            "content": " satisfies Definition C.4 with width "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
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+                                                504,
+                                                704
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "k"
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+                                            "bbox": [
+                                                116,
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+                                            ],
+                                            "type": "text",
+                                            "content": ", and any tree node "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                679,
+                                                504,
+                                                704
+                                            ],
+                                            "type": "text",
+                                            "content": " of odd depth has only one child;"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 23
+                        },
+                        {
+                            "bbox": [
+                                116,
+                                708,
+                                504,
+                                733
+                            ],
+                            "type": "text",
+                            "angle": 0,
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        116,
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+                                        {
+                                            "bbox": [
+                                                116,
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+                                            ],
+                                            "type": "text",
+                                            "content": "d) "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(F,T^{r})\\in S^{\\mathrm{Sub}(k)}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                708,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "text",
+                                            "content": " iff "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                708,
+                                                504,
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+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "(F,T^{r})"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                708,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "text",
+                                            "content": " satisfies Definition C.4 with width "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                708,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "k"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                708,
+                                                504,
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+                                            ],
+                                            "type": "text",
+                                            "content": ", and there exists a multiset "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                708,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "U\\subset V_{G}"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                708,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "text",
+                                            "content": " of size "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                708,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "|U| = k"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                708,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "text",
+                                            "content": " such that "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                708,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "U\\subset \\beta_T(t)"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                708,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "text",
+                                            "content": " for all "
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                708,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "t\\in V_T"
+                                        },
+                                        {
+                                            "bbox": [
+                                                116,
+                                                708,
+                                                504,
+                                                733
+                                            ],
+                                            "type": "text",
+                                            "content": "."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 24
+                        }
+                    ],
+                    "sub_type": "text"
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
+                        750,
+                        310,
+                        760
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                299,
+                                750,
+                                310,
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+                            ],
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+                                {
+                                    "bbox": [
+                                        299,
+                                        750,
+                                        310,
+                                        760
+                                    ],
+                                    "type": "text",
+                                    "content": "55"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 26
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 54
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        82,
+                        504,
+                        105
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                82,
+                                504,
+                                105
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        82,
+                                        504,
+                                        105
+                                    ],
+                                    "type": "text",
+                                    "content": "Then, we can analogously prove the following theorems. The proofs are almost the same as in Appendix C, so we omit them for clarity."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        106,
+                        505,
+                        143
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                106,
+                                505,
+                                143
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        505,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem E.5. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        505,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\mathsf{Sub}(k), \\mathsf{L}(k), \\mathsf{LF}(k), \\mathsf{F}(k)\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        505,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, any graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        505,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        505,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        505,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        505,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": " have the same representation under model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        505,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        505,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": " (i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        505,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^M(G) = \\chi_H^M(H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        505,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": ") iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        505,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{hom}(F, G) = \\mathsf{hom}(F, H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        505,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        505,
+                                        143
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        505,
+                                        143
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        144,
+                        506,
+                        196
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                144,
+                                506,
+                                196
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        144,
+                                        506,
+                                        196
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem E.6. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        144,
+                                        506,
+                                        196
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\mathrm{Sub}(k), \\mathsf{L}(k), \\mathsf{LF}(k), \\mathsf{F}(k)\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        144,
+                                        506,
+                                        196
+                                    ],
+                                    "type": "text",
+                                    "content": " be any model, and let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        144,
+                                        506,
+                                        196
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        144,
+                                        506,
+                                        196
+                                    ],
+                                    "type": "text",
+                                    "content": " be any graph such that no tree decomposition "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        144,
+                                        506,
+                                        196
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        144,
+                                        506,
+                                        196
+                                    ],
+                                    "type": "text",
+                                    "content": ". Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        144,
+                                        506,
+                                        196
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G(F)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        144,
+                                        506,
+                                        196
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        144,
+                                        506,
+                                        196
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H(F)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        144,
+                                        506,
+                                        196
+                                    ],
+                                    "type": "text",
+                                    "content": " be the Fürer graph and twisted Fürer graph with respect to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        144,
+                                        506,
+                                        196
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        144,
+                                        506,
+                                        196
+                                    ],
+                                    "type": "text",
+                                    "content": ". Then, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        144,
+                                        506,
+                                        196
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{hom}(F, G(F)) \\neq \\mathrm{hom}(F, H(F))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        144,
+                                        506,
+                                        196
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        144,
+                                        506,
+                                        196
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_{G(F)}^{M}(G(F)) = \\chi_{H(F)}^{M}(H(F))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        144,
+                                        506,
+                                        196
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        198,
+                        504,
+                        223
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                198,
+                                504,
+                                223
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        223
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem E.7. For any graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        223
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        223
+                                    ],
+                                    "type": "text",
+                                    "content": ", there is a tree decomposition "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        223
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        223
+                                    ],
+                                    "type": "text",
+                                    "content": " of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        223
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        223
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        223
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^{\\mathrm{Sub}(k)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        223
+                                    ],
+                                    "type": "text",
+                                    "content": " iff there exists "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        223
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "U \\subset V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        223
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        223
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "|U| \\leq k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        223
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        223
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F \\backslash U"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        223
+                                    ],
+                                    "type": "text",
+                                    "content": " is a forest."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        225,
+                        504,
+                        249
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                225,
+                                504,
+                                249
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        225,
+                                        504,
+                                        249
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem E.8. For any graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        225,
+                                        504,
+                                        249
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        225,
+                                        504,
+                                        249
+                                    ],
+                                    "type": "text",
+                                    "content": ", there is a tree decomposition "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        225,
+                                        504,
+                                        249
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        225,
+                                        504,
+                                        249
+                                    ],
+                                    "type": "text",
+                                    "content": " of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        225,
+                                        504,
+                                        249
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        225,
+                                        504,
+                                        249
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        225,
+                                        504,
+                                        249
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^{\\mathsf{L}(k)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        225,
+                                        504,
+                                        249
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        225,
+                                        504,
+                                        249
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        225,
+                                        504,
+                                        249
+                                    ],
+                                    "type": "text",
+                                    "content": " has a "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        225,
+                                        504,
+                                        249
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        225,
+                                        504,
+                                        249
+                                    ],
+                                    "type": "text",
+                                    "content": "-order strong NED."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        250,
+                        504,
+                        275
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                250,
+                                504,
+                                275
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        250,
+                                        504,
+                                        275
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem E.9. For any graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        250,
+                                        504,
+                                        275
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        250,
+                                        504,
+                                        275
+                                    ],
+                                    "type": "text",
+                                    "content": ", there is a tree decomposition "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        250,
+                                        504,
+                                        275
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        250,
+                                        504,
+                                        275
+                                    ],
+                                    "type": "text",
+                                    "content": " of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        250,
+                                        504,
+                                        275
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        250,
+                                        504,
+                                        275
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        250,
+                                        504,
+                                        275
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^{\\mathsf{F}(k)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        250,
+                                        504,
+                                        275
+                                    ],
+                                    "type": "text",
+                                    "content": " iff "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        250,
+                                        504,
+                                        275
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{tw}(F) \\leq k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        250,
+                                        504,
+                                        275
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        105,
+                        287,
+                        359,
+                        298
+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                287,
+                                359,
+                                298
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        287,
+                                        359,
+                                        298
+                                    ],
+                                    "type": "text",
+                                    "content": "E.4 EXPRESSIVITY GAP BETWEEN HIGHER-ORDER GNNS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        104,
+                        308,
+                        504,
+                        331
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                308,
+                                504,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                        331
+                                    ],
+                                    "type": "text",
+                                    "content": "In this subsection, we show how homomorphism expressivity can be used to build a complete expressiveness hierarchy for higher-order GNNs as shown in Corollary 4.2."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        335,
+                        505,
+                        403
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                335,
+                                505,
+                                403
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        505,
+                                        403
+                                    ],
+                                    "type": "text",
+                                    "content": "Gap between Subgraph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        505,
+                                        403
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        505,
+                                        403
+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN and Local "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        505,
+                                        403
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(k + 1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        505,
+                                        403
+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        505,
+                                        403
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(k \\geq 1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        505,
+                                        403
+                                    ],
+                                    "type": "text",
+                                    "content": ". The counterexample graph is a "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        505,
+                                        403
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(k + 1) \\times (2k + 2)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        505,
+                                        403
+                                    ],
+                                    "type": "text",
+                                    "content": " grid consisting "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        505,
+                                        403
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(k + 1) \\times (2k + 2)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        505,
+                                        403
+                                    ],
+                                    "type": "text",
+                                    "content": " vertices. First, it is easy to see that the graph is not in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        505,
+                                        403
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^{\\mathrm{Sub}(k)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        505,
+                                        403
+                                    ],
+                                    "type": "text",
+                                    "content": ", i.e., deleting "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        505,
+                                        403
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        505,
+                                        403
+                                    ],
+                                    "type": "text",
+                                    "content": " vertices of the graph cannot yield a forest. To see this, note that the graph consists of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        505,
+                                        403
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k \\times (2k + 1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        505,
+                                        403
+                                    ],
+                                    "type": "text",
+                                    "content": " \"squares\", and each vertex is related to at most four squares. Therefore, deleting "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        505,
+                                        403
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        505,
+                                        403
+                                    ],
+                                    "type": "text",
+                                    "content": " vertices cannot eliminate all squares when "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        505,
+                                        403
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k \\geq 2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        505,
+                                        403
+                                    ],
+                                    "type": "text",
+                                    "content": " (because "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        505,
+                                        403
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "4k < k \\times (2k + 1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        505,
+                                        403
+                                    ],
+                                    "type": "text",
+                                    "content": "). For the case of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        505,
+                                        403
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k = 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        335,
+                                        505,
+                                        403
+                                    ],
+                                    "type": "text",
+                                    "content": ", we clearly have that deleting one vertex cannot eliminate all squares."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        104,
+                        408,
+                        337,
+                        455
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                408,
+                                337,
+                                455
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        408,
+                                        337,
+                                        455
+                                    ],
+                                    "type": "text",
+                                    "content": "We next show that the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        408,
+                                        337,
+                                        455
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(k + 1)\\times (2k + 2)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        408,
+                                        337,
+                                        455
+                                    ],
+                                    "type": "text",
+                                    "content": " grid is in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        408,
+                                        337,
+                                        455
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^{\\mathsf{L}(k + 1)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        408,
+                                        337,
+                                        455
+                                    ],
+                                    "type": "text",
+                                    "content": ". This is also simple as shown in Figure 8, where each color represents a "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        408,
+                                        337,
+                                        455
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        408,
+                                        337,
+                                        455
+                                    ],
+                                    "type": "text",
+                                    "content": "-order ear. It can be seen that the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        408,
+                                        337,
+                                        455
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(k + 1)\\times (2k + 2)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        408,
+                                        337,
+                                        455
+                                    ],
+                                    "type": "text",
+                                    "content": " grid has a "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        408,
+                                        337,
+                                        455
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(k + 1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        408,
+                                        337,
+                                        455
+                                    ],
+                                    "type": "text",
+                                    "content": "-order strong NED."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "type": "image",
+                    "bbox": [
+                        348,
+                        407,
+                        500,
+                        477
+                    ],
+                    "blocks": [
+                        {
+                            "bbox": [
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+                                407,
+                                500,
+                                477
+                            ],
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                        407,
+                                        500,
+                                        477
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                348,
+                                                407,
+                                                500,
+                                                477
+                                            ],
+                                            "type": "image",
+                                            "image_path": "830e1ed6f48f52caf086dee6661a9f5214573c1207c8b1c416a1eb46510085f3.jpg"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 11,
+                            "angle": 0,
+                            "type": "image_body"
+                        },
+                        {
+                            "bbox": [
+                                342,
+                                483,
+                                504,
+                                507
+                            ],
+                            "lines": [
+                                {
+                                    "bbox": [
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+                                        483,
+                                        504,
+                                        507
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                342,
+                                                483,
+                                                504,
+                                                507
+                                            ],
+                                            "type": "text",
+                                            "content": "Figure 8: An 4-order strong NED of the "
+                                        },
+                                        {
+                                            "bbox": [
+                                                342,
+                                                483,
+                                                504,
+                                                507
+                                            ],
+                                            "type": "inline_equation",
+                                            "content": "4 \\times 8"
+                                        },
+                                        {
+                                            "bbox": [
+                                                342,
+                                                483,
+                                                504,
+                                                507
+                                            ],
+                                            "type": "text",
+                                            "content": " grid graph."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 12,
+                            "angle": 0,
+                            "type": "image_caption"
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        104,
+                        459,
+                        336,
+                        514
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                459,
+                                336,
+                                514
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        459,
+                                        336,
+                                        514
+                                    ],
+                                    "type": "text",
+                                    "content": "Relation between Subgraph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        459,
+                                        336,
+                                        514
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        459,
+                                        336,
+                                        514
+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        459,
+                                        336,
+                                        514
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        459,
+                                        336,
+                                        514
+                                    ],
+                                    "type": "text",
+                                    "content": "-FGNN ("
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        459,
+                                        336,
+                                        514
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k \\geq 2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        459,
+                                        336,
+                                        514
+                                    ],
+                                    "type": "text",
+                                    "content": "). To show that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        459,
+                                        336,
+                                        514
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^{\\mathrm{Sub}(k)} \\not\\subsetneq \\mathcal{F}^{\\mathrm{F}(k)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        459,
+                                        336,
+                                        514
+                                    ],
+                                    "type": "text",
+                                    "content": ", consider the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        459,
+                                        336,
+                                        514
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(k + 2)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        459,
+                                        336,
+                                        514
+                                    ],
+                                    "type": "text",
+                                    "content": "-clique. Clearly, deleting "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        459,
+                                        336,
+                                        514
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        459,
+                                        336,
+                                        514
+                                    ],
+                                    "type": "text",
+                                    "content": " vertices from the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        459,
+                                        336,
+                                        514
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(k + 2)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        459,
+                                        336,
+                                        514
+                                    ],
+                                    "type": "text",
+                                    "content": "-clique yields a graph consisting of two vertices linked by an edge, which is a tree. On the other hand, the"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        104,
+                        514,
+                        504,
+                        529
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                514,
+                                504,
+                                529
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        514,
+                                        504,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": "treewidth of a "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        514,
+                                        504,
+                                        529
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(k + 2)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        514,
+                                        504,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": "-clique is "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        514,
+                                        504,
+                                        529
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k + 1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        514,
+                                        504,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": " (Fact C.3). So the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        514,
+                                        504,
+                                        529
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(k + 2)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        514,
+                                        504,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": "-clique is in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        514,
+                                        504,
+                                        529
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^{\\mathrm{Sub}(k)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        514,
+                                        504,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": " but not in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        514,
+                                        504,
+                                        529
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^{\\mathsf{F}(k)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        514,
+                                        504,
+                                        529
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        104,
+                        533,
+                        505,
+                        579
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                533,
+                                505,
+                                579
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        505,
+                                        579
+                                    ],
+                                    "type": "text",
+                                    "content": "To show that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        505,
+                                        579
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^{\\mathsf{F}(k)} \\not\\subset \\mathcal{F}^{\\mathrm{Sub}(k)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        505,
+                                        579
+                                    ],
+                                    "type": "text",
+                                    "content": ", we can again use the grid graph, but this time consider the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        505,
+                                        579
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k \\times (2k + 2)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        505,
+                                        579
+                                    ],
+                                    "type": "text",
+                                    "content": " grid. On the one hand, a standard result in graph theory shows that the treewidth of a "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        505,
+                                        579
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "a \\times b"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        505,
+                                        579
+                                    ],
+                                    "type": "text",
+                                    "content": " grid graph is "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        505,
+                                        579
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\min(a, b)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        505,
+                                        579
+                                    ],
+                                    "type": "text",
+                                    "content": ". On the other hand, following the similar analysis above, we can prove that deleting "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        505,
+                                        579
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        505,
+                                        579
+                                    ],
+                                    "type": "text",
+                                    "content": " vertices from the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        505,
+                                        579
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k \\times (2k + 2)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        533,
+                                        505,
+                                        579
+                                    ],
+                                    "type": "text",
+                                    "content": " grid cannot eliminate all squares."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        104,
+                        583,
+                        505,
+                        665
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                583,
+                                505,
+                                665
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": "Gap between Local "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": "-FGNN ("
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k \\geq 2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": "). The counterexample graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": " is the union of the following graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "K_0 \\cup K_1 \\cup \\dots \\cup K_k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": ", where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "K_0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": " is a "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(k+1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": "-clique with vertex set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{u_1, \\dots, u_{k+1}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": " and for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i \\in [k+1]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "K_i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": " is a "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(k+1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": "-clique with vertex set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{u_1, \\dots, u_{i-1}, v_i, u_{i+1}, \\dots, u_{k+1}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v_i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": " is a new vertex (not in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "K_0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": ") and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v_i \\neq v_j"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": " for "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i \\neq j"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": ". Namely, each "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "K_i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": " has "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": " common vertices with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "K_0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": ". It is easy to construct a tree decomposition "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": " of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^{\\mathsf{F}(k)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": ". On the other hand, it is easy to see that the graph does not have a "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": "-order strong NED (equivalently, one can easily check that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": " does not admit a tree decomposition "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "T^r"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfying "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(F, T^r) \\in S^{\\mathsf{L}(k)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        583,
+                                        505,
+                                        665
+                                    ],
+                                    "type": "text",
+                                    "content": ")."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        104,
+                        669,
+                        504,
+                        693
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                669,
+                                504,
+                                693
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        669,
+                                        504,
+                                        693
+                                    ],
+                                    "type": "text",
+                                    "content": "Gap between "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        669,
+                                        504,
+                                        693
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        669,
+                                        504,
+                                        693
+                                    ],
+                                    "type": "text",
+                                    "content": "-FGNN and Local "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        669,
+                                        504,
+                                        693
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(k + 1)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        669,
+                                        504,
+                                        693
+                                    ],
+                                    "type": "text",
+                                    "content": "-GNN "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        669,
+                                        504,
+                                        693
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(k \\geq 2)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        669,
+                                        504,
+                                        693
+                                    ],
+                                    "type": "text",
+                                    "content": ". The counterexample graph is again the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        669,
+                                        504,
+                                        693
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(k + 1) \\times (2k + 2)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        669,
+                                        504,
+                                        693
+                                    ],
+                                    "type": "text",
+                                    "content": " grid. We have proved that the graph is in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        669,
+                                        504,
+                                        693
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^{\\mathsf{L}(k + 1)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        669,
+                                        504,
+                                        693
+                                    ],
+                                    "type": "text",
+                                    "content": " but not in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        669,
+                                        504,
+                                        693
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^{\\mathsf{L}(k)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        669,
+                                        504,
+                                        693
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        104,
+                        698,
+                        505,
+                        733
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                698,
+                                505,
+                                733
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        698,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "Regarding Local "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        698,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        698,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "text",
+                                    "content": "-IGN and Frasca et al. (2022). Finally, we remark that based on the above results, we essentially proved an open question raised in Frasca et al. (2022) regarding the expressive power of Local "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        698,
+                                        505,
+                                        733
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "k"
+                                },
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+                                    "type": "text",
+                                    "content": " proposed in their paper). Following Zhang et al. (2023a),"
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+                                            "content": "Figure 9: More illustration of NED. All NEDs in these example graphs are strong but not endpoint-shared. See Appendix F.2 for additional discussions."
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+                                    "content": "it is straightforward to see that Local "
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+                                    "type": "text",
+                                    "content": "-GNN. Therefore, Local "
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+                                    "type": "text",
+                                    "content": "-IGN is strictly more expressive than "
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+                                    "type": "inline_equation",
+                                    "content": "(k - 1)"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "-FGNN and strictly less expressive than "
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+                                    "type": "text",
+                                    "content": "-FGNN."
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+                                    "type": "text",
+                                    "content": "F ADDITIONAL DISCUSSIONS"
+                                }
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+                                    "type": "text",
+                                    "content": "F.1 REGARDING THE DEFINITION OF HOMOMORPHISM EXPRESSIVITY"
+                                }
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+                        }
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+                                    "type": "text",
+                                    "content": "In this paper, we have shown that homomorphism expressivity exists for a variety of popular GNNs. Unfortunately, due to the \"iff\" statement in Definition 3.1, homomorphism expressivity may not always be well-defined in the general case. We remark in this subsection that, there do exist pathological, intentionally designed GNNs such that the homomorphism expressivity is not well-defined."
+                                }
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+                                    "type": "text",
+                                    "content": "Consider a simple GNN "
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": " that outputs the representation of a graph "
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": " as follows. If "
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+                                    "type": "inline_equation",
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+                                    "bbox": [
+                                        104,
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+                                    "type": "text",
+                                    "content": " is a cycle of odd length, it outputs "
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+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "(1,L)"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " is the length of the cycle. Otherwise, it outputs "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "(0,\\chi_{G}^{\\mathrm{MP}}(G))"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": ", namely, running a MPNN on the graph. It follows that "
+                                },
+                                {
+                                    "bbox": [
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+                                        489
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+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " is strictly more powerful than MPNN, e.g., it can distinguish between the 9-cycle and three triangles, which MPNN fails to distinguish. As a result, "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " can count all trees under homomorphism. Moreover, it cannot count other patterns under homomorphism, as the counterexample graphs for MPNN (i.e., Fürer graphs) are not cycles of odd length, so they are still counterexample graphs for model "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ". Therefore, the homomorphism expressivity of "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " should be exactly the family of forests if it exists. However, the homomorphism information of forests cannot determine the representation of model "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " since "
+                                },
+                                {
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+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " is strictly more powerful than MPNN. So we conclude that "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^{M}"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " does not exist."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Note that the above GNN construction is inherently unnatural and hardly appears in practice. Actually, when a GNN is defined via the message-passing paradigm, we suspect that its homomorphism expressivity is always well-defined. As stated in Appendix B, we conjecture that, for any GNN characterized by a color refinement algorithm that outputs stable colors, the homomorphism expressivity always exists."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
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+                                    "type": "text",
+                                    "content": "F.2 REGARDING THE DEFINITION OF NED"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
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+                                    "type": "text",
+                                    "content": "In the main text, we have illustrated several types of NED with simple example graphs. To gain a deeper understanding of Definition 3.3, in this section we will present a few more complex examples (see Figure 9). Notably, unlike graphs in Figure 1(b), for all graphs in Figure 9 their NED contains ears such that only one endpoint is in its nested ear (e.g., ears 2 and 4 in Figure 9(a)). In other words, these ears have empty nested intervals."
+                                }
+                            ]
+                        }
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+                    "type": "text",
+                    "angle": 0,
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "The presence of ears with empty nested interval stems from the fact that the corresponding graph is not biconnected (Zhang et al., 2023b). Indeed, one can check that all graphs in Definition 3.3(b) do not have cut vertices, while all graphs in Figure 9 have cut vertices. Moreover, in these examples, the number of ears with empty nested interval always equals to the number of biconnected components minus one. In particular, for biconnected graphs, it can be proved that any NED does not contain an ear with empty nested interval."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                }
+            ],
+            "discarded_blocks": [
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
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+                                    "type": "text",
+                                    "content": "57"
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+                }
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "G HOMOMORPHISM AND SUBGRAPH COUNTING POWER"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
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+                    ],
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+                                105,
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "G.1 PROOF OF THEOREM 4.5"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                126,
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+                                160
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        126,
+                                        504,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": "Our proof draws inspiration from a recent technique developed in Seppelt (2023). We note that the original results in Seppelt (2023) are described for unlabeled graphs, but it is easy to extend these results to labeled graphs. To begin with, we define a concept called the graph categorical product."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        105,
+                        162,
+                        504,
+                        197
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                162,
+                                504,
+                                197
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        162,
+                                        504,
+                                        197
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition G.1. The categorical product of two graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        162,
+                                        504,
+                                        197
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G, H"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        162,
+                                        504,
+                                        197
+                                    ],
+                                    "type": "text",
+                                    "content": ", denoted as "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        162,
+                                        504,
+                                        197
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G \\times H"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        162,
+                                        504,
+                                        197
+                                    ],
+                                    "type": "text",
+                                    "content": ", is a graph where "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        162,
+                                        504,
+                                        197
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "V_{G \\times H} = V_G \\times V_H"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        162,
+                                        504,
+                                        197
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        162,
+                                        504,
+                                        197
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\ell_{G \\times H}(g, h) = (\\ell_G(g), \\ell_H(h))"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        162,
+                                        504,
+                                        197
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        162,
+                                        504,
+                                        197
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "g \\in V_G"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        162,
+                                        504,
+                                        197
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        162,
+                                        504,
+                                        197
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h \\in V_H"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        162,
+                                        504,
+                                        197
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        162,
+                                        504,
+                                        197
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "E_{G \\times H} = \\{ (g, h), (g', h') \\} : \\{ g, g' \\} \\in E_G, \\{ h, h' \\} \\in E_H"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        162,
+                                        504,
+                                        197
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        198,
+                        504,
+                        222
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                198,
+                                504,
+                                222
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        222
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma G.2 (Seppelt (2023)). For any graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        222
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F, G,"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        222
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        222
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        222
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        222
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{hom}(F,G\\times H) = \\operatorname{hom}(F,G)\\cdot"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        222
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{hom}(F,H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        198,
+                                        504,
+                                        222
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        233,
+                        505,
+                        278
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                233,
+                                505,
+                                278
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. We define a mapping "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": " from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{Hom}(F, G) \\times \\operatorname{Hom}(F, H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{Hom}(F, G \\times H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": " as follows. for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma_1 \\in \\operatorname{Hom}(F, G)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\sigma_2 \\in \\operatorname{Hom}(F, H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": ", define "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho = \\tau(\\sigma_1, \\sigma_2)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\rho(f) = (\\sigma_1(f), \\sigma_2(f))"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "f \\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": ". It is easy to see that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tau"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": " is a bijective mapping from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{Hom}(F, G) \\times \\operatorname{Hom}(F, H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        278
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{Hom}(F, G \\times H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        233,
+                                        505,
+                                        278
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        285,
+                        504,
+                        308
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                285,
+                                504,
+                                308
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        285,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition G.3. Given graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        285,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        285,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        285,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        285,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": ", denote by "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        285,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{Surj}(F, G)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        285,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": " the set of all homomorphisms from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        285,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        285,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": " to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        285,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        285,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": " that are surjective on both the vertices and edges of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        285,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        285,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": ", and define "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        285,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{surj}(F, G) = |\\operatorname{Surj}(F, G)|"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        285,
+                                        504,
+                                        308
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        104,
+                        316,
+                        390,
+                        327
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                316,
+                                390,
+                                327
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        316,
+                                        390,
+                                        327
+                                    ],
+                                    "type": "text",
+                                    "content": "The following proposition is straightforward (similar to Lemma C.25):"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        331,
+                        504,
+                        354
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                331,
+                                504,
+                                354
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        354
+                                    ],
+                                    "type": "text",
+                                    "content": "Proposition G.4. "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        354
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{hom}(G, H) = \\sum_{F} \\operatorname{surj}(G, F) \\cdot \\operatorname{sub}(F, H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        354
+                                    ],
+                                    "type": "text",
+                                    "content": ", where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        354
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        331,
+                                        504,
+                                        354
+                                    ],
+                                    "type": "text",
+                                    "content": " ranges over all non-isomorphic graphs."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        103,
+                        355,
+                        504,
+                        380
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                103,
+                                355,
+                                504,
+                                380
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        103,
+                                        355,
+                                        504,
+                                        380
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma G.5. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        355,
+                                        504,
+                                        380
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        355,
+                                        504,
+                                        380
+                                    ],
+                                    "type": "text",
+                                    "content": " be a GNN model such that its homomorphism expressivity "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        355,
+                                        504,
+                                        380
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        355,
+                                        504,
+                                        380
+                                    ],
+                                    "type": "text",
+                                    "content": " exists. Given a finite set of graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        355,
+                                        504,
+                                        380
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        355,
+                                        504,
+                                        380
+                                    ],
+                                    "type": "text",
+                                    "content": " and a function "
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        355,
+                                        504,
+                                        380
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha : \\mathcal{L} \\to \\mathbb{R} / \\{0\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        103,
+                                        355,
+                                        504,
+                                        380
+                                    ],
+                                    "type": "text",
+                                    "content": ", if"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        159,
+                        384,
+                        451,
+                        409
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                159,
+                                384,
+                                451,
+                                409
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        159,
+                                        384,
+                                        451,
+                                        409
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\chi_ {G} ^ {M} (G) = \\chi_ {H} ^ {M} (H) \\Longrightarrow \\sum_ {L \\in \\mathcal {L}} \\alpha (L) \\operatorname {h o m} (L, G) = \\sum_ {L \\in \\mathcal {L}} \\alpha (L) \\operatorname {h o m} (L, H),",
+                                    "image_path": "21409eba31588c26ce2d9a178a332fce53d11f20dc0e77873e5ea7e6d07704be.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        104,
+                        415,
+                        290,
+                        427
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                415,
+                                290,
+                                427
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        415,
+                                        290,
+                                        427
+                                    ],
+                                    "type": "text",
+                                    "content": "holds for all graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        415,
+                                        290,
+                                        427
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        415,
+                                        290,
+                                        427
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        415,
+                                        290,
+                                        427
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        415,
+                                        290,
+                                        427
+                                    ],
+                                    "type": "text",
+                                    "content": ", then "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        415,
+                                        290,
+                                        427
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L} \\subset \\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        415,
+                                        290,
+                                        427
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        104,
+                        440,
+                        505,
+                        535
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                440,
+                                505,
+                                535
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "text",
+                                    "content": " be the largest number of vertices for all graphs in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "text",
+                                    "content": ", and let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\mathcal{L}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "text",
+                                    "content": " be the set of all non-isomorphic graphs with no more than "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "text",
+                                    "content": " vertices. We can arrange all graphs in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\mathcal{L}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "text",
+                                    "content": " into a sequence "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "L_{1}, L_{2}, \\ldots, L_{N}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfying the following property: "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "|V_{L_{i}}| \\leq |V_{L_{i+1}}|"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "text",
+                                    "content": " or "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(|V_{L_{i}}| = |V_{L_{i+1}}|"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "|E_{L_{i}}| \\leq |E_{L_{i+1}}|)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i = 1, 2, \\ldots, N-1"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "text",
+                                    "content": ". We then define the matrices "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A^{\\text{hom}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A^{\\text{surj}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A^{\\text{sub}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "text",
+                                    "content": ", where the elements in the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i^{th}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "text",
+                                    "content": " row and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "j^{th}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "text",
+                                    "content": " column are "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\text{hom}(L_{i}, L_{j})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\text{surj}(L_{i}, L_{j})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "text",
+                                    "content": ", and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\text{sub}(L_{i}, L_{j})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "text",
+                                    "content": ", respectively. Proposition G.4 implies that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A^{\\text{hom}} = A^{\\text{surj}} \\cdot A^{\\text{sub}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "text",
+                                    "content": ". Since "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A^{\\text{surj}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "text",
+                                    "content": " is a lower triangular matrix with nonzero diagonal elements and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A^{\\text{sub}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "text",
+                                    "content": " is an upper triangular matrix with non-zero diagonal elements, the matrix "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "A^{\\text{hom}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        440,
+                                        505,
+                                        535
+                                    ],
+                                    "type": "text",
+                                    "content": " is invertible."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        104,
+                        540,
+                        505,
+                        603
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                540,
+                                505,
+                                603
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        505,
+                                        603
+                                    ],
+                                    "type": "text",
+                                    "content": "We next extend "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        505,
+                                        603
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        505,
+                                        603
+                                    ],
+                                    "type": "text",
+                                    "content": " to a function "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        505,
+                                        603
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\alpha}:\\tilde{\\mathcal{L}}\\to \\mathbb{R}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        505,
+                                        603
+                                    ],
+                                    "type": "text",
+                                    "content": " by setting "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        505,
+                                        603
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\alpha} (L) = \\alpha (L)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        505,
+                                        603
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        505,
+                                        603
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "L\\in \\mathcal{L}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        505,
+                                        603
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        505,
+                                        603
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\alpha} (\\tilde{L}) = 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        505,
+                                        603
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        505,
+                                        603
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{L}\\in \\tilde{\\mathcal{L}}\\backslash \\mathcal{L}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        505,
+                                        603
+                                    ],
+                                    "type": "text",
+                                    "content": ". Additionally, if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        505,
+                                        603
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^M (G) = \\chi_H^M (H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        505,
+                                        603
+                                    ],
+                                    "type": "text",
+                                    "content": ", then "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        505,
+                                        603
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{hom}(F,G) = \\mathrm{hom}(F,H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        505,
+                                        603
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        505,
+                                        603
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F\\in \\mathcal{F}^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        505,
+                                        603
+                                    ],
+                                    "type": "text",
+                                    "content": " (by definition of homomorphism expressivity). Then, given any graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        505,
+                                        603
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "K\\in \\tilde{\\mathcal{L}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        505,
+                                        603
+                                    ],
+                                    "type": "text",
+                                    "content": ", Lemma G.2 implies that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        505,
+                                        603
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{hom}(F,G\\times K) = \\mathrm{hom}(F,H\\times K)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        505,
+                                        603
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        505,
+                                        603
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F\\in \\mathcal{F}^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        505,
+                                        603
+                                    ],
+                                    "type": "text",
+                                    "content": ". This further implies that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        505,
+                                        603
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_{G\\times K}^{M}(G\\times K) = \\chi_{H\\times K}^{M}(H\\times K)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        540,
+                                        505,
+                                        603
+                                    ],
+                                    "type": "text",
+                                    "content": " by definition of homomorphism expressivity. Therefore, Lemma G.2 implies that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        156,
+                        608,
+                        504,
+                        634
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                156,
+                                608,
+                                504,
+                                634
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        156,
+                                        608,
+                                        504,
+                                        634
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\sum_ {L \\in \\mathcal {L}} \\alpha (L) \\operatorname {h o m} (L, G) \\cdot \\operatorname {h o m} (L, K) = \\sum_ {L \\in \\mathcal {L}} \\alpha (L) \\operatorname {h o m} (L, H) \\cdot \\operatorname {h o m} (L, K). \\tag {30}",
+                                    "image_path": "dee3dc28e12aa31a0f0906bae09f941a36f2220f73c1d618c4c82e31edb94fd2.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
+                        105,
+                        638,
+                        141,
+                        650
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                638,
+                                141,
+                                650
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        638,
+                                        141,
+                                        650
+                                    ],
+                                    "type": "text",
+                                    "content": "Namely,"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
+                        157,
+                        654,
+                        504,
+                        681
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                157,
+                                654,
+                                504,
+                                681
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        157,
+                                        654,
+                                        504,
+                                        681
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\sum_ {L \\in \\tilde {\\mathcal {L}}} \\tilde {\\alpha} (L) \\operatorname {h o m} (L, G) \\cdot \\operatorname {h o m} (L, K) = \\sum_ {L \\in \\tilde {\\mathcal {L}}} \\tilde {\\alpha} (L) \\operatorname {h o m} (L, H) \\cdot \\operatorname {h o m} (L, K). \\tag {31}",
+                                    "image_path": "4f5afb4390426f5b9d4884ea35e7d638c96fafe5345a3a55450d8dbd3fb55ad8.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
+                        104,
+                        687,
+                        504,
+                        711
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                687,
+                                504,
+                                711
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        711
+                                    ],
+                                    "type": "text",
+                                    "content": "Now define vectors "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        711
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{p}_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        711
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        711
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{p}_H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        711
+                                    ],
+                                    "type": "text",
+                                    "content": ", where the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        711
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "i^{th}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        711
+                                    ],
+                                    "type": "text",
+                                    "content": " element of vector "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        711
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{p}_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        711
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        711
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{p}_H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        711
+                                    ],
+                                    "type": "text",
+                                    "content": " is "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        711
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\alpha}(L_i) \\mathrm{hom}(L_i, G)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        711
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        711
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\alpha}(L_i) \\mathrm{hom}(L_i, H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        687,
+                                        504,
+                                        711
+                                    ],
+                                    "type": "text",
+                                    "content": ", respectively. We then have the following equation:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
+                        252,
+                        715,
+                        357,
+                        729
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                252,
+                                715,
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+                                729
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        252,
+                                        715,
+                                        357,
+                                        729
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\boldsymbol {A} ^ {\\text {h o m}} \\cdot \\boldsymbol {p} _ {G} = \\boldsymbol {A} ^ {\\text {h o m}} \\cdot \\boldsymbol {p} _ {H}.",
+                                    "image_path": "4a23d9101d525a9ed61dd40cc658789a9e3d5875c0ce2c83694b00bf58837252.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 19
+                }
+            ],
+            "discarded_blocks": [
+                {
+                    "bbox": [
+                        105,
+                        26,
+                        293,
+                        38
+                    ],
+                    "type": "header",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                105,
+                                26,
+                                293,
+                                38
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "bbox": [
+                        299,
+                        750,
+                        310,
+                        760
+                    ],
+                    "type": "page_number",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                299,
+                                750,
+                                310,
+                                760
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        299,
+                                        750,
+                                        310,
+                                        760
+                                    ],
+                                    "type": "text",
+                                    "content": "58"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 20
+                }
+            ],
+            "page_size": [
+                612,
+                792
+            ],
+            "page_idx": 57
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
+                        104,
+                        81,
+                        504,
+                        130
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                81,
+                                504,
+                                130
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": "Since "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{A}^{\\mathrm{hom}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": " is invertible, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pmb{p}_G^{\\mathrm{hom}} = \\pmb{p}_H^{\\mathrm{hom}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": ". Therefore, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{\\alpha}(L)\\mathrm{hom}(L,G) = \\tilde{\\alpha}(L)\\mathrm{hom}(L,H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "L \\in \\tilde{\\mathcal{L}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": ", namely, "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{hom}(L,G) = \\mathrm{hom}(L,H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "L \\in \\mathcal{L}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": ". To sum up, we have proved that for all graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G,H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^M (G) = \\chi_H^M (H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": " implies that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{hom}(L,G) = \\mathrm{hom}(L,H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "L \\in \\mathcal{L}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": ". By definition of homomorphism expressivity, we conclude that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L} \\subset \\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        81,
+                                        504,
+                                        130
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        135,
+                        504,
+                        160
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                135,
+                                504,
+                                160
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        135,
+                                        504,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": "Theorem G.6. Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        135,
+                                        504,
+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\mathrm{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        135,
+                                        504,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": " be any model. For any graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        135,
+                                        504,
+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        135,
+                                        504,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": ", if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        135,
+                                        504,
+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{Spasm}(F) \\backslash \\mathcal{F}^M \\neq \\emptyset"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        135,
+                                        504,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": ", then there exists a pair of graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        135,
+                                        504,
+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G, H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        135,
+                                        504,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": " such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        135,
+                                        504,
+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^M(G) = \\chi_H^M(H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        135,
+                                        504,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        135,
+                                        504,
+                                        160
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{sub}(F, G) \\neq \\mathrm{sub}(F, H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        135,
+                                        504,
+                                        160
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        171,
+                        504,
+                        210
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                171,
+                                504,
+                                210
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        504,
+                                        210
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Recall that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        504,
+                                        210
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{sub}(F, G) = \\sum_{\\tilde{F} \\in \\operatorname{Spasm}^{\\neq}(F)} \\alpha(F, \\tilde{F}) \\cdot \\operatorname{hom}(\\tilde{F}, G)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        504,
+                                        210
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        504,
+                                        210
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha(F, \\tilde{F}) \\neq 0"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        504,
+                                        210
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        504,
+                                        210
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F} \\in \\operatorname{Spasm}^{\\neq}(F)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        504,
+                                        210
+                                    ],
+                                    "type": "text",
+                                    "content": ". If the above theorem does not hold, then for all graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        504,
+                                        210
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G, H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        504,
+                                        210
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        504,
+                                        210
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^M(G) = \\chi_H^M(H)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        171,
+                                        504,
+                                        210
+                                    ],
+                                    "type": "text",
+                                    "content": " implies that"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        157,
+                        213,
+                        453,
+                        243
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        157,
+                                        213,
+                                        453,
+                                        243
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\sum_ {\\tilde {F} \\in \\operatorname {S p a s m} ^ {\\mathcal {Z}} (F)} \\alpha (F, \\tilde {F}) \\cdot \\operatorname {h o m} (\\tilde {F}, G) = \\sum_ {\\tilde {F} \\in \\operatorname {S p a s m} ^ {\\mathcal {Z}} (F)} \\alpha (F, \\tilde {F}) \\cdot \\operatorname {h o m} (\\tilde {F}, G).",
+                                    "image_path": "833df8c73a5dcdd2d8e9f74b75a46674aa3cd9b56333bbf815aa62b036d9a4c6.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
+                                        105,
+                                        247,
+                                        408,
+                                        259
+                                    ],
+                                    "type": "text",
+                                    "content": "Then, Lemma G.5 implies that "
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        247,
+                                        408,
+                                        259
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{Spasm}(F) \\subset \\mathcal{F}^M"
+                                },
+                                {
+                                    "bbox": [
+                                        105,
+                                        247,
+                                        408,
+                                        259
+                                    ],
+                                    "type": "text",
+                                    "content": ", yielding a contradiction."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "type": "image",
+                    "bbox": [
+                        494,
+                        247,
+                        504,
+                        255
+                    ],
+                    "blocks": [
+                        {
+                            "bbox": [
+                                494,
+                                247,
+                                504,
+                                255
+                            ],
+                            "lines": [
+                                {
+                                    "bbox": [
+                                        494,
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+                                        504,
+                                        255
+                                    ],
+                                    "spans": [
+                                        {
+                                            "bbox": [
+                                                494,
+                                                247,
+                                                504,
+                                                255
+                                            ],
+                                            "type": "image",
+                                            "image_path": "57ebffe1e90b2e72ff9a5e60fade4ea4cb2a805920423b6c74c3c01178cff917.jpg"
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 6,
+                            "angle": 0,
+                            "type": "image_body"
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        270,
+                        504,
+                        304
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                270,
+                                504,
+                                304
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        270,
+                                        504,
+                                        304
+                                    ],
+                                    "type": "text",
+                                    "content": "We next extend our analysis to the node/edge-level subgraph counting. Since the proof techniques are almost the same as the graph-level setting, we only present key definitions and lemmas below while omitting the detailed proofs for clarity."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        104,
+                        306,
+                        504,
+                        342
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                306,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "text",
+                                    "content": "Definition G.7. Given two rooted graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^{\\mathbf{u}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H^{\\mathbf{v}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "text",
+                                    "content": " with "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{u} \\in V_G^m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{v} \\in V_H^m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "text",
+                                    "content": " for some "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m \\in \\mathbb{N}_+"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "text",
+                                    "content": ", the categorical product of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^{\\mathbf{u}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H^{\\mathbf{v}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "text",
+                                    "content": ", denoted as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(G \\times H)^{(u_1, v_1), \\dots, (u_m, v_m)}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "text",
+                                    "content": ", is a rooted graph obtained from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G \\times H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "text",
+                                    "content": " by marking vertices "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "(u_1, v_1), \\dots, (u_m, v_m)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        306,
+                                        504,
+                                        342
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
+                        343,
+                        504,
+                        367
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                343,
+                                504,
+                                367
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        343,
+                                        504,
+                                        367
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma G.8 (Extension of Lemma G.2). For any rooted graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        343,
+                                        504,
+                                        367
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F^{\\mathbf{w}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        343,
+                                        504,
+                                        367
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        343,
+                                        504,
+                                        367
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^{\\mathbf{u}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        343,
+                                        504,
+                                        367
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        343,
+                                        504,
+                                        367
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H^{\\mathbf{v}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        343,
+                                        504,
+                                        367
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        343,
+                                        504,
+                                        367
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "|\\mathbf{u}| = |\\mathbf{v}| = | \\mathbf{w} | = m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        343,
+                                        504,
+                                        367
+                                    ],
+                                    "type": "text",
+                                    "content": ","
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
+                        156,
+                        370,
+                        453,
+                        384
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                156,
+                                370,
+                                453,
+                                384
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        156,
+                                        370,
+                                        453,
+                                        384
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\operatorname {h o m} \\left(F ^ {\\boldsymbol {w}}, (G \\times H) ^ {(u _ {1}, v _ {1}), \\dots , (u _ {m}, v _ {m})}\\right) = \\operatorname {h o m} \\left(F ^ {\\boldsymbol {w}}, G ^ {\\boldsymbol {u}}\\right) \\cdot \\operatorname {h o m} \\left(F ^ {\\boldsymbol {w}}, H ^ {\\boldsymbol {v}}\\right).",
+                                    "image_path": "34640b980515072612efaf1da8d4ed4ebf1f03b43eada3961ed7c270b0c68004.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
+                        104,
+                        387,
+                        504,
+                        422
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                387,
+                                504,
+                                422
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        387,
+                                        504,
+                                        422
+                                    ],
+                                    "type": "text",
+                                    "content": "Proposition G.9 (Extension of Proposition G.4). For any rooted graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        387,
+                                        504,
+                                        422
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        387,
+                                        504,
+                                        422
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        387,
+                                        504,
+                                        422
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H^v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        387,
+                                        504,
+                                        422
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        387,
+                                        504,
+                                        422
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "|\\boldsymbol{u}| = |\\boldsymbol{v}| = m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        387,
+                                        504,
+                                        422
+                                    ],
+                                    "type": "text",
+                                    "content": ", "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        387,
+                                        504,
+                                        422
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\operatorname{hom}(G^u, H^v) = \\sum_{F^w} \\operatorname{surj}(G^u, F^w) \\cdot \\operatorname{sub}(F^w, H^v)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        387,
+                                        504,
+                                        422
+                                    ],
+                                    "type": "text",
+                                    "content": ", where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        387,
+                                        504,
+                                        422
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F^w"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        387,
+                                        504,
+                                        422
+                                    ],
+                                    "type": "text",
+                                    "content": " ranges over all non-isomorphic rooted graphs marking "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        387,
+                                        504,
+                                        422
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "m"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        387,
+                                        504,
+                                        422
+                                    ],
+                                    "type": "text",
+                                    "content": " vertices."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
+                        104,
+                        430,
+                        504,
+                        452
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                430,
+                                504,
+                                452
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        430,
+                                        504,
+                                        452
+                                    ],
+                                    "type": "text",
+                                    "content": "We next present the main lemma for node-level subgraph counting. We omit the edge-level result for clarity."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "bbox": [
+                        104,
+                        454,
+                        504,
+                        488
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                454,
+                                504,
+                                488
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        454,
+                                        504,
+                                        488
+                                    ],
+                                    "type": "text",
+                                    "content": "Lemma G.10 (Extension of Lemma G.5). Let "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        454,
+                                        504,
+                                        488
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        454,
+                                        504,
+                                        488
+                                    ],
+                                    "type": "text",
+                                    "content": " be a GNN model such that its node-level homomorphism expressivity "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        454,
+                                        504,
+                                        488
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}_{\\mathfrak{n}}^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        454,
+                                        504,
+                                        488
+                                    ],
+                                    "type": "text",
+                                    "content": " exists. Given a finite set of rooted graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        454,
+                                        504,
+                                        488
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}_{\\mathfrak{n}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        454,
+                                        504,
+                                        488
+                                    ],
+                                    "type": "text",
+                                    "content": " and a function "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        454,
+                                        504,
+                                        488
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\alpha : \\mathcal{L}_{\\mathfrak{n}} \\to \\mathbb{R} / \\{0\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        454,
+                                        504,
+                                        488
+                                    ],
+                                    "type": "text",
+                                    "content": ", if"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "bbox": [
+                        134,
+                        491,
+                        474,
+                        518
+                    ],
+                    "type": "interline_equation",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                134,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        134,
+                                        491,
+                                        474,
+                                        518
+                                    ],
+                                    "type": "interline_equation",
+                                    "content": "\\chi_ {G} ^ {M} (u) = \\chi_ {H} ^ {M} (v) \\Longrightarrow \\sum_ {L ^ {w} \\in \\mathcal {L} _ {\\mathrm {n}}} \\alpha (L ^ {w}) \\operatorname {h o m} (L ^ {w}, G ^ {u}) = \\sum_ {L ^ {w} \\in \\mathcal {L} _ {\\mathrm {n}}} \\alpha (L ^ {w}) \\operatorname {h o m} (L ^ {w}, H ^ {v}),",
+                                    "image_path": "7f540a92a9949b678730fcf4cc4762a31d492dbd589c35d8fc079ac416d1db3e.jpg"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 14
+                },
+                {
+                    "bbox": [
+                        104,
+                        521,
+                        332,
+                        534
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                332,
+                                534
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        521,
+                                        332,
+                                        534
+                                    ],
+                                    "type": "text",
+                                    "content": "holds for all rooted graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        521,
+                                        332,
+                                        534
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G^{u}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        521,
+                                        332,
+                                        534
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        521,
+                                        332,
+                                        534
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H^{v}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        521,
+                                        332,
+                                        534
+                                    ],
+                                    "type": "text",
+                                    "content": ", then "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        521,
+                                        332,
+                                        534
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{L}_{\\mathfrak{n}} \\subset \\mathcal{F}_{\\mathfrak{n}}^{M}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        521,
+                                        332,
+                                        534
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 15
+                },
+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Theorem G.11. Let "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M \\in \\{\\text{Sub}, \\mathsf{L}, \\mathsf{LF}, \\mathsf{F}\\}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        536,
+                                        506,
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+                                    ],
+                                    "type": "text",
+                                    "content": " be any model. For any rooted graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        536,
+                                        506,
+                                        571
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F^w"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        536,
+                                        506,
+                                        571
+                                    ],
+                                    "type": "text",
+                                    "content": ", if "
+                                },
+                                {
+                                    "bbox": [
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+                                        536,
+                                        506,
+                                        571
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\text{Spasm}(F^w) \\backslash \\mathcal{F}_\\mathfrak{n}^M \\neq \\emptyset"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        536,
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+                                    ],
+                                    "type": "text",
+                                    "content": ", then there exist a pair of graphs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G, H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        536,
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+                                    ],
+                                    "type": "text",
+                                    "content": " and vertices "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u \\in V_G, v \\in V_H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": ", such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\chi_G^M(u) = \\chi_H^M(v)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\text{sub}(F^w, G^u) \\neq \\text{sub}(F^w, H^v)"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 16
+                },
+                {
+                    "bbox": [
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+                        594
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+                    "type": "title",
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+                                    "type": "text",
+                                    "content": "G.2 GRAPH STATISTICS AND EXAMPLES"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 17
+                },
+                {
+                    "bbox": [
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+                                    "type": "text",
+                                    "content": "In this section, we list the statistics of all moderate-size graphs that can/cannot be counted for each model in Section 2 at graph/node/edge-level. Table 4 presents the statistics under homomorphism count, while Table 5 presents the statistics under subgraph count. These tables offer a clear picture into how large the expressivity gaps are between different models. Several important findings are discussed below:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "bbox": [
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+                                            "type": "text",
+                                            "content": "- For homomorphism counting, graphs of 8 edges suffice to reveal the expressivity gaps between each pair of architectures (at edge-level)."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 19
+                        },
+                        {
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+                                            "type": "text",
+                                            "content": "- However, for subgraph counting, moderate-size graphs cannot reveal the gap between Local 2-GNN, Local 2-FGNN, and 2-FGNN. Moreover, even Subgraph GNN already matches the power of 2-FWL in counting small subgraphs at graph-level, but it is quite weak in counting subgraphs at node/edge-level."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 20
+                        }
+                    ],
+                    "sub_type": "text"
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+            ],
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
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+                },
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+                                    "content": "59"
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+                            ]
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+                }
+            ],
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+        {
+            "para_blocks": [
+                {
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+                    "lines": [
+                        {
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+                                    "type": "text",
+                                    "content": "- Subgraph counting is much more challenging than homomorphism counting. Intuitively, this is because the homomorphism image of a graph usually has 4-cliques, and any graph that contains a 4 clique as subgraph cannot be counted under homomorphism."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
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+                                    "type": "text",
+                                    "content": "In Table 6, we list all subgraphs at moderate size (no more than 6 vertices or 8 edges) that can/cannot be counted by each GNN model. Here, we only list the graph-level expressivity as the node/edge-level expressivity involves too many non-isomorphic rooted graphs and cannot be fully presented (see Tables 4 and 5). We believe these results is comprehensive enough to cover most substructures of interest in the GNN community."
+                                }
+                            ]
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+                                            "content": "Table 4: The number of (connected) graphs (or rooted graphs) of "
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+                                            "type": "text",
+                                            "content": " vertices or "
+                                        },
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+                                                240
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+                                            "type": "inline_equation",
+                                            "content": "m"
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+                                            "type": "text",
+                                            "content": " edges that can be counted under homomorphism by different models. These statistics can be viewed as a quantitative expressivity comparison between models."
+                                        }
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+                                }
+                            ],
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+                            "angle": 0,
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+                                            "html": "<table><tr><td rowspan=\"2\" colspan=\"2\"></td><td colspan=\"5\">Number of vertices n</td><td colspan=\"7\">Number of edges m</td><td></td></tr><tr><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td></tr><tr><td rowspan=\"6\">Graph level</td><td>MPNN</td><td>1</td><td>1</td><td>2</td><td>3</td><td>6</td><td>1</td><td>1</td><td>2</td><td>3</td><td>6</td><td>11</td><td>23</td><td>47</td></tr><tr><td>Subgraph GNN</td><td>1</td><td>2</td><td>5</td><td>15</td><td>51</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>29</td><td>76</td><td>210</td></tr><tr><td>Local 2-GNN</td><td>1</td><td>2</td><td>5</td><td>15</td><td>55</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>29</td><td>77</td><td>216</td></tr><tr><td>Local 2-FGNN</td><td>1</td><td>2</td><td>5</td><td>15</td><td>56</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>29</td><td>77</td><td>216</td></tr><tr><td>2-FGNN</td><td>1</td><td>2</td><td>5</td><td>15</td><td>56</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>29</td><td>77</td><td>216</td></tr><tr><td>All</td><td>1</td><td>2</td><td>6</td><td>21</td><td>112</td><td>1</td><td>1</td><td>3</td><td>5</td><td>12</td><td>30</td><td>79</td><td>227</td></tr><tr><td rowspan=\"6\">Node level</td><td>MPNN</td><td>1</td><td>2</td><td>4</td><td>9</td><td>20</td><td>1</td><td>2</td><td>4</td><td>9</td><td>20</td><td>48</td><td>115</td><td>286</td></tr><tr><td>Subgraph GNN</td><td>1</td><td>3</td><td>8</td><td>27</td><td>88</td><td>1</td><td>2</td><td>5</td><td>12</td><td>31</td><td>83</td><td>228</td><td>640</td></tr><tr><td>Local 2-GNN</td><td>1</td><td>3</td><td>10</td><td>44</td><td>215</td><td>1</td><td>2</td><td>5</td><td>13</td><td>37</td><td>113</td><td>361</td><td>1210</td></tr><tr><td>Local 2-FGNN</td><td>1</td><td>3</td><td>10</td><td>44</td><td>217</td><td>1</td><td>2</td><td>5</td><td>13</td><td>37</td><td>113</td><td>361</td><td>1210</td></tr><tr><td>2-FGNN</td><td>1</td><td>3</td><td>10</td><td>44</td><td>217</td><td>1</td><td>2</td><td>5</td><td>13</td><td>37</td><td>113</td><td>361</td><td>1210</td></tr><tr><td>All</td><td>1</td><td>3</td><td>11</td><td>58</td><td>407</td><td>1</td><td>2</td><td>5</td><td>13</td><td>37</td><td>114</td><td>367</td><td>1248</td></tr><tr><td rowspan=\"5\">Edge level</td><td>Subgraph GNN</td><td>1</td><td>4</td><td>18</td><td>77</td><td>340</td><td>1</td><td>3</td><td>10</td><td>33</td><td>107</td><td>347</td><td>1126</td><td>3664</td></tr><tr><td>Local 2-GNN</td><td>1</td><td>4</td><td>21</td><td>116</td><td>693</td><td>1</td><td>3</td><td>10</td><td>35</td><td>124</td><td>450</td><td>1665</td><td>6267</td></tr><tr><td>Local 2-FGNN</td><td>1</td><td>4</td><td>21</td><td>118</td><td>735</td><td>1</td><td>3</td><td>10</td><td>35</td><td>124</td><td>451</td><td>1678</td><td>6373</td></tr><tr><td>2-FGNN</td><td>1</td><td>4</td><td>21</td><td>118</td><td>735</td><td>1</td><td>3</td><td>10</td><td>35</td><td>124</td><td>451</td><td>1678</td><td>6374</td></tr><tr><td>All</td><td>1</td><td>4</td><td>23</td><td>162</td><td>1549</td><td>1</td><td>3</td><td>10</td><td>35</td><td>125</td><td>460</td><td>1747</td><td>6830</td></tr></table>",
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+                                            "type": "text",
+                                            "content": "Table 5: The number of (connected) graphs (or rooted graphs) of "
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+                                            "type": "inline_equation",
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+                                            "type": "text",
+                                            "content": " edges that can be subgraph-counted by different models. These statistics can be viewed as a quantitative expressivity comparison between models."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 5,
+                            "angle": 0,
+                            "type": "table_caption"
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+                                            "content": " that can be used to construct counterexample graphs (see Section 4.2)."
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m</td><td>F</td><td colspan=\"101\">Homonorphism Subgraph MP Sub L F F</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr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+                                            "image_path": "363789b35f253122d6b4338a02327fc097b6b836fbd0dff0fabacab310df8bec.jpg"
+                                        }
+                                    ]
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+                            "type": "table_body"
+                        }
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+            "discarded_blocks": [
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+                                    "bbox": [
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+                                        26,
+                                        293,
+                                        38
+                                    ],
+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
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+                },
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+                                    "type": "text",
+                                    "content": "68"
+                                }
+                            ]
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+                    ],
+                    "index": 2
+                }
+            ],
+            "page_size": [
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+                792
+            ],
+            "page_idx": 67
+        },
+        {
+            "para_blocks": [
+                {
+                    "bbox": [
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+                    "type": "title",
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+                                    "bbox": [
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+                                        94
+                                    ],
+                                    "type": "text",
+                                    "content": "H POLYNOMIAL EXPRESSIVITY"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
+                        104,
+                        106,
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": "Puny et al. (2023) proposed the equivariant graph polynomials, which are polynomials "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": " that take real squared matrices "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{X} \\in \\mathbb{R}^{n \\times n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": " as input and outputs "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P(\\mathbf{X}) \\in \\mathbb{R}^{n \\times n}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": ", such that "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": " is equivariant under permutations, i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P(\\pi \\cdot \\mathbf{X}) = \\pi \\cdot P(\\mathbf{X})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": " for all permutation "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\pi \\in S_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": ". The authors gave a concrete basis of equivariant polynomials, where each basis element "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{F^{uv}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": " corresponds to a rooted multi-graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F^{uv}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": " marking two vertices "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": ". Puny et al. (2023) showed that when "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathbf{X}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": " is restricted to be the adjacency matrix of an undirected simple graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": ", each "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F^{uv}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": " will also reduce to an undirected simple graph, and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{F^{uv}}(\\mathbf{X})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": " precisely computes the (unlabeled) homomorphism count "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{hom}(F^{uv}, G^{wx})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w, x \\in V_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        106,
+                                        506,
+                                        217
+                                    ],
+                                    "type": "text",
+                                    "content": ". Besides the original definition, Puny et al. (2023) also proposed the invariant graph polynomials and node-level equivariant graph polynomials, which are similarly related to the graph-level and node-level homomorphism count."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 2
+                },
+                {
+                    "bbox": [
+                        104,
+                        221,
+                        504,
+                        277
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                221,
+                                504,
+                                277
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        221,
+                                        504,
+                                        277
+                                    ],
+                                    "type": "text",
+                                    "content": "As a direct consequence, if a GNN model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        221,
+                                        504,
+                                        277
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        221,
+                                        504,
+                                        277
+                                    ],
+                                    "type": "text",
+                                    "content": " cannot count graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        221,
+                                        504,
+                                        277
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F^{uv} / F^u /F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        221,
+                                        504,
+                                        277
+                                    ],
+                                    "type": "text",
+                                    "content": " under homomorphism, it then cannot compute the equivariant/invariant graph polynomial "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        221,
+                                        504,
+                                        277
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{F^{uv}} / P_{F^u} / P_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        221,
+                                        504,
+                                        277
+                                    ],
+                                    "type": "text",
+                                    "content": ". Based on these connections, our results can be directly used to provide insights into which equivariant graph polynomials cannot be computed by model "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        221,
+                                        504,
+                                        277
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "M"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        221,
+                                        504,
+                                        277
+                                    ],
+                                    "type": "text",
+                                    "content": ". This recovers several results in their paper and answers an open problem shown below."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
+                        104,
+                        281,
+                        506,
+                        315
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                281,
+                                506,
+                                315
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        281,
+                                        506,
+                                        315
+                                    ],
+                                    "type": "text",
+                                    "content": "Corollary H.1. MPNN and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        281,
+                                        506,
+                                        315
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "PPGN++"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        281,
+                                        506,
+                                        315
+                                    ],
+                                    "type": "text",
+                                    "content": " are bounded by the Prototypical node-based model and Prototypical edge-based model (defined in Puny et al. (2023)) for computing node-level and edge-level equivariant graph polynomials, respectively."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
+                        104,
+                        327,
+                        504,
+                        417
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                327,
+                                504,
+                                417
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        327,
+                                        504,
+                                        417
+                                    ],
+                                    "type": "text",
+                                    "content": "Proof. Without loss of generality, we assume that the corresponding graphs of all equivariant polynomials are connected. According to Puny et al. (2023, Proposition H.2), the Prototypical node-based model can compute all "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        327,
+                                        504,
+                                        417
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{F^u}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        327,
+                                        504,
+                                        417
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        327,
+                                        504,
+                                        417
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F^u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        327,
+                                        504,
+                                        417
+                                    ],
+                                    "type": "text",
+                                    "content": " is a (rooted) tree and cannot compute other graph polynomials. If MPNN is not bounded by the Prototypical node-based model, then it can compute some "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        327,
+                                        504,
+                                        417
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{\\tilde{F}^u}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        327,
+                                        504,
+                                        417
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        327,
+                                        504,
+                                        417
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        327,
+                                        504,
+                                        417
+                                    ],
+                                    "type": "text",
+                                    "content": " is not a tree. However, this is impossible since MPNN can only count forests under homomorphism according to Theorem 3.7. Note that the MPNN defined in their paper is equivalent to our definition when only considering connected graphs (the extra global aggregation "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        327,
+                                        504,
+                                        417
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "11^\\top X"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        327,
+                                        504,
+                                        417
+                                    ],
+                                    "type": "text",
+                                    "content": " in their definition (9) does not improve the homomorphism expressivity)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        422,
+                        506,
+                        540
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                422,
+                                506,
+                                540
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "text",
+                                    "content": "We next turn to "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{PPGN}++"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "text",
+                                    "content": ", and the proof is similar (but more involved). We first show that the Prototypical edge-based model can compute any "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{F^{uv}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "text",
+                                    "content": " satisfying that the treewidth of the graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F} := (V_F, E_F \\cup \\{\\{u, v\\}\\}, \\ell_F)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "text",
+                                    "content": " is no more than 2. If "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{tw}(\\tilde{F}) \\leq 2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "text",
+                                    "content": ", it is a partial 2-tree. Thus, there is an ordering "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_1, \\dots, w_n"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "text",
+                                    "content": " of the vertex set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "text",
+                                    "content": " such that when deleting each vertex "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_i"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "text",
+                                    "content": " and all incident edges in turn, we only ever delete vertices of degree at most 2. Now we claim that we can always order the two vertices "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "text",
+                                    "content": " at the end, i.e., "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_{n-1} = u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_n = v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "text",
+                                    "content": ". Otherwise, there is a subset "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "U \\subset V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "text",
+                                    "content": " such that all vertices in the induced subgraph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F}[U]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "text",
+                                    "content": " are of degree at least 3 except "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "text",
+                                    "content": ". It follows that the "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{tw}(\\tilde{F}[U]) \\geq \\mathsf{tw}(H) \\geq 3"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "text",
+                                    "content": " where graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "H"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "text",
+                                    "content": " is the graph obtained from "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F}[U]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "text",
+                                    "content": " by contracting "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "text",
+                                    "content": " (or "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "text",
+                                    "content": ") if "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\deg_{\\tilde{F}[U]}(u) \\leq 2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "text",
+                                    "content": " (or "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\deg_{\\tilde{F}[U]}(v) \\leq 2"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "text",
+                                    "content": "). This yields a contradiction and verifies the claim that we can always set "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_{n-1} = u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w_n = v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        422,
+                                        506,
+                                        540
+                                    ],
+                                    "type": "text",
+                                    "content": "."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        544,
+                        506,
+                        639
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                544,
+                                506,
+                                639
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
+                                        544,
+                                        506,
+                                        639
+                                    ],
+                                    "type": "text",
+                                    "content": "Now, following the proof in Puny et al. (2023, Proposition H.3), the Prototypical edge-based model can contract "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        544,
+                                        506,
+                                        639
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F}^{uv}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        544,
+                                        506,
+                                        639
+                                    ],
+                                    "type": "text",
+                                    "content": " to a graph with only two vertices "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        544,
+                                        506,
+                                        639
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u, v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        544,
+                                        506,
+                                        639
+                                    ],
+                                    "type": "text",
+                                    "content": " and thus can compute the edge-level polynomial "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        544,
+                                        506,
+                                        639
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{\\tilde{F}^{uv}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        544,
+                                        506,
+                                        639
+                                    ],
+                                    "type": "text",
+                                    "content": ". If PPGN++ is not bounded by the Prototypical edge-based model, then it can compute some "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        544,
+                                        506,
+                                        639
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "P_{\\tilde{F}^{uv}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        544,
+                                        506,
+                                        639
+                                    ],
+                                    "type": "text",
+                                    "content": " where "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        544,
+                                        506,
+                                        639
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{tw}(\\tilde{F}) \\geq 3"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        544,
+                                        506,
+                                        639
+                                    ],
+                                    "type": "text",
+                                    "content": ". Therefore, it can count the graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        544,
+                                        506,
+                                        639
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        544,
+                                        506,
+                                        639
+                                    ],
+                                    "type": "text",
+                                    "content": " under homomorphism at graph-level (since it can already count "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        544,
+                                        506,
+                                        639
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\tilde{F}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        544,
+                                        506,
+                                        639
+                                    ],
+                                    "type": "text",
+                                    "content": " at edge-level). This implies that PPGN++ is strictly more expressive than 2-FGNN (2-FWL) because we have proved that all graphs in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        544,
+                                        506,
+                                        639
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathcal{F}^{\\mathrm{F}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        544,
+                                        506,
+                                        639
+                                    ],
+                                    "type": "text",
+                                    "content": " have a treewidth no more than 2. This yields a contradiction since PPGN++ is still bounded by 2-FWL in distinguishing non-isomorphic graphs."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "We also provide insights into the following results in their paper:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Corollary H.2. The Prototypical node-based model is not 3-node-polynomial-exact. The Prototypical edge-based model is not 6-node-polynomial-exact and not 5-edge-polynomial-exact."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
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+                                    "type": "text",
+                                    "content": "This is simply because the triangle is not a tree and the 4-clique does not have a NED (or equivalently, the treewidth of a 4-clique is 3). It is also clear why the degree of the edge-based polynomial is 6, which is one less than that of the node-based polynomial using the concept of NED."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                }
+            ],
+            "discarded_blocks": [
+                {
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+                    ],
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+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
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+                                    "type": "text",
+                                    "content": "69"
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+                    ],
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+                }
+            ],
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+                    ],
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+                                    "type": "text",
+                                    "content": "I EXPERIMENTAL DETAILS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
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+                                    "type": "text",
+                                    "content": "In this section, we provide all the experimental details in Section 5."
+                                }
+                            ]
+                        }
+                    ],
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+                },
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+                                    "type": "text",
+                                    "content": "I.1 DATASETS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "We conduct experiments on tive tasks: (i) graph homomorphism counting, (ii) subgraph counting, (iii) ZINC-subset (Dwivedi et al., 2020), (iii) ZINC-full (Dwivedi et al., 2020), and (iv) Alchemy (Chen et al., 2019a)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Homomorphism/Subgraph Counting. For both homomorphism and subgraph counting tasks, we use the standard synthetic graph dataset constructed in Zhao et al. (2022a) (which has been used in a number of papers, see e.g., Frasca et al., 2022; Huang et al., 2023; Zhang et al., 2023a). For homomorphism counting, we count the number of graph/node/edge-level homomorphisms for each pattern in Table 1 and normalize the value by the mean and variance across all graphs in the dataset. The evaluation metric of graph-level expressivity is chosen as the Mean Absolute Error (MAE). For node/edge-level expressivity, the error on each graph is defined to be the sum of absolute error over all vertices/edges. We then report the MAE across all graphs in the dataset. This ensures that graph/node/edge-level errors are roughly at the same scale (since "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        191,
+                                        506,
+                                        369
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\mathrm{hom}(F,G) = \\sum_{w\\in V_G}\\mathrm{hom}(F^u,G^w) = \\sum_{w,x\\in V_G}\\mathrm{hom}(F^{uv},G^{wx})"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        191,
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+                                    ],
+                                    "type": "text",
+                                    "content": " for all "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u,v\\in V_F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        191,
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+                                        369
+                                    ],
+                                    "type": "text",
+                                    "content": "). For subgraph counting, the data processing and evaluation metric is similar to homomorphism counting, but there is a slight difference in the node/edge-level setting: there are no marked vertices in the pattern graph "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        191,
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+                                        369
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        191,
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+                                        369
+                                    ],
+                                    "type": "text",
+                                    "content": " (see Table 3). Instead, given a graph "
+                                },
+                                {
+                                    "bbox": [
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+                                        191,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        191,
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+                                    ],
+                                    "type": "text",
+                                    "content": " in the dataset and a vertex "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w\\in V_G"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        191,
+                                        506,
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+                                    ],
+                                    "type": "text",
+                                    "content": ", we count the number of subgraphs containing "
+                                },
+                                {
+                                    "bbox": [
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+                                        191,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " that are isomorphic to "
+                                },
+                                {
+                                    "bbox": [
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+                                        191,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        191,
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+                                        369
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "w"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        191,
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+                                        369
+                                    ],
+                                    "type": "text",
+                                    "content": " can be mapped to any vertex in "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        191,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "F"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        191,
+                                        506,
+                                        369
+                                    ],
+                                    "type": "text",
+                                    "content": ". Due to this difference, in the node/edge-level setting, the error on each graph is defined to be the average of absolute error over all vertices/edges."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "bbox": [
+                        104,
+                        373,
+                        504,
+                        430
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                        430
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+                                    "type": "text",
+                                    "content": "ZINC. ZINC (Dwivedi et al., 2020) is a standard real-world dataset for benchmarking molecular property prediction. The dataset consists of 250K molecular graphs, and the task is to predict the constrained solubility of the given molecule. In addition to the full dataset (denoted as ZINC-full), ZINC-subset is a sampled dataset with 12k molecules from the ZINC-full dataset. We train and test our models on both datasets following the standard protocol from Dwivedi et al. (2020)."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 6
+                },
+                {
+                    "bbox": [
+                        104,
+                        434,
+                        506,
+                        479
+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
+                                104,
+                                434,
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+                                479
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                        479
+                                    ],
+                                    "type": "text",
+                                    "content": "Alchemy. Alchemy (Chen et al., 2019a) is another real-world dataset with 12 graph-level quantum mechanical properties. We follow the sampling and training protocol from Lim et al. (2023); Puny et al. (2023), using 100K samples for training, 10K samples for testing, and 10K samples for validation."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
+                        105,
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+                    ],
+                    "type": "title",
+                    "angle": 0,
+                    "lines": [
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+                                    "type": "text",
+                                    "content": "I.2 MODEL DETAILS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
+                        104,
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+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "All models are implemented using the PyTorch (Paszke et al., 2019) framework and the PyTorch Geometric library (Fey & Lenssen, 2019). We consider four types of GNNs defined in Section 2: MPNN, Subgraph GNN, Local 2-GNN, and Local 2-FGNN. For each GNN model, the feature initialization, message-passing layers, and final pooling operation are separately defined below."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                        563,
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+                                        693
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+                                    "type": "text",
+                                    "content": "Initialization. On both ZINC and Alchemy datasets, each graph node is an atom. We maintain a learnable atom embedding for each type of atom and use it to initialize features in GNN models. For MPNN, the initial feature "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
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+                                    "type": "inline_equation",
+                                    "content": "h^{(0)}(u)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " of node "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
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+                                    ],
+                                    "type": "text",
+                                    "content": " is simply the atom embedding, denoted as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        506,
+                                        693
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h^{(0)}(u) = e_{\\mathrm{atom}(u)}^{\\mathrm{N}}"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        506,
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+                                    ],
+                                    "type": "text",
+                                    "content": ". For other models, the initial feature "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        506,
+                                        693
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h^{(0)}(u,v)"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
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+                                    ],
+                                    "type": "text",
+                                    "content": " of node pair "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "(u,v)"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " consists of two parts: the first part is the node embedding of "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "v"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": ", and the second part is a distance encoding that embeds the shortest path distance between "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "u"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": " and "
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "v"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
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+                                    ],
+                                    "type": "text",
+                                    "content": ", as adopted in Zhang et al. (2023a). We note that while incorporating distance encoding does not increase the models' theoretical expressive power (see Zhang et al. (2023a)), it may add an inductive bias that can be helpful in real-world tasks. Formally, the initial feature can be written as "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
+                                        506,
+                                        693
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h^{(0)}(u,v) = [e_{\\mathrm{atom}(v)}^{\\mathrm{N}}, e_{\\mathrm{clip}(\\mathrm{dis}(u,v))}^{\\mathrm{D}}]"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        563,
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+                                    ],
+                                    "type": "text",
+                                    "content": ". Here, we clip the distance to a predefined value max_dis so that there are a finite number of distance embeddings, and distances greater than the hyper-parameter max_dis (including the disconnected case) share the embedding."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Propagation. On both ZINC and Alchemy datasets, each edge in a graph corresponds to a chemical bond and has a bond type. We maintain a learnable edge embedding for each type of edges in each layer and denote the embedding of edge "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        697,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "\\{u,v\\}"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": " in layer "
+                                },
+                                {
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "l"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " as "
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+                                    "type": "inline_equation",
+                                    "content": "g^{(l)}(u,v)"
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+                                    "type": "text",
+                                    "content": ". For MPNN, we use the"
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+                        }
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+                }
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+                                    "content": "Published as a conference paper at ICLR 2024"
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+                                    "type": "text",
+                                    "content": "standard GIN architecture proposed in Xu et al. (2019), which has the following form:"
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+                            ]
+                        }
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+                                    "content": "h ^ {(l + 1)} (u) = \\operatorname {R e L U} \\left(\\mathsf {B N} ^ {(l)} \\left(f ^ {(l)} (u)\\right)\\right), \\tag {32}",
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+                                    "content": "f ^ {(l)} (u) = \\mathsf {G I N} ^ {(l)} \\left(h ^ {(l)} (u), \\sum_ {v \\in N _ {G} (u)} \\operatorname {R e L U} \\left(\\mathsf {F C} ^ {(l)} \\left(h ^ {(l)} (v)\\right) + g ^ {(l)} (u, v)\\right)\\right), \\tag {33}",
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+                                    "type": "interline_equation",
+                                    "content": "\\operatorname {G I N} ^ {(l)} (\\boldsymbol {x}, \\boldsymbol {y}) = \\operatorname {M L P} ^ {(l)} \\left((1 + \\epsilon^ {(l)}) \\boldsymbol {x} + \\boldsymbol {y}\\right). \\tag {34}",
+                                    "image_path": "a3af74485321521403b37e480bf46b3fac872bc5a480759472caa17c9056625e.jpg"
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+                                    "type": "text",
+                                    "content": "Here, "
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+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{FC}^{(l)}"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " is a parameterized linear transformation, "
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+                                    "type": "inline_equation",
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+                                    "type": "text",
+                                    "content": " is a learnable parameter, "
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+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{BN}^{(l)}"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " is the batch normalization (Ioffe & Szegedy, 2015), and "
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+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "\\mathsf{MLP}^{(l)}"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " is a two-layer feed-forward network with another batch normalization in the hidden layer."
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+                                    "type": "text",
+                                    "content": "For other architectures, the "
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+                                    "type": "inline_equation",
+                                    "content": "l"
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+                                    "type": "text",
+                                    "content": "-th GNN layer analogously has the following form:"
+                                }
+                            ]
+                        }
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+                                    "type": "interline_equation",
+                                    "content": "h ^ {(l + 1)} (u, v) = \\operatorname {R e L U} \\left(\\mathsf {B N} ^ {(l)} \\left(f ^ {(l)} (u, v)\\right)\\right), \\tag {35}",
+                                    "image_path": "f6844b7d61d465170406f49d362e53016257485c07f216ce051b9a54b3526982.jpg"
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+                                    "type": "text",
+                                    "content": "where the term "
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+                                {
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+                                    "type": "inline_equation",
+                                    "content": "f^{(l)}(u,v)"
+                                },
+                                {
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+                                    "type": "text",
+                                    "content": " is defined separately for each model:"
+                                }
+                            ]
+                        }
+                    ],
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+                },
+                {
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+                                    "type": "text",
+                                    "content": "- Subgraph GNN:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
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+                                    "type": "interline_equation",
+                                    "content": "f ^ {(l)} (u, v) = \\mathsf {G I N} ^ {(l)} \\left(h ^ {(l)} (u, v), \\sum_ {w \\in N _ {G} (v)} \\operatorname {R e L U} \\left(\\mathsf {F C} ^ {(l)} \\left(h ^ {(l)} (u, w)\\right) + g ^ {(l)} (w, v)\\right)\\right). \\tag {36}",
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+                                    "type": "text",
+                                    "content": "- Local 2-GNN:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                },
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+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} f ^ {(l)} (u, v) = \\mathsf {G I N} ^ {(l, 1)} \\left(h ^ {(l)} (u, v), \\sum_ {w \\in N _ {G} (u)} \\operatorname {R e L U} \\left(\\mathsf {F C} ^ {(l, 1)} \\left(h ^ {(l)} (w, v)\\right) + g ^ {(l)} (u, w)\\right)\\right) \\\\ + \\operatorname {G I N} ^ {(l, 2)} \\left(h ^ {(l)} (u, v), \\sum_ {w \\in N _ {G} (v)} \\operatorname {R e L U} \\left(\\mathsf {F C} ^ {(l, 2)} \\left(h ^ {(l)} (u, w)\\right) + g ^ {(l)} (w, v)\\right)\\right). \\tag {37} \\\\ \\end{array}",
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+                                    "content": "- Local 2-FGNN:"
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+                                    "type": "interline_equation",
+                                    "content": "\\begin{array}{l} f ^ {(l)} (u, v) \\\\ = \\mathsf {G I N} ^ {(l, 1)} \\left(h ^ {(l)} (u, v), \\sum_ {w \\in N _ {G} (u)} \\mathsf {R e L U} \\left(\\mathsf {F C} ^ {(l, 1)} \\left(h ^ {(l)} (u, w)\\right) + \\mathsf {F C} ^ {(l, 2)} \\left(h ^ {(l)} (w, v)\\right) + g ^ {(l)} (u, w)\\right)\\right) \\\\ + \\mathsf {G I N} ^ {(l, 2)} \\left(h ^ {(l)} (u, v), \\sum_ {w \\in N _ {G} (v)} \\operatorname {R e L U} \\left(\\mathsf {F C} ^ {(l, 1)} \\left(h ^ {(l)} (w, v)\\right) + \\mathsf {F C} ^ {(l, 2)} \\left(h ^ {(l)} (u, w)\\right) + g ^ {(l)} (w, v)\\right)\\right). \\tag {38} \\\\ \\end{array}",
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+                                    "content": "For all the above GNN architectures, it can be seen that each layer only aggregates the local neighborhood of vertices or vertex pairs. This design will have shortcomings for disconnected graphs since a vertex cannot aggregate information from other connected components no matter how deep the model is (Barceló et al., 2020). Note that there do exist disconnected graphs in real-world datasets like ZINC. Therefore, on real-world datasets like ZINC and Alchemy, we also incorporate a global aggregation with the following form for each layer (similar to the global aggregation in Frasca et al. (2022); Zhang et al. (2023a)):"
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+                                    "type": "interline_equation",
+                                    "content": "\\operatorname {G I N} ^ {(l, G)} \\left(h ^ {(l)} (u, v), \\sum_ {w \\in V _ {G}} h ^ {(l)} (u, w)\\right) \\tag {39}",
+                                    "image_path": "8bb463948c2c1ebb161f6d522c1f72c98df91e737a265b549facef1fa3e80da3.jpg"
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+                                    "content": "Note that the global aggregation does not increase model's theoretical expressive power according to Zhang et al. (2023a)."
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+                                    "type": "text",
+                                    "content": "Pooling. Except for edge-level tasks, a final pooling layer is used to produce node-level features "
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "h(u)"
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+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " for all nodes "
+                                },
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+                                    "bbox": [
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+                                    "type": "inline_equation",
+                                    "content": "u"
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+                                    "type": "text",
+                                    "content": ". It is implemented as follows:"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 1
+                },
+                {
+                    "bbox": [
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+                                    "type": "interline_equation",
+                                    "content": "h (u) = \\operatorname {M L P} \\left(\\sum_ {v \\in \\mathcal {V}} h ^ {(L)} (u, v)\\right), \\tag {40}",
+                                    "image_path": "adbb77f818c4baacdf320294cbb05b615a3dd45b9537456f7f61588ff625fbcf.jpg"
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+                    "index": 2
+                },
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+                    ],
+                    "type": "text",
+                    "angle": 0,
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "where MLP is a 2-layer perceptron. For graph-level tasks, we further use a mean pooling layer to aggregate all "
+                                },
+                                {
+                                    "bbox": [
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+                                        149,
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+                                        172
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "h(u)"
+                                },
+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": " and obtain the graph representation."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "bbox": [
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+                    ],
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+                                    "type": "text",
+                                    "content": "I.3 TRAINING DETAILS"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 4
+                },
+                {
+                    "bbox": [
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+                        205,
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                        283
+                                    ],
+                                    "type": "text",
+                                    "content": "All experiments are run on a single NVIDIA Tesla V100 GPU. For all tasks, we use the distance encoding hyper-parameter max_dis = 5. To enable a fair comparison between models, for each task we keep the same depth for different models while varying the hidden dimension so that the number of model parameters is roughly the same. Note that the dimensions are chosen such that all models roughly obey the 500K parameter budget in ZINC and Alchemy. The hidden dimension size and the number of model parameters are listed as follows. All models are trained using the Adam optimizer."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "type": "table",
+                    "bbox": [
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+                    ],
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+                            "bbox": [
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+                                            ],
+                                            "type": "text",
+                                            "content": "Table 7: Model size in different tasks."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 6,
+                            "angle": 0,
+                            "type": "table_caption"
+                        },
+                        {
+                            "bbox": [
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+                                            "html": "<table><tr><td rowspan=\"2\">Task</td><td colspan=\"3\">Hidden dimension</td><td colspan=\"3\"># Parameters</td></tr><tr><td>Counting</td><td>ZINC</td><td>Alchemy</td><td>Counting</td><td>ZINC</td><td>Alchemy</td></tr><tr><td>MPNN</td><td>128</td><td>150</td><td>150</td><td>314,119</td><td>510,158</td><td>509,719</td></tr><tr><td>Subgraph GNN</td><td>128</td><td>120</td><td>120</td><td>314,759</td><td>503,774</td><td>503,425</td></tr><tr><td>Local 2-GNN</td><td>96</td><td>96</td><td>96</td><td>317,388</td><td>495,188</td><td>494,911</td></tr><tr><td>Local 2-FGNN</td><td>96</td><td>96</td><td>96</td><td>317,388</td><td>495,188</td><td>494,911</td></tr></table>",
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+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 7,
+                            "angle": 0,
+                            "type": "table_body"
+                        }
+                    ],
+                    "index": 7
+                },
+                {
+                    "bbox": [
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+                    "angle": 0,
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "type": "text",
+                                    "content": "Homomorphism/Subgraph Counting. We use a model depth of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        395,
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+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "L = 5"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        395,
+                                        504,
+                                        462
+                                    ],
+                                    "type": "text",
+                                    "content": " in all experiments. Following prior work (Huang et al., 2023), we remove all BN layers in all models. The initial learning rate is chosen as 0.001 and is decayed by a factor of 0.9 once the MAE on the validation set plateaus for 10 epochs. Each model is trained for 1200 epochs with a batch size of 512. We ran each experiment 4 times independently with different seeds and reported the average performance at the last epoch. We found that the standard deviation among different seeds is negligible."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 8
+                },
+                {
+                    "bbox": [
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+                    ],
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+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "ZINC. We use a model depth of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        467,
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+                                        544
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "L = 6"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        467,
+                                        504,
+                                        544
+                                    ],
+                                    "type": "text",
+                                    "content": " in all experiments for both ZINC-subset and ZINC-full. Following prior work (Zhang et al., 2023a; Frasca et al., 2022), The initial learning rate is chosen as 0.001 and is decayed by a factor of 0.5 once the MAE on the validation set plateaus for 20 epochs. Each model is trained for 400 epochs on ZINC-subset and 500 epochs on ZINC-full, both with a batch size of 128. We report the MAE for the model checkpoint with the best validation performance. We ran each experiment 10 times independently with different seeds and reported the average performance as well as the standard deviation."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 9
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "text",
+                    "angle": 0,
+                    "lines": [
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+                            "bbox": [
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+                                {
+                                    "bbox": [
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+                                    "type": "text",
+                                    "content": "Alchemy. We use a model depth of "
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        616
+                                    ],
+                                    "type": "inline_equation",
+                                    "content": "L = 6"
+                                },
+                                {
+                                    "bbox": [
+                                        104,
+                                        549,
+                                        504,
+                                        616
+                                    ],
+                                    "type": "text",
+                                    "content": " in all experiments. Following prior work (Lim et al., 2023; Puny et al., 2023), The initial learning rate is chosen as 0.002 and is decayed by a factor of 0.5 once the MAE on the validation set plateaus for 20 epochs. Each model is trained for 500 epochs with a batch size of 128. We report the MAE for the model checkpoint with the best validation performance. We ran each experiment 10 times independently with different seeds and reported the average performance as well as the standard deviation."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "bbox": [
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+                    ],
+                    "type": "title",
+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "I.4 PERFORMANCE OF BASELINE MODELS IN LITERATURE"
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "bbox": [
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+                    "angle": 0,
+                    "lines": [
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+                                    "type": "text",
+                                    "content": "For completeness, in this subsection we give a comprehensive list of the performance of GNN models in the literature on ZINC and Alchemy datasets. The numbers in each table below are directly taken from the original papers."
+                                }
+                            ]
+                        }
+                    ],
+                    "index": 12
+                }
+            ],
+            "discarded_blocks": [
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+                    "angle": 0,
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+                                    "type": "text",
+                                    "content": "Published as a conference paper at ICLR 2024"
+                                }
+                            ]
+                        }
+                    ],
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+                },
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+                                    "content": "72"
+                                }
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+                        }
+                    ],
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+                }
+            ],
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+        },
+        {
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+                    "type": "table",
+                    "bbox": [
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+                                            "type": "text",
+                                            "content": "Table 8: Performance of different GNN models on ZINC dataset reported in the literature."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 1,
+                            "angle": 0,
+                            "type": "table_caption"
+                        },
+                        {
+                            "bbox": [
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+                                            "type": "table",
+                                            "html": "<table><tr><td>Method</td><td>Model</td><td>Reference</td><td>Test MAE</td></tr><tr><td rowspan=\"8\">MPNN</td><td>GIN</td><td>(Xu et al., 2019)</td><td>0.526±0.051</td></tr><tr><td>GraphSAGE</td><td>(Hamilton et al., 2017)</td><td>0.398±0.002</td></tr><tr><td>GAT</td><td>(Veličković et al., 2018)</td><td>0.384±0.007</td></tr><tr><td>GCN</td><td>(Kipf &amp; Welling, 2017)</td><td>0.367±0.011</td></tr><tr><td>MoNet</td><td>(Monti et al., 2017)</td><td>0.292±0.006</td></tr><tr><td>GatedGCN-PE</td><td>(Bresson &amp; Laurent, 2017)</td><td>0.214±0.006</td></tr><tr><td>MPNN(sum)</td><td>(Gilmer et al., 2017)</td><td>0.145±0.007</td></tr><tr><td>PNA</td><td>(Corso et al., 2020)</td><td>0.142±0.010</td></tr><tr><td rowspan=\"4\">Higher-order GNN</td><td>RingGNN</td><td>(Chen et al., 2019b)</td><td>0.353±0.019</td></tr><tr><td>PPGN</td><td>(Maron et al., 2019a)</td><td>0.303±0.068</td></tr><tr><td>PPGN</td><td>(Puny et al., 2023)</td><td>0.079±0.005</td></tr><tr><td>PPGN++</td><td>(Puny et al., 2023)</td><td>0.076±0.003</td></tr><tr><td rowspan=\"7\">Subgraph GNN</td><td>NGNN</td><td>(Zhang &amp; Li, 2021)</td><td>0.111±0.003</td></tr><tr><td>GNN-AK</td><td>(Zhao et al., 2022a)</td><td>0.105±0.010</td></tr><tr><td>GNN-AK+</td><td>(Zhao et al., 2022a)</td><td>0.091±0.002</td></tr><tr><td>ESAN</td><td>(Bevilacqua et al., 2022)</td><td>0.102±0.003</td></tr><tr><td>SUN</td><td>(Frasca et al., 2022)</td><td>0.083±0.003</td></tr><tr><td>I2-GNN</td><td>(Huang et al., 2023)</td><td>0.083±0.001</td></tr><tr><td>ID-MPNN</td><td>(Zhou et al., 2023a)</td><td>0.083±0.003</td></tr><tr><td rowspan=\"4\">Local (F)GNN</td><td>SetGNN</td><td>(Zhao et al., 2022b)</td><td>0.075±0.003</td></tr><tr><td>GNN-SSWL</td><td>(Zhang et al., 2023a)</td><td>0.082±0.010</td></tr><tr><td>GNN-SSWL+</td><td>(Zhang et al., 2023a)</td><td>0.070±0.005</td></tr><tr><td>N2-GNN</td><td>(Feng et al., 2023)</td><td>0.059±0.002</td></tr><tr><td rowspan=\"4\">Substructure-based GNN</td><td>GSN</td><td>(Bouritsas et al., 2022)</td><td>0.101±0.010</td></tr><tr><td>CIN (Small)</td><td>(Bodnar et al., 2021a)</td><td>0.094±0.004</td></tr><tr><td>CIN</td><td>(Bodnar et al., 2021a)</td><td>0.079±0.006</td></tr><tr><td>CIN++</td><td>(Giusti et al., 2023)</td><td>0.077±0.004</td></tr><tr><td rowspan=\"6\">Graph Transformer</td><td>SAN</td><td>(Kreuzer et al., 2021)</td><td>0.139±0.006</td></tr><tr><td>K-Subgraph SAT</td><td>(Chen et al., 2022)</td><td>0.094±0.008</td></tr><tr><td>Graphormer</td><td>(Ying et al., 2021)</td><td>0.122±0.006</td></tr><tr><td>URPE</td><td>(Luo et al., 2022)</td><td>0.086±0.007</td></tr><tr><td>Graphormer-GD</td><td>(Zhang et al., 2023b)</td><td>0.081±0.009</td></tr><tr><td>GPS</td><td>(Rampasek et al., 2022)</td><td>0.070±0.004</td></tr><tr><td rowspan=\"6\">Other</td><td>PF-GNN</td><td>(Dupty et al., 2021)</td><td>0.122±0.010</td></tr><tr><td>KP-GIN</td><td>(Feng et al., 2022)</td><td>0.093±0.007</td></tr><tr><td>SignNet</td><td>(Lim et al., 2023)</td><td>0.084±0.006</td></tr><tr><td>PathNN</td><td>(Michel et al., 2023)</td><td>0.090±0.004</td></tr><tr><td>PPGN++(6)</td><td>(Puny et al., 2023)</td><td>0.071±0.001</td></tr><tr><td>PlanE</td><td>(Dimitrov et al., 2023)</td><td>0.076±0.003</td></tr></table>",
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+                                        }
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+                            "index": 2,
+                            "angle": 0,
+                            "type": "table_body"
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+                    ],
+                    "index": 2
+                },
+                {
+                    "type": "table",
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+                                            "type": "text",
+                                            "content": "Table 9: Performance of different GNN models on Alchemy dataset reported in the literature."
+                                        }
+                                    ]
+                                }
+                            ],
+                            "index": 3,
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+                                            "html": "<table><tr><td>Model</td><td>Reference</td><td>Test MAE</td></tr><tr><td>GIN</td><td>(Xu et al., 2019)</td><td>0.180±0.006</td></tr><tr><td>PF-GNN</td><td>(Dupty et al., 2021)</td><td>0.111±0.010</td></tr><tr><td>δ-2-GNN</td><td>(Morris et al., 2020)</td><td>0.118±0.001</td></tr><tr><td>Recon-GNN</td><td>(Cotta et al., 2021)</td><td>0.125±0.001</td></tr><tr><td>SpeqNet</td><td>(Morris et al., 2022)</td><td>0.115±0.001</td></tr><tr><td>SignNet</td><td>(Lim et al., 2023)</td><td>0.113±0.002</td></tr><tr><td>PPGN</td><td>(Puny et al., 2023)</td><td>0.113±0.001</td></tr><tr><td>PPGN++</td><td>(Puny et al., 2023)</td><td>0.111±0.002</td></tr><tr><td>PPGN++(6)</td><td>(Puny et al., 2023)</td><td>0.109±0.001</td></tr></table>",
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